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[ { "title": "Existence of Reissner-Nordstrom type black holes in f(R) gravity", "content": "S. Habib Mazharimousavi, ∗ M. Kerachian, † and M. Halilsoy ‡ Physics Department, Eastern Mediterranean University, G. Magusa north Cyprus, Mersin 10 Turkey. We investigate the existence of Reissner-Nordstrom (RN) type black holes in f(R) gravity. Our emphasis is to derive, in the presence of electrostatic source, the necessary conditions which provide such static, spherically symmetric (SSS) black holes available in f(R) gravity. We also study the thermodynamics of the black hole solution. Keywords: Reissner-Nordstr¨om; f(R) Gravity; Black hole. PACS: 04.50.Kd; 04.70.Bw; 04.40.Nr", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Due to a number of valid reasons f ( R ) gravity attracted much interest during the recent decade as an extension / modification of Einstein's general relativity [1-6] (for some review works see [7-10]). Here R stands for the Ricci scalar, the simplest among much complicated ones and f ( R ) is an analytic function of R . Herein we wish to look at f ( R ) gravity from a different angle which was introduced by Bergliaffa and Nunes in their novel paper [11, 12]. Since the black hole solutions in Einstein's f ( R ) = R , theory has already built enough prominence and play the leading role it should be wise to seek for similar solutions in the more general f ( R ) theories. This approach concerns directly the existence problem of black holes and it's associated necessary conditions for analog objects in the latter. The existence conditions may simply be dubbed as the ' near-horizon test ' in order to highlight the event horizon of a black hole as a physical reality. It is well-known that physically when the observer approaches the event horizon he / she feels nothing unusual except strong gravity, so this mathematically must reflect analytically on the event horizon. The analytic expansion of a metric function, say f ( r ), is developed in series of the form f ( r ) = f ( r 0 )+ f ' ( r 0 ) ( r -r 0 )+ O ( ( r -r 0 ) 2 ) where r 0 is the event horizon and ( r -r 0 ) stands naturally small. When these developed series are substituted back into the Einstein equations they will give conditions of zeroth,first and higher orders. These are precisely what we call the necessary conditions for the existence of certain / analog black hole types. To our amazement these necessary conditions emerge rather restrictive so that we can't propose arbitrarily any polynomial forms of f ( R ) as the representative black holes. For instance,what are the necessary conditions in order that it will admit Schwarzschildlike black hole solutions? This particular problem without external sources has already been considered and it's found that the possible f ( R ) must be of the form f ( R ) = α √ R + β , in which α and β are constants [11, 12]. An analytic expansion reveals that the first term retains the Einstein-Hillbert term with the addition of higher orders in R which seems to be the payoff in the enterprise of f ( R ) theory. Any f ( R ) theory is known to create it's own source from the inherent non-linearity of the theory. Beside these, however, additional external sources may be considered (some examples of f ( R ) black hole with charge are given in [13-19]), which makes the principal aim of the present paper. We consider an external static electric field as source and adopt the Reissner-Nordstrom (RN)-type black hole within f ( R ) gravity. Expectedly, the results for necessary conditions for the existence of a RN black hole are more complicated than the case of a Schwarzschild black hole. In this process we obtain an infinite series representation for the near-horizon behavior of our metric functions. The exact determination of the constant coefficients in the series is theoretically possible, at least in the leading orders. The addition of further external sources beside electromagnetism will naturally make the problem more complicated. An equally simple case is the extremal RN black hole which is also considered in our study. The paper is organized as follows. Section II investigates the necessary conditions for the existence of a RN-type black hole in f ( R ) gravity. Thermodynamics, and in particular the first law for such black holes are presented in Section III. Section IV is devoted to an extremal RN-type black hole. The paper is completed with Concluding Remarks, which appears in Section V.", "pages": [ 1 ] }, { "title": "II. ANALOG RN BLACK HOLES IN f ( R ) GRAVITY", "content": "The proper action in f ( R ) gravity coupled minimally with Maxwell source in 4 -dimensions is given by in which f ( R ) is a real function of the Ricci scalar R, F = 1 4 F µν F µν is the Maxwell invariant and κ = 8 πG where G is the Newton's constant. Our choice of the spacetime is a RN-type black hole solution whose line element can be written as where M and Q are two real constants which indicate the mass and the charge of the black hole respectively. Also Φ = Φ( r ) is an unknown real function which is well behaved everywhere and dies off at large r. The matter source which we shall consider in our consideration is a Maxwell electric field whose two-forms is given by where E ( r ) is the electric field. Following the line element (2) one finds the dual-Maxwell field as and in turn the Maxwell equation implies in which the integration constant q is to be identified with Q . Varying the action with respect to g µν provides the field equations where F = df dR , /square F = 1 √ -g ∂ µ ( √ -g∂ µ ( df dR )) and ∇ ν ∇ µ F = g αν [ F ,µ,α -Γ m µα F ,m ] . We obtain in which a prime denotes derivative with respect to r. Also in Eq. (7) the stress-energy tensor T ν µ reads as together with The first order equations admit another pair of equations together with which after considering the line element (2) and the Maxwell field (3) together with (6), one finds We note that another dependent equation is the vanishing trace condition which is obtained after knowing T = T µ µ = 0. The trace equation may be used to simplify the field equations and therefore Eq. (7) becomes From the metric given in (2) one finds the event horizon at r = r 0 = M + √ M 2 -Q 2 or consequently as the ADM mass. Based on the near horizon test introduced in Ref. [11, 12] we expand all the unknown functions about the horizon. This would lead to the expansions in which the sub zero implies the corresponding quantity evaluated at the horizon. After some manipulation, the field equations would develop as series in different orders of ( r -r 0 ) . In the zeroth order one finds two independent equations In these equations E = d 2 f dR 2 = dF dR and H = d 3 f dR 3 = dE dR and a sub '0' implies the value at the horizon. From these equations we find the possible solutions for the unknown coefficients. The following are the results: in which /epsilon1 = r -r 0 , β 1 , β 2 are constants and Let us note that Φ 0 remains unknown, but since it can be absorbed into the redefinition of time it can be set as Φ 0 = 0 . What we have here are some complicated relations between the forms of f, F and R in terms of β 1 and β 2 which are arbitrary. In the zeroth order the conditions on any f ( R ) can be written as and", "pages": [ 2, 3, 4 ] }, { "title": "A. Examples of f ( R ) = R and f ( R ) = R 2", "content": "For instance, in the case of R gravity we have f 0 = R 0 and F = 1 . The latter yields or consequently β 1 = 0 . Having β 1 = 0 implies which clearly leads to β 2 = 0. Therefore f ( R ) = R satisfies our general conditions with β 1 = 0 = β 2 . Next we test the case of f ( R ) = R 2 for which the above conditions become and These two conditions admit and which are only acceptable if 4 Q 2 -2 r 4 0 ≥ 0 . Specifically, once the equality holds one has while from (16) we have These together become or Also in this case we find r 4 0 2 r 2 0 - 2 Mr 0 + = 0 (41) and f 0 = R 0 = 1 while F 0 = 2 . These examples can further be extended to cover more general polynomial forms of f ( R ) to justify the validity of our existence conditions, however, we shall be satisfied with an extremal-RN example in the following section.", "pages": [ 4, 5 ] }, { "title": "B. Extremal RN-type black hole", "content": "An interesting case which can be considered here is the case for M = Q in (2). This will make the extremal RN-type black hole with the line element in which b 0 = Q. Taking this into account would lead from the general equations and in which β is an arbitrary, non-zero constant and /epsilon1 = ( r -r 0 ) . As before, we absorb Φ 0 into time. It is remarkable observe that R is zero at the horizon and so is f, but ( df dR ) = 1 . This is an indication that a proper candidate for such an f ( R ) is of the form in which the constant coefficients a i can be determined, using above conditions. For instance, if we restrict ourselves up to the third order we get f ( R ) ∼ R + r 2 0 12 R 2 + r 3 0 ( 5 72 r 0 + 19 108 β ) R 3 . One can easily check that this form of f ( R ) satisfies all the conditions given above up to the second order. Subsequent implication of the results found above is that f ( R ) ∼ R ν . Here any ν can not satisfy the conditions without choosing β = 0 , which is the case of ν = 1 or GR. Another example which at least satisfies the above conditions up to first order is f ( R ) = R 1 -R .", "pages": [ 5, 6 ] }, { "title": "III. THERMODYNAMICS OF THE ANALOG BLACK HOLE", "content": "After having the solution one may be curious about the thermodynamical properties of the solution. This is doable in exact form because of the metric function which is known about the horizon. First of all the horizon will remain as r = r 0 and the Hawking temperature is found by in which T ( RN ) H implies RN Hawking temperature. The form of Entropy is given by in which A| r = r 0 = 4 πr 2 0 is the surface area of the black hole at the horizon and F | r = r 0 = F 0 . We note that T H and S are both exact. Having T H and S one may find the heat capacity of the black hole in which where We comment that I in the RN limit (i.e., β i → 0) becomes unit as expected. Let us note also that the form of C q is exact. In order to study the thermodynamics of the extremal solution we use the general results found in the non-extremal RN type solution. One easily finds that T H = 0 , and C q = 0 .", "pages": [ 6 ] }, { "title": "A. First Law of Thermodynamics", "content": "Furthermore, in this section, we would like to show that in general, the above solution also satisfies the first law of thermodynamics. This is somehow a generalization of what was introduced in Ref. [20] to find a higher dimensional form of the Misner-Sharp (MS) energy [21] and was used in SSS black hole in f ( R ) gravity in Ref. [22]. To this end we rewrite the field equation in the following form in which G ν µ is the Einstein tensor, and for our later convenience we consider In turn, the tt component of the latter field equation would read in which and /square F = 2 3 f -1 3 RF. At the horizon (where the MS energy is introduced) U ( r 0 ) = 0 , which yields G 0 0 = U ' 0 r 0 -1 r 2 0 , ∇ 0 ∇ 0 F = 1 2 U ' 0 F ' 0 . A substitution in (40) and calculating everything at the horizon r = r 0 yields Next, we multiply both sides by the spherical volume element at the horizon i.e. dV 0 = A dr 0 to get Using A r 0 = 1 2 d dr 0 A and some manipulation one finds which is nothing but the first law of thermodynamics i.e., TdS = dE + PdV . This is due to the definition which we have for Hawking temperature T = U ' 0 4 π , entropy of the black hole S = 2 π A κ F 0 , the radial pressure P = T r r = T 0 0 and the MS energy as in which the integration constant is set to zero [20, 23-26] (also for a BH-like solutions see [27]). Here we comment that all quantities are calculated at the horizon and due to this the Hawking temperature becomes T = ( e -2Φ U ) ' 4 π ∣ ∣ ∣ r 0 = U ' 0 4 π . ∣ The above results imply that, using (65) as MS energy, the first law of thermodynamic is satisfied. Once more we wish to add that our results are exact.", "pages": [ 7 ] }, { "title": "IV. CONCLUDING REMARKS", "content": "In this paper we have applied the ' near-horizon test ' to the Reissner-Nordstrom (RN)-type black holes in f ( R ) gravity. Necessary conditions, not the sufficient ones that a RN-type black hole exists are derived. These are nothing but the regularity conditions of the metric functions in the vicinity of the event horizon. Our metric ansatz consists of a general static, spherically symmetric (SSS) case adopted from the Einstein's general relativity. We considered also the extremal case as an analog black hole in f ( R ) gravity and derived the underlying conditions. Due to their intricacy we didn't attempt to solve those equations in general. To the zeroth order, however, they can be obtained exactly while to the first order approximation is also tractable. Our analysis shows that a closed form of f ( R ) doesn't seem possible: With a given source we can determine f ( R ) implicitly as an infinite series in ( r -r 0 ) , since R ( r ) also is expressed in similar series. This is against the strategy adapted so far, namely, an explicit form of f ( R ) is assumed a priori to be tested whether it fits physical requirements. In our opinion, the ' near-horizon test ', introduced in [11, 12] and developed here further constitutes a more fundamental test than any other arguments in connection with black holes. We admit that since our necessary conditions for the existence of RN-type black holes are entirely local they don't involve the requirements for asymptotic flatness. Stability of such black holes must also be considered separately when one considers exact solutions. Our test must naturally be supplemented with d 2 f dR 2 > 0 and df dR > 0, for stability and no-ghost requirements [28, 29]. We have shown also that the thermodynamic of these analog black holes can be studied through the Misner-Sharp formalism to verify the validity of the first law. Finally, we remark that solution for f ( R ) gravity admitting an electromagnetic field with similar thermodynamics was reported before [30].", "pages": [ 8 ] } ]

[ { "title": "Dark Energy From Fifth Dimensional Brans-Dicke Theory", "content": "Amir F. Bahrehbakhsh 1 , 2 ∗ , Mehrdad Farhoudi 1 † and Hajar Vakili 3 ‡ 1 Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran 19839, Iran 2 Department of Physics, Faculty of Science, Payam-e-Nour University, Iran 3 Department of Physics, Sharif University of Technology, P.O. Box 11365-9161, Tehran, Iran June 4, 2013", "pages": [ 1 ] }, { "title": "Abstract", "content": "Following the approach of the induced-matter theory, we investigate the cosmological implications of a five-dimensional Brans-Dicke theory, and propose to explain the acceleration of the universe. After inducing in a four-dimensional hypersurface, we classify the energy-momentum tensor into two parts in a way that, one part represents all kind of the matter (the baryonic and dark) and the other one contains every extra terms emerging from the scale factor of the fifth dimension and the scalar field, which we consider as the energy-momentum tensor of dark energy. We also separate the energy-momentum conservation equation into two conservation equations, one for matter and the other for dark energy. We perform this procedure for different cases, without interacting term and with two particular (suitable) interacting terms between the two parts. By assuming the parameter of the state equation for dark energy to be constant, the equations of the model admit the power-law solutions. Though, the non-interacting case does not give any accelerated universe, but the interacting cases give both decelerated and accelerated universes. For the interacting cases, we figure out analytically the acceptable ranges of some parameters of the model, and also investigate the data analysis to test the model parameter values consistency with the observational data of the distance modulus of 580 SNe Ia compiled in Union2.1. For one of these interacting cases, the best fitted values suggest that the Brans-Dicke coupling constant ( ω ) is /similarequal -7 . 75, however, it also gives the state parameter of dark energy ( w X ) equal to /similarequal -0 . 67. In addition, the model gives the Hubble and deceleration parameters at the present time to be H · /similarequal 69 . 4 (km/s)/Mpc and q · /similarequal -0 . 38 (within their confidence intervals), where the scale factor of the fifth dimension shrinks with the time. PACS number: 04 . 50 . -h ; 04 . 50 .Kd ; 95 . 36 . + x ; 98 . 80 .Es Keywords: Brans-Dicke Theory; Induced-Matter Theory; FRW Cosmology; Dark Energy; Data Analysis of Observational Cosmology", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "While improving and searching for more suitable gravitational theory, attempts for a geometrical unification of gravity with other interactions has begun by employing extra dimensions beyond the conventional four-dimensional ( 4D ) space-time. At first, Nordstrøm [1] built a unified theory based on extra dimensions, then Kaluza [2] and Klein [3] established a 5 D version of general relativity ( GR ) in which electrodynamics rises from an extra fifth dimension. Till now, an intensive amount of researches have been performed on a similar idea either via different mechanism of compactification of extra dimension or generalizing it to non-compact scenarios [4] such as the Brane-World theories [5], the space-time-matter or induced-matter ( IM ) theories [6, 7], and higher dimensional cosmology, e.g. Ref. [8]. The latter theories are grounded on the Campbell-Magaard theorem that asserts every analytical N -dimensional Riemannian space can locally be embedded in an ( N + 1)-dimensional vacuum one [9]-[13]. The importance of this theorem is that the matter sources of 4 D space-times can be viewed as a manifestation of extra dimensions. In another word, 4 D field equations with matter sources can be achieved by inducing 5 D field equations without matter sources. This idea has been the core of the IM theory using GR as the underlying theory. On the other hand, Jordan [14] conceived a new kind of gravitational theory, known as the scalartensor theory, by embedding a curved 4 D space-time in a flat 5 D one. Following his idea, Brans and Dicke [15, 16] introduced a version of the scalar-tensor gravitational theory, as an alternative to GR, with a non-minimally scalar field coupled to the curvature, in which the weak equivalence principle is preserved and makes it to be more Machian than GR. Unfortunately, its weakness is mismatching with the solar system observations [17, 18], which is a generic difficulty of the scalar-tensor theories in the solar system constraint [19, 20]. However, it does not necessarily denote that the evolution of the universe, at all scales, should be close to GR, in which there are some debates on its tests on the cosmic scales as well [21, 22]. The measurements of anisotropies in the cosmic microwave background suggest that the ordinary 4 D universe is very close to a spatially flat universe [23]-[28], and the observations of type Iasupernovae indicate that the expansion of the universe presently is accelerating [29]-[33]. Hence, in one school of thought, the main component of the universe should be consist of what usually has been called dark energy [34, 35]. Since then, a considerable amount of work has been performed in the literature to explain the acceleration of the universe. Most of the dark energy models, such as the quintessence [36]-[39], the chaplygin gas [40, 41] and the k-essence [42, 43] models, involve minimally coupled scalar fields with different potentials. These scalar fields or the potentials have been added in priori by hand, and hence, their origins are not understood. Nevertheless, in the recent decades, explaining the accelerated expansion of the universe via fundamental theories has been a great challenge. Recently, a 5 D vacuum Brans-Dicke ( BD ) theory based on the idea of IM theory has been investigated [44], in which the role of GR has been replaced by the BD theory of gravitation as the fundamental underlying theory. It has been illustrated that 5 D vacuum BD equations, when reduced to four dimensions, give a modified version of the 4 D BD theory with an induced potential. Whereas in the literature, to obtain accelerating universes, such potentials has been added in priori by hand. A few applications of this approach have also been performed in Refs. [45]-[48]. Although one of the aims of higher dimensional theories is to obtain 4 D matter from pure geometry, but it is sometimes desirable to have a higher dimensional energy-momentum tensor or a scalar field, for example in compactification of the extra curved dimensions [49]. Indeed, the reduction procedure of a 5 D analogue of the BD theory with matter content, on every hypersurface orthogonal to an extra cyclic dimension (that recovers the modified BD theory described by a 4-metric coupled to two scalar fields) has previously been accomplished in the literature [50, 51]. Besides, the main idea of the brane models is the existence of a higher dimensional bulk in which the universe is sitting as a hypersurface and all the matter fields, except gravity, are confined to this brane [5]. Cosmological implications in the context of brane worlds have also been studied, e.g. it has been shown that the recent accelerated expansion of the universe can be explained by a geometrical originated dark energy caused by an addition of a brane curvature scalar term in the action [52, 53]. Similar to the idea of brane scenarios, but based on the IM theory with underlying the BD theory of gravitation, in this work, we employ a generalized Friedmann-Lemaˆıtre-Robertson-Walker ( FLRW ) type solution for a 5 D BD theory and investigate its cosmological implications. For this purpose, in the next section, we give a brief review of a 5 D BD theory following the idea of IM theory. Then, in a 4 D hypersurface, we gather the extra terms emerging from the scale factor of the fifth dimension and the scalar field as dark energy component of an energy-momentum tensor in addition to the usual energy-momentum tensor of all (the baryonic and dark) matter. In Section 3, we consider a generalized FLRW metric in a 5 D space-time, and derive the FLRW cosmological equations. Then, we manage and obtain the total energy conservation equation as separated into two energy conservation equations for all matter and dark energy, with non-interacting and two simple interacting cases. However, as the cosmological equations are strictly non-linear and do not have exact solution, in order to proceed, we assume some simplifications. In Section 4, we apply the observational constraints for the interacting cases of the model, and use the Markov Chain Monte Carlo ( MCMC ) method to fit the free parameters of the model with the SCP Union2.1 SN Ia compilation [54, 55] based on the Bayesian statistics [56]. A few tables and figures are also provided for a better view of the acceptable range and the best fitted values of parameters. Finally, conclusions are presented in the last section.", "pages": [ 2, 3 ] }, { "title": "2 Five-Dimensional Brans-Dicke Theory", "content": "Following the idea of IM theories, one can consider the BD theory of gravitation as the underlying theory instead of GR. In this respect, the action of 5 D BD theory in the Jordan frame can analogously be written as where c = 1, the capital Latin indices run from zero to four, (5) R is 5 D Ricci scalar, (5) g is the determinant of 5 D metric g AB , φ is a positive scalar field that describes the gravitational coupling in five dimensions, L m represents the matter Lagrangian and ω is a dimensionless coupling constant. Hence, the field equations obtained from action (1) are and where (5) G AB is 5 D Einstein tensor, (5) T AB is 5 D energy-momentum tensor, (5) T ≡ (5) T C C and (5) ✷ ≡ ; A A . In order to have a non-ghost scalar field in the conformally related Einstein frame, i.e. a field with a positive kinetic energy term in that frame, the BD coupling constant must be ω > -4 / 3 [57]-[59]. Though recently, some new ghost dark energy models have been suggested to explain the observed acceleration of the universe [60]. As a plausible assumption, we consider that (5) T AB exactly represents the same baryonic and dark matter sources of a 4 D hypersurface, i.e. T ( M ) αβ . Hence, in this case (5) T AB = diag ( ρ M , -p M , -p M , -p M , 0), where ρ M and p M are the energy density and the pressure of the matter, and the Greek indices run from zero to three. Also, cosmological purposes usually restrict attentions to 5 D metrics of the simple form in local coordinates x A = ( x µ , y ), where y represents the fifth coordinate and /epsilon1 2 = 1. By assuming the 5 D space-time is foliated by a family of hypersurfaces, say Σ, that are defined by fixed values of y , then, one can obtain the metric intrinsic to every generic hypersurface, e.g. Σ · ( y = y · ), by restricting the line element (4) to displacements confined to it. Thus, the induced metric on the hypersurface Σ · becomes in such a way that the usual 4 D space-time metric, g µν , can be recovered. Therefore, after some manipulations, equation (2) on the hypersurface Σ · can be written as where we consider T ( X ) αβ as dark energy component of the energy-momentum tensor that is defined by in which, analogous to the IM theory [47], and The prime denotes derivative with respect to the fifth coordinate. In the following section, we consider a generalized FLRW metric in a 5 D universe and investigate its cosmological properties.", "pages": [ 3, 4 ] }, { "title": "3 Generalized FLRW Cosmology", "content": "For a 5 D universe with an extra space-like dimension in addition to the three usual spatially homogenous and isotropic ones, metric (4), as a generalized FLRW solution, can be written as Generally, the scalar field φ and the scale factors a and b should be functions of t and y . However, for physical plausibility and simplicity, we assume that the hypersurface-orthogonal space-like is a Killing vector field in the underlying 5 D space-time [50, 51], i.e. the extra dimension to be a cyclic coordinate. Besides, the functionality of the scale factor b on y could be eliminated either by transforming to a new extra coordinate (if b be a separable function) or making no changes in the consequent equations if b is the only field that depends on y . In another word, in the compactified extra dimension scenarios, all fields can be Fourier-expanded around the fixed value y · , and hence, one can get the observable terms independent of y , i.e. physics would be effectively independent of the compactified fifth dimension, see, e.g. Ref. [4]. With this solution, we will show that the universe can accept both accelerating and decelerating expansion eras. By considering metric (10), the BD equations (2) reduce as follows. The time component A = 0 = B gives the spatial components A = B = 1 , 2 , 3 provide and the A = 4 = B component yields Also, the scalar field equation (3) becomes where H ≡ ˙ a/a , B ≡ ˙ b/b and F ≡ ˙ φ/φ . In the last part of equations (11) and (12), we have defined the energy density and pressure of dark energy as and Hence, the state equation of dark energy obviously yields In order to find the functionality of w X with the time, equations (11)-(14) must be exactly solved. However, as these equations are coupled non-linearly, one should apply numerical methods. As an alternative, in the following, by considering a few simplifications, we proceed to fit the parameters of the model by observational data analysis, and discuss the properties of the model. First of all, let us obtain the energy conservation equations. In this respect, one can take the time derivative of equation (11) and substitutes equation (12) into it, and finally gets where ˜ ρ ≡ ρ M + ρ X and ˜ p ≡ p M + p X . As the nature of dark energy is unknown, the detailed coupling form among it and matter is unclear. Hence, one can expect that their conservation equations should not be independent. In this case, let us apply a plausible simplification by separating equation (18) into two distinguished continuity equations for ρ X and ρ M as and where the Q term stands for interacting terms among matter and dark energy. Such a similar interacting term has also been used in the literature, e.g. Ref. [61]. To proceed further, we take w M = 0 as dust (the baryonic and dark) matter, and assume w X to be a constant. Hence, relation (17) imposes the power-law solutions and If one assumes the power-law solutions (21)-(23), equations (11)-(13) will restrict either the geometry to be spatially flat or α to be one, i.e. a free expanding universe. As the latter choice is not interested, we consider k = 0 in this work. This choice is also consistent with the measurements of anisotropies in the cosmic microwave background radiation that indicate the universe must be very close to spatially flat one [23]-[28]. Note that, there are four independent equations (11)-(14) that, in general, determine a , b , φ (or equivalently α , β , γ in the power-law solutions) and ρ M . These unknowns also depend on the BD parameter ω . In order that our ansatze do not lead to over-constraining equations, we manage the values of the parameters of the model including ω to be, by using the data analysis method, consistent with the observations. In the next two subsections, we find the relations of ρ X and ρ M with the scale factors and the scalar field for both the non-interacting and interacting cases. For interacting cases, we consider two simple and reasonable choices of Q ≡ Γ ρ M and Q ≡ Γ ρ X , where The symbol ψ is denoted as a general notation that represents either the scale factor of the ordinary spatial dimensions, a , or the scale factor of the fifth dimension, b , or the scalar field, φ . That is, Γ can be either H or B or F , however, by matching the model with the observations, we will figure out the best choice of ψ .", "pages": [ 4, 5, 6 ] }, { "title": "3.1 Non-Interacting Case Q = 0", "content": "Now, for the matter to be a dust one and the dark energy parameter of state to be time independent, if there is no interacting term, then equations (19) and (20) lead to and which for the power-law solutions (21)-(23) become and where ρ M · , ρ X · , a · and φ · are the energy density of matter, the energy density of dark energy, the scale factor and the scalar field at the present time, respectively. Substituting the power-law solutions in equation (11) (and also equation (14)) imposes α = 2 / 3 and w X = 0, which does not give an accelerated universe. Hence, in the following, we investigate different interacting cases to figure out any possible accelerating solutions. However, when w X is not constant (as a more general case), then the non-interacting choice may also provide some interesting results.", "pages": [ 6 ] }, { "title": "3.2 Interacting Cases Q ≡ Γ ρ M and Q ≡ Γ ρ X", "content": "With the same assumptions as in the non-interacting cases, to find out analytically the ranges of the parameters of our model for interacting cases Q ≡ Γ ρ M and Q ≡ Γ ρ X , we consider the power-law solutions (21)-(23). Thus, we assume Now, for the chosen interacting cases, we will show, in the following two parts, that the model can accept both acceleration and deceleration for the universe expansion eras.", "pages": [ 7 ] }, { "title": "A. Interacting Case Q ≡ Γ ρ M", "content": "Substituting Q ≡ Γ ρ M into equation (19) gives that for the power-law solutions (21)-(23), becomes Thus, an acceptable solution of (20), which satisfies equations (11)-(14), is Solutions (31) and (32) reveals that both the matter and dark energy evolutions are the same in this model. Substituting ρ M and ρ X into equation (11) or (12), imposes and then, using it into the second part of (32), gives where u · ≡ ρ M · /ρ X · . The acceptable ranges of w X , α and λ , for an expanding universe, are given in Table 1 and are depicted in Fig. 1. For this interacting case, equation (11) can also be presented as where Ω M · + Ω X · = 1. Relation (35) is the same as the result obtained in Ref. [62] for a 4 D BD model with a priori assumed (i.e. added-by-hand) potential term. However, in the next section, we also match the model consistency with the observational data of SN Ia, and search the best fitted parameters of the model. If one sets ψ ≡ a , i.e. Γ ≡ H and λ ≡ α , then from equation (33) it is obvious that α = 1, which gives a free expanding universe, thus, in the data analyzing of equation (35), we only consider the other two (more interesting) cases λ = β and λ = γ . Another suitable choice for non-interacting case is to take Q ≡ Γ ρ X . Thus, by substituting it into equation (20), one gets that for the power-law solutions (21)-(23), yields Therefore, an acceptable solution of (19) is Once again, the matter and dark energy evolve in the same manner in this model. Solutions (37) and (38) must also satisfy equations (11)-(14), that give and It is notable that equations (40) and (34) are exactly the same. Finally, one can rewrite equation (11) as where again Ω M · + Ω X · = 1. In the following, we highlight the conditions that correspond to a decelerating and an accelerating universe for both of the interacting cases.", "pages": [ 7, 8 ] }, { "title": "I. Deceleration and Free Expansion", "content": "The observational data reveals that when the universe was in the radiation or dust dominated phases, it was in a decelerating regime for a long time [63]. In our model, decelerating and free expanding universe can be obtained from equation (12) for w X ≥ -(1+ u ) / 3 (where u ≡ ρ M /ρ X ) and 0 < α ≤ 1 in the power-law solutions. The acceptable domains of w X , α and λ , in accord with this situation, are given in Table 1 and drown in Fig. 1.", "pages": [ 8 ] }, { "title": "II. Acceleration", "content": "Recent observations illustrate that the universe is in an accelerating phase at the present epoch [29]-[33]. An accelerating universe from equation (12) makes w X < -(1 + u ) / 3 and α > 1 for the power-law solutions. The acceptable values of w X , α and λ corresponding to this condition are also given in Table 1 and Fig. 1. Table 1 and Fig. 1 surprisingly show that both the decelerating and the accelerating solutions are acceptable in the range w X < -1 / 3. However, a fixed value for the parameter α for all epochs implies that the universe is always either decelerating or accelerating [62], which is the common property of the power-law solutions. In this respect, one should consider the model only for the late time acceleration of the universe. To analyze the nature of the acceleration part of the model, it is instructive to investigate the model consistency with the observational data. For this purpose, in the next section, we employ the MCMC method to fit the free parameters of the model with the SCP Union2.1 SN Ia compilation [54, 55] based on the Bayesian statistics [56].", "pages": [ 8 ] }, { "title": "4 Observational Constraints", "content": "The various data information from different observations are used to constrain the cosmological models. Among them, the SN Ia distance modulus declares the accelerated expansion of the universe [29]-[33]. We employ the Bayesian statistics to investigate the model consistency with the SCP Union2.1 SN Ia compilation that is an update of the Union2 compilation [64]. This compilation consists of nineteen data sets from 833 supernovae. However, it has been indicated [54, 55] that only 580 of these 833 supernovae can pass the usability cuts, which contain new data from the HST Cluster Survey. The HST Cluster Survey supplies the latest and the most complete data set for SN Ia observations till now. The most popular techniques for the parameter estimation in cosmology is the MCMC method. These methods were first employed in astrophysics [65], and since then, it has also been used in cosmology. We have used our own package, however, the standard packages for the cosmological parameter estimation are also publicly available [66, 67]. The likelihood function in the Bayesian statistics is defined to be proportional to exp ( -χ 2 / 2). Hence, the best fitted values for the parameters of the model are obtained by minimizing the χ 2 , which for the SN Ia is defined as In equation (42), { l } refers to the parameters of the model which can be estimated by the data analysis process and σ i stands for the 1 σ uncertainty associated to the i th data point. The symbol µ ob is the observed distance modulus and µ th is the theoretical distance modulus that is defined as where m is the apparent magnitude, M is the absolute magnitude and is the luminosity distance. We constrain the parameters of the model with the observational data from the supernovae type Ia, for both interacting cases A and B of the previous section. The analysis reveals that for the interacting case B, the parameters α and λ are strictly related and any change for each of them in the MCMC process yields different non-compatible results. By increasing the steps of the MCMC process, almost the whole parameter space will be covered. In other words, there are no preferred values for the parameters of the model in case B regarding to the SNe Ia observations, hence, we ignore this case. The result of the data analysis corresponding to the interacting case A is as follow. The best fitted values for α and H o are listed in Table 2, that suggest α /similarequal 1 . 61 and H · /similarequal 69 . 4 (km / s) / Mpc within their own confidence intervals. Fig. 2 illustrates the likelihood functions for them. The confidence intervals are also represented in Fig. 3 down. Also, the best fitted value for the deceleration parameter, q = -aa/ ˙ a 2 , within its confidence intervals is given in Table 2. The Hubble diagram (the distance modulus in terms of the redshift) simulated based on this case shows agreement between the model prediction and the observed data, Fig. 3 up. The Hubble time corresponding to the best fitted value of H · is t H · /similarequal 14 . 1 × 10 9 yr . Now, we can find the other parameters of the model with the best fitted value of α , namely α = 1 . 61. First of all, from relation (33), one gets λ = 2 . 83, thus, there are two choices, either λ = β or λ = γ (where for the latter choice, there are two solutions). Substituting α and β (or γ ) into equations (12) and (13) gives the best values of ω and γ (or β ) and then, from relation (17), one gets the best value for w X . The given value of w X from this procedure must fulfil the acceptable ranges of w X according to Table 1, which for α = 1 . 61, it reveals that one must have w X ≤ -0 . 58. Table 3 shows all the three groups of values for the parameters of the model. The results illustrate that the consistent values of the parameters of the model are those listed in the last column, which corresponds to λ ≡ γ case. This imposes that the best interacting term consists of the energy density of the matter multiplied by the ratio of the time derivative of the scalar field to itself. The consistent values of the parameters indicate that the scale factor of the fifth dimension shrinks with the time and the model has a ghost scalar field with ω /similarequal -7 . 75. Note that, as described in the Introduction, this mismatching value of ω with the values obtained from the solar system observations [17, 18] is expected [19]-[22]. Indeed, this value of the BD coupling constant actually corresponds to an imaginary conformal scalar field in the Einstein frame, where the issue of which conformal frame, the Jordan or the Einstein one, is physical is a matter of debate [68, 69]. In addition, in this model, the obtained dark energy state parameter, w X , is also /similarequal -0 . 67 > -1 which does not belong to a ghost one and the corresponding dark energy does not lead to big rip singularities.", "pages": [ 10, 11 ] }, { "title": "5 Conclusions", "content": "It is a general belief that the universe is in an accelerated expanding phase. Thus, the main content of it should be consisted of what usually has been called dark energy. Hence, a considerable amount of work has been performed to explain the acceleration of the universe, but until now, the origin and the nature of dark energy is unknown. Most of the dark energy models, such as the quintessence, the chaplygin gas and the k-essence models, involve minimally coupled scalar fields with different potentials which have been added by hand, and nothing has been asserted about their origins. Also in the recent decade, explaining the accelerated expansion of the universe by alternative theories of gravitation has been a great challenge. In this work, following the approach of the induced-matter theory, we have investigated the cosmological implications of a 5 D BD theory, in order to explain the acceleration of the universe. After inducing in a 4 D hypersurface, we have classified the energy-momentum tensor into two parts. One part represents all kind of the matter (the baryonic and dark), and the other one contains every extra terms emerging from the scale factor of the fifth dimension and the scalar field, which we have considered as the energy-momentum tensor of dark energy. As the cosmological equations of the model are extremely non-linearly coupled, we have assumed some simplifications in order to proceed the properties of the model. The total energy conservation has been separated into two equations, one for the matter conservation and the other for dark energy one. Of course, such a procedure has been performed without interacting term and with two particular interacting terms between the two parts. We consider the parameter of the state equation of dark energy to be constant. Hence, the equations of the model admit the power-law solutions which impose a spatially flat geometry. The non-interacting case gives decelerated universes, though, the interacting cases give both decelerated and accelerated universes. For the latter case, we have figured out analytically the acceptable ranges of parameters of the model, and have illustrated the results in a few tables and figures. Then after, for these interacting cases, we have employed the MCMC method based on the Bayesian statistics to investigate the consistency of the model parameters with the observational data from supernovae type Ia. For this purpose, we have used the data of the SCP Union2.1 SN Ia compilation. The data analysis process for the case, which its interacting term between the matter and dark energy is proportional to the energy density of dark energy, reveals that the involved parameters are strictly related and any change for each of them in the MCMC process yields different noncompatible results. By increasing the steps of the MCMC process, almost the whole parameter space will be covered. In other words, there are no preferred values for the parameters of the model in this interacting case. But for the other interacting case, which the interacting term is proportional to the energy density of the matter, the best fitted values suggest that H · /similarequal 69 . 4 (km/s)/Mpc, q · /similarequal -0 . 38 within their confidence intervals. The Hubble time corresponding to the best fitted value of H · is t H · /similarequal 14 . 1 × 10 9 yr . The best fitted values of the parameters of the model are also listed in a table, and the results suggest that the energy density of the matter multiplied by the ratio of the time derivative of the scalar field to itself plays the role of the best interacting term between the matter and dark energy. Also, the consistent values of the parameters indicate that the model has a ghost scalar field with ω /similarequal -7 . 75, while the scale factor of the fifth dimension shrinks with the time. Although the model has a ghost scalar field with this value for the BD coupling constant, the consistent values of the parameters reveals a non-ghost dark energy with the state parameter w X /similarequal -0 . 67 as well. That is, with this value of the dark energy state parameter, there should not be big rip singularities and ghost instabilities.", "pages": [ 11, 14 ] }, { "title": "Acknowledgements", "content": "We would like to thank Dr. M.S. Movahed for useful comments. M.F. also thanks the Research Office of Shahid Beheshti University G.C. for financial support.", "pages": [ 14 ] } ]

[ { "title": "Luca Fabbri", "content": "DIME Sez. Metodi e Modelli Matematici, Università di Genova INFN & Dipartimento di Fisica, Università di Bologna", "pages": [ 1 ] }, { "title": "Abstract", "content": "We shall consider the problem of Dark Matter in torsion gravity with Dirac matter fields; we will consider the fact that if WIMPs in a bath are allowed to form condensates then torsional effects may be relevant even at galactic scales: we show that torsionally-gravitating Dirac fields have interesting properties for the problem of DM. We discuss consequences.", "pages": [ 1 ] }, { "title": "Introduction", "content": "At the present stage, in the race between theories predicting phenomena that experiments must measure and experiments observing facts the theory has to explain, we are in a situation that is quite rare in the history of physics, because although on the one hand there is a vast phenomenology still far from being confirmed on the other hand there are only few things without a proper systematization: among them, one of the most intriguing is certainly Dark Matter. The problem of Dark Matter consists in the fact that the observed dynamics of the large scale universe, cluster of galaxies and galaxies themselves, seems to be well reproduced by simulations in which the gravitational force is stronger than what is expected to be; this could be due to two factors: a modified theory of gravitation or the same gravity of an exceeding matter distribution. Of these two approaches, the former might be able to describe some properties like galactic rotation curves, but it can say nothing about other phenomena such as the Bullet Cluster, which is the smallest of a couple of galaxies passing through each other, where during the crossing matter slows down due to the mutual gravitational attraction; however, gravitational lensing has been observed even out of the visible matter distribution, suggesting that there must be an invisible field very weakly-interacting which is nevertheless the source over large distances of a gravitational field: this implies that what causes the gravitational field outside the visible matter distribution cannot be an additional gravitational effect of that matter, because if this were true it would be impossible to have this matter distribution with its own gravitational field in leading-order residing within the visible matter distribution but with higher-order corrections dislodged out of the visible matter distribution itself, and so beside the visible matter another form of matter must be present [1]. Thus, it remains the latter approach, describing DM as a real although yet unknown form of matter. In terms of this approach, DM is a form of matter which must be neutral and very weakly-interacting so to justify why it is invisible and does not suffer the slowing process of the gravitational pull, and there are a few candidates possessing these features: the most relevant are Axions, ELKO and WeaklyInteracting Massive Particles WIMP; we shall briefly discuss them next. The basic idea of Axions has nothing to do with DM, as they were first postulated to solve problems related to chromodynamics, that is the so-called Peccei-Quinn model; it was only afterward that they have been recognized to have the character DM should have: however, this model in its most natural form is being restricted by observations in experiment such as ADMX and the consequently needed fine-tuning is diminishing its elegance. Both ELKO and WIMP are 1 2 -spin spinors, a form of matter well accepted. The ELKO fields are Majorana spinors solving the mass problem by postulating them to verify second-order derivative field equations [2, 3, 4, 5, 6]: these particles are a recent and promising attempt to furnish a candidate for DM, although the fact that they are spinors verifying higher-order derivative field equations may create issues for the torsional self-interaction [7, 8, 9, 10]; WIMP fields are 1 2 -spin fields verifying the Dirac equation, so they are both structurally and dynamically defined in terms of a commonly accepted framework. The WIMP field is a rather natural candidate for DM, but because neutrinos are massless, or at least, even if we believe that the existence of neutrino oscillations must necessarily be described in terms of neutrino masses, the hypothetical neutrino masses are not large enough, then neutrinos cannot be slow and therefore do not match some requisite to be WIMP, so that WIMP fields must be sought in some enlarged forms of the standard model of particle physics. We are not going to discuss here the extensions in which WIMP candidates can be found, since we shall focus on a different type of problem, that is assuming that WIMP can actually be found, then are there interesting properties that ought be investigated? To be more specific, let us assume that WIMP fields are the correct description of DM: as WIMP fields are 1 2 -spin spinor fields verifying the Dirac equation, then in a gravitational context they are described by the Sciama-Kibble completion of the Einstein theory for the Dirac matter, that is the Sciama-KibbleEinstein-Dirac SKED theory, where torsional contributions induce fermionic self-interactions in the matter field equation, as it is discussed for example in references [11] and [12, 13, 14, 15, 16]; these torsionally-induced fermion-fermion interactions can be equivalently rewritten in the form of Nambu-Jona-Lasinio NJL potentials [17, 18, 19]. The consequence of this fact is that WIMP fields permeating the galaxy are described by the SKED theory, therefore subject to a phenomenon of condensation analogous to the one happening in the NJL model; the fact that condensate fields may have a quantum but nevertheless macroscopic structure is expected, and if there is no fundamental interaction such as chromodynamics or electrodynamics confining the WIMP condensate field then it is not unreasonable that the macroscopic scale in this context may even be the galactic scale [20, 21]. As WIMP bath forming a single condensate at galactic scales is the most natural environment in which torsion may be relevant for the galactic rotation curves then we believe that condensates are the most natural systems in which to exploit those torsional effects that have been studied in a classical context in [22], already with intriguing results. On the other hand, in the usually accepted description of galactic rotation curves, the orbital velocity of a body within the matter distribution has the Newtonian behaviour linearly increasing with the distance before becoming constant as we move far from the center of the galaxy, which means that the density must scale according to 1 r for the visible matter contribution and according to 1 r 2 for the DM contribution; however, thinking at the 1 r behaviour as due to the Einsteinian gravitational effects of visible matter and at the 1 r 2 behaviour as due to the Einsteinian gravitational effects of WIMP is unsatisfactory since there is no reason why similar matter fields would have to behave so differently. Instead, if we think at the 1 r behaviour as still due to the Einsteinian gravitational effect of visible matter but at the 1 r 2 behaviour as now due to the torsionally-gravitating contribution of WIMP it is easy to see why they behave differently, and as a consequence we have that the correct behaviour is obtained without the impression of an accidental situation. In the present paper we will assume this point of view, eventually drawing some of its most relevant consequences.", "pages": [ 1, 2, 3 ] }, { "title": "1 WIMP Fields in SKED Theory", "content": "As we just mentioned, our starting point is to assume that WIMP fields exists in a galactic context, describing them in terms of the Sciama-Kibble torsional completion of Einstein gravity for Dirac matter fields: for the SKED theory we refer to [12, 13, 14, 15, 16] and [18, 19] for the fundamental definitions and the basic conventions; the formalism is the standard one but because we employ different notation, we will recall anyway some of them for the ease of the reader. In the paper, we will consider the metric tensors as g ασ and g ασ with connection Γ α µν defining a covariant derivative D µ for which Dg =0 and such that torsion tensor defined in terms of Q α µν =Γ α [ µν ] is taken to be completely antisymmetric [12, 13]: the metric-compatibility condition and complete antisymmetry of torsion make the connection decomposable according to the formula while the Riemann curvature tensor is given by antisymmetric in the first and second couple of indices, so with one independent contraction G α ρασ = G ρσ with G ρσ g ρσ = G called Ricci tensor and scalar, and in terms of the torsionless covariant derivative ∇ σ and torsionless curvature given by R µ ρσπ such that R α ρασ = R ρσ and R ρσ g ρσ = R as usual; the coordinate formalism can be translated in the tetrad formalism upon definition of the dual bases of orthonormal tetrads ξ a σ and ξ σ a such that they verify orthonormality conditions given by ξ σ a ξ ν b g σν = η ab and ξ a σ ξ b ν g σν = η ab in terms of the Minkowskian matrices, while the spin-connection Γ i jµ defining the covariant derivative D µ is such that it gives Dξ =0 and Dη =0 and for a connection with two different types of indices one cannot define torsion: these conditions imply that (1) is and it is antisymmetric in the two world indices while the curvature is antisymmetric in both the coordinate and the world indices and writable in terms of the Riemann curvature as G ab σπ = G µν σπ ξ a µ ξ b ν as obvious. The advantage of such change of formalism is that the most general coordinate transformations of the coordinate formalism (with Greek indices) are equivalently written in terms of the special Lorentz transformations of the tetrad formalism (with Latin indices) which admits a specific representation, suitable of being the usual real one but also a new complex one; in the tetrad formalism then, complex representations are definable and therefore we may proceed to the introduction of complex Lorentz transformations. These are called spinorial transformations. In such a geometrical background, spinor fields will be taken to be the simplest 1 2 -spin spinors, defined in terms of the 2 -dimensional sigma matrices /vector σ so that the most general Lorentz complex transformation can be written according to the expressions exp[( /vector ϕ + i /vector θ ) · /vector σ 2 ] or exp[( -/vector ϕ + i /vector θ ) · /vector σ 2 ] because of the sign ambiguity of the boosts: these can be merged into the reducible 4 -dimensional representation after introducing the γ µ matrices in chiral representation with sigma matrices 1 4 [ γ i , γ j ] = σ ij so that { γ i , σ jk } = iε ijkq γγ q in the complete Lorentz complex representation exp[ 1 2 θ ij σ ij ] as it is well-known, that is in the sought spinorial transformation in terms of which the 1 2 -spin spinors will be defined on a general spacetime background. Then it is possible to introduce the spinor-connection A µ defining the spinor-covariant derivative D µ containing the information about the dynamics of the spinor fields and for which the spinorial constancy of γ j is automatic: the spinor-connection A µ is given by in terms of the complex-valued spin-connection and the curvature is given by which is a tensorial spinor antisymmetric in the tensorial indices writable as in terms of the curvature of the spacetime, in a very compact form. This defines the basic formalism we are going to employ, in terms of which the kinematic background is now set up, and next point that needs to be settled is the implementation of the dynamics by requiring a link between the geometric fields on the one hand and the material quantities on the other hand by defining the fundamental Lagrangian: as it has been discussed in [18] when we develop the Lagrangian formalism we usually employ a geometric Lagrangian built on the torsional completion of the Ricci scalar, but this only includes torsion implicitly through the connection within the curvature while torsion in general should also be included explicitly in the action itself; since at the least-order derivative in the action the curvature appears linearly and torsion is squared, and because according to our restriction of having a completely antisymmetric torsion there is only one possible squared torsion term, consequently we have that the most general completely antisymmetric torsion completion of the gravitational least-order derivative dynamical action is given according to the following Lagrangian density L = a -16 πk 4 a 16 πk Q 2 + 1 16 πk G ≡-1 4 a Q 2 + 1 16 πk R in terms of the gravitational constant k and an additional torsional coupling constant that is in general different from the gravitational constant, and which can only be determined empirically. By varying this geometrical Lagrangian we get the system of field equations for the geometry coupling the completely antisymmetric torsion and curvature to the material quantities according to the expressions with completely antisymmetric spin S ρµν and energy T µν verifying the set of conservation laws given as usual by the following relationships which are such whenever the matter fields satisfy matter field equations. The material Lagrangian is given by the Dirac matter field Lagrangian as it is usually done; by complementing the geometrical Lagrangian with the Dirac matter field Lagrangian, the variation with respect to the matter field gives where the spin is completely antisymmetric and the energy is non-symmetric, and such that they verify the above conservation laws whenever are satisfied as matter field equations. Finally, when taken all together we have that the entire system of field equations is given by the equations and the matter field equations above given by as a direct calculation would show straightforwardly. Finally, it is worth noticing that our initial assumption of a completely antisymmetric torsion restrains the description to a completely antisymmetric spin allowing only the simplest spinor field to be defined without constraints, or equivalently, that such a restriction does not constitutes any loss of generality since we are interested in the simplest spinor field alone [14, 15]; thus this is the most general system of field equations we may have under the initial conditions with which we want to work, and so these are the field equations we will employ next: in this system of field equations, torsional quantities can be decomposed in terms of torsionless quantities and torsional contributions that can be converted through the torsion-spin coupling field equation (17) into spinorial potentials, so that the curvature-energy coupling field equations reduce to the form and the matter field equations after a Fierz rearrangement become where the gravitational field equations for the Ricci tensor are those we would have had in the torsionless case and the matter field equations are those we would have had if there were no torsion but Nambu-Jona-Lasinio potentials.", "pages": [ 3, 4, 5, 6 ] }, { "title": "1.1 Particle Condensate with Gravitational Corrections", "content": "We may proceed by specifying to the physical situation we want to study, namely that of WIMP fields forming a condensate over galactic distances: that quantum particles in the non-relativistic limit may condensate thus behaving as a single macroscopic field is known, as reviewed for example in [16, 19], and the idea that such macroscopic field may stretch to galactic scales has already been put forward, as it may be seen for instance in [20, 21]; the idea is that a bath of quantum particles would be a condensate of entangled entities behaving as a single macroscopic field filling galactic spaces. So the matter field we have will be interpreted as describing the particle condensate seen as a single macroscopic field with the extension of the galactic halo; in this article we will not discuss how this may occur but we will take it for given, developing its consequences for the galactic rotation curves. We will see that it will be possible with no further assumption to reproduce the observed behaviour of the rotating galaxy. As a first element, we have to specify the physical situation we will consider, that is the case of rotating galaxies: we will consider the case of stationary spherical symmetry, for which, in the frame centered in the center of the galaxy and with equatorial plane coincident with the rotation plane of the galaxy, described with coordinates ( t, r, θ, ϕ ) , it is widely known that the most general metric is given by ds 2 = A 2 dt 2 -[ B 2 dr 2 + r 2 ( dθ 2 +(sin θdϕ ) 2 )] in terms of two functions of the radial coordinate A ( r ) and B ( r ) to be determined; however in the weak-gravity slow-speed approximation used to study galaxies, it is also known that we may take A ≈ 1 B ≈ 1+2 V yielding for the curvature the approximated expression R t t ≈ R tt ≈ div grad V and we recall that the geodesic equation of motion /vectora + grad V ≈ 0 gives the acceleration felt by a test-body moving in the gravitational field. In this limit we have iψ γ ψ ≈ 0 and so with which temporal derivatives of the spinor are substituted with spatial derivatives of the spinor in the time-time component of the gravitational field equations so to remove the entire explicit temporal dependence, with /vector γ · /vector ∇ ψ denoting the spatial projection of the scalar product between the gamma matrices and the spinorial covariant derivatives: in slow-speed approximation, the stationary configurations of energy E verify E 2 -m 2 ≈ 2 m ( E -m ) and with the spinor in standard representation ψ =( φ † , -χ † ) the matter field equation splits as and the gravitational field equation is which is the Newton law we are seeking. The two matter field equations in their semi-spinorial form can be plugged into one another, after which one sees that the semi-spinor χ tends to vanish, justifying its usual name of small semi-spinor, while the semi-spinor φ is still present satisfying the field equation given by justifying its usual name of large semi-spinor satisfying the Schrödinger field equation, and so we also have that the term with spatial spinorial derivatives vanishes in the gravitational field equation leaving the simpler form with algebraic contributions of the large semi-spinorial component alone. Notice that the absence of any magnetic field allows us to consider the large semi-spinor's spin-up and spin-down projections as independent, so that we lose no generality in taking the semi-spinor field as φ † =( u ∗ , 0) with u verifying and with which now have to be solved, and this is what we are trying to do next.", "pages": [ 6, 7 ] }, { "title": "1.1.1 Large-density solutions for constant-valued curves", "content": "So far we have obtained the field equations with which we will work, that is the Schrödinger equation (28) and the Newton law (29): we must solve the Schrödinger equation, plugging the solution into the Newton law as to see what are the corrections to the galactic rotation curves in parallel to [22]; one point we need to remember is that the field is thought to represent condensates, which has high-density field distributions. In fact it is the high-density field distribution what makes relevant the non-linear potentials; these non-linear potentials can be as relevant or even more relevant than the linear term. We will see what happens to the field equations if the highest-order term overtakes all others. To proceed to the calculation, let us first take into account the Schrödinger equation and the Newton law written in stationary spherically symmetric coordinates and with the above high-density field distribution condition according to which we retain only the largest-power potential: according to such a condition the Schrödinger field equation is consequently given by the expression while the Newton law is given by for an approximately circular Keplerian orbit; this form of the Schrödinger equation has for possible solution the one given in the form which is square-integrable in the origin although not square-integrable at infinity, but since the condition of high-density field distribution forbids us to reach regions too far away then we are allowed not to care about regions too far outside the galactic halo. Inside the galactic halo such a solution is valid, and we plug it into the Newton law obtaining the following expression which has to be solved: writing the centrifugal acceleration in terms of the tangential velocity and taking for simplicity the equatorial plane, we get spelling that the tangential velocity is nearly constant, with constant value no longer containing any reference to universal constants apart from the purely geometrical ones. Notice that if the torsional constant is taken to be about the same as the Newton constant then this velocity approaches the speed of light, but for larger values of the torsional constant the velocity becomes smaller and for instance if it is about 10 8 times the Newton constant the tangential velocity's constant value becomes about 10 -3 times the speed of light, as measured. Now that we reached a result it is necessary to go back as to reconsider our assumptions and check their consistency with the result: the assumption that in the Schrödinger and Newton field equations (28-29) we retained only the largestdensity contributions so to get the Schrödinger and Newton approximated field equations (30-31) is condensed into a /planckover2pi1 2 u 2 /greatermuch m as a condition whose validity imposes the mass to be smaller than 10 -54 in Planck units; such a value is ridiculously small but it is not in contradiction with any known physical fact, since on the one hand light particles can energetically be produced easily but on the other hand WIMP particles are defined to have a low scattering amplitude allowing them to escape detection easily as well. Usually torsion is neglected and thus it is necessary to have a mass larger by 34 orders of magnitude for the WIMP particle bath to ensure the same physical effects: this enormous discrepancy in mass, while still retaining a similarly low capacity to interact with ordinary matter, should make the two types of WIMP particles quite easily distinguishable, so soon as experiments capable of measuring the mass of the WIMP candidate will be devised. When such experiments will be conceived they will be able to immediately rule out the present model if the masses are measured to be considerably larger than the presented upper limits, which means that this approach is clearly falsifiable and thus scientifically reliable. That the torsional coupling constant value a is in fact about 10 8 times the Newton constant may be much more difficult to check, because such a value has the property of being not too much larger then the Newton constant itself, which avoids a fine-tuning involving too many orders of magnitude, but at the same time this means that it is much smaller compared to the coupling constants of all other interactions, so that the torsionally-induced spin-contact interaction becomes relevant in particle physics much beyond the scale of the nuclear forces, and therefore beyond the possibility to measure it through scattering of elementary particles; this is a weird situation because in order to estimate the precise value of the torsional coupling constant one needs to test either the very large scale gravitational behaviour in cosmology or the very small scale scattering amplitude in particle physics, and while the former case is just reached in the case of Dark Matter the latter case is for the moment beyond the limits of the present accelerators. What this means is that for now we only have a single model to use the torsional interaction with such a value of the torsional coupling constant, and it is impossible to assess how likely this value really is by only studying a single physical effect.", "pages": [ 7, 8, 9 ] }, { "title": "Conclusions", "content": "So let us summarize what we have done in the present paper: if we were to distill all our hypotheses and assumptions we would find that: first of all, we have considered the hypothesis of existence of WIMP fields filling the underlying geometric background constituted by the most general torsional completion of gravity, that is the one in which we have that the spin-torsion and energy-curvature couplings have place in terms of different coupling constants, according to the most general SKED theory; secondly, we have assumed that the WIMP could form condensates retaining the torsionally-induced non-linear potentials as the most relevant ones, and that the weak-gravity slow-speed approximations were valid as it is usually done in studying galactic dynamics. On these bases, we merely have logically derived all the possible consequences: we have seen that the resulting dynamics of the WIMP condensate gives rise to gravitational corrections for which the galactic rotation curves have a tangential velocity that is shown to be nearly constant, as it is expected for Dark Matter. Because the value of the torsional constant can only be fixed empirically, and it has never been fixed so far, then we ignore what its value could actually be, but for a torsional constant of about 10 8 times the Newton constant the tangential velocity is not only constant but it also has the value it should have to match observations. It is important to stress it once more: all of the hypotheses and assumptions from which we decided to start, that is existence of WIMP, their possibility to condensate and the condition of having large torsional contributions compatible with a torsional constant of 10 8 in Planck units, are either accepted or seen as reasonable, as discussed in [20, 21] and [22]; also the methods of calculations that we have been employing here are those already commonly used in the known literature. There is no point, being it a principle or a computation, where we considered something never considered before: the novelty of our paper is that these principles have been considered together. And that these principles can stay together consistently is proven by the fact that our results provide a model for galactic rotation curves that fits observations adequately. The detailed description that comes out is that a WIMP bath that can condensate furnishes the condition to have torsional contributions with a constant of about 10 8 in Planck units as the source of relevant effects at galactic scales, and it does so in such a way that the tangential velocity turns out to be a constant with the measured value for galactic rotation curves: the fact there is no way in which such a tangential velocity might have been any different from constant makes this model more economic than the usual one where the additional hypothesis of DM density distribution with 1 r 2 behaviour had to be necessarily postulated, and the present model is also more predictive because its results are unavoidable while in the standard model the 1 r 2 behaviour is postulated with no reason other than eventually yielding the results that we already knew we should later obtain; both the present and standard model cannot say anything about the actual value of the constant velocity. The difference of the present and standard approaches is that if DM were not have been observed yet then the standard approach would have never been able to predict it while the present approach might have predicted it anyway; the fact that this approach does not predict the actual parameters of the problem should not be surprising since something must be set empirically in any approach to a description of the galactic rotation curves. Another observational difference between the present and standard model is that here the mass of the WIMP is some 34 orders of magnitude smaller, which is a dramatic discrepancy, but it may not necessarily be possible to test it in accelerators as high particle-production does not necessarily imply large cross-section 1 and thus cosmological indirect measures to discriminate these two mass values must be devised. A more serious experimental issue is that the torsional constant with a value of about 10 8 in Planck units has no impact on particle physics and in cosmology we have that the present treatment of Dark Matter constitutes its sole application, while on the other hand it would be desirable to have an alternative physical situation in which to see torsional effects for an independent evaluation of the torsional constant. On the other hand, Dark Matter is not only supported by galactic rotation curves, but also in terms of other galactic dynamics such as the Bullet Cluster, or more intriguing situations such as Tidal Galaxies, and even circumstances such as galactic formation, CMB and BAO, lying at the interface between cosmology and particle physics: further applications of the hypothesis presented here must also address all of these issues as well; clearly our primary interest here was a first application, so it is beyond the aim of this paper to address them at this stage, but in general what is expected to happen is that every time in which conditions arise for which the WIMP particles can condensate under gravitational pull then the torsionally-induced non-linear dynamics must be manifest. To be more specific is not possible because non-linear behaviours are too sensitive to the environmental conditions of each single specific application. To conclude, we would like to use the present study as a specific example of a more general problem: as already discussed, in the standard approach torsion is neglected and then additional assumptions have to be postulated; in general, this signifies that we are ready to refuse something we have been given as gift to buy extra assumptions in order to have the means to achieve what we would have accomplished had not we thrown anything away, and this could render lame a model that otherwise might have worked well instead. We believe this study has shown that too.", "pages": [ 9, 10, 11 ] } ]

[ { "title": "ABSTRACT", "content": "International Journal of Modern Physics D c © World Scientific Publishing Company", "pages": [ 1 ] }, { "title": "L. HAMOLLI, M. HAFIZI", "content": "Department of Physics, University of Tirana, Albania", "pages": [ 1 ] }, { "title": "A.A. NUCITA", "content": "Department of Mathematics and Physics Ennio De Giorgi and INFN, University of Salento, CP 193, I-73100 Lecce, Italy Received (received date) Revised (revised date) Free-floating planets are recently drawing a special interest of the scientific community. Gravitational microlensing is up to now the exclusive method for the investigation of free-floating planets, including their spatial distribution function and mass function. In this work, we examine the possibility that the future Euclid space-based observatory may allow to discover a substantial number of microlensing events caused by free-floating planets. Based on latest results about the free-floating planet mass function in the mass range [10 -5 , 10 -2 ] M /circledot , we calculate the optical depth towards the Galactic bulge as well as the expected microlensing rate and find that Euclid may be able to detect hundreds to thousands of these events per month. Making use of a synthetic population, we also investigate the possibility of detecting parallax effect in simulated microlensing events due to free-floating planets and find a significant efficiency for the parallax detection that turns out to be around 30%.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Gravitational microlensing is at present the only observational technique that allows the detection of extremely faint or even completely dark objects, when their gravitational field acts as a lens to magnify background source stars 1 . A gravitational lens is characterized by its Einstein ring radius, which is the radius of the ring image formed when the observer, the lens and the source are perfectly aligned. Here M is the mass of the lens and x = D l /D s is the normalized lens distance, whereas D s and D l are the source-observer and lensobserver distance, respectively. In a microlensing event, the image separation is too small to be resolved and the observable feature is the variation in time of the light magnification, as due to the lens-source relative motion. The key parameter of the microlensing light curve is the Einstein radius crossing time given by where v T is the relative transverse velocity between the lens and the source. In the simplest case (named standard case), when both the lens and the source can be considered as point-like objects and, additionally, their relative motion with resepect to the observer is assumed to be linear, the amplification of the source star follows the Paczy'nski profile 1 where is the separation between the lens and the line of sight in units of R E and u 0 is the minimum separation (impact parameter) obtained at the moment of the peak magnification t 0 . The light curve obtained by equation (3) is, evidently, symmetric around t 0 . When the projected source encounters the Einstein ring of the lens, i.e. when the projected separation is u = 1, the source amplification takes a specific value defined as the threshold amplification A th = 1 . 34. For space-based telescopes, due to the absence of seeing effects, the amplification threshold may be much smaller than 1.34, with a corresponding much larger value for the impact parameter. For A th = 1 . 001, as expected for the Euclid telescope, the maximum value of u in equation (3) turns out to be u max = 6 . 54. A standard microlensing light curve is described by three parameters, t 0 , t E and u 0 , but only one of them, the Einstein radius crossing time, t E , contains information about the lens. As can be seen from equations (1) and (2), the event duration is determined by three unknown parameters of the lens: its mass M , the transverse velocity v T and the distance D l . There are several methods that have been proposed for breaking the microlensing parameter degeneracy. Among them there is the parallax effects, occurring due to the motion of the Earth around the Sun and the effect of the relative accelerations among the observer, the lens and the source 2 . These second order effects induce small deviations in the light curve (with respect to the Paczy'nski profile), which may be extremely useful to break, at least partially, the parameter degeneracy problem in microlensing observations. Recently, it has been reported the observation by MOA-II of a galactic population of free objects with planetary masses, named free-floating planets (FFPs) 3 . Due to the intrinsic faintness of the FFPs, it is very hard to observe them directly. Hence, the gravitational microlensing method may be a suitable technique to detect such objects when they act as lenses for the background stars. Of course, as it will be clarified in the following, the small planetary masses imply microlensing events with very short time-scales. The purpose of this paper is the investigation of the traces of FFPs, among other galactic lens populations, in microlensing events that might be observed serendipitously during the planned observations of the Euclid satellite towards the Galactic bulge. In Section 2, we discuss the FFP mass function, their spatial distribution and the adopted velocity distribution. In Section 3, we review the microlensing method and its parameters in the case of Euclid observations. In Section 4, we show how the parallax effect may help in solving, at least partially, the degeneracy of parameters in the microlensing curves. Our main results are presented and discussed in Section 5, while in Section 6 we draw the main conclusions of this work.", "pages": [ 1, 2, 3 ] }, { "title": "2. Planetary population", "content": "In a recent survey of the Galactic bulge, the MOA-II collaboration 3 reported the discovery of planetary-mass objects either very distant from their host stars (more than 100 AU away) or entirely unbound. By analyzing the timescale distribution of all the observed microlensing events, they found a statistically significant excess of events with timescale t < 2 days as compared to the number of expected events from the standard Galactic model. The stellar mass-function for the standard Galactic model is generally expressed, for stars with mass ≤ 1 M /circledot and brown dwarfs (BDs), with three power laws of the form  A best-fit procedure to the observed microlensing events due to FFPs has also allowed Sumi et al. 3 to extend and constrain the power-law mass function at the low-mass regime of the FFPs The derived number of planetary mass objects per star turns out to be very large, although rather poorly constrained: N PL = 5 . 5 +18 . 1 -4 . 3 , mainly due to the poor precision of the lens mass estimate below 10 -4 M /circledot . The abrupt change from α BD = 0 . 49 to α PL = 1 . 3 favors the idea of a separate population, whose formation process is different from that of stars and BDs. These objects may have formed in proto-planetary disks and subsequently scattered into unbound or very distant orbits, becoming FFPs. From the likelihood contours of the power-law indices found in the brown dwarf and planetary-mass regime 3 , we get a useful correlation between α BD and α PL Regarding the spatial distribution of the FFPs, we assume that they are distributed as the stars in the Milky way 4 , 5 , 6 . Hence, we considered the following density distributions: 1. exponential thin disk, in cylindrical coordinates R (the galactocentric distance in the galactic plane) and z (the distance from the galactic plane). The scale parameters are H ∼ 0 . 30 kpc, h ∼ 3 . 5 kpc; R 0 = 8 . 5 kpc is the local galactocentric distance. 2. exponential thick disk, with H ∼ 1 kpc, h ∼ 3 . 5 kpc and R 0 = 8 . 5 kpc. 3. triaxial bulge 5 , 6 , 7 where a = 1 . 49 kpc, b = 0 . 58 kpc, c = 0 . 40 kp c. For the FFP velocity distribution, we assume for each coordinate the Maxwellian distribution 8 , 9 where the coordinates ( x, y, z ) have their origin at the galactic center and the x and z-axes point to the Sun and the north Galactic pole, respectively. We are interested only to the perpendicular velocity with respect to the line of sight, namely to y and z components. For lenses in the Galactic bulge we use the mean velocity components v y = v z = 0, with dispersion σ y = σ z = 100 km/s; for lenses in the Galactic disk we use the mean velocity components v y = 220 km/s, v z = 0, with dispersion velocity σ y = σ z = 30 km/s for the thin disk and σ y = σ z = 50 km/s for the thick disk.", "pages": [ 3, 4 ] }, { "title": "3. Microlensing events towards the Galactic bulge", "content": "Several microlensing surveys with relatively high image sampling have been undertaken until now towards the Galactic bulge by the MOA (Microlensing Observations in Astrophysics) Collaboration 10 and the OGLE (Optical Gravitational Lensing Experiment) Collaboration 11 (to cite only some of them), with the aim of searching for MACHOs (Massive Astrophysical Compact Halo Objects) and exoplanets. These surveys (undertaken since about two decades) have allowed the detection of several thousands of microlensing events, most of which are due to self-lensing (stars either in the Galactic disk and bulge). Ground-based observations may detect low-mass lenses (short time duration events) only with great difficulties, so to search for lens masses below 0 . 01 M /circledot , as for FFPs, space-based observations are needed. At present, there are two space-based missions which are planned for detecting microlensing events towards the Galactic bulge: the Wide-Field Infrared Survey Telescope (WFIRST) and Euclid. Euclid is a Medium Class mission of the ESA (European Space Agency), which is scheduled to be launched in 2017. For ten months, not necessarily consecutive 12 , it will perform microlensing observations towards the Galactic bulge. The galactic coordinates of the Euclid line of sight are b = -1 . 7 · , l = 1 . 1 · , the distance of observation can be considered D s = (7 -10) kpc, with mean value at D s = 8 . 5 kpc and the observing image rate (cadence) is expected to be about 20 min. In order to study the expected microlensing events that may be detected by the Euclid observatory we start by evaluating the microlensing optical depth and the event rate. The microlensing optical depth is defined as the probability that at any time a random star is magnified more than the threshold amplification A th = 1 . 34 by a lens belonging to a given population of lenses. It is given by (see e.g. 13 , 14 ) where ρ ( M,x ) is the mass density of the lens population; x = D l /D s . The microlensing rate is the number of events per unit time and per monitored star due to the lens population. It is given by (see e.g. 13 , 14 ) where v l , v s and v t are the lens, the source and the microlensing tube two-velocities in the plane transverse to the line of sight. The velocity distribution functions f ( v l ) and f ( v s ) are assumed to have Maxwellian forms 8 , 9 , with one-dimensional dispersion velocities different for each lens and source population. The tube velocity is given by where v /circledot is the local velocity transverse to the line of sight and θ is the angle between v /circledot and v s . In the case of observations towards the Galactic bulge, the source stars are mostly bulge stars which are distributed following, as usual, the same triaxial mass density model as given in eq. (10) with ρ Bulge 0 = M b / (8 πabc ), where M b /similarequal 2 × 10 10 M /circledot , a = 1 . 49 kpc, b = 0 . 58 kpc and c = 0 . 40 kpc. The limiting line flux of Euclid Telescope is estimated to be F l = 3 × 10 -19 Js -1 m -2 , 12 whereas the flux of a Sun-like star situated at the Galactic center is F /circledot = 4 . 44 × 10 -16 Js -1 m -2 . Based on the mass-luminosity relation L L /circledot = ( M M /circledot ) 2 . 4 for low-mass stars ( M < 0 . 8 M /circledot ), it can be directly shown that the telescope can observe all bulge stars. The mean mass for bulge stars is < M > = 0 . 27 M /circledot , found by using the Salpeter mass function 15 dN dM ∼ M -2 . 4 ; the Euclid's field of view is 0 . 54 square degree, hence the number of source stars in Euclid microlensing observations will be N ED = 2 . 3 × 10 8 . This number has to be multiplied by the microlensing rate (13) and the time of observation (in the following we take t obs = 1 month) to get an estimate of the number of microlensing event that we expect to be detectable by the Euclid telescope.", "pages": [ 4, 5, 6 ] }, { "title": "4. Parallax effect", "content": "The parallax effect due to the motion of the Earth around the Sun may leave in microlensing events some observable feature which can be used to break the degeneracy of microlensing parameters, or at least to constrain the microlensing parameter space. Here, we are focusing on the investigation of the parallax traces left on microlensing events that will possibly detected by the Euclid telescope. In order to estimate the parallax effects on the microlensing light curves we make use of the following useful geometrical relations 16 where ξ ( t ) is implicitly given by Here, a ⊕ is the semi-major axis of the Earth orbit around the Sun, /epsilon1 = 0 . 0167 is the Earth orbit eccentricity and ρ is the length of the semi-major axis projected onto the lens plane measured in Einstein radii. The position of the source stars is characterized by the parameters φ, χ and ψ in the relations (15) which give, respectively, the longitude measured in the ecliptic plane from perihelion towards the Earth motion, the latitude measured from the ecliptic plane towards the northern point of the ecliptic and the rotation angle in the lens plane which describes the relative orientation of velocity v T to the sun-earth system. We find that the deviations on the microlensing light curve due to the parallax effect depend substantially on the Earth position in its orbit at the time of the maximum amplification and get the largest value when ξ 0 = 165 · that happens in June. In the following calculations, we have assumed that the Euclid satellite is in the best position in its orbit in order to maximize the parallax effect on the microlensing event light curves. Using the usual transformation relations between coordinate systems, we find the following values for the Euclid's line of sight towards the Galactic bulge: φ /similarequal 167 . 8 · and χ /similarequal -5 . 4 · . To the aim of estimating the number of events for which the parallax feature affects the microlensing light curves as observed by Euclid, we make use of Monte Carlo numerical simulations by generating microlensing events towards the field of the sky planned to be observed by the Euclid telescope. Here, we briefly describe the adopted strategy. In particular, we draw In each case we choose the same position of the Earth, ξ 0 = 165 · , at the time t 0 of the light curve peak amplification. The parallax effect is estimated by calculating the residuals between the light curve A p ( t ) containing the parallax effect from eqs. (15) and the corresponding standard curve A s ( t ) (3), i.e. Res = | A s ( t ) -A p ( t ) | . As an example, in Fig.1 we show the standard curve, the parallax curve and residuals in the case of a freefloating planet with mass 10 -3 M /circledot at distance D l = 4 . 5 kpc from Earth. As one can see, in this case the residuals are up to /similarequal 12%.", "pages": [ 6, 7 ] }, { "title": "5. Results", "content": "We have calculated the microlensing optical depth from eq. (12) and the microlensing rate from eq. (13) for all the lens populations towards the Galactic bulge: FFPs, BDs and stars distributed in the thin disk, thick disk and the Galactic bulge. For their spatial distribution we use eqs. (8), (9), (10), coupled with the mass functions (5) and (6). For stars we assume the Salpeter mass function dN dM ∼ M -2 . 4 , while the relation between α BD and α PL is defined by equation (7). Brown dwarfs are faint objects distinguished only in Sun surroundings. Recently, the ratio R (in the Solar Neighbourhood) between the number of stars with mass in the range [0 . 08 , 1] M /circledot and BDs in the range [0 . 03 , 0 . 08] M /circledot , has been estimated by 17 . From Table 1 in 17 we adopt a mean value of R /similarequal 5 . 1 and assume that it applies to the whole Galaxy. In Table 1, we show the results of our calculations of the optical depth for FFPs and BDs considered as lenses. We perform separate calculations for each structure of the galaxy (bulge, thin disk and thick disk) and different values of α PL . The number of FFPs per star is chosen following Sumi et al. 3 : the lowest value 8 N PL = 1 . 2, mid value N PL = 5 . 5 and highest value N PL = 23 . 6. The optical depth yielded by stars is not dependent on α PL : our calculations give 2 . 59 × 10 -6 for bulge stars, 4 . 26 × 10 -7 for thin disk stars and 2 . 42 × 10 -7 for thick disk stars, respectively. We remark that the contribution of the bulge populations in the optical depth is the most important one. In Table 2, we show the results of our calculations for the microlensing rate (the probability to have a microlensing event per star during the observation time of one month), in the same conditions as above. The microlensing rate by stars is of course independent on α PL , its value being 2 . 41 × 10 -6 for bulge stars, 1 . 27 × 10 -7 for thin disk stars and 1 . 69 × 10 -7 for thick disk stars, respectively. The dominant contribution in the microlensing rate is again that of the bulge lens populations. We then estimate the number of microlensing events expected to be detectable by the Euclid telescope (taking A th = 1 . 001 and therefore u max = 6 . 54) multiplying the microlensing event rate by the number of source stars in the Euclid field of view N ED = 2 . 3 × 10 8 and by the observation time duration. In Table 3, we show the results of our calculations for the estimated number of microlensing events per month, where FFPs and BDs are considered as lenses. We perform separate calculations for each structure of the galaxy: bulge, thin disk and thick disk. The number of microlensing events per month produced by stars is not dependent on α PL , its value is 3657 for bulge stars, 193 for thin disk stars and 265 for thick disk stars. The bulge population contribution is the most important one also in this case. In Fig.2 we present our estimations for the total number of microlensing events due to BDs and FFPs per month, for different values of α PL . By numerical simulations we produce a large number of microlensing events caused by the population of the FFPs. We assume that a microlensing event can be detected if in its light curve there are at least 8 points in which the amplification is bigger than the threshold amplification A th = 1 . 001. The photometric error in this case is 0.1% In the case of Euclid telescope, the expected curve will contain points determined every 20 minutes, that means that any detectable event has to have a duration larger than 2.67 hours. For estimating the parallax effect on the observed light curves, we consider only those containing at least 8 points with Res > 0 . 001 inside Einstein ring We retain all synthetic events with residuals fulfilling the above-mentioned condition. Therefore, the efficiency for parallax effect detection is given by the ratio between the number of these events and the total number of detectable events. In Fig.3 we show our results for the parallax efficiency in microlensing events caused by FFPs and expected to be detectable by Euclid are shown. We consider three separate distributions, bulge, thin and thick disk lenses and show the parallax efficiency with respect to the value of α PL . For example, as one can see, in the case of α PL = 1 . 3 the parallax efficiency caused by bulge FFPs is 32%, by thin disk FFPs is 31% and by thick disk FFPs is 27%. So, approximately in 30% of the detectable events, the parallax effect due to the Earth motion may be detectable by the Euclid telescope, allowing to partially resolve the parameter degeneracy problem in this kind of observations and constrain the distance to the FFPs. This should allow to estimate not only the number of FFPs throughout the Milky Way, but also their spatial distribution.", "pages": [ 7, 8, 9, 10, 11, 12 ] }, { "title": "6. Conclusions", "content": "In this paper we investigate the possible observation of free-floating planets (in addition to normal stars and brown dwarfs) towards the Galactic bulge by the future Euclid space-based observatory, via detection of microlensing light curves. These events, either considered statistically or individually, are an important base of knowledge to better characterize the galactic populations of objects in addition to normal stars. For the calculation of the optical depth and the microlensing rate for brown dwarfs we assume that they are distributed like stars, spatially and in the velocity space. This should also be true for free-floating planets, based on the idea that these objects are most likely formed in proto-planetary disks and subsequently scattered into unbound or very distant orbits. The number of these objects per star is poorly constrained, as is also the slope of their mass distribution. We find that the optical depth and the microlensing rate for FFPs towards the Galactic bulge are much smaller than for stars, but slightly higher than for brown dwarfs. The highest contribution for the three object populations always comes from bulge objects. The theoretical optical depth and microlensing rate depend on the power law index of the FFP mass function, hence the corresponding observed values can be considered as sources of information for the still largely unknown mass function of brown dwarfs and FFPs. By theoretical calculations we predict that a considerably large number of microlensing events produced by free-floating planets towards the Galactic bulge are potentially observable by the Euclid satellite. We also take into account the deviations in the microlensing light curves due to FFPs induced by the Earth parallax effect. We find that these deviations depend substantially on the Earth position in its orbit around the Sun at the time of the event maximum amplification and get the largest value in June. By numerical simulations we also find that the efficiency (that is the ratio between the number of events due FFPs that fulfill equation (18) with respect to the total number of detectable events) of detecting the Earth parallax effect in the light curves due to FFPs is potentially interesting since the parallax effect turns out to be detectable in about 1/3 of all observable events (see Fig.3). We emphasize that the observation of this effect may allow to constrain the FFP distances, which is a fundamental information necessary to investigate how FFPs are distributed throughout the Milky Way. This, in turn, is an important issue in order to establish their origin. As a final remark we caution that the short time-scale microlensing features, such as those expected due to the Earth parallax, may be confused due to the socalled red-noise effect. Indeed, photometric observations are generally affected by the presence of the Earth atmosphere that is a source of correlated noise. A way to circumvent this problem is to use space-based telescopes, an opportunity that has clearly many advantages. However, the improved sensitivity of space-based observations have unveiled a new source of noise related to the intrinsic stellar variability that induces the red-noise, connected to the correlated time-series. This effect has been studied in connection to the transit technique when searching for exoplanets as observed by space telescopes such as CoRoT and Kepler 18 , 19 . The detailed study of this effect in connection to the microlensing lighcurves, in particular in connection to the searches for free-floating planets, is left to a following work. We would like to thank the colleagues who have discussed the subject of this paper with us and particularly Francesco De Paolis for guidance and useful comments.", "pages": [ 12, 13 ] } ]

[ { "title": "ABSTRACT", "content": "International Journal of Modern Physics D c © World Scientific Publishing Company", "pages": [ 1 ] }, { "title": "R. SHARMA", "content": "Department of Physics, P. D. Women's College, Jalpaiguri 735 101, India. E-mail: [email protected]", "pages": [ 1 ] }, { "title": "B. S. RATANPAL", "content": "Department of Applied Mathematics, Faculty of Technology and Engineering, The M. S. University of Baroda, Vadodara 390 001, Gujarat, India. E-mail: [email protected] Received Day Month Year Revised Day Month Year A class of solutions describing the interior of a static spherically symmetric compact anisotropic star is reported. The analytic solution has been obtained by utilizing the Finch and Skea ( Class. Quant. Grav. 6 (1989) 467) ansatz for the metric potential g rr which has a clear geometric interpretation for the associated background spacetime. Based on physical grounds appropriate bounds on the model parameters have been obtained and it has been shown that the model admits an equation of state (EOS) which is quadratic in nature. Keywords : General relativity; Exact solution; Compact star; Equation of state. PACS numbers:04.20.-q; 04.20.Jb; 04.40.Dg; 12.39.Ba", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "To construct models of relativistic compact stars, it is imperative to know the exact composition and nature of particle interactions at extremely high density regime. If the equation of state (EOS) of the material composition of a compact star is known, one can easily integrate the Tolman-Oppenheimer-Volkoff (TOV) equations to analyze the physical features of the star. The problem is that we still lack reliable information about physics of particle interactions at extremely high density that may be found in the 'natural laboratories' of compact astrophysical objects. The objective of the present paper is to construct models of equilibrium configurations of relativistic compact objects when no reliable information about the composition and nature of particle interactions are available. This can be achieved by generating exact solutions of Einstein's field equations describing the interior of a static spherically symmetric relativistic star. However, finding exact solutions of Einstein's field equations is extremely difficult due to highly non-linear nature", "pages": [ 1 ] }, { "title": "2 R. Sharma and B. S. Ratanpal", "content": "of the governing field equations. Consequently, many simplifying assumptions are often made to tackle the problem. Since General Relativity provides a mutual correspondence between the material composition of a relativistic star and its associated space-time, we will adopt a geometric approach to deal with such a situation. In this approach, a suitable ansatz for one of the metric potentials with a clear geometric characterization of the associated space-time metric will be prescribed to determine the other. Such a method was initially proposed by Vaidya and Tikekar 1 ; subsequently the method was utilized by many to generate and analyze physically viable models of compact astrophysical objects (see for example, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and references therein). In the present work, we shall utilize the Finch and Skea 12 ansatz for the metric potential g rr to determine the unknown metric potential g tt describing the interior space-time of a static spherically symmetric stellar configuration. Note that the t = constant hyper-surface of the background space-time corresponding to the Finch and Skea 12 ansatz is paraboloidal in nature 13 . In our work, we shall incorporate a general anisotropic term in the stress-energy tensor representing the material composition of the star. We would like to point out here that anisotropic matter is a very exotic choice for compact objects like neutron stars. Nevertheless, in the past, impacts of anisotropic stresses on equilibrium configurations of relativistic stars have been extensively investigated by Bowers and Liang 14 and Herrera and Santos 15 . Local anisotropy at the interior of an extremely dense object may occur due various factors such as the existence of type 3A superfluid 14 , 16 , 17 , phase transition 18 , presence of electromagnetic field 19 , etc. In 33, it has been shown that influences of shear, electromagnetic field etc. on self-bound systems can be absorbed if the system is considered to be anisotropic, in general. Mathematically, anisotropy provides an extra degree of freedom in our system of equations. Therefore, on top of Finch and Skea ansatz, we shall utilize this freedom to assume a particular pressure profile to solve the system. In the past, a large class of exact solutions corresponding to spherically symmetric anisotropic matter distributions have been found and analyzed (see for example, Ref. 20, 21, 22, 23, 24, 25, 26, 27). Maharaj and Chaisi 28 have prescribed an algorithm to generate anisotropic models from known isotropic solutions. Dev and Gleiser 29 , 30 , 31 have studied the effects of anisotropy on the properties of spherically symmetric gravitationally bound objects and also investigated stability of such configurations. It has been shown that if the tangential pressure p ⊥ is greater than the radial pressure p r of a stellar configuration, the system becomes more stable. Impact of anisotropy has also been investigated by Ivanov 32 . In an anisotropic stellar model for strange stars developed by Paul et al 34 , it has been shown that the value of the bag constant depends on the anisotropic parameter. For a charged anisotropic stellar model governed by the MIT bag model EOS, Rahaman et al 35 have shown that the bag constant depends on the compactness of the star. Making use of the Finch and Skea 12 ansatz, Tikekar and Jotania 13 have developed a two parameter family of solutions of Einstein's field equations and showed the relevance of the class of solutions for the description of strange stars. A core-envelope type model describing a gravitationally bound object with an anisotropic fluid distribution has been obtained in Ref. 36, 37, 38. In our work, following Finch and Skea 12 prescription, we have constructed a nonsingular anisotropic stellar model satisfying all the necessary conditions of a realistic compact star. Based on physical grounds, we have prescribed bounds on the model parameters and generated the relevant EOS for the system. An interesting feature of our model is that the solution admits a quadratic EOS. It is often very difficult to generate an EOS ( p = p ( ρ )) from known solutions of Einstein's field equations due to mathematically involved expressions. In fact, in most of the models involving an EOS, the EOS is prescribed a priori to generate the solutions. For example, Sharma and Maharaj 20 have obtained an analytic solution for compact anisotropic stars where a linear EOS was assumed. Thirukkanesh and Maharaj 39 have assumed a linear EOS to obtain solutions of an anisotropic fluid distribution. Feroze and Siddiqui 40 and Maharaj and Takisa 41 have separately utilized a quadratic EOS to generate solutions for static anisotropic spherically symmetric charged distributions. A general approach to deal with anisotropic charged fluid systems admitting a linear or non-linear EOS have been discussed by Varela et al 42 . In our model, we do not prescribe the EOS; rather the solution imposes a constraint on the EOS corresponding to the material composition of the highly dense system. The paper has been organized as follows. In Section 2, the relevant field equations describing a gravitationally bound spherically symmetric anisotropic stellar configuration in equilibrium have been laid down. We have solved the system of equations in Section 3 and analyzed bounds on the model parameters based on physical grounds. Physical features of the model have been discussed in Section 4. We have also generated an approximated EOS in this section which has been found to be quadratic in nature. In Section 5, we have concluded by pointing out some interesting features of our model.", "pages": [ 2, 3 ] }, { "title": "2. Field equations", "content": "We write the interior space-time of a static spherically symmetric stellar configuration in the standard form where ν ( r ) and λ ( r ) are yet to be determined. We assume that the material composition of the configuration is anisotropic in nature and accordingly we write the energy-momentum tensor in the form where ρ and p represent energy-density and isotropic pressure of the system and u i is the 4-velocity of fluid. The anisotropic stress-tensor π ij is assumed to be of the form", "pages": [ 3 ] }, { "title": "4 R. Sharma and B. S. Ratanpal", "content": "where S = S ( r ) denotes the magnitude of anisotropy and C i = ( 0 , -e -λ/ 2 , 0 , 0 ) is a radially directed vector. We calculate the non-vanishing components of the energy-momentum tensor as which implies that the radial pressure and the tangential pressure will take the form respectively. Therefore, magnitude of the anisotropy is obtained as The Einstein's field equations corresponding to the space-time metric (1) and the energy-momentum tensor (2) are obtained as (in relativistic units with G = c = 1) Defining the mass within a radius r as we rewrite the field equations (8)-(10) in the form", "pages": [ 4 ] }, { "title": "3. Interior solution", "content": "To solve the system of equations (12) - (14), we make use of the Finch and Skea 12 ansatz for the metric potential g rr as where R is a curvature parameter. The ansatz (15) has a geometric interpretation and was previously found to generate solutions for compact stellar objects 13 . Note that the t = constant hyper-surface of the metric (1) for the ansatz (15) represents a paraboloidal space-time immersed in 4-Euclidean space-time. The energy density and mass function are then obtained as Combining Eqs. (13) and (17), we get To integrate Eq. (18), we choose 8 πp r in the form where p 0 > 0 is a parameter such that p 0 R 2 denotes the central pressure. The particular form of the radial pressure profile assumed here is reasonable due to the following facts: For p 0 > 0, Eq. (20) implies that dp r /dr < 0, i.e., the radial pressure is a decreasing function of the radial parameter r . At a finite radial distance r = R the radial pressure vanishes which is an essential criterion for the construction of a realistic compact star. The curvature parameter R is then identified as the radius of the star. Substituting Eq. (19) in Eq. (18), we obtain which is integrable and yields where C is a constant of integration. Thus, the interior space-time of the configuration takes the form", "pages": [ 4, 5 ] }, { "title": "6 R. Sharma and B. S. Ratanpal", "content": "which is non-singular at r = 0. Making use of Eqs. (14), (16), (19) and (21), we determine the anisotropy as Note that anisotropy vanishes at the centre ( r = 0) as expected. The tangential pressure takes the form where,", "pages": [ 6 ] }, { "title": "3.1. Determination of the model parameters", "content": "Our model has three independent parameters, namely, p 0 , C and R . The requirement that the interior metric (23) should be matched to the Schwarzschild exterior metric across the boundary r = R of the star together with the condition that the radial pressure should vanish at the surface ( p r ( r = R ) = 0) help us to determine these constants. Note that the form of the radial pressure profile is such that the condition p r ( r = R ) = 0 itself becomes the definition of the radius R of the star in this construction. Matching the relevant metric coefficients across the boundary R then yields where M is the total mass enclosed within the boundary surface R . If radius R is known, Eq. (27) can be utilized to determine the total mass M of the star and vice-versa. For a given value of p 0 , Eq. (28) determines C . Note that in this model, p 0 /R 2 corresponds to the central pressure and, therefore, for a given mass ( M ) or radius ( R ), if the central pressure is specified the system is completely determined.", "pages": [ 6 ] }, { "title": "3.2. Bounds on the model parameters", "content": "Following Finch and Skea 43 and Delgaty and Lake 44 , we impose the following conditions on our system so that it becomes a physically acceptable model. Note that the requirements (i) and (ii) imply that the weak and dominant energy conditions are satisfied. Condition (iii) ensures regular behaviour of the energy density and two pressures while condition (iv) is invoked to ensure that the sound speed be causal. In addition, for regularity, we demand that the anisotropy should vanish at the centre, i.e., p r = p ⊥ at r = 0. From Eq. (24), we note that the anisotropy vanishes at r = 0 and S ( r ) > 0 for 0 < r < R . Interestingly, for a particular choice p 0 = 1, the anisotropy also vanishes at the boundary r = R in this construction. From Eq. (16), it is obvious that ρ > 0, and decreases radially outward. We have already stated that p 0 /R 2 corresponds to the central pressure which implies that p 0 > 0. From Eq. (25), it can be shown that for p ⊥ > 0, we must have p 0 < 1. Thus, a bound on p 0 is obtained as To obtain a more stringent bound on p 0 , we evaluate at two different points. At the centre of the star ( r = 0) and the boundary of the star ( r = R ), it takes the form which will be negative if p 0 > 1 3 . Therefore, a more stringent bound on the parameter p 0 is obtained as To verify whether the bound on p 0 satisfies the causality condition 0 < dp r dρ < 1, we combine Eqs. (20) and (29), to yield", "pages": [ 6, 7 ] }, { "title": "8 R. Sharma and B. S. Ratanpal", "content": "Now, at the centre of the star ( r = 0), dp r dρ < 1 if the condition p 0 < 1 . 6667 is satisfied and at the boundary of the star ( r = R ), dp r dρ < 1 if the condition p 0 < 3 is satisfied. Both these restrictions are consistent with the requirement given in (34). Similarly, we evaluate throughout the star. At the centre ( r = 0), the requirement dp ⊥ dρ < 1 puts a constraint on p 0 such that p 0 < 1 . 2250. At the boundary of the star the corresponding requirement is given by p 0 < 4 . 3333. Both these requirements are also consistent with the bound 1 3 < p 0 ≤ 1.", "pages": [ 8 ] }, { "title": "3.3. Stability", "content": "We now investigate the bound on the model parameters based on stability. To check stability of our model, we shall use Herrera's 45 overtuning technique which states that the region for which radial speed of sound is greater than the transverse speed of sound is a potentially stable region. The radial and tangential sound speeds in our model are obtained as Herrera's 45 prescription demands that we must have v 2 st -v 2 sr < 0 throughout the star. Now, at the centre of the star For ( v 2 st -v 2 sr ) ( r =0) < 0, it is required that -3 + 8 p 0 -p 2 0 < 0, i.e., p 0 < 0 . 3944. At the boundary of the star, we have which is obviously negative for 1 3 < p 0 < 0 . 3944. Therefore, our model is physically reasonable and stable if the following bound is imposed: 1 3 < p 0 < 0 . 3944.", "pages": [ 8 ] }, { "title": "4. Physical analysis", "content": "We now analyze the gross behaviour of the physical parameters of our model such as energy density and two pressures at the interior of the star. For a particular choice p 0 = 0 . 36 (consistent with the bound), plugging in c and G at appropriate places, we have calculated the mass M , central density ρ c and surface density ρ R of a star of radius R . This has been shown in Table 1. We note that the central density in each case (except V III , where we have assumed a comparatively larger radius which in turn has generated a bigger mass) lies above the deconfinement density 46 , 47 ∼ 700 MeV fm -3 which implies that quark phases may exist at the interiors of such configurations. Variations of the physical parameters for a particular case V I have been shown in Fig. (5)-(5). The figures clearly indicate that the physical parameters are well-behaved and all the regularity conditions discussed above are satisfied at all interior points of the star. Moreover, the assumed parameters generate a stable configuration as shown in Fig. (5).", "pages": [ 8, 9 ] }, { "title": "4.1. Generating approximated EOS", "content": "Having derived a physically acceptable model, question to be asked is, what kind of material composition can be predicted for the stellar configurations admissible in this model? In other words, what would be the EOS corresponding to the material compositions of the configurations constructed from the model? Though construction of an EOS is essentially governed by the physical laws of the system, one can parametrically relate the energy-density and the radial pressure from the mathematical model which may be useful in predicting the composition of the system. Making use of Eqs. (16) and (19), we have plotted variation of the radial pressure against the energy-density as shown by the solid curve in Fig. (5). Our intention now is to prescribe an approximate EOS which can produce similar kind of curve. Though, in principle, a barotropic EOS ( p r = p r ( ρ )) can be generated from Eqs. (16) and (19) by eliminating r , we assume that the relevant EOS has the form where ρ 0 , α and β are constants. We make use of this EOS to plot ρ vs p r which turns out to be almost identical to the curve generated from the analytic model if we set ρ 0 = -0 . 36, α = 9 . 6 × 10 -5 and β = 7 . 2 × 10 -8 (dashes curve in Fig. (5)). Though this has been shown to be true for a particular choice (case V I ), it can be", "pages": [ 9 ] }, { "title": "10 R. Sharma and B. S. Ratanpal", "content": "shown that the model admits the quadratic EOS (41) for different choices of the parameters as well.", "pages": [ 10 ] }, { "title": "5. Discussion", "content": "Making use of Finch and Skea 12 ansatz, we have generated exact solutions of Einstein's field equations representing a static spherically symmetric anisotropic stellar configuration. Bounds on the model parameters have been obtained on physical grounds and it has been shown that model is stable for 1 3 < p 0 < 0 . 3944. Note that p 0 /R 2 denotes the central density in this model and, therefore, the bound indicates that for a given radius or mass arbitrary choice of the central density is not permissible in this model. We have shown that the model admits an EOS which is quadratic in nature. Mathematically, this may be understood in the following manner. The ansatz (15), together with the assumption (19), generates an anisotropic stellar model whose composition may be described by the EOS of the form (41). Note that in Ref. 40, 41, quadratic EOS have been assumed a priori to obtain exact solutions of Einstein's field equations. In this paper, we have shown that such an assumption is consistent with an analytical model which has been constructed by making use of the Finch and Skea 12 ansatz having a clear geometrical representation. In cosmology, for an accelerating universe, a non-linear quadratic EOS has been shown to be relevant for the description of dark energy and dark matter 48 . What type of matter can generate such an EOS in the high density regime of an astrophysical object is a matter of further investigation and will be taken up elsewhere.", "pages": [ 10 ] }, { "title": "Acknowledgments", "content": "RS gratefully acknowledges support from the Inter-university Centre for Astronomy and Astrophysics (IUCAA), Pune, India, where a part of this work was carried out under its Visiting Research Associateship Programme. BSR is grateful to IUCAA, Pune, India, for providing facilities where the part of work was done. BSR also thanks V. O. Thomas for useful discussions.", "pages": [ 10 ] }, { "title": "References", "content": "Relativistic stellar model admitting a quadratic equation of state 11", "pages": [ 11 ] } ]

[ { "title": "Multidimensional gravitational model with anisotropic pressure", "content": "O. A. Grigorieva, G. S. Sharov 1, ∗ 1 Tver state university, 170002, Sadovyj per. 35, Tver, Russia (Dated: August 6, 2018) We consider the gravitational model with additional spatial dimensions and anisotropic pressure which is nonzero only in these dimensions. Cosmological solutions in this model include accelerated expansion of the Universe at late age of its evolution and dynamical compactification of extra dimensions. This model describes observational data for Type Ia supernovae on the level or better than the ΛCDM model. We analyze two equations of state resulting in different predictions for further evolution, but in both variants the acceleration epoch is finite. PACS numbers: 04.50.-h, 98.80.-k, 11.25.Mj", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The most important event of last 15 years in astrophysics is conclusion about accelerated expansion of our universe at late stage of its evolution. This conclusion was based on observations of luminosity distances and redshifts for the Type Ia supernovae [1, 2], cosmic microwave background [3], large-scale galaxy clustering [4], and other evidence [5, 6]. To explain accelerated evolution of the universe various mechanisms have been suggested, including the most popular cosmological model ΛCDM with a Λ term (dark energy) and cold dark matter (see reviews [5-8]). The ΛCDM model with 4% fraction of visible (baryonic) matter nowadays, 23% fraction of dark matter and 73% fraction of dark energy [3] describes Type Ia supernovae, data rather well and satisfies observational evidence, connected with rotational curves of galaxies, galaxy clusters and anisotropies of cosmic microwave background. However, the ΛCDM model (along with vague nature of dark matter and energy) has some problems with fine tuning of the observed value of Λ, which is many orders of magnitude smaller than expected vacuum energy density, and with different time dependence of dark energy Ω Λ and material Ω m fractions (we have Ω Λ /similarequal Ω m nowadays). Therefore a large number of alternative cosmological models have been proposed. They include theories with extra dimensions [9-21]; matter with nontrivial equations of state, for example, Chaplygin gas [22, 23]; scalar fields with a potential [24-26]; modified gravity with f ( R ) Lagrangian [27, 28] and many others [5-8]. In this paper we explore the cosmological model with anisotropic pressure and nontrivial equation of state in 1+3+ d dimensions, suggested by Pahwa, Choudhury and Seshadri in Ref. [10]. The authors omitted the important case d = 1, we include it into consideration. We also analyze how to modify the equation of state and to avoid 'the end of the world' (the finite-time future singularity) which is inevitable in the model [10]. In this model the 1 + 3 + d dimensional spacetime is symmetric and isotropic in two subspaces: in 3 usual spatial dimensions and in d extra dimensions. It has the following metric with two Robertson-Walker terms [10]: Here the signature is ( -, + , . . . , +), the speed of light c = 1, a ( t ) and k 1 is are the scale factor and curvature sign in usual dimensions, b ( t ) and k 2 are corresponding values for extra dimensions. It is supposed in Ref. [10] that the scale factor a ( t ) grows while b ( t ) diminishes, in other words, some form of dynamical compactification [10-20] takes place, a size of compactified b is small enough to play no essential role at the TeV scale. The authors of Ref. [10] develop the approach of Ref. [9] and suppose that the spacetime (1.1) is filled with a uniform density matter with anisotropic pressure and the following energy-momentum tensor: Here ρ is the energy density and P a ( P b ) is the pressure in normal (extra) dimensions. So in normal dimensions pressure is different from that in additional dimensions, while being isotropic within each subspace. In Ref. [10] matter in the form of a single fluid is supposed to behave like pressureless dust ( P a = 0) in usual dimensions, while in extra dimensions it has appreciable pressure P b depending on density ρ by a power law with a negative constant W . The latter equation of state resembles a generalized Chaplygin gas [23]. In this model matter (1.2) with anisotropic pressure plays a role of dark energy and source of accelerated expansion. So the following Einstein equation without usual Λ term is considered: To describe the late time acceleration of the universe many authors [9-17] used the similar approach, in particular, extra dimensions, a metric of the type (1.1) and the energy-momentum tensor (1.2). However, the cited authors used different equations of state. In particular, in Refs. [14-16] these equations were linear Under these conditions a set of cosmological solutions with power law dependence of a , b , ρ on t was obtained in Refs. [14, 15]. But for these solutions an acceleration for a and a dynamical compactification or stabilization for b are not possible simultaneously. The similar problem appears in Ref. [17], where the authors use the sum of two perfect fluids with densities ρ and ¯ ρ and the equations of state P a = w a ρ , P b = w b ¯ ρ . In this case for solutions with a ∼ t α an acceleration ( α > 1) suppresses any compactification or diminishing for b ( t ). The problem of dynamical compactification for the extra dimensions was solved in the paper by Mohammedi [11] under assumptions (1.1) with k 2 = 0, (1.2) and the following ansatz: Mohammedi constructed solutions with accelerated expansion without a predetermined equation of state. In his approach evolution of values ρ , P a , P b was calculated from the right hand sides if Eqs. (1.4) with a Λ term. Relations between these values correspond to equations of state, they appear at the last stage of this scheme. Application of the Mohammedi's solutions [11] to describing the observational data will be discussed below. Middleton and Stanley in Ref. [16] in the framework of the linear equations of state (1.5) deduced the relation generalizing Eq. (1.6). Here n = (3 w a -2 w b -1) / (1 -w b ). They obtained a set of cosmological solutions including a hypergeometric function of powers of a . However, for these solutions an accelerated expansion of a takes place only when the EoS parameters w a , w b in Eq. (1.5) are both negative, and also an accelerated expansion of a in the late universe is incompatible with dynamical compactification of b [16]. This conclusion corresponds to the findings in Refs. [14, 15]. It is worth noting that the cosmological acceleration with the dynamical compactification of extra dimensions may be achieved in scalar-tensor theories, in particular, in 5-dimensional Brans-Dicke models [19, 20]. But these models along with the extra metric component g 44 require the additional degree of freedom in the form of the scalar Brans-Dicke field φ . This paper is organized as follows. In Sec. II we show, that the model [10] not only for d ≥ 2 but also in the case d = 1 can describe the current acceleration of the universe with dynamical compactification of b . In Sec. III we apply this model for all d ≥ 1 to describing observational data for Type Ia supernovae and determine optimal model parameters. In Sec. IV we modify the model [10] to solve the above mentioned problem of 'the end of the world'.", "pages": [ 1, 2 ] }, { "title": "II. COSMOLOGICAL SOLUTIONS", "content": "For the considered metric (1.1) in the case k 2 = 0 the Einstein tensor components G µ ν ( µ, ν = 0 , 1 , . . . , d + 3, 1 ≤ i ≤ 3 < I ) are [10]: If we substitute these expressions into Eq. (1.4) and add the continuity condition T µ ν ; µ = 0 we obtain the system of cosmological equations. This system has the form in the case with d = 1 extra spatial dimension, that did not considered in Ref. [10]. Here pressure P a in 'usual' dimension equals zero, as mentioned above. Eq. (2.4) is the continuity condition for d = 1 and P a = 0. Using the Hubble constant H 0 /similarequal 2 . 28 · 10 -18 c -1 [3] and the critical density at the present time, we make the following substitutions and introduce dimensionless time τ , density ¯ ρ , pressure ¯ p b and logarithms A , B of the scale factors (here a 0 , b 0 are present time values of a and b ). We denote derivative with respect to τ as primes and rewrite the system (2.1) - (2.4) as follows: Here If we express from Eq. (2.7) and substitute it into three equations (2.8) - (2.10), one should note that Eq. (2.8) may be reduced to Eq. (2.9). So in the planar case we have the system of two independent equations If we fix an equation of state for pressure ¯ p b , for example, the above mentioned power law (1.3) we may consider the equations (2.13), (2.14) as a closed system of first order differential equations with respect to 2 unknown functions A ' ( τ ) and ¯ ρ ( τ ). The dependence (2.15) is used in Ref. [10], where parameters w and γ are chosen in accordance with observations. The Cauchy problem for the system (2.13), (2.14) requires two initial conditions. We refer them to the present epoch (here and below it corresponds to the value τ = 1) in the following form: ∣ ∣ The first condition results from definition of the Hubble constant ∣ In the second condition (2.16) we suppose that the energy density ρ = ¯ ρ · ρ c at the present time has the fraction Ω 0 in the critical density (2.5). In Ref. [10] this fraction equals matter density fraction in the ΛCDM model [6]: Note that in Ref. [10] the second condition (2.16) was used in the form ¯ ρ ∣ ∣ τ =1 = 1, but the value Ω 0 (2.17) was taken as the factor in the r.h.s. of Eq. (1.4). From our point of view, that approach introduces useless vagueness in physical sense of the value ρ . In our approach ρ in conditions (2.16) is density of all gravitating matter (visible and dark) with described above anisotropic pressure. Remind that we have no dark energy or Λ term in Eq. (1.4) in the model [10]. Anisotropic pressure in additional dimensions plays here the role of dark energy as a source of acceleration. The contribution of this source is the term Ω B = -B ' ∣ ∣ τ =1 in the equality Ω m +Ω B +Ω k = 1 , (2.18) that results from equation (2.7), if we fix it at the present time τ = 1. To obtain cosmological solutions for d = 1, k 1 = 0 in this model we are to solve numerically the Cauchy problem for the system (2.13), (2.14) with initial conditions (2.16) moving into the past for τ < 1 and into the future for τ > 1. Then we integrate functions A ' ( τ ) and B ' ( τ ) (2.12) keeping in mind Eqs. (2.6) and calculate dependence of the scale factors a = a 0 e A , b = b 0 e B and density ¯ ρ on dimensionless time τ . ρ The results of calculation for scale factors a ( τ ), b ( τ ), density ¯ ρ ( τ ) and the acceleration parameter ( q is the deceleration parameter) are presented in Fig. 1. Here k 1 = 0, Ω 0 = 0 . 27, γ = 0 . 9 and 3 scenarios for w = -1 . 6 (dash-dotted line), w = -1 . 8 (solid lines) and w = -2 (dashed lines) are shown. This evolution begins from infinite value of density ¯ ρ at some initial moment τ 0 . We can see here two different variants for this beginning. For solutions with w = -1 . 6 and w = -1 . 8 (we denote them as 'regular' solutions) the scale factor a expands from a = 0 like a ∼ √ τ -τ 0 at the initial stage whereas the scale factor b diminishes from initial infinite value up to values b /similarequal b 0 during some percent of total lifetime of this universe. This behavior of b ( τ ) looks like some variant of dynamical compactification, because the parameter b 0 is arbitrary one in this model, we may put b 0 to be sufficiently small. Another type of evolution ('singular' solutions) is represented with dashed lines in Fig. 1 for w = -2. For singular solutions infinite value of density ¯ ρ at τ = τ 0 corresponds to nonzero value of the scale factor a and b = 0. Obviously, these solutions are nonphysical and should be excluded. /negationslash ∣ ∣ For all reasonable values of four free parameters w , γ , Ω 0 , Ω k the stage of accelerated expansion appears to be finite, because density ¯ ρ inevitably vanishes in this model. In Fig. 1 this effect may be seen in the graphs ¯ ρ ( τ ) with logarithmic scale in Y-direction. We denote the moment of zero density by τ ∗ : ¯ ρ ( τ ∗ ) = 0. For τ > τ ∗ density ¯ ρ becomes negative and nonphysical, all energy conditions (in particular, the weak energy condition) are violated. All regular and singular solutions in Fig. 1 describe accelerated expansion (for the factor a ) at late stage of evolution. Beginning of this stage may be seen in the graph of the acceleration parameter -q ( τ ). Acceleration rate depends on the parameters w , γ , Ω 0 and the curvature fraction (2.11) Ω k = -k 1 ( a 0 H 0 ) -2 depending on the sign k 1 . If Ω k = 0 ( k 1 = ± 1), one should use the system (2.9) - (2.12) instead of Eqs. (2.13), (2.14). In this case we integrate numerically the function A ' ( τ ) simultaneously with solving the Cauchy problem for the system (2.9) - (2.12). We add here the natural initial condition A τ =1 = 0 to conditions (2.16). This finite-time future singularity may be classified as the Type IV singularity in accordance with the scheme from Refs. [7, 28]. For this singularity a ( τ ∗ ) is nonzero, ¯ ρ ( τ ∗ ) equals zero, the effective density and pressure remain nonzero, but higher derivatives of H diverge at τ → τ ∗ . Note that the main features of the considered cosmological solutions, in particular, the future singularity, finite lifetime τ 0 ≤ τ ≤ τ ∗ and negative density for τ > τ ∗ take place not only for d = 1, but also for higher dimensions d ≥ 2. In the case of d ≥ 2 additional dimensions after substituting the components G µ ν into Einstein equation (1.4) and substitutions (2.6) in these equations and Eq. (2.4) we have in the flat case k 1 = k 2 = 0 the following system [10], generalizing Eqs. (2.12) - (2.14): Solutions of the system (2.20) for d ≥ 2 were obtained in Ref. [10], but some features of them were not considered in that paper. For example, singular solutions with nonzero value a ( τ 0 ) (where ¯ ρ is infinite at the initial moment τ 0 ) also take place for d ≥ 2, if the value w is less than the critical value w cr ( γ, Ω 0 ). In Fig. 2 boundaries w = w cr separating domains of regular and singular solutions on the γ, w plane are presented for different d and Ω 0 . Singular solutions are described by the inequality w < w cr ( γ, Ω 0 ) and lie below corresponding lines in Fig. 2. Another important property of these cosmological solutions is their finite-time future singularity, in other words, inevitability of 'the end of the world' because of vanishing density at τ = τ ∗ for all d (see Fig. 4 below). The authors of Ref. [10] did not pay attention to this phenomenon, essential for their model. It is connected with the chosen equation of state (2.15) for pressure ¯ p b in extra dimensions. This drawback will be eliminated with modifying the model [10] in Sect. IV after application this model to describing observational data for Type Ia supernovae in the next section.", "pages": [ 2, 3, 4 ] }, { "title": "III. APPLICATION TO SUPERNOVAE OBSERVATIONS", "content": "To apply the model to describing the observational data it is convenient, following the authors of [10], to use Internet table [29] for Type Ia supernovae in distant galaxies. At the present moment this updated table contains redshifts z = z i , distance moduli µ i and errors σ i of µ i for N = 580 supernovae. Redshift is associated with the value of a at the time t of a supernova light emission. The distance modulus µ is the logarithmic function of the luminosity distance [6, 10]: To describe the data [29] of Type Ia supernovae, for given values d , w , γ , Ω 0 of this model we consider evolution of the scale factor a ( τ ) and dependence of the numerical integral (3.2) D L and µ on τ . For each value of redshift z i in the table [29] we calculate the corresponding τ = τ i with using Eq. (3.1) and linear approximation and the theoretical value µ th = µ ( τ i ) for τ i from Eq. (3.2). The measure of differences between these theoretical values µ th = µ th ( d, w, γ, Ω 0 , Ω k , z i ) and the measured values µ i is [10]: The authors of Ref. [10] calculated optimal parameters w and γ , minimizing the function (3.3) for the flat model ( k 1 = 0) with fixed Ω 0 = 0 . 27 (2.17) and d ≥ 2. In this approach for each d ≥ 2 they minimized the function χ 2 ( w, γ ) of two variables. We generalize their approach to the case d = 1 additional dimension. At the first step we fix k 1 = 0, Ω 0 = 0 . 27 in according with Ref. [10] and obtain the picture of level lines for the function χ 2 ( w, γ ), presented in Fig. 3 for d = 1 and d = 2. Here the dashed line is taken from Fig. 2 and separates regular and singular solutions. We see that for d = 1 and d = 2 the minimum of χ 2 lies above this line, that is in the domain of regular solutions. The same picture also takes place for d ≥ 3. For each d ≥ 1 we calculated minimums for the function of two variables χ 2 ( w, γ ) and coordinates w , γ of this minimum. They are represented in Table I. We compare these minimal values with the value of the function (3.3) for the flat ΛCDM model with the same parameters k 1 = 0, Ω m = Ω 0 = 0 . 27 (therefore, Ω Λ = 0 . 73) and the same supernova data [29]. We see that the predictions are rather close, and for d = 1 the model [10] fits the data better than the flat ΛCDM model. At the next step for more precise estimation of optimal model parameters we consider variations of fractions Ω 0 = Ω m and Ω k for matter density and curvature respectively. One should take into account these degrees of freedom in both models: the model [10] and ΛCDM. In the model [10] for each d ≥ 1 we minimize the function (3.3) of four variables: χ 2 ( w, γ, Ω 0 , Ω k ). We also compare this results with the same value of the ΛCDM model (where χ 2 depends on Ω m and Ω k ) and keep in mind the constraints on these parameters due to cosmic microwave background anisotropy, galaxy clustering and other factors [3]: Numerical search of this minimum includes a starting point (for example, the values from Table I), analysis of gradients or increments for χ 2 and the constraints (3.4). The results of calculation with optimal values of the model parameters are presented in Table II. We see that the ΛCDM model is more sensitive to variations of Ω m and Ω k and the better result for this model is achieved. Here optimal values of the model parameters are determined by the constraints (3.4). We impose these constraints on the model [10] though they are not strictly applicable to it. In this model min χ 2 weakly depends on Ω 0 and Ω k , so we can not diminish χ 2 appreciably if we slightly broaden the limitations (3.4). In Fig. 4 one can see evolution of the scale factor a ( τ ), (and b for the model [10]), the acceleration parameter -q ( τ ) and density ¯ ρ ( τ ) for the ΛCDM model and the model [10] with d = 1 (solid lines), d = 2 (dots) and d = 6 (dash-dotted lines). For all these models we use the optimal parameters from Table II. ρ τ Evolution of the scale factor a ( τ ) for the model [10] with different d and for the ΛCDM model is very close up to z /similarequal 1 . 5 ( a > 0 . 4 a 0 ), before this epoch the ΛCDM model demonstrates slower expansion. This difference is more visible for the acceleration graphs -q ( τ ). The scale factor b for the case [10] diminishes to b /similarequal b 0 according to the mentioned above compactification scheme (compare with the regular solutions in Fig. 1). Behavior of cosmological solutions in the future for both models is also different. The ΛCDM model demonstrates unlimited accelerated expansion whereas for the model [10] the acceleration turns into deceleration and inevitability results in the above mentioned zero density ¯ ρ at τ = τ ∗ with nonphysical values ¯ ρ < 0 for τ > τ ∗ . The finite lifetime of this universe depends on d , it is the smallest for d = 1. In the next section we discuss how to eliminate this essential drawback of the model.", "pages": [ 4, 5, 6 ] }, { "title": "IV. MODIFICATION OF THE MODEL", "content": "We have noted that all cosmological solutions in the model [10] have the finite-time future singularity. This inevitable 'end of the world' is connected with the chosen power law dependence (2.15) of pressure ¯ p b in extra dimensions on density ¯ ρ . The terms with the factor ¯ p b in equations (2.14) for d = 1 or (2.20) for d > 1 determine rate of density decreasing when ¯ ρ is small at the end of its evolution. In this case the leading terms in the mentioned equations are For ¯ ρ → 0 we have nonzero values A ' and B ' , so for the weak power law dependence (2.15) the approximate equation (4.1) ¯ ρ ' /similarequal -C ¯ ρ 1 -γ has the finite solution To avoid this finiteness we are to modify the equation of state (power law dependence) (2.15) of the model [10] for small ¯ ρ . In particular, a linear dependence for ¯ ρ close to zero ensures infinite evolution with positive density. The linear law (4.3) for all ¯ ρ does not describe the observed accelerated expansion. For good agreement with observations we are to search an equation of state ¯ p b (¯ ρ ) with slower growth of | ¯ p b | at high ¯ ρ similar to Eq. (2.15). We suggest the appropriate variant of this dependence with the linear law (4.3) for ¯ ρ /lessmuch ρ 0 (here w 0 = w 1 + w/ρ 0 ) and another linear law ¯ p b /similarequal w 1 ¯ ρ for ¯ ρ /greatermuch ρ 0 . The model (2.9) - (2.12) or (2.20) for d > 1 with the linear-fractional equation of state (4.4) makes it possible to avoid finite lifetime of the type (4.2) and to transform it into the exponential asymptotic behavior This behavior results from the equation ¯ ρ ' /similarequal -C ¯ ρ and may be observed in graphs ¯ ρ ( τ ) in Fig. 5. For the model with Eq. (4.4) we can find optimal values of parameters w , w 1 , ρ 0 , Ω 0 , Ω k presented in Table III and achieve better agreement with the supernovae data [29] than for the models ΛCDM and [10] with Eq. (2.15). Cosmological solutions for the model with Eq. (4.4) and parameters from Table III are shown in Fig. 5. We see in Table III that the accuracy of the model with Eq. (4.4) increases ( χ 2 diminishes) for large d , unlike in the case with Eq. (2.15) in Table II. We should note that the values χ 2 in Table III are not absolutely minimal, because we fixed the parameter ρ 0 = 0 . 005. It is interesting, that for all d we can achieve smaller values min χ 2 , if we take smaller values of ρ 0 . But if ρ 0 → 0, the factor C in the exponent (4.5) tends to infinity, the density ¯ ρ decreases too rapidly and the picture of vanishing ¯ ρ looks like in the finite case in Fig. 4. So we put the restriction ρ 0 ≥ 0 . 005 to exclude this almost instantaneous transition to the state with ¯ ρ /similarequal 0. Under this constraint we have the optimal value ρ 0 = 0 . 005 and also Ω 0 = 0 . 2815, Ω k = -0 . 0133 for all d . Fig. 5 demonstrates cosmological solutions for the model with Eq. (4.4) with the optimal values of parameters from Table III. For both models Eqs. (4.4) and (2.15) in Figs. 4 and 5 the acceleration epoch is finite and its duration depends on d in the same manner. But after this epoch for the model with Eq. (4.4) we see here infinite decelerated expansion. ρ Graphs of a ( τ ), -q ( τ ) and ¯ ρ ( τ ) here are more close to the dashed lines for the ΛCDM model during the acceleration epoch than the similar curves in Fig. 4. But after the mentioned epoch predictions of the ΛCDM model and the model (4.4) sharply diverge. In Fig. 6 we present how the model with Eq. (4.4) for d = 1, 2, 6 and the ΛCDM model describe the supernovae data from the site [29] (dots) in the z -D L plane. All these models were considered below in Fig. 5, we use the same optimal parameters from Table III and the same notations for the curves, in particular, the dashed line for the ΛCDM model. The presented curves are very close in the region z < 1, for larger z the ΛCDM line slightly diverges from others. These lines are result of optimal fitting to the observational data [29] (580 dots in Fig. 6). The values χ 2 in Table III show rather good results for the model [10] with Eq. (4.4) for pressure, but these values are not the best fit, because we fixed ρ 0 to avoid the mentioned above sharp transition to ¯ ρ /similarequal 0.", "pages": [ 6, 7 ] }, { "title": "V. CONCLUSION", "content": "The gravitational model of Pahwa, Choudhury and Seshadri [10] with additional spatial dimensions and anisotropic pressure provides accelerated expansion of the universe corresponding to observational data for Type Ia supernovae [29] simultaneously with dynamical compactification of d extra dimensions. It is important that such a behavior of solutions results from rather simple equations of state (2.15). This approach is more natural in comparison with the scheme of Mohammedi [11], where complicated equations of state are deduced from the constructed solutions, in particular, from the solution (5.1) described below. The authors of Ref. [10] did not consider the case d = 1, but we found that for the chosen in Ref. [10] power law equation of state (2.15) it is the case d = 1 that yields the best fit (see Tables I, II). Unfortunately, the model [10] with the power law dependence (2.15) inevitably predicts the finite-time future singularity of Type IV in classification from Refs. [7, 28]. It is connected with vanishing density ¯ ρ at finite time τ = τ ∗ (and negative density for τ > τ ∗ ). Evolution of this universe is broken at τ ∗ , the finite lifetime is shorter for small d (see Fig. 4). We demonstrated in Sect. IV, that this drawback has technical nature. It is connected with too weak dependence ¯ p b (¯ ρ ) for small ¯ ρ in the power law equation of state (2.15). If we modify this law and choose a linear dependence (4.3) for small ¯ ρ , we obtain an infinite cosmological evolution with positive density (but vanishing at τ → ∞ ). We suggest the linear-fractional variant (4.4) of dependence ¯ p b (¯ ρ ) to solve the following two problems: (a) to avoid 'the end of the world' of the type (4.2) and (b) to describe 580 Type Ia supernovae data points from the site [29]. The dependence (4.4) is a bit more complicated than Eq. (2.15), but it successfully conserves positive density ¯ ρ during infinite lifetime and fits the data [29] better than the ΛCDM model and the model from Ref. [10] with Eq. (2.15). Although the simplicity of the equations of state (1.3) is the important advantage of the model [10], we are to step back from this simplicity. But in our opinion, the dependence (4.4) is the minimal retreat that solves this problem. Our calculations should be compared with predictions of other multidimensional models [10-18]. We shall consider the models, describing the late time acceleration of a ( t ) together with a contraction of b ( t ), in particular, the Mohammedi model in Ref. [11] with the ansatz (1.6) b/b 0 = ( a/a 0 ) -n . It ensures a dynamical compactification, if n is positive and a ( t ) expands. For n satisfying the equality a set of solutions with accelerated expansion was obtained in Ref. [11] in the form Here ˜ τ = τ -1, the natural condition a ∣ ∣ t = t 0 = a 0 must be saticfied. We mentioned above, that the equation of state in the Mohammedi's approach may be determined at the last stage after substitution of the expressions (5.1) and (1.6) into Eqs. (1.4) with a Λ term. In particular, the relation between P a and ρ for solutions (5.1) results from the first two equations (1.4) (in our notations) if we exclude a 2 . This equation of state is mush more complicated than its analog P a = 0 for the model [10], in addition it has the negative limit of P a at a →∞ for the case Λ = 0. If we accept these complicated equations of state for the model [11], we can obtain the optimal solution (5.1), minimizing the sum χ 2 (3.3) for the same supernovae data [29]. For this purpose we use 2 fitting parameters of these solutions: ˜ C 1 and ˜ µ . The calculations result in the optimal values ˜ C 1 = 1 . 229, ˜ µ = 0 . 679 and the corresponding minimum χ 2 /similarequal 564 . 4. This minimum is close to the results of the considered model [10] in Tables II and III. So we may conclude that the solution (5.1) describes the supernovae data [29] rather successfully. It is interesting that solutions close to Eq. (5.1) appeared in Refs. [18] in the brane model. In Refs. [12, 13] Darabi obtained exponential solutions a = C 1 exp( µt ) for the model with varying Λ ∼ a -m . Such a solution with one fitting parameter is less adaptable in comparison with Eq. (5.1), the optimal sum (3.3) in this case χ 2 > 955. Note that in this paper for the model with Eq. (4.4) we practically used only two fitting parameters w and w 1 . The value ρ 0 was fixed because for very small ρ 0 we have better fit, but the sharp downfall of ¯ ρ ( τ ) to ¯ ρ /similarequal 0 looks like the mentioned 'end of the world'. The parameters Ω 0 and Ω k influence on minimum of χ 2 rather weakly for the model with Eq. (4.4). If we fix, for example, Ω 0 = 0 . 27 and Ω k = 0, minimums for χ 2 will differ from results in Table III less then 0.01 for all d . Cosmological solutions in the model with Eq. (4.4) are divided into regular and singular ones similarly to solutions with Eq. (2.15) shown in Fig. 1. However, Fig. 5 demonstrates that for the optimal values of parameters from Table III solutions with Eq. (4.4) are regular. It is interesting that the model [10] with both considered variants of dependence ¯ p b on ¯ ρ (2.15) and (4.4) predicts finiteness of the acceleration epoch. Its duration depends on d in the same manner (compare Figs. 4 and 5) and then acceleration sharply turns to deceleration. In the case (2.15) this evolution is broken at τ = τ ∗ with ¯ ρ ( τ ∗ ) = 0, but for the model with Eq. (4.4) the decelerated expansion is infinite and density ¯ ρ ( τ ) tends to zero in the exponential form (4.5).", "pages": [ 7, 8 ] } ]

[ { "title": "Cosmography of interacting generalized QCD ghost dark energy", "content": "Mohammad Malekjani ∗ 1, 2 1 Department of Physics, Faculty of Science, Bu-Ali Sina University, Hamedan 65178, Iran 2 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha, P. O. Box:55134-441, Iran Exploring the accelerated expansion of the universe, we investigate the generalized ghost dark energy (GGDE) model from the statefinder diagnosis analysis in a flat FRW universe. First we calculate the cosmological evolution and statefinder trajectories for non-interacting case and then extend this work by considering the interaction between dark matter and dark energy components. We show that in the non-interacting case the phantom line can not be crossed and also he evolutionary trajectories of model in s -r plane can not be discriminated. It has been shown that the present location of model in s -r plane would be close to observational value for negative values of model parameter. In the presence of interaction between dark matter and dark energy, the phantom regime is achieved, the accelerated phase of expansion occurs sooner compare with non-interacting case. The GGDE model is also discussed from the viewpoint of perturbation theory by calculating the adiabatic sound speed of the model. Finally, unlike the non-interacting case, the evolutionary trajectories in s -r plane can be discriminated in the interacting model. Like non-interacting model, in the interacting case the present location of GGDE model is closer to observational value for negative values of model parameter. PACS numbers: 98.80. - k, 95.36.+x Keywords: Cosmology, Dark energy", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The recent astronomical data from SNe Ia [1], WMAP [2], SDSS [3] and X-ray [4] experiments show that our universe experiences an accelerated expansion. The above observational data strongly suggest that the universe is spatially flat and dominated by an exotic component with negative pressure, the so-called dark energy [5-8]. Dark energy scenario has got a lot of attention in modern cosmology both from theoretical and observational point of view. Observationally, The result of WMAP experiment shows that dark energy occupies about 73% of the energy of our universe, dark matter about 23% and usual baryons occupies only about 4% of the total energy of the universe c2. Although the nature of dark energy is still un-known, but the ultimate fate of the current universe is determined by this mysterious component. Theoretically, the first and simplest model for dark energy is Einstein's cosmological constant with constant EoS parameter w Λ = -1 . Cosmological constant faces with the fine-tuning and cosmic coincidence problems [5-16]. In recent years a plenty theoretical models have been proposed to interpret the properties of dark energy [10-18]. Almost all of theoretical dark energy models need to introduce new degree(s) of freedom or modifying general relativity. However it would be better to consider a model of dark energy without a need of new degree(s) or new parameter in its theory. Recently the so-called QCD ghost dark energy has been proposed to interpret the dark energy without any new parameter or new degree of freedom [19-25]. The Veneziano ghost has been suggested to solve the U(1) problem in low energy effective theory of QCD [26-31]. The ghost field has no contribution to the vacuum energy density in the flat Minkowsi spacetime. However, in the case of curved spacetime it has a small energy density proportional to Λ 3 QCD H , where Λ QCD is QCD mass scale and H is the Hubble parameter [32-34]. This model does not encounter with some unwanted problems such as the violation of gauge invariance, unitarity and causality [32]. Since the Veneziano ghost field is totally embedded in standard model and general relativity, one needs not to introduce any new degree(s) of freedom or to modify the Einstein's general relativity. The present value of energy density of dark energy in this model is roughly of order Λ 3 QCD H 0 , with Λ QCD ∼ 100 MeV and H 0 ∼ 10 -33 eV which is in agreement with observed value (3 × 10 -3 eV ) 4 for energy density of dark energy [35]. This numerical coincidence is adsorbent and gets rid the model from fine tuning problem [19-25]. Observationally, the ghost dark energy model has been fitted by astronomical data including SnIa, BAO, CMB, BBN and Hubble parameter data [36]. The cosmological evolution of dark energy in the QCD ghost model has been calculated in [19, 20] and has been resulted that the universe begins to accelerate at redshift around z ∼ 0 . 6. Also the squared sound speed of the dark energy for this model is negative, indicating an instability of the model against perturbational theory [19-22]. In [37], the ghost dark energy (GDE,hereafter) has been extended in the presence of interaction between dark matter and dark energy in non-flat universe. The reconstructed potential and the dynamics of scalar fields according the exultation of GDE have been investigated in [38-42]. The reconstructed modified gravity for GDE which describes the late time accelerated expansion has been studied in [43]. The statefinder diagnostic of GDE has been presented in [44]. Form the statefinder viewpoint, the evolution of GDE model is similar to holographic dark energy model and present value of statefinder parameters in this model is in good agreement with observation [44]. In all above studies, the energy density of GDE is considered proportional to Hubble parameter as ρ d = αH . However, the energy density of Veneziano ghost field in QCD is generally in the form of H + O ( H 2 ) [45]. In this case the U (1) A problem in QCD can be solved. Although, up to now, only the leading term H has been assumed for energy density of GDE, but the sub-leading term H 2 can also be important in the early evolution of the universe [46]. Including the second term in the energy density of GDE results better agreement with observation in comparison with usual GDE model [47]. Like [48], we call this model as generalized ghost dark energy (GGDE). The energy density of GGDE model is written as ρ d = αH + βH 2 , where α and β are the constants of the model. It has been shown that the GGDE model can result a de-Sitter phase of expansion and also in the presence of interaction between dark matter and dark energy this model results the phantom regime of expansion ( w d < -1) [48]. The other features of GGDE model have been presented in [49, 50]. It is well known that in addition of dark energy component which describe the accelerated expansion of the universe, there exist another mysterious component in the universe so-called dark matter. The dark matter component can interpret the flat rotation curve of spiral galaxies and also the scenario of structure formation of universe [51-55]. Since the nature of these component are un known and they they have different gravitational treatment, therefore their evolution usually considered independent of each other. However recent observation from galaxy cluster Abell A596 indicates the interaction between these components [56]. Also the observational data from SNIa and CMB experiments is compatible with interacting forms of dark energy models [57]. However the strength of this interaction is not clearly identified [58]. From theoretical viewpoint it is also acceptable to consider the interaction between dark matter and dark energy. In the unified models of field theory dark matter and dark energy can be interpreted by a single scalar field in a minimally interaction. Also considering interaction between dark matter and dark energy can solve the coincidence problem [59-66] In this work our main task is to investigate the interacting GGDE model in statefinder diagnostic analysis. Different dynamical dark energy models obtain accelerated expansion at the present time ( q < 0), where q is deceleration parameter. Hence we need a diagnostic tool for discriminating these dark energy models. For this aim, Sahni et al. [67] and Alam et al. [68], by using the third time derivative of scale factor, introduced the statefinder pair { s, r } . These parameters in flat universe are given by The parameters s and r are geometrical, because they only depend on the scale factor. In statefinder analysis we plot the evolutionary trajectories of dark energy model is s -r plane. In recent years the various dark energy models such as quintessence, holographic,new holographic, phantom, tachyon, chaplygin gas, agegraphic, new agegraphic, , polytropic gas and ghost dark energy models have been studied in the statefinder analysis [67, 68, 7083]. These models have different evolutionary trajectories in { s, r } plane, therefore the statefinder tool can discriminate these models. The standard ΛCDM has no evolution in this plane and corresponds to the fixed point { s=0,r=1 } [67]. The present observational value for statefinder parameters are { s 0 = -0 . 006 , r = 1 . 02 } [69]. The distance of the current value of statefinder pair { s 0 , r 0 } of a given dark energy model from the observational value { s 0 = -0 . 006 , r = 1 . 02 } is a valuable criterion to examine of model. Here we see that the location of standard ΛCDM model in s -r plane is near to observational value. In [83], the evolution of original GDE has been calculated by statefinder diagnostic in s -r plane and shown that the GDE model mimics the ΛCDM at the late time. Also the behavior of GDE is similar to holographic dark energy in this plane [83]. In this work we first calculate the cosmological evolution of GGDE model and then investigate this model from statefinder diagnostic analysis. The paper is organized as follows: In sect.II, The GGDE model is presented in non-interacting universe. The interacting case of GGDE model is given in sect.III. In sect. IV we obtain the adiabatic sound speed for GGDE model. We calculate the numerical results in sect.V and conclude in sect.VI.", "pages": [ 2, 3, 4, 5 ] }, { "title": "II. NON-INTERACTING GGDE MODEL", "content": "A flat Friedmann-Robertson-Walker (FRW) universe dominated by dark matter and dark energy is given by where ρ m and ρ d are, respectively, the energy density of pressureless dark matter and dark energy and m p is the reduced planck mass. The energy density of GGDE is given by [48] where α and β are constants of model. The Friedmann equation (2) in terms of dimensionless parameters is written as where The conservation equations for pressureless dark matter and dark energy without interaction read the following equations Taking the time derivative of Friedmann equation (2) and using (4, 6, 7) obtains Differentiating Eq.(3) with respect to time yields Inserting (9) and (3) in conservation equation for dark energy (7) and using (8), the EoS parameter of GGDE model can be obtained as where ξ = 8 πGβ/ 3. In the limiting case ξ = 0, this relation reduces to its original form in [37]. The deceleration parameter q by using (8) and (10), in GGDE universe is obtained as The decelerated phase of expansion at the early time is indicated by q < 0 and accelerated phase is related to q > 0. Taking the time derivative of dimensionless dark energy density in (5) and using (3), (9), we obtain the equation of motion for the evolution of energy density of GGDE model as where prime is derivative with respect to ln a . Taking a derivative of (10) with respect to ln a , the equation of motion for EoS parameter can be calculated as Using the above relation, in this stage, we calculate the statefinder parameters s and r for GGDE model in non-interacting universe. In general form, relation (1) for a given dark energy model in flat universe can be written as and Inserting relations (12) and (13) in equations (14) and (15), we obtain the statefinder parameters for GGDE model in spatially flat universe In the limiting case of dark energy dominated universe (Ω d → 0) the parameters { s,r } tends to { 0 , 1 } , respectively. Hence the GGDE model mimics the ΛCDM model at the late time when Ω d → 0. In sect.V, we calculate numerically the evolution of GGDE model in non-interacting universe from the statefinder viewpoint.", "pages": [ 5, 6, 7 ] }, { "title": "III. INTERACTING GGDE MODEL", "content": "In this section we consider the interaction between dark matter and dark energy components. In this case the conservation equations for these components are: where Q in right hand side indicate the interaction term. The positive value of Q means the transition of energy from dark energy to dark matter component. It should be noted that the left side of (6) and (7) are inversely proportional to time. Therefore the parameter Q can be considered as a function of Hubble parameter H such as following forms: One can assume the above three forms as Q = Γ ρ d , where for case (i) Γ = 3 b 2 H , for case (ii) Γ = 3 b 2 H Ω m Ω d and for case (iii) Γ = 3 b 2 H 1 Ω d . The parameter b is a coupling constant indicating the strength of interaction between dark matter and dark energy [84-86]. In this work we assume the third form of interaction (i.e., Q = 3 Hb 2 ρ d Ω d ). Substituting Q in (19) and using (3), (8) and (9), the EoS parameter of interacting GGDE model is obtained as Inserting b = 0 recovers the EoS parameter of non-interacting case in previous section. Substituting (20) in (8) results the deceleration parameter q for interacting case as follows It has been shown that for selected parameters ( ξ = 0 . 03 , b = 0 . 15 , Ω d 0 = 0 . 72) the deceleration parameter at the present time is q 0 = -0 . 38 which is consistent with observation [48]. Taking the time derivative of dimensionless dark energy density in (5) and using (3), (9) and (20) , the equation of motion for the evolution of energy density of interacting GGDE model can be obtained as In the limiting non-interacting case ( b = 0) the respective relation in previous section is retained. Derivative of (20) with respect to ln a results Once again, the respective relation in previous section can be obtained by setting b = 0. Finally by substituting relations (22) and (23) in general relations (14) and (15), we obtain the statefinder parameters for interacting GGDE model as follows Setting b = 0 retains the relations for s and r in previous section. In section V we investigate the evolution of interacting GGDE model in s -r plane can calculate the effect of interaction parameter b 2 on the evolution of the model.", "pages": [ 7, 8 ] }, { "title": "IV. ADIABATIC SOUND SPEED", "content": "In linear perturbation theory, squared sound speed, c 2 is a crucial quantity. Stability or instability of a given perturbed mode can be calculated by determining the sign of c 2 . The positive sign (real value of sound speed) represents the periodic propagating mode for a density perturbation and in this case we have the stability. The negative sign (imaginary value of sound speed) indicates an exponentially growing mode for a density perturbation, meaning the instability [90, 91]. Here we obtain the squared sound speed for GGDE model both in non-interacting and interacting cases. The squared sound speed c 2 s is introduced as We now differentiate the equation of state, p d = w d ρ d with respect to time and find Inserting (27) in (26) and using Eq.(7), we obtain c 2 s for non-interacting GGDE model as follows where prime is the derivative with respect to ln a and w ' d = ˙ w d /H . In the case of interacting GGDE model by using Eq.(19), the parameter c 2 s can be obtained as Here same as previous section we used third form of interaction parameter Q = 3 Hb 2 ρ d / Ω d . In next section, we obtain the evolution of c 2 s as a function of cosmic redshift and discuss the stability or instability of GGDE model for both non-interacting and interacting universe.", "pages": [ 8, 9 ] }, { "title": "V. NUMERICAL RESULTS", "content": "Here we present numerical description for cosmological evolution and statefinder analysis of GGDE model in the flat FRW cosmology. In numerical procedure we fix the cosmological parameters at the present time as Ω 0 m = 0 . 3 and Ω 0 d = 0 . 7. We first consider non-interacting case and then interacting case of GGDE model.", "pages": [ 9 ] }, { "title": "A. non-interacting case", "content": "The EoS parameter of non-interacting GGDE model as a function of density parameter Ω d is given by (10). By solving coupled equations (12) and (10), the evolution of EoS parameter in terms of cosmic redshift z = 1 /a -1 and for different illustrative values of ξ is shown in Fig.(1). The cosmic redshift z = 0 represents the present time, z > 0 indicates the past times and z < 0 expresses the future. We see that for any value of ξ the non-interacting GGDE model can not enter the phantom regime ( w d < -1) at all. We also see that the EoS of GGDE model with ξ > 0 is larger than EoS of standard GDE model ( ξ = 0 . 0). At the future epoch, the EoS tends to -1 which implies that the GGDE model mimics the cosmological constant at that time. The deceleration parameter q which indicates the decelerated or accelerated phase of expansion for non-interacting case is given by (11). Solving coupled equations (11) and (12), the evolution of parameter q as a function of redshift parameter z has been shown in Fig.(2) for different values of model parameter ξ . The standard GDE model is indicated by solid line. one can see that for ξ < 0 the GGDE model enters the accelerated phase sooner and for ξ > 0 later compare with standard GDE model. The statefinder pair { s,r } for non-interacting GGDE model is given by relations(16) and (17). Solving these coupled equations together with (12), we obtain the evolution of parameters s and r in terms of redshift z . In Fig.(3), we plot the evolutionary trajectories of non-interacting GGDE model for different values of model parameter ξ in s -r plane. We see that the parameter r first decreases and then increases and also the parameter s decreases during the history of the universe from past to future. The important note is that in non-interacting case the GGDE model has been shown by single evolutionary trajectory for any value of ξ . Hence the evolutionary trajectories can not discriminated by model parameter ξ . The present value of statefinder pair { s 0 , r 0 } is indicated by colored circle on the figure. Also the location of ΛCDM model in s -r plane ,i.e., ( s = 0 , r = 1), has been shown by star symbol. The other feature is that the present value { s 0 , r 0 } is discriminated by model parameter ξ . In the case of ξ < 0, the distance of { s 0 , r 0 } from observational point { s 0 = -0 . 006 , r = 1 . 02 } (red star point on the figure) is shorter compare with standard GDE model (i.e., ξ = 0 . 0). While for ξ > 0 the distance is larger than GDE model. Finally we discuss numerically the stability or instability of non-interacting GGDE model form the viewpoint of pertrurbation theory based on Eq.(28). In Fig.(4), the evolution of adiabatic sound speed c 2 s is plotted as a function of redshift z for different values of GGDE model ξ . In the case of ξ ≥ 0, one can see c 2 s < 0 which indicates the instability of GGDE model against perturbation. For ξ < 0, we obtain c 2 s > 0 which represents the stability of model against perturbation.", "pages": [ 9, 10 ] }, { "title": "B. interacting case", "content": "/negationslash Here we calculate the numerical description of cosmological evolution and statefinder diagnosis for interacting GGDE model in spatially flat universe. First the evolution of EoS parameter in terms of cosmic redshift is plotted in Fig.(4). For this aim we solved the coupled equations (20) and (22). In left panel, by fixing interaction parameter as b = 0 . 2, the EoS parameter w d is plotted for different illustrative values of model parameter ξ as described in legend. The solid line represents the original GHDE model. For all cases of ξ , the interacting case of GGDE model can cross the phantom line ( w d = -1) form upper limit( w d < -1) to lower limit ( w d < -1). This behavior of interacting GGDE model in which the phantom line is crossed from up to below is in agrement with recent observations [ ? ]. In right panel the model parameter is fixed as ξ = 0 . 1 and the interaction parameter b is varied. We see that the non-interacting GGDE model (= ¯ 0 . 0) cannot enter the phantom regime (solid line), while in other cases ( ξ = 0) the phantom regime has been achieved. In Fig.(5), by numerical solving of relations (22) and (21), the evolution of deceleration parameter in terms of redshift has been shown in the context of interacting GGDE model. In left panel, by fixing b = 0 . 2, the parameter ξ is varied as indicated in legend. Same as non-interacting case, the parameter q starts from positive value at the earlier ( representing the decelerated phase at the past time) and ends to negative value later (indicating the accelerated phase at the present time). Like non-interacting case, transition from decelerated phase to accelerated phase for ξ < 0 takes place earlier and for ξ > 0 later, compare with original GDE model (see left panel of Fig.(5)). In right panel, for an illustrative value ξ = 0 . 1, the evolution of q has been shown for different values of interaction parameter b . The interaction parameter b can influence on the transition epoch from q > 0 to q < 0. We see that in the framework of interacting GGDE model the accelerated phase of expansion ( q < 0) can be achieved sooner for larger values of b . For all cases, q →-1 at the late time indicating that the GGDE model mimics the standard Λ CDM model at that time. Now, by solving relations (25) and 24), the evolution of interacting GGDE model in s -r plane is plotted in Fig.(6). In left panel, by fixing ξ = 0 . 1, the interaction parameter b is varied as indicated in legend. The evolutionary trajectories starts from right to left and /negationslash ended at the ΛCDM fixed point. The distance of present value { s 0 , r 0 } form the observational point { s 0 = -0 . 006 , r = 1 . 02 } (red star point on the figure) becomes larger by increasing b . In right panel, by fixing b = 0 . 2, the trajectories have been plotted for different illustrative values of ξ . The important note is that, contrary with non-interacting case, in the presence of interaction between dark matter and dark energy ( b = 0) the evolutionary trajectories in s -r plane are discriminated by parameter ξ . Also the distance of { s 0 , r 0 } from observational point { s 0 = -0 . 006 , r = 1 . 02 } is shorter for ξ < 0 and larger for ξ > 0 compare with original GDE model (i.e., ξ = 0 . 0). Finally, same as non-interacting case, we investigate the interacting GGDE model from the viewpoint of perturbation theory. The adiabatic sound speed c 2 s for interacting case is given by Eq.(29). In Fig.(8), we calculate the evolution of c 2 s as a function of cosmic redshift for different values of model parameter ξ as well as interaction parameter b . In left panel, by fixing model parameter ξ = 0 . 5, we obtain the evolution of c 2 s for different values of interaction parameter b . Here one can interpret that same as non-interacting case, the interacting GGDE model for ξ > 0 is instable ( c 2 s < 0) against perturbation. In right panel, by fixing interaction parameter b = 0 . 1, the evolution of c 2 s has been shown for different values of model parameter ξ . Like non-interacting case, we conclude that the interacting GGDE model is stable for ξ < 0 and instable for ξ ≥ 0.", "pages": [ 11, 12 ] }, { "title": "VI. CONCLUSION", "content": "In summary, we considered the generalized version of QCD ghost dark energy (GGDE) model in both non-interacting and interacting universe. The cosmological evolution and also the statefinder diagnosis of the model have been calculated. We showed that: (i). In the non-interacting GGDE model the phantom regime can not be achieved and the EoS parameter reaches to asymptotic value w d = -1 at the late time. We also showed that for negative values of model parameter ξ the transition from decelerated to accelerated phase takes place sooner compare with original ghost dark energy (GDE) model. The statefider analysis was also performed for non-interacting GGDE model. In this case, in the absence of interaction between dark matter and dark energy, the evolutionary trajectories of model in s -r plane can not discriminated. However, the present value of statefinder parameter { s 0 , r 0 } of the model is diagnosed by parameter ξ . We concluded that { s 0 , r 0 } is closer to observational value { s 0 = -0 . 006 , r = 1 . 02 } for negative values of ξ (see Fig.(3)). We also obtained the stability or instability of the model against perturbation by calculating the adiabatic sound speed and showed that the non-interacting case of GGDE model is instable for ξ ≥ 0 and stable for ξ < 0", "pages": [ 12, 13 ] }, { "title": "Acknowledgements", "content": "This work has been supported financially by Research Institute for Astronomy & Astrophysics of Maragha (RIAAM) under research project 1/2782-51.", "pages": [ 13 ] } ]

[ { "title": "ABSTRACT", "content": "International Journal of Modern Physics D c © World Scientific Publishing Company", "pages": [ 1 ] }, { "title": "REMO RUFFINI", "content": "Dip. di Fisica, Sapienza University of Rome and ICRA Piazzale Aldo Moro 5, I-00185, Rome, Italy ICRANet, Piazzale della Repubblica 10, I-65122 Pescara, Italy Universit'e de Nice Sophie Antipolis, Nice, CEDEX 2 Grand Chˆateau Parc Valrose ∗ E-mail: [email protected] Received ?? ?? 2013 Revised 8 August 2013 We review recent progress in our understanding of the nature of gamma ray bursts (GRBs) and in particular, of the relationship between short GRBs and long GRBs. The first example of a short GRB is described. The coincidental occurrence of a GRB with a supernova (SN) is explained within the induced gravitational collapse (IGC) paradigm, following the sequence: 1) an initial binary system consists of a compact carbon-oxygen (CO) core star and a neutron star (NS); 2) the CO core explodes as a SN, and part of the SN ejecta accretes onto the NS which reaches its critical mass and collapses to a black hole (BH) giving rise to a GRB; 3) a new NS is generated by the SN as a remnant. The observational consequences of this scenario are outlined. Keywords : Black Hole; Supernova; Gamma Ray Burst.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "While supernovae (SNe) have been known and studied for a long time, from 1054 A.D. to the classic work of Baade and Zwicky in 1939, observations of GRBs only date from the detection by the Vela satellites in the early 1970s, see e.g. Ref. 1. It has only been after the observations by the Beppo-Sax satellite and the optical identification of GRBs that their enormous energetics, 10 3 -10 4 times larger than those of SNe, have been determined: energies of the order of 10 54 erg, equivalent to the release of ∼ M /circledot c 2 in few tens of seconds. This situation has become even more interesting after the observation of a temporal coincidence between the emission of a GRB and a SN, see e.g. GRB 980425 2 and SN 1998bw. 3 The explanation of this coincidence has led to a many-cosmic-body-interaction and therefore to the introduction of a cosmic matrix: a C-matrix. This totally unprecedented situation has lead to the opening of a new understanding of a vast number of unknown domains of physics and astrophysics.", "pages": [ 1 ] }, { "title": "1.1. CRAB - pulsars and NS rotational energy", "content": "Of all the objects in the sky none has been richer in results for physics, astronomy and astrophysics than the Crab Nebula. Although a result of the 1054 A.D. supernova observed by Chinese, Japanese and Korean astronomers, the nebula itself was not identified till 1731, and not associated with that supernova until the last century, but it has been of interest to astronomers, and later astrophysicists and theoretical physicists ever since, even very recently, see e.g. the discovery by Agile of the giant flare discovered in September 2010. 4 It was only in 1968 that a pulsar was discovered at its center following the predicted existence of rapidly rotating NSs in that decade then soon after observed as pulsars. However, there still remains to explain an outstanding physical process needed to model this object: the expulsion of the shell of the SN during the process of gravitational collapse to a NS. We are currently gaining some understanding of the physical processes governing NSs, motivated by the research on GRBs and BH formation which is being fully exploited to this end at the present time. Paradoxically the study of BHs was started by the discovery of the NS in the Crab Nebula. This study and the understanding of BH formation and consequently of the emission of GRBs is likely to lead, in this Faustian effort to learn the laws of nature, to the understanding of the process of NS formation and the expulsion of the remnant in the SN explosion. That NSs exist in nature has been proven by the direct observation of pulsars. The year 1967 marked the discovery of the first pulsar, observed at radio wavelengths in November 28, 1967 by Jocelyn Bell Burnell and Antony Hewish. 5 Just a few months later, the pulsar NP0532 was found in the center of the Crab Nebula (see Fig. 1) and observed first at radio wavelengths and soon after at optical wavelengths (see Fig. 2). The discovery of NSs led our small group working around John Wheeler in Princeton to direct our main attention to the study of continuous gravitational collapse introduced by Oppenheimer and his students (see Fig. 3). The work in Princeton addressed the topic of BHs, gravitational waves (GWs) and cosmology. A summary of that work can be found in Refs. 7, 8, where a vast number of topics of relativistic astrophysics was reconsidered, including the cross-sections of GW detectors, the possible sources of GWs and especially, an entirely new family of phenomena occurring around BHs.", "pages": [ 2, 3 ] }, { "title": "1.2. The BH mass-energy formula", "content": "The most important result in understanding the physics and astrophysics of BHs has been the formulation of the BH mass-energy formula. From this formula, indeed, it became clear that up to 50% of the mass-energy of a BH could be extracted by using reversible transformations. 9 It then followed that during the formation of a BH, some of the most energetic processes in the universe could exist, releasing an energy of the order of ∼ 10 54 erg for a 1 M /circledot BH.", "pages": [ 3 ] }, { "title": "1.3. VELA satellites and GRBs", "content": "In Ref. 10 I described how the observations of the Vela satellites were fundamental in discovering GRBs, see Fig. 4. Initially it was difficult to model GRBs to understand their nature since their distance from the Earth was unknown, and thousands of models were presented 11 attempting to explain the mystery they presented. Just a few months after the public announcement of their discovery, 1 with T. Damour, a collaborator at Princeton, I formulated a theoretical model based on the extractable 5 energy of a Kerr-Newmann BH through a vacuum polarization process as the origin of GRBs, see Fig. 5. In our paper, 12 we pointed out that vacuum polarization occurring in the field of electromagnetic BHs could release a vast e + e -plasma which self-accelerates and gives origin to the GRB phenomenon. Energetics for GRBs all the way up to ∼ 10 55 ergs was theoretically predicted for a 10 M /circledot BH. The dynamics of this e -e + plasma was first studied by J.R. Wilson and myself with the collaboration of S.-S Xue and J.D. Salmonson. 13, 14", "pages": [ 4, 5 ] }, { "title": "1.4. The BATSE detectors and short and long GRBs", "content": "The launching of the Compton satellite with the BATSE detectors on-board (see Fig. 6) led to the following important discoveries: The crucial contribution to interpreting GRBs came from the Beppo-Sax satellite which led to a much more precise definition of their position in the sky obtained using a wide field X-ray camera and narrow field instrumentation. This enabled the optical identification of GRBs and the determination of their cosmological redshifts, and consequently of their energetics, which turned out to be up to ∼ 10 55 erg, precisely the value predicted by Damour and myself in Ref. 12. Since that time no fewer than ten different X- and γ -ray observatory missions and numerous observations at optical and radio wavelengths have allowed us to reach a deeper understanding of the nature of GRBs. After reviewing in the next paragraphs some recent theoretical progress motivated by the study of GRBs, I will turn to the first example of a genuine short GRB 090227B. 16 Then I will describe the analysis of the GRB 090618 in the fireshell scenario 17 and illustrate the first application of the IGC paradigm to it. 18 Finally I will indicate some recent results on a possible distance indicator inferred from a GRB-SN connection within the IGC paradigm, 19 then giving some additional evidence coming from the identification of the NS created by the SN and its use as a cosmological candle.", "pages": [ 5, 6, 7 ] }, { "title": "1.5. Some recent theoretical progress", "content": "I would like just to present some key images and cite corresponding references to articles documenting some crucial progress we have made that is propedeutic for understanding the physics and astrophysics of GRBs.", "pages": [ 7 ] }, { "title": "1.5.1. Mass, charge and angular momentum in a Kerr-Newman BH: the dyadotorus", "content": "Fig. 8 summarizes the profound difference in analyzing the Kerr-Newman BH between the original paper of B. Carter 20 and our current approach to the physics of the dyadotorus. In Carter's approach attention was focused on geodesics crossing through the horizon of an eternally existing BH and reaching either the BH singularity or analytic extensions to other asymptotically flat space-times. Instead our approach is directed to the fundamental physical processes occurring outside the horizon of a BH and to their possible detection in the dynamical phases of BH formation. Our major focus is to understand the quantum processes leading to vacuum polarization and pair creation and the resulting dynamical expansion to infinity. This mechanism is essential to extract energy from the BH, an amount which can be as high as 50% of its total mass energy as already mentioned above. To reach a theoretical understanding of this problem, it was necessary to introduce the dyadotorus, see Fig. 8.", "pages": [ 7 ] }, { "title": "1.5.2. Thermalization of an electron-positron plasma", "content": "A key result was obtained by analyzing the evolution of the e + e -plasma created in the dyadotorus by vacuum polarization. Cavallo and Rees 22 envisaged that the sudden annihilation of the e + e -pairs and the expansion of the thermal radiation in the circumburst medium (CBM) would lead to an explosion very similar to an H-bomb, a scenario identified as the fireball model. By considering the essential role of three-body interactions, we have proven that the e + e -pairs do not annihilate all at once as claimed by Cavallo and Rees 22 but they thermalize with the photons 23 and keep expanding in a shell until transparency of the e + e -plasma is reached, 24 a new paradigm for GRBs called the fireshell model.", "pages": [ 7 ] }, { "title": "1.5.3. The new approach to analyzing NS equilibrium configurations in an unified approach encompassing all fundamental interactions", "content": "A completely new approach to NS equilibrium configurations was advanced in recent years and has evolved into a much more complicated model, fulfilling the cri- teria needed conceptually for the description of NSs. 25,26 The first model for a NS was given by Gamow as a system entirely composed of neutrons governed by both Fermi statistics and Newtonian gravity. The extension of this model to general relativity was made by Oppenheimer and his students, leading to the classic Tolman-Oppenheimer-Volkoff (TOV) equilibrium equations. 27,28 This was then extended to a system of three degenerate gases of neutrons, protons and electrons and solved by John Wheeler and his students and collaborators. 29 However, they assumed local charge neutrality for mathematical convenience. It was later realized that a more complete description was needed, since the previous analyses violated basic thermodynamic and general relativistic conditions required for conservation of the Klein potential. 30 A new much more complete treatment appeared to be needed involving in a self-consistent way all the fundamental forces. A new model has since emerged, extending the general relativistic Thomas-Fermi equations to the strong and weak interactions throughout the entire NS 26 (see Fig. 10-12). This complete model satisfies instead global charge neutrality of the entire configuration and not strict local charge neutrality, an erroneous assumption usually made in the existing literature on NS models.", "pages": [ 7, 8, 9 ] }, { "title": "Generalizing to the Strongly Interacting Case (Rueda", "content": "With this short summary of the most relevant conceptual and theoretical is-", "pages": [ 9 ] }, { "title": "Neutron Star Equilibrium Configurations", "content": "(Belvedere, Pugliese, Rueda, Ruffini, Xue, Nucl. Phys. A 883, 1, 2012) sues, I now briefly summarize how some of them have allowed us to reach a new understanding of the short GRBs and the SN-GRB connection.", "pages": [ 10 ] }, { "title": "2.1. Introduction", "content": "Using the data obtained from the Fermi-GBM satellite, 32 Ref. 16 has proven the existence of yet another class of GRBs theoretically predicted by the fireshell model 24,33 which we define here as the 'genuine short GRBs.' This canonical class of GRBs is characterized by extremely small values of the Baryon Load B /lessorsimilar 10 -5 (see Fig. 13). The energy emitted in the proper GRB (P-GRB) described below is predominate with respect to the extended afterglow and its characteristic duration 16 is expected to be shorter than a fraction of a second (see Sec. 2.2.8). A search has begun for these genuine short GRBs among the bursts detected by the Fermi-GBM instrument during the first three years of its mission. The initial list of short GRBs was reduced by requiring that no prominent X-ray or optical afterglow be observed. The GRB 090227B has been identified among the remaining bursts. A spectral analysis of its source has been performed from its observed light curves, and its cosmological redshift and all the basic parameters of the burst, as well as the isotropic energy, the Lorentz Γ factor at transparency, and the intrinsic duration, have all been inferred from theory. In Sec. 2.2 the relevant properties of the fireshell model are summarized. In Sec. 2.3 the observations of GRB 090227B by various satellites and their data analysis are reviewed. In Sec. 2.4 all the parameters characterizing this GRB within the fireshell scenario, including the redshift, are determined. In the conclusions we show that this GRB is the missing link between the genuine short and the long GRBs, with some common characteristics of both classes. Further analysis of genuine short GRBs with a smaller value of B should lead to a P-GRB with an even more pro- nounced thermal component. The progenitor of GRB 090227B is identified as a symmetric binary system of two neutron stars, each of ∼ 1 . 34 M /circledot , see e.g. Ref. 34.", "pages": [ 10, 11 ] }, { "title": "2.2.1. The GRB prompt emission in the fireball scenario", "content": "A variety of models have been developed to theoretically explain the observational properties of GRBs, among which the fireball model 35 is one of those most often used. In Refs. 22, 36, 37 it was proposed that the sudden release of a large quantity of energy in a compact region can lead to an optically thick photon-lepton plasma and to the production of e + e -pairs. The sudden initial total annihilation of the e + e -plasma was assumed by Cavallo and Rees, 22 leading to an enormous release of energy pushing on the CBM: the 'fireball.' An alternative approach, originating in the gravitational collapse to a BH, is the fireshell model, see e.g. Refs. 38, 39. Here the GRB originates from an optically thick e + e -plasma in thermal equilibrium, with a total energy of E e ± tot . This plasma is initially confined between the radius r h of a BH and the dyadosphere radius where α is the usual fine structure constant, /planckover2pi1 the Planck constant, c the speed of light, and m e the mass of the electron. The lower limit of E e ± tot is assumed to coincide with the observed isotropic energy E iso emitted in X-rays and gamma rays alone in the GRB. The condition of thermal equilibrium assumed in this model 23 distinguishes this approach from alternative ones, e.g. Ref. 22.", "pages": [ 11 ] }, { "title": "12 Remo Ruffini", "content": "In the fireball model, the prompt emission, including the sharp luminosity variations, 40 are caused by the prolonged and variable activity of the 'inner engine'. 35, 41 The conversion of the fireball energy to radiation originates in shocks, either internal (when faster moving matter overtakes a slower moving shell, see Ref. 41) or external (when the moving matter is slowed down by the external medium surrounding the burst, see Ref. 42). Much attention has been given to synchrotron emission from relativistic electrons in the CBM, possibly accompanied by Self-Synchrotron Compton (SSC) emission, to explain the observed GRB spectrum. These processes were found to be consistent with the observational data of many GRBs. 43,44 However, several limitations have been reported in relation with the low-energy spectral slopes of time-integrated spectra 45-48 and the time-resolved spectra. 48 Additional limitations on SSC emission have also been pointed out in Refs. 49, 50. The latest phases of the afterglow are described in the fireball model by assuming an equation of motion given by the Blandford-McKee self-similar power-law solution. 51 The maximum Lorentz factor of the fireball is estimated from the temporal occurrence of the peak of the optical emission, which is identified with the peak of the forward external shock emission 52, 53 in the thin shell approximation. 54 Several partly alternative and/or complementary scenarios have been developed distinct from the fireball model, e.g. based on quasi-thermal Comptonization, 55 Compton drag emission, 56, 57 synchrotron emission from a decaying magnetic field, 58 jitter radiation, 59 Compton scattering of synchrotron self-absorbed photons, 60, 61 and photospheric emission. 62-68 In particular, it was pointed out in Ref. 67 that photospheric emission overcomes some of the difficulties of purely non-thermal emission models.", "pages": [ 12 ] }, { "title": "2.2.2. The fireshell scenario", "content": "In the fireshell model, the rate equation for the e + e -pair plasma and its dynamics (the pair-electromagnetic pulse or PEM pulse for short) have been described in Ref. 14. This plasma engulfs the baryonic material left over from the process of gravitational collapse having a mass M B , still maintaining thermal equilibrium between electrons, positrons, and baryons. The baryon load is measured by the dimensionless parameter B = M B c 2 /E e + e -tot . Ref. 69 showed that no relativistic expansion of the plasma exists for B > 10 -2 . The fireshell is still optically thick and self-accelerates to ultrarelativistic velocities (the pair-electromagnetic-baryonic pulse or PEMB pulse for short 69 ). Then the fireshell becomes transparent and the P-GRB is emitted. 24 The final Lorentz gamma factor at transparency can vary over a wide range between 10 2 and 10 4 as a function of E e + e -tot and B , see Fig. 14. For its final determination it is necessary to explicitly integrate the rate equation for the e + e -annihilation process and evaluate, for a given BH mass and a given e + e -plasma radius, at what point the transparency condition is reached 14 (see Fig. 15). The fireshell scenario does not require any prolonged activity of the inner engine. After transparency, the remaining accelerated baryonic matter still expands ballistically and starts to slow down from collisions with the CBM of average den- sity n CBM . In the standard fireball scenario, 70 the spiky light curve is assumed to be caused by internal shocks. In the fireshell model the entire extended-afterglow emission is assumed to originate from an expanding thin shell, which maintains energy and momentum conservation during its collision with the CBM. The condition of a fully radiative regime is assumed. 24 This in turn allows one to estimate the characteristic inhomogeneities of the CBM, as well as its average value. It is appropriate to point out another difference between our treatment and others in the current literature. The complete analytic solution of the equations of motion of the baryonic shell were developed in Refs. 71, 72, while elsewhere the Blandford-McKee self-similar approximate solution is almost always adopted without justification. 64,73-81 The analogies and differences between the two approaches have been explicitly explained in Ref. 82. In our general approach, a canonical GRB bolometric light curve is composed of two different parts: the P-GRB and the extended afterglow. The relative energetics of these two components and the observed temporal separation between the corresponding peaks is a function of the above three parameters E e + e -tot , B , and the average value of the n CBM . The first two parameters are inherent to the accelerator characterizing the GRB, i.e., the optically thick phase, while the third one is inherent to the environment surrounding the GRB which gives rise to the extendedafterglow. For the observational properties of a relativistically expanding fireshell model, a crucial concept has been the introduction of the EQui-Temporal Surfaces (EQTS). Here too our model differs from those in the literature by having deriving an analytic expression of the EQTS obtained from the solutions to the equations of motion. 82", "pages": [ 12, 13 ] }, { "title": "2.2.3. The emission of the P-GRB", "content": "The lower limit for E e + e -tot is given by the observed isotropic energy E iso emitted in the GRB. The identification of the energy of the afterglow and of the P-GRB determines the baryon load B and from these it is possible to determine the value of the Lorentz Γ factor at transparency, the observed temperature as well as the temperature in the comoving frame and the laboratory radius at transparency, see Fig. 15. We can indeed determine from the spectral analysis of the P-GRB candidate the temperature kT obs and the energy E P -GRB emitted at the point of transparency. The relation between these parameters cannot be expressed analytically, only through numerical integration of the entire set of fireshell equations of motion. In practice we need to perform a trial-and-error procedure to find a set of values that fits the observations. The direct measure of the temperature of the thermal component at transparency offers very important new information on the determination of the GRB parameters. Two different phases are present in the emission of the P-GRB: one corresponding to the emission of the photons when transparency is reached and another corresponding to the early interaction of the ultra-relativistic protons and", "pages": [ 13 ] }, { "title": "14 Remo Ruffini", "content": "electrons with the CBM. A spectral energy distribution with both a thermal and a non-thermal component should be expected to result from these two phases.", "pages": [ 14 ] }, { "title": "2.2.4. The extended afterglow", "content": "The majority of articles in the current literature have analyzed the afterglow emission as the result of various combinations of synchrotron and inverse Compton processes. 35 It appears, however, that this description is not completely satisfactory. 48-50 We adopted a pragmatic approach in our fireshell model by making full use of the knowledge of the equations of motion, of the EQTS formulations, 72 and of the correct relativistic transformations between the comoving frame of the fireshell and the observer frame. These equations, which relate four distinct time variables, are necessary for interpreting the GRB data. They are: a) the comoving time, b) the laboratory time, c) the arrival time, and d) the arrival time at the detector corrected for cosmological effects. This is the content of the relative space-time transformation paradigm, essential for the interpretation of GRB data. 83 This paradigm requires a global rather than a piecewise description in time of the GRB phenomenon 83 and has led to a new interpretation of the burst structure paradigm. 24 As mentioned in the introduction, a new conclusion arising from the burst structure paradigm has been that emission by the accelerated baryons interacting with the CBM is indeed occurring already in the prompt emission phase, just after the P-GRB emission. This is the extended-afterglow emission, which exhibits in its 'light curve' a rising part, a peak, and a decaying tail. Following this paradigm, the prompt emission phase consists therefore of the P-GRB emission and the peak of the extended afterglow. Their relative energetics and observed time separation are functions of the energy E tot e + e -, of the baryon load B , and of the CBM density distribution n CBM (see Fig. 16). In particular, fordecreasing B , the extended afterglow light curve 'squeezes' itself on the P-GRB and the P-GRB peak luminosity increases (see Fig. 17). To evaluate the extended-afterglow spectral properties, we adopted an ansatz for the spectral properties of the emission in the collisions between the baryons and the CBM in the comoving frame. We then evaluated all observational properties in the observer frame by integrating over the EQTS. The initial ansatz of a thermal spectrum 24 has recently been modified to where α is a phenomenological parameter defined in the comoving frame of the fireshell, 84 determined by the optimization of the simulation of the observed data. It is well known that in the ultrarelativistic collision of protons and electrons with the CBM, collective processes of ultrarelativistic plasma physics are expected, which are not yet fully explored and understood (e.g. the Weibel instability, see Ref. 85). Promising results along this line have already been obtained in Refs. 86, 87 and may lead to the understanding of the physical origin of the α parameter in Eq. 2. To take into due account the filamentary, clumpy and porous structure of the CBM, we introduced the additional parameter R , which describes the fireshell surface-filling factor. It is defined as the ratio between the effective emitting area of the fireshell A eff and its total visible area A vis , see e.g. Refs. 33, 88. One of the main features of the GRB afterglow has been the observation of hardto-soft spectral variation, which is generally absent in the first spike-like emission, and which we have identified as the P-GRB. 89-92 An explanation of the hard-to-soft spectral variation has been advanced on the grounds of two different contributions: the curvature effect and the intrinsic spectral evolution. In particular, Ref. 93 used the model developed in Ref. 94 for the spectral lag analysis, taking into account an intrinsic band model for the GRBs and using a Gaussian profile for the GRB pulses to take into account angular effects, and they found that both provide a very good explanation for the observed time lags. Within the fireshell model we can indeed explain a hard-to-soft spectral variation in the extended-afterglow emission very naturally. Since the Lorentz Γ factor decreases with time, the observed effective temperature of the fireshell will drop as the emission goes on, and consequently the peak of the emission will occur at lower energies. This effect is amplified by the curvature effect, which originates from the EQTS analysis. Both these observed features are considered to be responsible for the time lag observed in GRBs.", "pages": [ 14, 15, 16, 17 ] }, { "title": "2.2.5. The simulation of a GRB light curve and spectra of the extended afterglow", "content": "The simulation of a GRB light curve and the respective spectrum also requires the determination of the filling factor R and of the CBM density n CBM . These extra parameters are extrinsic and they are just functions of the radial coordinate from the source. The parameter R , in particular, determines the effective temperature in the comoving frame and the corresponding peak energy of the spectrum, while n CBM determines the temporal behavior of the light curve. Particularly important is the determination of the average value of n cbm . Values on the order of 0 . 1-10 particles/cm 3 have been found for GRBs exploding inside star-forming region galaxies, while values on the order of 10 -3 particles/cm 3 have been found for GRBs exploding in galactic halos. 89, 90, 92 It is found that the CBM is typically formed of 'clumps'. This clumpy medium, already predicted in pioneering work by Fermi on the theoretical study of interstellar matter in our galaxy, 95, 96 is by now well-established both from the GRB observations and by additional astrophysical observations, see e.g. the CBM observed in SNe, 97 or by theoretical considerations involving a super-giant massive star clumpy wind. 98 The determination of the parameter R and n CBM depends essentially on the reproduction of the shape of the extended-afterglow and of the respective spectral emission in a fixed energy range. Clearly, the simulation of a source within the fireshell model is much more complicated than simply fitting the photon spectrum N ( E ) of the burst (number of photons at a given energy) with analytic phenomeno-", "pages": [ 17 ] }, { "title": "18 Remo Ruffini", "content": "logical formulas for a finite temporal range of the data. It is a consistent picture, which has to find the best value for the parameters of the source, the P-GRB, 24 its spectrum, its temporal structure, as well as its energetics. For each spike in the light curve the parameters of the corresponding CBM clumps are computed, taking into account all the thousands of convolutions of comoving spectra over each EQTS that leads to the observed spectrum. 72,82 It is clear that, since the EQTSs encompass emission processes occurring at different comoving times weighted by their Lorentz and Doppler factors, the 'fitting' of a single spike is not only a function of the properties of the specific CBM clump but of the entire previous history of the source. Any mistake at any step of the simulation process affects the entire evolution that follows and conversely, at any step a fit must be made consistently with the entire previous history: because of the nonlinearity of the system and the EQTSs, any change in the simulation produces observable effects up to a much later time. This leads to an extremely complicated trial and error procedure in the data simulation, in which the variation of the parameters defining the source are increasingly narrowed down, reaching uniqueness very quickly. Of course, we cannot expect the last parts of the simulation to be very accurate, since some of the basic hypotheses about the equations of motion and possible fragmentation of the shell can affect the procedure. In particular, the theoretical photon number spectrum to be compared with the observational data is obtained by an averaging procedure over instantaneous spectra. In turn, each instantaneous spectrum is linked to the simulation of the observed multiband light curves in the chosen time interval. Therefore, the simulation of the spectrum and of the observed multiband light curves have to be performed together and have optimized simultaneously.", "pages": [ 18 ] }, { "title": "2.2.6. The canonical long GRBs", "content": "According to the fireshell model theory, the canonical long GRBs are characterized by a baryon load varying in the range 3 . 0 × 10 -4 /lessorsimilar B ≤ 10 -2 and they occur in a typical galactic CBM with an average density 〈 n CBM 〉 ≈ 1 particle/cm 3 . As a result the extended afterglow is predominant with respect to the P-GRB (see Fig. 13).", "pages": [ 18 ] }, { "title": "2.2.7. The disguised short GRBs", "content": "After the observations by Swift of GRB 050509B, 99 which was declared in the literature as the first short GRB with an extended emission ever observed, it has become clear that all such sources are actually disguised short GRBs. 92 It is conceivable and probable that also a large fraction of the declared short duration GRBs in the BATSE catalog, observed before the discovery of the afterglow, are members of this class. In the case of the disguised short GRBs the baryon load varies in the same range of the long bursts, while the CBM density is of the order of 10 -3 particles/cm 3 . As a consequence, the extended afterglow results in a 'deflated' emission that can be exceeded in peak luminosity by the P-GRB. 89-92,100 Indeed the integrated emission in the extended afterglow is much larger than the one of the P-GRB (see Fig. 13), as expected for long GRBs. With these understandings long and disguised short GRBs are interpreted in terms of long GRBs exploding, respectively, in a typical galactic density or in a galactic halo density. This interpretation has been supported by direct optical observations of GRBs located in the outskirts of the host galaxies. 101-107", "pages": [ 18, 19 ] }, { "title": "2.2.8. The class of genuine short GRBs", "content": "The canonical genuine short GRBs occur in the limit of very low baryon load, e.g. B /lessorsimilar 10 -5 with the P-GRB predominant with respect to the extended afterglow. For such small values of B the afterglow peak emission shrinks over the P-GRB and its flux is lower than that of the P-GRB (see Fig. 17). Since the baryon load is small but not zero, in addition to the predominant role of the P-GRB, which has a thermal spectrum, a nonthermal component originating from the extended afterglow is expected. The best example of a genuine short GRB is GRB 090227B (see details in Ref. 16).", "pages": [ 19 ] }, { "title": "2.3. Observations and data analysis of GRB 090227B", "content": "At 18:31:01.41 UT on February 27, 2009, the Fermi-GBM detector 108 triggered and located the short and bright burst GRB 090227B (trigger 257452263/090227772). The on-ground calculated location, using the GBM trigger data, was (RA, Dec)(J2000)=(11 h 48 m 36 s , 32 o 10 ' 12 '' ), with an uncertainty of 1.77 o (statistical only). The angle from the Fermi LAT boresight was 72 o . The burst was also located by IPN 109 and detected by Konus-Wind, 110 showing a single pulse with duration ∼ 0 . 2 s (20 keV - 10 MeV). No X-rays or optical observations were reported on the GCN Circular Archive, so the redshift of the source is unknown. To obtain the Fermi-GBM light-curves and the spectrum in the energy range 8 keV - 40 MeV, we made use of the RMFIT program. For the spectral analysis, we have downloaded from the gsfc website a the TTE (Time-Tagged Events) files, suitable for short or highly structured events. We used the light curves corresponding to the NaI-n2 (8 - 900 keV) and the BGO-b0 (250 keV - 40 MeV) detectors. The 64 ms binned GBM light curves show one very bright spike with a short duration of 0 . 384 s, in the energy range 8 keV - 40 MeV, and a faint tail lasting up to 0 . 9 s after the trigtime T 0 in the energy range 10 keV - 1 MeV. After the subtraction of the background, we have proceeded with the time-integrated and time-resolved spectral analyses.", "pages": [ 19 ] }, { "title": "2.3.1. Time-integrated spectral analysis", "content": "We have performed a time-integrated spectral analysis in the time interval from T 0 -0 . 064 s to T 0 + 0 . 896 s, which corresponds to the T 90 duration of the burst. We have fit the spectrum in this time interval considering the following models: comptonization (Compt) plus power-law (PL) and band 111 plus PL, as outlined, e.g. in Ref. 112, as well as a combination of black body (BB) and band (see Fig. 18). Within the T 90 time interval, the BB+Band model improves the fit with respect to the Compt+PL model at a confidence level of 5%. The comparison between Band+PL and Compt+PL models is outside such a confidence level (about 8%). The direct comparison between BB+Band and Band+PL models, which have the same number of degrees of freedom, provides almost the same C-STAT values for the BB+Band and Band+PL models (∆C-STAT ≈ 0 . 89). This means that all three models are viable. For the BB+Band model, the ratio between the fluxes of the thermal component and the non-thermal (NT) component is F BB /F NT ≈ 0 . 22. The BB component is important for the determination of the peak of the νF ν spectrum and has an observed temperature kT = (397 ± 70) keV. We have then focused our attention on the spike component, namely the time interval from T 0 -0 . 064 s to T 0 +0 . 192, which we indicate in the following as the T spike . We have repeated the time-integrated analysis considering the same spectral models of the previous interval (see Fig. 19). Within the T spike time interval, both the BB+Band and Band+PL models marginally improve the fit of the data with respect to the Compt+PL model within a confidence level of 5%. Again, the C-STAT values of the BB+Band and Band+PL models are almost the same (∆C-STAT ≈ 0 . 15) and they are statically equivalent in the T spike . For the BB+Band model, the observed temperature of the thermal component is kT = (515 ± 28) keV and the flux ratio between the BB and NT components increases up to F BB /F NT ≈ 0 . 69. We have performed a further analysis in the time interval from T 0 +0 . 192 s to T 0 + 0 . 896 s, which we indicate as T tail , by considering the BB+PL, Compt and PL models (see Fig. 20). The statistical comparison shows that the best fit is the Compt model and a thermal component is ruled out. For details, see Ref. 16. In view of the above, we have focused our attention on the fit of the data of the BB+Band model within the fireshell scenario, being equally probable from a mere statistical point of view with the other two choices, namely the Band+PL and Compt+PL. According to the fireshell scenario (see Sec. 2.2.3), the emission within the T spike time interval is related to the P-GRB and is expected to be thermal. In addition the transition between the transparency emission of the P-GRB and the extended afterglow is not sharp. The time separation between the P-GRB and the peak of the extended afterglow depends on the energy of the e + e -plasma E tot e + e -, the baryon load B and the CBM density n CBM (see Fig. 17). As shown in Figs. 16 and 17, for decreasing values of B an early onset of the extended afterglow in the P-GRB spectrum occurs and thus an NT component in the T spike is expected. As a further check, the theory of the fireshell model indeed predicts in the early part of the prompt emission of GRBs a thermal component due to the transparency of the e + e -plasma (see Sec. 2.2), while in the extended afterglow no thermal component is expected (see Sec. 2.2.4), as observed in the T tail time interval. Our theoretical interpretation has shown to be consistent with the observational data and the statistical analysis. From an astrophysical point of view the BB+Band model is preferred over the other two models, statistically equivalent in view of the above theoretical considerations.", "pages": [ 20, 21, 22 ] }, { "title": "2.3.2. Time-resolved spectral analysis", "content": "We have performed a time-resolved spectral analysis on selected shorter time intervals of 32 ms (see Fig. 21) in order to correctly identify the P-GRB, namely finding out in which time interval the thermal component exceeds or at least has a comparable flux with respect to the NT one due to the onset of the extended afterglow. In this way we can single out the contribution of the NT component in the spectrum of the P-GRB. Within the first time-resolved interval the BB+PL model has a thermal flux (11 . 2 ± 3 . 4) times bigger than the PL flux; the fit with the BB+Band provides F BB = (0 . 50 ± 0 . 26) F NT , where the NT component is in this case the band model. In the second and fourth intervals, the BB+Band model provides an improvement at a significance level of 5% in the fitting procedure with respect to the simple band model. In the third time interval as well as in the remaining time intervals up to T 0 +0 . 192 s the band spectral models provide better fits with respect to the BB+NT ones. This is exactly what we expect from our theoretical understanding: from T 0 -0 . 032 s to T 0 +0 . 096 s we have found the edge of the P-GRB emission, in which the thermal components have fluxes higher or comparable to the NT ones. The third interval corresponds to the peak emission of the extended afterglow (see Fig. 24). The contribution of the extended afterglow in the remaining time intervals increases, while the thermal flux noticeably decreases. We have then explored the possibility of a further rebinning of the time interval T spike , taking advantage of the large statistical content of each time bin. We have plotted the NaI-n2 light curve of GRB 090227B using time bins of 16 ms (see Fig. 22, left panels). The re-binned light curves show two spike-like substructures. The duration of the first spike is 96 ms and it is clearly distinct from the second spike. In this time range the observed BB temperature is kT = (517 ± 28) keV and the ratio between the fluxes of the thermal and non-thermal components is F BB /F NT ≈ 1 . 1. Consequently, we have interpreted the first spike as the P-GRB and the second spike as part of the extended afterglow. Their spectra are shown in Fig. 22, right panels.", "pages": [ 22, 23, 24 ] }, { "title": "2.4. Analysis of GRB 090227B in the fireshell model", "content": "The identification of the P-GRB is fundamental in order to determine the baryon load and the other physical quantities characterizing the plasma at the transparency point (see Fig. 15). It is crucial to determine the cosmological redshift, which can be derived by combining the observed fluxes and the spectral properties of the P-GRB and of the extended afterglow with the equation of motion of our theory. From the cosmological redshift we derive E tot e + e -and the relative energetics of the P-GRB and of the extended afterglow components (see Fig. 15). Having so derived the baryon load B and the energy E tot e + e -, we can constrain the total energy and simulate the canonical light curve of the GRBs with their characteristic pulses, modeled by a variable number density distribution of the CBM around the burst site.", "pages": [ 24, 25 ] }, { "title": "2.4.1. Estimation of the redshift of GRB 090227B", "content": "Having determined the redshift of the source, the analysis consists of equating E tot e + e -≡ E iso (namely E iso is a lower limit on E tot e + e -) and inserting a value of the baryon load to complete the simulation. The right set of E tot e + e -and B is determined when the theoretical energy and temperature of the P-GRB match the observed ones of the thermal emission [namely E P -GRB ≡ E BB and kT obs = kT blue / (1 + z )]. In the case of GRB 090227B we have estimated (see Ref. 16) the ratio E P -GRB /E tot e + e -from the observed fluences where d l is the luminosity distance of the source and S = F ∆ t are the fluences. The fluence of the BB component of the P-GRB is S BB = (1 . 54 ± 0 . 45) × 10 -5 erg/cm 2 . The total fluence of the burst is S tot = (3 . 79 ± 0 . 20) × 10 -5 erg/cm 2 and has been evaluated in the time interval from T 0 -0 . 016 s to T 0 + 0 . 896 s. This interval differs slightly from T 90 because of the new time boundaries defined after the rebinning of the light curve at a resolution of 16 ms. Therefore the observed energy ratio is E P -GRB /E tot e + e -= (40 . 67 ± 0 . 12)%. As is clear from the bottom right diagram in Fig. 15, for each value of this ratio we have a range of possible parameters B and E tot e + e -. In turn, for each of their values we can determine the theoretical blue-shifted toward the observer temperature kT blue (see the top right diagram in Fig. 15). Correspondingly, for each pair of values of B and E tot e + e -we estimate the value of z by the ratio between kT blue and the observed temperature of the P-GRB kT obs , In order to remove the degeneracy [ E tot e + e -( z ) , B ( z )], we have made use of the isotropic energy formula in which N ( E ) is the photon spectrum of the burst and the integrals are due to the bolometric correction on S tot . By imposing the condition E iso ≡ E tot e + e -, we have found the values z = 1 . 61 ± 0 . 14 for B = (4 . 13 ± 0 . 05) × 10 -5 and E tot e + e -= (2 . 83 ± 0 . 15) × 10 53 ergs. The complete quantities determined in this way are summarized in Table 1.", "pages": [ 25 ] }, { "title": "2.4.2. The analysis of the extended afterglow and the observed spectrum of the P-GRB", "content": "As mentioned in Sec. 2.2, the arrival time separation between the P-GRB and the peak of the extended afterglow is a function of E tot e + e -and B and depends on the detailed profile of the CBM density. For B ∼ 4 × 10 -5 (see Fig. 16) the time separation is ∼ 10 -3 -10 -2 s in the source cosmological rest frame. In this light, there is an interface between reaching transparency in the P-GRB and the early part of the extended afterglow. This connection has already been introduced in the literature, see e.g. Refs. 113, 17, 114. From the determination of the initial values of the energy E tot e + e -= 2 . 83 × 10 53 ergs, the baryon load B = 4 . 13 × 10 -5 , and the Lorentz factor Γ tr = 1 . 44 × 10 4 , we have simulated the light curve of the extended afterglow by deriving the radial distribution of the CBM clouds around the burst site (see Table 2 and Fig. 23). In particular, each spike in Fig. 23 corresponds to a CBM cloud. The error boxes on the number density on each cloud is defined as the maximum possible tolerance to ensure agreement between the simulated light curve and the observed data. The average value of the CBM density is 〈 n 〉 = (1 . 90 ± 0 . 20) × 10 -5 particles/cm 3 with an average density contrast 〈 δn/n 〉 = 0 . 82 ± 0 . 11 (see also Table 1). These values are typical of the galactic halo environment. The filling factor varies in the range 9 . 1 × 10 -12 ≤ R ≤ 1 . 5 × 10 -11 , up to 2 . 38 × 10 17 cm away from the burst site, and then drops to the value R = 1 . 0 × 10 -15 . The value of the α parameter has been found to be -1 . 99 along the entire duration of the GRB. In Fig. 24 we show the NaI-n2 simulated light curve (8-1000 keV) of GRB 090227B and in Fig. 25 (left panel) the corresponding spectrum in the early ∼ 0 . 4 s of the emission, using the spectral model described by Eq. 2. The simulation of the extended afterglow starts T a -T 0 ∼ 0 . 017 s after the trigtime T 0 . At the 13 th Marcel Grossmann Meeting in 2012, G. Vianello suggested extending our simulations from 1 MeV all the way s) 2 photons/(keV cm to 40 MeV, since significant data are available from the BGO detector. Without changing the parameters used in the theoretical simulation of the NaI-n2 data, we have extended the simulation up to 40 MeV and have compared the results with the BGO-b0 data (see Fig. 25, right panel). The theoretical simulation we performed, optimized on the NaI-n2 data alone, is perfectly consistent with the observed data all over the entire range of energies covered by the Fermi-GBM detector, both NaI and BGO. We turn now to the emission of the early 96 ms. We have studied the interface between the P-GRB emission and the on-set of the extended afterglow emission. In Fig. 26 we have plotted the thermal spectrum of the P-GRB and the fireshell simulation (from T 0 + 0 . 015 s to T 0 + 0 . 080 s) of the early interaction of the extended afterglow. The sum of these two components is compared with the observed spectrum from the NaI-n2 detector in the energy range 8-1000 keV (see Fig. 26, left panel). Then again, from the theoretical simulation in the energy range of the NaIn2 data, we have verified the consistency of the simulation extended up to 40 MeV with the observed data all over the range of energies covered by the Fermi-GBM detector, both NaI and BGO. The result is shown in Fig. 26 (right panel).", "pages": [ 26, 27, 28 ] }, { "title": "2.5. Conclusions", "content": "The comprehension of this short GRB has been improved by analyzing the different spectra in the T 90 , T spike and T tail time intervals. We have shown that within the T 90 and the T spike all the considered models (BB+Band, Band+PL, Compt+PL) are viable, while in the T tail time interval, the presence of a thermal component is ruled out. The result of the analysis in the T tail time interval gives an additional correspondence between the fireshell model (see Sec. 2.2.4) and the observational data. According to this picture, the emission within the T spike time interval is related to the P-GRB and it is expected to have a thermal spectrum with in addition an extra NT component due to an early onset of the extended afterglow. In this time interval a BB with an additional band component has been observed and we have shown that it is statistically equivalent to the Compt+PL and the Band+PL models. Our theoretical interpretation is consistent with the observational data and statistical analysis. From an astrophysical point of view the BB+Band model is preferred over the Compt+PL and the Band+PL models, being described by a consistent theoretical model. GRB 090227B is the missing link between the genuine short GRBs, with the baryon load B /lessorsimilar 5 × 10 -5 and theoretically predicted by the fireshell model, 24,83,115 and the long bursts. From the observations, GRB 090227B has an overall emission lasting ∼ 0 . 9 s with a fluence of 3 . 79 × 10 -5 erg/cm 2 in the energy range 8 keV - 40 MeV. In absence of an optical identification, no determination of its cosmological redshift and of its energetics was possible. Thanks to the excellent data available from Fermi-GBM, 32 it has been possible to probe the comparison between the observations and the theoretical model. In this sense, we have then performed a more detailed spectral analysis on the time scale as short as 16 ms of the time interval T spike . As a result we have found in the early 96 ms a thermal emission which we have identified with the theoretically expected P-GRB component. The subsequent emission of the time interval T spike has been interpreted as part the extended afterglow. Consequently, we have determined the cosmological redshift z = 1 . 61 ± 0 . 14, as well as the baryon load B = (4 . 13 ± 0 . 05) × 10 -5 , its energetics, E tot e + e -= (2 . 83 ± 0 . 15) × 10 53 ergs, and the extremely high Lorentz Γ factor at the transparency Γ tr = (1 . 44 ± 0 . 01) × 10 4 . We are led to the conclusion 34 that the progenitor of this GRB is a binary neutron star, which for simplicity we assume to have the same mass, by the following considerations:", "pages": [ 28, 29 ] }, { "title": "30 Remo Ruffini", "content": "particles/cm 3 ; this fact points to two compact objects in a binary system that have spiraled out in the halo of their host galaxy; 89-92,100,116 We can then generally conclude the existence of three different possible structures for the canonical GRBs (see Fig. 27 and Table 3): regions; Finally, if we turn to the theoretical model within a general relativistic description of the gravitational collapse to a 10 M /circledot BH, see e.g. Refs. 117, 118 and Fig. 2 in Ref. 119, we find it necessary to use time resolutions on the order of a fraction of a ms, possibly down to µ s, in order to follow such a process. One would need new space missions with larger collecting area to prove with great accuracy the identification of a thermal component. It is likely that an improved data acquisition with high signal to noise ratio on a shorter time scale would show more clearly the thermal component as well as distinguish more effectively different fitting procedures.", "pages": [ 30, 31 ] }, { "title": "3.1. Introduction", "content": "Until recently, all the X- and γ -ray activities of a signal sufficiently short in time, less than 10 2 -10 3 s, and of extragalactic origin have been called a GRB. A new situation has occurred with the case of GRB 090618 120 in which the multi-component nature of GRBs has been illustrated. This GRB is a member of a special class of bursts associated with a SN. It is now clear from the detailed analysis that there are at least three different components in the nature of this GRB: episode 1 which corresponds to the early emission of the SN event with Lorentz factor Γ ∼ 1; episode 2 which corresponds to the GRB with Lorentz factor 10 2 /lessorsimilar Γ /lessorsimilar 10 4 ; and episode 3 which appears to be related to the activities of the newly born NS. I will describe a few key moments in the recent evolution of our understanding of this system which is very unique within physics and astrophysics.", "pages": [ 31 ] }, { "title": "3.2. The case of GRB 090618", "content": "GRB 090618 represents the prototype of a class of energetic ( E iso ≥ 10 52 erg) GRBs, characterized by the presence of a supernova observed 10 (1+z) days after the trigger time, and the observation of two distinct emission episodes in their hard X-ray light curve (see details in Ref. 17). It was discovered by the Swift satellite. 121 The BAT light curve shows a multipeak structure, whose total estimated duration is ∼ 320 s and whose T 90 duration in the (15-350) keV range was 113 s. 122 The first 50 s of the light curve shows a smooth decay trend followed by a spiky emission, with three prominent peaks at 62, 80, and 112 s after the trigger time, respectively, and each have the typical appearance of a FRED pulse, 123 see Fig. 3.2.1. The time-integrated spectrum, (t 0 - 4.4, t 0 + 213.6) s in the (15-150)keV range, was found to agree with a power-law spectral model with an exponential cut-off, whose photon index is γ = 1.42 ± 0.08 and a cut-off energy E peak = 134 ± 19 keV. 124 The XRT observations started 125 s after the BAT trigger time and lasted ∼ 25.6 ks 125 and reported an initially bright uncatalogued source, identified as the afterglow of GRB 090618. Its early decay is very steep, ending at 310 s after the trigger time, when it starts a shallower phase, the plateau. Then the light curve breaks into a steeper late phase. GRB 090618 was observed also by the Gamma-ray Burst Monitor (GBM) on board the Fermi satellite. 32 From a first analysis, the time-integrated spectrum, ( t 0 , t 0 + 140) s in the (8-1000)keV range, was fit by a band 111 spectral model, with a peak energy E peak = 155.5 keV, α = -1 . 26 and β = -2 . 50, 126 but with strong spectral variations within the considered time interval. The redshift of the source is z = 0 . 54 and it was determined thanks to the identification of the MgII, Mg I, and FeII absorption lines using the KAST spectrograph mounted at the 3 m Shane telescope at the Lick observatory. 127 Given the redshift and the distance of the source, we computed the emitted isotropic energy in the 8 - 10000 keV energy range, with the Schaefer formula: 128 using the fluence in the (8-1000 keV) as observed by Fermi-GBM, S obs = 2.7 × 10 -4 , 126 and the Λ Cold Dark Matter (CDM) cosmological standard model H 0 = 70 km/s/Mpc, Ω m = 0.27, Ω Λ = 0.73, we obtain for the emitted isotropic energy the value of E iso = 2.90 × 10 53 erg. This GRB was observed also by Konus-WIND, 129 Suzaku-WAM, 108 and by the AGILE satellite, 130 which detected emission in the (18-60) keV and in the MCAL instrument, operating at energies greater than 350 keV, but it did not observe highenergy photons above 30 MeV. GRB 090618 was the first GRB observed by the Indian payloads RT-2 on board the Russian satellite CORONAS-PHOTON. 131-133 Thanks to the complete data coverage of the optical afterglow of GRB 090618, the presence of a supernova underlying the emission of its optical afterglow was reported. 134 The evidence of a supernova emission came from the presence of several bumps in the light curve and by the change in R c -i color index over time: in the early phases, the blue color is dominant, typical of the GRB afterglow, but then the color index increases, suggesting a core-collapse SN. At late times, the contribution from the host galaxy was dominant.", "pages": [ 32, 33 ] }, { "title": "3.2.1. Data analysis", "content": "We have analyzed GRB 090618, considering the BAT and XRT data of the Swift satellite together with the Fermi-GBM and RT2 data of the Coronas-PHOTON satellite (see Fig. 28). The data reduction was made with the Heasoft v6.10 packages b for BAT and XRT, and the Fermi-Science tools for GBM. The details of the data reduction and analysis are given in Ref. 17. In Table 4 we give the results of our spectral analysis. The time reported in the first column corresponds to the time after the GBM trigger time t trig = 267006508 s, where the β parameter was not constrained, we used its averaged value, β = -2.3 ± 0.10, as delineated in Ref. 135. We considered the chi-square statistic for testing our data fitting procedure. The reduced chi-square ˜ χ 2 = χ 2 /N , where N is the number of degrees of freedom (dof), which is N = 82 for the NaI dataset and N = 121 for that of the BGO.", "pages": [ 33 ] }, { "title": "34 Remo Ruffini", "content": "For the last pulse of the second episode, the band model is not very precise (˜ χ 2 = 2.24), but a slightly better approximation is given by a power-law with an exponential cut-off, whose fit results are shown for the same intervals in the last two columns. From these values, we built the flux light curves for both detectors, which are shown in Fig. 3.2.1.", "pages": [ 34 ] }, { "title": "3.2.2. Spectral analysis of GRB 090618", "content": "We proceed now to the detailed spectral analysis of GRB 090618. We divide the emission into six time intervals, shown in Table 4, each one identifying a significant feature in the emission process. We then fit for each time interval the spectra by a band model and a blackbody with an extra power-law component, following Ref. 136. In particular, we are interested in estimating the temperature kT and the observed energy flux φ obs of the blackbody component. The specific intensity of emission of a thermal spectrum at energy E in energy range dE into solid angle ∆Ω is The source of radius R is seen within a solid angle ∆Ω = πR 2 /D 2 , and its full luminosity is L = 4 πR 2 σT 4 . What we are fitting, however, is the backgroundsubtracted photon spectra A ( E ), which is obtained by dividing the specific intensity I ( E ) by the energy E : where h , k and σ are the Planck, Boltzmann, and Stefan-Boltzmann constants respectively, c is the speed of light and φ obs = L/ (4 πD 2 ) is the observed energy flux of the blackbody emitter. The great advantage of Eq. (7) is that it is written in terms of the observables φ obs and T , so from a spectral fitting procedure we can obtain the values of these quantities for each time interval considered. To determine these parameters, we must perform an integration of the actual photon spectrum A ( E ) over the instrumental response R ( i, E ) of the detector that observes the source, where i denotes the different instrument energy channels. The result is a predicted count spectrum where E min ( i ) and E max ( i ) are the boundaries of the i -th energy channel of the instrument. Eq. (8) must be compared with the observed data by a fit statistic. The main parameters obtained from the fitting procedure are shown in Table 5. We divide the entire GRB into two main episodes, as proposed in Ref. 120: one lasting the first 50 s and the other from 50 to 151 s after the GRB trigger time, see Fig. 30. Clearly, the first 50 s of emission, corresponding to the first episode, are well-fit by a band model as well as a blackbody with an extra power-law model, Fig. 31. The same happens for the first 9 s of the second episode (from 50 to 59 s after the trigger time), Fig. 32. For the subsequent three intervals corresponding to the main peaks in the light curve, the blackbody plus a power-law model does not provide a satisfactory fit. Only the band model fits the spectrum with good accuracy, with the exception of the first main spike (compare the values of χ 2 in the table). We find also that the last peak can be fit by a simple power-law model with a photon index γ = 2.20 ± 0.03, better than by a band model. The result of this analysis points to a different emission mechanism in the first 50 s of GRB 090618 and in the next 9 s. A sequence of very strong pulses follows, whose spectral energy distribution is not attributable either to a blackbody or a blackbody and an extra power-law component. Good evidence for the transition is shown by the test of the data fitting, whose indicator is given by the changing of ˜ χ 2 ( N dof = 169)", "pages": [ 34, 35 ] }, { "title": "36 Remo Ruffini", "content": "for the blackbody plus a power-law model for the different time intervals, see Table 5. Although the band spectral model is an empirical model without a clear physical origin, we checked its validity in all time-detailed spectra with the sole exception of the first main pulse of the second episode. The χ 2 corresponding to the band model for this main pulse, although better than that corresponding to the blackbody and power-law case, is unsatisfactory. We now directly apply the fireshell model to make the above conclusions more stringent and reach a better understanding of the source.", "pages": [ 36 ] }, { "title": "3.3. Analysis of GRB 090618 in the fireshell scenario: from a single GRB to a multi-component GRB", "content": "3.3.1. Attempt for a single GRB scenario: the role of the first episode We first approach the analysis of GRB 090618 by assuming that we observe a single GRB and attempt identification of the P-GRB emission of a canonical GRB within the fireshell scenario (see panel A in Fig. 32 and Table 5). This has been shown to be inconsistent (see details in Ref. 17). We then turn to a multicomponent emission.", "pages": [ 36 ] }, { "title": "3.3.2. The multi-component scenario: the second episode as an independent GRB", "content": "The identification of the P-GRB of the second episode. We now proceed to the analysis of the data between 50 and 150 s after the trigger time as a canonical GRB in the fireshell scenario, namely the second episode, 120 see Fig. 30. We proceed to identify the P-GRB within the emission between 50 and 59 s, since we find a blackbody signature in this early second-episode emission. Considerations based on the time variability of the thermal component bring us to conclude that the first 4 s of this time interval to due to the P-GRB emission. The corresponding spectrum (8-440 keV) is well fit (˜ χ 2 = 1 . 15) with a blackbody of a temperature kT = 29 . 22 ± 2 . 21 keV (norm = 3.51 ± 0.49), and an extra power-law component with photon index γ = 1.85 ± 0.06, (norm = 46.25 ± 10.21), see Fig. 33. The fit with the band model is also acceptable (˜ χ 2 = 1 . 25), which gives a low-energy power-law index α = -1 . 22 ± 0 . 08, a high-energy index β = -2 . 32 ± 0 . 21 and a break energy E 0 = 193 . 2 ± 50 . 8, see Fig. 33. In view of the theoretical understanding of the thermal component in the P-GRB (see Section 3.2), we focus below on the blackbody + power-law spectral model. The isotropic energy of the second episode is E iso = (2.49 ± 0.02) × 10 53 ergs. The simulation within the fireshell scenario is made assuming E e + e -tot ≡ E iso . From the observed temperature, we can then derive the corresponding value of the baryon load. The observed temperature of the blackbody component is kT = 29 . 22 ± 2 . 21, so that we can determine a value of the baryon load of B = 1 . 98 ± 0 . 15 × 10 -3 , and deduce the energy of the P-GRB as a fraction of the total E e + e -tot . We therefore obtain a value of the P-GRB energy of 4.33 +0 . 25 -0 . 28 × 10 51 erg. Now we can derive the radius of the transparency condition, to occur at r tr = 1.46 × 10 14 cm. From the third panel we derive the bulk Lorentz factor of Γ th = 495. We compare this value with the energy measured only in the blackbody component of E BB = 9.24 +0 . 50 -0 . 58 × 10 50 erg, and with the energy in the blackbody plus the power-law component of E BB + po = 5.43 +0 . 07 -0 . 11 × 10 51 erg, and verify that the theoretical value is in between these observed energies. We have found this result to be quite satisfactory: it represents the first attempt to relate the GRB properties to the details of the BH responsible for the overall GRB energetics. The above theoretical estimates were based on a nonrotating BH of 10 M /circledot , a total energy of E e + e -tot = 2.49 × 10 53 erg and a mean temperature of the initial e + e -plasma of 2.4 MeV, derived from the expression for the dyadosphere radius, Eq. 1. Any refinement of the direct comparison between theory and observations will have to address a variety of fundamental problems such as 1) the possible effect of rotation of the BH, leading to a more complex dyadotorus structure, 2) a more detailed analysis of the transparency condition of the e + e -plasma, simply derived from the condition τ = ∫ R dr ( n e ± + n b e -) σ T = 0 . 67, 69 and 3) an analysis of the general relativistic, electrodynamical, strong interaction descriptions of the gravitational core collapse leading to BH formation. 21, 69, 137 The analysis of the extended afterglow of the second episode. The extended afterglow starts at the above given radius of the transparency, with an initial value of the Lorentz Γ factor of Γ 0 = 495. To simulate the extended-afterglow emission, we need to determine the radial distribution of the CBM around the burst site, which we assume for simplicity to be spherically symmetric, from which we infer a characteristic size of ∆ R = 10 15 --16 cm. We already described above how the simulation of the spectra and of the observed multi-band light curves have to be performed together and need to be jointly optimized, leading to the determination of the fundamental parameters characterizing the CBM medium. 138 This radial distribution is shown in Fig. 35 and is characterized by a mean value of 〈 n 〉 = 0.6 part/cm 3 and an average density contrast with a 〈 δn/n 〉 ≈ 2, see Fig. 35 and Table 7. The data up to 8.5 × 10 16 cm are simulated with a value for the filling factor R = 3 × 10 -9 , while the data from this value on with R = 9 × 10 -9 . From the radial distribution of the CBM density, and considering the 1 / Γ effect on the fireshell visible area, we found that the CBM clumps causing the spikes in the extended-afterglow emission have masses on the order of 10 22 --24 g. The value of the α parameter was found to be -1 . 8 along the total duration of the GRB. In Fig. 36 we show the simulated light curve (8-1000 keV) of the GRB and the corresponding spectrum, using the spectral model described in Refs. 71, 84. We focus our attention on the structure of the first spikes. The comparison between the spectra of the first main spike (t 0 +59, t 0 +66 s) of the extended afterglow of GRB 090618 obtained with three different assumptions is shown in Fig. 37: in", "pages": [ 37, 38, 39 ] }, { "title": "40 Remo Ruffini", "content": "the upper panel we show the fireshell simulation of the integrated spectrum (t0+59, t0+66 s) of the first main spike, in the middle panel we show the best fit with a blackbody and a power-law component model and in the lower panel the best fit using a simple power-law spectral model. We can see that the fit with the last two models is not satisfactory: the corresponding ˜ χ 2 is 7 for the blackbody + power-law and ∼ 15 for the simple power-law. We cannot give the ˜ χ 2 of the fireshell simulation, since it is not represented by an explicit analytic fitting function, but it originates in a sequence of complicated highly nonlinear procedures. It is clear from a direct scrutiny that it correctly reproduces the low-energy emission, thanks in particular to the role of the α parameter, which was described previously. At higher energies, the theoretically predicted spectrum is affected by the cut-off induced by the thermal spectrum. The temporal variability of the first two spikes is well simulated. We are not able to accurately reproduce the last spikes of the light curve, since the equations of motion of the accelerated baryons become very complicated after the first interactions of the fireshell with the CBM. 138 This happens for various reasons. First, a possible fragmentation of the fireshell can occur. 138 Moreover, at larger distances from the progenitor, the fireshell visible area becomes larger than the transverse dimension of a typical blob of matter, consequently a modification of the code for a three-dimensional description of the interstellar medium will be needed. This is unlike the early phases in the prompt emission, which is the main topic we address at the moment, where a spherically symmetric approximation applies. The fireshell visible area is smaller than the typical size of the CBM clouds in the early phases of the prompt radiation. 139 The second episode, lasting from 50 to 151 s, agrees with a canonical GRB in the fireshell scenario. Particularly relevant is the problematic presented by the PGRB. It interfaces with the fundamental physics problems, related to the physics of the gravitational collapse and the BH formation. There is an interface between reaching transparency of the P-GRB and the early part of the extended afterglow. This connection has already been introduced in the literature, see e.g. Ref. 113. We studied this interface in the fireshell by analyzing the thermal emission at the transparency with the early interaction of the baryons with the CBM matter, see Fig. 34. We now aim to reach a better understanding of the meaning of the first episode, between 0 and 50 s of the GRB emission. To this end we examine the two episodes with respect to 1) the Amati relation, 2) the hardness variation, and 3) the observed time lag.", "pages": [ 40 ] }, { "title": "3.3.3. A different emission process in the first episode", "content": "The time-resolved spectra and temperature variation. One of the most significant outcomes of the multi-year work of Felix Ryde and his collaborators Ref. 140 has been the identification and the detailed analysis of the thermal plus power-law features observed in time-limited intervals in selected BATSE GRBs. Similar features have also been observed in the data acquired by the Fermi satellite. 140,141 We propose to divide these observations into two broad families. The first family presents a thermal plus power-law(s) feature, with a temperature changing in time following a precise power-law behavior. The second family is also characterized by a thermal plus power-law component, but with the blackbody emission generally varying without a specific power-law behavior and on shorter time scales. It is our goal to study these features within the fireshell scenario to possibly identify the underlying physical processes. We have already shown in Sec. 2.2.3 that the emission of the thermal plus power-law component characterizes the P-GRB emission. We have also emphasized that the P-GRB emission is the most relativistic regime occurring in GRBs, uniquely linked to the process of BH formation, see Sec. 2.2.3. This process appears to belong to the second family considered above. Our aim here is to see if the first episode of GRB 090618 can lead to the identification of the first family of events: those whose temperature changes with time following a power-law behavior on time scales from 1 to 50 s. We have already pointed out in the previous section that the hardness-ratio evolution and the long time lag observed for the first episode 133 points to a distinct origin for the first 50 s of emission, corresponding to the first episode. We made a detailed time-resolved analysis of the first episode, considering different time bin durations to obtain good statistics in the spectra and to take into account the sub-structures in the light curve. We then used two different spectral models to fit the observed data, a classical band spectrum, 111 and a blackbody with a power-law component. To obtain more accurate constraints on the spectral parameters, we made a joint fit considering the observations from both the n4 NaI and the b0 BGO detectors, covering a wider energy range in this way, from 8 keV to 40 MeV. To avoid some bias from low-photon statistics, we considered an energy upper limit of the value of 10 MeV. In the last three columns of Table 8 we report the spectral analysis performed in the energy range of the BATSE LAD instrument (20-1900 keV), as analyzed in Ref. 67 as a comparison tool with the results described in that paper. Our analysis is summarized in Figs. 38 and 39, and in Table 8, where we report the residual ratio diagram and the reducedχ 2 values for the spectral models. We conclude that both the band and the proposed BB+PL spectral models fit the observed data very well. Particularly interesting is the clear evolution in the time-resolved spectra, which corresponds to the blackbody and power-law component, see Fig. 38. In particular the kT parameter of the blackbody shows a strong decay, with a temporal behavior well-described by a double broken power-law function, see the upper panel in Fig. 39. From a fitting procedure we find that the best fit (R 2 -statistic = 0.992) for the two decay indexes for the temperature variation are a kT = -0.33 ± 0.07 and b kT = -0.57 ± 0.11. In Ref. 67 an average value for these parameters on a set of 49 GRBs is given: 〈 a kT 〉 = -0.07 ± 0.19 and 〈 b kT 〉 = -0.68 ± 0.24. The results presented in Figs. 38 and 39, and Table 8 point to a rapid cooling of the thermal emission with time of the first episode. The evolution of the corresponding power-law spectral component also appears to be strictly related to the change of the temperature kT . The power-law γ index falls, or softens, with temperature, see Fig. 38. An interesting feature appears to occur at the transition of the two power-laws describing the observed decrease of the temperature. The long time lag observed in the first episode has a clear explanation in the power-law behavior of the temperature and corresponding evolution of the photon index γ (see Figs. 38 and 39). The radius of the emitting region. We turn now to estimate an additional crucial parameter for identifying the nature of the blackbody component: the radius of the emitter r em . We proved that the first episode is not an independent GRB and not part of a GRB. We can therefore provide the estimate of the emitter radius from nonrelativistic considerations, just corrected for the cosmological redshift z . In fact we find that the temperature of the emitter T em = T obs (1 + z ), and that the luminosity of the emitter, due to the blackbody emission, is where r em is the emitter radius and σ is the Stefan-Boltzmann constant. From the luminosity distance definition, we also have that the observed flux φ obs is given by We then obtain The above radius differs from the radius r ph given in Eq. (1) of Ref. 67, which was also clearly obtained by interpreting the early evolution of GRB 970828 as belonging to the photospheric emission of a GRB and assuming a relativistic expansion with a Lorentz gamma factor Γ where ˆ R = ( φ obs / ( σT 4 ob ) ) 1 / 2 and the prefactor 1.06 arises from the dependence of r ph on the angle of the line of sight. 142 Typical values of r ph are at least two orders of magnitude higher than our radius r em . Assuming a standard cosmological model ( H 0 = 70 km/s/Mpc, Ω m = 0 . 27 and Ω Λ = 0 . 73) for estimating the luminosity distance D , and using the values for the observed flux φ obs and the temperature kT obs , we give in Fig. 40 the evolution of the surface radius that emits the blackbody r em as a function of time. Assuming an exponential evolution with time t δ of the radius in the comoving frame, we obtain the value δ = 0 . 59 ± 0 . 11 from a fitting procedure, which is well compatible with δ = 0 . 5. We also notice a steeper behavior for the variation of the radius with time corresponding to the first 10 s, which corresponds to the emission before the break of the double power-law behavior of the temperature. We estimate an average velocity of ¯ v = 4067 ± 918 km/s, R 2 = 0.91 in these first 10 s of emission. In episode 1 the observations lead to a core of an initial radius of ∼ 12000 km expanding in the early phase with a higher initial velocity of ∼ 4000 km/s. The effective Lorentz Γ factor is very low, Γ -1 ∼ 10 -5 . I propose to identify this first episode as the early phases of the SN explosion in the IGC scenario which I discuss in the next paragraph.", "pages": [ 40, 41, 42, 44, 45 ] }, { "title": "4.1. Induced gravitational collapse of a NS to a BH by a type Ib/c SN", "content": "The systematic and spectroscopic analysis of GRB-SN events, following the pioneering discovery of the temporal coincidence of GRB 980425 2 and SN 1998bw, 3 has revealed evidence for the association of other nearby GRBs with Type Ib/c SNe (see Ref. 143 for a recent review of all the GRB-SN systems). It has also been clearly understood that SN Ib/c lack Hydrogen (H) and Helium (He) in their spectra, and the most likely explanation is that the SN progenitor star is in a binary system with a compact companion, a neutron star (see e.g. Refs. 144, 145, 146, for details). In the current literature there has been an attempt to explain both the SN and the GRB as two aspects of the same astrophysical phenomenon. Hence, GRBs have been assumed to originate from a specially violent SN process, a hypernova or a collapsar (see e.g. Ref. 147 and references therein). Both of these possibilities imply a very dense and strong wind-like CBM structure. Such a dense medium appears to be in contrast with the CBM density found in most GRBs (see e.g. Fig. 10 in Ref. 18). In fact, the average CBM density, inferred from the analysis of the afterglow, has been shown to be in most of the cases of the order of 1 particle cm -3 (see e.g. Ref. 39). The only significant contribution to the baryonic matter component in the GRB process is the one represented by the baryon load. 14 In a GRB, the electronpositron plasma, loaded with a certain amount of baryonic matter, is expected to expand at ultra-relativistic velocities with Lorentz factors Γ /greaterorsimilar 100. 73, 148, 149 Such an ultra-relativistic expansion can actually occur if the amount of baryonic matter, quantifiable through the baryon load parameter, does not exceed the critical value B ∼ 10 -2 (see Ref. 14, for details). In our approach we have consistently assumed that the GRB has to originate from the gravitational collapse to a BH. The SN follows instead the complicated pattern of the final evolution of a massive star, possibly leading to a NS or to a complete explosion but never to a BH. There is a further general argument in favor of our explanation, namely the extremely different energetics of SNe and GRBs. While the SN energy range is 10 49 -10 51 erg, the GRBs are in a larger and wider range of energies 10 49 -10 54 erg. It is clear that in no way a GRB, being energetically dominant, can originate from the SN. We explain the temporal coincidence of the two phenomena, the SN explosion and the GRB, within the concept of induced gravitational collapse . 115,150 In recent years we have outlined two different possible scenarios for the GRB-SN connection. In the first version, 115 we have considered the possibility that GRBs may have caused the trigger of the SN event. For this scenario to occur, the companion star has to be in a very special phase of its thermonuclear evolution (see Ref. 115 for details). More recently, I have proposed in Ref. 150 a different possibility occurring at the final stages of the evolution of a close binary system: the explosion in such a system of a Ib/c SN leads to an accretion process onto the NS companion. The NS will reach the critical mass value, undergoing gravitational collapse to a BH. The process of gravitational collapse to a BH leads to the emission of the GRB (see Figs. 41 and 42). Here we evaluate the accretion rate onto the NS and give the explicit expression of the accreted mass as a function of the nature of the components and the binary parameters following Ref. 151. We turn now to the details of the accretion process of the SN material onto the NS. In a spherically symmetric accretion process, the magnetospheric radius is 152 where B , M NS , R are the NS magnetic field, mass, radius, and ˙ M ≡ dM/dt is the mass-accretion rate onto the NS. We now estimate the relative importance of the NS magnetic field for the accretion process. At the beginning of a SN explosion, the ejecta moves at high velocities v ∼ 10 9 cm s -1 and the NS will capture matter at a radius approximately given by R sph cap ∼ 2 GM/v 2 . For R m << R sph cap , we can neglect the effects of the magnetic field. It is already clear from Eq. (13) that a high accretion rate might reduce the magnetospheric radius drastically. In Fig. 43 we plot the ratio between the magnetospheric radius and the gravitational capture radius as a function of the mass accretion rate onto a NS of B = 10 12 Gauss, M NS = 1 . 4 M /circledot , R = 10 6 cm, and for a flow with velocity v = 10 9 cm s -1 . It can be seen that for high accretion rates the influence of the magnetosphere will be negligible. We therefore assume for simplicity hereafter that the NS is nonrotating and neglect the effects of the magnetosphere. The NS captures the material ejected from the core collapse of the companion star in a region delimited by the radius R cap from the NS center where M NS is the initial NS mass and v rel , ej is the velocity of the ejecta relative to the orbital motion of the NS around the supernova progenitor star with v ej the ejecta velocity in the frame of the supernova progenitor star with mass M SN -prog and v orb is the orbital velocity of the NS, given by where a is the binary separation, and thus the orbital period of the binary system is The NS accretes the material that enters into its capture region defined by Eq. (14). The mass-accretion rate is given by 153 where the parameter ξ is lies in the range 1 / 2 ≤ ξ ≤ 1, ρ ej is the density of the accreted material, and in the last equality we have used Eqs. (14) and (15). The upper value ξ = 1 corresponds to the Hoyle-Lyttleton accretion rate. 154 The actual value of ξ depends on the properties of the medium in which the accretion process occurs, e.g. vacuum or wind. The velocity of the SN ejecta v ej will be much larger than the sound speed c s of the already existing material between the C+O star and the NS due to the prior mass transfer, namely the Mach number of the SN ejecta will certainly satisfy M = v ej /c s >> 1. Thus in practical calculations we can assume the value ξ = 1 in Eq. (18) and the relative velocity v rel , ej of the SN ejecta with respect to the NS companion is given only by the NS orbital velocity and the ejecta velocity as given by Eq. (15). In Fig. 42 we have sketched the accreting process of the supernova ejected material onto the NS. The density of the ejected material can be assumed to decrease in time following the simple power-law 155 where without loss of generality we have assumed that the radius of the SN ejecta expands as r ej = σt n , with σ and n constants. Therefore the velocity of the ejecta obeys v ej = nr ej /t . One can integrate Eq. (18) to obtain the accreted mass in a given time interval where with k = v 2 orb / ( nσ ) 2 and 2 F 1 ( a, b ; c ; z ) is the hypergeometric function. The integration constant is computed with the condition ∆ M ( t ) = 0 for t ≤ t acc 0 , where t acc 0 is the time at which the accretion process starts, namely the time at which the SN ejecta reaches the NS capture region (see Fig. 42). We discuss now the problem of the maximum stable mass of a NS. Nonrotating NS equilibrium configurations have been recently constructed taking into proper account the strong, weak, electromagnetic, and gravitational interactions within general relativity. The equilibrium equations are given by the general relativistic Thomas-Fermi equations coupled with the Einstein-Maxwell equations to form the Einstein-Maxwell-Thomas-Fermi system of equations, which must be solved under the condition of global charge neutrality. 26 These equations supersede the traditional Tolman-Oppenheimer-Volkoff ones that impose the condition of local charge neutrality throughout the configuration. The maximum stable mass M crit = 2 . 67 M /circledot of nonrotating NSs has been obtained in Ref. 26. The high and rapid accretion rate of the SN material can lead the NS mass to reach the critical value M crit = 2 . 67 M /circledot . This system will undergo gravitational collapse to a BH, producing a GRB. The initial NS mass is likely to be rather high due to the highly nonconservative mass transfer during the previous history of the evolution of the binary system (see e.g. Refs. 144, 145, 146, for details). Thus the NS could reach the critical mass in just a few seconds. Indeed we can see from Eq. (18) that for an ejecta density 10 6 g cm -3 and velocity 10 9 cm s -1 , the accretion rate might be as large as ˙ M ∼ 0 . 1 M /circledot s -1 . The occurrence of a GRB-SN event in the scenario depends on some specific conditions satisfied by the binary progenitor system, such as a short binary separation and an orbital period < 1 h. This is indeed the case with GRB 090618 and 110709B that we have already analyzed within the context of this scenario in Refs. 18, 156, respectively (see below in the next subsections). In addition to offering an explanation for the GRB-SN temporal coincidence, the considerations presented here lead to an astrophysical implementation of the concept of proto-BH, generically introduced in our previous works on GRBs 090618, 970828, and 101023 (see Refs. 18, 157, 114). The proto-BH represents the first stage 20 /lessorsimilar t /lessorsimilar 200 s of the SN evolution. It is appropriate now to discuss the possible progenitors of such binary systems. A viable progenitor is represented by X-ray binaries such as Cen X-3 and Her X1. 1, 158-163 The binary system is expected to follow an evolutionary track: 144-146 the initial binary system is composed of main-sequence stars 1 and 2 with a mass ratio M 2 /M 1 /greaterorsimilar 0 . 4. The initial mass of the star 1 is likely M 1 /greaterorsimilar 11 M /circledot , leaving a NS through a core-collapse event. The star 2, now with M 2 /greaterorsimilar 11 M /circledot after some almost conservative mass transfer, evolves filling its Roche lobe. It then starts a spiralling in of the NS into the envelope of the star 2. If the binary system does not merge, it will be composed of a helium star and a NS in close orbit. The helium star expands filling its Roche lobe and a nonconservative mass transfer to the NS takes place. This scenario naturally leads to a binary system composed of a C+O star and a massive NS, as the one considered here. We point out that the systems showing a temporal GRB-SN coincidence form a special class of GRBs: (2) There are GRBs that do not show the presence of an associated SN. This is certainly the case of GRBs at large cosmological distances z /greaterorsimilar 0 . 6 when the SN is not detectable even by the current high power optical telescopes. This is likely the case of GRB 101023. 114 (3) There is the most interesting case of GRBs that do not show a SN, although it would be detectable. This is the case of GRB 060614 90 in which a possible progenitor has been indicated in a binary system formed of a white dwarf and a NS, which clearly departs from the considered binary class. Finally there are systems giving rise to genuinely short GRBs which have been proved to have their progenitors in binary NSs, and clearly do not have an associated SN, e.g. GRB 090227B. 16, 34 It is clear that after the occurrence of the SN and the GRB emission, the outcome is represented, respectively, by a NS and a BH. A possible strong evidence of the NS formation is represented by the observation of a characteristic late ( t = 10 8 -10 9 s) X-ray emission (called URCA sources, see Ref. 166) that has been interpreted as originating from the young ( t ∼ 1 minute-(10-100) years), hot ( T ∼ 10 7 -10 8 K) NS, which we have called neo-NS (see Ref. 167, for details). This has been indeed observed in GRB 090618 17 and also in GRB 101023. 114 If the NS and the BH are gravitationally bound they give rise to a new kind of binary system, which can lead itself to the merging of the NS and the BH and consequently to a new process of gravitational collapse of the NS into the BH. In this case the system could originate yet another process of GRB emission and possibly a predominant emission in gravitational waves.", "pages": [ 46, 47, 48, 49, 50, 51, 52 ] }, { "title": "4.2. The application to GRB 090618", "content": "We apply the previous considerations of Ref. 151 to the specific case of GRB 090618 and its associated SN (see Ref. 18, for details). We have shown that GRB 090618 18 is composed of two sharply different emission episodes. A time-resolved spectral analysis showed that the first episode, which lasts ∼ 32 s in the rest frame, is characterized by a black-body emission that evolves due to a temperature decreasing with time (see Fig. 17 in Ref. 17). Associated to the decreasing black-body temperature, the radius of the emitter has been found to increase with time (see Fig. 18 in Ref. 17). From the evolution of the radius of the black-body emitter, we find that it expands at nonrelativistic velocities (see Eq. (22), below). Consequently, the first episode cannot be associated to a GRB. Because it happens prior to the GRB and therefore to the BH formation, this first episode emission has been temporally called a proto-BH, from the ancient Greek πρ ˜ ωτoς , meaning before in space and time. We here identify the proto-BH of the first episode as the first stages of the SN expansion. The black-body-emitting surface in the first episode evolves during the first ∼ 32 s, as observed in the rest frame, following a power-law behavior where σ = 8 . 048 × 10 8 cm s -n , n ≈ 3 / 5 as shown in Fig. 40, and v SN = dr SN /dt is the corresponding early SN velocity of the SN, so ∼ 4 × 10 8 cm s -1 at the beginning of the expansion. When the mass accreted onto the NS triggers the gravitational collapse of the NS into a BH, the authentic GRB emission is observed in the subsequent episode at t -t 0 /greaterorsimilar 50 s (observer frame). The characteristics of GRB 090618 are shown in Table 3 of Ref. 17 and we refer to that reference for more details on the GRB light curve and spectrum simulation. We now turn to the details of the accretion process of the SN material onto the NS. The NS of initial mass M NS accretes mass from the SN ejecta at a rate given by 151 where r 3 SN ( t ) given by Eq. (22) and M ej ( t ) = M ej , 0 -M acc ( t ) is the available mass to be accreted by the NS as a function of time, with M ej , 0 the mass ejected in the SN. v rel , ej = √ v 2 orb + v 2 SN is the velocity of the ejecta relative to the NS, where v SN is the SN ejecta velocity given by Eq. (22) and v orb = √ G ( M core + M NS ) /a is the orbital velocity of the NS. Here M core is the mass of the SN core progenitor and a the binary separation. Hereafter we assume a = 9 × 10 9 cm, a value higher than the maximum distance traveled by the SN material during the total time interval of Episode 1, ∆ t /similarequal 32 s, ∆ r ∼ 7 × 10 9 cm (see Fig. 40). If the accreted mass onto the NS is much smaller than the initial mass of the ejecta, i.e., M acc /M ej , 0 << 1, the total accreted mass can be obtained from the formula given by Eq. (8) of Ref. 151, which for GRB 090618 leads to ∣ where k = v 2 orb / ( nσ ) 2 and t acc 0 is the time at which the accretion process starts, namely the time at which the SN ejecta reaches the NS capture region, R cap = 2 GM NS /v 2 rel , ej , so for t ≤ t acc 0 we have M acc ( t ) = 0. The accretion process leads to the gravitational collapse of the NS onto a BH when it reaches the critical mass value. Here we adopt the critical mass M crit = 2 . 67 M /circledot computed recently in Ref. 26. Eq. (24) is more accurate for massive NSs since the amount of mass needed to reach the critical mass by accretion is much smaller than M ej , 0 . In general, the total accreted mass must be computed from the numerical integration of Eq. (23), which we present below for GRB 090618. The occurrence of a GRB-SN event in the accretion induced collapse scenario is subject to some specific conditions of the binary progenitor system such as a short binary separation and orbital period. The orbital period in the present case is We denote by ∆ t acc the total time interval since the beginning of the SN ejecta expansion all the way up to the instant where the NS reaches the critical mass. In Fig. 45 we plot ∆ t acc as a function of the initial NS mass and for different masses of the SN core progenitor mass. The mass of the SN ejecta is assumed to be M ej , 0 = M core -M rem , where M rem is the mass of the central compact remnant (NS) left by the SN explosion. Here we assumed M core = (3-8) M /circledot at the epoch of the SN explosion, and M rem = 1 . 3 M /circledot , following some of the type Ic SN progenitors studied in Refs. 144, 145, 146. We can see from Fig. 45 that, for GRB 090618, the mass of the NS companion that collapses onto a BH should be in the range 1 . 8 /lessorsimilar M NS /M /circledot /lessorsimilar 2 . 1 corresponding to the SN Ic progenitors 3 ≤ M core /M /circledot ≤ 8. The massive NS companion of the evolved star is in line with the binary scenario proposed in Ref. 150. These results also agree with the well-understood Ib/c nature of the SN associated with GRBs. The most likely explanation for SN Ib/c, which lack H and He in their spectra, is that the SN progenitor star is in a binary system with an NS; see also Refs. 144, 145, 146 and also 168, 169. It is also interesting to compare the results on the IGC of an NS to a BH by a type Ib/c SN 151 with the results of Chevalier 155 on the accretion of a supernova material by the central NS generated by the supernova. A total accreted mass of up to 0 . 1 M /circledot in a time of a few hours was obtained there for a normal type II SN. Thus a similar amount of mass can be accreted in the two cases, but in the latter the accretion occurs over a longer time. To reach a high accretion rate of the inner SN material onto the central NS, a mechanism is needed that helps to increase the density of the NS surrounding layers, which is decreasing due to the expansion after being unbound by the SN explosion. Ref. 155 analyzed the possibility of having a reverse shock wave as this mechanism while it moves back through the SN core. The reverse shock is formed in the interaction of the mantle gas with the low-density envelope. The time scale of the accretion process is thus determined by the time it takes the reverse shock to reach the vicinity of the central newly born NS, which is a few hours in the case of SN II progenitors. However, the existence of a low-density outer envelope, e.g. H and He outer layers, is essential for the strength of the reverse shock. Fall-back accretion onto the central NS is expected to be relevant only in SN II but not in SN Ic like those associated to GRBs, where H and He are absent. The argument presented in 151 naturally explains the sequence of events: SN explosion - IGC-BH formation - GRB emission. Correspondingly, the accretion of the material ejected by the SN into the nearby NS of the IGC model presented here occurs almost instantaneously. Indeed for the SN expansion parameters obtained from the observations of episode 1 in GRB 090618 (see Eq. (22), the accretion of the SN material onto the nearby NS occurs in a few seconds (see Figs. 44 and 45). The binary parameters are such that the ejecta density does not decrease too much (from 10 6 to ∼ 10 4 g cm -3 ) before reaching the capture region of the NS, leading to a high accretion rate. As pointed out in Ref. 155, radiative diffusion will lower the accretion rate up to the Eddington limit (and then to even lower rates) when the trapping radius of the radiation in the flow r tr = κ ˙ M acc / (4 πc ), 155 where κ is the opacity, is equal to the Bondi radius r B = GM NS /v 2 rel , ej , the gravitational capture radius. The radius r tr is located where the outward diffusion luminosity is equal to the inward convective luminosity. It can be checked that for the parameters of our system given by Eqs. (22)-(24), the equality r tr = r B occurs in a characteristic time ∼ 200 days, where we used κ = 0 . 2 cm 2 g -1 . Thus, this regime is not reached in the present case since the NS is brought to its critical mass just in a few seconds. In the case analyzed by Ref. 155, it happens in a time ∼ 8 days. In conclusion, the IGC binary scenario applied here to the specific case of GRB 090618 naturally leads to understanding the energetics and the temporal coincidence of SN and GRBs, as well as their astrophysical scenario and their origins. It also provides new predictions of the final outcome, originating from a binary system composed of an evolved core and an NS. It is clear, however, that these GRBs and their associated SNe form a special class of long GRBs and of SNe Ib/c. There are in fact SNe Ib/c that are not associated to a GRB, e.g. SN 1994I 164 and SN 2002ap. 165 Their observations refer to late phases of the SN evolution typically ∼ 15-20 days after the original collapse process. The existing descriptions of these late phases after 15-20 days from the original explosion make use of a Sedov-type behavior r ∝ t 2 / 5 , see Refs. 170, 171. In the present case of the IGC we present here for the first time, the first ∼ 30 s of the very early evolution of an SN Ib/c associated to a GRB (see Eq. (22). The energetic of this SN Ib/c, as shown from episode 1, appears to be much higher than the ones of the usual SNe Ib/c not associated to GRBs, E iso,Epi 1 ∝ 10 52 erg. 17 The reason for this marked difference is certainly due to the accretion process during an SN explosion into the companion NS and consequent gravitational collapse of the NS onto a BH. The description of this challenging process, although clear from a general energetic point of view, has still to be explored in detail theoretically and certainly does not show any relation to the Sedov-type solution.", "pages": [ 52, 53, 54, 55, 56 ] }, { "title": "5. On a Possible Distance Indicator from GRB-SN-IGC", "content": "It is appropriate to remember an important selection effect occurring in the study of the IGC scenario. Only for systems with cosmological redshift z /lessorsimilar 1 does the current optical instrumentation allow the observation of the related SN Ib/c. A particularly challenging analysis is that of the system GRB 101023 114 in which the SN is not detectable but the IGC nature of the source is clearly recognized by the two different episodes in the GRB sources and the spectral features of the first episode. Following the case of GRB 101023, we have found and analyzed the X-ray emission of a sample of 8 GRBs having E iso ≥ 10 52 erg and satisfying at least one of the following three requirements: The characteristics of the 8 GRBs are the following: GRB 060729 . In this source a SN bump was observed in the optical GRB afterglow. 172 It is at the same redshift z = 0 . 54 of GRB 090618 and shows a small precursor plus a main event in the prompt light curve and a peculiar prolonged duration for the X-ray afterglow. 173 The isotropic energy emitted in this burst is E iso = 1 . 6 × 10 52 erg. GRB 061007 . This GRB has no associated SN but is characterized by the presence of an almost long precursor where a clear evolving thermal emission was reported. 174 With an energetic of E iso = 1 . 2 × 10 54 erg at z = 1 . 261, it is the farthest GRB in our sample. The large distance directly implies difficulties in the detection of a SN from this GRB. GRB 080319B . A debatable SN was reported also for GRB 080319B, well known as the naked-eye GRB, whose prompt emission shows also a possible double emission episode. 175 Its measured redshift is z = 0 . 937. This is one of the most energetic GRBs with E iso = 1 . 4 × 10 54 and its X-ray light curve is well described by a simple decaying power-law. GRB 090618 . This GRB is the prototype of the IGC GRB-SN subclass. Its prompt emission shows a clear episode 1 plus episode 2 structure in light curve and spectrum. The measured redshift is z = 0 . 54 and the isotropic energy emitted by the burst is E iso = 2 . 7 × 10 53 erg. There is a clear identification in the afterglow light curve of GRB 090618 of a late ∼ 10 day optical bump associated to the SN emission. 172 The characteristic parameters of this GRB, including baryon load ( B = 1 . 98 × 10 -3 ), the Lorentz gamma factor at trasparency (Γ tr = 495) and the nature of the CBM ( 〈 n CBM 〉 = 0 . 6 part/cm 3 ) have been estimated. 18 GRB 111228 . A SN feature is reported in the literature also for GRB 111228, 178 which shows a multiply peaked prompt light curve in the Fermi-GBM data. The measured redshift of this GRB is z = 0 . 713, its isotropic energy is E iso = 2 . 3 × 10 52 erg and a dedicated analysis of this GRB will be presented elsewhere. The detection of a SN in GRB 111228 is debatable, since the eventual optical bump has the same flux than the host galaxy of the source, but SN features were observed in the differential photometry between the last epochs of observations, where a transient component was detected unrelated to the afterglow and consequently associated to the SN. GRB 091127 . GRB 091127 is associated with SN 2009nz at a distance of z = 0 . 49. 176 The isotropic energy emitted in this burst is E iso = 1 . 4 × 10 52 erg. 177 GRB 101023 . This GRB shows clear episode 1 plus episode 2 emission in the prompt light curve and spectrum, but there is no detection of a SN and no measured redshift because of the lack of optical observations at late times. We have estimated the redshift of this source as z = 0 . 9 in analogy with the late X-ray afterglow decay observed in the 6 GRBs with a measured redshift. This leads to the estimation of an isotropic energy of E iso = 1 . 3 × 10 53 erg, a baryon load of B = 3 . 8 × 10 -3 , a Lorentz gamma factor at transparency of Γ tr = 260, and an average density for the CBM of ( 〈 n CBM 〉 ≈ 16 part/cm 3 . 114 GRB 110709B . Like GRB 101023, this GRB shows a clear episode 1 plus episode 2 emission in the prompt light curve and spectrum, but there is no detection of a SN. This can be explained by the fact that it is a dark GRB, so its emission is strongly influenced by absorption. Particularly interesting is the detection of a clear radio emission from GRB 110709B. 179 There is no measure for the redshift but, as for the case of GRB 101023, we have estimated it as z = 0 . 75 in analogy with the late X-ray afterglow decay observed in the 6 GRBs with measured redshifts. This leads to the estimation of an isotropic energy of E iso = 2 . 43 × 10 52 erg, a baryon load of B = 5 . 7 × 10 -3 , a Lorentz gamma factor at transparency of Γ tr = 174 and an average density of the CBM of 〈 n CBM 〉 ≈ 76 part/cm 3 . 156 We have focused our attention on the analysis of all the available XRT data of these sources. 19 Characteristically, XRT follow-up starts only about 100 seconds after the BAT trigger (typical repointing time of Swift after the BAT trigger). Since the behavior was similar in all the sources, we have performed an analysis to compare the XRT luminosity light curve L rf for the six GRBs with measured redshift z in the common rest frame energy range 0 . 3 - 10 keV. To perform this computation, the first step is to convert the observed XRT flux f obs to the one in the 0 . 3 - 10 keV rest frame energy range. In the detector frame, the 0 . 3 - 10 keV rest frame energy range becomes [0 . 3 / (1 + z )] - [10 / (1 + z )] keV where z is the redshift of the GRB. We assume a simple power-law function as the best-fit for the spectral energy distribution of the XRT data c : We can then write the flux light curve f rf in the 0 . 3 - 10 keV rest frame energy range as: Then, we have to multiply f rf by the luminosity distance to get L rf : where we assume a standard cosmological model ΛCDM with Ω m = 0 . 27 and Ω Λ = 0 . 73. Clearly, this luminosity must be plotted as a function of the rest frame time t rf , namely: The X-ray luminosity light curves of the six GRBs with measured redshift in the 0 . 3-10 keV rest frame energy band are plotted together in Fig. 46. What is most striking is that these six GRBs, with redshift in the range 0 . 49 - 1 . 261, show a remarkably common behavior of the late X-ray afterglow luminosity light curves (episode 3) despite that their prompt emissions (episode 1 and 2) are very different and that their energetics spans more than two orders of magnitude. Such a common behavior starts between 10 4 - 10 5 s after the trigger and continues up to when the emission falls below the XRT threshold. This standard behavior of episode 3 represents strong evidence of very low or even the absence of beaming in this particular phase of the X-ray afterglow emission process. We have proposed that this late time X-ray emission in episode 3 is related to the process of the SN explosion within the IGC scenario, possibly emitted by the newly born NS, and not by the GRB itself. 167 This scaling law, when confirmed in sources presenting the episode 1 plus the episode 2 emissions, offers a powerful tool to estimate the redshift of GRBs belonging to this subclass of events. As an example, we present in Fig. 47 the rest frame X-ray luminosity (0.3 10 keV) light curve of GRB 090618 (considered as a prototype for the common behavior shown in Fig. 46) with the rest frame X-ray luminosity light curves of GRB 110709B estimated for selected values of its redshifts, z = 0 . 4 , 0 . 6 , 0 . 8 , 1 . 0 , 1 . 2, and similarly the correspondent analysis for GRB 101023 for selected values of the redshift, z = 0 . 6 , 0 . 8 , 1 . 0 , 1 . 2 , 1 . 5. We then find that GRB 101023 should have been located at z ∼ 0 . 9 and GRB 110709B at z ∼ 0 . 75. These redshift estimations are within the range expected using the Amati relation as shown in Ref. 114, 156. This is an important independent confirmation of validity for this new redshift estimator we propose for the family of IGC GRB-SN systems. It should be stressed, however, that the determination of the redshift is done assuming the validity of the standard ΛCDM cosmological model for sources with redshift in the range z = 0 . 49 1 . 216. We are currently testing the validity of this assumption for sources at larger cosmological redshifts. Concerning the nature of the late X-ray emission discussed in 19, I am currently exploring the possibility that the emission process is linked to the decay of transuranic elements produced by the interaction of the GRB with the SNe through the r -process 180 and accreted onto the newly-formed NS.", "pages": [ 56, 57, 58, 59, 60 ] }, { "title": "5.1. Conclusions", "content": "The nature of GRBs is presenting itself as one of the richest diagnostics ever encountered within physics and astrophysics. It is clear that phenomena never before explored in this domain can now be submitted to theoretical and observational scrutiny. In the GRB-SN connection we have introduced, in analogy with the Smatrix of particle physics, a cosmic matrix (C-matrix) in which the in-states are a NS and an evolved core undergoing a SN explosion in a binary system, and the out-states are a BH and a newly-born NS. With the same spirit, the C-matrix of a genuine short GRB has as in-states two NSs and as out-states GW emission and the formation of a BH.", "pages": [ 61 ] }, { "title": "64 Remo Ruffini", "content": "Wells, N. E. White and R. A. M. J. Wijers, Nature 437 , 851 (October 2005).", "pages": [ 65 ] }, { "title": "66 Remo Ruffini", "content": "landri, K. Wiersema, A. Pozanenko, A. J. van der Horst, G. G. Pooley, A. FernandezSoto, A. J. Castro-Tirado, A. D. U. Postigo, M. Im, A. P. Kamble, D. Sahu, J. AlonsoLorite, G. Anupama, J. L. Bibby, M. J. Burgdorf, N. Clay, P. A. Curran, T. A. Fatkhullin, A. S. Fruchter, P. Garnavich, A. Gomboc, J. Gorosabel, J. F. Graham, U. Gurugubelli, J. Haislip, K. Huang, A. Huxor, M. Ibrahimov, Y. Jeon, Y.-B. Jeon, K. Ivarsen, D. Kasen, E. Klunko, C. Kouveliotou, A. Lacluyze, A. J. Levan, V. Loznikov, P. A. Mazzali, A. S. Moskvitin, C. Mottram, C. G. Mundell, P. E. Nugent, M. Nysewander, P. T. O'Brien, W.-K. Park, V. Peris, E. Pian, D. Reichart, J. E. Rhoads, E. Rol, V. Rumyantsev, V. Scowcroft, D. Shakhovskoy, E. Small, R. J. Smith, V. V. Sokolov, R. L. C. Starling, I. Steele, R. G. Strom, N. R. Tanvir, Y. Tsapras, Y. Urata, O. Vaduvescu, A. Volnova, A. Volvach, R. A. M. J. Wijers, S. E. Woosley and D. R. Young, MNRAS 413 , 669 (May 2011).", "pages": [ 67 ] } ]

[ { "title": "ABSTRACT", "content": "International Journal of Modern Physics D c © World Scientific Publishing Company", "pages": [ 1 ] }, { "title": "BLAZAR ANTI-SEQUENCE OF SPECTRAL VARIABILITY FOR INDIVIDUAL TeV BLAZARS", "content": "Jin Zhang 1 , 2 , Shang-Nan Zhang 3 , 1 , En-Wei Liang 4 1 National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100012, China; [email protected] 2 College of Physics and Electronic Engineering, Guangxi Teachers Education University, Nanning, 530001, China 3 Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China 4 Department of Physics and GXU-NAOC Center for Astrophysics and Space Sciences, Guangxi University, Nanning, 530004, China Received Day Month Year Revised Day Month Year Communicated by Managing Editor We compile from literature the broadband SEDs of twelve TeV blazars observed simultaneously or quasi-simultaneously with Fermi /LAT and other instruments. Two SEDs are available for each of the objects and the state is identified as a low or high state according to its flux density at GeV/TeV band. The observed SEDs of BL Lac objects (BL Lacs) are fitted well with the synchrotron + synchrotron-self-Compton (syn+SSC) model, whereas the SEDs of the two flat spectrum radio quasars (FSRQs) need to include the contributions of external Compton scattering. In this scenario, it is found that the Doppler factor δ of FSRQs is smaller than that of BL Lacs, but the magnetic field strength B of FSRQs is larger than that of BL Lacs. The increase of the peak frequency of the SEDs is accompanied with the increase of the flux for the individual sources, which seems opposite to the observational phenomena of the blazar sequence. We refer this phenomenon to blazar anti-sequence of spectral variability for individual TeV blazars. However, both the blazar sequence from FSRQs to BL Lacs and blazar anti-sequence of the spectral variability from low state to high state are accompanied by an increase of the break Lorentz factor of the electron's spectrum γ b and a decrease of B . We propose a model in which the mass accretion rate ˙ M is the driving force behind both the blazar sequence for ensembles of blazars and the blazar anti-sequence for individual blazars. Specifically we suggest that the differences in 〈 ˙ M 〉 of different blazars produce the observed blazar sequence, but ∆ ˙ M in each blazar results in the observed blazar anti-sequence. Keywords : Radiation mechanisms: non-thermal; BL Lacertae objects: general; quasars: general; gamma-rays: theory", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Blazars, a subsample of active galactic nuclei (AGNs), are composed of BL Lac objects (BL Lacs) and flat spectrum radio quasars (FSRQs). It is well known that the observed emission from blazars is jet dominated and the observed broadband spectral energy distributions (SEDs) are bimodal. Generally, the bump at the IRoptical-UV band is explained with the synchrotron process of relativistic electrons accelerated in the jets and the bump at the GeV-TeV gamma-ray band is due to the inverse Compton (IC) scattering of the same electron population. The seed photon fields may be from the synchrotron radiation themselves (the so-called SSC model) 1 , 2 or from external radiation fields (EC), such as the broad line region (BLR) 3 . Broadband SEDs obtained simultaneously or quasi-simultaneously are critical to investigate the radiation mechanisms and the physical properties of the emitting regions for the two kinds of blazars. With a large sample of different types of blazars, Fossati et al. (1998) reported a spectral sequence of FSRQ-LBL-HBL 4 , i.e., along with this sequence, observationally, an increase in the peak frequency of synchrotron radiation ( ν s ) corresponds to the decreases of bolometric luminosity and the ratio ( L IC /L syn ) of the luminosities for the high- and low-energy spectral components. Theoretically, an anti-correlation between the break Lorentz factor of electrons ( γ b ) and the energy density of radiation regions ( U ' ) in the comoving frame was found. So this blazar sequence was interpreted by Ghisellini et al. (1998) with the cooling of the external photon fields (such as the BLRs) of the blazars 5 . The discoveries of 'blue quasars' posed a challenge to the previous interpretation for blazar sequence. To address this problem, Ghisellini & Tavecchio (2008) reported that, more physically, the sequence is due to the different black hole (BH) masses and accretion rates of the sources 6 . According to this interpretation, the 'blue quasars' should have large BH masses and intermediate accretion rates and thus the emission region is beyond the BLR. Moreover, the red low-luminosity blazars should exist, which have small BH masses and relatively large accretion rates. More recently, Chen & Bai (2011) extended this sequence to narrow line Seyfert 1 galaxies 7 , which are similar to the low-peak-frequency and low-luminosity blazars. Observations show that the SED peak frequency of a TeV blazar increases with its flux, indicating a tendency that a brighter TeV emission corresponds to a harder spectrum for the emission at the X-ray and gamma-ray bands 8 , 9 . This phenomenon seems to be opposite to the blazar sequence; we thus refer this phenomenon to blazar anti-sequence of spectral variability for individual TeV blazars. The spectral shift at different states of TeV emission for the individual sources may be due to the different γ b at different states 9 . Thus, it seems that both the blazar sequence and anti-sequence are caused by the change of γ b . However it remains unclear whether there are any connections between the two phenomena. So far, about forty AGNs have been detected at the TeV gamma-ray band. Except for two radio galaxies, all of the confirmed TeV AGNs are blazars. Moreover, most of the confirmed sources by Fermi /LAT are also blazars. The abundant observation data provide an excellent opportunity to reveal the physical connections between the blazar sequence and anti-sequence.", "pages": [ 1, 2 ] }, { "title": "2. Sample Selection", "content": "In order to obtain well-sampled SEDs, only the TeV blazars, which have positive Fermi /LAT detections and two or more observed broadband SEDs, are considered. Twelve blazars are included in our sample, two FSRQs (3C 279 and PKS 1510089) and ten BL Lacs (W Com, Mkn 421, Mkn 501, PKS 2155-304, 1ES 1101-232, BL Lacertae, 1ES 2344+514, 1ES 1959+650, PKS 2005-489, S5 0716+714). We compile their broadband SEDs that were simultaneously or quasi-simultaneously observed with Fermi /LAT and other instruments from literature. The two wellsampled SEDs for a source are identified as a low or high state according to the observed or extrapolated flux density at 1 TeV, except for PKS 1510-089. Because the SEDs of PKS 1510-089 are cut off at 1 TeV, the high and low states of this source are defined by the flux density at 10 GeV. The observation data of the broadband SEDs for 3C 279 and PKS 1510-089 are from literature 10 , 11 , 12 .", "pages": [ 3 ] }, { "title": "3. Models and Results", "content": "The observed SEDs for those sources are double-peaked. For BL Lacs the external photon fields from their BLRs, if exist, are very weak and negligible, compared with the synchrotron radiation photon fields. Therefore, only the syn+SSC model is considered to fit the observed broadband SEDs of the ten BL Lacs. The detailed calculation process for the model and the strategy for parameter constraints can be found in Ref. 9 . Following the methods described in Ref. 9 , the SEDs for the ten BL Lacs are fitted well by the single-zone syn+SSC model; the fitting results and parameters can be found in Fig. 1 and table 1 of Ref. 9 . For the two FSRQs, the contributions of external field photons from their BLRs need to be considered. The total luminosities from the BLRs for the two FSRQs are taken from Ref. 13 and then the BLR sizes are calculated using the BLR luminosity with formula (23) given in Ref. 14 . The radiation from a BLR is assumed to be a blackbody spectrum and the corresponding energy densities seen in the comoving frame are U ' BLR = 5 . 37 × 10 -3 Γ 2 erg cm -3 and U ' BLR = 4 . 98 × 10 -3 Γ 2 erg cm -3 for 3C 279 and PKS 1510-089, respectively, where we take Γ = δ . Thus, the syn+SSC+EC/BLR model is used to fit the observed SEDs of the two FSRQs and the model can explain the observed SEDs well as shown in Fig. 1. The range of δ for these BL Lacs is from 8 to 50 and is clustered at 11 ∼ 30. On average, δ of BL Lacs is larger than that of the FSRQs, but the magnetic field strength B of BL Lacs is smaller than that of the FSRQs, consistent with the results of some previous works 15 , 16 . The values of γ b for these BL Lacs vary from 10 3 to 10 6 , but the values of B are clustered around 0 . 1 ∼ 0 . 6 G. No correlation between γ b and B for these BL Lacs is found.", "pages": [ 3 ] }, { "title": "4.1. The Blazar Anti-Sequence", "content": "As shown in Fig. 1 and Fig. 1 in Ref. 9 , there are clear spectral shifts accompanying the flux variations for these TeV blazars. The SEDs of the twelve TeV blazars in our sample are obtained in the low and high TeV states, which is defined according to the flux density at 1 TeV except for PKS 1510-089, for which the state is defined according to its flux density at 10 GeV. We compare ν s and ν c between the high and low states in Fig. 2(a). It is clear that the SEDs in the high states shift to a higher energy band than that in the low state. As reported in our previous paper 9 , to investigate what may be responsible for the spectral shift in the low and high states, we derive the ratios of the flux density at 1 TeV ( R 1 TeV ) and the physical parameters ( R x ) in the high state to that in the low state for these sources, where x stands for L bol , B , δ , γ b , or P jet . The results on the SEDs of PKS 1510-089 are not included here. The values of γ b , L bol , and P jet of the high states are systematically higher than that of the low state. A tentative correlation between R γ b and R 1 TeV is found with a correlation coefficient r = 0 . 55 and a chance probability p = 0 . 077 as shown in Fig. 2(b). Therefore, it is possible that the spectral shift at different states is due to the different γ b of these sources. In order to compare the differences of parameters for the high and low states, we also calculate the magnetic field energy density ( U B ), the available photon energy density ( U ' ph , U ' ph = U ' BLR + U ' syn , avail for FSRQs and U ' ph = U ' syn , avail for BL Lacs) for IC process in the comoving frame, the luminosities of the synchrotron radiation ( L syn ) and the IC process ( L IC , L IC = L SSC for BL Lacs and L IC = L SSC + L EC for FSRQs) in high and low states. The ratios of those parameters for the two states ( R U B , R U ' ph , R L syn , and R L IC ) as a function of the ratio of γ b ( R γ b ) are shown in Fig. 3. No correlations between the ratios of those parameters and the ratio of γ b are found. As presented in Fig. 3(a), for most of the sources U B becomes smaller in the high state than in the low state. However, the ratios of U ' ph , L syn , and L IC are larger than unity for most of the sources as presented in Fig. 3(a), (b).", "pages": [ 4 ] }, { "title": "4.2. The Blazar Sequence", "content": "As described in section 1, according to the observational phenomenon of the blazar sequence 5 , one can expect that both L bol and L IC /L syn are anti-correlated with ν s . L bol and L IC /L syn as a function of ν s , are shown in Fig. 4(a) and (b). A weak correlation is found for ν s -L bol . The Spearman correlation analysis yields a correlation coefficient r = -0 . 58 and a chance probability p = 0 . 05 for the high state data, r = -0 . 61 and p = 0 . 04 for low state data, respectively. Excluding the two FSRQs, no correlation would be found. However, the ratio of L IC /L s is indeed anticorrelated with ν s , especially in their low states. The Spearman correlation analysis yields a correlation coefficient r = -0 . 71 and a chance probability p = 0 . 009 for the high state data, r = -0 . 94 and p < 10 -4 for the low state data, respectively. Because most of the sources in our sample are BL Lacs, the external photon fields outside their jets are much weaker than the synchrotron radiation photon field and the EC process is thus not considered for these sources. As ν s increases, the SEDs shift to the higher frequency end and the KN effect should be more significant. According to equation (20) in Ref. 17 , U ' syn , avail = U ' syn ( 3 mc 2 δ 4 hγ b ν s ) 1 -α 1 , the available photon energy density of synchrotron radiation for IC process decreases with the increase of γ b . For L SSC L syn ∼ U ' syn , avail U B , the ratio of L SSC /L syn would decrease as ν s increases, since U B is almost constant and U ' syn , avail decrease along with the increase of γ b for the BL Lacs in our sample. Therefore, the anti-correlation of L IC /L syn -ν s may be also due to the KN effect, especially for BL Lacs. Because the interpretation for blazar sequence is cooling of the external photon fields, a more 'theoretical' scenario than the purely phenomenological sequence is the anti-correlation between the break Lorentz factor of electrons γ b and the energy density of radiation regions U ' in the comoving frame 5 , 15 . γ b as a function of U ' ( U ' = U ' syn , avail + U ' B for BL Lacs, U ' = U ' syn , avail + U ' BLR + U ' B for FSRQs) is also shown in Fig. 4(c). Although the correlation between γ b and U ' of our sample sources has large scatters, especially for BL Lacs, comparing our results with that in Ref. 15 , the two results are consistent and the BL Lacs included in our sample distribute in the left top of Fig. 4(c), where the EC process is not important. If the different γ b is totally due to the different external photon field, there should be a correlation between the luminosity of the BLR and the peak frequency of synchrotron emission. With a FSRQs sample 18 , however, no correlation between L BLR and ν s is found as shown in Fig. 5(a). Sometimes, for simplicity, the luminosity and the radius of a BLR is assumed as L BLR = 0 . 1 L disk and R BLR = 10 17 L 1 / 2 disk , 45 . So the comoving energy density of a BLR is given by U ' BLR = 3 . 76 × 10 -2 Γ 2 erg cm -3 under these assumptions 16 and is totally decided by the value of Γ = δ . Nevertheless, no correlation between Γ 2 and γ b is found either, as presented in Fig. 5(b). γ b is correlated with the total energy density of the emitting regions in the comoving frame as shown in Fig. 4(c), but not correlated with the energy density of the BLR. Using the sample data in Ref. 16 and our sample data, it is found that γ b is correlated with the magnetic field energy density, as shown in Fig. 5(c). Some works indeed demonstrate that the magnetic field strength B of FSRQs is different from and larger than that of BL Lacs 15 , 16 , also consistent with our results. So the blazar sequence may be due to the different γ b and B of these sources.", "pages": [ 4, 5 ] }, { "title": "4.3. Implications for Blazar Sequence and Anti-Sequence", "content": "As shown in Fig. 1 of this paper and Fig. 1 of Ref. 9 , the characteristics of spectral evolution at different states for a given source are opposite to the blazar sequence. In order to investigate the physical connections of the two phenomena, firstly, we need to define another physical parameter ( R Y ), the ratio of physical parameters in the high state to that in the low state for the twelve sources, where Y stands for ν s , L bol , L IC /L syn , γ b , and U ' . Comparing the spectral evolution for a given source with the blazar sequence, we find the following: As described above, both the blazar sequence from FSRQs to BL Lacs and antisequence of the individual sources from low state to high state are accompanied by an increase of γ b and a decrease of B , as shown in Fig. 3(a) and Fig. 5(c). Ghisellini & Tavecchio (2010) reported that the blazar sequence is linked to the different BH masses and accretion rates of different sources 15 . We propose here that the different states of the individual objects are also linked to the variations of the accretion rate for each source. So both the blazar sequence (change of source type) and the blazar anti-sequence of spectral variability of the individual objects (change of source state) are linked to the change of accretion rate. The flow chart illustrating how the accretion rate drives the blazar sequence and the anti-sequence is shown in Fig. 7, which is explained as follows: that the emission lines of BL Lacs are very weak, thus the second bump of their SEDs are not contributed by IC/BLR ( L EC ). Therefore, the values of L IC /L syn for FSRQs are larger that of BL Lacs as presented in Fig. 4(b).", "pages": [ 5, 6, 7 ] }, { "title": "5. Summary", "content": "We have compiled the broadband SEDs of twelve TeV blazars that were simultaneously or quasi-simultaneously observed with Fermi /LAT and other instruments from literature. Each of those sources has two broadband SEDs available, which are identified as a high or a low state according to its flux density at GeV/TeV band. We found that the syn+SSC model can well represent the observed SEDs for BL Lacs, whereas the EC/BLR contribution needs to be considered for explaining the observed SEDs of the two FSRQs. The magnetic field strength B of the two FSRQs is larger, but their Doppler factor δ is smaller than that of the ten BL Lacs. Significant spectral shift to high energies accompanying with the flux increase is observed for each individual source, which seems opposite to the observational phenomenon of the blazar sequence. We refer this phenomenon to the blazar anti-sequence. However, it is found that both the blazar sequence from FSRQs to BL Lacs and the anti-sequence of the individual sources from low to high states are accompanied by an increase of γ b and a decrease of B . We propose here that the blazar sequence and the anti-sequence of spectral variability of the individual sources are driven by decreasing accretion rate. A simple flow chart, which describes qualitatively how the accretion rate contributes to the blazar sequence and anti-sequence, is also given in this paper.", "pages": [ 7 ] }, { "title": "Acknowledgments", "content": "This work is supported by the National Natural Science Foundation of China (Grants 11078008, 11025313, 10873002, 11133002, 10821061, 10725313), the National Basic Research Program (973 Programme) of China (Grant 2009CB824800), China Postdoctoral Science Foundation, Guangxi Science Foundation (2011GXNSFB018063, 2010GXNSFC013011), and Guangxi SHI-BAI-QIAN project (Grant 2007201). /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32", "pages": [ 8, 9 ] }, { "title": "References", "content": "/s32 /s32 /s76 /s47 /s76 /s98 /s110 /s121 /s115 /s67 /s73 /s48/s46/s49 /s32 /s32 /s32 /s32 /s32 /s98 /s98 /s32 /s32 /s32 /s32 /s32", "pages": [ 9 ] }, { "title": "10 Zhang et al.", "content": "/s32 /s32 /s32 /s32", "pages": [ 10 ] } ]

[ { "title": "ABSTRACT", "content": "International Journal of Modern Physics D c © World Scientific Publishing Company", "pages": [ 1 ] }, { "title": "XIAOYU LAI and RENXIN XU", "content": "School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing, 100871, China; { laixy, r.x.xu } @pku.edu.cn The nature of pulsar-like compact stars is still in controversy although the first pulsar was found more than 40 years ago. Generally speaking, conventional neutron stars and nonmainstream quark stars are two types of models to describe the inner structure of pulsars, with the former composed mainly of hadrons and the latter of a peculiar kind of matter whose state equation should be understood in the level of quarks rather than hadrons. To construct a more realistic model from both theoretical and observational points of view, we conjecture that pulsars could be 'quark-cluster stars' which are composed of quarkclusters with almost equal numbers of up, down and strange quarks. Clustering quark matter could be the result of strong coupling between quarks inside realistic compact stars. The lightest quark clusters could be of H -dibaryons, while quark clusters could also be heavier with more quarks. Being essentially related to the non-perturbative quantumchromo dynamics (QCD), the state of supra-nuclear condensed matter is really difficult to obtain strictly by only theoretical QCD-calculations, and we expect, nevertheless, that astrophysical observations could help us to have a final solution. Keywords : pulsars; dense matter; quark matter PACS numbers: 97.60.Gb, 26.60.Kp, 21.65.Qr", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The subjects which are attractive and of great importance are usually those ones that are beyond our comprehension, and the nature of pulsar-like compact stars is of such a kind. Our understanding towards pulsars is still developing, thanks to the developments both theoretical and observational, but the real state of matter is still uncertain. Neutron stars and quark stars, as two types of models for the nature of pulsar, have been debated for a very long time, but which model is more realistic remains to be seen. The research history of extremely dense matter goes back to the very early time of stellar compact objects. Astronomers were not be able to understand the compactness of white dwarfs until the British physicist Ralph Howard Fowler (1889 -1944) recognized the quantum pressure of degenerate electrons there, who for the first time discussed also dense matter at his maximum possible density in the same seminal paper (1926) 1 as ' The density of such 'energetic' matter is then only limited a priori by the 'sizes' of electrons and atomic nuclei. The 'volumes' of these are perhaps 10 -14 times of the volume of the corresponding atoms, so that densities up to 10 14 times that of terrestrial materials may not be impossible ' after Ernest Rutherford", "pages": [ 1 ] }, { "title": "2 Xiaoyu Lai and Renxin Xu", "content": "constructed the nucleus model of atom in 1911. What if the matter density becomes so high? The year 1932 was special. (1) Neutron (or 'neutral doublet' in Rutherford's words) was experimentally discoveries by James Chadwick although it had been speculated to exist for a long time and for a variety of reasons. (2) Landau 2 conjectured a condensed core with nuclear matter densities inside a star where protons and electrons combined tightly forming the 'neutronic' state in order to explain the origin of stellar energy. In addition to advanced and detailed calculations, authors are modeling normal neutron stars generally along Landau's line although Landau did make two mistakes 3 80 years ago because of the historical limitations. The inner structure of quark stars which are totally composed of quark matter (with u , d and s quarks) was first calculated by Itoh 4 in 1970, because it was realized previously that there could be deconfined quarks inside neutron stars. From then on, neutron stars are defined as such a kind of compact objects that mainly composed of neutrons, with hyperons or even quark matter in their innermost cores, and quark stars as a kind of compact objects composed of pure (strange) quark matter. It is worth mentioning that, quark stars are characterized by soft equations of state, because the asymptotic freedom of QCD tells us that as energy scale goes higher, the interaction between quarks becomes weaker. The recent discovery 5 of a ∼ 2 M /circledot neutron star seems to be evidence against quark stars unless the coupling between deconfined quarks is still very strong. 6 Anyway, a working model with the coupling parameter as high as α s /greaterorsimilar 0 . 6 -0 . 7 could be possible in principle, 7 but such a strong interaction may also favour a kind of condensation in position space rather than in momentum space as was already noted in 2003. 8 Neutron stars and quark stars respectively correspond to two distinct regions in the QCD phase diagram, the hadron phase in the low density region and the quarkgluon plasma phase in the high density region. In other words, inside neutron stars the highly non-perturbative strong interaction makes quarks grouped into neutrons, whereas inside quark stars the perturbative strong interaction makes quarks to be almost free if the coupling is weak. At a few nuclear matter densities and extremely low temperature, the quark degrees of freedom should be significant, and there is possible observational evidence that pulsars could be quark stars (see reviews, e.g. Ref 9, 10, 11). However, in cold quark matter at realistic baryon densities of compact stars ( ρ ∼ 2 -10 ρ 0 ), the energy scale is far from the region where the asymptotic freedom approximation could apply. The strong coupling between quarks even exists in the hot quark-gluon plasma 12 , then it is reasonable to infer that quarks could be coupled strongly also in the interior of quark stars, which could make quarks to condensate in position space to form quark clusters. The quark matter inside compact stars could in the 'quark-clustering phase', where the energy scale could be high enough to allow the restoration of light flavor symmetry, but may not be high enough to make the quarks really deconfined. Quark-cluster stars are treated here to assort with the type of quark stars since (1) they manifest in a similar way of quark star with self-bound rather gravity-bound of neutron stars, (2) their equation of state should be understood in the level of quarks rather than hadrons (i.e., the quark degree of freedom would play a significant role in determining the equation of state and during the formation of quark-cluster stars), and (3) the term of 'quark-cluster star' might be abbreviated simply as 'quark star'. The observational tests from polarization, pulsar timing and asteroseismology have been discussed, 8 and it is found that the idea of clustering quark matter could provide us a way to understand different manifestations of pulsars. The realistic quark stars could then be actually 'quark-cluster stars'. An interesting suggestion is that quark matter could be in a solid state 13 , 14 , 15 , and for quark-cluster stars, solidification could be a natural result if the kinetic energy of quark clusters is lower than the interaction energy between the clusters. To calculate the interaction between quarks and to predict the state of matter for quark stars by QCD calculations is a difficult task; however, it is still meaningful for us to consider phenomenologically some models to explore the properties of quarks at the low energy scale. In this paper we show two models for clustering quark matter. In the LennardJones model 16 we take the number of quarks inside each quark-cluster N q to be a free parameter, and in the H -stars model 17 we take H -cluster as a specific kind of quark-clusters. Under a wide range of parameter-space, the maximum mass of quark-cluster stars could be well above 2 M /circledot . The asymptotic freedom of QCD makes the perturbative theory applicable to study the systems under strong interaction, but it cannot describe the systems with vast assemblies of particles under strong interaction that exist in the Universe. Quark matter at high density and low temperature is difficult to be created in laboratories as well as difficult to be study along with QCD calculations, and the observational tests should play an important role to constrain the properties of QCD at low energy scales.", "pages": [ 2, 3 ] }, { "title": "2. Clustering Quark Matter", "content": "Due to QCD's asymptotic freedom, cold dense quark matter would certainly be of Fermi gas or liquid if the baryon density is extremely high. However, perturbative QCD would work reasonably well only for quark chemical potentials above 1 GeV, while the quark chemical potential for a typical quark stars is about 0.4 GeV. We can make an estimate on the chemical potential and the interaction energy of quarks inside quark stars. The strong interaction between quarks in compact stars may result in the formation of quark clusters, with a length scale l q and an interaction energy E q . An estimate from Heisenberg's relation, if quarks are dressed with mass m q /similarequal 300 MeV, gives l q ∼ 1 α /planckover2pi1 c m q c 2 /similarequal 1 α fm, and E q ∼ α 2 mc 2 /similarequal 300 α 2 s MeV. The strong coupling constant α can be estimated from non-perturbative QCD as a function of baryon number density, and at a few nuclear density in compact stars, the coupling could be very strong rather than weak, with α /similarequal 2 18 . This means that a weakly coupling treatment could be dangerous for realistic cold quark matter, and quarks would be", "pages": [ 3 ] }, { "title": "clustered.", "content": "Quark-clusters could emerge in cold dense matter because of the strong coupling between quarks. The quark-clustering phase has high density and the strong interaction is still dominant, so it is different from the usual hadron phase, and on the other hand, the quark-clustering phase is also different from the conventional quark matter phase which is composed of relativistic and weakly interacting quarks. The quark-clustering phase could be considered as an intermediate state between hadron phase and free-quark phase, with deconfined quarks grouped into quarkclusters, and that is another reason why we take quark-cluster stars as a special kind of quark stars. It is worth noting that, whether the chiral symmetry broken and confinement phase transition happen simultaneously inside compact stars is a matter of debate (see 19 and references therein), but here we assume that the chiral symmetry is broken in quark-clustering phase. What are quark-clusters explicitly? There is no clear answer, and we could only have some candidates. A 18-quark cluster, called quark-alpha 20 , could be completely symmetric in spin, light flavor and color space. Λ particles, with structure uds , is the light particle with light flavor symmetry. There could probably attraction between two Λs 21 , 22 , so H -clusters with structure uuddss could emerge. If the light flavor symmetry is ensured, then the dominant components inside the stars is very likely to be H -clusters. In the following, we will show that the degree of light flavor symmetry breaking could be small in the macroscopic strange quark matter. About light flavor symmetry. It is well know that there is an asymmetry term to account for the observed tendency to have equal numbers of protons ( Z ) and neutrons ( N ) in the liquid drop model of the nucleus. This nuclear symmetry energy (or the isospin one) represents a symmetry between proton and neutron in the nucleon degree of freedom, and is actually that of up and down quarks in the quark degree 23 . The possibility of electrons inside a nucleus is negligible because its radius is much smaller than the Compton wavelength λ c = h/m e c = 0 . 24 ˚ A . The lepton degree of freedom would then be not significant for nucleus, but electrons are inside a large or gigantic nucleus, which is the case of compact stars. Now there is a competition: isospin symmetry favors Z = N while lepton chemical equilibrium tends to have Z /lessmuch N . The nuclear symmetry energy ∼ 100( Z -N ) 2 /A MeV, where A = Z + N , could be around 100 MeV per baryon if N /greatermuch Z . Interesting, the kinematic energy of an electron is also ∼ 100 MeV if the isospin symmetry keeps in nuclear matter. However, the situation becomes different if strangeness is included: no electrons exist if the matter is composed by equal numbers of light quarks of u , d , and s with chemical equilibrium. In this case, the 3-flavor symmetry, an analogy of the symmetry of u and d in nucleus, may results in a ground state of matter for gigantic nuclei. Certainly the mass different between u , d and s quarks would also break the symmetry, but the interaction between quarks could lower the effect of mass differences and try to restore the symmetry. Although it is hard for us to calculate how strong the interaction between quarks is, the non-perturbative nature and the energy scale of the system make it reasonable to assume that the degree of the light flavor symmetry breaking is small, and there is a few electrons (with energy ∼ 10 MeV). The above argument could be considered as an extension of the Bodmer-Witten's conjecture. Possibly it doesn't matter whether three flavors of quarks are free or bound. We may thus re-define strange matter as cold dense matter with light flavor symmetry of three flavors of u , d , and s quarks.", "pages": [ 4, 5 ] }, { "title": "3. The Global Structure of Quark-cluster Stars", "content": "We propose that pulsar-like compact stars could be quark-cluster stars which are totally composed of quark clusters. Quark-cluster stars could have different properties from neutron stars and conventional quark stars, such as the radiation properties, cooling behavior and global structure. In this paper, we only focus on the global structure of quark-cluster stars, deriving the mass-radius relation based on the equation of state, under the Lennard-Jones model and H star model respectively.", "pages": [ 5 ] }, { "title": "3.1. Lennard-Jones quark matter model", "content": "In the Lennard-Jones quark matter model, the interaction potential u between two quark-clusters as the function of their distance r is assumed to be described by the Lennard-Jones potential, similar to that between inert molecules, where U 0 is the depth of the potential and r 0 can be considered as the range of interaction. It is worth noting that the property of short-distance repulsion and longdistance attraction presented by Lennard-Jones potential is also a characteristic of the interaction between nuclei. Under the interaction, quark-clusters could be localized and behave like classical particles, and in this case the tatal interaction potential of one quark-cluster could be written as where R is the nearest distance between two quark-clusters, and A 12 and A 6 are the coefficients depending on the lattice structure. The localization is natural for clustering quark matter, with the following reasons. One quark-cluster with mass m is under the composition of interaction from its neighbor quark-clusters, which forms a potential well. The energy fluctuation makes this quark-cluster oscillate about its equilibrium position with the deviation ∆ x , ∆ E /similarequal /planckover2pi1 2 / ( m ∆ x 2 ) /similarequal k ∆ x 2 , where k /similarequal ∂ 2 V ( r ) /∂r 2 , and r is the distance of two neighbor H -clusters. We use the inter-cluster interaction in Eq(2), and estimate ∆ x at density ρ = 10 ρ 0 , ∆ x /similarequal ( /planckover2pi1 2 /mk ) 1 / 4 /similarequal 0 . 2 fm(18 /N q ) 1 / 4 , where N q is the number of quarks inside each quark-cluster. On the other hand, the distance between two nearby quark-clusters at density ρ = 10 ρ 0 is d = n -1 / 3 /similarequal 1 . 5 fm ( N q / 18) 1 / 3 , with n the number density", "pages": [ 5 ] }, { "title": "6 Xiaoyu Lai and Renxin Xu", "content": "of quark-clusters. Consequently, the interaction would localize H -clusters in the potential well at the stellar center, since ∆ x < d . On the stellar surface, ρ /similarequal 2 ρ 0 , we have ∆ x = 0 . 4 fm and d ∼ 2 . 6 fm. Therefore, under the interaction, quarkclusters could be localized and behave like classical particles, and the quantum effect would be negligible. Under such potential, we can get the equation of state for quark-cluster stars. Because of the strong interaction, the surface density ρ s should be non-zero, and r 0 can be derived at the surface where the pressure vanishes. Choosing ρ s = 2 ρ 0 to ensure the deconfinement of quarks, we can derive the mass and radius of a quark star by combining the equation of state with the hydrostatic equilibrium condition. The dependence of maximum mass of quark stars on U 0 and N q is shown in Figure 2 24 . We can see that there is enough parameter space for the existence of quark stars with mass larger than 2 M /circledot . The case N q > 10 4 should be ruled out by the discovery of PSR J1614-2230.", "pages": [ 6 ] }, { "title": "3.2. H -cluster Stars", "content": "In Lennard-Jones model, quark-clusters are analogized to electric neutral molecules; however, quark clusters may also be analogized to hadrons. A dihyperon with quantum numbers of ΛΛ ( H dibaryon) was predicted to be stable state or resonance 25 , and recent lattice QCD simulations show some evidence that the H -dibaryons (with structure uuddss ) are bound states 21 , 22 . Motivated by both the theoretical prediction and numerical simulations, we consider a possible kind of quark-clusters, H -particles, that could emerge inside quark stars during their cooling, as the dominant building blocks 17 . To study quark stars composed of H -matter, i.e. H stars, we assume that the interaction between H -particles is mediated by σ and ω mesons and introduce the Yukawa potential to describe the H -H interaction 26 , and then derive the dependence of the maximum mass of H stars on the depth of potential well, taking into account the in-medium stiffening effect. Using the similar argument as that in the Lennard-Jones model, H -clusters could be localized and behave like classical particles, and Bose condensate would not take place even though they are bosons. On the other hand, although H -clusters could be weakly bound particles which would decay to lighter baryons, such as the reaction of H → 2 n +2 π , the decay could hardly happen inside compact stars. At density ρ larger than 2 ρ 0 , the fermi energy of neutrons is larger than 100 MeV, which makes H -clusters difficult to decay into neutrons and pions. Moreover, H -clusters could be safe under the high momentum fluctuation ∆ p at high densities inside compact stars, because the energy fluctuation ∆ E is not so high due to their high mass. We can make the estimation of ∆ E ∼ ∆ p 2 / 2 m H /similarequal 7 MeV( ρ/ 10 ρ 0 ) 2 / 3 ( m H / 2210 MeV) -5 / 3 , where we set the mass of H -cluster, m H = 2 m Λ -20 MeV = 2210 MeV, and m Λ the mass of Λ 0 . The energy of ∆ E could be not much lower than the binding energy of H -clusters and potential drop of interaction between H -clusters, but it could be reasonable to ensure the existence of H -clusters with large enough mass fraction of the star. Compact stars composed of pure H -clusters are electric neutral, but in reality there could be some flavor symmetry breaking that leads to the non-equality among u , d and s , usually with less s than u and d . The positively charged quark matter is necessary because it allows the existence of electrons that is crucial for us to understand the radiative properties of pulsars. The pressure of degenerate electrons is negligible compared to the pressure of H -clusters, so the contribution of electrons to the equation of state is negligible. We adopt the Yukawa potential with σ and ω coupling between H -particles 26 , where g ωH and g σH are the coupling constants of H -particles and meson fields. In dense nuclear matter, the in-medium stiffening effect, i.e., the Brown-Rho scaling effect, should be considered 27 , and then the effective meson masses m ∗ M satisfy the scaling law m ∗ M /similarequal m M (1 -α BR n/n 0 ), where α BR is the coefficient of the scaling and m M is the meson mass in free space. In the problem we are now considering, however, a quark star is at supra-nuclear density, and we then use a modified scaling law of which also shows the in-medium effect that stiffens the inter-particle potential by reducing the meson effective masses, and approximately the same as the usual scaling law at the nuclear matter density. From the potential, we can get the equation of state, and derive the total mass M and radius R of an H star by numerical integration. Figure 3 17 shows the", "pages": [ 6, 7 ] }, { "title": "8 Xiaoyu Lai and Renxin Xu", "content": "dependence of M max on the depth of the potential well V 0 and the Brown-Rho scaling coefficient α BR , in the case ρ s = 2 ρ 0 . To make comparison, we also plot the result when α BR = 0, and the discrepancy between different values of non-zero α BR is not very significant. 0 H stars could have stiff equation of state, and under a wide range of parameterspace, the maximum mass of H stars can be well above 2 M /circledot , providing a possible way to explain the observed high mass of the newly discovered pulsar PSR J16142230. Furthermore, if we know about the properties of pulsars from observations, we can get information on H -H interaction; for example, if a pulsar with mass larger than 3 M /circledot is discovered, then we can constrain -V 0 to be larger than 60 MeV.", "pages": [ 8 ] }, { "title": "4. Conclusions and Discussions", "content": "Pulsars could be either neutron stars or quark stars. a Although the state of cold quark matter at a few nuclear densities is still an unsolved problem in low energy QCD, various pulsar phenomena would give us some hints about the properties of elemental strong interaction, complementary to the terrestrial experiments. Pulsarlike compact stars provide high density and relatively low temperature conditions where quark matter with quark-clusters could exist, and we have discussed some possible kinds of models to describe such kind of quark matter which could be tested by observations. We apply the Lennard-Jones model and H star model, where the quark-clusters are treated as molecules in the former and dibaryons in the latter. The 2 M /circledot pulsar puts constraints on the number of quarks in one quark-cluster N q to be less than 10 4 in the Lennard-Jones model. To put any constraint on the H -matter model with in-medium stiffening effect, some more massive pulsars (e.g. M > 3 M /circledot ) should be found in the future. After addressing a lot about modeling quark-cluster stars, it could be interesting to compare Landau's 'giant nucleus', neutron star and quark-cluster star. Landau conjectured a 'neutronic' state core with nuclear matter densities inside a star in order to solve the origin problem of stellar energy. In Landau's scenario, the 'neutronic' state core and the surrounding ordinary matter are in chemical equilibrium at the boundary, which is very similar to the neutron star picture where the inner and outer cores and the crust keep chemical equilibrium at each boundary. Landau's giant nucleus is then bound by gravity. The 'neutronic' core should have a boundary and is in equilibrium with the ordinary matter because the star has a surface composed of ordinary matter. There is, however, no clear observational evidence for a neutron star's surface, although most of authors still take it for granted that there should be ordinary matter on surface, and consequently a neutron star has different components from inner to outer parts. Being similar to traditional quark stars, quark-cluster stars have almost the same composition from the center to the surface, and the quark matter surface could be natural for understanding some different observations. As an analog of neutrons, quark-clusters are bound states of several quarks, so to this point of view a quark-cluster star is more similar to a real giant nucleus of self-bound (not that of Landau), rather than 'giant hadron' which describes traditional quark stars. It is also worth noting that, although composed of quark-clusters, quark-cluster stars are self-bound. They are bound by the residual interaction between quarkclusters. This is different from but similar to the traditional MIT bag scenario. The interaction between quark-clusters could be strong enough to bind the star, and on the surface, the quark-clusters are just in the potential well of the interaction, leading to non-vanishing density but vanishing pressure. It has been 80 years since Landau proposed the idea of 'neutron' stars, and more than 40 years since the first pulsar was discovered, but the interior structure of pulsar-like compact stars is still in controversy. The nature of pulsars is essentially a non-perturbative QCD problem, corresponding the region between hadron phase and quark-gluon plasma phase in the QCD phase diagram. Although the state of cold quark matter at a few nuclear densities is still an unsolved problem in low energy QCD, various pulsar phenomena would give us some hints about the properties of elemental strong interaction, 18 complementary to the terrestrial experiments. Pulsar-like compact stars provide high density and relatively low temperature conditions where quarks may not be free but would be clustered to form quark-cluster matter. Whether this quark matter composed of quark-clusters could achieve at supra-nuclear density is still unknown, and on the other hand, the nature of pulsar-like stars also depends on the physics of condensed matter. These prob- lems are essentially related to the non-perturbative QCD, and we hope that future astrophysical observations would test the existence of quark-cluster stars.", "pages": [ 8, 9, 10 ] }, { "title": "Acknowledgements", "content": "We would like to thank useful discussions at our pulsar group of PKU. This work is supported by the National Natural Science Foundation of China (Grant Nos. 10935001, 10973002), the National Basic Research Program of China (Grant Nos. 2009CB824800, 2012CB821800), the John Templeton Foundation, and China Postdoctoral Science Foundation Project.", "pages": [ 10 ] } ]

[ { "title": "SPECIAL RELATIVITY OVER THE FIELD OF RATIONAL NUMBERS", "content": "JUDIT X. MADAR ' ASZ AND GERGELY SZ ' EKELY Abstract. We investigate the question: what structures of numbers (as physical quantities) are suitable to be used in special relativity? The answer to this question depends strongly on the auxiliary assumptions we add to the basic assumptions of special relativity. We show that there is a natural axiom system of special relativity which can be modeled even over the field of rational numbers.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "In this paper, we investigate, within an axiomatic framework, the question: what structures of numbers (as physical quantities) are suitable to be used in special relativity? There are several reasons to investigate this kind of questions in the case of any theory of physics. First of all, we cannot experimentally verify whether the structure of quantities is isomorphic to the field of real numbers. Moreover, the fact that the outcome of every measurement is a finite decimal suggests that rational numbers (or even integers) should be enough to model physical quantities. Another reason is that these investigations lead to a deeper understanding of the connection of the mathematical assumptions about the quantities and the other (physical) assumptions of the theory. Hence these investigations lead to a deeper understanding of any theory of physics, which may come handy if we have to change some of the basic assumptions for some reason. For a more general perspective of this research direction, see [4]. So in general we would like to investigate the question 'What structure can numbers have in a certain physical theory?' To introduce the central concept of our investigation, let Th be a theory of physics that contains the concept of numbers (as physical quantities) together with some algebraic operations on them (or at least these concepts are definable in Th ). In this case, we can introduce notation Num ( Th ) for the class of the possible quantity structures of theory Th : Num ( Th ) = { Q : Q is a structure of quantities In this paper, we investigate our question only in the case of special relativity. However, this question can be investigated in any other physical theory the same way. over which Th has a model. } (1) We will see an axiom on observers implying that positive numbers have square roots. Therefore, we recall that Euclidean fields , which got their names after their role in Tarski's first-order logic axiomatization of Euclidean geometry [18], are ordered fields in which positive numbers have square roots. Our axiom system for d -dimensional special relativity ( SpecRel d , see p.5) captures the kinematics of special relativity perfectly if d ≥ 3, see Theorem 3.1. Without any extra assumptions SpecRel d has a model over every ordered field, i.e.,", "pages": [ 1, 2 ] }, { "title": "Num ( SpecRel d ) = { Q : Q is an ordered field } ,", "content": "see Remark 3.3. Therefore, SpecRel has a model over Q (the field of rational numbers), too. However, if we assume that inertial observes can move with arbitrary speed less than that of light, see AxThExp on p.6, then every positive number has to have a square root if d ≥ 3 by Theorem 3.2, i.e., Num ( SpecRel d + AxThExp ) = { Q : Q is a Euclidean field } if d ≥ 3 ,", "pages": [ 2 ] }, { "title": "Q /negationslash∈ Num ( SpecRel d + AxThExp ) if d ≥ 3 .", "content": "see [4]. In particular, the number structure cannot be the field of rational numbers if AxThExp is assumed and d ≥ 3, i.e., Theorem 3.4, the main result of this paper, shows that our axiom system SpecRel has a model over Q (in any dimension) if we assume axiom AxThExp only approximately, i.e.,", "pages": [ 2 ] }, { "title": "Q ∈ Num ( SpecRel d + AxThExp -) if d ≥ 2 ,", "content": "see the precise formulation of AxThExp -on p.6. Assuming AxThExp -instead of AxThExp is reasonable because we cannot be sure in anything perfectly accurately in physics. Theorem 3.4 implies that SpecRel + AxThExp can be modeled over every subfield of the field of real numbers ( R ), see Corollary 3.5; and we conjecture that this axiom system has a model over every ordered field, see Conjecture 3.6. An interesting and related approach of Mike Stannett introduces two structures one for the measurable numbers and one for the theoretical numbers and assumes that the set of measurable numbers is dense in the set of theoretical numbers, see [16]. We chose first-order predicate logic to formulate our axioms because experience (e.g., in geometry and set theory) shows that this logic is the best logic for providing an axiomatic foundation for a theory. A further reason for choosing first-order logic is that it is a well defined fragment of natural language with an unambiguous syntax and semantics, which do not depend on set theory. For further reasons, see, e.g., [1, § Why FOL?], [5], [17, § 11], [19], [20].", "pages": [ 2, 3 ] }, { "title": "2. The language of our theories", "content": "To our investigation, we need an axiomatic theory of special relativity. Therefore, we will recall our axiom system SpecRel d in Section 3. To write up any axiom system, we have to choose the set of basic symbols of its language, i.e., what objects and relations between them will be used as basic concepts. Here we will use the following two-sorted 1 language of first-order logic (FOL) parametrized by a natural number d ≥ 2 representing the dimension of spacetime: where B (bodies) and Q (quantities) are the two sorts, IOb (inertial observers) and Ph (light signals) are one-place relation symbols of sort B , + and · are two-place function symbols of sort Q , ≤ is a two-place relation symbol of sort Q , and W (the worldview relation) is a d +2place relation symbol the first two arguments of which are of sort B and the rest are of sort Q . Relations IOb ( m ) and Ph ( p ) are translated as ' m is an inertial observer ,' and ' p is a light signal ,' respectively. To speak about coordinatization of observers, we translate relation W ( k, b, x 1 , x 2 , . . . , x d ) as ' body k coordinatizes body b at space-time location 〈 x 1 , x 2 , . . . , x d 〉 ,' (i.e., at space location 〈 x 2 , . . . , x d 〉 and instant x 1 ). To make them easier to read, we omit the outermost universal quantifiers from the formalizations of our axioms, i.e., all the free variables are universally quantified. Quantity terms are the variables of sort Q and what can be built from them by using the two-place operations + and · , body terms are only the variables of sort B . IOb ( m ), Ph ( p ), W ( m,b, x 1 , . . . , x d ), x = y , and x ≤ y where m , p , b , x , y , x 1 , . . . , x d are arbitrary terms of the respective sorts are so-called atomic formulas of our first-order logic language. The formulas are built up from these atomic formulas by using the logical connectives not ( ¬ ), and ( ∧ ), or ( ∨ ), implies ( → ), if-and-only-if ( ↔ ) and the quantifiers exists ( ∃ ) and for all ( ∀ ). We use the notation Q n for the set of all n -tuples of elements of Q . If ¯ x ∈ Q n , we assume that ¯ x = 〈 x 1 , . . . , x n 〉 , i.e., x i denotes the i -th component of the n -tuple ¯ x . Specially, we write W ( m,b, ¯ x ) in place of W ( m,b, x 1 , . . . , x d ), and we write ∀ ¯ x in place of ∀ x 1 . . . ∀ x d , etc. We use first-order logic set theory as a meta theory to speak about model theoretical terms, such as models, validity, etc. The models of this language are of the form where B and Q are nonempty sets, IOb M and Ph M are subsets of B , + M and · M are binary functions and ≤ M is a binary relation on Q , and W M is a subset of B × B × Q d . Formulas are interpreted in M in the usual way. For the precise definition of the syntax and semantics of first-order logic, see, e.g., [7, § 1.3], [8, § 2.1, § 2.2].", "pages": [ 3, 4 ] }, { "title": "3. Axioms for special relativity", "content": "Now having our language fixed, we can recall axiom system SpecRel d , as well as two theorems on SpecRel d related to our investigation. The key axiom of special relativity states that the speed of light is the same in every direction for every inertial observers. AxPh : For any inertial observer, the speed of light is the same everywhere and in every direction (and it is finite). Furthermore, it is possible to send out a light signal in any direction (existing according to the coordinate system) everywhere: where space 2 ( ¯ x , ¯ y ) := ( x 2 -y 2 ) 2 + . . . + ( x d -y d ) 2 and time ( ¯ y , ¯ y ) := x 1 -y 1 . To get back the intended meaning of axiom AxPh (or even to be able to define subtraction from addition), we have to assume some properties of numbers. In our next axiom, we state some basic properties of addition, multiplication and ordering true for real numbers. AxOField : The quantity part 〈 Q , + , · , ≤〉 is an ordered field, i.e., Using axiom AxOFiled instead of assuming that the structure of quantities is the field of real numbers not just makes our theory more flexible, but also makes it possible to meaningfully investigate our main question. Another reason for using AxOField instead of R is that we cannot experimentally verify whether the structure of physical quantities are isomorphic to R . Hence the assumption that the structure of quantities is R cannot be empirically supported. The two properties of real numbers which are the most difficult to defend from empirical point of view are the Archimedean property, see [11], [12, § 3.1],[13], [14], and the supremum property. 3 We also have to support AxPh with the assumption that all observers coordinatize the same 'external' reality (the same set of events). By the event occurring for observer m at point ¯ x , we mean the set of bodies m coordinatizes at ¯ x : AxEv : All inertial observers coordinatize the same set of events: These three axioms are enough to capture the essence of special relativity. However, let us assume two more simplifying axioms. From now on, we will use ev m ( ¯ x ) = ev k ( ¯ y ) to abbreviate the subformula ∀ b [ W ( m,b, ¯ x ) ↔ W ( k, b, ¯ y )] of AxEv . AxSelf : Any inertial observer is stationary relative to himself: Our last axiom on inertial observers is a symmetry axiom saying that they use the same units of measurement. AxSymD : Any two inertial observers agree as to the spatial distance between two events if these two events are simultaneous for both of them; furthermore, the speed of light is 1 for all observers: Let us introduce an axiom system for special relativity as the collection of the five simple axioms above:", "pages": [ 4, 5 ] }, { "title": "SpecRel d := AxPh + AxOField + AxEv + AxSelf + AxSymD .", "content": "To show that the five simple axioms of SpecRel d capture special relativity well, let us introduce the concept of worldview transformation between observers m and k (in symbols, w mk ) as the binary relation on Q d connecting the coordinate points where m and k coordinatize the same events: Map P : Q d → Q d is called a Poincar'e transformation iff it is an affine bijection having the following property for all ¯ x , ¯ y , ¯ x ' , ¯ y ' ∈ Q d for which P ( ¯ x ) = ¯ x ' and P ( ¯ y ) = ¯ y ' . Theorem 3.1 shows that our axiom system SpecRel d captures the kinematics of special relativity since it implies that the worldview transformations between inertial observers are Poincar'e transformations. Theorem 3.1. Let d ≥ 3. Assume SpecRel d . Then w mk is a Poincar'e transformation if m and k are inertial observers. For the proof of Theorem 3.1, see [4]. For a similar result over Euclidean fields, see, e.g., [2, Thms. 1.4 & 1.2], [3, Thm. 11.10], [17, Thm.3.1.4]. Let us now introduce a further auxiliary axiom about the possibility of motion of inertial observers. AxThExp : Inertial observers can move along any straight line with any speed less than the speed of light: Theorem 3.2 below shows that axiom AxThExp implies that positive numbers have square roots if SpecRel d is assumed. Num ( SpecRel d + AxThExp ) = { Q : Q is a Euclidean field } . Remark 3.3. Axiom AxThExp cannot be omitted from Theorem 3.2 since SpecRel d has a model over every ordered field, i.e., for all d ≥ 2, for all d ≥ 2. Moreover, SpecRel d also has non trivial models in which there are several observers moving relative to each other. We conjecture that there is a model of SpecRel d over every ordered field such that the possible speeds of observers are dense in interval [0 , 1], see Conjecture 3.6 on p.7. Since our measurements have only finite accuracy, it is natural to assume AxThExp only approximately. AxThExp -: Inertial observers can move roughly with any speed less than the speed of light roughly in any direction: By Theorem 3.4, a model of SpecRel d + AxThExp -has a model over the field of rational numbers in any dimension. We use the notation Q ∈ Num ( Th ) for algebraic structure Q the same way as the model theoretic notation Q ∈ Mod ( AxField ), e.g., Q ∈ Num ( Th ) means that Q , the field of rational numbers, can be the structure of quantities in theory Th . Theorem 3.4. For all d ≥ 2, For the proof of Theorem 3.4, see Section4. An ordered field is called Archimedean field iff for all a , there is a natural number n such that ︸ ︷︷ ︸ holds. By Pickert-Hion Theorem, every Archimedean field is isomorphic to a subfield of the field of real numbers, see, e.g., [9, § VIII], [10, C.44.2]. Consequently, the field of rational numbers is dense in any Archimedean field since it is dense in the field of real numbers. Therefore, the following is a corollary of Theorem 3.4.", "pages": [ 5, 6, 7 ] }, { "title": "Corollary 3.5. For all d ≥ 2,", "content": "{ Q : Q is an Archimedean field } /subsetornotdbleql Num ( SpecRel d + AxThExp -) . The question 'exactly which ordered fields can be the quantity structures of theory SpecRel d + AxThExp -?' is open. By LovenheimSkolem Theorem it is clear that Num ( SpecRel d + AxThExp -) cannot be the class of Archimedean fields since it has elements of arbitrarily large cardinality while an Archimedean field has at most the cardinality of continuum since Archimedean fields are subsets of the field of real numbers by Pickert-Hion Theorem. We conjecture that there is a model of SpecRel d + AxThExp -over every ordered field in any dimension, i.e.: Conjecture 3.6 . For all d ≥ 2, Num ( SpecRel d + AxThExp -) = { Q : Q is an ordered field } .", "pages": [ 7 ] }, { "title": "4. Proof of Theorem 3.4", "content": "In this section, we are going to prove our main result. To do so, let us recall some concepts and theorems from the literature. The following theorem is well-known, see, e.g., [15, Thm.2.1]. Theorem 4.1. The unit sphere of R n has a dense set of points with rational coordinates. The Euclidean length of ¯ x ∈ Q n if n ≥ 1 is defined as: Let us recall that the norm of linear map A : R d → R d , in symbols || A || , is defined as follows: Linear bijection A is called orthogonal transformation if it preserves the Euclidean distance. Theorem 4.1 implies Theorem 4.2, see [15, Thm.3.1]. Theorem 4.2. For all orthogonal transformation T : R n → R n and any ε > 0, there is a orthogonal transformaion A : Q n → Q n such that || T -A || < ε . Using Theorem 4.2, let us prove that its statement also holds for Poincar'e transformations. Theorem 4.3. For every Poincar'e transformation L : R d → R d and positive real number ε , there is a Poincar'e transformation L ∗ : Q d → Q d such that || L -L ∗ || < ε . We are going to prove Theorem 4.3 by using the fact that every Poincar'e transformation is a composition of a Lorentz boost and two orthogonal transformations. Lorentz boost corresponding to velocity v ∈ [0 , 1), in symbols B v , is defined as the following linear map: Lemma 4.4. For all Lorentz boost B v : R d → R d and positive number ε , there is a Lorentz boost B w : Q d → Q d such that || B v -B w || < ε . Proof. Since, by Theorem 4.1, the set of rational points are dense in the unit circle, we have that, for all δ > 0 and v ∈ [0 , 1), there is a w ∈ Q ∪ [0 , 1) such that | v -w | < δ and √ 1 -w 2 ∈ Q , i.e., B w takes rational point to rational ones. So we have to show that || B v -B w || < ε if δ is small enough. Since in a finite-dimensional vector space all norms are equivalent, see [6, § 8.5], it is enough to show that the norm of B v -B w can be less than any positive real number according to the Euclidean norm, which is ∣ ∣ ∣ ∣ By the continuity of functions v ↦→ (1 -v 2 ) -1 2 and v ↦→ v (1 -v 2 ) -1 2 , the Euclidean norm of B v -B w is less than any fixed positive real number if | v -w | is small enough. Therefore, there is a Lorentz boost B w such that B w maps rational points to rational ones and || B w -B v || < ε . /squaresolid Lemma 4.5. Let A and B be linear bijections of R d . Let A ' and B ' linear maps such that || A -A ' || < ε 1 and || B -B ' || < ε 2 . Then || BA -B ' A ' || ≤ ε 1 || B || + ε 1 ε 2 + ε 2 || A || . Proof. First let us note that || A ' || = || A ' -A + A || ≤ || A ' -A || + || A || = ε 1 + || A || (12) by the triangle inequality. Let ¯ x ∈ R d such that | ¯ x | = 1. We have to show that By the triangle inequality and the fact that | M ¯ y | ≤ || M || · | ¯ y | , we have /squaresolid and this is what we wanted to prove. Proof of Theorem 4.3. Every Poincar'e transformation is a composition of a translation, a Lorentz-boost B v and an orthogonal transformation. Therefore, Lemmas 4.4 and 4.5, together with Theorem 4.2 imply our statement. /squaresolid Now we are going to prove Theorem 3.4. Let Id be the identity map of Q d . We denote the origin of Q n by ¯ o , i.e., Let the time-axis be defined as the following subset of Q d : Let H be a subset of Q d and let f : Q d → Q d be a map. The f -image of set H is defined as: The so-called worldline of body b according to observer m is defined as follows: Proof. We are going to construct a model of SpecRel d + AxThExp -over Q . So let 〈 Q , + , · , ≤〉 be the ordered field of rational numbers. Let IOb := { m : m is a Poincar'e transformation from Q d to Q d } , (20) and let B = IOb ∪ Ph . First we are going to give the worldview of observer Id . Let for any other inertial observer m , let and for any light signal p ∈ Ph , let Now the worldview of observer Id is given. From the worldview of Id , we construct the worldview of another inertial observer m as follows: Now we have given the model. Let us see why the axioms of SpecRel d and AxThExp -are valid in it. for all body b ∈ B , see Figure 1. By the above definition of W , if m and k are inertial observers, then and if m ∈ IOb and p ∈ Ph , then The worldview transformations between inertial observers m and Id is m , i.e., w m Id = m by equation (24). Therefore, the worldview transformation between inertial observers m and k is k -1 · m , i.e., since w mk = w Id k · w m Id and w Id k = ( w k Id ) -1 by the definition of the worldview transformation (5). Specially, the worldview transformations between inertial observers are Poincar'e transformations in these models (as Theorem 3.1 requires it). Hence Axiom AxPh is valid for observer Id by the definition of Ph and that of his worldview. It is also clear that the speed of light is 1 for observer Id . Axiom AxPh is valid for the other observers since Poincar'e transformations take lines of slope one to lines of slope one. This also show that the speed of light is 1 according to every inertial observer, which is the second half of AxSymD . Axiom AxOField is valid in this model since Q is an ordered field. Axiom AxEv is valid in this model since Poincar'e transformations are bijections. Axiom AxSelf is valid in this model since Any Poincar'e transformation P preserves the spatial distance of points ¯ x , ¯ y for which x 1 = y 1 and P ( ¯ x ) 1 = P ( ¯ y ) 1 . Therefore, inertial observers agree as to the spatial distance between two events if these two events are simultaneous for both of them. We have already shown that the speed of light is 1 for each inertial observers in this model. Hence axiom AxSymD is also valid in this model. Now we are going to show that AxThExp -is valid in this model. The ∃ h IOb ( h ) part of axiom AxThExp is valid, since there are Poincar'e transformations (e.g., Id is one). To show that the rest of axiom AxThExp -is valid, let m be an inertial observer and let us fix an ε > 0 and a ¯ v ∈ Q d for which v 2 2 + . . . v 2 d < 1 and v 1 = 1. Let ¯ 1 be vector 〈 1 , 0 , . . . , 0 〉 . Let L be a Lorentz transformation (i.e., linear Poincar'e transformation) for which Let 0 < δ < 1 be such that ∣ ∣ for any x for which | L ( ¯ 1 ) 1 -x | < δ . By Theorem 4.3, there is a Lorentz transformation L ∗ which takes rational points to rational ones and || L -L ∗ || < δ . Then since | ¯ 1 | = 1. We have | L ( ¯ 1 ) 1 -L ∗ ( ¯ 1 ) 1 | < δ since | x 1 | < | ¯ x | for all ¯ x ∈ Q d . By triangle inequality, we also have Let By triangle inequality, we have Let ¯ x , ¯ y ∈ Q d such that there is a λ ∈ Q such that ¯ y -¯ x = λ ¯ w . To finish the proof of AxThExp -, we have to show that there is an inertial observer k such that W ( m,k, ¯ x ) and W ( m,k, ¯ y ), i.e., ¯ x , ¯ y ∈ wl m ( k ). Let P ∗ = L ∗ + ¯ x . P ∗ is a Poincar'e transformation taking rational points to rational ones. Therefore, there is an inertial observer k such that w km = P ∗ . Since wl m ( k ) = w km [ t -axis ], we have that w km ( ¯ o ) = ¯ x ∈ wl m ( k ) and that ¯ y = aL ∗ ( ¯ 1 ) + ¯ x = w km ( a ¯ 1 ) ∈ wl m ( k ), where a = λ/L ∗ ( ¯ 1 ) 1 . This shows that AxThExp -is also valid in our model. /squaresolid", "pages": [ 8, 9, 10, 11, 12 ] }, { "title": "5. Acknowledgments", "content": "This research is supported by the Hungarian Scientific Research Fund for basic research grants No. T81188 and No. PD84093, as well as by a Bolyai grant for J. X. Madar'asz.", "pages": [ 12 ] }, { "title": "References", "content": "Alfr'ed R'enyi Institute of Mathematics, Hungarian Academy of Sciences, Re'altanoda utca 13-15, H-1053, Budapest, Hungary E-mail address : { madarasz.judit, szekely.gergely } @renyi.mta.hu", "pages": [ 13 ] } ]

[ { "title": "Stability of Non-asymptotically flat thin-shell wormholes in generalized dilaton-axion gravity", "content": "Ayan Banerjee ∗ , Farook Rahaman † , Surajit Chattopadhyay ‡ , Sumita Banerjee § October 31, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "We construct a new type of thin-shell wormhole for non-asymptotically flat charged black holes in generalized dilaton-axion gravity inspired by low-energy string theory using cut-and-paste technique. We have shown that this thin shell wormhole is stable. The most striking feature of our model is that the total amount of exotic matter needed to support the wormhole can be reduced as desired with the suitable choice of the value of a parameter. Various other aspects of thin-shell wormhole are also analyzed.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The study of traversable wormholes and thin-shell wormholes are very interesting subject in recent years as it is known that wormholes represent shortcut in space-time, though the observational evidence of wormholes has not been possible still now. At first, Morris and Thorne [1] elaborated the structure of traversable wormhole with two mouths and a throat. The traversable wormhole was obtained as a solution of Einstein's equations having two asymptotically flat regions. These flat regions are connected by a minimal surface area, known as the throat satisfying the flare-out condition [2]. This type of wormhole would allow the travel between two parts of the same universe or between two different universes. Morris and Thorne [1] had the idea to make the conversion of a wormhole traversing space into a wormhole traversing time. In the Morris and Thorne model of wormhole the violation of positive energy condition is unavoidable. The throat of the wormhole is held open with exotic matter. Since it is very difficult to deal with the exotic matter (violation of positive energy condition) Visser [3], proposed a way to minimize the usage of exotic matter by applying cut-and-paste technique on a black hole to construct a new class of spherically symmetric wormholes, known as thin-shell wormhole in which the exotic matter is concentrated within the joining shell. This shell can act as the wormhole throat. Using the Darmois-Israel formalism[4], the surface stress-energy tensor components within the shell i.e. at the throat could be determined. Visser's approach is very simple for theoretical construction of wormhole and perhaps also practical because it minimizes the amount of exotic matter required and therefore this approach was adopted by various authors to construct thin shell wormholes [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 16, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. Recently, Mazharimousavi et al [32] have studied d-dimensional non-asymptotically flat thin-shell wormholes in Einstein-Yang-Mills-Dilaton gravity where as Rahaman et al [33] have obtained a new class of stable (2+1) dimensional non-asymptotically flat thin-shell wormhole. In this investigation, we are searching stable non-asymptotically flat thin-shell wormholes in generalized dilaton-axion gravity. In 2005, Sur, Das and SenGupta[34] discovered new black hole solutions considering gravity coupled with dilaton ( ϕ ), Kalb-Ramond axion ( ζ ) scalar fields and a Maxwell field with arbitrarily coupled to the scalars. They considered the generalized action in four dimensions as: where κ = 8 πG , R is the curvature scalar and F µν is the Maxwell field tensor where two massless scalar or pseudo scalar fields are represented by ϕ and ζ which are coupled to the Maxwell field with the functional dependence by the functions α ( ϕ, ζ )and β ( ϕ, ζ ). Using the above action, Sur,Das and SenGupta[34] have found two types of black-hole solutions, classified as asymptotically flat and asymptotically non-flat. For our thin-shell wormhole solution, we have considered non asymptotically flat metric given by where and The parameter n is a dimensionless constant lies in the range 0 < n < 1. The various parameters for non asymptotically flat metric are given by where m is the mass of the black hole and the inner, outer event horizons and curvature singularity represented by the parameters r + , r -and r = r 0 respectively. The parameters obey the restriction r 0 < r -< r + . Employing the above black hole solution, we have constructed a new thin-shell wormhole. We have shown that this thin shell wormhole is stable. The most striking feature of our model is that the total amount of exotic matter needed to support the wormhole can be reduced as desired by choosing the parameter n very close to unity. Various other aspects of thin-shell wormhole have also been analyzed. We have organized the work as follows: In section 2, we have provided mathematical formulation for constructing a thin-shell wormhole from charged black hole in generalized dilaton-axion gravity. In section 3, we have discussed the effect of gravitational field on wormhole i.e. attractive or repulsive nature of the wormhole. In section 4, we have discussed about the equation of state relating pressure and density at the wormhole throat and in section 5, we have calculated total amount of exotic matter. Linearized stability analysis has been discussed in section 6. Finally, we have made a conclusion about the work.", "pages": [ 1, 2, 3 ] }, { "title": "2 Construction of thin-shell wormhole", "content": "For the construction of thin-shell wormhole, at first, we remove two identical copies from four-dimension spacetime of the black hole given in equation (2) region with radius r ≤ a . Here we have assumed a > r + to avoid all types of singularity. Thus we have two copies of regions: Now, we stick them together at the hypersurface, Σ = Σ ± , to get a geodesically complete manifold M = M + ⋃ M -. Here, two regions are connected by minimal surface area, called throat with radius a. The induced 3-metric on the hypersurface is given by where τ ,represents the proper time on the junction surface. To obtain the surface stressenergy tensor S i j = diag ( -σ, P, P ), we use the Lancozos equations [4], which reduce to and √ where σ and P represents the surface energy density and surface pressures respectively. To understand the dynamics of the wormhole ,we consider that the throat as function of proper time i.e. a = a ( τ ) and over dot and prime denote the derivatives with respect to τ and a. For a static configuration of radius a (assuming ˙ a = 0 and a = 0), we obtain the values of the surface energy density and the surface pressures as where From Eqs.(15) and (16), we see that the energy density σ is negative, however the pressure p is positive depending on the position of the throat and on the parameters r 0 , r -and r + defining on wormhole. Here, matter distribution within the shell of the wormhole violates the weak energy condition. We plots σ and P versus a in Figs.(1-2) for such wormholes whose radii fall within the range of 6 to 16 km, keeping the restriction on the parameters r + > r -> r 0 . The sensitivity are given with respect to n, for both the figures σ and P as described in the caption of the figures. Note that the weak energy condition requires that σ > 0 and σ + P > 0. These state that the energy density is positive and the pressure is not too large compared to the energy density. The null energy condition σ + P > 0 is a special case of the latter and implies that energy density can be negative if there is a compensating positive pressure. The figure 3 indicates in the present case that the null energy condition is satisfied. σ a a a", "pages": [ 3, 4, 5, 6 ] }, { "title": "3 The gravitational field", "content": "In this section we have analyzed the nature of the gravitational field of the wormhole constructed from charged black holes in generalized dilaton-axion gravity and calculate the observer's four-acceleration A µ = u µ ; ν u ν , where u ν = dx ν /dτ = (1 / √ f ( r ) , 0 , 0 , 0). Only non-zero component for the line element in Eq.(1), is given by The equation of motion of a test particle is given by when it is radially moving and initially at rest. For A r = 0, we get the geodesic equation. From the figure 4, we note that the wormhole is attractive as A r > 0 out side the event horizon.", "pages": [ 6 ] }, { "title": "4 An equation of state", "content": "To know the exact nature of the matter distribution in the shell, we calculate equation of state (EoS) which is given by The plot for w (fig 5) indicates that matter distribution in the shell is phantom energy type. For a →∞ i.e. if the wormhole throat is very very large, then w →-1 2(1 -n ) . In that case, EoS solely depends on the parameter n only. One can note that for the following range of n, 1 2 < n < 3 2 , w lies on -1 < w < -1 3 . This indicates the mater distribution is quentessence like. However, for n > 3 2 , w < -1 i.e. the matter distribution is phantom energy type as w < -1.", "pages": [ 7 ] }, { "title": "5 The total amount of exotic matter", "content": "We observe that matter distribution at throat is phantom energy type i.e. matter at throat is exotic. Here, we evaluate the total amount of exotic matter which can be quantified by the integral [10, 11, 12, 13, 14, 15, 17] where g represents the determinant of the metric tensor. Now, by using the radial coordinate R = r -a , we have For the infinitely thin shell it does not exert any radial pressure and using ρ = δ ( R ) σ ( a ) we have, where K is given in Eq.(17). It is to be noted that the total amount of exotic matter needed can be reduced as small as desired by choosing the parameter n very close to unity (see figure 6). Also, the exotic matter can be minimized by taking the value of a very close to the location of the outer event horizon (see figure 7).", "pages": [ 7, 8 ] }, { "title": "6 Linearizing method", "content": "Now, we are trying to find the local stability of the configuration under small perturbation around the static solution at a = a 0 . So, rearranging the Eq.(13) we can write n a which represents the equation of motion of the shell, where the potential V ( a ) is define as Using the Taylor series expansion for V ( a ) around the static solution a 0 , we get where the prime denotes the derivative with respect to a. For our stability analysis, we start with the energy conservation equation. Now using Eqs.(15) and (16), preserving the symmetry of linearizing radial perturbations, the energy conservation equation defined as where A = 4 πh ( a ) denotes the area of the wormhole throat. From Eq.(25), one can get which finally gives The second order derivative for the potential can be defined as Now, using the parameter β 2 = dP dσ , which is normally interpreted as the speed of sound, the above expression can be written as The solution gives a stable configuration if V ( a ) process a local minima at a 0 , in other words, V '' ( a 0 ) > 0. Now using the condition V ( a 0 ) = 0 and V ' ( a 0 ) = 0, we solve for β 2 as Now, we are trying find the stable region with the help of graphical representation due to complexity of the expression of V '' ( a 0 ) > 0. In Figs. 8-11, we find the possible range of a 0 , where V ( a 0 ) = 0 possess a local minima. We plot the figures 8 and 9 for the parameters r 0 = 2, r -= 3 and r + = 4 and different values of n=0.5 and 0.7, which show that graphs don't possess any stable configuration as β 2 lies outside (0,1). However, if we choose the the parameters r 0 = 2, r -= 1, r + = 6 and n=0.5, respectively, then the stable region is more closer to the normal range of β 2 . This indicates that the increase of the difference between inner and outer event horizons, stable region may fall to the normal range of β 2 . Note that this criteria holds when we are dealing with real matter. However, according Poisson and Visser [5] the interpretation of β 2 should be relaxed when dealing with exotic matter. As a result, the values of β 2 may fall out side (0,1). Since the stability criteria is V '' ( a 0 ) > 0 and our figures 8-11 indicate the stable regions where V '' ( a 0 ) > 0, therefore, our thin shell wormholes are stable.", "pages": [ 8, 9, 10, 11 ] }, { "title": "7 Conclusions", "content": "In this work, we have provided a new type of thin-shell wormhole applying the cut-andpaste technique on charged black hole in generalized dilaton-axion gravity. The metric which we have used is not asymptotically flat, therefore, the the wormholes are not asymptotically flat. The matter confined within the shell of the wormhole violates the weak energy condition. The matter distribution in the shell is of phantom energy type. However, when the wormhole throat is very very large, then for the following range of n, 1 4 < n < 3 4 , mater distribution is quentessence like. It is interesting to note that the total amount of exotic matter needed can be reduced as small as desired by choosing the parameter n very close to unity. Finally, to show the stability, we have performed linearized stability analysis around the static solution. Since the matter distribution within the shell is exotic type, therefore, according to Poisson and Visser [5], we can relax the range of β 2 . Here the stability regions are shown graphically.", "pages": [ 13 ] }, { "title": "Acknowledgments", "content": "FR is thankful to IUCAA, Pune, India for providing research facility. FR is also grateful UGC for providing financial support.", "pages": [ 13 ] } ]

[ { "title": "John G. Hartnett", "content": "School of Physics, the University of Western Australia, 35 Stirling Hwy, Crawley 6009 WA Australia [email protected] October 17, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "The solution of Einstein's field equations in Cosmological General Relativity (CGR), where the Galaxy is at the center of a finite yet bounded spherically symmetrical isotropic gravitational field, is identical with the unbounded solution. This leads to the conclusion that the Universe may be viewed as a finite expanding white hole. The fact that CGR has been successful in describing the distance modulus verses redshift data of the high-redshift type Ia supernovae means that the data cannot distinguish between unbounded models and those with finite bounded radii of at least cτ . Also it is shown that the Universe is spatially flat at the current epoch and has been at all past epochs where it was matter dominated. Keywords: Cosmological General Relativity, high redshift type Ia supernovae, dark matter", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "In an interview with Scientific American George Ellis once said [9] 'People need to be aware that there is a range of models that could explain the observations, . . . For instance, I can construct you a spherically symmetrical universe with Earth at its center, and you cannot disprove it based on observations. . . . You can only exclude it on philosophical grounds. In my view there is absolutely nothing wrong in that. What I want to bring into the open is the fact that we are using philosophical criteria in choosing our models. A lot of cosmology tries to hide that.' This paper proposes a model where the Galaxy is at the center of a spherically symmetrical finite bounded universe. It contends that fits to the magnituderedshift data of the highz type Ia supernovae (SNe Ia) [12, 13, 1], are also consistent with this model. That is, providing that the radius of the Universe (a spherically symmetrical matter distribution) is at least cτ where c is the speed of light and τ ≈ 4 . 28 × 10 17 s (or 13 . 54 Gyr ).[8] Here τ is the Hubble-Carmeli time constant, or the inverse of the Hubble constant evaluated in the limits of zero gravity and zero distance. This model is based on the Cosmological General Relativity (CGR) theory [5] but explores the motion of particles in a central potential. In this case the central potential is the result of the Galaxy being situated at the center of a spherically symmetrical isotropic distribution comprising all matter in the Universe. This paper is preceded by Hartnett [10] that forms the basis of the work presented here. Also Oliveira and Hartnett [8] progressed the work by developing a density function for higher redshifts. Those paper assumed the unbounded model. The reader should be familiar with Hartnett [10] at least before reading this.", "pages": [ 1, 2 ] }, { "title": "1.1 Cosmological General Relativity", "content": "The metric [2, 3, 5] used by Carmeli (in CGR) in a generally covariant theory extends the number of dimensions of the Universe by the addition of a new dimension - the radial velocity of the galaxies in the Hubble flow. The Hubble law is assumed as a fundamental axiom for the Universe and the galaxies are distributed accordingly. The underlying mechanism is that the substance of which space is built, the vacuum, is uniformly expanding in all directions and galaxies, as tracers, are fixed to space and therefore the redshifts of distant first ranked galaxies quantify the speed of the expansion. In determining the large scale structure of the Universe the usual time dimension is neglected ( dt = 0) as observations are taken over such a short time period compared to the motion of the galaxies in the expansion. It is like taking a still snap shot of the Universe and therefore only four co-ordinates x µ = ( x 1 , x 2 , x 3 , x 4 ) = ( r, θ, φ, τv ) are used - three of space and one of velocity. The parameter τ , the Hubble-Carmeli constant, is a universal constant for all observers. Here the CGR theory is considered using a Riemannian four-dimensional presentation of gravitation in which the coordinates are those of Hubble, i.e. distance and velocity. This results in a phase space equation where the observables are redshift and distance. The latter may be determined from the high-redshift type Ia supernova observations.", "pages": [ 2 ] }, { "title": "1.2 Phase space equation", "content": "The line element in CGR [6] represents a spherically symmetrical isotropic universe, that is not necessarily homogeneous. It is fundamental to the theory that ds = 0. In the case of Cosmological Special Relativity (see chap.2 of [5]), which is very useful pedagogically, we can write the line element as ignoring θ and φ co-ordinates for the moment. By equating ds = 0 it follows from (2) that τdv = dr assuming the positive sign for an expanding universe. This is then the Hubble law in the small v limit. Hence, in general, this theory requires that ds = 0. Using spherical coordinates ( r, θ, φ ) and the isotropy condition dθ = dφ = 0 in (1) then dr represents the radial co-ordinate distance to the source and it follows from (1) that where ξ is a function of v and r alone. This results in where the positive sign has been chosen for an expanding universe.", "pages": [ 2, 3 ] }, { "title": "2 Solution in central potential", "content": "Carmeli found a solution to his field equations, modified from Einstein's, (see [10] and [2, 5, 6]) which is of the form with R ' = 1, which must be positive. From the field equations and (5) we get a differential equation where f ( r ) is function of r and satisfies the condition f ( r )+1 > 0. The prime is the derivative with respect to r . Here κ = 8 πG/c 2 τ 2 and ρ eff = ρ -ρ c where ρ is the averaged matter density of the Universe and ρ c = 3 / 8 πGτ 2 is the critical density. The solution of (6), f ( r ), is the sum of the solution (2 GM/c 2 r ) to the homogeneous equation and a particular solution (κ 3 τ 2 ρ eff r 2 ) to the inhomogeneous equation. In [5] Carmeli discarded the homogeneous solution saying it was not relevant to the Universe, but the solution of a particle at the origin of coordinates, or in other words, in a central potential. Now suppose we model the Universe as a ball of dust of radius ∆ with us, the observer, at the center of that ball. In this case the gravitational potential written in spherical coordinates that satisfies Poisson's equation in the Newtonian approximation is for the vacuum solution, but inside an isotropic matter distribution where it is assumed the matter density ρ is uniform throughout the Universe. At the origin ( r = 0) Φ(0) = -2 Gπρ m ∆ 2 , where ρ = ρ m the matter density at the present epoch. In general ρ depends on epoch. Because we are considering no time development ρ is only a function of redshift z and ρ m can be considered constant. From (8) it is clear to see that by considering a finite distribution of matter of radial extent ∆, it has the effect of adding a constant to f ( r ) that is consistent with the constant 2 Gπρ ∆ 2 in (8), where f ( r ) is now identified with -4Φ /c 2 . Equation (5) is essentially Carmeli's equation A.19, the solution to his equation A.17 from p.122 of [5]. More generally (5) can be written as where K is a constant. This is the most general form of the solution of Carmeli's equation A.17. So by substituting (9) into Carmeli's A.18, A.21 becomes instead Therefore (9) is also a valid solution of the Einstein field equations (A.12 A.18 [5]) in this model. Making the assignment R = r in (10) yields a more general version of (6), that is, The solution of (11) is then From a comparison with (8) it would seem that the constant K takes the form K = 8 πGρ eff (0)∆ 2 /c 2 . It is independent of r and describes a non-zero gravitational potential of a finite universe measured at the origin of coordinates. There is some ambiguity however as to which density to use in Carmelian cosmology since it is not the same as Newtonian theory. Here ρ eff is used and evaluated at r = 0. In the above Carmelian theory it initially assumed that the Universe has expanded over time and at any given epoch it has an averaged density ρ , and hence ρ eff . The solution of the field equations has been sought on this basis. However because the Carmeli metric is solved in an instant of time (on a cosmological scale) any time dependence is neglected. In fact, the general time dependent solution has not yet been found. But since we observe the expanding Universe with the coordinates of Hubble at each epoch (or redshift z ) we see the Universe with a different density ρ ( z ) and an effective density ρ eff ( z ). Carmeli arrived at his solution with the constant density assumption. I have made the implicit assumption that the solution is also valid if we allow the density to vary as a function of redshift, as is expected with expansion. Now it follows from (4), (9) and (12) that where Ω = ρ/ρ c . This compares with the solution when the central potential is neglected (i.e. ∆ → 0). In fact, the result is identical as we would expect in a universe where the Hubble law is universally true. Therefore (13) may be integrated exactly and yields the same result as Carmeli, Since observations in the distant cosmos are always in terms of redshift, z , we write (14) as a function of redshift where r is expressed in units of cτ and v/c = ((1 + z ) 2 -1) / ((1 + z ) 2 +1) from the relativistic Doppler formula. The latter is appropriate since this is a velocity dimension. What is important to note though is that regardless of the geometry of the Universe, provided it is spherically symmetrical and isotropic on the large scale, (14) is identical to that we would get where the Universe has a unique center, with one difference which is explored in the following section. For an isotropic universe without a unique center, one can have an arbitrary number of centers. However if we are currently in a universe where the Galaxy is at the center of the local isotropy distribution this means the Universe we see must be very large and we are currently limited from seeing into an adjacent region with a different isotropy center.", "pages": [ 3, 4, 5 ] }, { "title": "3 Gravitational Redshift", "content": "In Hartnett [10] the geometry in the model is the usual unbounded type, as found in an infinite universe, for example. In a finite bounded universe, an additional effect may result from the photons being received from the distant sources. The gravitational redshift ( z grav ) resulting from the Galaxy sitting at the unique center of a finite spherically symmetrical matter distribution must be considered. In this case we need to consider the difference in gravitational potential between the points of emission and reception of a photon. Now the 00th metric component, the time part of the 5D metric of coordinates x k = t, r, θ, φ, v ( k = 0 -4), is required but it has never been determined for the cosmos in the Carmelian theory. In general relativity we would relate it by g 00 = 1 -4Φ /c 2 where -4Φ is the gravitational potential. The factor 4 and minus sign arise from the Carmelian theory when (12) and (8) are compared. So the question must be answered, 'What is g 00 metric component for the large scale structure of the universe in CGR?' First note from (5) and (6) the g 11 metric component (considered in an unbounded universe for the moment) in CGR we can write a scale radius Hence we can define an energy density from the curvature which, when we use (16), becomes This quantifies the energy in the curved spacevelocity . In the FRW theory the energy density of the cosmological constant is defined ρ Λ = Λ / 8 πG hence Even though the cosmological constant is not explicitly used in CGR, it follows from the definition of the critical density that when the cosmological constant Λ is identified with 3 /τ 2 . Therefore in CGR it follows that This means that in CGR the vacuum energy ρ vac = Λ / 8 πG is encoded in the metric via the critical density since ρ eff = ρ -ρ c principally defines the physics. So Ω Λ = 1 identically and at all epochs of time. (The determination of Ω Λ in [10] was flawed due to an incorrect definition.) Also we can relate Ω Λ to the curvature density by which becomes at the present epoch ( z ≈ 0). Here Ω = Ω m (1 + z ) 3 and hence Ω K → Ω k as z → 0. Finally we can write for the total energy density, the sum of the matter density and the curvature density, which means the present epoch value is trivially This means that the 3D spatial part of the Universe is always flat as it expands. This explains why we live in a universe that we observe to be identically geometrically spatially flat. The curvature is due to the velocity dimension. Only at some past epoch, in a radiation dominated universe, with radiation energy density Ω R (1 + z ) 4 , would the total mass/energy density depart from unity. Now considering a finite bounded universe, from (12), using Ω = ρ/ρ c , I therefore write g 00 as where r and ∆ are expressed in units of cτ . Equation (26) follows from g 00 = 1 -4Φ /c 2 where Φ is taken from the gravitational potential but with effective density, which in turn involves the total energy density because we are now considering spacetime . Clearly from (24) it follows that g 00 ( r ) = 1 regardless of epoch. Thus from the usual relativistic expression and the gravitational redshift is zero regardless of epoch. As expected if the emission and reception of a photon both occur in flat space then we'd expect no gravitational effects. In an unbounded universe, though no gravitational effects need be considered, the result g 00 = 1 is also the same. Therefore we can write down the full 5D line element for CGR in any dynamic spherically symmetrical isotropic universe, The θ and φ coordinates do not appear due to the isotropy condition dθ = dφ = 0. Due to the Hubble law the 2nd and 3rd terms sum to zero leaving dt = ds/c , the proper time. Clocks, co-moving with the galaxies in the Hubble expansion, would measure the same proper time. Since it follows from (26) that g 00 ( r ) = 1 regardless of epoch, g 00 ( r ) is not sensitive to any value of ∆. This means the above analysis is true regardless of whether the universe is bounded or unbounded. The observations cannot distinguish. In an unbounded or bounded universe of any type no gravitational redshift (due to cosmological causes) in light from distant source galaxies would be observed. However inside the Galaxy we expect the matter density to be much higher than critical, ie Ω galaxy /greatermuch 1 and the total mass/energy density can be written because Ω k ≈ 1, since it is cosmologically determined. Therefore this explains why the galaxy matter density only is appropriate when considering the Poisson equation for galaxies.[11] As a result inside a galaxy we can write in terms of densities at some past epoch. Depending on the mass density of the galaxy, or cluster of galaxies, the value of g 00 here changes. As we approach larger and larger structures it mass density approaches that of the Universe as a whole and g 00 → 1 as we approach the largest scales of the Universe. Galaxies in the cosmos then act only as local perturbations but have no effect on Ω K . That depends only on the average mass density of the whole Universe, which depends on epoch ( z ). Equation (30) is in essence the same expression used on page 173 of Carmeli [5] in his gravitational redshift formula rewritten here as involving a cosmological contribution (Ω K r 2 /c 2 τ 2 ) and R S = 2 GM/c 2 , a local contribution where the mass M is that of a compact object. The curvature (Ω K ) results from the averaged mass/energy density of the whole cosmos, which determines how the galaxies 'move' but motions of particles within galaxies is dominated by the mass of the galaxy and the masses of the compact objects within. Therefore when considering the gravitational redshifts due to compact objects we can neglect any cosmological effects, only the usual Schwarzschild radius of the object need be considered. The cosmological contributions in (31) are generally negligible. This then leads back to the realm of general relativity.", "pages": [ 5, 6, 7, 8 ] }, { "title": "4 White Hole", "content": "Now if we assume the radial extent of a finite matter distribution at the current epoch is equal to the current epoch scale radius, we can write expressed in units of cτ . In such a case, ∆ = 1 . 02 cτ if Ω m = 0 . 04 and ∆ = 1 . 01 cτ if Ω m = 0 . 02. It is important to note also that in Carmeli's unbounded model (14) describes the redshift distance relationship but there is no central potential. In Hartnett [10] and in Oliveira and Hartnett [8] equation (14) was curve fitted to the SNe Ia data and was found to agree with Ω m = 0 . 02 -0 . 04 without the inclusion of dark matter or dark energy. Therefore the same conclusion must also apply to the finite bounded model suggested here. In order to achieve a fit to the data, using either the finite bounded or unbounded models, the white hole solution of (6) or (11) must be chosen. The sign of the terms in (12) means that the potential implicit in (12) is a potential hill, not a potential well. Therefore the solution describes an expanding white hole with the observer at the origin of the coordinates, the unique center of the Universe. Only philosophically can this solution be rejected. Using the Carmeli theory, the observational data cannot distinguish between finite bounded models ( ∞ > ∆ ≥ cτ ) and finite (∆ = 0) or infinite (∆ = ∞ ) unbounded models . The physical meaning is that the solution, developed in this paper, represents an expanding white hole centered on the Galaxy. The galaxies in the Universe are spherically symmetrically distributed around the Galaxy. The observed redshifts are the result of cosmological expansion alone. Moreover if we assume ∆ ≈ cτ and Ω m = 0 . 04 then it can be shown [8] that the Schwarzschild radius for the finite Universe Therefore for a finite universe with ∆ ≈ cτ it follows that R s ≈ 0 . 04 cτ ≈ 200 Mpc . Therefore an expanding finite bounded universe can be considered to be a white hole. As it expands the matter enclosed within the Schwarzschild radius gets less and less. The gravitational radius of that matter therefore shrinks towards the Earth at the center. This is similar to the theoretical result obtained by Smoller and Temple [14] who constructed a new cosmology from the FRW metric but with a shock wave causing a time reversal white hole. In their model the total mass behind the shock decreases as the shock wave expands, which is spherically symmetrically centered on the Galaxy. Their paper states in part '...the entropy condition implies that the shock wave must weaken to the point where it settles down to an Oppenheimer Snyder interface, (bounding a finite total mass), that eventually emerges from the white hole event horizon of an ambient Schwarzschild spacetime.' This result then implies that the earth or at least the Galaxy is in fact close to the physical center of the Universe. Smoller and Temple state [15] that 'With a shock wave present, the Copernican Principle is violated in the sense that the earth then has a special position relative to the shock wave. But of course, in these shock wave refinements of the FRW metric, there is a spacetime on the other side of the shock wave, beyond the galaxies, and so the scale of uniformity of the FRW metric, the scale on which the density of the galaxies is uniform, is no longer the largest length scale'[emphasis added]. Their shock wave refinement of a critically expanding FRW metric leads to a big bang universe of finite total mass. This model presented here also has a finite total mass and is a spatially flat universe. It describes a finite bounded white hole that started expanding at some time in the past.", "pages": [ 8, 9, 10 ] }, { "title": "5 Conclusion", "content": "Since the Carmeli theory has been successfully analyzed with distance modulus data derived by the high-z type Ia supernova teams it must also be consistent with a universe that places the Galaxy at the center of an spherically symmetrical isotropic expanding white hole of finite radius. The result describes particles moving in both a central potential and an accelerating spherically expanding universe without the need for the inclusion of dark matter. The data cannot be used to exclude models with finite extensions ∆ ≥ cτ .", "pages": [ 10 ] } ]

[ { "title": "ABSTRACT", "content": "We examine the question of energy localization for an exact solution of Einstein's equations with a scalar field corresponding to the phantom energy interpretation of dark energy. We apply three different energy-momentum complexes, the Einstein, Papapetrou and Møller prescriptions, to the exterior metric and determine the energy distribution for each. Comparing the results, we find that the three prescriptions yield identical energy distributions. Published in ISRN Astronomy and Astrophysics, Vol. 2013. Subject headings: black hole; phantom energy; Schwarzschild solution; dark energy; energy-momentum complex", "pages": [ 2 ] }, { "title": "The Localized Energy Distribution of Dark Energy Star Solutions", "content": "Paul Halpern and Michael Pecorino Department of Mathematics, Physics and Statistics, University of the Sciences in Philadelphia, 600 S. 43rd St., Philadelphia, PA. 19104, USA Received ; accepted", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The question of energy localization has been a pressing issue in general relativity since Einstein formulated his field equations. Einstein sought to include the energy and momenta of gravitational fields along with those of matter and non-gravitational fields as a well-defined locally conserved quantity. To that end, he proposed an energy-momentum complex that follow conservation laws for those quantities (Einstein 1915). One concern about the Einstein energy-momentum is that its value depends on the coordinate system used. Specifically, it favors the use of quasi-Cartesian (perturbations of a flat background) coordinates. Accordingly, it does not transform as a tensor. Because it is asymmetric in its indices, it does not conserve angular momentum. To address these issues, a number of other theorists have proposed alternative definitions for energy-momentum complexes. These include prescriptions by Papapetrou ( 1948), Landau and Lifshitz ( 1962), Weinberg ( 1972) and others, each designed to conserve angular momentum in addition to energy-momentum. A prescription by Møller ( 1958, 1961) offers the special advantage of being coordinate-system independent. While at first the multiplicity of energy-momentum complexes discouraged theorists from using them, important results in the 1990s suggested an underlying unity that served to revive interest. In 1990, Virbhadra found consistency in the application of distinct energymomentum prescriptions to the same metric (Virbhadra 1990). Six years later, a paper by Aguirregabiria, Chamorro and Virbhadra revealed that all metrics of the Kerr-Schild class, including the Schwarzchild, Reissner-Nordstrom and other solutions, yield identical energy and momentum distributions for a variety of prescriptions (Aguirregabiria, et. al. 1996(@). Then in 1999, a paper by Virbhadra indicated the energy-momentum complexes of Einstein, Papapetrou, Weinberg, and Landau and Lifshitz produced similar results for metrics more general than the Kerr-Schild class if they are constructed with Kerr-Schild Cartesian coordinates (Virbhadra 1999). There have subsequently been many attempts to apply the various energy-momentum complexes to black holes and related astronomical objects. In recent years, for example, Radinschi, along with various colleagues, has investigated the energy-momentum distribution for stringy black holes (Radinschi and Yang 2005; Radinschi and Ciobanu 2006), HoˆravaLifshitz black holes (Radinschi, et. al. 2011) , for charged black holes in generalized dilaton-axion gravity (Radinschi, et. al. 2010), for asymptotically de Sitter spacetimes (Radinschi 2011), and in a Schwarzschild-quintessence spacetime (Radinschi, et. al. 2012). Vagenas has examined the energy distribution in the dyadosphere of a Reissner-Nordstrom black hole (Vagenas 2006). Berman has looked at the energy and angular momentum of dilation black holes (Berman 2008). Sharif has examined the energy of rotating spacetimes in teleparallel gravity (Sharif and Jawad 2011). Halpern has investigated the energy distribution of a charged black hole with a minimally coupled scalar field (Halpern 2008). In this paper, we consider yet another scenario: a static, electrically-neutral, spherically-symmetric massive object (such as a star or black hole) embedded in a space filled with phantom energy. Phantom energy is an especially potent type of dark energy proposed by Caldwell ( 2002) that possesses an equation of state parameter w < -1. It and other forms of dark energy have been hypothesized as possible agents for the acceleration of the universe, as recorded in supernova surveys (Riess, et. al. 1998; Perlmutter, et. al. 1999). The interior and exterior metrics of a dark energy star were calculated by Yazadjiev ( 2011). The metrics depend on a mass parameter m , the radius r and an additional parameter β . From these, Yazadjiev determined an overall mass M (that includes the rest mass as well as dark energy) and a dark charge D , through the following relationships: Note that β = 0 corresponds to the situation of a conventional Schwarzschild solution without phantom energy. The conventions that c = 1 and G = 1 are followed. Applying the prescriptions of Einstein, Papapetrou and Møller to the exterior metric derived by Yazadjiev we will examine the localized energy of the dark energy star.", "pages": [ 3, 4, 5 ] }, { "title": "2. Applying Einstein's prescription to dark energy stars", "content": "We now apply Einstein's prescription to Yazadjiev's dark energy star solution. The exterior metric has the following form: where: Einstein's prescription defines the local energy-momentum density as: where the superpotentials H i kl are given by: This complex has the antisymmetric property that: The energy-momentum components can be found by integrating the energy-momentum density over the volume under consideration: Through Gauss's theorem we can express this as a surface integral: where µ α is the outward unit vector normal to the spherical surface element: The localized energy E = P 0 can thereby be expressed as: Therefore, the relevant superpotentials to determine the localized energy are H 0 0 α with α ranging from 1 to 3. Taking i = k = 0 we find that (6) reduces to: Note that g 0 n = 0 for all values of n except n = 0 Therefore, the only non-zero components of (12) are the ones for which n = 0, namely: We substitute the metric (3) into expression (13) and find that the relevant superpotential values are: We insert the super potentials (14-16) into the double integral (11). Evaluating this double integral over the full coordinate range, we find the total energy of s solution within a sphere of radius r to be: It is interesting that the localized energy precisely matches the dark energy star's total mass M . This indicates that the Einstein complex well-encompasses the complete energy of the star, including the energy associated with its rest mass m as well as its dark energy. Note that if the dark charge D is zero, κ becomes 1 and the localized energy reverts to the Schwarzschild value of m .", "pages": [ 5, 6, 7 ] }, { "title": "3. Determining the energy by use of Papapetrou's prescription", "content": "We now turn to a second method for determining the local energy, Papapetrou's prescription. Unlike Einstein's prescription, Papapetrou's energy-momentum complex offers the advantage of being symmetric in its indices. Hence it permits the precise definition of local conservation laws. The Papapetrou energy-momentum complex is defined as: where the functions N ikab are given by: and the η ab terms represent the components of a Minkowski metric of signature -2. Again, we use Gauss's theorem to express the total energy as a surface integral: with the superpotentials χ 00 α defined as: We determine the relevant values of the superpotentials to be: Substituting equations (20-22) into (18) yields: This is identical to the expression obtained using Einstein's prescription. It is instructive to see that both complexes produce the same result.", "pages": [ 7, 8, 9 ] }, { "title": "4. Determining the energy by use of the Møller prescription", "content": "We now turn to a third method for determining the local energy, the Møller prescription, which has the marked advantage of being coordinate-system-independent. The complex defined by Møller can be expressed as: where: Inserting the metric components for Yazadjiev's solution (3) we obtain: One more time, we employ Gauss's theorem to express the total energy as a surface integral: Integrating over the full range of coordinates, we find the total energy using the Møller complex to be: This is identical to the expression obtained using Einstein's and Papapetrou's prescriptions.", "pages": [ 9, 10 ] }, { "title": "5. Conclusion", "content": "We have determined the localized energy distribution for Yazadjiev's solution representing a static, electrically-neutral, spherically-symmetric massive object with phantom energy. In applying the Einstein, Papapetrou and Møller energy-momentum complexes to Yazadjiev's metric, we have found that each yields an identical localized energy equal to the mass M . This extends earlier results for Kerr-Schild objects such as the Schwarzschild solution to an interesting case that could bear upon the dark energy question. Our findings help broaden the applicability of energy-momentum complexes, augmenting their use in general relativity as consistent, well-defined ways of describing the local distribution of energy. Thanks to K. S. Virbhadra for his helpful advice throughout the years about energy localization and other aspects of general relativity.", "pages": [ 10, 11 ] }, { "title": "REFERENCES", "content": "Aguirregabiria, J. M., Chamorro, A. and K. Virbhadra, K.S. (1996) Gen. Rel. Grav. 28, 1393. Berman, M.S. (2008) Revista Mex. Astron. Astrof. 44, 285. Caldwell, R. (2002) Phys. Lett. B 545, 23. Einstein, A. (1915) Preuss. Akad. Wiss. Berlin 47, 778. Halpern, P. (2008) Astrophys. and Space Sci. 313, 4, 357. Landau, L.D. and Lifshitz, E.M. (1962) The Classical Theory of Fields, Pergamon Press, Oxford, 341. M φ ller, C. (1958) Ann. Phys. 4, 347. M φ ller, C. (1961) Ann. Phys. 12, 118. Papapetrou, A. (1948) Proc. R. Ir. Acad. A 52, 11. Perlmutter, S. et al. [Supernova Cosmology Project Collaboration] (1999) Astrophys. J. 517, 565. Radinschi, I. (2011) Cent. Eur. Jour. of Phys 9, 5, 1173. Radinschi, I. and Ciobanu, B. (2006) 'Weinberg Energy-Momentum Complex for a Stringy Black Hole Solution,' gr-qc/0608029. Radinschi, I., Grammenos, T. and Spanou, A.( 2012) 'Distribution of EnergyMomentum in a Schwarzschild-Quintessence Space-time Geometry,' http://arxiv.org/abs/1204.1663. Radinschi, I., Rahaman F. and Banerjee, A. (2011) Int. J. Theor. Phys. 50, 9, 2906. Radinschi, I., Rahaman, F. and Ghosh, A. (2010) Int. J. Theor. Phys 49, 943. Radinschi I. and Yang, I.C. 'On the Energy of String Black Holes,' (2005) New Developments in String Theory Research, ed. Susan A. Grece, New York: Nova Science. Riess, A.G., et al. [Supernova Search Team Collaboration] (1998) Astron. J. 116, 1009. Sharif, M. and Jawad, A. (2011) Astrophys. and Space Sci. 331, 1, 321. Vagenas, E.C. (2006) Mod. Phys. Lett. A21, 1947. Virbhadra, K.S. (1990) Phys. Rev. D41, 1086. Virbhadra, K.S. (1999) Phys. Rev. D60, 104041. Weinberg, S. (1972) Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, Wiley, New York, 165. Yazadjiev, S. (2011) Phys. Rev. D. 83, 127501.", "pages": [ 12, 13 ] } ]

[ { "title": "The Smallest Particles in Saturn's A and C Rings", "content": "Rebecca A. Harbison ∗† , Philip D. Nicholson /star & Matthew M. Hedman /star November 1, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "Radio occultations of Saturn's main rings by spacecraft suggest a power law particle size-distribution down to sizes of the order of 1 cm (Marouf et al. , 1983), (Zebker et al. , 1985). The lack of optical depth variations between ultraviolet and near-IR wavelengths indicate a lack of micron-sized particles. Between these two regimes, the particle-size distribution is largely unknown. A cutoff where the particle-size distribution turns over must exist, but the position and shape of it is not clear from existing studies. Using a series of solar occultations performed by the VIMS instrument on-board Cassini in the near-infrared, we are able to measure light forward scattered by particles in the A and C rings. With a model of diffraction by ring particles, and the previous radio work as a constraint on the slope of the particle size distribution, we estimate the minimum particle size using a truncated power-law size distribution. The C Ring shows a minimum particle size of 4 . 1 +3 . 8 -1 . 3 mm, with an assumed power law index of q = 3 . 1 and a maximum particle size of 10 m. The A Ring signal shows a similar level of scattered flux, but modeling is complicated by the presence of self-gravity wakes, which violate the assumption of a homogeneous ring, and higher optical depths, which require multiple-order scattering. If q < 3, our A Ring model requires a minimum particle size below one millimeter ( < 0 . 34 mm for an assumed q = 2 . 75, or 0 . 56 +0 . 35 -0 . 16 mm for a steeper q = 2 . 9) to be consistent with VIMS observations. These results might seem to contradict previous optical (Dones et al. , 1993) and infrared (French & Nicholson, 2000) work, which implied that there were few particles in the A Ring smaller than 1 cm. But, because of the shallow power law, relatively little optical depth (between 0.03 and 0.16 in extinction, or 0.015 - 0.08 in absorption) is provided by these particles. NOTICE: this is the authors version of a work that was accepted for publication in Icarus . Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Icarus , [volume 226, issue 2, 11/2013] 10.1016/j.icarus.2013.08.015", "pages": [ 1, 2 ] }, { "title": "1 Introduction", "content": "The vast majority of particles that make up Saturn's main rings cannot be seen individually, but as an aggregate they become one of the most striking objects in the Solar System. From past observations, we know that the ring particles come in a variety of sizes. The Voyager radio science experiment used radio occultations at 3.6 and 13 cm to probe the ring particles by two methods. Direct inversion of the radio signal forward-scattered by meter-sized particles produced a size distribution showing a sharp drop-off above a radius of ∼ 5 m, while the differential optical depth measured between the two bands used in the occultation allowed a power law to be fit between particle radii of 1 m and 1 cm (Marouf et al. , 1983; Zebker et al. , 1985). However, the Voyager radio science experiment was insensitive to particles smaller than 1cm; smaller ring particles does not absorb even the shorter 3.6 cm radio waves from Voyager. French & Nicholson (2000) used the 28 Sagitarii (28 Sgr) stellar occultation, as observed from Earth in July, 1989 at wavelengths between 1 and 4 µ m, to look for light forward-scattered by ring particles. Unlike the monochromatic radio-science experiments, they could not separate forwardscattered light from light directly transmitted through the rings. However, the differing range and acceptance angle between this and the Voyager PPS stellar occultation allowed a gross measurement of forward-scattering. This measurement could then be modeled with a truncated power law. The 28 Sgr occultation, like Voyager, had limited sensitivity to particles under 1 centimeter, but for a different reason: the scattering angles of such small material was larger than the photometric aperture size, so could not be measured. Previous radio occultation and stellar occultation experiments were thus most sensitive to particles in the centimeter to meter range. This situation changed with the arrival of the Cassini spacecraft at Saturn in 2004. As Saturn was near its northern winter solstice in 2004, the rings were more open than when Voyager observed them, reducing the effective optical depth as seen from Earth and increasing the signal to noise for occultations by dense rings. In addition to the 3.6 and 13 cm radio bands, Cassini can also transmit at 1.3 cm. Not only does a shorter wavelength probe smaller particle sizes, but three measurements of the optical depth at different wavelengths allow for more exact models to constrain both the effective minimum particle size and effective power law index. The C ring minimum particle size was estimated at 4 mm, while the data for the A ring suggest a larger minimum particle size (Marouf et al. , 2008). A full discussion of these results can be found in Cuzzi et al. (2009). While some micrometer-sized particles have been seen within the main rings, they are either found in transient spoke features (D'Aversa et al. , 2010; Mitchell et al. , 2013), probably dislodged from the surfaces of larger ring particles (Mitchell, 2006), or are confined to narrow, diffuse regions of the rings such as the Encke Gap ringlets (Hedman et al. , 2007b) and the 'Charming Ringlet' in the Laplace Gap (Hedman et al. , 2010). Differential optical depth, phase-function, and microwave emissivity measurements all show that very little dust persists within the main rings on a large scale in both space and time (Dones et al. , 1993; French & Nicholson, 2000; Spilker et al. , 2005). Theoretical work by Bodrova et al. (2012) also has shown that, under unperturbed main ring conditions, particles smaller than a few centimeters will adhere onto larger ring particles. When Cassini entered Saturn orbit in 2004, its wide range of orbital geometries not only allowed for multiple radio and (space-based) stellar occultations, but also permitted the first solar occultations by the rings to be observed. The Visual and Infrared Mapping Spectrometer (VIMS) onboard Cassini can accept light through a special solar port, which has the attenuation needed to safely observe the Sun with the VIMS detector array. Given the 0.5 milliradian pixel size of VIMS and its near infrared (0.9 to 5.2 microns) spectral range, the VIMS data are most sensitive to the previously- unsampled size regime of 100 microns to a few millimeters. In this work, we will use the VIMS solar occultations to examine this neglected regime, with the goal of setting an effective minimum radius on the ring particle size distribution in different regions. Following a description of the VIMS solar port and the data taken during solar occultations, we will present our method for reducing the solar port data and separating the component of light scattered at small angles from the direct solar image. Once this component is separated, it can be compared to a model of particle diffraction to estimate an effective minimum particle size for the C ring. This model is then refined to account for the self-gravity wakes and higher optical depths observed within the A ring - which violate several simplifying assumptions made at first - and applied to that ring.", "pages": [ 2, 3, 4 ] }, { "title": "2.1 Basic Processing", "content": "As of February 2010, Cassini has observed eleven solar occultations of the rings: see Tables 1 and 2 for a list. The procedure for observing solar occultations involves taking a series of 12 pixel by 12 pixel (6 x 6 milliradians) multispectral images of the area of the sky around the Sun using the VIMS solar port, which has an attenuation on the order of 10 5 . The instrument's visible channel is turned off, as the visible-light spectra, even through the solar port, saturate within a few milliradians of the Sun. Thus, data obtained through the solar port have a wavelength coverage of 0.9 to 5.2 µ m. A single VIMS 'cube' of two spatial and one spectral dimensions is constructed pixel by pixel, using a 2D scanning mirror. Each pixel has an exposure time of 40 ms, and approximately 5 cubes of 144 pixels each are obtained for every minute of the occultation. Each occultation data set is thus a time series of cubes - one temporal dimension, two spatial, and one spectral. For full details of the VIMS instrument, see Brown et al. (2004). The onboard VIMS signal processing electronics perform automatic background subtraction. At the end of each line of 12 pixels, VIMS takes a spectrum of the thermal background signal by closing off the spectrometer from outside light and taking a measurement. Four measurements of this dark spectrum are averaged together, then subtracted from the last four lines of pixels taken. As a result, each cube has a slightly uneven background sub- traction, as there is some shot-noise variance even after averaging over four measurements. In most cases, these three background spectra are within a data number (DN) or two of one another 1 , but a cosmic ray can hit the detector during a background measurement, producing an artifictially high background in one or more spectral channels. In order to correct this, the background was re-added to the signal, returning the data to its raw form, and then the median of the three dark current spectra recorded for each cube was used as the background instead. The slight temperature change when Cassini moves into the rings' shadow lowers the dark current by approximately 2 DN. Hence the dark background subtracted is slightly dependent on the position of Cassini, so further timeaveraging of the background was not done. The cubes showing the unocculted Sun were used as a reference to define transmission of the rings, and all measurements are reported either in units of transmission or in 'raw' data numbers (DN), rather than absolute flux. The position of the Sun within the image varied by well under a single pixel in each occultation, making any variable response due to a slightly different beam path within the solar port or the spectrometer minimal.", "pages": [ 4, 5 ] }, { "title": "2.2 Instrumental Effects", "content": "The VIMS solar port is designed to attenuate the Sun enough to make it safe to observe with the VIMS instrument. However, the optics that do this also produce abundant stray light within the instrument. As a result, in addition to the normal solar image that can be fit to a two-dimensional Gaussian point-spread function (PSF), there is also a diffuse component that extends at least 6 solar diameters from the Sun (Figure 1). To first order, this diffuse component is flat over the 12 by 12 pixel images. At approximately 1/10th of the peak of the solar signal, the diffuse signal is ∼ 10 times larger than the flux within the nominal solar image when integrated over the entire cube (Figure 2). In addition, the diffuse component is spectrally different from the direct component, being distinctly 'redder'. This greatly complicates any attempt to look for scattered light from millimeter-sized ring particles, but a method to exploit the stray light will be discussed in Section 5.2.", "pages": [ 5 ] }, { "title": "2.3 Data Selection", "content": "Of the eleven solar occultations taken before equinox in 2009 and observed by VIMS, nine cover the A ring, and six extend into the C ring. The A ring occultations (Table 1) are mixed between nearly-radial occultations for which the Sun passed behind all of the rings (and then behind Saturn itself), and chord occultations for which the Sun passed behind one of the ansae, giving two 'cuts' across the outer rings. For the A ring, both the radial and chord occultations sample nearly uniformly in the radial direction as well as sampling only a limited range of longitude ( /lessorsimilar 5 · ). The six occultations clearly covering the C Ring (Table 2) are also a mix of chord occultations and radial occultations. As all of the chord occultations 'turn around' in the C ring, the data here have variable radial sampling, with the inner parts of the occultation (near the turnaround point) sampled more than outer parts. The criteria considered when deciding which data sets to use include the opening angle of the rings and the number of cubes within each ring. For the A ring, occultations done later in the mission are almost opaque due to the low opening angle of the rings. The C ring has the opposite problem; the large opening angles at the beginning of the mission meant that most of the sunlight is transmitted without interacting with the ring at all. ∗ The data set from the Rev. 43 occultation ingress starts near the inner edge of the A Ring, meaning the A Ring ingress was omitted from this table.", "pages": [ 6, 8 ] }, { "title": "2.4 Transmission Spectra", "content": "Transmission spectra of the main rings can be produced by summing the cubes over their spatial dimensions and normalizing to the solar spectrum as measured outside of the A ring. This offers a high signal-to-noise spectrum of the ring's transmission properties in the near infrared, given the brightness of the Sun. Combining repeated measurements at slightly different locations in the ring (sampled as the occultation progressed), we can increase signal-tonoise further at the expense of spatial resolution. This gives a transmission spectrum with errors between 0.005 and 0.022 (in units of transmission). In Figure 3, we plot mean transmission spectra of the three main rings and the F Ring. The spectra were constructed by fitting a gaussian curve to the image of the Sun in each wavelength, then taking the integral over that curve to find the total flux at that wavelength. Then an 'average' spectrum for each area of the ring was produced by taking the mean over each cube 'on' the rings, and normalizing to a solar spectrum obtained by taking the mean of cubes outside of the ring system. The main rings' transmission spectra show no obvious bands, and are remarkably flat in the region of 2 to 4 microns (the region from 4 to 5 microns is not plotted due to a much lower signal to noise ratio). This is in marked contrast to the reflection spectra of the main rings, which show strong water ice bands in this region (see Nicholson et al. (2008) for a fuller discussion of the rings' reflectance spectra). This indicates that the vast majority of ring particles are so large as to be opaque in the near infrared. However, not all regions of Saturn's rings behave in this matter. Free ring particles in the tens of microns (or smaller) size range do show prominent features in transmission, as is seen in our mean F Ring spectrum in Figure 3, and described by Hedman et al. (2011) in transmission spectra of the F Ring taken during stellar occultations. Most visible in F Ring spectra is a strong increase in transmission at ∼ 2.9 µ m due to the Christensen effect: the optical properties of water ice at this wavelength minimize absorption and internal reflection. (Hedman et al. , 2011; Vahidinia et al. , 2011) Other features, such as the peaks and dips near the order-sorting filters, are likely artifacts due to a lack of signal. However, the slight 'blue' slope around 1 to 1.5 microns may be a real measure of ring properties and will be discussed later in this paper.", "pages": [ 9, 10 ] }, { "title": "3 Transmission Spectra Analysis", "content": "Hedman et al. (2011) introduce the ratio ρ to measure the ratio in optical depth in and out of the 2.9 µ m feature in stellar occultations. In order to avoid contamination from reflected sunlight in addition to the transmitted starlight, they define ρ as the ratio of optical depths at 2.9 and 3.2 µ m, as the rings are dark in reflection at both wavelengths. As solar occultations focus entirely on the dark sides of ring particles, the choice of a reference wavelength out of the 2.9 µ m feature is less constrained. We define ρ 2 . 5 as or the ratio between the optical depth of the 2.9 micron band (defined as the integrated optical depth from 2.82 to 2.93 µ m) and the optical depth at 2.5 microns (defined as the integrated optical depth from 2.45 to 2.56 µ m), with optical depths found the conventional way, from the transmission, T = exp τ/µ . 2.5 µ m was chosen as a reference wavelength based on the high signal to noise in this region of the solar spectrum as measured by VIMS. Figure 4 plots the composite spectra of the A, C and F rings from the Rev. 09 solar occultation in terms of the optical depth normalized to the optical depth at 2.5 µ m. In Figure 4, the 2.9 µ m peak in the F Ring transmission spectrum is seen as a dip, while the A and C ring spectra continue to appear flat. The measurements of ρ 2 . 5 from six solar occultations (Revs. 09, 43, 55, 59, 62 and 65) are included in Table 3. From the table, the F ring shows a ρ 2 . 5 of between 0.77 and 0.86, with a mean value of 0.82 ± 0.03. The A and C rings, however, yield values consistent with unity. If we assume the A and C rings are a mixture of F ring-like material, with a ρ 2 . 5 equal to the mean F ring value of 0.82, and 'large ring particles' with a ρ 2 . 5 of 1, we can set a limit on the amount of dusty or F-ring-like material. From the measured values of ρ 2 . 5 , we conclude that neither the A nor the C Ring shows a significant difference from a flat spectrum. The A Ring can contain less than 5.5% (1 σ ) by cross sectional area of F-ring-like material, while the C Ring can contain less than 1.4% of F-ring-like material. From this, we can infer that free-floating ice grains in the tens of microns size range, capable of producing the Christiansen effect(Hedman et al. , 2011), are quite rare within the main rings, unlike within the F Ring.", "pages": [ 10, 11, 13 ] }, { "title": "4.1 Introduction", "content": "While, in the previous sections, we excluded a significant population of particles smaller than 100 µ m in the main rings, somewhat larger particles can produce observable effects by diffraction, while being opaque. It is to observe this diffraction that the spatial data taken by VIMS become useful. To first order, sunlight diffracted by ring particles of radius a will scat- ter into a cone of angular radius θ /similarequal λ/ 2 a . Given VIMS's pixel size (0.5 milliradians) (see Figure 5), the solar diameter at Saturn ( ≈ 1 milliradian) and operating wavelengths (1 - 5 µ m), VIMS should be able to best image diffracted light from ring particles with a radius of several millimeters and less: A full model of the diffraction of sunlight by ring particles will be presented in the following section.", "pages": [ 13, 14 ] }, { "title": "4.2 General Expression", "content": "The model of French & Nicholson (2000) was chosen as a simple representation of forward scattering and absorption in a ring. French & Nicholson (2000) assume a simple truncated power-law size distribution and, for simplicity, neglect any contribution from multiple scattering - which is a valid assumption for τ/ 2 µ /lessorsimilar 1. We accept this for now, but in Section 6.3, we extend our analysis to include multiple-scattering for higher optical depths. As higher-order scattering broadens the phase function, ignoring higher-order effects will, in general, result in underestimates of the minimum particle size. (French & Nicholson, 2000) This model states that the flux incident on the detector from light scattered by a uniform sheet of particles as a function of scattering angle, F ( θ ) is where F 0 is the solar flux incident on the rings, µ is the cosine of the incidence angle, 〈 /pi1 0 〉 is the single scattering albedo, assumed to be 0.5 for particles much larger than the wavelength of light being studied 2 , A is the solid angle of the detector (in this case, one VIMS pixel), and P ( θ ) is the mean phase function of the diffracted light, normalized such that the integral over all solid angles is 4 π (thus the flux from scattered light integrated over all solid angles is F 0 τ/ 2 µ exp ( -τ/µ )). P ( θ ) depends on the distribution of particle sizes assumed. Note that the optical depth, τ , used in Equation 3 and for the rest of the paper (unless otherwise noted) is the extinction optical depth, which, for particles much larger than the wavelength of light, is twice that of the geometric optical depth, τ = 2 τ geo , where τ geo is typically used in optical and near-infrared studies of the rings, including French & Nicholson (2000). For a full derivation of the model, please see A. We chose a truncated power law with particles between a min and a max in size, and with a power law index of -q . To speed computational time over many orders of magnitude, we implement this in our code by two approximations valid over different angular regimes: the medium-angle case and the large-angle case, which are defined by the characteristic diffraction angle of the smallest particles in the size distribution, θ 2 = λ/ 2 a min . These cases are also useful in understanding the behavior of the model. The value of θ 2 is unknown, because the minimum particle size is the quantity we are trying to measure. Given that the size of one VIMS pixel and coincidentally the solar radius at 9 AU - is 0.5 milliradians on the sky, our data will be most sensitive to diffraction by particles with x /lessorsimilar 6000, or, at 2 microns wavelength, particle sizes of 2 millimeters or less. Barring a much-lower-than-expected minimum size cutoff, the large-angle scattering case will be most relevant, though we will include the medium-angle case in our calculations to account for the possibility of free-floating particles from ∼ 100 µ m to ∼ 2 millimeters. The large angle case, where the scattering angle, θ is much larger than the characteristic diffraction angle of the smallest particles ( θ /greatermuch θ 2 ), has a phase function of approximately where the dimensionless size parameter x min = 2 πa min /λ and α is a normalization factor given by /negationslash The medium angle case, where θ is in between the characteristic diffraction angle of the smallest and largest particles ( θ 2 /greatermuch θ /greatermuch θ 1 ; θ 1 = λ/ 2 a max ), has a phase function of approximately The J ∞ 0 ( q ) in Equation 6 is shorthand for ∫ ∞ 0 z 2 -q J 1 ( z ) 2 dz . It is nearly constant over the range of 2 ≤ q ≤ 5, except when q approaches 2 or 5. Previous studies indicate that q is between 2.7 and 3.1 within the main rings, giving J ∞ 0 ≈ 0 . 5 (Zebker et al. 1985, French & Nicholson 2000, Cuzzi et al. 2009). At this point, we remind the reader that most of the light diffracted by the centimeter to meter-sized particles which dominate the main rings at near-infrared wavelengths is not detectable, since it is confined to angles much less than the solar radius. Our only hope is to detect the 'tail' of the scattering function, due primarily to millimeter and smaller sized particles, if they exist in sufficient numbers. Finally, we note that for the large-angle case, the slope of ¯ P ( θ ) is independent of q , but the absolute level depends on q and x min (or a min ), while for the medium-angle case, the slope of ¯ P ( θ ) depends on q , but the absolute level depends only weakly on q (via J ∞ 0 ) or x min (via α ).", "pages": [ 15, 16, 17 ] }, { "title": "5.1 Simple Attempts", "content": "Our goal is to detect and measure a faint 'halo' of diffracted light around the image of the occulted sun, in the presence of a much brighter background of instrument-scattered light and fit this by the method described in the previous section. Our first approach was to attempt to create a template from the data of the unocculted Sun as seen through the solar port to serve as our comparison for cubes containing the occulted Sun. We selected cubes outside of the F ring or inside the C ring to construct the template. As observations were structured to give such windows on either side of solar occultations, these data were available for all occultations. While the shape and spectrum of the diffuse background does vary depending on where in the field the Sun is, based on solar calibrations performed in flight, Cassini is a very stable platform for observations, and the movements of the Sun within the field during any single occultation are much smaller than a VIMS pixel. As a result, the diffuse background changes little during an occultation, other than to scale with the total solar flux transmitted by the rings. Once we have a template, we can divide data cubes within the rings by that template. Given the tiny levels of diffracted signal expected, data cubes from nearby radii in the rings were summed, creating composite data cubes for the average A Ring between the radii of 122,000 and 133,000 km and for the average C ring between 75,000 and 92,000 km. Due to previous results which showed that the trans-Encke region of the A ring has a different particle size distribution than the middle and inner A ring (French & Nicholson, 2000; Zebker et al. , 1985), all A ring cubes outside of the Encke Gap were omitted from the average. Cubes near ring edges were also omitted. The number of cubes fitting these criteria from each occultation are listed in Tables 1 and 2 above. Figures 6 (the A ring, from Rev. 43) and 7 (the C ring, from Rev. 65) show data from such ratio images, plotted in units of transmission (found by dividing the composite image by the templates constructed for each occultation using cubes containing an unocculted Sun). The data from the ratio cubes were sorted by distance from the center of each pixel to the center of the Sun's image, and then binned in 0.25 milliradian (0.5 pixels) increments. Due to the non-zero instrumental background of diffusely-scattered light, transmission measurements can be recorded even far from the Sun itself, and are not themselves a sign of diffraction from ring particles. Although at first glance the transmission profiles are 'flat', closer inspection does show some evidence for diffracted light. Figure 7 shows a significant increase of a fraction of a percent in transmission at around 1 milliradian. If a similar peak is present in Figure 6, it is invisible compared to the pixel-to-pixel variation (as seen in the error bars, which mark the standard error). Also note that the innermost datapoints (seen most clearly in Figure 6.c and Figure 7.b, but present in others) show a significant decrease in transmission relative to the 'far-field' mean transmission measured from 1 to 4 milliradians (and plotted as a horizontal line). These pixels are within the 0.5 milliradians that define the solar angular radius as seen by VIMS, indicating that the transmission as measured by looking directly at the Sun is lower than that measured from the diffusely-scattered background, which is also produced by sunlight shining through the rings. This apparent contradiction can be reconciled by remembering that when sunlight is diffracted into a halo, it has to come from somewhere. The individual pixels 'on' the Sun will show some additional attenuation due to light scattered out of the beam. However a wider-angle measurement - like that of the stray light scattered within the VIMS instrument - will collect both the direct and scattered light. Were the Sun a point source and a VIMS pixel small enough to exclude all scattered light, the difference in transmission would correspond to a factor of two in optical depth. Since neither is the case here, the difference is much more modest. However, this difference in transmission can be measured, and, thus, can allow the amount of light diffracted at angles larger than a VIMS pixel to be measured, even if a clear diffraction halo is not seen (as in the A Ring measurements shown in Figure", "pages": [ 17, 18 ] }, { "title": "5.2 Quantifying Transmission Differences", "content": "Let us construct a simple model for imaging the Sun with VIMS. The Sun is not a point source, so even an unocculted Sun will take up several VIMS pixels. For simplicity, we assume that the direct solar flux - that not scattered by the solar port's optics or diffracted outside the sun's disc by small ring particles - is confined to an area of N s pixels, which can be measured from the unocculted template we created in the previous section. When the Sun is behind the rings, there is an additional halo of diffracted light from small particles (defined as those capable of scattering light outside of the solar disc), covering N pixels. This is shown in Figure 8. Let the total direct (i.e., excluding that scattered within the VIMS solar port optics) unocculted solar signal be S s , measured in DN per integration at a specific wavelength, λ . In addition, as mentioned in Section 2.2 and illustrated in Figure 1, there is a diffusely-scattered background signal, spatially non-uniform, denoted by which we assume scales in brightness with the direct solar signal, but is spectrally different than the direct signal S s (Figure 2). The total integrated signal within the unocculted solar image (the N s pixels that are 'on' the Sun) can be written schematically as where 〈 σ ob 〉 is a mean of σ ob ( x, y ). We can estimate the values of S s and 〈 σ ob 〉 by fitting a two-dimensional Gaussian profile, plus a constant offset, to the central part of the template image. With the sun occulted by the rings, its total flux (direct plus diffracted) is reduced by a factor T = e -τ/ 2 µ . The total flux from the sun is thus TS s . A portion of this flux has been diffracted by the rings at angles θ /lessorsimilar θ 2 . We denote the fraction of the full solar flux diffracted into the range θ s ≤ θ ≤ θ 2 by f , where θ s is the effective radius of the solar image in the VIMS cubes, or about 0.5 mrad. Therefore, N s ≈ πθ 2 s . The diffracted flux is then fS s , which is assumed to be spread uniformly over an area of N ≈ πθ 2 2 pixels, centered on the solar image. Three measured quantities are of interest in the cubes obtained during the occultation: 1. the background signal outside the diffraction halo, 2. the background signal within the halo (i.e. in the annulus described by θ s ≤ θ ≤ θ 2 ), which has a mean value of 3. the total signal within the solar image (defined as the same area of N s pixels above) of or Given that we can measure S 0 and 〈 σ ob 〉 by fitting a two-dimensional Gaussian curve plus a constant to the data of the unocculted sun, as mentioned above, the only remaining unknown is S s . We can use Equation 8 to rewrite Equation 12 in terms of observables, rather than the unknown S s , and we can normalize this by S 0 to get an effective transmission, T s , measured only within the solar image: Note that all quantities in this expression, with the exception of f (which is the measure of scattering by 'small' particles in the ring, and which it is our goal to quantify), and N (which is set by θ 2 and thus the size of the smallest particles) can be directly measured from the data. If we estimate a minimum particle size of a min ≈ 0 . 5 mm, this gives θ 2 ≈ 2 mrad at 2 µ m. Since θ s ≈ 0 . 5 mrad, the quantity N s /N ≈ ( θ s /θ 2 ) 2 is 1/16. We can thus assume that 1 -N s /N ≈ 1, and solve Equation 13 for f : T is most readily obtained from Equation 9, using the measurements of the instrument-scattered background. In reality, σ rb ( x, y ) is spatially variable, so we use a Gaussian plus constant offset fit to occulted and unocculted cubes to find the local mean background in each image. Then we obtain T by dividing the constants of the two fits: T s is also obtained by fitting offset gaussians to the occulted and unocculted solar images and integrating over the solar disk: ∫ While gaussian curves are bounded at infinity, the constant background needed to properly fit the images are not. We chose to assume the background under the solar image covers an area equivalent to the ellipse described by the fitted standard deviations of the gaussian function. This 'footprint' was chosen instead of a circle of angular radius θ s (or the angular radius of the Sun at Saturn) to account for the distortion in the solar image: while the width of the gaussian in the x direction matches the angular size of the Sun, the image appears stretched in the z direction, as is clearly visible in Figure 1. Effectively what these calculations do is to estimate f not from the diffracted light itself, but via its removal from the direct solar flux. The advantage of this somewhat indirect method is that the diffracted light is spread over N pixels, while the solar image covers only N s pixels, where N s /N /lessmuch 1 for a min /lessorsimilar 0 . 5 mm. A secondary benefit is that the derived value of f is almost independent of the unknown quality N , so long as N s /N /lessmuch 1. Our first method (as explained in the previous section and shown in Figures 6 and 7) amounts to trying to measure the difference between the σ ' rb /σ rb ≈ 1+ fS s / ( NTσ ob ), which dwarfs the quantity of interest, f , by other factors, rather than measure it directly as we do here. French & Nicholson (2000) define a similar measure of the observed scattered light, Q occ : the ratio of the observed optical depth, including some fraction of scattering, to the geometric (or absorption) optical depth, as defined in their equation 15; τ obs = Q occ τ geo . They define the total scattered flux measured in their equation 18, which can be written in our notation as As a test of concept, we can refer back to Figure 3, which plots the direct solar signal (as measured by a Gaussian fit) in terms of transmission and as a function of wavelength. We would expect that shorter-wavelength light would have less light scattered at angles large enough to be removed from the direct signal, producing a slightly blue slope as the redder regions of the spectrum had some light removed. In a qualitative sense, this can be seen in Figure 3's spectrum of the C Ring (and possibly the A Ring): the region of the spectrum blueward of ∼ 1.6 µ m has slightly increased transmission than the rest of the spectrum. We also would expect that this effect would be somewhat dependent on optical depth - at low or high optical depths, such a signal would not be as prominent as the intermediate optical depths that contain enough material to scatter, but not so much as to absorb the scattered light.", "pages": [ 21, 23, 24, 25 ] }, { "title": "5.3 Measuring the Diffracted Light", "content": "Given the indications in Figures 6 and 7 that T s is indeed slightly less than T , we can calculate f numerically as described in Section 5.2, by using a gaussian fit to both the template and individual cubes to calculate T s and T - and thus, f , the fraction of light diffracted out of the solar image. As in the simple test performed above, it was necessary to take the mean of f over the entire A or C ring - with the same caveats of avoiding the trans-Encke region and the edges of the ring - in order to achieve a satisfactory signal to noise level. In addition, the data were binned by wavelength, taking the median of f over 10 channels, with error bars calculated from the standard errors within each bin. Based on those error bars, we focus on the region from 1.8 to 2.8 microns. Note that this bins data far more than in the simple plots we did in Section 5.1. While Figures 6 and 7 were means over wide ring regions, as are these measurements of f , here we bin ten adjacent wavelength channels as opposed to examining a single channel, and reduce an entire 144-pixel image into a single measurement (while in Figures 6 and 7 each bin contains roughly a half-dozen points). Thus, we should expect a corresponding reduction of noise. In order to predict f for a particular assumed size distribution, the model described in Section 4 is used and integrated over an annulus centered on the Sun. A circle of radius 0.5 milliradians (1 pixel) was chosen as the inner boundary for the model's integral. However, as a result of the optics, the data show an clearly elliptical image of the Sun, and our measurements of f (derived from the Gaussian fits to the data) take the apparent ellipticity into account. The distortion from the optics that produced an elliptical solar image could introduce a systematic difference between model and observation, but attempting to fit the image with a circular solar image would also introduce or exclude light. Without a better mapping of the distortions caused by the boresight optics, an empirical measurement seems the best guess as defining the difference between 'Sun' and 'sky'. As the amount of scattered light drops off sharply with increasing angle, we assume an outer radius of infinity. This introduces a negligible increase in the modeled value of f for a given a min compared to what we measure. Thus, f is simply an integral over the intensity function, as specified in Equation 3, divided by the unocculted solar flux: As a test of robustness, we integrated a hypothetical C ring model of τ/µ = 0 . 5, q = 3 . 1, a max = 10m and several lower particle size cutoffs for varying inner radii. The results are shown in Figure 9. Expanding the inner radius to an unphysical two times the solar angular radius in the image can reduce the minimum particle size by a factor of 2. Consequently, any plausible error in estimating the 'correct' annulus for the model would result in an overestimate of the particle size (as it seems unlikely that the most appropriate annulus would have an inner radius smaller than the solar radius). The C Ring occultations yield three data sets (those from Revs. 9, 62, and 65) which show a significant fraction of scattered light over the full spectral range considered (2 to 2.8 microns), and one more (Rev. 59) which shows a significant non-zero fraction of scattered light over part of this range. Table 2 includes the mean transmissions ( T ) and opening angles. The Rev. 11 occultation does not give a significant detection; of the occultations, it has the highest background and it could be that statistical noise overwhelmed the signal. Figure 10 shows our three positive and one marginal detections. As many of the C Ring solar occultations are non-uniformly sampled by radius, direct comparisons between occultations may be misleading if there are variations in particle size within the C Ring. There are a mix of nearly radial occultations (Revs. 9 and 11), which sample all parts of the C Ring evenly, and occultations that cut across the ansae, which sample the innermost portions of the occultation more heavily (Revs. 59, 62, and 65). Given a model of the scattered light as discussed above and using values of a max = 10 meters and q = 3 . 1(Zebker et al. , 1985), the three positive detections (Rev. 09, Rev 62 and Rev. 65 occultations) yield a minimum particle size of between 0.2 and 2 cm. This range is of the same order as that derived by Marouf et al. (2008) for the C Ring. The slightly lower signal in the Rev. 65 and 62 occultations could indicate a slightly larger particle size cutoff in the inner portions of the C ring, as these two occultations oversample the inner regions, but the result is not at the 3 σ level given the error bars, especially those of the Rev. 62 occultation. The minimum particle size derived is somewhat dependent on the other model parameters a max and q - a steeper power law or a smaller maximum particle size will increase the fraction of optical depth in smaller particles, and increase the amount of scattering at angles greater than θ s . In our angular regime, namely that of large-angle scattering, the strongest effect is with q : a steeper power law (larger q ) implies more particles with sizes small enough to scatter at the relevant angles. Consequently, for a given value of f , a steeper power law leads to a larger a min . a max has only a weak effect; a larger maximum size reduces the number of particles per unit area for a given optical depth, slightly lowering the minimum size for the same value of f . However, for q > 3, as has been derived for the C ring (Zebker et al. , 1985), most of the cross-sectional area is in small particles, so a modest increase in the number of large particles produces an inconsequential effect on scattering at this angular scale. Figure 11 plots a min vs. q for the C ring. The function was calculated by taking the scattering fraction from the Rev. 09, Rev. 62 and Rev. 65 occultations (the three in which a clear positive detection was made) at a wavelength of 2.3 µ m, and calculating the a min for a given q needed to produce the observed scattered light. The line plotted in Figure 11 are then a mean of the values of a min calculated from each of the three occultations. At q = 3 . 1, corresponding to previous estimates of the C ring power-law index (Zebker et al. , 1985; French & Nicholson, 2000; Marouf et al. , 2008), we find a value of a min = 4 . 1 +3 . 8 -1 . 3 mm. The C ring shows a robust value of a min somewhere between 0.3 and 1 cm for values of q between 2.95 and 3.5. Completing the same analysis on the A ring - shown in Figure 12 shows a significant fraction of scattered light in five occultations (Revs. 9, 43, 55, 59, and 62) over the same wavelength range. Rev. 65 shows a partial detection over some of the range. Note that comparing the far-field signal to the decrease in signal, and binning by wavelength, produces a far clearer detection in Rev. 43 than seen in Figure 6. However, unlike the C ring, which is homogenous and optically thin ( τ/µ /lessorsimilar 1), the A Ring is neither. Those complicating factors, self-gravity wakes and the possibility of multiple scattering, are examined below and our simple model modified appropriately.", "pages": [ 25, 26, 28, 30, 31 ] }, { "title": "6.1 Introduction to Self-Gravity Wakes", "content": "Re-examining the assumptions made in the model described in Section 4, we see that one stands out. The model assumes that the ring in question is made up of a thick slab with a homogenous distribution of particles. However, the A ring is not well described by this model. Observations show that the A ring has an azimuthally-dependent optical depth, which varies by up to a factor of a few depending on the longitude relative to the planet-to-star direction that is sampled by the occultation (Colwell et al. , 2006; Hedman et al. , 2007a). The accepted explanation for this variation, based on numerical simulations of this ring, is the presence of self-gravity wakes (Salo, 1992). Because the self-gravity wakes are long aggregates of particles with a characteristic trailing orientation with respect to the radial direction, they change the optical depth depending on the cross section they present to the beam of light, which depends on the observed longitude with respect to the stellar direction, φ . They also don't show the simple 1/ µ dependence of τ on opening angle, instead following a more complicated relation. Our next step in modeling ring scattering within the A ring is to account for the wakes within the ring.", "pages": [ 31 ] }, { "title": "6.2 Scattering with Opaque Wakes", "content": "Following Hedman et al. (2007a), we assume the A ring consists of a parallel series of cylindrical wakes of characteristic width W , height H , spacing λ and alignment φ W measured relative to the radial direction. The wakes themselves are opaque, but the interwake 'gaps' have a finite optical depth τ G due to particles outside the wakes. Tiscareno et al. (2010) show that this is not an exact description of the wake behavior in dynamical simulations, but that this simple model reproduces optical depth measurements for opening angles larger than ∼ 10 · . As none of the occultations we use for the A ring measurements are less than ≈ 8 · , Hedman et al. 's wake model should be sufficient for our purposes. We assume that the particles within the gaps form a homogenous layer so that the same model used earlier applies within the gaps. If f W is the fraction of ring covered by the wakes as viewed by VIMS (which depends both on the opening angle of the rings ( B ), and the longitude ( φ ), as well as the parameters W , H , λ and φ W ), then the fraction of scattered light can be written as where the integrand is calculated as before. Not that for f W = 0 (and τ G , the extinction optical depth of the interwake material, equal to τ ), this equation reduces to the simpler case used in the previous section. A full derivation for f W is given by Hedman et al. (2007a), resulting in the expression ∣ ∣ The values of φ , the longitude of the area sampled, and B are known for each occultation and are listed in Table 1. φ changes slightly as the occultation progresses, as none are totally radial occultations, but for the regions of the A ring sampled, this change is small. Repeated stellar occultations suggest that τ G is between 0.3 and 0.6, H/λ is between 0.09 and 0.12, and W/λ is between 0.3 and 0.65 for the A ring(Nicholson & Hedman, 2010). Note that Nicholson & Hedman's values for optical depth ( τ g ) correspond to absorption, so we have used the equation τ G = 2 τ g to derive values for extinction optical depths within the gaps. These and other studies of A ring photometry show that the wakes are oriented to have a peak transmission at φ W ≈ 70 · and 250 · longitude (with 0 · being the direction to the Sun (or star) from Saturn)(Nicholson & Hedman, 2010). The effect of the wakes on f is not simple. While f W decreases the ring area which provides the scattered signal, replacing τ by the much smaller τ G increases the amount of scattered light available when τ/µ > 1, which is usually the case in solar occultations by the A Ring.", "pages": [ 31, 33 ] }, { "title": "6.3 Effects of Multiple Scattering", "content": "Our first model assumed that all light interacts with a ring particle once and is absorbed, (singly-)diffracted or transmitted. However, in reality the ring particles we are considering are far smaller than the thickness of the ring, so multiple scattering is possible. For τ/ 2 µ /lessmuch 1, the contribution from light diffracted more than once is small. However, even when we consider an expected normal optical depth of 0.3 to 0.65 (the estimated extinction optical depth between self-gravity wakes in Hedman et al. (2007a)), only the Rev. 9 occultation at B = 21 . 5 · (and the lowest optical depth estimate of the gap material) satisfies τ/ 2 µ < 1. Zebker et al. (1985), in analyzing the low incidence Voyager radio occultations, developed a scheme for handling multiple scattering in a thin ring. They treat the ring as N layers of optical depth τ 1 = τ/N , where τ 1 /lessmuch 0 . 5, so that within each layer, the single scattering approximation holds. This model allows for multiple scattering (to degree N ) by calculating the fraction of absorption, scattering or transmission through each layer and treating it as a sum of terms to produce the intensity function. The phase function for multiple scattering is treated of a convolution of single scattering, as Zebker et al. (1985) do in their Equation 7. In the notation used in this paper, we can write their equation as where I 1 ( θ ) is the intensity distribution from single scattering within a layer of ring, as calculated from Equation 3 (but without the solid angle that changes an intensity into a flux), using τ 1 as the optical depth. [ I 1 ] k , represents the k th convolution of I 1 with itself, so each term of the sum represents the contribution of k th order scattering to the whole, with an attenuation factor to account for absorption by the N -k other layers, and a combinatoric factor to account for the k layers chosen from N to scatter photons. We can also re-write Equation 21 in terms of the phase function for single scattering, P ( θ ), to remove quantities not dependent on θ , and to bring out the 'hidden' τ 1 in the intensity distribution: N is an approximation for the number of particles thick the ring is. As mentioned earlier, the Voyager radio occultation was most sensitive to suprameter particles, with smaller particles sensed only as a differential optical depth between the two wavelengths of radio waves transmitted through the rings. As the rings are thin relative to meter-sized particles, even when considering the slant-path at low incidence angles, Zebker et al. could assume N was small and searched for the value of N which best agreed with the data. However, in the case of millimeter-sized particles, the rings are no longer physically thin relative to the particle diameter, even at normal incidence angles. Thus, rather than N being a few, it becomes on the order of a thousand. If we let N become large, then the equation becomes We will be using this equation to include the effects of double- and tripleparticle scattering. Higher order terms are small relative to these terms, so were omitted. From Figure 13, we can see that double-particle scattering produces the dominant effect at angles larger than the ∼ 0.5 milliradians that marks the size of the solar image (and, thus, the minimum angle required to remove light from the signal), confirmation of the necessity of accounting for multiple-particle scattering. Conceptually, this can be explained as the more times a photon is scattered, the broader the diffraction cone becomes. If little light is being singly scattered at a certain angle, doubly scattered light will dominate if the ring is optically thick enough. The decrease in intensity from single and double-particle scattering to triple-particle scattering justifies our neglect of higher-order terms.", "pages": [ 34, 35 ] }, { "title": "6.4 Measuring Diffracted Light in the A Ring", "content": "Now that we have discussed the complicating effects of multiple-order scattering and self-gravity wakes, we can add them to the model. Note that the two effects to an extent work against each other: multiple-order scattering will increase the amount of scattered light for a given optical depth, while self-gravity wakes will lower the material available to scatter light, which will decrease the scattered light in general (as well as add a longitude-dependent term). It is not obvious which (if either) effect will dominate at the scales we are interested in for this problem. To model the A Ring, we had to choose parameters to represent the selfgravity wakes. The wake dimensions of W/λ =0.5 and H/λ =0.1 were chosen as representative parameters from the stellar occultation data discussed in Section 6.2. Individual values of τ G for each cube were calculated based on those numbers and assuming T = (1 -f W ) e -τ G / 2 sin B , with T being the calculated transmission in that cube and f W calculated from Equation 20. As before, the values of f were averaged over the entire A ring, and binned spectrally. Figure 14 shows the binned and rescaled measurements of f for five occultations, with representative models. For a comparison, a wakeless model using the full observed optical depth (but including multiple scattering), is also shown in Figure 15. Of the five clear positive detections mentioned in Section 5.3 the diffracted light measurements were larger than we'd expect from models for the Rev. 59 and Rev. 62 occultations. Below a min ≈ 100 microns, the fraction of light removed from the direct signal becomes nearly constant, as the models are no longer dominated by the large-angle 'tails' of diffraction from the millimetersized and larger particles in the ring. Rev. 62's measurements only allow an upper limit on a min to be set, rather than having a value that best agrees with the data, and the data from Rev. 59 are inconsistent with the model entirely for the value of q used. Omitting the effects of self-gravity wakes, as in Figure 15, changes the minimum particle size corresponding to a given value of f , but still cannot reproduce the Rev. 59 observations. To better quantify our results, we again calculated the mean a min over the three occultations (Revs. 9, 43 and 55) for which a clear detection (rather than an upper bound) was observed, as a function of q from the fraction of scattered light observed at 2.3 microns, just as we did for the C ring. The results are shown in Figure 16. Using the diffraction model that accounts for both the effects of self-gravity wakes on optical depth and double and triple particle scattering, we infer that the minimum particle size is 0 . 56 +0 . 35 -0 . 16 mm at a power law index of 2.9, the index inferred by the Voyager Radio Science experiment (Zebker et al. , 1985). The shallower q = 2 . 75 power law index observed by French & Nicholson (2000) lowers the minimum particle size to an upper limit of < 0 . 18 mm. Including the Rev 59 and 62 occultations in the mean a min lowers these values further to 0 . 38 +0 . 27 -0 . 12 mm at q = 2 . 9, but cannot replicate all the observations using q = 2 . 75. Both the homogenous ring and wake model give a minimum particle size somewhat smaller for expected values of q (between 2.7 and 3.0) than those seen by the Cassini RSS measurements and French & Nicholson's observation of few sub-centimeter-sized particles in the 28 Sgr occultation (Marouf et al. , 2008; French & Nicholson, 2000). Zebker et al. (1985) note that the difference in optical depth between that measured at λ =3.6 cm by Voyager and that measured at 0.5 µ mis large enough to suggest the existence of a substantial population of sub-centimeter sized particles, but a significant difference in optical depth between the 3.6 and 0.9 cm bands in the A Ring was not seen by Cassini RSS occultations(Marouf et al. , 2008), implying few particles in the centimeter size range. A major caveat to all of these studies is that none of them accounted for the effects of self-gravity wakes, though Zebker et al. (1985) and Marouf et al. (2008) both included analysis of multiple scattering effects. French & Nicholson (2000) even notice what could have been a longitudinal asymmetry in optical depth in the A Ring between the δ Sco and 28 Scr optical depths, but, without a model, chose to adopt a 'fudge factor' to scale the two occultations as best they could. A model of the A Ring that include self-gravity wakes would lower expected differential optical depths between all wavelengths smaller than the wake size, as a fraction of the optical depth would be caused by the wakes themselves, rather than the continuum of ring particles. Therefore, a wakeless model would find larger minimum particle sizes for a given differential optical depth than a model that included self-gravity wakes. It is also worth mentioning that our (and others') observations derive distributions for the material in-between the wakes, which may be different in size distribution from the ring as a whole. Using the three-occultation mean, our model requires < 12 . 1 % of the interwake optical depth to be from particles smaller than 1 cm at q = 2 . 75, which increases to 20 . 1 +4 . 2 -1 . 2 %for q = 2 . 9. For typical interwake optical depths used earlier ( τ G between 0.3 and 0.65 in extinction), this gives extinction optical depths due to such small particles of between 0.03 and 0.16, within Zebker et al. 's range.", "pages": [ 35, 37, 41 ] }, { "title": "7 Conclusions", "content": "When analyzing the solar occultation data recorded by Cassini-VIMS, we observed a small excess of forward-scattered light, once instrumental effects were taken into account. We believe this to be due to diffraction by small particles in the rings and have used it to estimate minimum particle sizes, assuming a power law index, q , and maximum particle size from previous work (Zebker et al. , 1985; French & Nicholson, 2000; Marouf et al. , 2008). Among the three C Ring solar occultations in which a clear positive excess was measured, a minimum particle size of 4 . 1 +3 . 8 -1 . 3 mm is inferred for a canonical value of q = 3 . 1. For a wider range of likely q s, the data still indicate a minimum particle size between 3 and 10 mm. This is somewhat larger than the a min ≈ 4 mm measured by Marouf et al. (2008) using the Cassini Radio Science experiment, and it's possible this could be due to a radial variation of minimum particle size in the C Ring, as the chord occultations (Rev. 62 and 65) show a larger minimum than the Rev. 09 radial occultation. Further work would be required to confirm such a variation. In the A Ring observations, multiple-particle scattering produces a nonnegligible effect due to the larger optical depths involved, and must be taken into account to explain the larger-than-expected amount of scattered light seen. The effects of the A Ring's self-gravity wakes on the amount of scattering are more complicated, but are clearly seen in optical depth measurements of the A Ring from both these solar occultations and other data sets (such as stellar occultations). The shallow power law indices of q = 2 . 75 found by French & Nicholson (2000) and Marouf et al. (2008) require a very small a min of < 0 . 34 mm to explain our observations, even accounting for multiple scattering and self-gravity wakes. Raising the power law index to q = 2 . 9 as measured by the Voyager radio occultations (Zebker et al. , 1985) still requires particles of 0 . 56 +0 . 35 -0 . 16 mm to explain the amount of scattered light measured by our solar occultation observations. These numbers appear to be inconsistent with estimates of a lack of material smaller than one centimeter advanced by French & Nicholson (2000), but the shallow power law and amount of material sequestered in self-gravity wakes may mean the optical depth required in particles smaller than 10 mm could be as small as τ = 0 . 03 in extinction. This may render our data consistent with this lack of optical depth variation with wavelength seen in radio occultations, especially when the effects of self-gravity wakes are taken into account. We were also able to constrain the fraction of free-floating ice grains smaller than 100 µ m in the A ring to be ≤ 5%, assuming a dust size distribution similar to the F Ring. The fraction within the C ring was even smaller; ≤ 1 . 4%. Regardless of their minimum particle sizes, it is clear that the A and C Rings lack the persistent icy dust that is a strong feature of the F Ring.", "pages": [ 41, 42 ] }, { "title": "A Phase Functions", "content": "For a single-size particle distribution, the forward-scattering, or diffraction, phase function is given by (Liou, 1980) where we introduce the dimensionless variable z = 2 πa sin θ/λ , a being the radius of the particles and λ being the wavelength observed. J 1 ( z ) is the first-order Bessel function of the first kind. Integrating Equation 24 over a truncated power law distribution of particle sizes, dn/da = n 0 ( a/a 0 ) -q , where a min ≤ a ≤ a max and n 0 and a 0 are constants that can be folded into the value of τ , we find /negationslash The usual dimensionless size parameter x is defined by x = 2 πa/λ , with subscripts denoting the limiting values of a . The mean phase function (Equation 25) can be conveniently approximated in different limiting cases, as the full function can be computationally expensive to integrate. The limiting cases are set by the relevant angles in the problem, which are determined by the ratio of particle size to wavelength (as quantified by x ). Let the minimum characteristic diffraction angle - the angle where the largest particles will be diffracting light - be θ 1 = πx -1 max . Similarly, we define the maximum characteristic diffraction angle (where the smallest particles will be diffracting light) as θ 2 = πx -1 min . Two angles give us three cases to consider, but only two are of real interest in this case. Small-angle diffraction - where the angles we observe at are all smaller than θ 1 - isn't relevant here, as the upper boundary of the ring particle size-distribution in the A and C Rings extends to 5m in radius (Zebker et al. , 1985), and at near infrared wavelengths (0.9 to 5.2 µ m), this corresponds to a θ 1 of tenths of microradians. Thus we either have a case of where medium-angle diffraction (the angles we observe are between θ 1 and θ 2 ) or large-angle diffraction (all angles observed are larger than θ 2 ). The value of θ 2 is unknown, because the minimum particle size is the quantity we are trying to measure. Given that the size of one VIMS pixel and coincidentally the solar radius at 9 AU - is 0.5 milliradians on the sky, our data will be most sensitive to diffraction by particles with x /lessorsimilar 6000, or, at 2 microns wavelength, particle sizes of 2 millimeters or less. Barring a much-lower-than-expected minimum size cutoff, the large-angle scattering case will be most relevant, though we will include the medium-angle case in our calculations to account for the possibility of free-floating particles from ∼ 100 µ m to ∼ 2 millimeters. For the large-angle case, (i.e. θ /greatermuch θ 2 ), all particles are scattering most of their light at angles smaller than those we are measuring. Thus the bounds on the integral of Equation 25 are both much larger than unity. We can then use the approximation J 1 ( z ) ≈ √ 2 /πz cos( z -3 π/ 4), giving Because the particle size distribution is very broad (remember we're dealing with particles with radii from millimeters to meters in size), we also know that x max /greatermuch x min , and both are very large. So, a further approximation is to drop the x 2 -q max term (which will be very small as long as q > 2), which leaves the simpler expression In the case of medium angle diffraction (i.e. θ 1 /lessmuch θ /lessmuch θ 2 ), we again use a broad particle size distribution to approximate a phase function. Because of this distribution and an angle ( θ ) that is between the minimum and maximum characteristic diffraction angle, we are mostly sampling light neither from the smallest nor the largest particles, but from medium-sized ring particles that have that characteristic diffraction angles. Because θ is much smaller than the maximum ( θ 2 ), we can assume that z min = π sin θ/θ 2 is much less than unity, and because θ is much larger than the minimum ( θ 1 ), we can assume that z max = π sin θ/θ 1 is much greater than unity. We can then approximate the integral in Equation 25, as covering the full range of positive values of z , from zero to infinity, as most of the power is around z ≈ 1. This leads to a constant that is only dependent on q , allowing the integral to be calculated once per q . Thus, we have the approximation The J ∞ 0 ( q ) in Equation 29 is shorthand for ∫ ∞ 0 z 2 -q J 1 ( z ) 2 dz . It is nearly constant over the range of 2 ≤ q ≤ 5, except when q approaches 2 or 5. Previous studies indicate that q is between 2.7 and 3.1 within the main rings, giving J ∞ 0 ≈ 0 . 5 (Zebker et al. 1985, French & Nicholson 2000, Cuzzi et al. 2009).", "pages": [ 43, 44, 45 ] }, { "title": "References", "content": "Bodrova, Anna, Schmidt, Jurgen, Spahn, Frank, & Brilliantov, Nikolai V. 2012. Adhesion and collisional release of particles in dense planetary rings. Icarus , 218 (1), 60-68. Brown, Robert H, Baines, Kevin H, Bellucci, G, Bibring, J-P, Buratti, Bonnie J, Capaccioni, F, Cerroni, P, Clark, Roger N, Coradini, Angioletta, Cruikshank, Dale P, Drossart, Pierre, Formisano, V, Jaumann, Ralf, Langevin, Y, Matson, Dennis L, McCord, Thomas B, Mennella, V, Miller, E, Nelson, Robert M, Nicholson, Philip D, Sicardy, Bruno, & Sotin, C. 2004. The Cassini Visual and Infrared Mapping Spectrometer (VIMS) investigation. Space Science Reviews , 115 , 111-168. Colwell, Joshua E, Esposito, Larry W, & Sremˇcevi'c, Miodrag. 2006. Selfgravity wakes in Saturn's A ring measured by stellar occultations from Cassini. Geophysical Research Letters , 33 (7), 7201-7204. Cuzzi, Jeffrey N. 1985. Rings of Uranus - Not so thick, not so black. Icarus , 63 (Aug.), 312-316. Cuzzi, Jeffrey N, Clark, Roger N, Filacchione, Gianrico, French, Richard G, Johnson, Robert E, Marouf, Essam A, & Spilker, Linda J. 2009. Ring Particle Composition and Size Distribution. Saturn from Cassini-Huygens , 459-509. D'Aversa, E, Bellucci, G, Nicholson, Philip D, Hedman, Matthew M, Brown, Robert H, Showalter, Mark R, Altieri, F, Carrozzo, F G, Filacchione, Gianrico, & Tosi, F. 2010. The spectrum of a Saturn ring spoke from Cassini/VIMS. Geophysical Research Letters , 37 (1), 1203-1207. Dones, Luke, Cuzzi, Jeffrey N, & Showalter, Mark R. 1993. Voyager photometry of Saturn's A ring. Icarus , 105 (Sept.), 184-215. French, Richard G, & Nicholson, Philip D. 2000. Saturn's Rings II. Particle sizes inferred from stellar occultation data. Icarus , 145 , 502-523. Hedman, Matthew M, Nicholson, Philip D, Salo, Heikki J, & Wallis, B D. 2007a. Selfgravity wake structures in Saturn's A ring revealed by Cassini VIMS. The Astrophysical Journal , 133 (June), 2624-2629. Hedman, Matthew M, Burns, Joseph A, Tiscareno, Matthew S, & Porco, Carolyn C. 2007b. The heliotropic rings of Saturn. American Astronomical Society , 39 (Oct.), 427-427. Hedman, Matthew M, Burt, Jennifer A, Burns, Joseph A, & Tiscareno, Matthew S. 2010. The shape and dynamics of a heliotropic dusty ringlet in the Cassini Division. Icarus , 210 (1), 284-297. Hedman, Matthew M, Nicholson, Philip D, Showalter, Mark R, Brown, Robert H, Buratti, Bonnie J, Clark, Roger N, Baines, Kevin H, & Sotin, C. 2011. The Christiansen Effect in Saturn's narrow dusty rings and the spectral identification of clumps in the F ring. Icarus , 215 (2), 695-711. Liou, K N. 1980. An introduction to atmospheric radiation. An introduction to atmospheric radiation. Marouf, Essam A, Tyler, G L, Zebker, H A, Simpson, R A, & Eshleman, V R. 1983. Particle size distributions in Saturn's rings from Voyager 1 radio occultation. Icarus , 54 , 189-211. Marouf, Essam A, French, R, Rappaport, N, Wong, K, McGhee, C, & Anabtawi, A. 2008. Physical properties of Saturn's rings from Cassini radio occultations. Page 429 of: AAS/Division for Planetary Sciences Meeting Abstracts . Mitchell, C J. 2006. Saturn's spokes: Lost and found. Science , 311 (5767), 1587-1589. Mitchell, C J, Porco, Carolyn C, Dones, H L, & Spitale, J N. 2013. The behavior of spokes in Saturn's B ring. Icarus , 225 (1), 446-474. Nicholson, Philip D, & Hedman, Matthew M. 2010. Self-gravity wake parameters in Saturn's A and B rings. Icarus , 206 (Apr.), 410-423. Nicholson, Philip D, Hedman, Matthew M, Clark, Roger N, Showalter, Mark R, Cruikshank, Dale P, Cuzzi, Jeffrey N, Filacchione, Gianrico, Capaccioni, F, Cerroni, P, Hansen, Gary B, Sicardy, Bruno, Drossart, Pierre, Brown, Robert H, Buratti, Bonnie J, Baines, Kevin H, & Coradini, Angioletta. 2008. A close look at Saturn's rings with Cassini VIMS. Icarus , 193 , 182-212. Roques, F, Moncuquet, M, & Sicardy, B. 1987. Stellar occultations by small bodies - Diffraction effects. Astronomical Journal , 93 (June), 1549-1558. Salo, Heikki J. 1992. Gravitational wakes in Saturn's rings. Nature , 359 (Oct.), 619-621. Spilker, Linda J, Edgington, S G, Wallis, B D, Brooks, S M, Pearl, J C, & Flasar, F M. 2005. Cassini CIRS observations of a roll-off in Saturn ring spectra at submillimeter wavelengths. Earth , 96 (June), 149-163. Tiscareno, Matthew S, Perrine, Randall P, Richardson, Derek C, Hedman, Matthew M, Weiss, John W, Porco, Carolyn C, & Burns, Joseph A. 2010. An analytic parameterization of self-gravity wakes in Saturn's rings, with application to occultations and propellers. Astronomical Journal , 139 (2), 492-503. Vahidinia, Sanaz, Cuzzi, Jeffrey N, Hedman, Matt, Draine, Bruce, Clark, Roger N, Roush, Ted, Filacchione, Gianrico, Nicholson, Philip D, Brown, Robert H, Buratti, Bonnie J, & Sotin, Christophe. 2011. Saturn's F ring grains: Aggregates made of crystalline water ice. Icarus , 215 (2), 682-694. Zebker, H A, Marouf, Essam A, & Tyler, G L. 1985. Saturn's rings - Particle size distributions for thin layer model. Icarus , 64 , 531-548.", "pages": [ 45, 46, 47 ] } ]

[ { "title": "Temperature-dependent molecular absorption cross sections for exoplanets and other atmospheres", "content": "Christian Hill, Sergei N. Yurchenko and Jonathan Tennyson Department of Physics and Astronomy, University College London, Gower Street, WC1E 6BT London, UK", "pages": [ 1 ] }, { "title": "Abstract", "content": "Exoplanets, and in particular hot ones such as hot Jupiters, require a very significant quantities of molecular spectroscopic data to model radiative transport in their atmospheres or to interpret their spectra. This data is commonly provided in the form of very extensive transition line lists. The size of these line lists is such that constructing a single model may require the consideration of several billion lines. We present a procedure to simplify this process based on the use of cross sections. Line lists for water, H + 3 , HCN /HNC and ammonia have been turned into cross sections on a fine enough grid to preserve their spectroscopic features. Cross sections are provided at a fixed range of temperatures and an interpolation procedure which can be used to generate cross sections at arbitrary temperatures is described. A web-based interface (www.exomol.com/xsec) has been developed to allow astronomers to download cross sections at specified temperatures and spectral resolution. Specific examples are presented for the key water molecule. Keywords: Atmospheres, composition, Extra-solar planets, Infrared observations, Radiative transfer", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "With the growing realization that many, probably most, stars support exoplanets, developing the means to systematically characterize the atmospheres of these planets has become a major scientific priority (Tinetti et al., 2012). Given the likely complex chemistry of these atmospheres and the elevated temperature that is found in the most observable planets, there is a significant demand for spectroscopic data on the probable exoplanet atmospheric constituents. Recently we have launched a new project, called ExoMol (see www.exomol.com), with the aim of providing molecular transition data appropriate for exoplanet models which is reliable over a wide range of temperatures (Tennyson and Yurchenko, 2012). The ExoMol project involves constructing line lists of spectroscopic transitions for key molecules which are valid over the entire temperature and wavelength domain that is likely to be astrophysically important for these species. Especially for polyatomic molecules, these line lists can become very large with hundreds of millions (Harris et al., 2006; Barber et al., 2006; Voronin et al., 2010; Tashkun and Perevalov, 2011) or even billions (Yurchenko et al., 2011) of individual transitions needing to characterized and stored. A complete linelist for methane, for which so far only a preliminary version is available (Warmbier et al., 2009), can be expected to be even larger. Indeed potential line lists for larger species, such as higher hydrocarbons, for which spectroscopic data is needed for exoplanetary research, are likely to be so large as to potentially make their use impractical. Molecular line lists are being actively used to model the spectra of exoplanets (eg Beaulieu et al. (2011)) and cool brown dwarfs with similar tem- peratures (eg Lucas et al. (2010); Cushing et al. (2011)). However, sampling billions of individual transitions to model relatively low resolution astronomical spectra is probably not necessary in many cases. An alternative approach is to represent the molecular absorptions as cross sections generated at an appropriate resolution and temperature. The advantage of this approach is that the data handling issues related to dealing with large data sets largely disappear. The disadvantage is that cross sections are inflexible - a particular cross section set is only valid for a single state of temperature and pressure. Cross sections are therefore often regarded a second choice compared to maintaining a full line list (Rothman et al., 2009). In this paper we develop a strategy whereby cross sections are provided for the user in a flexible fashion without loss of accuracy or the specificity of using a complete line list. To this end we have provided a web application which, starting from very high resolution cross sections generated for each molecule at a range of temperatures, can provide cross sections at a temperature and resolution specified by the user. Of course this approach is based on the implicit assumption of local thermodynamic equilibrium (LTE) and any nonLTE treatment will continue to have to rely on the explicit use of transition line lists. So far, these cross sections do not consider collisional broadening effects and are therefore, at their highest resolution, appropriate for the zero pressure limit only. The line lists for water (Barber et al., 2006; Voronin et al., 2010), H + 3 (Neale et al., 1996; Sochi and Tennyson, 2010), HCN /HNC (Harris et al., 2002, 2006, 2008) and ammonia (Yurchenko et al., 2011) were used to generate cross sections for these species. For concreteness, this work uses the main water isotopologue, H 2 16 O, as its working example. Water is known to be a key species in exoplanetary atmospheres and the BT2 line list has been used in studies of exoplanets (Tinetti et al., 2007; Swain et al., 2009; Tinetti et al., 2010a; Baraffe et al., 2010; Tinetti et al., 2010b; Shabram et al., 2011; Barman et al., 2011; Tessenyi et al., 2012) as well in a large variety of planetary (Bykov et al., 2008; Chesnokova et al., 2009; Bailey, 2009), astrophysical (Warren et al., 2007; Dello Russo et al., 2004, 2005; Burgasser et al., 2008; Barber et al., 2009; Lyubchik et al., 2007; Banerjee et al., 2005) and, indeed, engineering (Kranendonk et al., 2007; Lindermeir and Beier, 2012) studies which generally focus on the radiative transport by hot water. The BT2 line list was used as part of the recently updated HITEMP database (Rothman et al., 2010). In that work the size of the line list was reduced using a technique based upon importance sampling at a range of key temperatures. In practice the number of water lines in HITEMP remains large, over 100 million. The calculation of opacities and other spectral properties due to water vapour at these elevated temperatures can therefore become onerous, and so we present here pre-calculated absorption cross sections for a range of temperatures between 296 K and 3000 K, binned to different resolutions. The highest resolution cross sections are suitable for modelling low-density environments where only Doppler broadening contributes to the line width whereas by binning to a wavenumber grid spacing significantly larger than the pressure-broadened half-width, higher-density environments are described well by the calculated cross sections. However, no attempt is made to include contributions to the opacity from the water vapour continuum or water dimer absorption.", "pages": [ 2, 3, 4, 5 ] }, { "title": "2. Method", "content": "The high-resolution cross section is calculated on an evenly-spaced wavenumber grid, ˜ ν i , defining bins of width ∆˜ ν . Only Doppler broadening is considered so each absorption line has a Gaussian shape: where the line centre position is ˜ ν 0; j and the Doppler half-width at halfmaximum, at temperature T for a molecule of mass m . The contribution to the cross section within each bin is a sum over contributions from individual lines: where where erf is the error function and are the scaled limits of the wavenumber bin centred on ˜ ν i relative to the line centre, ˜ ν 0; j , and the line intensity in units of cm -1 / (molecule cm -2 ) is Here, g ' j and E j '' are the upper-state degeneracy and lower-state energy respectively, A j is the Einstein A coefficient for the transition and c 2 ≡ hc/k B is the second radiation constant. For H 2 16 O, the molecular partition func- ( T ), was obtained from the tabulated values of Vidler and Tennyson (2000). Note that in the limit of ∆˜ ν glyph[greatermuch] α j , eqn (4) reduces to whereas for ∆˜ ν glyph[lessmuch] α j , However, the exact expression in all calculations of the cross sections presented in this work.", "pages": [ 5, 6, 7 ] }, { "title": "3. Results", "content": "The absorption cross section of H 2 16 O was calculated between 10 cm -1 and 30000 cm -1 across the temperature range 296 K - 3000 K (Table 1), using the wavenumber grid-spacing given in Table 2. For comparison with experimental spectra, low-resolution cross sections were produced by binning the high-resolution cross sections to the following fixed grid spacing across the entire wavenumber range: ∆˜ ν = 0 . 01, 0.1, 1, 10, 100 cm -1 . At these resolutions, the structure due to individual lines is lost and direct comparison can be made with, for example, the experimental water vapour cross sections of the PNNL database (Sharpe et al., 2004). Such a comparison for the ∆˜ ν = 10 cm -1 resolution spectra is shown in Figure 2.", "pages": [ 7, 8 ] }, { "title": "4. Interpolation of cross sections between temperatures", "content": "For use in the web-based application described below, cross sections for the molecules given in Table 5 have been calculated using a wavenumber grid spacing of 0 . 01 cm -1 at a range of temperatures in 100 K intervals below 1000 K and 200 K intervals above 1000 K. A cross section at some intermediate temperature between the values at which the stored cross sections have been calculated may be obtained by interpolation. Suppose that σ i ( T 1 ) and σ i ( T 2 ) are calculated cross sections at temperatures which bracket the temperature of the desired cross section: T 1 < T < T 2 (we consider interpolation using only σ i calculated at the two temperatures closest to T ). One possible interpolation strategy is the linear interpolation However, we find a more accurate approach is to estimate the temperature dependence to be of the form where the coefficients a i and b i at each wavenumber bin may be calculated from The largest values of the interpolation residual error in the region 1000 20000 cm -1 , calculated as δσ i = σ i, calc -σ i, interp , are found to be associated with the ν 2 band - as an illustration, this is plotted in Figure 3 at 350 K. The maximum value of the interpolation residual across this wavenumber region at a range of temperatures and wavenumber binning intervals is given in Table 3, expressed as a percentage of the corresponding absorption cross section: In all cases, δσ % max is found to be less than the estimated uncertainty in the ab initio line intensities that the cross section calculation is based on. Interpolation is performed on the 0.01 cm -1 grid before binning to a coarser wavenumber grid, if required. Finally we note that Hargreaves et al. (2012) recently presented an experimental ammonia spectrum recorded at a range of temperatures at 100 K intervals. We suggest that our proposed interpolation scheme would also be appropriate for interpolating their data.", "pages": [ 8, 9, 10 ] }, { "title": "5. Web based application", "content": "Calculated absorption cross sections can be obtained from the interface at the url http://www.exomol.com/xsecs. The user of this web-based interface can select a wavenumber range, temperature and wavenumber grid spacing; using these parameters the interface software first obtains a high-resolution cross section at the desired temperature by the interpolation procedure described in the previous section on the pre-calculated spectra, and then bins this interpolated cross section to the requested wavenumber grid. Cross sections are returned as a list of floating point numbers in a text file, separated by the Unix-style newline character, LF (' \\ n', 0x0A ). The wavenumber grid can be generated from the linear sequence where the total number of points in the requested cross section is We also provide an XML file in XSAMS format (Dubernet et al., 2009), compatible with the standards of the VAMDC project (Dubernet et al., 2010). This file may be thought of as a 'wrapper' to the cross section data, providing contextual metadata such as the molecular identity and structure, temperature of the calculation, and wavenumber limits and grid spacing. An example of the format is given in Table 4. Cross section files have been generated for the polyatomic line lists currently available on the ExoMol website. These are listed in Table 5. The table also specifies the maximum wavenumber (˜ ν max ) and maximum temperature ( T max ) for each species; we strongly caution against relying on the cross sections or indeed the underlying line lists at temperatures greater than those given. Further cross sections will be provided as line lists for new species as they become available.", "pages": [ 10, 11 ] }, { "title": "6. Conclusion", "content": "High resolution absorption cross sections have been calculated for a number of molecules likely to be important in the atmospheres of exoplanets. The online interface provided at the ExoMol website (www.exomol.com) allows customized cross sections for a given molecular species to be returned at a specified temperature and resolution. Cross sections are only available for those species for which extensive line lists exist. New cross sections will be provided as further species are added to the ExoMol database, see Tennyson and Yurchenko (2012) for example. Table 4: Sample XSAMS format (Dubernet et al., 2009) XML wrapper for a cross section for H 2 16 O generated from 1000 to 20000 cm -1 in steps of 1 cm -1 at a temperature of 296 K. In this example, the cross section itself is provided in the file H2O 1000-20000 296K-10.0.sigma as a column of values, here in cm 2 , one for each of 901 grid points. The absorption cross section for H2O at 296.0 K, calculated at Sun Mar 11 19:50:45 2012, retrieved from www.exomol.com/xsecs H2O_1000-20000_296K-10.0.sigma H2O_1000-20000_296K-10.0.sigma XEXOMOL-XLYOFNOQVPJJNP-FNDQEIABSA-N XEXOMOL-XLYOFNOQVPJJNP-FNDQEIABSA-N It is our intention to make the cross section facility in ExoMol fully interoperable with other spectroscopic databases as part of the VAMDC (Virtual Atomic and Molecular Data Centre) project (Dubernet et al., 2010). Work in this direction will be reported in due course.", "pages": [ 11, 12, 13 ] }, { "title": "Acknowledgements", "content": "We thank Giovanna Tinetti and Bob Barber for many helpful discussions during the course of this work. This work was performed as part of ERC Advanced Investigator Project 267219 and the project VAMDC which is funded by the European Union INFRA-2008-1.2.2 Scientific Data Infrastructure program under Grant Agreement number 239108.", "pages": [ 13 ] }, { "title": "References", "content": "Bailey, J., 2009. A comparison of water vapor line parameters for modeling the venus deep atmosphere. Icarus 201, 444 - 453. Banerjee, D. P. K., Barber, R. J., Ashok, N. K., Tennyson, J., 2005. Nearinfrared water lines in V838 Monocerotis. Astrophys. J. 672, L141-L144. Baraffe, I., Chabrier, G., Barman, T., 2010. The physical properties of extrasolar planets. Rep. Prog. Phys. 73, 016901. Barber, R. J., Miller, S., Dello Russo, N., Mumma, M. J., Tennyson, J., Guio, P., 2009. Water in the near IR spectrum of comet 8P/Tuttle. Mon. Not. R. Astr. Soc. 398, 1593-1600. Barber, R. J., Tennyson, J., Harris, G. J., Tolchenov, R. N., 2006. A high accuracy computed water line list. Mon. Not. R. Astr. Soc. 368, 1087-1094. Barman, T. S., Macintosh, B., Konopacky, Q. M., Marois, C., 2011. Clouds and chemistry in the atmosphere of extrasolar planet hr8799b. Astrophys. J. 733, 65. Beaulieu, J. P., Tinetti, G., Kipping, D., Ribas, I., Barber, R. J., Cho, J. Y.-K., Polichtchouk, I., Tennysson, J., Yurchenko, S. N., Griffith, C. A., Waldmann, I., Miller, S., Carey, S., Mousis, O., Fossey, S. J., Aylward, A., 2011. Probing the atmosphere of the transiting hot Neptune GJ436b. Astrophys. J. 731, 18. Burgasser, A. J., Liu, M. C., Ireland, M. J., Cruz, K. L., Dupuy, T. J., 2008. Subtle signatures of multiplicity in late-type dwarf spectra: The unresolved m8.5+t5 binary 2mass j03202839-0446358. Astrophys. J. 681, 579-593. Bykov, A. D., Lavrientieva, N. N., Mishina, T. P., Sinitsa, L. N., Barber, R. J., Tolchenov, R. N., Tennyson, J., 2008. Water vapor line width and shift calculations with accurate vibration-rotation wave functions. J. Quant. Spectrosc. Radiat. Transf. 109, 1834-1844. Chesnokova, T. Y., Voronin, B. A., Bykov, A. D., Zhuravleva, T. B., Kozodoev, A. V., Lugovskoy, A. A., Tennyson, J., 2009. Calculation of solar radiation atmospheric absorption with different H 2 O spectral line data banks. J. Mol. Spectrosc. 256, 41-44. Cushing, M. C., Kirkpatrick, J. D., Gelino, C. R., Griffith, R. L., Skrutskie, M. F., Mainzer, A., Marsh, K. A., Beichman, C. A., Burgasser, A. J., Prato, L. A., Simcoe, R. A., Marley, M. S., Saumon, D., Freedman, R. S., Eisenhardt, P. R., Wright, E. L., 2011. The discovery of y dwarfs using data from the wide-field infrared survey explorer (WISE). Astrophys. J. 743. Dello Russo, N., Bonev, B. P., DiSanti, M. A., Gibb, E. L., Mumma, M. J., Magee-Sauer, K., Barber, R. J., Tennyson, J., 2005. Water production rates, rotational temperatures and spin temperatures in comets C/1999 H1 (Lee), C/1999 S4 and C/2001 A2. Astrophys. J. 621, 537-544. Dello Russo, N., DiSanti, M. A., Magee-Sauer, K., Gibb, E. L., Mumma, M. J., Barber, R. J., Tennyson, J., 2004. Accurate rotational temperature retrievals from hot-band line near 2.9 µ m. Icarus 168, 186-200. Dubernet, M. L., Boudon, V., Culhane, J. L., Dimitrijevic, M. S., Fazliev, A. Z., Joblin, C., Kupka, F., Leto, G., Le Sidaner, P., Loboda, P. A., Mason, H. E., Mason, N. J., Mendoza, C., Mulas, G., Millar, T. J., Nunez, L. A., Perevalov, V. I., Piskunov, N., Ralchenko, Y., Rixon, G., Rothman, L. S., Roueff, E., Ryabchikova, T. A., Ryabtsev, A., Sahal-Brechot, S., Schmitt, B., Schlemmer, S., Tennyson, J., Tyuterev, V. G., Walton, N. A., Wakelam, V., Zeippen, C. J., 2010. Virtual Atomic and Molecular Data Centre. J. Quant. Spectrosc. Radiat. Transf. 111, 2151-2159. Dubernet, M. L., Humbert, D., Ralchenko, Y., Roueff, E., Schultz, D. R., Chung, H.-K., Braams, B., 2009. XSAMS: XML Schema for Atomic, Molecular and Solid Data, version 0.1. Internation Atomic Energy Authority, http://www-amdis.iaea.org/xsams/documents/ . Hargreaves, R. J., Li, G., Bernath, P. F., 2012. Ammonia line lists from 1650 to 4000 cm -1 . J. Quant. Spectrosc. Radiat. Transf. 113, 670-679. Harris, G. J., Larner, F. C., Tennyson, J., Kaminsky, B. M., Pavlenko, Y. V., Jones, H. R. A., 2008. A H 13 CN/HN 13 C linelist, model atmospheres and synthetic spectra for carbon stars. Mon. Not. R. Astr. Soc. 390, 143-148. Harris, G. J., Polyansky, O. L., Tennyson, J., 2002. Opacity data for HCN and HNC from a new ab initio linelist. Astrophys. J. 578, 657-663. Harris, G. J., Tennyson, J., Kaminsky, B. M., Pavlenko, Y. V., Jones, H. R. A., 2006. Improved HCN/HNC linelist, model atmsospheres syntheic spectra for WZ Cas. Mon. Not. R. Astr. Soc. 367, 400-406. Kranendonk, L. A., An, X., Caswell, A. W., Herold, R. E., Sanders, S. T., Huber, R., Fujimoto, J. G., Okura, Y., Urata, Y., 2007. High speed engine gas thermometry by fourier-domain mode-locked laser absorption spectroscopy. Opt. Express 15, 15115-15128. Lindermeir, E., Beier, K., 2012. Hitemp derived spectral database for the prediction of jet engine exhaust infrared emission using a statistical band model. J. Quant. Spectrosc. Radiat. Transf. 113, 1575 - 1593. URL http://www.sciencedirect.com/science/article/pii/ S0022407312001847 Lucas, P. W., Tinney, C. G., Burningham, B., Leggett, S. K., Pinfield, D. J., Smart, R., Jones, H. R. A., Marocco, F., Barber, R. J., Yurchenko, S. N., Tennyson, J., Ishii, M., Tamura, M., Day-Jones, A. C., Adamson, A., Allard, F., Homeier, D., 2010. The discovery of a very cool, very nearby brown dwarf in the Galactic plane. Mon. Not. R. Astr. Soc. 408, L56-L60. Lyubchik, Y., Jones, H. R. A., Pavlenko, Y. V., Martin, E., McLean, I., Prato, L., Barber, R. J., Tennyson, J., 2007. Spectral analysis of high resolution near-infrared spectra of ultracool dwarfs. Astron. Astrophys. 473, 257-264. Neale, L., Miller, S., Tennyson, J., 1996. Spectroscopic properties of the H + 3 molecule: a new calculated linelist. Astrophys. J. 464, 516-520. Rothman, L. S., Gordon, I. E., Barbe, A., Benner, D. C., Bernath, P. E., Birk, M., Boudon, V., Brown, L. R., Campargue, A., Champion, J. P., Chance, K., Coudert, L. H., Dana, V., Devi, V. M., Fally, S., Flaud, J. M., Gamache, R. R., Goldman, A., Jacquemart, D., Kleiner, I., Lacome, N., Lafferty, W. J., Mandin, J. Y., Massie, S. T., Mikhailenko, S. N., Miller, C. E., Moazzen-Ahmadi, N., Naumenko, O. V., Nikitin, A. V., Orphal, J., Perevalov, V. I., Perrin, A., Predoi-Cross, A., Rinsland, C. P., Rotger, M., Simeckova, M., Smith, M. A. H., Sung, K., Tashkun, S. A., Tennyson, J., Toth, R. A., Vandaele, A. C., Vander Auwera, J., 2009. The HITRAN 2008 molecular spectroscopic database. J. Quant. Spectrosc. Radiat. Transf. 110, 533-572. Rothman, L. S., Gordon, I. E., Barber, R. J., Dothe, H., Gamache, R. R., Goldman, A., Perevalov, V. I., Tashkun, S. A., Tennyson, J., 2010. HITEMP, the High-Temperature Molecular Spectroscopic Database. J. Quant. Spectrosc. Radiat. Transf. 111, 2139-2150. Shabram, M., Fortney, J. J., Greene, T. P., Freedman, R. S., 2011. Transmission spectra of transiting planet atmospheres: Model validation and simulations of the hot neptune GJ 436b for the James Webb Space Telescope. Astrophys. J. 727, 65. Sharpe, S. W., Johnson, T. J., Sams, R. L., Chu, P. M., Rhoderick, G. C., Johnson, P. A., 2004. Gas-phase databases for quantitative infrared spectroscopy. Appl. Spectrosc. 58, 1452-1461. Sochi, T., Tennyson, J., 2010. A computed line list for the H 2 D + molecular ion. Mon. Not. R. Astr. Soc. 405, 2345-2350. Swain, M. R., Tinetti, G., Vasisht, G., Deroo, P., Griffith, C., Bouwman, J., Chen, P., Yung, Y., Burrows, A., Brown, L. R., Matthews, J., Rowe, J. F., Kuschnig, R., Angerhausen, D., 2009. Water, methane, and carbon dioxide present in the dayside spectrum of the exoplanet HD 209458b. Astrophys. J. 704, 1616-1621. Tashkun, S. A., Perevalov, V. I., JUN 2011. Cdsd-4000: High-resolution, high-temperature carbon dioxide spectroscopic databank. J. Quant. Spectrosc. Radiat. Transf. 112 (9), 1403-1410. Tennyson, J., Yurchenko, S. N., 2012. ExoMol: molecular line lists for exoplanet and other atmospheres. Mon. Not. R. Astr. Soc. Tessenyi, M., Ollivier, M., Tinetti, G., Beaulieu, J. P., du Foresto, V. C., Encrenaz, T., Micela, G., Swinyard, B., Ribas, I., Aylward, A., Tennyson, J., Swain, M. R., Sozzetti, A., Vasisht, G., Deroo, P., 2012. Characterising the Atmospheres of Transiting Planets with a Dedicated Space Telescope. Astrophys. J. 746, 45. Tinetti, G., Aylward, A., van Boekel, R., Coradini, A., Encrenaz, T., Snellen, I., Zapatero-Osorio, M. R., Bouwman, J., Cho, J. Y.-K., Coud'e du Foresto, V., Guillot, T., Lopez-Morales, M., Mueller-Wodarg, I., Palle, E., Selsis, F., Sozzetti, A., Tennyson, J., et al., 2012. Exoplanet Characterisation Observatory. Experimental Astronomy. Tinetti, G., Deroo, P., Swain, M. R., Griffith, C. A., Vasisht, G., Brown, L. R., Burke, C., McCullough, P., 2010b. Probing the terminator region atmosphere of the hot-jupiter XO-1b with transmission spectroscopy. Astrophys. J. 712, L139-L142. Tinetti, G., Griffith, C. A., Swain, M. R., P. Deroo, J. P. B., Vasisht, G., Kipping, D., Waldmann, I., Tennyson, J., Barber, R. J., Bouwman, J., Allard, N., Brown, L., 2010a. Exploring extrasolar worlds: from gas giants to terrestrial habitable planets. Faraday Discuss. 147, 369-377. Tinetti, G., Vidal-Madjar, A., Liang, M.-C., Beaulieu, J.-P., Yung, Y., Carey, S., Barber, R. J., Tennyson, J., Ribas, I., Allard, N., Ballester, G. E., Sing, D. K., Selsis, F., 2007. Water vapour in the atmosphere of a transiting extrasolar planet. Nature 448, 169-171. Vidler, M., Tennyson, J., 2000. Accurate partition function and thermodynamic data for water. J. Chem. Phys. 113, 9766-9771. Voronin, B. A., Tennyson, J., Tolchenov, R. N., Lugovskoy, A. A., Yurchenko, S. N., 2010. A high accuracy computed line list for the HDO molecule. Mon. Not. R. Astr. Soc. 402, 492-496. Warmbier, R., Schneider, R., Sharma, A. R., Braams, B. J., Bowman, J. M., Hauschildt, P. H., 2009. Ab initio modeling of molecular ir spectra of astrophysical interest: application to ch4. Astron. Astrophys. 495, 655661. Warren, S. J., Mortlock, D. J., Leggett, S. K., Pinfield, D. J., Homeier, D., Dye, S., Jameson, R. F., Lodieu, N., Lucas, P. W., Adamson, A. J., Allard, F., Barrado y Navascues, D., Casali, M., Chiu, K., Hambly, N. C., Hewett, P. C., Hirst, P., Irwin, M. J., Lawrence, A., Liu, M. C., Martin, E. L., Smart, R. L., Valdivielso, L., Venemans, B. P., 2007. A very cool brown dwarf in UKIDSS DR1. Mon. Not. R. Astr. Soc. 381, 1400-1412. Yurchenko, S. N., Barber, R. J., Tennyson, J., 2011. A variationally computed hot (up to T=1500 K) line list for NH 3 . Mon. Not. R. Astr. Soc. 413, 1828-1834.", "pages": [ 14, 15, 16, 17, 18, 19, 20, 21 ] } ]

[ { "title": "Mayank Vahia", "content": "Tata Institute of Fundamental Research, Homi Bhaba Road, Colaba, Mumbai 400 005, India, and Manipal Advanced Research Group, Manipal University, Manipal, Karnataka 576 104, India. Email: [email protected] and", "pages": [ 1 ] }, { "title": "Srikumar M. Menon", "content": "Manipal School of Architecture and Planning, Manipal University, Manipal, Karnataka 576 104, India. Abstract: Astronomy arises very early in a civilisation and evolves as the civilisation advances. It is therefore reasonable to assume that a vibrant knowledge of astronomy would have been a feature of a civilisation the size of the Harappan Civilisation. We suggest that structures dedicated to astronomy existed in every major Harappan city. One such city was Dholavira, an important trading port that was located on an island in what is now the Rann of Kutch during the peak of the Harappan Civilisation. We have analysed an unusual structure at Dholavira that includes two circular rooms. Upon assuming strategically-placed holes in their ceilings we examine the internal movement of sunlight within these rooms and suggest that the larger structure of which they formed a part could have functioned as an astronomical observatory. Keywords: Harappan Culture; Dholavira, astronomical observatory", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Harappan civilisation is probably the largest and the most sophisticated of the Bronze Age civilisations in the world (Agrawal, 2007; Joshi, 2008; Possehl, 2009). During its peak period, between 2500 BC to 1900 BC, it covered an area of more than 1.5 million square km and traded over several thousand kilometres to western Asia and the horn of Africa (Wright, 2010). The Civilisation itself was settled along the banks and upper reaches of two major rivers east of the Thar Desert in what is now Pakistan and India (Figure 1). One of its most interesting features is several large and medium-sized settlements in the present day Gujarat region in what is called the Kutch (Chakrabarti, 2004; Rajesh and Patel, 2007). Studies of the sites in the Kutch region suggest that the Little Rann of Kutch was covered with water with a few scattered islands. Several Harappan settlements have been found along the higher points in this region reinforcing the idea that the sites in Gujarat were used as t t rading outposts from which the Harappans traded with West Asia. This is further reinforced by the nature of the settlements, ports and industries found in this area. Several of these were urban centres but in addition there were villages, craft centres, camp sites, fortified places etc. (see Ratnagar, 2006).", "pages": [ 1, 2 ] }, { "title": "2 DHOLAVIRA", "content": "The largest Harappan site in this region of Gujarat was the city of Dholavira (Bisht, 1999), which was on the banks of two seasonal rivulets, and close to a port from which extensive trading is believed to have occurred. Dholavira was divided into several functional sectors, in keeping with other Harappan cities at this time (Bisht, 2000; Joshi, 2008), and Figure 2 shows a town plan, while dimensions of different parts of the city are listed in Table 1. For the purpose of this paper, the area of interest is the region referred to as the 'Bailey' in Figure 2, which lies immediately to the west of the Citadel.", "pages": [ 2 ] }, { "title": "2.1 The Bailey", "content": "In the Bailey region of the city is a structure with a plan-form that is markedly different from all of the other structures in the city and from Harappan plan-forms in general. It consists of the plinth and the foundations of what was probably a 13-room rectangular structure, which included two circular rooms. It is located west of the 'citadel' and is near the edge of the terrace forming the Bailey, which drops off to the west. The flat featureless horizons to the north, west and south are visible without any obstruction, while to the east the mound of the citadel obscures the horizon to a large extent. It is possible that buildings-of which only foundations are visible today-may have obscured the northern horizon to some extent when the city was occupied. The ground slopes down to the south, so it is unlikely that any structures would have obscured the southern horizon (assuming that they were all only single-storied). In Figure 3 is a photograph of the Bailey. As can be seen, the structure consists of rooms of circular and square shape. Since most other residential and workshop buildings in Dholavira are rectangular, it generally has been assumed that the Bailey structure also dates to the Late Harappan Period or an even later Period. However, we suggest that the Bailey structure dates to an earlier period, because of its unique combination of rectangular and circular rooms. Furthermore, we suggest that the two non-circular rooms were designed for non-residential purposes, because: 1) The rectangular rooms adjacent to the circular rooms had bathing and other utilitarian areas, but these were missing from the circular rooms. 2) Each rectangular room typically was connected to one or more other rooms, but each of the circular rooms had only one entrance. 3) Each of the circular rooms was far too small to have served as a residence. It also is clear that the Bailey structure was built on top of an earlier Harappan structure. We suggest that the entire Bailey area was infilled and reconstructed at the peak period of the city, thereby acquiring its present shape.", "pages": [ 2, 3 ] }, { "title": "2.1.1 Survey of the Bailey Structure", "content": "In December 2010 we surveyed the remains of the Bailey structure (see Figure 4), and noticed a number of unique features of the construc- tion. Firstly, at three places where E-W oriented cross walls met a N-S oriented wall, they were offset by the thickness of the wall. Since the obvious common sense approach would have been to carry on the cross walls in the same line, this misalignment must have been deliberate (although the reason for this remains obscure). As Figure 4 indicates, there are two circular rooms-one in the north and one in the west, and details of these are presented in Figures 5 and 6. The western room is perfectly circular with a mean internal diameter of 3.4 m and a wall thickness of 0.75 m, while the northern circular room is like a spiral in plan such that the line of the outer surface of its wall comes in line with its inner line at its northernmost point as it completes 360 (presumably without letting too much diffused light into the room). A 'straight wall' 0.75 m in thickness extends N-S into the room at this point for 4.0 m. A wedge-shaped segment 1.5 m on two sides and bounded by the curvature of the circular wall of the room is situated in the southwestern quadrant of the room. In the course of the survey we noticed the following unusual aspects of the entire area: 1) Unlike all other regions of the settlement, the land surface in the Bailey area rises from south to north with an inclination of nearly 23.5 , which corresponds exactly to the latitude of the site. Hence, for someone standing at the southern end of the Bailey, the North Pole would be at the top of the slope, and all stars seen would be circumpolar. 2) While the city of Dholavira is aligned 6 0.5 from true north, features associated with the two circular rooms pointed exactly to the west (270 0.5 ; see Figure 5) and the north (0 0.5 ; see Figure 6). 3) The circular room at the northern end of the Bailey has a small platform in the southwestern part of the room. 4) At the southern end of the Bailey are two deep rectangular pits (shown in Figure 4). These lack any entry/exit stairs and while their function is not obvious, they possibly could have been used for observing stars at and near the zenith. Let us now investigate whether the Bailey structure could have had an astronomical function.", "pages": [ 3, 4 ] }, { "title": "2.1.2 Our Simulation", "content": "Assuming a wall height of 2.5 m (see Bisht, 2000; Danino, 2008; Joshi, 2008) for the Bailey structure and entry to these northern and western circular rooms from the north and west respectively, we simulated the response of the rooms to solar geometry for the latitude of Dholavira. The assumptions and the simulation procedure are detailed below. For the northern circular room we assumed that the entry point was via a break in the 2.5 m high circular wall where the 'straight wall' penetrates from the north. The width of the entry was taken as 0.50 m-which is the thickness of the 'straight wall'. The 'straight wall' was interpreted as a walkway just 0.60 m high. In keeping with our knowledge of Harappan architecture (ibid.) a flat roof was assumed for the room, with a circular opening 0.50-m in diameter located directly above the termination point of the walkway (see Figure 7). Upon simulating the summer solstice day, the circle of light cast by the aperture in the roof slides down the circular wall in the west and across the floor and, at local solar noon, falls directly upon the extreme south portion of the walkway (Figure 8) before continuing across the floor and up the eastern portion of the circular wall. This is expected since we have deliberately positioned the aperture over the southern end of the walkway and the Sun is directly overhead at local solar noon at the time of the summer solstice for the latitude of Dholavira. But what is particularly exciting and caught our attention is that when we simulated the movement of the Sun at the time of the winter solstice, using this same geometry, the circle of sunlight travels down the N-W part of the circular wall and when it is on the top surface of the walkway, its northern edge grazes the bottom edge of the circular wall (see Figure 9). Similarly, for the western room, we assumed that the entry was via a break in the 2.5 m high circular wall where the straight wall joins from the west. The width of the entry is taken as 1.30 m, which is the thickness of the straight wall. The straight wall is once again taken as a walkway just 0.60-m high. A flat roof was assumed for the structure, with a circular opening 0.50m in diameter at the southern extreme. Upon simulating for the summer solstice day, the circle of light cast by the aperture in the roof slides down the circular wall in the S-W and is on the floor at local solar noon, its southern edge grazing the bottom edge of the southern wall before continuing up the S-E portion of the circular wall (see Figure 10). This is expected since we have deliberately positioned the aperture over the southern extreme and the Sun is directly overhead at local solar noon on the summer solstice at Dholavira, as mentioned earlier. Simulating the Sun's movement on the day of the winter solstice and using this same geometry, the circle of light travels down the N-W part of the circular wall and when it is on the straight wall, its northern edge passes close to the bottom edge of the circular wall (Figure 11). In addition, it is seen that the two sections of E-W oriented walls to the west of the west circular room frame the extreme points of the setting Sun as seen from the 1.30 m wide slit in the circular wall. In other words, the shadow of the northern of these walls touches the northern extremity of the slit at sunset on the summer solstice day and that of the southern of these walls touches the southern extremity of the slit on the winter solstice day (Figures 12 and 13).", "pages": [ 4, 5 ] }, { "title": "3 DISCUSSION", "content": "The city of Dholavira is located on the Tropic of Cancer. Thus the shadows of the Bailey structure would always be to the north of that structure, except at noon on the day of the summer solstice when the Sun would be at the zenith and no shadows would be cast. This is clearly something that the Harappan astronomers would have noticed. The Bailey structure at Dholavira is unusual in several ways. It was built on what seems to be an intentional incline that points to the region of sky where stars always would be circumpolar. The structure also included two circular rooms, a rare occurrence for the 'rectangle-loving' Harappans. However, the workmanship of these two anomalous rooms and their inter-connection with neighbouring ones indicates that they all were contemporaneous. While structures erected by the Harappans normally did not have stone walk- ways leading to their entrances, these two circular rooms had such walkways. While the whole city was inclined 6 to the west of north, the two circular rooms in the Bailey structure had openings that faced due north and due west respectively. In addition, the west-pointing room had two walls to the west that were constructed so that their shadows would just touch the entrance to the room on the winter and summer solstice days. In seeking to explain the function(s) of the two circular rooms we made assumptions about the superstructure, such as the height of the walls (2.5 meters), the presence flat roofs and the presence of an aperture of a certain size (which was not crucial for our simulations) and positioning. Note that none of these parameters is at variance with what is currently known about Harappan architecture (e.g. see Joshi, 2008; Possehl, 2002). Adopting these assumptions, we then simulated the movement of the circles of light cast by the holes in the ceilings of these two rooms. In the course of the year these circles of light clearly illuminated specific spots within these rooms, and important days in the solar calendar could easily be identified. The narrow beams of light entering these rooms (which had unusually narrow entrances compared to other rooms at Dholavira) would also have accentuated the movement of the Sun over the course of a year. In the case of the northern circular room, what is interesting is that by positioning the aperture in the roof above the southern extremity of the straight wall, the northern and southern edges of the straight wall marked the points where the circle of light was cast at noon on the solstices. But, in addition, if a marked wooden plank was laid along the walkway, the position of the Sun at noon would vary systematically throughout the year (see Figure 14), and in this way the room also could function as a calendrical observatory. In the case of the western circular room, once again by positioning the aperture in the roof directly above the southern boundary of the circular wall, the extremes of the N-S diameter of the room marked the points where the circle of light was cast at noon on the solstices. Meanwhile, shadows cast by the two E-W oriented walls to the west of the entrance to the circular room just reached the entrance at sunset on the solstice days, also allowing these specific days to be easily identified. Meanwhile, if a N-S oriented marked wooden plank was laid across the room, this also would show the changing position of the Sun at noon during the year (Figure 15), so this room, too, could serve as a calendrical observatory.", "pages": [ 5, 6, 7 ] }, { "title": "4 CONCLUSIONS", "content": "It can be safely assumed that astronomers in the intellectually-advanced Harappan Civilization had detailed knowledge of positional astronomy. However, apart from some stray references (e.g. see Maula, 1984; Vahia and Menon, 2011), up to now there has been no positive identification of any structure or artefact with obvious celestial associations at any of the 1500 or so known Harappan archaeological sites. The Bailey structure at Dholavira is the first Harappan structure that seems to have been constructed specifically in response to the solar geometry at the site, and it is highly probable that the two circular rooms in the structure were designed for solar observations. If this supposition is correct, then this is the first identified Harappan example of a building that was used specifically for observational astronomy. We would argue, however, that similar structures must have existed at all major Harappan cities, and the identification of other examples is simply a matter of time. Finally, we should mention that since the Dholavira was an important centre of trade and commerce, keeping track of time would have been crucial, but to date no structures that ob- viously served this purpose have been identified.", "pages": [ 7 ] }, { "title": "5 ACKNOWLEDGEMENTS", "content": "The authors wish to acknowledge the funding for the project from the Jamsetji Tata Trust under the programme, 'Archaeo-astronomy in Indian Context'. We also wish to gratefully acknowledge the permission given to us by the Archaeological Survey of India to survey Dholavira in 2007, 2008 and 2010. Without this it would have been impossible to carry out our research. We also wish to thank our friends Mr Kishore Menon and others whose endless discussions greatly helped with this work. We wish to thank Professor Vasant Shinde for his continuing encouragement; Professor Sir Arnold Wolfendale for useful suggestions; and Nisha Yadav for her helpful input during this research. Finally, we are particularly grateful to Professor Wayne Orchiston for all of the effort he took to make the contents of this paper precise and clear.", "pages": [ 7 ] }, { "title": "6 REFERENCES", "content": "Agrawal, D.P., 2007. The Indus Civilisation . New Delhi, Aryan Books International. International Conference on Oriental Astronomy . Tokyo, National Astronomical Observatory of Japan. Pp. 27-36. Vahia, M.N., and Yadav, N., 2011. Reconstructing the history of Harappan Civilisation. Journal of Social Evolution and History , 10, 67-86. Wright, R.P., 2010. The Ancient Indus: Urbanism, Economy and Society . Cambridge, Cambridge University Press. Mayank Vahia has B.Sc. and Master of Physics degrees from the University of Mumbai (India). He is currently a Professor at the Tata Institute of Fundamental Research in Mumbai. He has worked on several projects involving Indian satellites flown on Indian, Russian and American missions to study high energy emis- sion from the Sun and other objects. He has more than 200 publications in most of the major journals in astronomy and astrophysics as well as computer science. Mayank is a member of the IAU Commissions 41 (History of Astronomy) and 44 (Space & High Energy Astrophysics). For the past six years he has been researching the origin and growth of astronomy in the Indian subcontinent and has published about 30 papers on the subject, several of which have appeared in earlier issues of this Journal . Srikumar M. Menon is an architect with a B. Arch. degree from University of Kerala, India. He has a Ph.D. in archaeoastronomy from Manipal University. He currently teaches in the Faculty of Architecture, Manipal University, India. His research interests are prehistoric architecture of India and early temple architecture of the same region. He is the author of the book Ancient Stone Riddles: Megaliths of the Indian Bubcontinent (2013, Manipal, Manipal University Press).", "pages": [ 7, 8 ] } ]

[ { "title": "Reheating in non-minimal derivative coupling model", "content": "H. Mohseni Sadjadi ∗ and Parviz Goodarzi Department of Physics, University of Tehran, P. O. B. 14395-547, Tehran 14399-55961, Iran November 8, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "We consider a model with non-minimal derivative coupling of inflaton to gravity. The reheating process during rapid oscillation of the inflaton is studied and the reheating temperature is obtained. Behaviors of the inflaton and produced radiation in this era are discussed.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "To solve some problems of standard model of cosmology and particle physics, such as the horizon problem, flatness, isotropy and homogeneity of the universe, the absence of magnetic monopoles and so on, and to make the bing bang cosmology more consistent with astrophysical data, the inflation theory, which considers an epoch of accelerated expansion for the early universe, was introduced [1]. One straightforward way to describe this era, is to consider a slowly rolling scalar field , dubbed inflaton, whose energy density was dominated by its potential during inflation [2, 3]. At the end of inflation the universe was cold, so there must be a procedure trough which the scalar field decayed to particles which became thermalized and reheated the universe. This could be realized by decaying of the scalar field, e.g. during coherent oscillation in the bottom of the potential [4]. At first sight, it seems that a candidate for this scalar field may be the Higgs boson. But parameters of the standard model do not agree with those required for inflaton [3]. So to reconciliate the slow roll inflation with standard model parameters, a framework in which the Higgs boson is non minimally coupled to Ricci scalar has been introduced in [5]. Recently a model comprising a non minimal coupling between the derivatives of the Higgs boson and Einstein tensor has been proposed, which besides its capacity to explain the inflationary phase, is also safe of quantum corrections and unitary violation problem [6]. In this context, the nonminimal derivative coupling may allow the model to describe the acceleration as well as the super-acceleration of the universe [8]. Coupling the Einstein tensor to kinetic term of inflaton, as we will see later, enhances the gravitational friction during slow-roll and this allows us to consider more general steep potentials, such as Higgs potential, without contradiction with the CMB observations or collider experimental bound [6]. In [7], this model was employed to study the natural inflation, where the inflaton is assumed to be a pseudo-Nambu-Goldstone boson. In this framework, the global shift symmetry is broken at a scale f , giving rise to the inflaton mass. For small field values, the potential is stable against radiative corrections, but slow roll requires that f becomes much larger than the Planck scale, giving rise to eta problem. In [7], it was shown that nonminimal derivative coupling allows to take f /lessmuch M P , without introducing new degrees of freedom, and protects the tree-level shift invariance of the scalar field as well as the perturbative aspects of the theory. Similar models including nonminial derivative coupling between a scalar field and gravity have also been used to study the late time evolution of the universe by considering the scalar field as the dark energy [9]. In this manuscript, following [6] we assume that the inflation is implemented by a scalar field with non-minimal derivative coupling to gravity, with a power law potential, and study the reheating process in this model, which to our knowledge, was not studied before. First, we review briefly the inflationary epoch and then study quasi-periodic motion of the inflaton at the end of the slow roll. We consider the decay of the scalar field to ultra-relativistic particles (radiation) via a phenomenological source during coherent rapid oscillation and find the reheating temperature.", "pages": [ 1, 2 ] }, { "title": "2 Non-minimal derivative coupling model and inflation", "content": "An action describing a scalar field coupled non-minimally to gravity via its kinetic term is given by 1 [6]: where G µν is the Einstein tensor, w is a constant with the dimension of inverse mass squared, and M P = 2 . 4 × 10 18 GeV is the reduced Planck mass. In the absence of terms containing more than two time derivatives, additional degrees of freedom are not produced in this theory. In the spatially flat Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime, the Friedmann equation is and the scalar field equation of motion is given by where 'dot' denotes derivative with respect to time and 'prime' denotes derivative with respect to ϕ . Using the energy momentum tensor derived from (1), the energy density and the pressure of the scalar field are obtained as respectively. ρ ϕ and P ϕ satisfy the continuity equation In the presence of another component, with the energy density ρ R and the pressure P R , interacting with the scalar field via the source term Q , the continuity equation becomes In the inflationary era, we take the field ϕ as the inflaton, and assume that the universe is dominated by only this scalar field. But in the subsequent epochs, one must also take into account the presence of other components such as radiation. In the following we adopt the high friction condition [10] The slow roll regime is characterized by yielding The slow-roll conditions are satisfied provided that In comparison with minimal models, conditions (10) may be satisfied by more steep potentials in the high friction limit (7) [10]. For the potential these relations require ϕ q +2 /greatermuch M 4 P wλ . In the slow-roll era, where ˙ H /lessmuch H 2 holds, the requiring that the model be outside of the quantum gravity regime implies that R /similarequal 12 H 2 /lessmuch M 2 P 2 . This condition, in terms of the potential, is rewritten as The number of e-folds is given by where ϕ e ( ϕ 0 ) is the value of the scalar field at the end (the beginning) of the slow-roll inflation. By assuming ϕ e /lessmuch ϕ 0 , one can estimate the e-folds number for a chaotic inflation with potential (11) as When the conditions (10) cease to be valid the slow-roll inflation ends. For power law potential (11) and q ∼ O (1), (15) gives", "pages": [ 2, 3, 4 ] }, { "title": "3 Quasi-periodic evolution", "content": "In this part we try to study the dynamics of the inflaton after the end of slow-roll. In this era, conditions (8,10) cease to be valid and quasi-periodic evolution of the scalar field about the bottom of the potential begins. In fig.(1), using numerical methods, oscillation of the field ϕ is depicted for quadaratic potential This figure shows that, at the first stages after the slow-roll, the amplitude of the scalar field drops down rapidly. Later, during a phase of rapid oscillation of the field, the rate of decrease of the amplitude becomes much less than the oscillation frequency. To get an an insight about the solutions in this epoch, we proceed in the same way as [11] and consider the power law potential (11), with an even integer q . During the quasi-periodic evolution where the amplitude and the frequency of oscillation are time dependent, the scalar field is presented as [11] where, the amplitude Φ( t ) is given by This equation means that the potential, when is evaluated at the amplitude, gives all the energy. A ( t ) is some function of time which may be determined as follows: By taking a time derivative of (18), we easily obtain (20), and (18) result in The continuity equation and eqs. (4) lead to: By substituting ¨ ϕ from the equation of motion (3) into the above equation we arrive at By using ˙ Φ Φ = 1 q ˙ ρ ϕ ρ ϕ derived from (19), we deduce ∣ ∣ ∣ ∣ where +( -) corresponds to ˙ ϕ > 0( < 0). This expression is much less than unity, ∣ ∣ ∣ ϕ ˙ ϕ ˙ Φ Φ ∣ ∣ ∣ /lessmuch 1, whenever the slow-roll is ceased and becomes valid. This is opposite to slow-roll conditions (see eqs. (8,10) and their subsequent discussion). In this regime ∣ ∣ ∣ ˙ Φ Φ ∣ ∣ ∣ /lessmuch ∣ ∣ ∣ ˙ ϕ ϕ ∣ ∣ ∣ is the stage of rapid oscillation or high frequency regime, i.e. when ∣ ∣ ∣ ˙ Φ Φ ∣ ∣ ∣ /lessmuch A . In this epoch we have ∣ ∣ ∣ ∣ showing that the amplitude, the Hubble parameter and the energy density decrease slowly during one period of oscillation. The time average of energy density over an oscillation cycle (from t to t + T ) is where T is the period. As the amplitude decreases very slowly, we deduce 〈 ρ ϕ ( t ) 〉 = V (Φ( t )). This result is in agreement with (19), because from (27), it is clear that ρ ϕ changes insignificantly during a period. During the rapid oscillation of the scalar field, the parameter γ , defined by γ = 1 + 〈 P ϕ 〉 〈 ρ ϕ 〉 , is determined as follows To obtain the above equation we have used (4). Using (4), (29) can be rewritten as In the above computations, we have converted time integration to ϕ integration, and the variable change x = ϕ Φ was applied. The same method for w = 0, results in γ = 2 q q +2 [11]. This constant is the effective value of γ in rapid oscillation era, i.e., in this era the scalar field behaves as a barotropic fluid. To elucidate this point we proceed as [11]. Taking time average of (5), and by using (30), we obtain Using the definition of time average, we obtain ˙ 〈 ρ ϕ 〉 = δρ ϕ T , where δρ ϕ is the change of ρ ϕ over the period T . Hence In the high frequency regime δρ ϕ T ≈ ˙ ρ ϕ , and we arrive at In the following, as in (33), we will not use the symbol <> for the oscillating scalar field, e.g., by ρ ϕ we mean the time averaged of the scalar field energy density in the sense explained above. From eq. (19) we get an approximate equation for Φ evolution From (34) and (27) it is obvious that H /lessmuch A . This means that the expansion rate is much less than the oscillation frequency. Effectively the scale factor is a ( t ) ∝ t q +2 q and the Hubble parameter is given by H = q +2 q t -1 . ρ satisfies corresponding to ρ ϕ ∝ t -2 which is q independent.", "pages": [ 5, 6, 7, 8 ] }, { "title": "4 Particle production", "content": "In this part we try to study inflaton decay to ultra-relativistic particles (radiation) in the rapid oscillatory phase. This decay is due to the interaction between the inflaton and produced particles. To study this decay, a phenomenological interaction (see(6)), where Γ is a positive constant, was proposed by [12]. Afterwards, in [13], a Lagrangian, including bosonic and fermionic fields and their interactions with the inflaton, was introduced, and it was shown that the effect of particle production can be explained by adding a polarization operator to the inflaton mass term. There was shown that, for quadratic potential, the role of polarization operator may be mimicked by the phenomenological friction term (36), in high frequency regime. However, by taking under consideration the back reaction of the quantum effects on the evolution of the inflaton field [14], and also considering possible decays of the inflaton to other particles [15], (36) can no more be deduced from a Lagrangian. Other phenomenological models with temperature and field dependent friction coefficient term, Q = Γ( T, ϕ ) ˙ ϕ 2 , have been also considered in the literature [16]. However, as the nature of the inflaton and also primordial produced particles are unknown, we have not yet an exact expression for the form of interaction. In our study, we adopt the widely used phenomenological interaction (36), which reduces significantly computational complexity. In the presence of the source term (36), the inflaton evolution is given by In the following we consider the power law potential (11). Using (4) and with the same method used in (30), one can obtain Inserting the above relation in the first equation of (6), and by replacing P ϕ + ρ ϕ with its time average over an oscillation cycle, we arrive at We note again that in the above equation, all the values must be regarded as the time averaged values over one oscillation. This relation is only valid on large time with respect to the period of fast oscillation. The second term (friction term) describes dilution resulted from the universe expansion, while the third term corresponds to particle production during the coherent oscillation of the inflaton. Comparing (39) with the corresponding minimal model relation where γ m = 2 q q +2 , shows that, in the limit wH 2 /greatermuch 1, the decrease of ρ ϕ (due to particle production) and as we will see the reheating temperature are very less than the corresponding values in the minimal model. Note (39) is true only for wH 2 /greatermuch 1, so (40) cannot be derived from (39) by simply setting w = 0. The solution of (39) is where t osc. is the time when the oscillation commences. In the beginning of particle production, the universe is dominated by the scalar field, and we assume Γ /lessmuch wH 3 , but H is decreasing and this approximation ceases to be valid later, when the second and the third term in (39) acquire the same order of magnitude (we will denote this time by t rh ). So, with our assumptions, and in the scalar field dominated era it is safe to use the approximation H = 2 3 γt until t = t rh . The behavior of the Hubble parameter, during rapid oscillation, can also be investigated via numerical method. E.g. in fig.(2), H is plotted for the quadratic potential (17) and by using Friedmann and continuity equations. So let us write (41) as where Ξ = t 2 osc. ρ osc. e Γ γ 3 4 w t 3 osc. and ρ osc. = ρ ( t osc. ). The decrease of ρ ϕ is due to particle production which is encoded in the exponential term and also due to the term t -2 corresponding to dilution via the universe expansion. The dilution term is independent of q as was explained after eq. (35). For larger values of Γ wM 3 the decay rate is faster. P Comparing this result with what was obtained before for { w = 0, γ = 0 . 5 } model [17], shows that the decrease rate of ρ ϕ , due to the expansion of the universe, is the same. But in the presence of w , the rate of particle production is decreased. We note again that our result (42) is true only for wH 2 /greatermuch 1, so we cannot obtain (43) from (42) by setting w = 0. The radiation satisfies To study the evolution of ρ R during scalar field dominated epoch, i.e., when H 2 /similarequal 1 3 M P ρ ϕ we write (44) in the form whose approximate solution is given by To derive the above equation we have assumed that after the slow-roll, the universe was cold: ρ R ( t = t osc. ) = 0. In fig. (3), ρ R is plotted for the quadratic potential (17), showing that the radiation density increases monotonically in the rapid oscillation phase. However this behavior is valid only for ρ ϕ /greaterorsimilar ρ R , and in the radiation dominated era we expect that ρ R decreases. Based on Friedmann and continuity equations, the behavior of ρ R is also depicted in fig.(4), via numerical method for the quadratic potential (17). This figure shows that the increase of ρ R continues until ρ R /similarequal ρ ϕ . After t = t rh , i.e. from the beginning of radiation dominated era, ρ R begins to decrease. Relativistic particles interact quickly (with respect to the expansion rate of the universe) with each other to become in a thermal equilibrium characterized by temperature T r , given by where g ∗ is the total number of effectively massless degrees of freedom. If T r is greater than the electroweak scale T r > 300 GeV then g > 106 . 75 [18]. The reheating time, t rh , is given by i.e. , when the second and third terms of (39) acquire a same order of magnitude, as it was mentioned before. At this order of time, ρ R ( t rh ) has also the same order of magnitude as ρ ϕ ( t rh ). At the reheating time we have H /similarequal ( Γ 9 w ) 1 3 , so the use high friction condition (7) during the reheating era is safe only when For t /greaterorsimilar t rh , almost all the energy of the inflaton is transferred to newly produced particles and the universe becomes radiation dominated. Now, we can estimate the reheating temperature defined by T rh = T ( t rh ). From (46), and t osc. t rh /lessmuch 1, we obtain Note that the above equation could also be obtained by using H 2 /similarequal 1 3 M 2 P ρ R , and (47). This temperature is specified only by parameters of the system and is independent of initial conditions. In the absence of non-minimal derivative coupling, i.e. for w = 0, and for γ = 0 . 5, the radiation energy density is obtained as [17] ρ R increases rapidly from ρ R = 0 to its maximum value and then decreases again, so the maximum temperature is occurred in the beginning of ϕ oscillation before reheating (see fig.(5), depicted for the potential (17)). This is in contrast to our model, where as it can be seen from (46), ρ R increases continuously in ϕ oscillation epoch until ρ R /similarequal ρ ϕ . In w = 0 model, reheating temperature is determined by [17] Therefore T rh ( wH 2 /greatermuch 1) /lessmuch T rh ( w = 0), provided that a same Γ is taken into account for both theories. Note that as the reheating process is realized after inflation, T rh must be below the GUT scale: T rh < 10 16 GeV . Based on astrophysical data, we are able to estimate the relation of reheating temperature and the number of e-folds. To do so, we proceed as follows: Consider a length scale l which at time t = t ∗ , in the inflation era, left the Hubble radius: The number of e-folds from t ∗ to the end of the slow-roll inflation, denoted by subscript e , may be expressed as where 0, rh , and eq subscripts denote present, reheating, and matter-radiation equality densities epochs respectively. As we have seen, during rapid oscillation (from t e until t rh ), the universe is dominated by a scalar field whose the effective equation of state parameter is given by γ -1, whence [19] The right hand side expressions are determined as follows: By setting a 0 = 1, and adopting the WMAP (pivot) scale l = 1 k = 500 Mpc [20], we arrive at The present Hubble parameter is H 0 = 7 30000 Mpc -1 [20]. To obtain V ∗ , we consider the scalar perturbations. The spectral index n s is [10] The quantities in the right hand side must computed at the time of horizon crossing c s k = aH during slow-roll inflation, where k is the comoving wavenumber and c s is the sound speed of scalar perturbation. In the limit wH 2 /greatermuch 1 we have c s /similarequal 1 [21], and for the potential (11), we deduce Based on WMAP data [20], From t ∗ until t e , the slow-roll approximation is valid, hence V e may be estimated with the help of (15), as: ρ rh is specified by (46) or equivalently by (47) and (50): In the slow-roll regime, as expected, we have V 1 4 ∗ /similarequal V 1 4 e . V 1 4 ∗ /lessorsimilar 10 16 GeV also holds [19] (10 16 GeV is the GUT scale). The reheating is provided by the inflaton energy, hence ρ rh /lessorsimilar V e . So finally we expect to have N ∗ < 60. Only for a prompt reheating, N ≈ 60 may be possible. As an example, for the quadratic potential V ( ϕ ) = 1 2 m 2 ϕ 2 where N ∗ depends on the parameters of the model, i.e. Γ, w and m , e.g. choosing wm 2 /similarequal 10 8 , Γ m /similarequal 10 -2 (in agreement with (49)), and setting m = 10 -6 M P [2], we find N ∗ = 43 . 41.", "pages": [ 8, 9, 10, 11, 12, 13, 14 ] }, { "title": "5 Conclusion", "content": "As a summary, we briefly discussed inflation in the framework of a nonminimal derivative coupling model proposed in [6], and then studied a gap in the literature: the reheating process in this framework. We investigated inflaton evolution in quasi-periodic oscillation at the end of slow-roll. We allowed the scalar field to decay to ultra-relativistic particles (radiation) via We showed that the energy density of radiation, during oscillatory era and when it is smaller than the energy density of inflaton, increases monotonically. This behavior is in contrast to ordinary inflation theory where the maximum temperature occurs before reheating. We confirmed our results via numerical methods.", "pages": [ 14, 15 ] } ]

[ { "title": "ABSTRACT", "content": "ICRR-Report-633-2012-22 IPMU12-0190 TU-922", "pages": [ 1 ] }, { "title": "Non-Gaussianity from Axionic Curvaton", "content": "Masahiro Kawasaki, a,b 1 Takeshi Kobayashi, c,d 2 and Fuminobu Takahashi b,e 3 We study non-Gaussianity of density perturbations generated by an axionic curvaton, focusing on the case that the curvaton sits near the hilltop of the potential during inflation. Such hilltop curvatons can generate a red-tilted density perturbation spectrum without invoking large-field inflation. We show that, even when the curvaton dominates the Universe, the non-Gaussianity parameter f NL is positive and mildly increases towards the hilltop of the curvaton potential, and that f NL = O (10) is a general and robust prediction of such hilltop axionic curvatons. In particular, we find that the non-Gaussianity parameter is bounded as f NL glyph[lessorsimilar] 30 - 40 for a range of the scalar spectral index, n s = 0 . 94 - 0 . 99, and that f NL = 20-40 is realized for the curvaton mass m σ = 10-10 6 GeV and the decay constant f = 10 12 - 10 17 GeV. One of the plausible candidates for the axionic curvaton is an imaginary component of a modulus field with mass of order 10 - 100 TeV and decay constant of 10 16 - 17 GeV. We also discuss extreme cases where the curvaton drives a second inflation and find that f NL is typically smaller compared to non-inflating cases.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Several theoretical difficulties of the standard big bang cosmology such as the horizon and flatness problems can be elegantly solved by inflation [1]. In fact, the existence of the inflationary era in the early Universe is strongly supported by the observations [2]; the density perturbations extending beyond the horizon at the last scattering surface can be interpreted as the evidence for the accelerated expansion in the past. The study of density perturbations such as isocurvature perturbations, non-Gaussianity, tensormode, and their effects on the cosmic microwave background (CMB) power spectrum is a powerful diagnostic of the mechanism that laid down the primordial density fluctuations, but it is not enough at present to pin down the model. This is partly because of our ignorance of thermal history of the Universe beyond the standard big bang cosmology, especially concerning how the Universe was reheated. Whereas one of the plausible explanations for the density perturbations is the quantum fluctuations of the inflaton from the minimalistic point of view, it may be that there are many other light scalars in nature, one of which is responsible for the observed density perturbation via the curvaton [3, 4, 5, 6] (or its variant, e.g. modulated reheating [7, 8]) mechanism. In fact, there are many moduli fields that necessarily appear at low energies through compactifications in string theory. Most of them must be stabilized in order to have a sensible low-energy theory, but some of them may remain relatively light, and therefore are a candidate for the curvaton. Interestingly, there is an argument that string theory contains a plenitude of axions, the so called 'string axiverse.'[9] We shall see later that the axion is indeed a plausible candidate for the successful curvaton. 4 One of the distinguishing features of the curvaton mechanism is that it can generate the density perturbation with large non-Gaussianity. If any primordial non-Gaussianity is found by the Planck satellite, it would exclude a simple class of inflation models as the origin of the entire density perturbation, and therefore, it has a tremendous impact on our understanding of the early Universe. Recently, the present authors studied non-Gaussianity generated by the curvaton mechanism in great detail, and developed a formalism to calculate the density perturbation for a generic curvaton potential [10]. We pointed out that the curvaton should be located at a potential with negative curvature during inflation, and in particular it must be close to the local maximum ('hilltop') of the potential, in order to generate a red-tilted density perturbation spectrum which is strongly favored by the recent observations [2]. Interestingly, we found that, even if the curvaton dominates the Universe, the non-Gaussianity parameter f NL is positive and gets enhanced logarithmically in the hilltop limit, and therefore f NL of O (10) is a robust prediction of the hilltop curvaton. Applying our formalism to the axionic (or pseudo-Nambu-Goldstone) curvaton with a sinusoidal potential, we found that f NL can be as large as about 30, which is realized for the curvaton mass of order 10 TeV and the decay constant of order the GUT scale. In this analysis we fixed the scalar spectral index n s = 0 . 96 for simplicity. The mild increase of the non-Gaussianity in the hilltop limit is originated from the fact that the density perturbation generated by the curvaton is enhanced. This enhancement is due to non-uniform onset of curvaton oscillations [11, 10]. This result should be contrasted to a simple curvaton model with a quadratic potential, which predicts a negative f NL of order unity in the case that it dominates the Universe. In this paper, we extend our previous work on the non-Gaussianity generated by the axionic curvaton with the hilltop initial condition. We will discuss its dependence on the scalar spectral index, and also scan the curvaton parameters, namely, the mass and the decay constant. Interestingly, we find that f NL is bounded as f NL glyph[lessorsimilar] 30 for n s = 0 . 94 - 0 . 99, and the maximal non-Gaussianity is realized for the curvaton mass 10 TeV and the decay constant of order the GUT scale. (If reheating happens prior to the curvaton oscillation, then the bound becomes f NL glyph[lessorsimilar] 40.) Furthermore, f NL = 20-40 is realized for a wide range of parameters, the curvaton mass m σ = 10-10 6 GeV and the decay constant f = 10 12 - 10 17 GeV. One of the plausible candidates for such an axionic curvaton is an imaginary component of the moduli (i.e., axions) with mass of order 10 - 100 TeV and decay constant of 10 16 - 17 GeV. The moduli fields are stabilized by the non-perturbative effect and the supersymmetry (SUSY) breaking, and it is plausible that the moduli mass is closely related to the SUSY breaking scale in the visible sector. Intriguingly, such several tens TeV SUSY breaking scale is consistent with the recently discovered Higgs boson mass of 125 - 126 GeV [12, 13]. The rest of the paper is organized as follows. After briefly reviewing density perturbations from general curvatons in Section 2, then in Section 3 we discuss axionic curvatons in detail. We then give discussions and conclusions in Section 4 and 5, respectively. The appendix discusses an extreme case where the curvaton drives a second inflationary period. After analytically computing density perturbations from inflating curvatons in general, we then apply the discussions to axionic curvatons. We find that the non-Gaussianity turns out to be rather small when the axionic curvaton drives a second inflation.", "pages": [ 2, 3 ] }, { "title": "2 Review of Curvatons with a General Potential", "content": "In the curvaton mechanism, the light curvaton field acquires super-horizon field fluctuations during inflation. The density perturbations are produced in the post-inflationary era, as the curvaton oscillates and its energy density relatively grows compared to other radiation components. In this section we give a brief review of density perturbations generated by a curvaton σ with a generic effective potential V ( σ ). We refer the reader to [10] for detailed derivation of the following results.", "pages": [ 4 ] }, { "title": "2.1 Density Perturbations from Curvatons", "content": "The density perturbations generated by curvatons depend on the curvaton dynamics during and after inflation. In the simple curvaton model with a quadratic potential, the curvaton dynamics is determined by the curvaton mass and the initial deviation from the origin. If the mass is much smaller than the Hubble parameter during inflation, the curvaton hardly evolves until it starts to oscillate, and the resultant density perturbation is given in a rather simple form. However this is no longer the case for a general curvaton potential. In particular, the curvature of the potential should be negative and non-negligible in order to account for the observationally favoured red-tilted perturbation spectrum, then the curvaton significantly evolves after inflation, affecting the density perturbation. If the curvaton potential V ( σ ) has no explicit dependence on time, then the curvaton dynamics prior to the oscillation can be tracked by the attractor solution which is a good approximation while | V '' / ˆ cH 2 | glyph[lessmuch] 1. Here, a prime denotes a derivative with respect to σ , an overdot a time derivative, and H = ˙ a/a . Setting the minimum of the potential about which the curvaton oscillates to σ = 0, the onset of the oscillation can be defined as when the time scale of the curvaton rolling becomes comparable to the Hubble time, i.e. Then the Hubble parameter at the time is obtained as where the subscript 'osc' denotes values at the onset of the curvaton oscillation, and c is a constant depending on whether reheating (= inflaton decay, at t reh ) is earlier/later than the onset of the curvaton oscillation (corresponding to ˆ c in the attractor (2.1) right before the oscillation): The absolute value sign in (2.3) can be removed by supposing the curvaton potential to be monotonically increasing (decreasing) for σ > ( < )0, so that the curvaton can roll down to the origin. Let us here summarize simplifying assumptions concerning the evolution of the energy densities of the curvaton and the inflaton. We assume the curvaton potential to be well approximated by a quadratic one around its minimum so that the curvaton oscillations are sinusoidal. 5 Then the curvaton energy density redshifts similarly to nonrelativistic matter after the onset of the oscillations until the curvaton decays into radiation. On the other hand, we consider the inflaton to behave as matter from the end of inflation until reheating when it decays into radiation. The energy density of the curvaton before the beginning of its oscillation is assumed to be negligibly tiny compared to the total energy of the Universe, having little effect on the expansion history. Supposing the curvaton field fluctuations to be nearly Gaussian with P δσ ( k ) = ( H | k = aH / 2 π ) 2 at the time when the comoving wave mode k exits the horizon, then using the δ N -formalism [19, 20, 21, 22], the power spectrum of the density perturbations at the CMB scale is expressed as [10] with Here, the subscript ∗ denotes values when the CMB scale exits the horizon, and r is the energy density ratio between the curvaton and radiation (which originates from the inflaton) upon curvaton decay The function X denotes effects due to the non-uniform onset of the curvaton oscillations (which are absent for a purely quadratic curvaton potential), defined as follows: where the constant c is given in (2.4). From the above expressions, the spectral index of the linear order perturbations follows as (note that the scale-dependence of (2.6) shows up only through σ ∗ , since σ osc and r are independent of the comoving wave number) The recent observations strongly suggest that the density perturbation power spectrum is red-tilted, n s = 0 . 968 ± 0 . 012 [2]. This requires that the curvaton potential be tachyonic and the size of the curvature must be of order 10 % of the Hubble parameter during inflation, unless the inflaton is allowed to take super-Planckian field values, or some special configurations are arranged in the inflationary setup (cf. Footnote 8.) Curvatons also generate local-type 6 bispectrum, whose amplitude is represented by the nonlinearity parameter f NL . This is given by A quadratic potential V ∝ σ 2 realizing X ( σ osc ) = 0 reproduces the known result for quadratic curvatons whose f NL is determined only by r . Let us also rewrite the energy density ratio r (2.7) in terms of the inflaton and curvaton parameters: where M p glyph[similarequal] 2 . 4 × 10 18 GeV is the reduced Planck mass, and the first and second terms in the Max. parentheses correspond to the curvaton being subdominant and dominant at its decay, respectively, while the Min. parentheses are due to whether the onset of oscillation is after or before reheating. Γ φ and Γ σ are constants that denote, respectively, the decay rates of the inflaton and the curvaton. We note that in obtaining the above results, we have adopted the sudden decay approximation where the scalar fields suddenly decay into radiation when H = Γ. Finally, the curvaton field value at the onset of the oscillations σ osc is obtained by integrating (2.1), which can be solved for σ osc as a function of σ ∗ . 7 Here, N ∗ is the number of e-folds during inflation between the horizon exit of the CMB scale and the end of inflation, c is given in (2.4), and H inf is the inflationary Hubble scale (we are assuming a nearly constant Hubble parameter during inflation, thus H inf glyph[similarequal] H ∗ ). Therefore by combining the above expressions, one can compute the density perturbations from a curvaton with a generic potential V ( σ ), given the curvaton field value at the CMB scale horizon exit σ ∗ , the decay rates of the inflaton Γ φ and curvaton Γ σ , the inflationary scale H inf , and the duration of inflation N ∗ .", "pages": [ 4, 5, 6, 7 ] }, { "title": "2.2 Case Study: Hilltop Curvatons", "content": "As an example that will be relevant for analyzing axionic curvatons in the next section, here let us apply the above generic results to a curvaton located at the hilltop, whose potential around σ osc and σ ∗ is well approximated by where m , σ 0 , and V 0 ( > 0) are constants. Without loss of generality, we assume 0 < σ osc < σ ∗ < σ 0 . Then one can check that when the curvaton is close enough to the hilltop to satisfy then the resulting power spectrum (2.5) and the non-Gaussianity (2.10) take the form with spectral index (2.9) The equation (2.12) which relates σ ∗ and σ osc gives where we dropped the H 2 inf contribution on the right hand side from the condition (2.14) and also by assuming m 2 /H 2 inf glyph[lessorsimilar] 10 -2 . As the initial value σ ∗ is shifted towards the hilltop, σ osc approaches σ 0 much slower than σ ∗ does since the left hand side is logarithmic. Therefore as one approaches the hilltop, P ζ (2.15) blows up due to the enhancement factor ( σ 0 -σ osc ) / ( σ 0 -σ ∗ ), while f NL (2.16) increases slowly. We also note that the value of f NL is greater than one even when r glyph[greatermuch] 1, from (2.14). The extreme amplification of the linear perturbations corresponds to the curvaton taking longer time to start its oscillation when starting closer to the hilltop. Before ending this section, we should remark that in the extreme hilltop limit, the approximation (2.1) for the curvaton dynamics mildly breaks down before the curvaton starts to oscillate. This gives rise to errors of O (1) for the above results in this limit. However, the above analytic expressions suffice for our order of magnitude estimations on axionic curvatons in the next section. We will also carry out numerical computations when further accuracy is required, e.g., when calculating predictions on f NL .", "pages": [ 7 ] }, { "title": "3 Axionic Curvatons", "content": "Now let us move on to the investigation of axionic curvatons, which is the main topic of this paper. As was explained in the introduction, we focus on the case where the curvaton is a pseudo-NambuGoldstone boson of a broken U(1) symmetry, possessing a periodic potential of the form where f and Λ are mass scales. Without loss of generality, we restrict the initial field value to lie within the range 0 < σ ∗ < πf . The curvaton's effective mass at the potential minimum is denoted by Then supposing that the coupling of the axionic curvaton with its decay product is suppressed by the symmetry breaking scale f , the curvaton decay rate takes the value where the constant β is naively of order unity. In the following, we ignore the time-variation of the Hubble parameter during inflation, and especially, neglect the ˙ H contribution to the spectral index (2.9). In other words, we do not consider inflationary models with rather large | ˙ H/H 2 | which requires super-Planckian field ranges or some special configurations. 8 Hence the axionic curvaton need to be located beyond the inflection point during inflation, i.e. 0 . 5 < σ ∗ /πf < 1, in order to source a red-tilted power spectrum. The axionic curvaton with σ ∗ glyph[lessmuch] πf whose potential is well approximated by a quadratic was studied in [29], and the whole potential including the hilltop region was investigated in [10]. There it was shown along the line of discussion in Section 2.2, that unless the axionic curvaton is initially located close to the hilltop, both the inflation and reheating scales need to be very high. For e.g., for σ ∗ /πf = 0 . 75 to satisfy both the WMAP normalization P ζ ≈ 2 . 42 × 10 -9 and the spectral index n s ≈ 0 . 96, then H inf glyph[greaterorsimilar] 10 13 GeV and ρ 1 / 4 reh glyph[greaterorsimilar] 10 13 GeV are required, where ρ reh represents the radiation energy density at the reheating. This is because the spectral index of order 1 -n s ∼ 0 . 01 requires a rather large curvaton mass m σ ∼ 0 . 1 H inf , forcing the curvaton to start its oscillation soon after the end of inflation. Hence without high inflation and reheating scales, the curvaton cannot even come close to dominating the Universe to source measurable density perturbations. 9 The story where M p is the reduced Planck mass and N the e-folding number. Thus | ˙ H/H 2 | as large as to give sizable contribution to the spectral index (2.9) whose typical value is n s ≈ 0 . 968 (WMAP central value) normally requires a super-Planckian field range for the inflaton. The field range bound may be alleviated by inflaton potentials giving sudden changes to dφ/d N during inflation [28]. 9 The curvaton's effective mass during inflation is decoupled from the mass at the potential minimum (3.2) when the curvaton is close to the inflection point, i.e. σ ∗ /πf ≈ 0 . 5, however in such case even higher inflation/reheating scales are required. is quite different for an axionic curvaton in the hilltop region, where the onset of the oscillation is delayed and curvaton domination is allowed with lower inflation/reheating scales. This, together with the amplification of the linear perturbations in the hilltop limit (cf. discussions around (2.18)), makes axionic curvatons compatible with many orders of magnitude of the inflation and reheating scales. In light of the above considerations, in this section we elaborate on axionic curvatons in the hilltop region, which dominate the Universe before decaying into radiation. We will find that this particular limit of axionic curvatons has interesting predictions, especially in terms of the non-Gaussianity.", "pages": [ 8, 9 ] }, { "title": "3.1 Parameter Space in the Hilltop Regime", "content": "The axionic curvaton model has five free parameters, which are the symmetry breaking scale f , the effective mass m σ = Λ 2 /f , the curvaton field value at CMB scale horizon exit σ ∗ , the inflationary scale H inf , and the inflaton decay rate Γ σ . However, since we are focusing on a curvaton that dominates the Universe before it decays, as long as there exists a parameter window which allows r glyph[greatermuch] 1, the cosmological observables do not depend on the explicit value of r or Γ σ . In this sense, the dominant axionic curvaton is actually a four parameter model. Strictly speaking, there are three more parameters: the e-folding number N ∗ between the CMB scale horizon exit and the end of inflation, the constant c (2.4) representing whether t reh ≷ t osc (though this is determined when the other parameters such as Γ φ are fully given), and β in (3.3) parameterizing the curvaton decay rate. N ∗ determines how much the curvaton rolls during inflation (cf. (2.12)), however such rolling is negligible compared to that in the post-inflationary era as seen in (2.18), and thus has little effects on the model. Hence we simply fix the e-folding number to N ∗ = 50 in the following discussions. As for c , whether reheating happens before/after the onset of the curvaton oscillations do not affect the allowed parameter window for f and m σ , but give slightly different predictions on f NL . This will be discussed in Section 3.2. The parameter β for the decay rate is set to unity in the following, and implications of β taking other values are also discussed later. Out of the four parameters, H inf and σ ∗ /f can be fixed from the WMAP normalization and also from requiring the spectral index to be consistent with the WMAP bound Later on we will see that the detailed value of the spectral index, as long it is not so close to unity, only have minor effects on axionic curvatons. Hence we are left with two parameters for the axionic curvaton f and m σ . Order of magnitude constraints on these parameters can be obtained using the analytic formulae in Section 2 (or in Section 2.2), which we present in Figure 1. The yellow region corresponds to the allowed window for a dominant axionic curvaton in the hilltop. H inf and σ ∗ /f are fixed to appropriate values by the observational constraints (3.4) and (3.5) at each point in the window, as indicated in the upper figures showing their contour lines. This also fixes the curvaton decay rate via (3.3), cf. lower left figure. (The relativistic degrees of freedom is fixed to g ∗ = 100 upon drawing the T dec contours). On the other hand, the inflaton decay rate Γ φ is not fixed at each point but is allowed to take values within a certain range, as we will soon explain. The lower right figure shows contour lines for the non-Gaussianity f NL , which is typically a few tens. Here we note that since the analytic formulae in the previous section can contain O (1) errors (cf. discussions at the end of Section 2.2), the f NL values have been computed numerically. We have shown the f NL contours inside the allowed window where the constraints described in the following are well satisfied, but the values can be modified at regions very close to the boundaries. We find that the allowed window is constrained by the following four conditions: The upper edge (green line) is set by the requirement that the curvaton initially lies in the hilltop regime, Recall that a non-hilltop axionic curvaton can work only with very high inflation/reheating scales. The right edge (blue line) denotes the requirement that the curvaton be subdominant until it starts its oscillation, When going beyond this boundary, the curvaton starts to drive a secondary inflation. A rather strict relationship between H in and σ ∗ /f is required for such inflating curvatons to work, as is discussed in Appendix A. The lower edge (orange line) requires the curvaton to decay at temperatures higher than 5 MeV in order not to ruin Big Bang Nucleosynthesis (BBN) [30, 31, 32, 33], i.e. with the relativistic degrees of freedom g ∗ = 10 . 75. Finally, the left edge (red line) follows from the dominant condition 10 These conditions give the most stringent constraints on the hilltop axionic curvaton, and other requirements for a consistent curvaton scenario are satisfied in the window bordered by (3.6) - (3.9). Let us also lay out such satisfied conditions: Firstly, the curvaton energy density is negligibly small during inflation, Moreover, quantum fluctuations during inflation should not make the curvaton jump over its potential minimum in order to avoid the resulting density perturbations from being highly non-Gaussian, or over the maximum to avoid domain walls, In the hilltop region, the classical rolling becomes suppressed, which can compete with the quantum fluctuations during inflation. 11 The curvaton's classical rolling dominates over the quantum fluctuations if where the curvaton is considered to slow-roll due to (3.10) and the lightness condition that follows from the spectral index (3.5). Furthermore, the curvaton decay should happen after reheating and the onset of the oscillations, The mass m σ is required to be larger than the curvaton decay temperature, in order to avoid possible backreaction effects to the curvaton's perturbative decay (see e.g. [34, 35, 36]). Assuming instant thermalization, this condition is written roughly as As for the inflaton sector, the energy scale of reheating (= inflaton decay) is lower than that of inflation, while an upper bound on the inflationary scale is given by constraints on primordial gravitational waves. The 7-year WMAP+BAO+ H 0 gives P T / P ζ < 0 . 24 (95% CL), which translates into 12 Let us repeat that all the requirements (3.10) - (3.15) are satisfied in the allowed window of Figure 1. We should also remark on the constraints on the reheating scale Γ φ before ending this subsection. As we have noted above, dominant axionic curvatons are insensitive to the explicit value of Γ φ . The only constraints on Γ φ are that the inflaton should decay after the end of inflation (3.15) but before the curvaton decay (3.13). For the case of t osc < t reh , the dominant condition (3.9) sets an additional lower bound on Γ φ , cf. (2.11). 13 The inflaton decay rate should take values within these bounds at each point of the allowed window in Figure 1. We note that the contour lines of various quantities in the figures are obtained assuming that the values of Γ φ at each point do not saturate the lower/upper bounds set by the above requirements. If, for example, Γ φ takes lowest possible values saturating the dominant condition (3.9) for the t osc < t reh case, then r becomes as small as ≈ 10, slightly modifying f NL from the shown values.", "pages": [ 9, 10, 11 ] }, { "title": "3.2 Dependence on Various Parameters", "content": "The spectral index n s -1 = -2 m 2 σ / 3 H 2 inf fixes the inflationary scale H inf proportional to the curvaton mass m σ , as is shown in the upper right figure. The rather wide range allowed for m σ is translated into the axionic curvaton being compatible with inflationary scales with many orders of magnitude. 13 For the case of t osc < t reh , the left red edge actually denotes where the upper bound on Γ φ set by Γ φ < H osc and the lower bound from the dominant condition (3.9) take the same values. In other words, t osc < t reh and (3.9) are incompatible beyond the red line. On the other hand, for the t osc < t reh case, r is independent of Γ φ , which allows one to set the dominant condition (3.9) independently of Γ φ . However we note that (3.9) produces the left red edge at the same place on the f -m σ plane for both t reh ≷ t osc cases. 11 12 13 14 15 16 17 18 /RParen1 /RParen1 /RParen1 /RParen1 /RParen1 /RParen1 /RBracket1 /RBracket1 /RBracket1 /RBracket1 /RBracket1 /RBracket1 /LBracket1 /LBracket1 /LBracket1 /LBracket1 /LBracket1 /LBracket1 /LParen1 /LParen1 /LParen1 /LParen1 /LParen1 /LParen1 11 12 13 14 15 16 17 18 /RParen1 /RParen1 /RParen1 /RParen1 /RParen1 /RParen1 /RBracket1 /RBracket1 /RBracket1 /RBracket1 /RBracket1 /RBracket1 /LBracket1 /LBracket1 /LBracket1 /LBracket1 /LBracket1 /LBracket1 /LParen1 /LParen1 /LParen1 /LParen1 /LParen1 /LParen1 11 12 13 14 15 16 17 18 /RParen1 /RParen1 /RParen1 /RParen1 /RParen1 /RParen1 /RBracket1 /RBracket1 /RBracket1 /RBracket1 /RBracket1 /RBracket1 /LBracket1 /LBracket1 /LBracket1 /LBracket1 /LBracket1 /LBracket1 /LParen1 /LParen1 /LParen1 /LParen1 /LParen1 /LParen1 11 12 13 14 15 16 17 18 /RParen1 /RParen1 /RParen1 /RParen1 /RParen1 /RParen1 /RBracket1 /RBracket1 /RBracket1 /RBracket1 /RBracket1 /RBracket1 /LBracket1 /LBracket1 /LBracket1 /LBracket1 /LBracket1 /LBracket1 /LParen1 /LParen1 /LParen1 /LParen1 /LParen1 /LParen1 10 10 10 10 10 /LParen1 /LBracket1 /RBracket1/RParen1 /LParen1 /LBracket1 /RBracket1/RParen1 /LParen1 /LBracket1 /RBracket1/RParen1 /LParen1 /LBracket1 /RBracket1/RParen1 /LParen1 /LBracket1 /RBracket1/RParen1 /LParen1 /LBracket1 /RBracket1/RParen1 /LParen1 /LBracket1 /RBracket1/RParen1 /LParen1 /LBracket1 /RBracket1/RParen1 /LParen1 /LBracket1 /RBracket1/RParen1 /LParen1 /LBracket1 /RBracket1/RParen1 /LParen1 /LBracket1 /RBracket1/RParen1 /LParen1 /LBracket1 /RBracket1/RParen1 The upper left figure shows the curvaton's initial value σ ∗ , which comes closer to the hilltop πf towards the lower right corner of the allowed region. However when so close to the hilltop such that σ ∗ /πf glyph[greaterorsimilar] 1 -10 -11 , then the axionic curvaton either ruins BBN, or drives a secondary inflation. For dominant curvatons r glyph[greatermuch] 1, the non-Gaussianity f NL is determined by how close the curvaton initially is to the hilltop (cf. (2.16) and (2.18)), thus the f NL contours run parallel to those of ( πf -σ ∗ ) /πf . Here, recall that when shifting σ ∗ towards the hilltop, σ osc increases much slower than σ ∗ does. This leads to a mild increase of f NL , whose largest possible value is ∼ 30 for the case of t osc < t reh , and ∼ 40 for t osc > t reh . We also note that the regime beyond the right blue edge corresponds to axionic curvatons driving a second inflationary stage. However, such inflating axionic curvatons produce rather small non-Gaussianity, as discussed in detail in Appendix A. Now let us discuss the model dependence on other parameters.", "pages": [ 11, 12, 13 ] }, { "title": "Reheating Before/During Curvaton Oscillations", "content": "Whether reheating happens before or during the curvaton oscillation only slightly modify the curvaton velocity prior to the oscillation. This does not affect our order of magnitude estimation on the allowed window in the f -m σ plane, except for that the case of t reh < t osc restricts the inflaton decay rate Γ φ to lie within a rather narrow range [10]. The σ ∗ /πf , H inf , and T dec contours are also nearly the same for the two cases, however we note that f NL can be slightly larger when t reh < t osc . This is because a radiation dominated Universe allows σ to roll less compared to when dominated by matter, and thus slightly makes σ osc closer to the hilltop. The lower right figure shows the f NL contours for t reh > t osc , but the case of t reh < t osc increases the f NL values by up to ∼ 10.", "pages": [ 13 ] }, { "title": "Spectral Index", "content": "The allowed window and non-Gaussianity are insensitive to the explicit value of n s (whether n s is, say, 0 . 94 or 0 . 98). However, if the spectral index is as close to unity as n s > 0 . 99, then the curvaton potential is required to be so flat such that the quantum fluctuations can dominate over the curvaton's classical rolling during inflation in the hilltop regime. When increasing n s beyond 0.99 towards unity, the condition (3.12) is violated first in the lower right corner of the allowed window in the f -m σ plane, and eventually in the entire window at n s glyph[greaterorsimilar] 0 . 999. Normally the model loses precise predictions if the quantum fluctuations dominate over the classical rolling. However we expect that the predictions for hilltop curvatons are not affected much by such quantum jumps during inflation, since it is the non-uniform onset of the curvaton oscillation that mainly generates the linear and second order perturbations, and also because (3.11) is satisfied even for n s ≈ 0 . 999, i.e., the quantum jumps (even when they dominate the curvaton dynamics) do not drastically change the curvaton position during inflation. We leave this question for future work, and let us close this paragraph by stating that as long as n s glyph[lessorsimilar] 0 . 99, the detailed values of the spectral index has little effect on axionic curvatons.", "pages": [ 13 ] }, { "title": "Curvaton Decay Rate", "content": "We have been setting β as unity in the curvaton decay rate (3.3). A further suppressed Γ σ (i.e. smaller β ) delays the curvaton decay, thus makes the BBN constraint (3.8) more stringent, while making it easier for the curvaton to dominate the Universe and relaxes the dominant condition (3.9). For example, β = 10 -3 tightens the lower edge (orange boundary) of the allowed window in Figure 1 by ∆(log 10 m σ ) ∼ 1, but pushes out the left edge (red) slightly (i.e. does not change the order of f ). It should be noted that the tightening of the BBN constraint results in decreasing the largest possible value for f NL , as can be seen in the lower figure. For β = 10 -3 , the maximum f NL is about 27.", "pages": [ 13, 14 ] }, { "title": "4 Discussion", "content": "The upshot of our results is that an axionic curvaton generating density perturbations consistent with current observations also generically produce non-Gaussianity f NL of O (10), even when the curvaton dominates the Universe. In particular, as one can see from Fig. 1, f NL = 20-40 is realized for the curvaton mass m σ = 10-10 6 GeV and the decay constant f = 10 12 - 10 17 GeV. What is the plausible candidate for the axionic curvaton? Interestingly, there are many moduli fields ( T ) in the string theory, and they are massless at the perturbative level because of the shift symmetry, where α is a real transformation parameter. After the moduli fields are stabilized by non-perturbative effects and SUSY breaking, the imaginary components of the moduli fields, namely the (string) axions, acquire a sinusoidal potential like Eq. (3.1). The symmetry breaking scale f is naively expected to be of order the GUT or Planck scale. Thus, the string axion is one of the plausible candidates for the axionic curvatons. 14 Recently, the standard-model like Higgs boson was discovered by the ATLAS and CMS experiments [12, 13]. The observed Higgs boson mass is about 125 - 126 GeV, which can be explained if SUSY is realized at a relatively high scale [37, 38], ranging from 10 TeV up to several tens PeV depending on the ratio of the up- and down-type Higgs boson VEVs. While the axion mass crucially depends on the stabilization mechanism, it is related to the gravitino mass in a KKLT-type stabilization [39], and so, it is conceivable that the axion mass is not many orders of magnitude different from the suggested SUSY breaking scale in the visible sector. It is intriguing that the axion with mass of this order can generate a large non-Gaussianity within the reach of the Planck satellite. The initial position of the curvaton must be very close to the hilltop of the potential. If some symmetries are restored at the maximum of the potential, the curvaton sits initially very close to the hilltop without any fine-tuning. This is possible if one considers a moduli space spanned by multiple scalar fields [40]. To be concrete, we consider a supersymmetric theory with the superpotential, Here S , χ and φ are chiral superfields, and µ is a mass scale that is real. We assume that both χ and φ parameterize D-flat directions so that their origins are enhanced symmetry points where the corresponding gauge fields become massless. In the supersymmetric limit, there is a moduli space characterized by where it should be noted that both χ and φ are complex scalar fields. The scalar potential vanishes in the moduli space. There are two special symmetry-enhanced points, i.e., χ = 0 and φ = 0. The degrees of freedom orthogonal to the moduli space are heavy, and can be integrated out. For instance, χ is heavy at φ ≈ 0, one of the symmetry enhanced points, and we can erase χ by using (4.3). In order to see that the potential has extrema at those symmetry enhanced points, let us introduce a soft SUSY breaking mass, m 2 | χ | 2 , which lifts the moduli space. A similar soft SUSY breaking mass can be introduced for φ , but it does not change the argument. Since it is φ that is light at φ ≈ 0, the effective potential can be written as where we have supposed m 2 > 0 and minimized the angular component of φ , and defined ϕ ≡ | φ | . Thus, φ = 0 is the local maximum. Note that one should write the effective potential in terms of χ at ϕ ≈ µ , since φ becomes heavy and it is χ that is light. Then the potential is simply given by m 2 | χ | 2 , which clearly shows that the potential is minimized at χ = 0 (or ϕ = µ ). Now let us discuss other cosmological issues. In order to have successful cosmology, it is necessary to generate a right amount of baryon asymmetry and dark matter. Since the baryonic/CDM isocurvature density perturbation is tightly constrained by observations, it also limits possible baryogenesis and dark matter candidates [41]. If the baryon asymmetry is generated (or dark matter density is fixed) before the curvaton dominates the Universe, too large isocurvature perturbation will be produced. Thus, both baryon asymmetry and dark matter must be generated after the curvaton domination. The Hubble parameter at the curvaton domination H dom depends on the reheating temperature as well as on the curvaton parameters, hence the value of H dom is not uniquely determined at each point in Figure 1. Largest values for H dom at each point are realized when t reh ≤ t osc , 15 in such case H dom increases as m σ and f . For m σ = 10TeV -100PeV and f ∼ 10 17 GeV, it ranges from 10 GeV to 100 TeV. There are several baryogenesis mechanisms which work at a Hubble parameter below H dom . For instance, in the Affleck-Dine mechanism [42, 43], the baryon number is generated and fixed when the Hubble parameter is comparable to the soft mass of the flat direction in the MSSM. For the sfermion masses of order 10 - 100 TeV, it is possible that the AD field starts to oscillate after the curvaton domination. Since the mass of the AD field at large field value has rather large uncertainty, H dom below TeV may be also allowed; for instance, this is the case if the potential of the AD field becomes flatter at large fields values. There are many dark matter candidates. Since the curvaton decays just before BBN for the case of our interest, there is an entropy dilution. One of the plausible dark matter candidates is the QCD axion, which starts to oscillate when the plasma temperature drops down to the QCD scale. There may be other ultralight axions which contribute to the dark matter density. Also, the thermal relic abundance of the WIMPs as well as WIMPs non-thermally produced by the curvaton decays are candidates for the dark matter. We have assumed that the curvaton is responsible for the observed density perturbation. From the minimalistic point of view, of course, the quantum fluctuation of the inflaton is the leading candidate. However, requiring both an extremely flat potential for sufficiently long inflation and the normalization of density perturbation may be too strong constraint on the inflation sector. If there are many other light scalars in nature, it might be more probable that there are two scalars, namely, the inflaton and the curvaton, responsible for the inflationary expansion and the origin of density perturbations, respectively.", "pages": [ 14, 15, 16 ] }, { "title": "5 Conclusions", "content": "In this paper we have studied non-Gaussianity of the density perturbation generated by the axionic curvaton, focusing on the case that the curvaton initially sits near the hilltop of the potential during inflation, and dominates the Universe before it decays. Interestingly, we have found that the nonGaussianity parameter f NL is positive and gets enhanced up to 30 (or 40 for early reheating) in the hilltop limit, even when the curvaton dominates the Universe. We have confirmed that this conclusion holds for n s = 0 . 94 - 0 . 99. It was also shown that in extreme cases where the axionic curvaton drives a secondary inflation, then the produced non-Gaussianity is typically f NL glyph[lessorsimilar] 10 and is smaller than non-inflating cases. Note that, as long as the curvaton dominates the Universe, f NL cannot be larger than 30 - 40; this should be contrasted to other scenarios which can generate arbitrarily large non-Gaussianity, and some parameters must be tuned to realize f NL = O (10). We have also pointed out that one of the plausible candidates for the axionic curvaton is the string axion with mass of order 10 - 100 TeV and decay constant of 10 16 - 17 GeV. If there are many axions in the Universe, one of them may be indeed responsible for the origin of the density perturbation.", "pages": [ 16 ] }, { "title": "Acknowledgements", "content": "This work was supported by the Grant-in-Aid for Scientific Research on Innovative Areas (No.24111702[FT], No. 21111006[FT,MK], and No.23104008[FT]) , Scientific Research (A) (No. 22244030 and No.21244033 [FT]), Scientific Research (C) (No. 14102004 [MK]) and JSPS Grant-in-Aid for Young Scientists (B) (No. 24740135) [FT]. This work was also supported by World Premier International Center Initiative (WPI Program), MEXT, Japan.", "pages": [ 16 ] }, { "title": "A Inflating Curvatons", "content": "In this appendix we consider the possibility that the curvaton drives a second inflationary stage before it starts to oscillate. After giving general discussions on density perturbations sourced by such inflating curvatons, we study the case for axionic curvatons.", "pages": [ 16 ] }, { "title": "A.1 Density Perturbations from Inflating Curvatons", "content": "The case studied in this appendix is illustrated in Figure 2: The curvaton initially (i.e. during the first inflationary era) has negligibly tiny energy density compared to the total energy, however dominates the universe before it starts its oscillation. We suppose that this second inflationary period is not so long, and the CMB scale exits the horizon during the first inflation. The curvaton's field fluctuations obtained from the first inflation lead to slight difference in the lengths of the second inflationary periods among different patches of the universe, thus generate the density perturbations. We also note that the second inflation is not necessarily a slow-roll one, but may be a rapid-roll inflation [44, 45, 46, 47, 48], depending on the curvaton potential. 16 log Upon calculating the density perturbations using the δ N -formalism, we assume that the second inflation lasts long enough (say, more than one e-fold) such that during this period the inflaton energy density becomes negligibly tiny and then the universe is well described as composed only of the curvaton. This assumption allows us to choose the final uniform energy hypersurface to possess energy density ρ f equal to or larger than that at the end of the second inflationary period, cf. Figure 2. In other words, we set the final hypersurface to be before the end of the curvaton-driven inflation, but late enough so that after which the inflaton can be ignored and no further δ N is produced. The energy density of the inflaton φ is considered to redshift as ρ φ ∝ a -3 after the first inflation, and we first study the case where the inflaton decay happens after the curvaton domination. Moreover, the curvaton σ is assumed to drive slow/rapid-roll inflation after dominating the universe. Hereafter we use the subscripts ∗ to denote values when the CMB scale exits the horizon (during the first inflation), 'end' for values at the end of inflation, 'dom' for when the curvaton starts to dominate the universe (i.e. ρ φ dom = ρ σ dom ), and ' f ' at the final constant energy density hypersurface. Then in order to compute the resulting density perturbations, we would like to obtain the σ ∗ -dependence of the e-folding number from the end of inflation until the final surface ( ρ σ is negligibly tiny during the first inflation, thus the curvaton has little effect on the expansion history before t end ): where Here H = ˙ a/a , with an overdot denoting a time-derivative. We take V ( σ ) to be the energy potential of the curvaton, which we assume to have no explicit time dependence. Then using ρ σ dom glyph[similarequal] V ( σ dom ), one finds thus where a prime denotes a derivative in terms of σ . However, we will soon see that δ N a only gives a minor contribution to the density perturbations. After the curvaton domination, for simplification, we ignore ρ φ and describe the second inflation as a single-component slow/rapid-roll inflation. 17 Then the inflationary dynamics is approximated by (cf. appendix of [48]), which are stable attractors under the condition and for a nearly constant η satisfying η ≤ 3 / 4. Here, note that η is not bounded from below, and that ˜ c ≥ 3 / 2. The familiar slow-roll approximations are recovered when | η | glyph[lessmuch] 1. To be precise, inflation can happen even for glyph[epsilon1] > 1 given a large ˜ c (i.e. largely negative η ). 18 However in this appendix we limit our studies to curvaton potentials satisfying (A.6) at σ = σ dom , as the hilltop potentials which are discussed in the next section satisfy this condition. Then, since the curvaton field value is monotonically increasing or decreasing in terms of time, we can use σ as a clock, Here, note that H/ ˙ σ is a function of σ , and since ρ f is a constant among different patches of the universe, so is σ f . 19 Hence by partially differentiating both sides in terms of σ ∗ , one obtains In order to compute ∂σ dom /∂σ ∗ , we make use of the slow-roll approximation 3 H ˙ σ glyph[similarequal] -V ' while t ≤ t end . During t end ≤ t ≤ t dom , for simplification we treat the universe as a matter dominated one and adopt 9 2 H ˙ σ glyph[similarequal] -V ' (cf. (2.1), see also Footnote 17). Also using ˙ H/H 2 = -3 / 2 for t end ≤ t ≤ t dom , then one can check that Partially differentiating both sides by σ ∗ , and using 3 M 2 p H 2 dom glyph[similarequal] 2 V ( σ dom ), 20 one obtains where we have dropped the contribution from the right hand side of (A.9) from the condition (A.6) satisfied at σ = σ dom . Combining the above results, we can calculate the density perturbation spectrum: where (again using (A.6)) The spectral index follows as taking the same form as for non-inflating curvatons (2.9). The non-Gaussianity parameter can also be calculated: where · · · denotes terms proportional to ∂ ˜ c ( σ dom ) /∂σ dom . One immediately sees that the first term in the { } parentheses is much smaller than unity from the condition (A.6), while the second term can be larger than unity for rapid-roll inflation, i.e. | η | glyph[greaterorsimilar] 1. In the above discussion, we have considered the inflaton to decay after the curvaton domination. Similar computations can be carried out also for the case where the inflaton decays between the first and second inflationary periods, by approximating the universe as matter dominated while t end ≤ t ≤ t reh (here the subscript 'reh' denotes values at H = Γ φ , when the inflaton is assumed to suddenly decay), and then radiation dominated while t reh ≤ t ≤ t dom . Further assuming the condition (A.6), and also that the tilt of the curvaton potential at σ reh to be not much greater than at σ dom , i.e., then one obtains the same results (A.12), (A.13), and (A.14). Whether the inflaton decays before or after the curvaton domination has little effect since the density perturbations are sourced mainly through different patches of the universe experiencing slightly longer/shorter periods of the second inflation. In summary, independently of whether the inflaton decays before/after the curvaton domination, under the condition (A.6) (and also (A.15) for t reh < t dom ), the linear perturbation sourced by inflating curvatons is of the form (A.11) with (A.12), the spectral index is (A.13), and the nonlinearity parameter is given by (A.14).", "pages": [ 16, 17, 18, 19, 20 ] }, { "title": "A.2 Inflating at the Hilltop", "content": "As an example, let us consider inflating curvatons with a hilltop potential where V 0 , m , and σ 0 are constants. Given that the curvaton is located sufficiently close to the hilltop σ 0 such that V 0 glyph[greatermuch] m 2 ( σ -σ 0 ) 2 and V 2 0 glyph[greatermuch] M 2 p m 4 ( σ -σ 0 ) 2 until the curvaton starts driving inflation, then one finds and where Note especially that η is (almost) a constant which is negative in this example. We also remark that we have dropped the contribution on f NL from the first term in the parentheses of (A.14) which is clearly smaller than unity.", "pages": [ 20 ] }, { "title": "A.3 Axionic Curvatons", "content": "In this final subsection, we look into axionic curvatons inflating at the hilltop of the potential (3.1). (We do not consider axionic curvatons away from the hilltop driving large-field inflation with superPlanckian decay constants f .) The discussions in Section A.2 can be applied to this case by simply substituting Then one can see that the negative η parameter is determined merely by the symmetry breaking scale f as Non-Gaussianities from inflating axionic curvatons are in general smaller compared to noninflating cases, which can be seen from (A.18): Large f NL requires a large, negative η , however a large | η | substantially accelerates the curvaton thus makes it challenging for the curvaton to drive inflation in the first place. We show this explicitly in Figure 3, which investigates the parameter space of axionic curvatons beyond the allowed window for non-inflating axionic curvatons discussed in Section 3. Here we fix the curvaton mass to m σ = Λ 2 /f = 10 8 GeV, and investigate large f values beyond the right blue edge (corresponding to the requirement that the curvaton be subdominant until it starts oscillating) in Figure 1. Using (A.13), we fix the (first) inflation scale H inf from the spectral index n s ≈ 0 . 96 assuming constant H during inflation, and also fix the curvaton position at CMB scale horizon exit σ ∗ /f from P ζ ≈ 2 . 4 × 10 -9 using (A.17). Furthermore, we set the number of e-foldings between the CMB scale horizon exit and the end of the (first) inflation as N ∗ = 50, the curvaton decay rate Γ σ by (3.3) with β = 1, and the inflaton decay rate Γ φ small enough such that the inflaton decays after the curvaton domination. (The explicit value of Γ φ is irrelevant for the density perturbations, however whether the inflaton decays before/after the curvaton domination slightly affects the number of e-folds obtained in the second inflation.) The resulting non-Gaussianity f NL is plotted as a function of f in Figure 3, where the blue solid line denotes the analytic calculation (A.18) with (A.21). We have also numerically computed f NL , whose results are shown as blue dots in the figures. One sees that the analytic and numerical results match well. In the right figure, we also show the number of e-folds N sec obtained in the second inflationary period driven by the axionic curvaton. N sec here is defined as the e-folding number from the curvaton domination until when the curvaton starts oscillating, i.e. (2.2). When the second inflationary period is very short, the analytic estimations derived in this appendix are invalid, which sources the slight difference between the analytic and numerical computations of f NL at f ≈ 10 17 . 4 GeV. When further increasing f beyond the plotted regime, f NL becomes further suppressed while N sec rapidly increases, soon making the axionic curvaton responsible for driving most of the inflationary e-folds after the CMB scale horizon exit. In summary, non-Gaussianity in the region beyond the right blue edge in Figure 3 decreases for larger f , taking values smaller than ∼ 10 in most of the region. Larger non-Gaussianity is generated when closer to the edge, i.e. when the second inflationary period is very short and the situation is close to the familiar non-inflating curvatons. Let us also note that the power spectrum (A.17) is now written as For a sub-Planckian f (i.e. f ≤ M p ), the prefactor κ can only take values within 0 . 04 glyph[lessorsimilar] κ glyph[lessorsimilar] 0 . 12. Therefore, once the initial position of the curvaton σ ∗ /πf is given, the inflationary scale H ∗ needs to be tuned to a rather narrow scale range in order for an inflating axionic curvaton to source the linear perturbation with an appropriate amplitude. This is in contrast to non-inflating axionic curvatons, which can work with a wide range of inflationary scales for each value of σ ∗ /πf [10]. We remark that the inflating curvaton was studied in Ref. [49], however their results differ from ours. In particular, they claimed that the non-Gaussianity parameter f NL is negative and of order unity 21 , while we have shown that f NL from a hilltop curvaton is positive and can take values larger as well as smaller than order unity. Moreover we have confirmed our results for the case of axionic curvatons by numerical calculations as shown in Fig. 3, where one sees that f NL varies from 9 to 1 as the e-folding number in the second inflation increases from 1 to 10.", "pages": [ 20, 21, 22 ] } ]

[ { "title": "Mark Hindmarsh a,c D. R. Timothy Jones b", "content": "a E-mail: [email protected], [email protected] Abstract. We consider a class of supersymmetric inflation models, in which minimal gauged F-term hybrid inflation is coupled renormalisably to the minimal supersymmetric standard model ( MSSM ), with no extra ingredients; we call this class the 'minimal hybrid inflationary supersymmetric standard model' ( MHISSM ). The singlet inflaton couples to the Higgs as well as the waterfall fields, supplying the Higgs µ -term. We show how such models can exit inflation to a vacuum characterised by large Higgs vevs, whose vacuum energy is controlled by supersymmetry-breaking. The true ground state is reached after an intervening period of thermal inflation along the Higgs flat direction, which has important consequences for the cosmology of the F-term inflation scenario. The scalar spectral index is reduced, with a value of approximately 0.976 in the case where the inflaton potential is dominated by the 1-loop radiative corrections. The reheat temperature following thermal inflation is about 10 9 GeV, which solves the gravitino overclosure problem. A Higgs condensate reduces the cosmic string mass per unit length, rendering it compatible with the Cosmic Microwave Background constraints without tuning the inflaton coupling. With the minimal U(1) ' gauge symmetry in the inflation sector, where one of the waterfall fields generates a right-handed neutrino mass, we investigate the Higgs thermal inflation scenario in three popular supersymmetry-breaking schemes: AMSB , GMSB and the CMSSM , focusing on the implications for the gravitino bound. In AMSB enough gravitinos can be produced to account for the observed dark matter abundance through decays into neutralinos. In GMSB we find an upper bound on the gravitino mass of about a TeV, while in the CMSSM the thermally generated gravitinos are sub-dominant. When Big Bang Nucleosynthesis constraints are taken into account, the unstable gravitinos of AMSB and the CMSSM must have a mass O(10) TeV or greater, while in GMSB we find an upper bound on the gravitino mass of O(1) TeV. Keywords: Supersymmetry, Higgs, inflation, cosmic strings", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Inflation is the accepted paradigm for the very early universe, thanks to its power to account accurately for cosmological data in one simple framework. However, it raises a number of theoretical problems, principally the identity of the inflaton, the flatness of its potential, and how it is coupled to the Standard Model. A technically natural way of achieving a flat potential is through supersymmetry ( SUSY ). However, the flatness is generically spoiled in supergravity [1], which must be taken into account if the inflaton changes by an amount of order the Planck scale or more ('large-field' inflation). Given the large parameter space of supergravity theories, this motivates starting the search for a supersymmetric theory of inflation with small-field inflation, in the context of a renormalisable theory. At the same time, low energy supersymmetry remains an attractive theoretical framework in which to understand the smallness of the electroweak scale relative to the Planck scale. The Minimal Supersymmetric Standard Model ( MSSM ) is the most economical possibility to combine low energy SUSY with the phenomenological triumph of the Standard Model (although the high Higgs mass and the absence of positive results from the Tevatron and LHC increases the amount of parameter tuning required). Indeed, the MSSM itself can realise inflation along one of the many flat directions [2] with the addition of non-renormalisable couplings. Inflation takes place near an inflection point in the potential, where trilinear and soft mass terms are balanced against each other, although the amount of tuning required [3] reduces the attractiveness of the scenario. The tuning can be reduced by extending the MSSM [4, 5]. The simplest class of renormalisable supersymmetric inflation models is minimal F-term hybrid inflation, by which we mean the first supersymmetric model of Ref. [1], characterised by the superpotential General theoretical considerations of small-field inflation drive one towards this model [6], which works without a Planck-scale inflaton field, non-renormalisable operators, or supersymmetry-breaking terms. It invokes an inflaton sector of (at least) 3 chiral superfields, consisting of the inflaton itself, S , and two waterfall [7] fields, Φ , Φ, with an optional gauge superfield. 1 Because the the inflaton field appears linearly in the superpotential, it does not suffer from the generic supergravity problem of Hubble-scale mass terms during inflation [1]. In its standard form, however, F-term hybrid inflation suffers from a number of problems which reduce its power to fit cosmological data. First and foremost is the gravitino problem, which limits the reheat temperature to be unnaturally small compared with the inflation scale. Of less severity is the spectral index problem. If the inflaton potential is dominated by the 1-loop radiative corrections, F-term hybrid inflation predicts that the spectral index of cosmological perturbations N e-foldings before the end of inflation is n s = 1 -1 /N . For the canonical 60 e-foldings, this is more than 1 σ above the WMAP7 value n s = 0 . 963 ± 0 . 012. Finally, many models generate cosmic strings, and the CMB constraints on their mass per unit length forces one to very weak inflaton couplings, where n s → 1 [8]. There also remains the question of how the inflaton sector is coupled to the MSSM . If we restrict ourselves to renormalisable theories combining minimal U(1) ' -gauged Fterm hybrid inflation with the MSSM , with no other fields, and preserving all the symmetries, the choices are limited. The singlet inflaton S can couple in the superpotential only to the product of the Higgs fields or the square of the right-handed neutrino fields (which we take to be included the MSSM ). If the MSSM fields have non-trivial charge assignments under the U(1) ' of F-term inflation, the coupling of S to the neutrinos is forbidden, and its place taken by one of the waterfall fields. This has the nice feature of generating a see-saw mechanism, with the neutrino masses also controlled by the vev of the waterfall fields. Neutrino masses are also allowed if the waterfall fields are U(2) triplets, with SU(2) R as a subgroup. We will refer to the minimal case where the symmetry of the waterfall fields is U(1) ' as the Minimal Hybrid Inflationary Supersymmetric Standard Model ( MHISSM ). In the model, it is very natural that the gauge singlet inflaton S should be coupled both to the waterfall fields and to the Higgs fields, which mixes the standard MSSM Higgs flat direction with the hybrid inflation waterfall direction. If the coupling of the inflaton to the Higgs is smaller than to the waterfall fields, inflation ends with the development of vevs for the Higgs multiplets, h 1 , 2 , breaking the electroweak symmetry. Soft terms lift the flat direction, and if certain constraints are satisfied, the Higgs fields will finally reach the standard vacuum after a period of thermal inflation, with a reheat temperature of about 10 9 GeV. This solves the gravitino overclosure problem, and Big Bang Nucleosynthesis constraints can be satisfied with massive (O(10) TeV or more) or stable gravitinos [9-12]. We call this second period of accelerated expansion Higgs thermal inflation. It is a natural consequence of the coupling of the F-term hybrid inflaton to the Higgs fields, and offers a generic solution to the gravitino problem. At the same time, a TeVscale vacuum expectation value for the inflaton generates an effective µ -term. The model was first introduced in Ref. [13] in the context of Anomaly-Mediated Supersymmetry Breaking ( AMSB ). We termed the version of AMSB there deployed strictly anomaly mediated supersymmetry breaking ( sAMSB ), because D-terms associated with the U(1) ' symmetry resolve the AMSB tachyonic slepton problem, without requiring an additional explicit source of supersymmetry breaking. In this paper we demonstrate that the interesting cosmological consequences, in particular Higgs thermal inflation, are a result of the structure of the model at the inflation scale, and not of the particular supersymmetry-breaking scenario. We derive the effective potential for the combination of fields driving thermal inflation, and the constraints on the soft breaking parameters for a phenomenologically acceptable ground state, in three popular supersymmetry-breaking scenarios: anomaly-mediated ( AMSB ), gauge-mediated ( GMSB ) and the constrained minimal supersymmetric standard model ( CMSSM ). We find that the lower reheat temperature following thermal inflation solves the gravitino problem in the CMSSM , while in AMSB enough gravitinos can be produced to account for the observed dark matter abundance through decays into neutralinos. In GMSB we find an upper bound on the gravitino mass of about a TeV, derived from constraints on NLSP decays during and after Big Bang Nucleosynthesis ( BBN ). F-term models with Higgs thermal inflation have other important features. The spectral index of scalar Cosmic Microwave Background fluctuations n s is reduced, as fewer e-foldings of F-term inflation are required. In the range of couplings for which the 1-loop radiative corrections dominate the inflaton potential, we find n s = 0 . 976(1), where the uncertainty comes from the spread of reheat temperatures in that range. The cosmic string mass per unit length is greatly reduced by the presence of a Higgs condensate at the string core, and is rendered independent of the inflaton coupling. Finally, thermal inflation sweeps away the gravitinos generated at the first stage of inflation, and any GUT-scale relics such as magnetic monopoles. There are other models which renormalisably couple F-term hybrid inflation to the MSSM . F D hybrid inflation [14, 15] has the same field content as ours, but the MSSM has no U(1) ' charges; and it requires a Fayet Iliopoulos term. Also potentially in the class is the B -L model of Refs. [16-18], although there is no explicit discussion of the coupling of the inflaton to the Higgs fields. In the model of Ref. [19] the waterfall fields are SU(2) R triplets. The authors identified a flat direction involving the Higgs, without pursuing its consequences. The original F-term inflation model [1] had a spontaneously broken global U(1) symmetry, and models based on coupling it to the MSSM have recently been explored in [20], again without the possibility of Higgs thermal inflation being noticed. The same field content can also produce a promising superconformal D-term inflation model [21]. Further afield, it is also possible to construct renormalisable models of inflation in the Next-to-Minimal Supersymmetric Standard Model using soft terms to generate the vacuum energy [22]. Inflation along a flat direction which mixes a singlet with an MSSM flat direction has also been investigated recently in Ref. [23]. In that work, a single stage of inflation was envisaged, and in order to supply a satisfactory spectral index, the coupling to the inflaton has to be non-renormalisable. The spectral index problem can also be solved with a non-minimal Kahler potential [24], or tuning the inflaton coupling to be small enough that the linear soft term dominates its potential [25]. In this paper we will restrict ourselves to the case where radiative corrections dominate the inflaton potential, and the Kahler potential is canonical.", "pages": [ 2, 3, 4, 5 ] }, { "title": "2 Coupling F-term inflation and the MSSM", "content": "Our guiding principle is to couple minimal F-term hybrid inflation and the MSSM (which we take to include 3 families of right-handed neutrinos) in a renormalisable way, preserving all symmetries including supersymmetry (while allowing soft breaking terms in both sectors). Hence the superpotential will take the form where W I is the standard linear F-term hybrid inflation superpotential of Eq. (1.1), W A is the MSSM Yukawa superpotential and W X is the coupling between the inflaton sector and the MSSM superpotential, containing renormalisable terms only. We will assume that the U(1) ' symmetry of the waterfall fields is gauged. The inflaton S must be a gauge singlet, and so q Φ = -q Φ . The mass scale M sets the inflation scale and the vevs of Φ and Φ. Given that the inflation scale is of order 10 14 GeV, the waterfall fields must be SU(3) ⊗ SU(2) ⊗ U(1) Y singlets. Note that W I has a global U(1) R-symmetry, which forbids the terms S 2 , S 3 and ΦΦ. In order to preserve the flat potential for the inflaton, we must preserve this symmetry; we will discuss more of its implications in a moment. /negationslash The form of W X is now tightly constrained by symmetry and anomaly cancellation. Possible anomaly-free U(1) ' charge assignments for the MSSM fields are shown in Table 1. The SM gauged U(1) Y is q L = -1 , q E = 2. U(1) B -L is q E = -q L = 1; in the absence of N this would have U(1) 3 and U(1)-gravitational anomalies. The diagonal subgroup of SU(2) R is q L = 0 , q E = 1. Note that quite generally q H 1 = -q H 2 , so we will write q H 2 = -q H 1 = q H . We will assume that the MSSM fields couple to a U(1) ' distinct from U(1) Y , i.e. that 2 q L + q E = 0, and moreover that in the AMSB case the values of q L and q E result in a solution to the AMSB tachyonic slepton problem [26]. For the resulting sparticle spectra in this case, see Ref. [13]. (Note that if the U(1) ' does not couple to MSSM fields, we are driven to F D inflation [14, 15]). Three SU(3) ⊗ SU(2) ⊗ U(1) Y singlets quadratic in the MSSM fields are available for W X , namely H 1 H 2 , LH 2 and NN [27]. The U(1) ' charge assignments, combined with the global R-symmetry, with superfield charges now uniquely specify the coupling term as where we have set q Φ , Φ = ± (4 q L +2 q E ) to permit the first term. All renormalisable B, L violating interactions and the NN and LH 2 mass terms are forbidden by the U(1) ' gauge invariance, and the superpotential Eq. (2.1) contains all renormalisable terms consistent with U(1) ' and the R-symmetry. Note in particular that the R-symmetry forbids the Higgs µ -term H 1 H 2 . Moreover, the R-symmetry forbids the quartic superpotential terms QQQL and UUDE , which are allowed by the U(1) ' symmetry, and give rise to dimension 5 operators capable of causing proton decay [28, 29]. In fact the charges in Eq. (2.4) disallow B-violating operators in the superpotential of arbitrary dimension. Soft terms break the continuous R-symmetry to the usual R-parity. The lightest supersymmetric particle ( LSP ) is therefore stable. (From Eq. (2.4), the LSP is a scalar quark or lepton, or a gaugino, or a fermionic Higgs, S , Φ or Φ.) To summarise the assumptions which force us to this unique class of theories, we require a theory with : Note that if Φ and Φ are gauged under a larger symmetry group, the coupling NN Φ is not allowed, unless they are triplets of SU(2) R and ( N,E ) are doublets [19]. The parameters M,λ 1 , λ 3 are real and positive and λ 2 is a symmetric 3 × 3 matrix which we will take to be real and diagonal. The sign of the λ 3 term above is chosen because with our conventions, in the electroweak vacuum we have H 1 H 2 →-1 2 v 1 v 2 . In the following we will denote the SU(3) ⊗ SU(2) ⊗ U(1) Y gauge couplings by g 3 , g 2 and g 1 , and the U(1) ' gauge coupling by g ' . The normalisation of the U(1) Y gauge coupling corresponds to the usual SM convention, not that appropriate for SU(5) unification. We will denote the soft parameters for the gaugino masses M a , for a cubic interaction with Yukawa coupling λ h λ , and for a mass term φ ∗ φ (where φ denotes a scalar field), m 2 φ . For the one mass term of the form φ 2 in the MSSM ( H 1 H 2 ) we will use m 2 3 .", "pages": [ 5, 6, 7 ] }, { "title": "3 The Higgs potential and its extrema", "content": "In this section we explore the important extrema of the Higgs potential, and demonstrate that there is a 1-parameter family of supersymmetric ground states with non-zero vevs for φ, φ and h 1 , 2 before supersymmetry-breaking is taken into account. We will assume that M , the scale of inflation and U(1) ' symmetry-breaking, is much larger than the scale of supersymmetry-breaking. The existence of the one-parameter family (before thermal effects and soft terms are taken into account), is demonstrated as follows. The minimum of the scalar potential is determined by the requirement that both the F- and D-terms vanish. The vanishing of the D-terms ensures that | φ | = | φ | , | h 1 | = | h 2 | and h † 1 h 2 = 0, while the vanishing of the F-term is assured by λ 1 φφ -λ 3 h 1 h 2 = M 2 . The minimum can therefore be parametrised by an SU(2) gauge transformation and angles χ, ϕ defined by The ϕ angle can always be removed by a U(1) '' gauge transformation (where the residual symmetry unbroken by the Higgs vevs alone is U(1) em × U(1) '' ), so the physical flat direction just maps out the interval 0 ≤ χ ≤ π/ 2. At the special point χ = 0 the U(1) '' symmetry is restored, and at χ = π/ 2 the SU(2) ⊗ U(1) Y is restored. Away from these special points only U(1) em is unbroken. The degenerate minima have been noted before [19] in a model with gauge group SU(3) ⊗ SU(2) L ⊗ SU(2) R ⊗ U(1) B -L . However, the important cosmological consequences which follow was first explored in Ref. [13]. Let us first consider the limiting cases where either h 1 , 2 or φ, φ vanish.", "pages": [ 7, 8 ] }, { "title": "3.1 The φ, φ, s extremum ( φ -vacuum)", "content": "In the φ, φ, s subspace (lower case fields denote the scalar component of the superfields) the scalar potential (including soft supersymmetry-breaking terms) is: We will assume that the term linear in s is small enough not to be important for inflation (and quantify this smallness in Section 5). In AMSB there are arguments [30] to show that, without a quadratic term S 2 in the superpotential, the only RG invariant solution for ρ is ρ = 0. Let us establish the minimum in this subspace, under the assumption that m 3 2 /lessmuch M . We shall call this the φ -vacuum. With the notation 〈 φ 〉 = v φ / √ 2, 〈 φ 〉 = v φ / √ 2 and 〈 s 〉 = v s / √ 2, we find From Eqs. (3.3), (3.4) we find Then from Eqs. (3.6), (3.7), to leading order in an expansion in m 3 2 /M we have and from Eq. (3.5) that v s is O ( m 3 2 ). It follows from Eq. (3.7) that and from Eq. (3.5) that From now on we neglect ρ , assuming that ∣ ∣ Substituting back from Eqs. (3.8), (3.10) into Eq. (3.2), we obtain to leading order and from Eq. (3.10) a Higgs µ -term naturally of the same order as the supersymmetry-breaking scale. The theory is approximately supersymmetric at the scale M , so the U(1) ' gauge boson, the Higgs boson, the gaugino and one combination of ψ φ,φ form a massive supermultiplet with mass m ∼ g ' √ v 2 φ + v 2 φ , while the remaining combination of φ and φ and the other combination of ψ φ,φ form a massive chiral supermultiplet, with mass m ∼ λ 1 √ v 2 φ + v 2 φ . 2 The large vev for φ generates inflation-scale masses for the N triplet, thus naturally implementing the see-saw mechanism.", "pages": [ 8, 9 ] }, { "title": "3.2 The h 1 , 2 , s extremum ( h -vacuum)", "content": ". In the h 1 , 2 , s subspace, the scalar potential is Note that we assume there is no h 1 h 2 mass term; its absence follows from the absence of the corresponding term in the superpotential (which is forbidden by the R-symmetry) when the source of supersymmetry breaking can be represented by a non-zero vev for a spurion (or conformal compensator) field. The structure is similar to Eq. (3.2), with the addition of SU(2) and U (1) Y Dterms. Without loss of generality the SU(2) D-term vanishes with the choice h 1 = ( v 1 / √ 2 , 0) and h 2 = (0 , v 2 / √ 2), and v 1 = v 2 . The values of the fields at the minimum (which we term the h -vacuum) and the value of the potential at this extremum can then be recovered from the result of the previous section with the replacement λ 1 → λ 3 ), leading to a potential energy density", "pages": [ 9, 10 ] }, { "title": "3.3 Potential along the φ , φ , h 1 , h 2 flat direction", "content": "As we outlined at the beginning of the section, the supersymmetric minima are parametrised by an angle χ , defined in (3.1). Soft terms lift this degeneracy, and the leading terms in the effective potential for χ can be found in an expansion in m 2 3 2 /M 2 . After solving for s , it is found that where we have defined", "pages": [ 10 ] }, { "title": "4 Supersymmetry-breaking and the true minimum", "content": "In this section we investigate under which conditions the phenomenologically acceptable largeφ solution is the true minimum, in three popular supersymmetry-breaking scenarios. Hence we are looking for constraints on the soft supersymmetry-breaking parameters such that We will also check that the false vacuum at χ = 0 is a local maximum, from the sign of V '' (0), which can be recovered from V '' ( π/ 2) by the replacements 1 ↔ 3 and ¯ m 2 φ ↔ ¯ m 2 h . A metastable false vacuum, as we will demonstrate in Section 7, would lead to the universe remaining trapped in an inflating phase. We assume that the U(1) ' symmetry is broken by a vev of order v ' ∼ M/ √ λ 1 , 3 , and evaluate the soft terms at this scale, rather than running down to the electroweak scale. This is the appropriate renormalisation scale to investigate a potential with vevs of order v ' , whose important radiative corrections are from particles of mass of order g ' v ' and M . Note that in inflation models, with inflaton couplings λ 1 and λ 3 are generally small, and so the U(1) ' gauge boson mass m A = g ' √ v 2 φ + v 2 φ is much greater than M , unless g ' is also small.", "pages": [ 10, 11 ] }, { "title": "4.1 Anomaly-mediated supersymmetry-breaking", "content": "With anomaly mediation, the soft breaking parameters take the generic renormalisation group invariant form Here µ is the renormalisation scale, and m 3 2 is the gravitino mass; β g a are the gauge β -functions and γ is the chiral supermultiplet anomalous dimension matrix. Y U,D,E,N are the 3 × 3 Yukawa matrices, µ h is the superpotential Higgs µ -term, κ and k are constants, and Y ' i are charges corresponding to the U(1) ' symmetry. In the MSSM , κ is an arbitrary parameter, which in practice is fixed by minimising the Higgs potential at the electroweak scale. The parameter k is generated by the breaking of the U(1) ' symmetry at a large scale, and forms the basis of the solution to the tachyonic slepton problem within the framework of AMSB , as explained in [13], whence the name strictly anomaly-mediated supersymmetry-breaking ( sAMSB ) originates. The Higgs µ -term, µ h , is generated by the the vev of the inflation s , which in turn is triggered by the U(1) ' symmetry-breaking. Hence the parameter k , and the equation for m 2 3 , are relevant only below the U(1) ' symmetry-breaking scale v ' . As a first approximation, we will assume that the g ' terms dominate throughout, as q H and q Φ are generally large, in which case the h λ 1 and h λ 3 trilinear soft terms are given from Eq. (4.4) as: while the mass soft terms are The one loop g ' β -function is where for n G = 3. Hence Thus the difference in the energy densities between the two vacua is, in this approximation, The coefficient Q is in general large, and larger than both q 2 Φ and q 2 H , so the condition for V φ to be the true minimum may be written It is not hard to check from Eq. (4.2)) that under the same assumptions, the φ -vacuum is a minimum and the h -vacuum is a maximum. Hence no further constraints on the parameters are generated. In the next section we will see that if λ 3 > λ 1 , then inflation ends with φ, φ developing non-zero vevs, whereas if λ 3 < λ 1 it is 〈 h 1 , 2 〉 which become non-zero; this statement is independent of the nature of the soft breaking terms. Now is easy to show that ( q H q Φ ) 2 < 1 unless However, the domain defined by Eq. (4.19) does not permit a satisfactory electroweak vacuum in the AMSB case [26]. For example, for the specific choice q L = 0, which can lead to an acceptable electro-weak vacuum [13], the condition V h > V φ becomes (from Eq. (4.17)) or λ 1 /lessorsimilar 4 λ 3 from the approximation Eq. (4.18). We see, therefore, that there will generally be a domain such that the universe exits to the false high Higgs vev h -vacuum, evolving subsequently to the true vacuum as we shall describe later. In the Appendix we include a more accurate computation of the vacuum energy difference, taking into account the SM gauge couplings and the top Yukawa coupling.", "pages": [ 11, 12, 13 ] }, { "title": "4.2 Gauge-mediated supersymmetry-breaking", "content": "In the GMSB framework (see e.g. [31]), supersymmetry-breaking is communicated by a set of messenger fields C which have SM gauge charges in a vector-like representation, which should be complete GUT multiplets if gauge unification is to be preserved. The messenger fields are supposed to have a large mass, given by the vev of the scalar component of a chiral superfield X , which also has a non-zero F-term F X , the source of the supersymmetry breaking. Although there are many possibly choices for the field representations of the messenger fields, we can adapt the simple model described in [31] to study our model. We introduce the following superpotential for the extra fields assuming that some extra dynamics at a higher scale gives both the scalar component of X and F X a vev. We will assume that 〈 X 〉 /greatermuch M . Radiative corrections from the messenger particles then induce masses for the gauginos at one loop, where Λ g = N mi 〈 F X 〉 /M X , M X = λ 5 〈 X 〉 , and N mi is the messenger index, equal to twice the sum of the Dynkin indices of the messenger fields. Scalars acquire masses from 2-loop corrections of where Λ 2 s = N mi ( 〈 F X 〉 /M X ) 2 , C a ( i ) is the quadratic Casimir associated with the a th gauge group for the i th scalar, and the sum over a includes the four gauge couplings g 1 → 3 , g ' . Trilinear terms are also induced at 2 loops, and so are of order Λ g ( α a / 4 π ) 2 . They are small compared with the gaugino masses, and it is a reasonable approximation to take them to vanish at the messenger scale M X . We assume that Λ g,s are of the correct order of magnitude for supersymmetry-breaking. We thus have Thus the difference between the vacuum energies is so that, if we assume dominance of the g ' terms, the condition that V φ < V h becomes This is precisely the opposite condition to that in AMSB , Eq. (4.18). As in AMSB , the condition that V φ < V h is sufficient to ensure that V φ is a minimum and V h a maximum. Now in GMSB , we do not have the constraint on the domain ( q L , q E ) that we described in the AMSB case. Inflation will end in the Higgs phase unless in which case it ends directly in the true φ -vacuum.", "pages": [ 13, 14 ] }, { "title": "4.3 Constrained minimal supersymmetric standard model", "content": "At the high scale we will have the CMSSM pattern of soft breaking parameters, and hence Hence if λ 3 < λ 1 (so that inflation ends in the h -vacuum) then for V h > V φ we require (4.31) It is easy to check from Eq. (4.2) that this is again a sufficient condition that V '' ( π/ 2) be positive. On the other hand, there is then a range for which the h -vacuum is also a local minimum. We will see that this scenario is not consistent with a graceful exit from Higgs thermal inflation, and hence for a cosmologically acceptable potential, we must demand", "pages": [ 14, 15 ] }, { "title": "5.1 F-term inflation", "content": "We assume that the vevs of MSSM fields apart from the Higgs are negligible, in which case the relevant tree potential is The soft terms in V soft are those appearing in Eqs. (3.2), (3.14), and are all suppressed by at least one power of m 3 2 . The most important soft term for inflation is one linear in s , the effect of which we assume is small compared with the radiative correction. We will see in Eq. (5.7) that this implies tuning below O(1) only if the couplings λ 1 , 3 are very small. We also assume that the higher order terms in the Kahler potential do not contribute significantly. At large s , and with all other fields vanishing, the potential is approximately where ∆ V 1 represents the one-loop corrections, which dominate the soft terms. As S is coupled only to Φ, Φ and H 1 , 2 , the contribution to the one-loop scalar potential is [32] For large s (meaning λ 1 , 3 s 2 /greatermuch M 2 ) the potential can be written as where an O( α ) correction to M 4 has been dropped, and We will neglect supergravity contributions in the potential, which will require a small coupling c of the quartic term c | s | 4 /m 2 P in the Kahler potential, and impose a constraint [33] There are also potentially important contributions from the linear soft term ρM 2 m 3 2 s + c.c.. These are negligible provided We will shortly see that s c ∼ 10 16 GeV, so assuming m 3 2 ∼ 10 5 GeV, a soft term with ρ ∼ 1 is negligible provided Henceforth we will assume that the Kahler potential is canonical and that λ is in the range given by Eqs. (5.6), (5.8). We note, however, that interesting consequences for the spectral index flow from a non-canonical Kahler potential [24] and from couplings small enough for the soft term to contribute [25].", "pages": [ 15, 16 ] }, { "title": "5.2 Perturbation amplitudes", "content": "The scalar and tensor power spectra P s , P t and the scalar spectral index n s generated on a scale k equal to the co-moving Hubble scale aH at N k e-foldings before the end of inflation are given by the standard formulae (see e.g. [34]), The WMAP7 best-fit values for P s ( k 0 ) and n s at a pivot scale k = k 0 = 0 . 002 h Mpc -1 in the standard ΛCDM model are [35] From this data we infer that showing approximately a 2 σ discrepancy with the standard Hot Big Bang result N k 0 /similarequal 58 + ln( T rh / 10 15 GeV) (assuming only MSSM degrees of freedom at T rh ). We will see shortly that the reheat temperature lies in a range around 10 14 GeV, and in Section 7 that there are N θ /similarequal 15 e-foldings of thermal inflation at a lower scale. Therefore one can estimate N Fti /similarequal 42(1) e-foldings of F-term inflation while the pivot scale k 0 is outside the horizon, where the uncertainty comes from the range of reheat temperatures, given in Eq. (5.19). The scalar spectral index is thereby reduced to Lower values of the spectral index are possible if λ drops below the limit (5.8) and the linear soft term comes into play [25].", "pages": [ 16, 17 ] }, { "title": "5.3 End of inflation and reheating", "content": "F-term inflation ends when one set of scalar fields becomes unstable. If λ 3 > λ 1 , the φ , φ pair become unstable first, and inflation ends at the critical value s 2 c 1 = M 2 /λ 1 . The fields φ , φ gain vevs and the universe makes a transition to the U(1) ' -broken phase described by Eq. (3.3)-Eq. (3.5). On the other hand, if λ 3 < λ 1 , the Higgs fields become unstable first, the critical value of s is s 2 c 3 = M 2 /λ 3 , and the universe makes a transition to a phase where h 1 and h 2 develop vevs of order the unification scale rather than φ, φ . In this phase the SU (2) L symmetry is broken. At first sight, this would appear to rule out the model with λ 3 < λ 1 . However, provided the correct (small Higgs vev) vacuum has the lowest energy density at zero temperature, the universe can seek the true vacuum when thermal corrections become sub-dominant. We will establish in Section 7 that the evolution to the true ground state proceeds by a period of inflation. Assuming that λ 3 < λ 1 , inflation exits to the h -vacuum, with symmetry-breaking Here, U(1) '' is generated by the linear combination of hypercharge and U(1) ' generators which leaves the Higgses invariant: There are still two Abelian symmetries, and SU(2) is completely broken with no discrete subgroup. Hence cosmic strings are not formed at this transition. We expect reheating to be very rapid [36-41], as the period of oscillation of the fields is of order M -1 , which is much less than a Hubble time, and the couplings of the Higgs field are not all small. Hence the universe regains a relativistic equation of state almost immediately, and thermalises at a temperature T rh1 given by where g rh1 is the effective number of relativistic degrees of freedom at temperature T rh1 . From (5.13), and taking g rh1 = 915 / 4 (a slight overestimate), we find Hence the range of reheat temperatures corresponding to the range of couplings defined by Eq. (5.6) and Eq. (5.8) is Finally, we note that large vevs of other fields along supersymmetric flat directions can lead to blocking of particle production during reheating [27]. On the other hand, radiative corrections during inflation generically generate masses of order y 2 H 2 [42], where y is a combination of Yukawa couplings, and so we expect that other vevs besides that of the inflaton will be generally small. We leave a detailed examination of the flat directions for another work, assuming for now that any flat directions which do not have y of order 1 are small.", "pages": [ 17, 18 ] }, { "title": "5.4 High temperature ground state", "content": "As the universe reheats, it will seek a minimum of the finite temperature effective potential, or equivalently the free energy density. To discuss the free energy, it is convenient to define a dimensionful field X = v + χ , with v + = √ 2 v φ = 2 M/ √ λ 1 . The free energy density can then be expressed as where g eff ( X,T ) is the effective number of relativistic degrees of freedom at temperature T . At weak coupling, g eff ( X,T ) can be calculated in the high-temperature expansion for a particle of mass m /lessmuch T [43], where there are contributions to c 0 of 1 , 7 8 and to c 1 of 1 24 , 1 48 for bosons and fermions respectively. For particles with m>T , g eff is exponentially suppressed. We can see that X = 0 is a local minimum for temperatures m 3 2 /lessmuch T /lessorsimilar M , because away from that point the U(1) '' gauge boson develops a mass proportional to 〈 φ 〉 , and so g eff decreases. For similar reasons X φ = v + π/ 2 is also a local minimum: away from that point the MSSM particles develop masses and again reduce g eff . In fact, by counting relativistic degrees of freedom at temperatures m 3 2 /lessmuch T /lessorsimilar M one finds that X φ is the global minimum at high temperature. In the h -vacuum the relativistic species are the Φ , Φ chiral multiplets and the U(1) '' gauge multiplet. In the φ -vacuum, the particles of the MSSM are all light relative to T . Hence The minima of the free energy density are separated by a free energy barrier of height ∼ T 4 . The transition rate can be calculated in the standard way [44] by calculating the free energy of the critical bubble E c . The transition rate per unit volume is then The critical bubble is a solution to the equation where r is the radial distance from the bubble centre, and we have neglected O(1) complications in the kinetic term from the non-linear field transformation. An orderof-magnitude estimate can be given, recognising that X has to change by an amount ∆ X ∼ v + from the inside to the outside of the bubble, while negotiating a local free energy bump of order ∆ V T eff ∼ T 4 . Neglecting the damping term, one can translate the equation into a harmonic oscillator problem, finding that the critical bubble radius is approximately and so the critical bubble energy The universe will stay in the wrong ground state if the transition rate per unit volume is significantly below the Hubble rate per Hubble volume, or Γ < H 4 . Hence the reheat temperature T rh1 should be parametrically Recalling that T rh /similarequal M and v + = 2 M/ √ λ 1 , we see that if inflation exits to the h -vacuum it is likely that the universe continues to evolve with large (inflation-scale) Higgs vevs, provided λ 1 /lessmuch 1.", "pages": [ 18, 19 ] }, { "title": "6 Review of gravitino constraints", "content": "There are strong constraints on the gravitino mass and lifetime from cosmology [912]. If the gravitinos are unstable, they can conflict with Big Bang Nucleosynthesis ( BBN ) by photodissociating light elements, or they can decay directly into the LSP , which in turn produces a limit from the known density of dark matter in the standard cosmological model. The gravitino may also be the LSP , in which case the dark matter constraint applies directly. Gravitinos are produced by collisions of high-energy particles in the thermal bath, principally gluons and gluinos, with an abundance of approximately [45] where M ˜ g is the gaugino mass at the GUT scale. We include an O(1) factor ω ˜ G to take into account the theoretical uncertainties [46-48], arising from the strong dynamics of the coloured plasma. BBN constraints [45] are not easily summarised, but are much tighter for lighter gravitinos which decay during or after nucleosynthesis, as relevant for the CMSSM . For gravitino masses less than about O(10) TeV, the reheat temperature is bounded above by T rh /lessorsimilar (0 . 2 -1) × 10 6 GeV. For higher gravitino masses, the dark matter density provides a bound, and so it is appropriate use Eq. (6.1) in the limit M 2 ˜ g /m 2 3 2 → 0.. Given that the LSP density parameter arising from a particular relic abundance in the MSSM is the LSP density parameter from (high mass) thermally produced gravitinos can be found as This must be less than or equal to the dark matter abundance inferred from the CMB [35] The presence of cosmic strings in our model, although affecting the CMB power spectrum, does not significantly affect this inferred value [49]. In our model, we will see that the gravitinos generated by the first stage of reheating are diluted by a period of thermal inflation. The constraint therefore applies to reheating after thermal inflation. We will also see that the second reheat temperature is about 10 9 GeV, and so we can only tolerate unstable gravitinos of mass greater than about 10 TeV in order not to spoil BBN . This is natural in AMSB , problematic in GMSB , while the CMSSM keeps m 3 as a separate parameter. 2 There are also non-thermal production mechanisms from coherent oscillations of the inflaton [50, 51] and from ordinary perturbative decay [52], whose rates depend on the inflaton mass and vev. We will see in the next section that the relevant inflaton mass and vev will be those of the Higgs. However the BBN constraints mean that the gravitino, when it is not the LSP , must be much more massive than the Higgs and so cannot be produced by direct decays. Hence only thermal production is relevant.", "pages": [ 20, 21 ] }, { "title": "7 Higgs thermal inflation and gravitinos", "content": "As the temperature falls, the energy density difference between the vacua becomes comparable to thermal energy density, and the universe can seek its true ground state, which is χ = π/ 2, the φ -vacuum. At zero temperature we can write the difference in energy density between the h -vacuum and the φ -vacuum as (see Eqs. (4.17), (A.13), (4.26) and (4.30)) where we recall that v 2 + = 4 M 2 /λ 1 , and we have defined an effective SUSY -breaking scale m sb . In the supersymmetry-breaking scenarios under consideration  A period of thermal inflation [53] starts at where g i is the effective number of degrees of freedom at temperature T i . The CMB normalisation (5.13) for N e-foldings of standard hybrid inflation gives ( v + /m P ) /similarequal 5 × 10 -3 (40 /N ) 1 4 . Using the number of degrees of freedom for a U(1) em ⊗ U(1) '' theory with two light chiral multiplets Φ and Φ, g i = 15, we have (on dropping the unimportant dependence on N ) The h -vacuum must be a local maximum at zero temperature, i.e. the soft mass terms m 2 φ | φ | 2 + m 2 φ | φ | 2 must be negative. If the h -vacuum were a local minimum, one can estimate that the tunnelling rate per Hubble time per Hubble volume [54] would be where S E is the action of the Euclidean tunnelling solution. This ratio must be of order unity for the universe not to remain trapped in the false vacuum [55], and since the prefactor is much less than unity, we see that we cannot allow a metastable h -vacuum for a graceful exit from thermal inflation. Thermal inflation continues until the quadratic term in the thermal potential g ' 2 T 2 ( | φ | 2 + | φ | 2 ) becomes the same size as the negative soft mass terms. Near the false vacuum, the high temperature effective potential for the field X breaking the U(1) '' symmetry can be written [44, 56] where γ , δ and λ X are dimensionless constants, and T 0 /similarequal | m φ | /g ' . The cubic term arises from the gauge boson, and the transition is first order provided λ X < e 4 , where e is the effective U(1) '' gauge coupling [44, 56]. Hence the transition which ends thermal inflation takes place at T e ∼ m sb , and the number of e-foldings of thermal inflation is Thus gravitinos will be diluted to unobservably low densities, as will any baryon number generated prior to thermal inflation, and any other dangerous GUT-scale relics such as monopoles. After thermal inflation ends, there is another period of reheating as the energy of the modulus X is converted to particles. Around the true vacuum, the X is mostly Higgs, and so its large amplitude oscillations will be quickly converted into the particles of the MSSM . The natural oscillation frequency around χ = π/ 2 is of order m 3 2 , while the Hubble rate is of order m 3 2 M/m P . Hence in much less than an expansion time, the vacuum energy will be efficiently converted into thermal energy. The reheat temperature following thermal inflation is thus where g rh2 is the effective number of relativistic degrees of freedom at T rh2 , given its MSSM value g rh2 = 915 / 4. This second reheating regenerates the gravitinos, and we may apply the gravitino density formula Eq. (6.3), finding We can convert the relic density into a constraint on the effective SUSY -breaking scale m sb , requiring that the LSP density is less than or equal to the observed dark matter abundance, Ω dm h 2 /similarequal 0 . 11. The parameter m sb is directly related to physical observables differently in the different SUSY -breaking schemes, for which we can derive constraints.", "pages": [ 21, 22 ] }, { "title": "7.1 Gravitino constraint in AMSB", "content": "Using Eq. (7.10) and the expression for m sb in Eq. (7.2), we find Hence AMSB -based models requires a high gravitino mass in order to saturate the bound and generate the dark matter. We can be a bit more precise if we use use the phenomenological relations derived in [13]. Firstly, in order to fit µ h we have (using Eqs. (3.13), (4.7)) while we can use a phenomenological formula for the LSP mass Hence We also have a constraint (Eq. (4.21)) on λ 3 /λ 1 from requiring the exit to a false h -vacuum. Hence in order for the LSP in this model to comprise all the dark matter, we have For example, taking q L = 0 as in [13], we find that m 3 2 is independent of q E and in the range where we recall from the discussion around Eq. (6.1) that ω ˜ G is O(1). It was noted in [13] that a Higgs of mass 125 GeV demands a gravitino mass of about 140 TeV in sAMSB , which is compatible with an LSP produced by gravitino decays being the dark matter.", "pages": [ 23 ] }, { "title": "7.2 Gravitino constraint in GMSB", "content": "In the GMSB framework the LSP is usually the gravitino, whose mass is given by where k < 1 parametrises the fraction of the total F-term contained in the messenger sector, and we recall that M X is the messenger scale. It is more convenient to phrase the dark matter constraint in terms of the larger electroweak gaugino mass M 2 , which dominates in the equation for the SUSY -breaking scale Λ g , as in Eq. (4.23), and therefore Eq. (7.2) can be rewritten Hence The bound can be saturated for a TeV-scale gravitino with TeV-scale gaugino masses without special tuning of the ratio λ 3 /λ 1 . A lighter gravitino forces the gaugino mass upwards. There is a separate constraint from decays of the NLSP (which is generally a neutralino for unless M X is small), which may interfere with Big Bang Nucleosynthesis (see e.g. [31]). A careful analysis of the nucleosynthesis constraints [57] shows that a messenger mass of up to about 10 14 GeV is allowed, before hadronic jets injected after 10 4 s results in the overproduction of 7 Li. The combination of the dark matter and BBN constraints M X /lessorsimilar 10 14 GeV may be written", "pages": [ 23, 24 ] }, { "title": "7.3 Gravitino constraint in the CMSSM", "content": "In the CMSSM , the gravitino bound Eq. (7.10) can be expressed in terms of the soft scalar masses m 0 and the trilinear parameter A from Eq. (7.2), as This is a very weak bound, unless the ratio λ 3 /λ 1 is very small: hence there is generically a very low density of LSP dark matter generated by decays of gravitinos. Instead, the CMSSM can generate an acceptable dark matter density through the standard freeze-out scenario [58] (see [59-61] for recent analyses of the the CMSSM parameter space in the light of recent Higgs results). This requires that the gravitino mass is larger than about 10 TeV to avoid BBN constraints [45]. We conclude that the Higgs thermal inflation solution generally has no effect on the gravitino problem in the CMSSM , beyond determining the reheat temperature and hence the standard BBN -induced lower bound gravitino mass.", "pages": [ 24 ] }, { "title": "8 Conclusions", "content": "In this paper we have shown how models which couple F-term hybrid inflation with the MSSM without extra ingredients naturally realise a period of thermal inflation, with a reheat temperature of around 10 9 GeV, while generating the Higgs µ -term. The inflation is driven by the relaxation of the Higgs fields to zero in a potential generated by the Higgs and waterfall field soft terms. This second period of intermediate scale inflation, which we have called Higgs thermal inflation, has a number of beneficial effects. It solves the gravitino overabundance problem of supersymmetric cosmology, while still maintaining the possibility of leptogenesis. It reduces the cosmic string mass per unit length so that CMB bounds are satisfied, and renders it independent of the inflaton couplings. Hence the scalar spectral index is not driven to unity in the effort to make the strings light, from which the tight constraints on standard F-term hybrid inflation are generated [8]. The period of thermal inflation means a reduced number of e-foldings of F-term inflation are required, and the scalar spectral index is reduced: in the range of inflaton couplings where the inflaton potential is dominated by the radiative corrections, we find n s /similarequal 0 . 976(1). The MHISSM is the simplest, attractive, formulation of this scenario, and generates right-handed neutrino masses as well as the Higgs µ -term. We found constraints on the couplings and soft terms in order for the scenario to work: i.e. for F-term inflation to exit towards a vacuum with inflation-scale vevs for the Higgs field, and for that vacuum to be unstable. We investigated the implications of these constraints in three popular supersymmetry-breaking scenarios: AMSB (where the model coincides with strictly anomaly-mediated supersymmetry-breaking [13]), GMSB , and the CMSSM . We found constraints on the ratio of the inflaton couplings in AMSB (Eq. (4.18)) and GMSB (Eq. (4.27)), and that in the CMSSM the soft scalar mass must be greater than the half the magnitude of the soft trilinear term, multiplied by the square root of the ratio of the inflaton couplings (Eq. (4.33)). In AMSB , the gravitino problem becomes the gravitino solution: the observed dark matter density can be generated by the decays of gravitinos which are produced thermally following Higgs thermal inflation. In GMSB , the gravitino is the LSP , and a weak upper bound on its mass of about 1 TeV follows from the combined requirement that it supply the dark matter without NLSP decays spoiling nucleosynthesis. In the CMSSM , the density of thermally-produced gravitinos is generally sub-dominant, and the standard freeze-out scenario must do the work of making neutralino dark matter. However, the gravitinos must decay early enough not to spoil nucleosynthesis, meaning that the gravitino mass must be O(10) TeV or greater. A reheat temperature of 10 9 GeV is broadly consistent with thermal leptogenesis, provided at least one right-handed neutrino is light enough to be thermally produced. We leave the details for a future publication. The MHISSM predicts the formation cosmic strings, with dimensionless mass per unit length estimated as Gµ s /similarequal 10 -7 [13]. While satisfying current CMB bounds, there are tight bounds on the GeV-scale cosmic γ -ray spectrum [62], so strings should have a very small branching fraction into γ . Strings may instead decay into gravitational waves, but there are also increasingly strict bounds on the stochastic gravitational wave background from pulsar timing [63, 64]. Should the bounds ultimately fall below the predicted value of Gµ s , this will rule out the MHISSM but not the Higgs thermal inflation scenario in general, which remains a possibility whenever the inflaton is coupled to a set of waterfall fields which include the Higgs.", "pages": [ 24, 25 ] }, { "title": "Acknowledgments", "content": "This research was supported in part by the Science and Technology Facilities Council [grant numbers ST/J000477/1 and ST/J000493/1], the Project of Knowledge Innovation Program (PKIP) of Chinese Academy of Sciences, Grant No. KJCX2.YW.W10, and the National Science Foundation under Grant No. NSF PHY11-25915. MH and TJ gratefully acknowledge the hospitality of the Kavli Institutes for Theoretical Physics, in China and Santa Barbara respectively, and also the role of the UK Particle Cosmology workshop in the development and dissemination of this research.", "pages": [ 26 ] }, { "title": "A sAMSB soft parameters and inflaton coupling constraints", "content": "In this appendix we give formulae for a more accurate calculation of the sAMSB soft parameters in Section 4.1, relevant for the calculation of the vacuum energies and hence the constraint on λ 3 /λ 1 . The h λ 1 and h λ 3 trilinear soft terms are determined in accordance with Eq. (4.4): while the mass soft terms are In the following, we include the SM gauge couplings and the top Yukawa coupling, which we shall denote y t . We also retain the neutrino Yukawas Y N , since their magnitude is model dependent. We will assume that the value of tan β is such that it is a good approximation to neglect all the other Yukawas, and we will also neglect λ 1 and λ 3 . The relevant soft breaking parameters are then m (16 π 2 φ m = - 3 2 2 ) 2 We have assumed above for simplicity that, like λ 2 , Y N is real and diagonal. The difference in the energies between the large-Higgs and largeφ vacua can be written where The condition on the couplings deriving from the vacuum energies can therefore be written where Taking the values of the SM couplings at the U(1) ' breaking scale to be the values at gauge coupling unification, we find using the renormalisation group analysis of Ref. [13] (where we have taken tan β = 16). Hence, at this level of accuracy, We can derive successive approximations. Firstly, neglecting terms of order q 2 H /Q and q 2 φ /Q , which is a good approximation given Eq. (4.14), we have Secondly, we can neglect terms of order β H,φ and higher (which is not necessarily a good approximation), to obtain In the case q L = 0, we obtain Eq. (4.20). We can also expand in powers of β H,φ while still neglecting terms of order Y 2 N , and bearing in mind that an acceptable electroweak vacuum requires Tr λ 2 2 /similarequal 4 β H [13]), we find to second order", "pages": [ 26, 27, 28 ] } ]

[ { "title": "Galaxy correlations and the BAO in a void universe: structure formation as a test of the Copernican Principle", "content": "Sean February 1 , Chris Clarkson 1 , and Roy Maartens 2 , 3 1 Astrophysics, Cosmology & Gravity Centre, Department of Mathematics and Applied Mathematics, University of Cape Town, Cape Town 7701, South Africa 2 Department of Physics, University of Western Cape, Cape Town 7535, South Africa 3 Institute of Cosmology & Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK (Dated: November 2, 2018) A suggested solution to the dark energy problem is the void model, where accelerated expansion is replaced by Hubble-scale inhomogeneity. In these models, density perturbations grow on a radially inhomogeneous background. This large scale inhomogeneity distorts the spherical Baryon Acoustic Oscillation feature into an ellipsoid which implies that the bump in the galaxy correlation function occurs at different scales in the radial and transverse correlation functions. We compute these for the first time, under the approximation that curvature gradients do not couple the scalar modes to vector and tensor modes. The radial and transverse correlation functions are very different from those of the concordance model, even when the models have the same average BAO scale. This implies that if void models are fine-tuned to satisfy average BAO data, there is enough extra information in the correlation functions to distinguish a void model from the concordance model. We expect these new features to remain when the full perturbation equations are solved, which means that the radial and transverse galaxy correlation functions can be used as a powerful test of the Copernican Principle.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The homogeneous and isotropic Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) universe is based on the assumptions that the Copernican Principle holds true and that the universe is isotropic about our location. The high degree of isotropy of the Cosmic Microwave Background (CMB) provides strong support for the second assumption. The Copernican assumption cannot be directly tested, although null tests of this assumption have been devised (for reviews, see [1, 2]). A spatially flat FLRW universe containing 20% cold dark matter (CDM), 5% baryons and 75% dark energy in the form of the cosmological constant (Λ) provides an excellent fit to the wide range of observations to date. However, there is still no satisfactory theoretical explanation from fundamental physics for the observed value of Λ. This has prompted many authors to consider alternatives to ΛCDM, such as modified gravity [3], backreaction [4], and spherically symmetric but inhomogeneous exact solutions to Einstein's equations known as Lemaˆıtre-Tolman-Bondi (LTB) models (see [2, 5] for recent reviews). In this work we focus on the latter alternative. Hubble-sized LTB void models break the Copernican Principle by placing our galaxy in a special position, at the centre of an underdense region of O (Gpc) scale. The simplest of these models are able to fit the distance-redshift data from type Ia supernovae (SNIa) and the CMB, though this gives considerable tension with the locally measured value of H 0 (see e.g., [6]). Attempts to overcome this require the relaxation of assumptions on the homogeneity of the early universe in the form of isocurvature modes [7] or a change to the primordial power spectrum [8]. It remains an open issue whether they can fit the Baryon Acoustic Oscillation (BAO) data, and the more general data on the growth and distribution of large scale structure. Previous papers [6, 9-12] have computed the BAO scales in a geometric approximation, using the anisotropic expansion rates of the background model but ignoring any effects from the anisotropic growth of structure in void models. This is not surprising, since structure formation on an LTB background has not yet been calculated, because it is much more complicated than in the standard model [13-18]. Here we calculate for the first time the 2-point correlation function on an LTB background, and use it to extract the radial and transverse BAO scales. This incorporates the effects of the evolution of density perturbations on an LTB background, using the perturbation formalism developed in [18]. We neglect the coupling of scalar to vector and tensor modes in the metric potentials. This is expected to be a good approximation for the simplest LTB models, in which the background shear -responsible for the coupling of the modes in the first place -is typically of the order of a few percent [11]. (The accuracy of this approximation is under investigation via numerical solutions [19].) LTB models have enough freedom to always fit the average BAO scale. In these models the proper radius of the sound horizon at the drag epoch is approximately given by [2, 7] where N eff = 3 . 04, f b = Ω b / Ω m is the local baryon fraction, η = 10 -10 η 10 is the baryon-photon ratio at that time, T d is the temperature at the drag epoch and it is assumed that during the process of recombination, the scale of the void inhomogeneity is much larger than the horizon size at that time ( ∼ 100 Mpc). In general, f b and η 10 have radial degrees of freedom in them, and are no longer measured by the CMB at the radial scales of interest for the BAO. Consequently, d s is not constant spatially, and can vary over the scale of the model. Therefore, measuring the mean BAO scale in some shell around us cannot place constraints on late-time inhomogeneity without some other measurement of f b and η in the same shell at early times - which lie inside our past lightcone. In fact, the models can even be fine-tuned to have the same radial and angular BAO scales, by altering the bang time function appropriately. As we shall show, however, the radial and transverse 2-point correlation functions contain much more information than the peak positions which determine the BAO scales. In real space, the radial and transverse correlation functions are typically very different from each other. To make them close to those of the ΛCDM model with the same BAO scale would require high levels of fine-tuning in either d s or the primordial power spectrum. Effectively, the radial and transverse 2-point correlation functions can thus be used as an important probe of the Copernican assumption. This paper is organized as follows. In Section II we recap the standard geometric approximation for computing BAO scales and describe our method for determining the background dynamics of the LTB spacetime. In Section III we provide an overview of perturbation theory in LTB via the 2+2 decomposition approach. This is followed by a derivation of the anisotropic two-point correlation function for the gauge-invariant matter density perturbation in Section IV, where we present the computation of the correlations and the BAO scales. Finally, in Section V we discuss the consequences of our results.", "pages": [ 1, 2 ] }, { "title": "II. BACKGROUND MODEL AND EVOLUTION OF THE BAO SCALES", "content": "The background void model is described by the LTB metric, where d A is the angular diameter distance, and a prime indicates ∂/∂r . In the FLRW limit, a ‖ ( t, r ) = a ⊥ ( t, r ) = a ( t ) and κ ( r ) = K . The expansion rates transverse to and along the line-of-sight are The past lightcone of the central observer has null geodesics that are given by We use the notation F ( z ) ≡ F ( t ( z ) , r ( z )) to denote evaluation on the past lightcone. The anisotropic expansion rates (3) act on the acoustic sphere of proper radius L ∗ at an initial high redshift z ∗ , so that by redshift z it has evolved into an axisymmetric ellipsoid with semi-axes [10, 20] However, this geometric approximation does not give the correct BAO feature in the galaxy distribution - because it neglects the anisotropic effects of perturbations in LTB and their impact on the correlation function. Previous work [6, 8, 10-12, 21] on comparing the BAO scales in LTB with observations has all neglected the effects of LTB perturbations. Below we fill this gap by computing the correlation functions associated with the density perturbation and then extracting the BAO scales from the correlation functions. The observable quantities of the BAO feature are its redshift extent δz ( z ) and angular size δθ ( z ). These are converted to the physical radial and transverse length scales via for small δz and δθ . Note that we neglect redshift space distortions. The quantity [6] encodes an average of the two observable scales of the sound ellipsoid. In an FLRW model it reduces to where L ∗ (1 + z ∗ ) is the comoving sound horizon and D V is the standard volume-averaged BAO scale [22]. The LTB analogue of the Friedmann equation is The observed Hubble constant is H in 0 ≡ H ⊥ 0 (0) = 100 h km/s/Mpc, where 'in' indicates evaluation at the centre. For open LTB models, the parametric solution is where we choose a simultaneous big bang (uniform bang time function). Setting t = t 0 in (10) and (11) gives Thus H ⊥ 0 ( r ) is determined when Ω m ( r ) and t 0 are chosen. Then (11) determines u ( t, r ) and a ⊥ ( t, r ) follows from (10). For the purposes of this study, we choose a simple Gaussian void profile for the dimensionless density parameter, where 'out' refers to the asymptotic Einstein-de Sitter region, and σ characterizes the size of the void (see [24] for details). The physical matter density is then", "pages": [ 2, 3 ] }, { "title": "III. SCALAR PERTURBATIONS ON AN LTB BACKGROUND", "content": "The full perturbation theory on an LTB background is developed in [18] via a 2+2 split of the spacetime, which makes explicit the coupling of vector and tensor modes to scalar modes at linear order. A first approximation is to neglect this mode-mixing, and focus only on 'scalar' modes which occur in the even parity sector. Then the perturbed metric in Regge-Wheeler gauge is ([18], with notational change, ϕ →-2Φ) where ¯ g ij is the spatial part of (2). The Newtonian potential obeys a simple generalization, without gradients, of the FLRW evolution equation for the Newtonian potential [18]: 0 Because there are no spatial gradients, Φ evolves independently in each r =const shell, as if in a separate dust FLRW model. This does not mean that there is no dependence on spatial gradients: density fluctuations depend on spatial gradients of Φ which couple to the anisotropic expansion of the model. The gauge-invariant matter density perturbation ∆ is found via the equivalent of the Poisson equation in LTB [18]: (Recall that the LTB model contains only CDM and baryons.) In FLRW, we recover the standard Poisson equation: Here /lscript is the angular wave number in a spherical harmonic expansion, and similarly for ∆. We set initial conditions for Φ at a high redshift, z ∗ = 100, where we assume the background is effectively FLRW. We write The subsequent time evolution of φ ( t, r ) is then determined by (16) for each r . Using (10) and (11), this implies Now, Φ ∗ /lscriptm ( r ) can be written as which is related to the power spectrum via The initial power spectrum of the Newtonian potential is given by where P R ( k 0 ) = 2 . 41 × 10 -9 is the amplitude of the primordial curvature perturbation on the scale k 0 = 0 . 002 Mpc -1 , and T ( k ) is the matter transfer function, with T ( k 0 ) ≈ 1. The concordance parameters (49) are used in the fitting formula of [25] to compute T ( k ), which is employed in all of the models. The result is where ∂ 2 r j /lscript ( kf ) = Note that when using a flat FLRW initial power spectrum, we need to use the flat FLRW comoving coordinate r F in (25) at t ∗ , as opposed to the LTB coordinate r . Proper radial distance is independent of coordinates: d p ( t ∗ , r F ) = d p ( t ∗ , r ). Since d p ( t ∗ , r F ) = a ( t ∗ ) r F , we find that where f ( r ) ≈ (1 + z ∗ ) a ⊥ ( t ∗ , r ) r since √ 1 -κ ( r ) r 2 ≈ 1 for all r and a ‖ = ∂ r ( a ⊥ r ). Then (25) becomes", "pages": [ 3, 4, 5, 6 ] }, { "title": "IV. CORRELATION FUNCTIONS AND THE BAO SCALES", "content": "The two-point correlation function (2PCF) for the density perturbation ∆, as observed by a central observer down their past lightcone, is The second line follows from statistical isotropy, which applies for central observers. We neglect redshift space distortions for simplicity, since we are not testing the void models against data but only comparing them with the concordance model. We also neglect all complications from bias for the same reason. Using the Poisson equation (17), the correlation function (30) becomes where a subscript i = 1 , 2 on a function of ( t, r ) means the quantity is evaluated at ( t ( z i ) , r ( z i )). Using (26) and standard identities in (31), we get where J /lscript ( z, k ) = L [ φ ( t ( z ) , r ( z )) j /lscript ( kf ( z )) ] (33) To evaluate (33), we use (18) and the following identities for the spherical Bessel function [ /lscript f '' f + /lscript ( /lscript - 1) f ' 2 f 2 - k f ] j /lscript - ( f - 2 f ' 2 f ) kj /lscript +1 . (35) 2 ' 2 '' and In the flat FLRW case, (36) becomes The usual flat FLRW correlation function may then be obtained using the following identity: where s ≡ s ( z 1 , z 2 , δθ ) = √ r 2 1 + r 2 2 -2 r 1 r 2 cos δθ . In LTB, the real-space radial and transverse BAO scales are different, and are given by the peaks in the radial and transverse correlation functions. These we define as: where the radial and transverse coefficients follow from (32): Equations (32), (36) and (45)-(48) summarize our new results that derive the correlation function of matter density perturbations on a radially inhomogeneous background.", "pages": [ 6, 7 ] }, { "title": "A. Computation of the correlation functions and extraction of the BAO scales", "content": "We can now compute the correlation functions for two specific LTB models, and compare with the standard case (see e.g. [26-31] for various approaches to compute these quantities from galaxy surveys in the standard homogeneous framework.) We consider the following models: This is our benchmark model which we use to compare with void models of the type given by (13). ) θ δ , 1 (z ⊥ ξ 4 3 2 1 0 -1 -2 -3 x 10 Transverse 2PCF: z δθ -3 20 40 60 80 (degrees) In the two void models, we choose FLRW initial conditions to ensure that the effects we find arise from the evolution of structure on the inhomogeneous background. We take the early-time parameters f b and η in (1) to be those derived from the same WMAP 5-year values used for the fΛCDM model. This fixes the initial proper BAO scale to be the same in all models. The background density Ω m and expansion rates H ‖ , H ⊥ are shown for these 3 models in Fig. 1 (upper panels). We also show (lower panels) the geometric approximations to the radial and transverse scales, L geo ‖ and L geo ⊥ , and the average BAO scale d z calculated from them. =0.05 1 Figure 2 shows the current profile and the redshift evolution of the gravitational potential for the 3 models. Note the greater decay in the amplitude of φ for the void models, due to the presence of curvature, which explains the decrease in the overall amount of clustering relative to ΛCDM. The normalized density perturbation is illustrated in Fig. 3 for the 3 models. For small-scale modes (large /lscript ), ∆ scales approximately as (1 + z ) -1 . For the large-scale mode /lscript = 2, the 'decaying' behaviour at high redshift is due to the mode entering the Hubble-scale at low redshift. We calculate the correlation functions by smoothing away power on scales below 1 Mpc, via P Φ ( k ) → P Φ ( k ) exp[ -k 2 / (1 Mpc -1 ) 2 ]. This makes the sums over /lscript in the correlation functions (45), (46) converge relatively quickly (typically we require /lscript max ( z ) /lessorsimilar 10 r ( z ) / Mpc), but without altering the resulting correlation function. Figure 4 shows the angular power spectrum /lscript ( /lscript +1) C ⊥ /lscript for the BV void model compared to the concordance model. The drop in power for high /lscript results from the smoothing. The correlation functions (45), (46) for the two void models are shown in Figs. 5 and 6. The radial correlation function ξ ‖ , starting at various redshifts z 1 and extending to z 2 = z 1 + δz , shows the correlation of structure along a line of sight, as the observer looks into higher density regions. The redshift extent of the radial BAO feature is δz peak , which is given by the location of the bump in ξ ‖ . The transverse correlation function ξ ⊥ describes the correlation across the sky in a sphere at redshift z 1 . The angular size of the BAO is δθ peak , given by the bump in ξ ⊥ . It is apparent from Fig. 5 that for the SV model, the radial correlation function is very different from the concordance one. This is due to the large curvature gradients at low redshift, compared to void models that fit SN1a data. We neglect redshift space distortions - but curiously, the effect of the void is qualitatively similar to the effect of redshift space distortions in FLRW (see [33]). We determined δz peak and δθ peak numerically from the local maxima in the correlation functions. The results are shown in Figs. 7 and 8. In these figures we also show the geometric approximations (i.e., without incorporating the effect of perturbations), where L geo ‖ , L geo ⊥ are given by (5). Our results show that the geometric formulas commonly used for constraining LTB with BAO fail at the percent level. While current data are not able to resolve such differences, this may be possible with future surveys such as SKA and Euclid. Furthermore, note that the size of these corrections are of a similar order to the corrections from redshift space distortions in FLRW [33]. Note that the geometric formulas in (52) give the correct observed scales for fΛCDM - except for large δθ , for which the small-angle formula in (52) breaks down.", "pages": [ 7, 8, 9, 10 ] }, { "title": "V. CONCLUSIONS", "content": "We have derived for the first time the anisotropic real-space two-point correlation function for the gauge-invariant matter density perturbation, in an LTB universe with radial inhomogeneity in the background - summarized in (32), (36) and (45)-(48). For this we neglected the coupling of scalar modes with vector and tensor modes - which should be a good approximation, at least on the large scales relevant for the BAO. An analysis of the effects of mode-coupling, which would entail the integration of partial differential equations, is currently underway [19]. We also neglected bias and redshift space distortions, since our primary focus was a comparison with the concordance model, not to test void models against data. Redshift space distortions in LTB void models deserve further investigation, in particular to check whether the FLRW formula provides a useful approximation. We computed the radial and angular correlation functions for two void models, one relatively small (SV) and one Hubble-sized void that fits the average BAO data (BV) - see Figs. 5 and 6. We used the peaks of the computed correlation functions to extract the radial and transverse BAO scales. The results were compared with the geometric approximation that has been used in all previous work, showing that the geometric approximation to the BAO scales in LTB fails at the percent level - see Figs. 7 and 8. Future large-volume surveys, such as SKA and Euclid, may thus be able to rule out the void models on the basis of their BAO scales. However, even if void models can be fine-tuned to reproduce the radial and transverse BAO scales, these scales represent only one feature in the galaxy correlation functions. The void correlation functions differ significantly from those of the concordance model (Figs. 5 and 6). In particular, the void radial correlation can become negative (anticorrelation) before, and even at, the BAO peak, where the concordance correlation is positive. The void transverse correlation may be positive for all scales, unlike the concordance one. These features resemble the effect of redshift space distortions in FLRW (see Figs. 4 and 6 in [33]), since the anisotropic expansion rate in LTB can mimic the effect of radial peculiar velocities in FLRW. However, there are significant further differences between the two models which arise from the effect of LTB perturbations. This leads to our key final result: even if the radial and transverse BAO scales match observations, the radial and transverse correlation functions contain direct signatures of the anisotropic growth of perturbations in a non-FLRW model. These correlation functions can thus be used as direct tests of the Copernican Principle.", "pages": [ 10, 11 ] }, { "title": "Acknowledgments", "content": "We thank Bruce Bassett, Marco Regis and Miguel Zumalac'arregui for useful discussions. SF and RM were funded by the South African Square Kilometre Array (SKA) Project. CC and RM were supported by the National Research Foundation (South Africa). RM was supported by the Science & Technology Facilities Council (UK) (grant no. ST/H002774/1). All authors were supported by a Royal Society (UK)/ NRF (SA) exchange grant. Computations were performed using facilities provided by the University of Cape Town's ICTS High Performance Computing team.", "pages": [ 11 ] } ]

[ { "title": "Modulated curvaton decay", "content": "Hooshyar Assadullahi 1 , 2 , ∗ Hassan Firouzjahi 3 , † Mohammad Hossein Namjoo 4 , 5 , ‡ and David Wands 1 § 1 Institute of Cosmology and Gravitation, University of Portsmouth, Dennis Sciama Building, Burnaby Road, Portsmouth PO1 3FX, United Kingdom 2 School of Earth and Environmental Sciences, University of Portsmouth, Burnaby Building, Burnaby Road, Portsmouth PO1 3QL, United Kingdom 3 School of Astronomy, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran 4 Yukawa Institute for theoretical Physics, Kyoto University, Kyoto 606-8502, Japan and 5 School of Physics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran (Dated: April 26, 2018)", "pages": [ 1 ] }, { "title": "Abstract", "content": "We study primordial density perturbations generated by the late decay of a curvaton field whose decay rate may be modulated by the local value of another isocurvature field, analogous to models of modulated reheating at the end of inflation. We calculate the primordial density perturbation and its local-type non-Gaussianity using the sudden-decay approximation for the curvaton field, recovering standard curvaton and modulated reheating results as limiting cases. We verify the Suyama-Yamaguchi inequality between bispectrum and trispectrum parameters for the primordial density field generated by multiple field fluctuations, and find conditions for the bound to be saturated.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Inflation in the very early universe provides a classical cosmology that drives the universe towards spatial flatness and homogeneity. It also provides an origin for primordial density perturbations through the quantum fluctuations of light scalar fields, stretched by the inflationary expansion to super-Hubble scales. Originally structure was assumed to originate from fluctuations in the inflaton field driving inflation, but more recently it has been realised that there are many possible mechanisms through which scalar field fluctuations could generate the observed primordial density perturbations [1]. Examples incude the late decay (some time after inflation has ended) of a curvaton field [2-5], modulated reheating at the end of inflation, where the rate of inflaton decay is modulated by the local vacuum expectation value (VEV) of an isocurvature field [6, 7], or an inhomogeneous end of inflation [8, 9]. All of these mechanisms for the origin of the primordial density perturbations may be distinguished from adiabatic perturbations in the inflaton field driving inflation as they introduce local-type primordial non-Gaussianity [10], most simply characterised by the non-linearity parameter, f NL [11], which typically becomes large when the transfer efficiency becomes small [12]. In this paper we go beyond the simplest curvaton models to include the possible modulation of the curvaton decay by a modulator field, χ , in analogy with modulated reheating at the end of inflation. In modulated reheating scenarios the rate of decay of a massive field σ is assumed to be dependent on the VEV of a modulator field, χ . Consider a Lagrangian which describes the decay of the σ -field oscillations into ψ -particles If the field, σ , is displaced from its minimum during inflation, then the field oscillates once H < m σ where m 2 σ = V '' ( σ ) at its minimum. Note that the potential, V ( σ ), may deviate from a simple quadratic potential, but we will assume it can be described by a simple quadratic potential once it begins to oscillate about its minimum, with effective mass m σ . If the effective mass of the χ field remains small while the σ field oscillates, | U '' ( χ ) | < H 2 < m 2 σ , then the decay rate of massive σ -particles into light ψ -particles is given by Γ ∝ λ 2 which is assumed to be a function of the local VEV of χ . For example, if the coupling λ is a linear function of χ then the decay rate is a quadratic function Γ ∝ χ 2 . Any light field during inflation acquires an almost scale-invariant distribution of perturbations due to small-scale vacuum fluctuations being stretched up and frozen-in on superHubble scales. In particular, we will assume that light fields acquire a Gaussian distribution of perturbations at Hubble-exit ( k = ( aH ) ∗ ) with power spectrum Local variations in the χ field change the local rate of reheating and hence the primordial density perturbations, which can be represented by ζ , the metric perturbation on uniformdensity hypersurfaces [13] in the primordial radiation-dominated era, some time after inflation has ended. In the original modulated reheating scenario σ is the inflaton whose oscillations dominate the energy density of the universe immediately after inflation comes to an end. In this work we will consider the case where σ is a curvaton field whose energy density is sub-dominant during inflation. We assume its effective mass is small compared to the Hubble scale during inflation and hence it also acquires an almost scale-invariant spectrum of fluctuations. The curvaton mass becomes larger than the Hubble scale either immediately at the end of inflation or some time after, when H = m σ and the curvaton field eventually decays. Fluctuations in the local VEV of the curvaton during inflation lead to perturbations in the amplitude of oscillations and the density of curvaton particles when the field oscillates after inflation. These density perturbations are transferred to the radiation when the curvaton decays. If both the modulator field and the curvaton field are light during inflation then their vacuum fluctuations can generate a primordial density perturbation, in addition to any adiabatic density perturbations produced from adiabatic fluctuations in the inflaton field, φ , during inflation. The outline of this paper is as follows. In section II we describe how we can estimate the primordial density perturbations produced by the modulated curvaton decay in terms of the inhomogeneous densities and metric on an instantaneous decay hypersurface, generalising previous results for non-linear perturbations from curvaton decay with an unmodulated decay rate [14]. We calculate observable quantities including the primordial power spectrum, tensor-scalar ratio, bispectrum and trispectrum. We show that the Suyama-Yamaguchi inequality between the tree-level bispectrum and trispectrum holds and consider the condition for this inequality to be saturated. We present our conclusions and discussion in section III.", "pages": [ 2, 3, 4 ] }, { "title": "II. SUDDEN-DECAY APPROXIMATION", "content": "We will work in the sudden-decay approximation where the curvaton decay is modelled as an instantaneous transfer of energy from the curvaton field oscillations, ρ σ , into radiation, ρ γ . This has been shown to be a good approximation to the full numerical results in the usual curvaton scenario [14]. If the curvaton decay rate is a constant, Γ = ¯ Γ, then the decay hypersurface, H dec = Γ, corresponds to a uniform-density hypersurface (on super-Hubble scales at the decay time) with In the presence of fluctuations in the local VEV of χ , the modulator field, and therefore local fluctuations of the decay rate, we have We will allow for the existence of inhomogeneous perturbations of the radiation density, ζ γ , after inflation but before the curvaton decays, due to adiabatic inflaton field fluctuations during inflation, ζ γ = ζ φ = -Hδφ/ ˙ φ . Before the curvaton decay, the curvature perturbation on uniform-radiation-density hypersurfaces, ζ γ , and uniform-curvaton-density hypersurfaces, ζ σ , are independently conserved [15]. After the decay, assuming the decay products are relativistic, the curvature perturbation on uniform-total-density hypersurfaces, ζ , is conserved. On the decay hypersurface itself, we therefore have [14, 16] where ζ dec is the curvature perturbation ( δN from a flat hypersurface) on the decay hypersurface. Note that we neglect the energy density of the modulator field χ throughout. Matching ρ γ + ρ σ = ρ at the decay time yields where", "pages": [ 4 ] }, { "title": "A. Linear perturbations", "content": "At linear order, one can expand the above relation (6) to give Using the above relation to eliminate ζ dec in (5) and also using (4), results in where we have defined the transfer parameter The change in the curvature perturbation, ζ -ζ γ in Eq. (9), due to the modulated curvaton decay is seen to arise from the relative entropy perturbation between the curvaton density and the radiation density, ζ σ -ζ γ , and the perturbed decay rate, δ Γ /similarequal Γ ' δχ . In the case of homogeneous curvaton decay rate, δ Γ = 0, we reproduce the standard curvaton result. The inhomogeneous decay rate adds an extra term in the primordial curvature perturbation (9) proportional to f . Note that since χ remains overdamped throughout the decay, we assume that we can neglect its background evolution and hence its perturbation on the decay hypersurface correspond directly to its perturbation on spatially flat hypersurfaces during inflation. The curvaton, σ , is overdamped during inflation, but starts to oscillate when H = m σ . We assume the curvaton density is still negligible at this time, so the curvaton field fluctuations on this surface correspond to curvaton density perturbations on a uniform radiation-density hypersurface Allowing for possible evolution of the curvaton field from the end of inflation up until the point at which it starts oscillating we write At linear order we thus have where ¯ σ + δσ describes the local curvaton VEV at the end of inflation.", "pages": [ 5 ] }, { "title": "B. Non-linear perturbations", "content": "Expanding (4) and (5) to second order yields Eliminating ζ dec in (6) and solving for ζ order by order yields where we define the entropy isocurvature perturbation", "pages": [ 6 ] }, { "title": "C. Observables", "content": "When calculating observables, such as the power spectrum and higher-order correlators of the primordial density perturbations, it will be convenient to express the non-linear perturbation ζ in terms of the perturbed logarithmic expansion, N = ∫ Hdt , from an initial spatially flat hypersurface where the scalar field perturbations originate during inflation and a final uniform-density hypersurface during the subsequent radiation-dominated era [16, 17] we note that the total curvature perturbation can be rewritten by where δφ a are the three Gaussian fields δφ , δχ and δσ , N a = N a ≡ ∂N/∂φ a and N ab = N ab ≡ ∂ 2 N/∂φ a φ b . Comparing Eq. (17) with (15) we identify in which Note that higher-derivatives with respect to the inflaton field φ can be neglected during slow-roll inflation. From Eqs.(18-19) we identify This will allow us to calculate all higher derivatives of N starting from Eqs. (18-19). One can verify this is consistent with Eqs. (20-21). The power spectrum is given, at leading order, by in which, in analogy with the inflaton contribution, we have defined [18] The relative contribution of each field to the power spectrum (25) is given by the weights w a defined via P ζ a = w a P ζ in which Note that w φ + w σ + w χ = 1. The spectral index is then The tensor to scalar ratio is also given by The local-type primordial bispectrum is characterised at leading order in term of f NL [17] which we can write as where we write the different contributions as where we have used Eqs. (18-21). In the limit w σ → 1 then f NL → f σ NL and we recover the standard result for the curvaton [14, 17, 19]. In the opposite limit where w χ → 1 and f → 1 we recover the standard result for modulated reheating [20] The primordial trispectrum is composed of distinct two terms [21] which in our case we can write as where we identify the different contributions to g NL as where we have used Eqs. (20-24). In the curvaton limit w σ → 1 then g NL → g σ NL and we recover the standard result for the curvaton [14]. In the opposite limit, w χ → 1, we recover the result for modulated reheating when f → 1 [10, 22]", "pages": [ 6, 7, 8, 9, 10 ] }, { "title": "D. The Suyama-Yamaguchi inequality", "content": "Here we verify that the Suyama-Yamaguchi (SY) inequality [22], stating that τ NL ≥ ( 6 5 f NL ) 2 , holds at tree-level in our model 1 . A direct analysis shows that in which /negationslash We wish to determine the sign of K in Eq. (45) which is a quadratic function of f χ NL for c 1 = 0. Hence we re-write Eq. (45) as where Both the coefficient, c 1 in Eq. (46), and the discriminator, ∆ in Eq. (50) are non-negative, since the weights, w σ and w χ , and their sum, w σ + w χ , are bounded between zero and one. Hence K given in Eq. (49) is non-negative and we conclude that τ NL ≥ ( 6 5 f NL ) 2 as required. One may ask under what conditions the SY inequality is saturated. Firstly this can occur if c 1 = 0 and c 2 f χ NL + c 3 = 0, which requires either w σ = 1 or w χ = 1, i.e., either the curvaton fluctuations or the modulator fluctuations dominate the primordial density perturbations corresponding to effectively a single source for the primordial density field, or w φ = 1 corresponding to the Gaussian case, τ NL = f NL = 0 /negationslash For c 1 = 0, the SY equality is saturated when ∆ = 0 and 2 c 1 f χ NL + c 2 = 0 in Eq. (49). This either requires which implies τ NL = f NL = 0, or requires w σ + w χ = 1 and or, equivalently, Note that this is possible only when f σ NL and f χ NL are either both less than f σχ NL or both greater than f σχ NL , i.e.,", "pages": [ 10, 11 ] }, { "title": "III. DISCUSSION", "content": "The curvaton and modulated reheating scenarios have previously been studied as distinct models for the origin of structure from quantum field fluctuations during inflation. Here we have considered a curvaton scenario where the curvaton decay rate may be modulated by a second scalar field. Thus fluctuations in two independent fields may be responsible for both the primordial power spectrum and primordial non-Gaussianity described by higher-order correlations. The relative contributions to the primordial power spectrum (25) are determined by the weights, w σ and w χ defined in (27). The overall contribution of both curvaton and modulator fluctuations to the first-order primordial density perturbation is proportional to the fractional density in the curvaton at the time of decay, f defined in (10). The relative contribution depends on the fractional perturbations in the curvaton amplitude of oscillation, g , versus the decay rate, Γ: We recover (i) previous results for the curvaton scenario in the limit that curvaton field fluctuations dominate the primordial power spectrum ( w σ /greatermuch w χ ) and (ii) previous results for modulated reheating in the limit that the curvaton dominates the total energy when it decays ( f → 1) and modulated fluctuations dominate ( w χ /greatermuch w σ ) . We also allow for an adiabatic density perturbation produced due to inflaton perturbations during inflation, ζ φ . This is required to be Gaussian, hence the inflaton contributes to the first-order density perturbation, but not to higher-order correlators. Any detection of local-type primordial non-Gaussianity would be evidence of non-adiabatic field fluctuations playing a role in the origin of large-scale structure, see however [25]. Non-Gaussianity can become very large, either for f /lessmuch 1 or due to non-linear evolution, g '' g /greatermuch g ' 2 , in the curvaton scenario. Indeed it is bounded by current observations [26] Such a large value is difficult to achieve in standard modulated reheating unless the decay rate is strongly dependent upon the modulator field, Γ '' Γ / Γ ' 2 ∼ 8. On the other hand one can easily have large non-Gaussianity in a modulated curvaton reheating when f /lessmuch 1 even when the modulator dominates the primordial fluctuations, w χ → 1. In the simplest scenario of a quadratic curvaton potential, leading to linear evolution of the curvaton field, g '' = 0, and a linear coupling, λ ( χ ) in Eq. (1), leading to a quadratic decay rate, Γ ∝ χ 2 , the primordial non-Gaussianity is a function of f , w σ and w χ . Hence a measurement of the three lowest-order non-Gaussian correlators, f NL , τ NL and g NL , would be required to determine f , w σ and w χ . There are some general predictions for the simplest model of a quadratic curvaton potential with g '' = 0. The contributions to the primordial bispectrum, given in Eq. (31), from f χ NL and f σχ NL , defined in Eqs. (34) and (33), are non-negative for any Γ '' Γ ≤ Γ ' 2 , which includes a quadratic decay rate. Thus in the modulated curvaton decay when g '' = 0, we find a lower bound f NL ≥ f σ NL ≥ -5 / 4, generalising the result found previously for a single curvaton [14], and multiple curvaton decays [27]. This bound is saturated, f NL = -5 / 4, for w σ = 1 and f = 1. /negationslash For this simple quadratic curvaton scenario with linear evolution, such that g '' and g ''' can be neglected, we find the third-order trispectrum parameter g σ NL ∝ f -1 ∝ f NL whereas the second-order trispectrum parameter τ NL ∝ f -2 ∝ f 2 NL . Hence τ NL is much larger than g σ NL if f NL is large. On the other hand g σσχ NL ∝ f -2 ∝ f 2 NL even for g '' /similarequal 0, and hence g NL /τ NL is not necessarily suppressed for a simple quadratic curvaton in a modulated curvaton scenario with δ Γ = 0. We generally expect g NL ∝ τ NL ∝ f 2 NL due to non-linear evolution of the curvaton field with a self-interaction potential [28-32]. On the other hand the self-interacting curvaton can give rise to strongly scale-dependent non-Gaussianity [33], while the modulated curvaton decay with a quadratic curvaton potential gives rise to non-Gaussianity which is scaleindependent. We have verified that the trispectrum parameter τ NL obeys the Suyama-Yamaguchi inequality [22] τ NL ≥ (36 / 25) f 2 NL . It is saturated when the curvaton perturbations ( w σ → 1) or modulator perturbations ( w χ → 1) dominate, or in the trivial case of Gaussian perturbations when w φ → 1. It can also be satisfied for particular parameter values even when both curvaton and modulator fluctuations contribute to the primordial density field, w σ + w χ = 1 given by Eq. (53). This emphasizes that a single-source for the primordial density field is a sufficient condition for the SY inequality to be saturated, but not a necessary condition [22]. Throughout this work we have assumed that the decay products of the curvaton rapidly thermalise leaving no residual isocurvature perturbations. If the matter asymmetry inherits a different density perturbation from the overall radiation density (because it comes exclusively from either the curvaton decay products or the pre-existing radiation before the curvaton decay, but not both) then it may leave a residual matter isocurvature perturbation [34-36]. This would be further evidence of the origin of structure from non-adiabatic field perturbations during inflation, and measurements of the relative amplitude of residual isocurvature perturbations and their correlations with the adiabatic density perturbation could give independent constraints on model parameters. It would be interesting to investigate whether these could then distinguish modulated curvaton decay from standard curvaton or modulated reheating scenarios. Although we have assumed that the modulator field has negligible energy density, it would be interesting to consider possible observational signatures of the eventual decay of the modulator field, assuming that it too eventually decays into standard model particles [37], analogous to our recent study of the effect of the late-time decay of a field responsible for the inhomogeneous end of hybrid inflation [38]. We leave this for future work. Note added: While completing this work, we became aware of related work by Langlois and Takahashi [39]. Both papers should appear on the arXiv on the same day.", "pages": [ 11, 12, 13, 14 ] }, { "title": "Acknowledgments", "content": "HF would like to thank the ICG, Portsmouth, for hospitality while this work was initiated and later finalized. HF, MHN and DW are grateful to the organisers of the YITP long-term workshop on Gravitation and Cosmology 2012, YITP-T-12-03. MHN is in part supported by Yukawa Institute for Theoretical Physics (YITP), Kyoto University, under the Exchange Program for Young Researchers of YITP. DW is supported by STFC grant ST/H002774/1.", "pages": [ 14 ] } ]

[ { "title": "J.J. Gómez-Cadenas, J. Martín-Albo, J. Muñoz Vidal and C. Peña-Garay", "content": "Instituto de Física Corpuscular (IFIC), CSIC & Universitat de Valencia Calle Catedrático José Beltrán, 2, 46090 Paterna, Valencia, Spain E-mail: [email protected], [email protected], [email protected], [email protected] Abstract. The South Pole Telescope (SPT) has probed an expanded angular range of the CMB temperature power spectrum. Their recent analysis of the latest cosmological data prefers nonzero neutrino masses, ∑ m ν = 0 . 32 ± 0 . 11 eV. This result, if confirmed by the upcoming Planck data, has deep implications on the discovery of the nature of neutrinos. In particular, the values of the effective neutrino mass m ββ involved in neutrinoless double beta decay ( ββ 0 ν ) are severely constrained for both the direct and inverse hierarchy, making a discovery much more likely. In this paper, we focus in xenon-based ββ 0 ν experiments, on the double grounds of their good performance and the suitability of the technology to large-mass scaling. We show that the current generation, with effective masses in the range of 100 kg and conceivable exposures in the range of 500 kg · year, could already have a sizable opportunity to observe ββ 0 ν events, and their combined discovery potential is quite large. The next generation, with an exposure in the range of 10 ton · year, would have a much more enhanced sensitivity, in particular due to the very low specific background that all the xenon technologies (liquid xenon, high-pressure xenon and xenon dissolved in liquid scintillator) can achieve. In addition, a high-pressure xenon gas TPC also features superb energy resolution. We show that such detector can fully explore the range of allowed effective Majorana masses, thus making a discovery very likely.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Neutrinos, unlike the other Standard Model fermions, could be Majorana particles, that is, indistinguishable from their antiparticles. The existence of Majorana neutrinos would have profound implications in particle physics and cosmology. If neutrinos are Majorana particles, there must exist a new scale of physics, the level of which is inversely proportional to neutrino masses, that characterises new underlying dynamics beyond the Standard Model. The existence of such a new scale provides the simplest explanation of why neutrino masses are so much lighter than the charged fermions. Understanding the new physics that underlies neutrino masses is one of the most important open questions in particle physics. It could have profound implications in our understanding of the mechanism of symmetry breaking, the origin of mass and the flavour problem [1]. Furthermore, the existence of Majorana neutrinos would imply that lepton number is not a conserved quantum number which could be the origin of the matter-antimatter asymmetry observed in the Universe. The new physics related to neutrino masses could provide a new mechanism to generate the asymmetry, called leptogenesis. Although the predictions are model dependent, two essential ingredients must be confirmed experimentally: 1) the violation of lepton number and 2) CP violation in the lepton sector. The only practical way to establish experimentally that neutrinos are their own antiparticle is the detection of neutrinoless double beta decay ( ββ 0 ν ). This is a postulated very slow radioactive process in which a nucleus with Z protons decays into a nucleus with Z +2 protons and the same mass number A , emitting two electrons that carry essentially all the energy released ( Q ββ ). The process can occur if and only if neutrinos are massive, Majorana particles. Several underlying mechanisms - involving, in general, physics beyond the Standard Model - have been proposed for ββ 0 ν , the simplest one being the virtual exchange of light Majorana neutrinos. Assuming this to be the dominant process at low energies, i.e., there are only three light neutrino mass eigenstates, the half-life of ββ 0 ν can be written as In this equation, G 0 ν is an exactly-calculable phase-space integral for the emission of two electrons; M 0 ν is the nuclear matrix element (NME) of the transition, which has to be evaluated theoretically; and m ββ is the effective Majorana mass of the electron neutrino: where m i are the neutrino mass eigenstates and U ei are elements of the neutrino mixing matrix. The matrix elements U ei have been stablished by neutrino oscillation experiments, which also measure the mass differences δm 2 = m 2 2 -m 2 1 and ∆ m 2 = m 2 3 -m 2 2 . Instead, the two Majorana phases α 1 and α 2 are unknown. On the other hand, cosmological observations, probe the sum of the three neutrino masses: Combining equations (1.2) and (1.3) one can solve for the individual values of the masses, by imposing an additional constrain, namely ∆ m 2 > 0 (the so-called 'normal hierarchy') or ∆ m 2 < 0 (the so-called 'inverse hierarchy'). The analysis of recent cosmological observations, including the South Pole Telescope (SPT) observations, do indeed show the evidence of a finite value for the neutrino cosmological mass, ∑ m ν = 0 . 32 ± 0 . 11 [2]. This important result does not seem to be verified by the analysis including the new Atacama Cosmology Telescope (ACT) observations [3], which nevertheless correspond to a much smaller sky coverage than the SPT ones. The cosmological analysis including Planck [4] results, coming in a few months, will clarify the evidence of nonzero neutrino cosmological mass claimed by the SPT team. In this paper we explore how this result affects the discovery potential of ββ 0 ν experiments. In previous works [5, 6], we have made detailed comparisons between the discovery potential of all the major ββ 0 ν experiments in the field. In this paper we focus exclusively in xenon-based experiments. Our main argument for doing so is the fact that 136 Xe is, by far, the cheapest ββ 0 ν decaying isotope, so much so, that a ton of enriched xenon is already available in the world. In addition, the best current sensitivity to m ββ is obtained by two xenon experiments: EXO-200 [7], a liquid xenon TPC, and KamLAND-Zen [8], a large calorimeter where the xenon is dissolved in liquid scintillator. The combination of both results, all but excludes the long-standing claim of a positive observation [8-10]. Furthermore, recent results from the NEXT experiment [11-14], a high-pressure xenon gas TPC with electroluminescent readout [15, 16] show excellent resolution and a very low expected background rate, due to the availability of a topological signature (the observation of the two electrons emitted in the decay) which allows a powerful discrimination between signal and background. This work is organised as follows: we first summarise the measurement of the total neutrino mass inferred by the analysis of the latest cosmological data and then explore the implication of this result on the predictions of the neutrinoless double beta effective mass. Next, we discuss the current generation of neutrinoless double beta decay experiments and concentrate particularly on the xenon-based experiments, KamLAND-Zen, EXO and NEXT. Finally, we discuss the sensitivity of current and future ton-scale xenon experiments to the effective mass and conclude with the impact of the total neutrino mass measurement in the vicinity of 300 meV on the resolution of the crucial question in neutrino physics by xenon-based experiments: are neutrinos their own antiparticles?", "pages": [ 2, 3, 4 ] }, { "title": "2 Cosmological observations and the neutrino mass", "content": "Cosmological observations can test the sum of the neutrino masses ( ∑ m ν ), due to the impact of these on the rate of expansion and on the growth of perturbations. In fact, a tight upper bound of about 0.3 eV (95 % CL) is derived when the analysis includes cosmic microwave background (CMB) [17], baryonic acoustic oscillations (BAO) [18-20] and Hubble constant (H 0 ) [21, 22] data combined with either abundances of Sunyaev-Zeldovich selected galaxy clusters [23] or galaxy clustering data [24, 25]. Measurements of the damping tail of the CMB, through the effect of gravitational lensing, are sensitive to the low-redshift universe and break the geometric degeneracy present in the large-scale CMB data. The South Pole Telescope (SPT) has probed an expanded angular range of the CMB temperature power spectrum and confirms a trend for a decreasing power at high multipoles [26] relative to the expectation of the Λ CDM model determined by the CMB data at lower multipoles [27, 28]. This trend can be accommodated by a scale-dependent tilt that becomes increasingly red at higher multipoles. Cosmological data can not single out the extension of the model needed to accommodate the data. In particular, nonzero neutrino masses, smaller helium abundance than predicted by Big Bang nucleosynthesis, running of the scalar spectral index, extra relativistic species or nonzero early dark energy (and possible combinations of these) are extensions that could explain the combined set of data. Let us concentrate on the sensitivity of cosmological data to the total neutrino mass, which is known to be larger than ∼ 58 meV by neutrino oscillation data [29]. The combined analysis of CMB, BAO, H 0 data and Sunyaev-Zeldovich selected galaxy clusters abundances prefer nonzero neutrino masses, ∑ m ν = 0 . 32 ± 0 . 11 eV [2]. The significant improvement in the CMB data by SPT leads to a better determination of the spectral index, which is anticorrelated with the total mass of neutrinos. The addition of the other probes, particularly BAO and cluster abundances, further improve the constraints. Other cosmological parameters may decrease the significance (spatial curvature, running of the scalar spectral index), increase the significance (equation of state parameter of dark energy and effective number of neutrino species) or be insensitive (Helium abundance). In fact, the model with extra effective neutrinos (N eff ) and nonzero total neutrino mass is the best model of the CMB+BAO+H 0 data. The combined analysis of CMB, BAO, H 0 data and Sunyaev-Zeldovich selected galaxy clusters abundances prefer nonzero neutrino masses, ∑ m ν = 0 . 51 ± 0 . 15 eV and N eff = 3 . 86 ± 0 . 37 Recently, the ACT team has presented their analysis of cosmological observations including the small angular scale CMB observations by the Atacama telescope [3]. Their observations are consistent with a zero neutrino mass and correctly point to the fact that the nonzero mass determined by the SPT team requires a rather low amplitude of the linear power spectrum on the scale of 8 Mpc/h, σ 8 , which is in tension with their cluster and skewness measurements. Nevertheless, SPT has a lot more sky coverage. Certainly, the analysis of these recent observations needs more discussion and will be further tested by including observations of the upcoming cosmological Planck CMB data [4].", "pages": [ 4, 5 ] }, { "title": "3 Neutrinoless double beta rate derived by neutrino data", "content": "The determination of the total mass of neutrinos has an important impact on a crucial question in neutrino physics, i.e, whether the neutrino is a Dirac or a Majorana fermion. The question is resolved if neutrinoless double beta decay is observed. On the other hand, long lifetimes may not be accessible experimentally and the question would remain unsolved. We will show next, that the total neutrino mass derived by cosmological data [2] leads to an upper bound on the lifetime, which can be reached experimentally, in particular by (multi)ton xenon-based experiments. The neutrinoless double beta decay rate is proportional to the effective mass m ββ equation (eq:Tonu) , which is given by the sum of three terms which may have partial cancellation among them, equation (1.2). The total neutrino mass derived by cosmological observations with the mass squared differences measured by reactor neutrino experiments determine the masses of the free neutrinos m i . The moduli of the mixing matrix elements are well measured by solar and reactor neutrinos experiments, where | U e 1 | = cos θ 12 cos θ 13 , | U e 2 | = sin θ 12 cos θ 13 and | U e 3 | = sin θ 13 . The relative phases between the three terms are free unknown parameters. In the set of measured neutrino parameters, the total mass of neutrinos is the most uncertain. We have explored the predictions of m ββ derived by the measured neutrino oscillations parameters shown in the most up-to-date analysis [29]: sin 2 θ 12 = 0 . 302 ± 0 . 013 , sin 2 θ 13 = 0 . 023 ± 0 . 002 , δm 2 = 75 ± 2 meV 2 , ∆ m 2 = 2470 ± 70 meV 2 ( ∆ m 2 = -2427 +42 -65 meV 2 ) for the normal (inverted) neutrino mass hierarchy. Figure 1 shows the 1 σ allowed region for two degrees of freedom (dof) in the observables space [30], m ββ and ∑ m ν , both in meV. The regions are calculated with the assumption of gaussian uncorrelated errors in the neutrino parameters derived by the global analysis of neutrino oscillation and cosmological data, and varying the Majorana phases within their physical range. The two regions correspond to the normal and inverted mass hierarchy scenarios. We can see that the mass determined by cosmological observations, large compared to the mass splittings, leads to a quasi-degenerate spectrum. Normal and inverted hierarchy lead to similar neutrinoless double beta rate predictions. The effective mass m ββ is smaller than a maximum value close to a third of the sum of neutrino masses, as expected in a degenerate spectrum. More importantly, m ββ is also bound from below. The variation of the unknown Majorana phases can not completely cancel the effective mass, which is larger than ∼ 20 meV. The 1 σ allowed range of the m ββ parameter by present neutrino oscillation and cosmological data is i", "pages": [ 5, 6 ] }, { "title": "· 28 ≤ m ββ ≤ 145 for the inverted hierarchy.", "content": "We find that, irrespectively of the (quasi-degenerated) hierarchy, the 1 σ range of m ββ is [26 , 145] . Therefore, a confirmation of the results presented in [2] are very important to validate the 20-meV target sensitivity for neutrinoless double beta decay experiments, needed to identify the nature of neutrinos irrespectively of the mass hierarchy.", "pages": [ 6 ] }, { "title": "4 The current generation of ββ 0 ν experiments", "content": "The detectors used to search for ββ 0 ν are designed, in general, to measure the energy of the radiation emitted by a ββ 0 ν source. In a neutrinoless double beta decay, the sum of the kinetic energies of the two released electrons is always the same, and equal to the mass difference between the parent and the daughter nuclei: Q ββ ≡ M ( Z, A ) -M ( Z +2 , A ) . However, due to the finite energy resolution of any detector, ββ 0 ν events would be reconstructed within a given energy range centred around Q ββ and typically following a gaussian distribution. Other processes occurring in the detector can fall in that region of energies, thus becoming a background and compromising drastically the sensitivity of the experiment [6]. All double beta decay experiments have to deal with an intrinsic background, the standard two-neutrino double beta decay ( ββ 2 ν ), that can only be suppressed by means of good energy resolution. Backgrounds of cosmogenic origin force the underground operation of the detectors. Natural radioactivity emanating from the detector materials and surroundings can easily overwhelm the signal peak, and hence careful selection of radiopure materials is essential. Additional experimental signatures, such as event topological information, that allow the distinction of signal and background are a bonus to provide a robust result. Besides energy resolution and control of backgrounds, several other factors such as detection efficiency and scalability to large masses must be taken into consideration in the design of a double beta decay experiment. The simultaneous optimisation of all these parameters is most of the time conflicting, if not impossible, and consequently many different experimental techniques have been proposed. In order to compare them, a figure of merit, the experimental sensitivity to m ββ , is normally used [6]: where ε is the signal detection efficiency, M is the ββ isotope mass used in the experiment, t is the data-taking time, δE is the energy resolution and b is the background rate in the region of interest around Q ββ (expressed in counts per kg of ββ isotope, year and keV, henceforth abbreviated as ckky ). The status of the field has been the subject of several recent reviews [5, 31-33]. Among the on-going and planned experiments, many different experimental techniques are utilised, each with its pros and cons. The time-honored approach of emphasising energy resolution and detection efficiency is currently spearhead by germanium calorimeters like GERDA [34] and Majorana [35], as well as tellurium bolometers such as CUORE [36]. A different, and powerful approach, the main topic of this paper, proposes the use of xenon-based experiments. Two of them, EXO-200 [37] and KamLAND-Zen [38] are already operating, while NEXT [11] is in the initial stages of construction, and foresees to start taking data in 2015. Other experiments that will operate in the next few years are the SuperNEMO demonstrator [32], a tracker-calorimeter approach which provides a powerful topological signal (the observation of the two electrons emitted in a ββ decay) but is hard to extrapolate to larger masses (the demonstrator itself will have a mass of less than 10 kg of isotope), and SNO+ [5], a large liquid scintillator calorimeter (the same approach that KamLAND-Zen), in which natural Neodymium is dissolved in the scintillator. Neodymium is a very interesting isotope, whose large Q ββ suppresses many of the low-energy background than other experiments have to deal with, but the ββ 0 ν decaying isotope, 150 Nd is only 5% of the natural Neodymium, limiting the total mass that the experiment can deploy.", "pages": [ 6, 7 ] }, { "title": "5 Xenon experiments", "content": "Xenon is an almost-optimal element for ββ 0 ν searches, featuring many desirable properties, both as a source and as a detector. It has two naturally occurring-isotopes that can decay via the ββ process, 134 Xe ( Q ββ = 825 keV) and 136 Xe ( Q ββ = 2458 keV). The latter, having a higher Q value, is preferred since the decay rate is proportional to Q 5 ββ and the radioactive backgrounds are less abundant at higher energies. Moreover, the ββ 2 ν mode of 136 Xe is slow ( ∼ 2 . 3 × 10 21 years [39, 40]) and hence the experimental requirement for good energy resolution to suppress this particular background is less stringent than for other ββ sources. The process of isotopic enrichment in the isotope 136 Xe is relatively simple and cheap compared to that of other ββ isotopes. Xenon has no long-lived radioactive isotopes and is therefore intrinsically clean. As a detector, xenon is a noble gas, and therefore one can build a time projection chamber (TPC) with pure xenon as detection medium. Both a liquid xenon (LXe) TPC and a (high-pressure) gas (HPXe) TPC are suitable technologies, chosen by the EXO-200 and the NEXT experiment respectively. Being a noble gas, xenon can also be dissolved in liquid scintillator. This is the approach of the KamLAND-Zen experiment.", "pages": [ 7, 8 ] }, { "title": "5.1 KamLAND-Zen", "content": "The KamLAND-Zen experiment is a modification of the well-known KamLAND neutrino detector [41]. A transparent balloon, ∼ 3 m diameter, containing 13 tons of liquid scintillator loaded with 320 kg of xenon (enriched to 91% in 136 Xe ) is suspended at the centre of KamLAND. The scintillation light generated by events occurring in the detector is recorded by an array of photomultipliers surrounding it. From the detected light pattern, the position of the event vertex is reconstructed with a spatial resolution of about 15 cm / √ E (MeV) . The energy resolution is (6 . 6 ± 0 . 3)% / √ E (MeV) , that is, 9.9% FWHM at the Q value of 136 Xe . The signal detection efficiency is ∼ 0 . 42 due to the tight fiducial cut introduced to reject backgrounds originating in the balloon. The achieved background rate in the energy window between 2.2 MeV and 3.0 MeV is 10 -3 counts / (keV · kg · y) . KamLAND-Zen has searched for ββ 0 ν events with an exposure of 89.5 kg · year. They have published a limit on the half-life of ββ 0 ν of T 0 ν 1 / 2 ( 136 Xe) > 1 . 9 × 10 25 years [8].", "pages": [ 8 ] }, { "title": "5.2 EXO", "content": "The EXO-200 detector [37] is a symmetric LXe TPC deploying 110 kg of xenon (enriched to 80.6% in 136 Xe ). In EXO-200, ionisation charges created in the xenon by charged particles drift under the influence of an electric field towards the two ends of the chamber. There, the charge is collected by a pair of crossed wire planes which measure its amplitude and transverse coordinates. Each end of the chamber includes also an array of avalanche photodiodes (APDs) to detect the 178-nm scintillation light. The sides of the chamber are covered with teflon sheets that act as VUV reflectors, improving the light collection. The simultaneous measurement of both the ionisation charge and scintillation light of the event may in principle allow to reach a detector energy resolution as low as 3.3% FWHM at the 136 Xe Q-value, for a sufficiently intense drift electric field [42]. The EXO-200 detector achieves currently an energy resolution of 4% FWHM at Q ββ , and a background rate measured in the region of interest (ROI) of 1 . 5 × 10 -3 counts / (keV · kg · y) . The experiment has also searched for ββ 0 ν events. The total exposure used for the published result is 32.5 kg · year. They have published a limit on the half-life of ββ 0 ν of T 0 ν 1 / 2 ( 136 Xe) > 1 . 6 × 10 25 years [7]. The combination of the KamLAND-Zen and EXO results yields a limit T 0 ν 1 / 2 ( 136 Xe) > 3 . 4 × 10 25 years, which essentially excludes the long-standing claim of Klapdor-Kleingrothaus and collaborators [9] [8, 10].", "pages": [ 8 ] }, { "title": "5.3 NEXT", "content": "The NEXT experiment [11] will search for the neutrinoless double beta decay of 136 Xe using an asymmetric high pressure gas xenon (HPXe) TPC, filled with 100-150 kg of xenon (enriched to 91% in 136 Xe ) gas at 15-20 bar pressure. NEXT offers two major advantages for the search of neutrinoless double beta decay, namely: a) excellent energy resolution , with an intrinsic limit of about 0.3% FWHM at Q ββ and 0.5-0.7% demonstrated by the large-scale prototypes NEXT-DBDM and NEXT-DEMO [13, 14], b) tracking capabilities that provide a powerful topological signature to discriminate between signal (two electron tracks with a common vertex) and background (mostly, single electrons). The topological signature, combined with a radio clean detector results in a very low specific background rate. The combination of radio purity and the additional rejection power provided by the topological signature of the two electrons results in an expected background rate of 10 -4 --5 × 10 -4 counts / (keV · kg · y) , depending of the level of background of the energy plane PMTs. There are only upper limits for those PMTs. The most sensitive measurement, performed by the LUX collaboration, quotes am upper limit in the background of each PMT of less than 700 µ Bq, and corresponds to the lowest limit of the background rate, while the XENON collaboration quotes a less sensitive limit that results in the upper limit of the background rate. The NEXT collaboration is currently screening all the PMTs entering the detector energy plane. While the measurement program is going on, they quote the upper limit of their background level, 5 × 10 -4 counts / (keV · kg · y) , as reference [11, 43]. The construction of the detector is underway at the Laboratorio Subterráneo de Canfranc (LSC), in Spain. NEXT owns 100 kg of enriched xenon, and foresees to start a physics run in 2015.", "pages": [ 8, 9 ] }, { "title": "6.1.1 Experimental parameters", "content": "The experimental parameters of the three xenon experiments described here, as defined in equation (4.1), are collected in Table 1. The parameters of EXO-200 and KamLAND-Zen are those published by the collaborations [7, 8]. The resolution in NEXT corresponds to the most conservative result obtained by their prototypes [14], and the predicted background rate and efficiency comes from the full background model of the collaboration [11, 43], assuming a conservative background level for the dominant source of background (the energy-plane PMTs, see discussion in the previous section). A caveat is in order concerning NEXT. Although the resolution is solidly established by the NEXT-DEMO and NEXT-DBDM prototypes, and the different components that will enter the detector have been carefully screened [12], to construct the background model, the predictions of the Monte Carlo have not been validated with actual data from the operating detector, as the other two experiments have already done. In this sense, the comparisons in this work are intended to show the potential of the technology, rather than its demonstrated performance. This said, the availability of a topological signature (also clearly established by the prototypes), the excellent resolution, and the on-going campaign to screen every component that goes into the detector, builds a strong case in favour of the HPXe technology.", "pages": [ 9, 10 ] }, { "title": "6.1.2 Nuclear matrix elements", "content": "In order to compute a sensitivity plot, one has to choose a given set of nuclear matrix elements (NME). In the last few years the reliability of the calculations has been addressed, with several techniques being used (see [6] and references therein), namely: the Interacting Shell Model (ISM); the Quasiparticle Random Phase Approximation; the Interacting Boson Model (IBM); and the Generating Coordinate Method (GCM). In most cases the results of the ISM calculations are the smallest ones, while the largest ones come often from IBM. Each one of the major methods has some advantages and drawbacks. The clear advantage of the ISM calculations is their full treatment of the nuclear correlations, while their drawback is that they may underestimate the NMEs due to the limited number of orbits in the affordable valence spaces. It has been estimated [6] that this effect can be of the order of 25%. On the contrary, the QRPA variants, the GCM and the IBM are bound to underestimate the multipole correlations in one or another way. As it is well established that these correlations tend to diminish the NMEs, these methods should tend to overestimate them. With this considerations in mind, a physics-motivated range (PMR) of theoretical values for the NMEs of different isotopes was proposed in [6]. In the case of 136 Xe the PMR range extends from a lower limit, defined by the ISM model, with a NME of 2.2 to an upper limit, defined by the IBM model, with a NME of ∼ 4. The central PMR value, used for all plots in this paper is 2.9. See [6] for further discussion.", "pages": [ 10 ] }, { "title": "6.1.3 Sensitivity", "content": "Figure 2, shows the expected performance of the three experiments, assuming the parameters described in Table 1 and the central value of the PMR described above. We consider a run of five years for NEXT (2015 to 2020) and a longer run of eight years for EXO-200 and KamLAND-Zen (2012 to 2020). A total dead-time of 10% a year for all experiments is assumed. Observe the following features: It follows that all the three experiment will have a chance of making a discovery if m ββ is in the upper part of its allowed range, see Figure 1. The fact that the experiments are based in different experimental techniques, with different systematic errors makes their simultaneous 300 running even more attractive. The combination of the three can reach a sensitivity of about 65 meV, which covers a significant fraction of the phase space. This result is affected by uncertainties in the values of the NME. Taking the lower bound of the PMR we find a sensitivity of 87 meV for the combined limit, while taking the IBM model as upper bound of the PMR we find a sensitivity of 48 meV.", "pages": [ 10, 11 ] }, { "title": "6.2 Sensitivity of ton-scale xenon experiments", "content": "To study the projected sensitivity of future xenon experiments, we consider three hypothetical detectors of the same mass (1 ton) running for the same total exposure (up to 10 ton · year) based in the three technologies discussed above: liquid xenon (LXe), xenon-liquid scintillator (XeSci) and high pressure gas xenon (HPXe). Our choice of one ton as the reference mass for these studies is motivated by the following reasons: Furthermore, considering the same mass for the three technologies and running their sensitivity as a function of the exposure allows to compare their potential in the same level ground.", "pages": [ 11, 12 ] }, { "title": "6.2.1 Resolution of one-ton xenon detectors", "content": "For HPXe, the intrinsic limit dictated by the Fano factor in xenon gas is 0.3% FWHM at Q ββ , but we consider safer to quote the actual resolution measured by the NEXT-DBDM prototype [13], which obtains 0.5 % FWHM at Q ββ . For LXe, we use the best projected resolution for the technology, 3.3% FWHM at Q ββ [42], also near the intrinsic limit in liquid xenon. For SciXe, we consider that a liquid scintillator calorimeter can be upgraded (by adding more PMTs) to improve the energy resolution. As a reference we consider SNO+ detector, which boasts the best energy resolution of all liquid scintillator calorimeters, 6.5% FWHM at Q ββ .", "pages": [ 12 ] }, { "title": "6.2.2 Background rate of one-ton xenon detectors", "content": "For HPXe we take the best case of the NEXT background model, which predicts an specific background rate of 10 -4 counts / (keV · kg · y) when using the most sensitive limits measured the energy plane PMTs [43]. For LXe, the current, very low background rate, achieved by the EXO-200 detector, is obtained with only marginal self-shielding. The reason for that is that EXO-200 is a small apparatus, and leaving part of the LXe as a shield has a large cost in efficiency. The situation, however, improves dramatically for a larger detector. For the sake of simplicity, consider an spherical LXe detector, with one ton mass and a radius of 43 cm. Leaving a shell of 10 cm of LXe as a shield reduces the specific background by a factor 1/15, and keeps 43% of the enriched xenon as the fiducial mass of the experiment. Assuming that the selection efficiency of the future LXe experiment will be similar to that of EXO-200 we find that a one-ton LXe detector could reach a background rate of 10 -4 counts / (keV · kg · y) with an overall efficiency of 38%. Concerning the liquid scintillator calorimeter, we assume that destilation of the 110 m Ag will result in about one order of magnitude reduction in the specific background, as discussed in [44]. For simplicity, we also consider an specific background rate of 10 -4 counts / (keV · kg · y) and leave the efficiency unchanged with respect to KamLAND-Zen.", "pages": [ 12 ] }, { "title": "6.2.3 Experimental parameters", "content": "Table 2 summarises our projections of the experimental parameters for the three technologies. Notice that, while we believe that the parameters displayed in Table 2 are reasonable, we are not claiming that they represent any specific design. We assume a resolution near the practical limit for the three technologies, and use reasonable assumptions to predict their achievable background rate, which turns out to be, both very small and quite similar.", "pages": [ 12, 13 ] }, { "title": "6.2.4 Sensitivity", "content": "Figure 3, shows the expected performance of the three technologies, assuming the parameters described in Table 2, up to a total exposure of 10 ton · year. Although we have used a reference mass of one ton, the actual detector designs could consider, of course, larger masses. The tradeoff between total detector mass and exposure time needs to be done taking into account detector design and the cost of enriched xenon. At the maximum exposure, the LXe and and XeSci detectors reach a draw at 40 meV, while the HPXe detector reaches 25 meV. Each one of the experiments covers a large fraction of the available phase space, with HPXe covering practically all the range of allowed values. The combination of the three experiments is 19 meV, fully covering the phase space, while the combination of HPXe and one of the other two is 21 meV. This result is, of course, affected by uncertainties in the values of the NME. Taking the lower bound of the PMR we find a sensitivity of 25 meV for the combined limit, while taking the IBM model as upper bound of the PMR we find a sensitivity of 14 meV. Notice that the HPXe (using the central value of the PMR) fully covers the one-sigma range of m ββ values ( [26 , 145] meV, see Section 3) , while the combination of the three experiments covers the range even for the lower bound of the PMR (that is the ISM, which gives the lowest NME of all the available models).", "pages": [ 13, 14 ] }, { "title": "7 Discussion", "content": "In this work, we have addressed the question whether present and ton scale xenon-based double beta decay experiments can fully answer the quest on the nature of neutrinos or not. We find a positive answer, based on the evidence of a nonzero value for the neutrino cosmological mass, ∑ m ν = 0 . 32 ± 0 . 11 , determined by the SPT team [2] and the assumption of the standard model extended by only three light neutrinos. This measured total neutrino mass of light neutrinos implies a quasi-degenerate neutrino mass spectrum, what leads to important consequences. Our findings are summarized in Figure 4, where we use the parameter space of neutrinoless double beta effective mass and lowest neutrino mass. The 1 σ allowed regions by neutrino oscillation and cosmological data show that m ββ is bound from below, at the level of 20 meV, independently of the neutrino mass hierarchy. In particular, the one-sigma range of m ββ is [26 , 145] meV. The free Majorana phases are unable to produce full cancellation of the effective mass due to the degeneracy of the neutrino masses. We have considered xenon-based ββ 0 ν experiments, on the double grounds of their good performance, and the suitability of the technology to large-mass scaling. Firstly we discuss the current generation of experiments, KamLAND-Zen, EXO-200 and NEXT, with effective masses in the range of 100 kg and conceivable exposures in the range of 500 kg · year. The expected sensitivity of the three experiments to m ββ in 2020 is shown by blue horizontal lines in Figure 4. All three experiments have sensitivity to some of the effective mass predicted by neutrino oscillation and total mass measurements, and the combination of the three, with sensitivity of 65 meV, have the potential to test about half of the allowed effective mass parameter space. The uncertain value of the NME modifies the sensitivity of the combination, from 48 meV to 87 meV. More importantly, the lower bound in m ββ , implies the potential to distinguish whether neutrinos are Dirac or Majorana particles, under the quoted assumptions. The next generation experiments, with an exposure in the range of 10 ton · year, would have a much more enhanced sensitivity, as shown by red horizontal lines in Figure 4. Al three technologies have the potential to explore most of the m ββ allowed region. The high pressure gas xenon TPC, due to the excellent energy resolution, can cover the full range of m ββ predictions under reasonable NME assumptions (the central value of the PMR). The combination of the three technologies would cover the full range of allowed m ββ values even for the smallest NME. In summary, xenon experiments may be the tool to demonstrate that neutrinos are Majorana particles in the next few years.", "pages": [ 15 ] }, { "title": "Acknowledgments", "content": "We warmly acknowledge C. González-García and P. Hernández for discussions, help and insight. This work was supported by the Ministerio de Economía y Competitividad of Spain under grants CONSOLIDER-Ingenio 2010 CSD2008-0037 (CUP), FPA2009-13697-C04-04 and FPA2011-29678, and by the Generalitat Valenciana grant PROMETEO/2009/116 and the ITN INVISIBLES (Marie Curie Actions, PITN-GA-2011-289442).", "pages": [ 15 ] }, { "title": "References", "content": "[1] P. Hernández, Neutrino physics , arXiv:1010.4131 .", "pages": [ 15 ] } ]

[ { "title": "Density Perturbations from Modulated Decay of the Curvaton", "content": "David Langlois 1 and Tomo Takahashi 2 1 APC (CNRS-Universit'e Paris 7), 10, rue Alice Domon et L'eonie Duquet, 75205 Paris Cedex 13, France 2 Department of Physics, Saga University, Saga 840-8502, Japan", "pages": [ 1 ] }, { "title": "Abstract", "content": "We study density perturbations, including their non-Gaussianity, in models in which the decay rate of the curvaton depends on another light scalar field, denoted the modulaton. Although this model shares some similarities with the standard curvaton and modulated reheating scenarios, it exhibits interesting predictions for f NL and g NL that are specific to this model. We also discuss the possibility that both modulaton and curvaton fluctuations contribute to the final curvature perturbation. Our results naturally include the standard curvaton and modulated reheating scenarios as specific limits and are thus useful to present a unified treatment of these models and their variants.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Cosmological observations are increasingly precise, providing us with a lot of information on the origin of cosmic structure, i.e. on primordial density fluctuations. Although the fluctuations of the inflaton field are considered as the main candidate for their origin, other possibilities have also been investigated (see e.g. [1] for introductory lectures), especially in the light of recent constraints on primordial non-Gaussianity. The degree of non-Gaussianity in primordial density fluctuations is usually characterized by the nonlinearity parameter f NL which represents the amplitude of the bispectrum. In the case of standard slow-roll single-field inflation, f NL is predicted to be too small to be observable. On the other hand, the present constraints on f NL for local type non-Gaussianity obtained from cosmic microwave background (CMB) and large scale structure are respectively f (local) NL = 37 . 2 ± 19 . 9 (1 σ C.L.) from WMAP9 [2] and f (local) NL = 62 ± 27 (1 σ C.L.) from NRAO VLA Sky Survey [3], which may give some hints that the value of f NL is away from zero. In this context, other candidates for primordial fluctuations, in particular those giving significant f NL , have been extensively discussed such as the curvaton model [4-6], modulated reheating scenario [7, 8], inhomogeneous end of hybrid inflation [9-12], modulated trapping [13,14] and so on. Even if we limit ourselves to the curvaton and modulated reheating scenarios, various extensions of them have been proposed and studied, for example mixed inflaton-curvaton model [15-19], mixed inflaton-modulated reheating model [20], multi-curvaton [21,22], modulated curvaton [23,24] and so on #1 . In most of those scenarios, a light scalar field (degree of freedom) other than the inflaton is involved in some way, and the final values of density perturbations depend on how the initial fluctuations are converted to the final ones during the evolution of the early Universe. This consideration provides a strong motivation to treat models involving curvatons and/or modulatons in a unified formalism (see [34]). In the present work, we wish to focus our attention on a scenario that interpolates between the modulated reheating and the curvaton models: that of the modulated curvaton decay, in which the decay of the curvaton field is modulated by the dependence of its decay rate Γ on another fluctuating scalar field, which we will call the modulaton #2 . In this paper, we focus on this new mechanism of generating primordial density fluctuations and derive its predictions, paying particular attention to non-Gaussianities by computing the bispectrum and trispectrum. Although the model we propose here is, in some sense, a straightforward extension of the curvaton and modulated reheating scenarios, we find that it leads to a rich phenomenology, with interesting observational implications for primordial non-Gaussianity. We also provide general formulas, which could be applied to other similar types of scenarios (see [34] for a systematic approach, including isocurvature perturbations). The structure of this paper is as follows. In the next section, we derive a general expression describing the final curvature perturbation, up to third order in the perturbations. Focussing then, in Section 3, on the modulaton fluctuations, we analyse the density perturbation and compute the non-linear parameters such as f NL and g NL . In Section 4, we investigate models in which the curvaton fluctuations also contribute to the observed perturbations in addition to those from the modulaton. The final section contains a summary of this paper.", "pages": [ 2, 3 ] }, { "title": "2 Computing the post-decay perturbation", "content": "The present scenario relies on the presence of three scalar fields: an inflaton field φ which drives the inflationary expansion; a curvaton (or modulus) field σ , with an energy density negligible during inflation, which, well after inflation, oscillates at the bottom of its potential before decaying; and, finally, a modulaton field χ , which is light during inflation and thus acquires fluctuations from the amplification of quantum fluctuations. The crucial assumption here is that the decay rate of the curvaton σ depends on the modulaton χ . Therefore, fluctuations of χ directly lead to a varying decay rate and eventually produce density fluctuations. In practice, we will not need any detail about the inflaton field. Its role will be simply to drive inflation so that the modulaton field can acquire some fluctuations. In our scenario, σ can be either light during inflation ( m σ /lessmuch H ), in which case it will also acquire some fluctuations, or be massive ( m σ /greatermuch H ) in which case its fluctuations are suppressed. Strictly speaking, the curvaton scenario assumes a light field during inflation, but there exist models where the additional scalar fields, such as moduli, are not necessary light during inflation. Whereas the original curvaton scenario would not apply to scalar fields of this type, our model does. In the following, although σ is a scalar field, we will be interested in the cosmological phase where it oscillates at the bottom of its potential and can be effectively described as a fluid, which is pressureless if the potential is quadratic. Note that our formalism also applies to the decay of the inflaton oscillating in a quadratic potential at the end of inflation, if σ is simply replaced by the inflaton φ . Our formalism thus includes automatically the modulated reheating scenario. For each fluid characterized by an equation of state parameter w i ≡ P i /ρ i , which is assumed here to be constant, it is convenient to introduce the non-linear curvature perturbation ζ i [35] (see also [36-38] for a covariant definition) where δN denotes the local perturbation of the number of e-folds and a barred quantity must be understood as homogeneous. From the above formula, the nonlinear energy density of the species i can be written locally as In our case, we will consider only two species: radiation ( w r = 1 / 3) and the curvaton field, treated as a pressureless fluid ( w σ = 0). Using the instantaneous decay approximation, the value of the Hubble parameter at the decay of the curvaton σ (or, alternatively, of the inflaton to describe inhomogeneous reheating) is given by where the decay rate Γ is a function of the modulaton χ . Because of the modulaton fluctuations δχ = H/ (2 π ) generated during inflation, the decay hypersurface characterized by the above relation is inhomogeneous . Using Friedmann's equations #3 , this implies for the local energy density where t D ( /vectorx ) represents the local decay time. Substituting in the relation (4), both for the matter contents just before and just after decay, we find where we have introduced the curvaton fraction of the total energy density (just before the decay) Ω σ ≡ ¯ ρ σ / ¯ ρ tot , as well as the (nonlinear) relative fluctuations of the decay rate The first equality in (6) gives us the expression of δN D as a function of the two pre-decay curvature perturbations ζ r and ζ σ . And the second equality in (6) yields the expression of the post-decay curvature perturbation ζ (carried by the only-remaining radiation fluid) as a function of δN D , namely There is no general nonlinear expression for δN D given in terms of ζ r and ζ σ , but by expanding the first equality of (6) order by order in the perturbations, one can iteratively obtain an explicit expression for δN D valid up to any order. Computing δN D up to third order with this method and substituting in (8), we finally get the following expression for the post-decay curvature perturbation: where we have introduced, for convenience, the curvaton isocurvature perturbation and the parameter r , defined by ∣ Although isocurvature fluctuations can also be generated in principle, we restrict our analysis to adiabatic perturbations in the present work (see [34] for an analysis including isocurvature modes). Note that the perturbations ζ r , ζ σ and δ Γ are related to the fluctuations of the inflaton, curvaton and modulaton, via the expressions #4 where a prime denotes the derivatives with respect to χ . Substituting the above expressions in (9) would thus give the curvature perturbation ζ as a function of the fluctuations δφ , δσ and δχ . Once the curvature perturbation has been computed, here up to third order, it is useful, in order to confront the model with observations, to calculate the power spectrum P ζ , bispectrum B ζ and trispectrum T ζ . They correspond, respectively, to the 2-point, 3-point and 4-point correlation functions in Fourier space: In the case of local non-Gaussianity, it is convenient to express the bispectrum and trispectrum in terms of the power spectrum and to introduce the so-called non-linearity parameters f NL for the bispectrum, τ NL and g NL for the trispectrum: Quite generically, if the curvature perturbation can be written in the form where the ϕ a denotes any number of light scalar fields, labelled by the index a , with statistical independent fluctuations generated during inflation #5 , the power spectrum is given by and the non-linearity parameters by the simple expressions [39-41] where we have used the Kronecker symbols to raise the scalar field indices, e.g. N a ≡ δ ab N b . Equipped with the general formalism presented in this section, we now discuss in more details various scenarios based on the modulated decay of the curvaton in the following sections.", "pages": [ 3, 4, 5, 6 ] }, { "title": "3 Modulaton dominated case", "content": "In this section, we restrict ourselves to the simplest case where only the modulaton fluctuations are relevant, i.e. we neglect the fluctuations of the curvaton and of the pre-decay radiation fluid. Note that the curvaton fluctuations can be ignored in two distinct situations: firstly, if their contribution turns out to be numerically negligible in the expression for ζ ; secondly, if the curvaton field is massive during inflation (i.e. m σ /greatermuch H ), in which case its fluctuations are suppressed. From the general formula (9), the curvature perturbation in this case reduces to Substituting the expansion of δ Γ in terms of δχ , given in (12), one obtains, up to third order, the expression As mentioned earlier, our analysis includes the original modulated reheating scenario if σ is identified with the inflaton. One can indeed check that the above formula, in the limit r = 1 (since the inflaton dominates the total energy density), reduces to the third-order expression obtained for ζ in the modulated reheating scenario [42]. Since the expression (25) is exactly of the form (18) with χ as a unique scalar field, one can readily use the general formulas (20-22) to determine the power spectrum and the non-linearity parameters. The power spectrum is thus while we obtain for the non-linearity parameters In the limit r = 1, one recovers the predictions of the modulated reheating scenario. In the limit r /lessmuch 1, one finds that the dominant terms in f NL and g NL , barring some special cancellation, scale like 1 /r and 1 /r 2 , respectively. This is different from the standard curvaton model, where both f NL and g NL scale like 1 /r #6 . This specific feature of our model can be expressed by the relation where the numerical value of the coefficient C , and in particular its sign, depends on the functional form of Γ: To go further and determine quantitatively the parameters f NL and g NL , taking into account the observed amplitude of the power spectrum, we need to assume specific expressions for the function Γ( χ ). We consider below three possibilities for Γ( χ ). By imposing the CMB normalization for the power spectrum P ζ = 2 . 4 × 10 -9 [2], Eq. (26) determines the parameter r in terms of the inflationary Hubble parameter H inf and χ : Furthermore, r should be less than unity by definition, which limits the parameter range of H inf and χ to satisfy Regarding the non-linearity parameters f NL and g NL , the dependence on the functional form of Γ only appears in the combinations ΓΓ '' / (Γ ' ) 2 and ΓΓ ''' / (Γ ' ) 3 . By substituting the functional form, we obtain In the limit r /lessmuch 1, one thus finds that f NL is always positive and can become large if r is small enough. In the same limit, g NL is positive and enhanced by the factor 1 /r 2 , as already mentioned. In Fig. 1, we show contours of f NL (left plot) and g NL (right plot) in the χ -H inf plane. The value of r is fixed by the CMB normalization as described above. In many models, the coupling can be written as a Taylor expansion of χ , where M represents some high energy scale and χ/M /lessmuch 1 is assumed. The coefficients a and b are parameters of order one, which are supposed to depend on the details of some explicit model of high energy physics. The non-linearity parameters are then given by where we have used χ/M /lessmuch 1. As easily read off from the above expressions, the signs of f NL and g NL can be positive or negative, depending on a and b . Since a and b are assumed to be O (1), the amplitude of f NL and g NL is mainly controlled by the value of r . In the latter case, we consider the possibility that the expansion starts with a higher order polynomial. Once again, M is some high energy scale characterizing the underlying physics, and we assume χ/M /lessmuch 1. The non-linearity parameters are now given by Since χ/M /lessmuch 1, one sees that f NL can be large even with r = 1, but the sign of f NL is then negative, which is in contradiction with the current constraints from WMAP [2]. On the other hand, when r /lessmuch 1, f NL and g NL can be both positive, which is similar to the case I.", "pages": [ 6, 7, 8, 9, 10 ] }, { "title": "4 Hybrid curvaton-modulaton case", "content": "In this section, we consider the general situation where both the curvaton and modulaton fluctuations contribute to the final density perturbation. We also take into account the inflaton contribution in the power spectrum. The curvature perturbation in this case is given by (9), with the substitutions of (12). This leads to an expression of the form (18), which contains now three scalar fields φ , σ and χ . It is convenient to introduce two dimensionless parameters that characterize the relative contributions of σ and χ to the total power spectrum, defined by The inflaton contribution in the power spectrum is thus given by 1 -Ξ σ -Ξ χ . The bispectrum parameter ˜ f NL can then be decomposed into three terms (the inflaton does not contribute to the non-Gaussianity here) where we have defined Their explicit expressions are where the first equation corresponds to the standard curvaton expression [39], while the second one is the modulaton contribution calculated in the previous section. The final one is a mixed contribution. Similarly, the trispectrum coefficient ˜ g NL can be decomposed into four terms: where By using (9) and (12), we can explicitly write down these quantities as where one can identify the usual curvaton contribution [43], the pure modulaton contribution calculated in the previous section, as well as two mixed curvaton-modulaton contributions. Finally, the τ NL coefficient can be decomposed as In the r /lessmuch 1 limit, one finds that the dominant terms for the non-Gaussianity coefficients are Here it is interesting to notice that the leading term of g NL vanishes when Ξ χ = 0. Thus the enhancement by the factor of 1 /r 2 comes from the modulaton fluctuations, which is absent in the standard curvaton model. This property of the trispectrum could be useful to discriminate this model from other ones.", "pages": [ 10, 11 ] }, { "title": "5 Summary", "content": "We have investigated the density perturbations in a scenario based on the modulated decay of the curvaton, where the curvature perturbation is generated via the modulation of the curvaton decay rate due to its dependence on another scalar field, called the modulaton χ . We have paid special attention to non-Gaussianity, since it is potentially the best way to discriminate among various scenarios, and we have computed specifically the non-linearity parameters f NL and g NL in this class of models. As discussed in Section 3, this model shares some similarities with the usual curvaton and modulated reheating models: the size of these parameters are mainly determined by r , which is similar to the curvaton model. On the other hand, the signs of these parameters highly depend on the functional form of Γ( χ ), which is the same as usual modulated reheating model. However, the model also exhibits interesting predictions coming from a hybrid nature of this model. When the curvaton is a subdominant component of the Universe at its decay, the non-linearity parameters are related as g NL ∝ f 2 NL where the proportionality factor depends on the functional form of Γ( χ ). Thus this model generically predicts enhanced g NL , which is different from the standard curvaton and modulated reheating models, predicting g NL ∼ f NL . We have also investigated a more general case where fluctuations of the curvaton itself also contribute to density fluctuations. We have presented the formulas of the curvature perturbation up to the 3rd order, and the non-linearity parameters. The formulas given in such a case naturally also include the standard curvaton and modulated reheating models, which provides a unified treatment of those kind of models and their variants. Although we have considered only adiabatic perturbations in the present work, the modulated decay of the curvaton could also produce isocurvature perturbations, similarly to the standard curvaton scenario [44]. It would be interesting to study these possible isocurvature modes and their non-Gaussianities following the analysis introduced in [45, 46]. Isocurvature modes, possibly correlated with adiabatic modes, lead to very specific signatures in the CMB non-Gaussianities and Planck or future CMB data could detect these isocurvature non-Gaussianities [47,48] #7 . In the near future, one can hope that new cosmological data of unprecedented precision, in particular from Planck, will enable to test the model presented here, together with various other mechanisms that generate primordial density perturbations. In this respect, the unified treatment proposed in this paper should be useful for a simplified confrontation of a large class of models with cosmological data. Note added: While completing this manuscript, we became aware that the authors of [52] were working on a very similar topic.", "pages": [ 12 ] }, { "title": "Acknowledgments", "content": "T.T would like to thank APC for the hospitality during the visit, where part of this work has been done. The authors would also like to thank the Yukawa Institute for Theoretical Physics at Kyoto University, where this work was completed during the Longterm Workshop YITP-T-12-03 on 'Gravity and Cosmology 2012'. D.L. is partly supported by the ANR (Agence Nationale de la Recherche) grant STR-COSMO ANR-09-BLAN0157-01. The work of T.T. is partially supported by the Grant-in-Aid for Scientific research from the Ministry of Education, Science, Sports, and Culture, Japan, No. 23740195.", "pages": [ 13 ] } ]

[ { "title": "Cosmological constraints on spontaneous R-symmetry breaking models", "content": "Yuta Hamada and Tatsuo Kobayashi Department of Physics, Kyoto University, Kyoto 606-8502, Japan", "pages": [ 1 ] }, { "title": "Kohei Kamada", "content": "Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, D-22607 Hamburg, Germany Yutaka Ookouchi The Hakubi Center for Advanced Research & Department of Physics, Kyoto University, Kyoto 606-8302, Japan (Dated: August 28, 2018)", "pages": [ 1 ] }, { "title": "Abstract", "content": "We study general constraints on spontaneous R-symmetry breaking models coming from the cosmological effects of the pseudo Nambu-Goldstone bosons, R-axions. They are substantially produced in the early Universe and may cause several cosmological problems. We focus on relatively long-lived R-axions and find that in a wide range of parameter space, models are severely constrained. In particular, R-axions with mass less than 1 MeV are generally ruled out for relatively high reheating temperature, T R > 10 GeV.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Supersymmetry (SUSY) has been considered to be the strongest candidate of the physics beyond the standard model (BSM). Although the recent data from the Large Hadron Collider (LHC) has not shown any evidence for SUSY but discovered a Standard Model Higgs-like particle with a mass of around 125 GeV [1], it still remains a strong candidate of BSM because it suggests the gauge coupling unification, it guarantees proton stability sufficiently, and it provides a reasonable dark matter candidate. Moreover, in string theories, which are the most powerful candidates of the quantum theory of gravity, it plays a crucial role for consistency and must be broken at a scale between the electroweak scale and the Planck scale. Therefore, it is important to investigate SUSY-breaking models in the light of LHC data [2]. R-symmetry, which is a specific symmetry of supersymmetric models, is a key ingredient for SUSY breaking and its application to model building. Recent drastic progresses on SUSY breaking by exploiting a metastable state (see [3-5] for reviews and references therein) gives us a better understanding of the role of the R-symmetry in realistic model building [6-20]. Nelson-Seiberg's argument [21] beautifully demonstrates a connection between metastability and R-symmetry in the context of generalized Wess-Zumino models with a generic superpotential. If R-symmetry is preserved, there is no SUSY vacuum in a finite distance in field space. On the other hand, if a gaugino mass has Majorana mass, R-symmetry has to be broken to generate the gaugino mass. Thus, there is a tension between stability of vacuum and generating gaugino mass. A simple solution to this problem is to introduce an approximate R-symmetry. One of the interesting ways to break R-symmetry is spontaneous breaking. In Ref. [22], D. Shih revealed a quite fascinating condition for spontaneous R-symmetry breaking in the context of generalized O'Raifeartaigh models: For R-symmetry breaking, there must be a field with R-charge different from 0 or 2. Such models were applied to gauge mediation [23] and some classes of the models successfully generated large gaugino masses. According to the general argument by Komargodski and Shih [24], large gaugino mass is related to a tachyonic direction at a point in pseudo moduli space toward the messenger direction. In the R-symmetric model, such tachyonic direction exists at the origin of the pseudo moduli space. When the spontaneous breaking of U (1) R symmetry occurs, cosmic R-strings are formed by the Kibble-Zurek mechanism [25, 26]. Plugging the structure of the pseudo-moduli space mentioned above and R-string forming, we will meet a quite dangerous possibility. It is known as a 'roll-over' process of vacuum through inhomogeneous energy distribution by an impurity such as a cosmic string [27, 28]. In the core of the R-string, the system can easily slide down to the lower vacuum via the tachyonic direction at the origin and form a sort of 'R-tube' in which the core sits in the lower energy vacuum. Thus, if the tube is unstable, by rapid expansion of the radius, the universe can be filled by the unwanted SUSY vacuum. As discussed in Ref. [29] this gives a constrain for model building. However, as emphasized in Ref. [30, 31], when a D -term contribution is not negligible, it can lift the tachyonic direction and stabilize the pseudo-moduli space. In such models, the roll-over process does not occur. Also, when the amplitude of (tachyonic) messenger mass at the origin is sufficiently smaller than that of R-symmetry breaking field, the vacuum selection is successfully realized. As we will see, R-strings are unstable due to the explicit R-symmetry breaking term in the superpotential and hence the roll-over process can be circumvented if the life-time due to the explicit R-symmetry breaking is shorter than that for the roll-over process. In this paper, we assume such an early stage scenario and study general cosmological constraints for the models. In this sense, the results shown in the present paper is complementary to the ones studied in Ref. [29]. In spontaneous R-symmetry breaking models, there exists a pseudo Nambu-Goldstone boson, called R-axion, as well as the modulus field called R-saxion. They are copiously produced in the early Universe from scattering of thermal plasma, coherent oscillation, Rstring decay and so on, and may cause other cosmological problems. Note that although we commented on the importance of R-strings, there are many other sources of R-axions and we should take into account all the contributions at the same time. Model parameters on spontaneous R-symmetry breaking model can be constrained from such cosmological considerations. Note that, unlike the QCD-axion, R-axions receive relatively heavy mass from gravitational coupling with explicit R-symmetry breaking constant term in the superpotential and its lifetime can be much shorter than the cosmic age. Thus, we can impose not only constraints from the R-axion overclosure problem but also that from R-axion decay. In this paper, we investigate their cosmological constraints focusing on relatively long-lived parameter range. We show that the model parameter space is severely constrained and many parameter space of R-axion is ruled out from the cosmological consideration. This paper is organized as follows. In section II, we explain the general feature of spontaneous R-symmetry breaking models. In section III, we evaluate the R-axion abundance produced in the early Universe. Here we assume that cosmic R-string is produced in some earlier epoch. We list the cosmological effects induced by R-axions in section IV. We also evaluate the constraint on the parameter space from these effects. Section V is devoted to conclusion and discussion.", "pages": [ 2, 3, 4 ] }, { "title": "II. SPONTANEOUS R-SYMMETRY BREAKING MODEL", "content": "In spontaneous R-symmetry breaking models, the SUSY-breaking field with a finite Rcharge acquires nonvanishing vacuum expectation value. The phase of the SUSY-breaking field is almost massless and identified as the Nambu-Goldstone boson. It acquires a small mass from explicit R-symmetry breaking term in the superpotential and called R-axions. In order to see its cosmological consequeces, we should first investigate their properties and interactions. Here, we review a simple but general R-symmetry breaking model focusing on R-axions and read off their interactions with several modes.", "pages": [ 4 ] }, { "title": "A. R-symmetry breaking model", "content": "Let us consider a simple effective superpotential for the R-charged SUSY-breaking field, X , integrating out the messenger fields, Here Λ eff gives the nonvanishing F -term for the SUSY-breaking field and R-symmetry breaking constant W 0 is introduced for the cosmological constant to vanish. Note that from the flat Universe condition, they are related as Λ 4 eff = 3 W 2 0 /M 2 pl with M pl being the reduced Planck mass. Assuming a noncanonical Kahler potential, X can be destabilized at the origin [29]. Here we consider the effective potential for X , where we have defined X = ( χ/ √ 2) e ia/ √ 2 f a . The second term that breaks U (1) R symmetry comes from the R-symmetry breaking constant term in the superpotential that couples to X field through the Planck suppressed interaction in supergravity 1 . The R-axion mass is related to the parameters in the potential as where m 3 / 2 = W 0 /M 2 pl is the gravitino mass. Let us investigate the model further. We here expand X around X = f a as follows, so that the fields a and s have canonical kinetic terms. Here the phase part a and the radius part s are identified as R-axion and R-saxion, respectively. Note that the mass of R-saxion is related to the R-symmetry breaking scale as In the last equality, we assumed that the Kahler metric is given by with ˜ λ being a numerical constant of order of the unity, and λ is related to the model parameters as λ /similarequal m 2 a M pl /f 3 a . The fermionic partner of X field, 'R-axino,' is the goldstino for the SUSY-breaking and absorbed in the gravitino. Thus, we just have to consider cosmology of gravitinos instead of R-axinos.", "pages": [ 4, 5 ] }, { "title": "B. Interactions of R-axions", "content": "We now investigate interactions of the R-axion with several modes as well as its cross sections and decay rates. As we will see, they are useful for the cosmological constraints on R-axion abundance. First of all, the R-axion to R-saxion interaction can be read off from the kinetic term of X , From this interaction, we can evaluate the decay rate of R-saxion to 2 R-axions as We can assign R-charges to the supersymmetric Standard Model fields such that the R-symmetry is consistent with all of the interactions. After SUSY and R-symmetry are broken, the R-axion appears in the gaugino mass terms as well as the so-called B-term and A-terms. In addition, the R-axion couplings with the gauge bosons appear through the anomaly coupling terms. That is, the coupling between the R-axion and the photon is given by where F µν is the field strength tensor of U (1) em and C em is the anomaly coefficient, i.e. Tr U (1) R U (1) 2 em , which is model-dependent. Then, the decay width of the R-axion into two photons is given by Similarly, the R-axion coupling with the gluon is given by where G µν is the SU(3) field strength tensor and C g is the anomaly coefficient, i.e. Tr U (1) R SU (3) 2 . This interaction is effective in thermal production of R-axions. The anomaly coefficients are typically numerical factors of the order of the unity. It slightly changes our result but basic features do not change according to the choice of the coefficients. In the following, we assume C em = C g = 2 unless we explicitly note. The interactions of the R-axion with the Higgs fields appear through the B-term. Then, the R-axion and the Higgs fields mix each other in their mass terms (see for its detail Appendix A.). The eigenstate corresponding to the low-energy R-axion ˜ a includes the axial parts of the up and down-sector Higgs fields, ξ u and ξ d [32], where r = v/ ( √ 2 f a ), v = 246 GeV, κ = (1+ r 2 sin 2 2 β ) -1 / 2 . Note that a denotes the R-axion at high energy beyond the electroweak symmetry breaking. Since the coefficients of ξ u,d are very small, it is found that ˜ a ∼ a . Hereafter, we denote the low-energy R-axion by a instead of ˜ a . However, because of this mixing, the R-axion can couple with the quarks and leptons through their Yukawa couplings. That is, the couplings of the R-axion with the up-type quarks, the down-type quarks and the charged leptons, λ u , λ d and λ /lscript , are given by respectively, where y f and m f with f = u, d, /lscript are their Yukawa couplings and masses. Through these couplings, the R-axion can decay to a pair of the SM fermions, if m a > 2 m f . Its decay width is given by For example, the decay rate into the electron pair is given by The decay rate into the µ pair is enhanced by its mass as Γ( a → µ + µ -) = ( m µ /m e ) 2 × Γ( a → e + e -), but such a decay occurs for m a > 2 m µ . Similarly, we can compute the couplings between the R-axion and the neutrinos. For the neutrinos, we consider the Weinberg operator in the superpotential, y ν ( LH u ) 2 /M R , instead of the Yukawa couplings terms. Then, similar to the above, the coupling of the R-axion with neutrinos is given by Thus, the decay rate of the R-axion into the neutrino pair is suppressed because it is proportional to the neutrino mass squared, i.e. Therefore, the branching ratio of R-axions into pair of neutrinos are small enough even in the case where the decay channel into electron is closed, m a /lessorsimilar MeV. Note that R-axion decay associated with QCD jet production occurs when it is heavier than at least the proton mass, m a /greaterorsimilar 1 GeV, which is beyond our interest. Thus, we do not consider it here. The lifetime of R-axions is given by τ a ≡ Γ -1 . In Fig. 1, we show its m a dependence with each choice of f a = 10 6 , 10 8 , 10 10 and 10 12 GeV. We can see that the lifetime of R-axions becomes longer for smaller m a and larger f a . We can also see that the decay channels to electrons opens at m a /similarequal 1 MeV and to muons at m a /similarequal 200 MeV and the R-axion lifetime becomes shorter.", "pages": [ 5, 6, 7, 8 ] }, { "title": "III. R-AXION PRODUCTION IN THE EARLY UNIVERSE", "content": "Let us consider the cosmology of the spontaneous R-symmetry breaking model focusing on the R-axion production and evaluate the R-axion abundance. We consider the case where U (1) R is restored due to some additional mass terms such as the Hubble induced mass or thermal mass in the early Universe 2 . After some epoch, X field is destabilized as the additional mass term decreases and acquires vacuum expectation value χ = f a . Since the approximate U (1) R symmetry breaks spontaneously at that time, (unstable) cosmic strings are formed by the Kibble-Zurek mechanism. The long cosmic strings in a Hubble volume intersect each other and generates closed string loops 3 . These closed string loops shrink with emitting R-axions. As a consequence, the cosmic string network enters the scaling regime. As the Hubble parameter decreases, the explicitly R-symmetry breaking term in the potential becomes no longer irrelevant to the dynamics of the system and the string network turns to the string-wall system where domain walls are attached to cosmic strings [33, 34]. The string-wall networks are unstable and annihilate when the domain wall tension becomes comparable to that of cosmic strings. The energy stored in the string-wall system turns to R-axion particles. The fate of R-axions produced from the cosmic string loops and the string-wall system as well as the scattering of thermal plasma and the vacuum misalignment is determined by the lifetime of R-axions, which, then, constrain the model parameters of spontaneous R-symmetry breaking models 4 . In the following, we estimate the R-axion abundance from each source. We will examine the cosmological constraints in Sec. IV.", "pages": [ 8, 9 ] }, { "title": "A. R-axion production from vacuum misalignment", "content": "First we evaluate the energy density of the coherent oscillation of the R-axion field [35]. After the spontaneous R-symmetry breaking phase transition, the R-axion field acquires some initial value, a i , and keeps its position after a while due to large Hubble friction. When the Hubble parameter decreases to the R-axion mass, the R-axion field starts to oscillate. Here the subscription 'osc' indicates that the parameter or variable is evaluated at the onset of the R-axion oscillation. The energy density of the oscillating R-axion ρ a, o is given by If the R-symmetry is broken after inflation, the initial value of R-axion distributes randomly from -√ 2 πf a to √ 2 πf a since the correlation length of R-axion becomes much shorter than the Hubble length at the onset of the R-axion oscillation. Therefore, we estimate the mean value of a i as Since the energy density of R-axion oscillation decreases as a -3 , the quantity ρ a /s is conserved as long as there are no entropy production, where s is the entropy density. Therefore, we characterize the axion abundance by this quantity as  where g ∗ and g ∗ s are (effective) relativistic degrees of freedom for energy density and entropy, respectively, and the subscript ' R ' represents that the parameter or variable is evaluated at reheating. Note that T osc is given by Here we assume that the scale factor increases like matter dominated era during inflaton oscillation dominated era and take into account the dilution until the inflaton decay or reheating when H osc > H R .", "pages": [ 9, 10 ] }, { "title": "B. R-axion production from global cosmic strings", "content": "Next we evaluate the energy density of R-axions radiated from the cosmic string loops [36, 37] following the discussion in Appendix B of Ref. [38]. When the R-string network enters the scaling regime, the energy density of the long R-strings are estimated as Here the scaling parameter ξ /similarequal 0 . 9 [38, 39] represents the mean number of strings in a Hubble volume and d string /similarequal λ -1 / 2 f -1 a represents the core width of R-string. Note that the line energy density or the tension of R-string is given by [40] Assuming all the energy loss of long R-strings is converted into R-axion particles through the string loops, we obtain the evolution equations where the energy emission rate from the string loops,  Here we assume that R-axion particles released from cosmic string loops are relativistic. Since the mean comoving momentum of radiated R-axion can be evaluated as with the constant /epsilon1 /similarequal 0 . 25 [37, 38], we can estimate the number density of radiated R-axions as Here t ∗ is the time when the R-string network enters the scaling regime. When the Hubble parameter becomes comparable to the R-axion mass and R-symmetry breaking mass term becomes no longer irrelevant, t = t osc , string-wall system forms and R-axion emission from R-string loops stops. We can evaluate the resultant number density of R-axions from the R-string loops as The radiated R-axions become nonrelativistic after some epoch. Therefore, we can approximate the R-axion energy density as ρ a, str = m a n a, str and the R-axion energy-to-entropy ratio as", "pages": [ 10, 11, 12 ] }, { "title": "C. R-axion production from string-wall system", "content": "Let us evaluate the energy density of R-axions from the string-wall system annihilation [33, 34]. At t /similarequal t osc , the explicitly R-symmetry breaking term in the potential (2) becomes no longer irrelevant, and string-wall system forms. The surface mass density of domain walls are estimated as [40] When the tension of domain walls dominates that of strings, the string-wall system annihilates. As following the discussion in Ref. [34], we assume that the energy stored in the string-wall system released to R-axion particles. Thus, we evaluate the number density of R-axions as where ω a = α w m a is the average energy of radiated axions and A ≡ ρ wall t/σ wall /similarequal 0 . 5 [34] is the area parameter of domain walls. The radiated R-axions become eventually nonrelativistic and hence we can evaluate the energy-to-entropy ratio as Noting that the logarithmic factor is evaluated as ln(1 /m a √ ξd string ) /similarequal ln( √ λ/ξ ( f a /m a )) = 30 for f a /similarequal 10 10 GeV and m a /similarequal 1 MeV, hereafter we approximate the R-axion abundance from R-axion dynamics, i.e. , the coherent oscillation, the decay of cosmic string loops, and the decay of the string-wall system, where K 1 /similarequal O (1) and K 2 /similarequal O (10) are numerical parameters.", "pages": [ 12, 13 ] }, { "title": "D. R-axion production from thermal bath", "content": "We have estimated the abundance of R-axions generated from their dynamics. We should also take into account that generated from other sources. Here we evaluate the R-axion abundance from thermal bath. The R-axion abundance from R-saxion decay is discussed in Appendix B and is generally negligible. R-axions are produced in the thermal plasma from (mainly) gluon scattering, gg → ag . Since the gluon-axion interaction comes from the anomaly term, with C g being the model dependent anomalous coefficient and g s being the strong gauge coupling, the R-axion abundance is calculated as [41-43], Note that R-axions are thermalized once if the reheating temperature is high enough, where T D is R-axion decoupling temperature. In this case, the R-axion abundance is evaluated as Note that R-axion is produced thermally only if T R /greaterorsimilar m a is satisfied. As a result, the total R-axion abundance in the early Universe is evaluated by the sum of these contributions and given by In Fig. 2, we show the theoretical predictions for the R-axion to entropy ratio with f a = 10 6 GeV, 10 8 GeV, 10 10 GeV, and 10 12 GeV. Here the solid lines represent contribution from the thermal production (Eqs. (38) and (40)) and dashed ones represent the R-axion dynamics (Eq. (36)) with K 1 = 1, K 2 = 20, respectively. Black, blue, green and red lines correspond to T R = 10 -2 GeV, 1GeV, 10 3 GeV, and 10 6 GeV, respectively. In the case of T R > T D , R-axion abundance from thermal production is independent of T R . We can see that the contribution from the R-string dynamics and other R-axion dynamics generally dominates for f a /greaterorsimilar 10 12 GeV and smaller m a . Vice versa, thermal R-axion production dominates for f a /lessorsimilar 10 12 GeV. Anyway, we will compare the total R-axion abundance expressed in Eq. (41), including those from R-string dynamics and thermally produced ones, to the cosmological constraints discussed in the next section and will give the constraints on the model parameters.", "pages": [ 13, 14 ] }, { "title": "IV. COSMOLOGICAL CONSTRAINTS FROM R-AXION", "content": "Now we consider the generic constraints of the R-symmetry breaking model from cosmology. One may think that the model with long-lived R-axions is safe if they never dominate the energy density of the Universe or R-axions are responsible for the dark matter in the present Universe. However, even if they are subdominant component of the Universe, their (partial) decay is constrained by several cosmic/astrophysical observations depending on their abundance [44]. Since we have evaluated the R-axion abundance and its lifetime, we can constrain the model from various observations. As we will see, strong constraints for the model parameters are imposed.", "pages": [ 14, 15 ] }, { "title": "A. Cosmological constraints on axion abundance", "content": "Let us see the various constraints of R-axion abundance from cosmology and astrophysical observations. We will compare all these constraints on the R-axion abundance to that evaluated in the previous section, especially in Eq. (41) and translate them in the constraints on the R-axion model parameters in the next subsection. Note that our cosmological constraints are basically irrelevant to what is the dominant source of R-axions, but relevant to the total R-axion abundance in Eq.(41).", "pages": [ 15 ] }, { "title": "1. Big Bang Nucleosynthesis", "content": "The R-axion decay into photon or electron (radiative decay) after the Big Bang Nucleosynthesis (BBN) epoch may break the light elements and the R-axion abundance is constrained [45]. The radiative decay of R-axion causes photo-dissociation process of light elements and changes the light elements abundance. We can read off the constraint on the R-axion abundance at its decay from Ref. [45] as where B r is the radiative branching ratio 5 . Note that this effect is negligible if the energy of the injected photons is so small that they cannot destroy the light elements. Thus, we here impose a condition for this constraint to be effective, which corresponds to the threshold energy for the deuteron destruction process, D + γ → n + p .", "pages": [ 16 ] }, { "title": "2. Cosmic microwave background distortion", "content": "The radiative decay of R-axion before the recombination may distort the blackbody spectrum of CMB. After the double-Compton scattering freezes out at t /similarequal 10 6 s, energy injections generate nonzero chemical potential µ of the CMB spectrum, which imposes the constraint from the blackbody spectrum distortion of CMB. Energy injections after t /similarequal 10 9 s, when the Compton scattering is no longer in thermal equilibrium, thermalize electron, which causes the Sunyaev-Zel'dovich (SZ) effect. Since the SZ effect is constrained by the Compton y -parameter, we can impose a constraint on the R-axion abundance. The COBE FIRAS measurement [46] constrains the CMB distortion as Since the injected energy is related to these parameters as [47, 48] the constraints on the R-axion abundance is given by depending on its life time. Note that µ and y -parameters impose almost the same constraint on the R-axion abundance at its decay.", "pages": [ 16, 17 ] }, { "title": "3. Diffuse X-ray and γ -ray background", "content": "The R-axion decay to photons after recombination, t > 10 13 s, may be constrained from the diffuse X-ray and γ -ray background observation. Photons with energy 1keV < E γ < 1TeV rarely scatter with the CMB photons and intergalactic medium. Therefore, the photons produced from the R-axion decay in the 'transparency window' [49], propagate through the Universe and can be detected as diffuse background. The flux of the extragalactic diffuse photons is roughly given by Here we applied the observational results of ASCA [50] for 0.25-10 keV, HEAO [51] for 25 keV- 800 keV, COMPTEL [52] for 800 keV-30 MeV, EGRET [53] for 30 - 100 MeV, and Fermi [54] for 100 MeV-100 GeV. Note that we have taken into account the resolved source of diffuse X-ray background [55, 56] and used the fitting formula derived in Ref. [57]. The flux of photons produced from the R-axion decay can be approximated as  where the subscriptions '0' and 'dec' indicate that the parameter or variable is evaluated at the present and the R-axion decay time, respectively, and B γ is the branching ratio to photons. Note that the energy of photons should be evaluated at E = m a / 2 for τ a > t 0 and E = (3 H 0 τ a √ Ω m / 2) 2 / 3 ( m a / 2) for τ a < t 0 , taking into account of the redshift of the photons. Then, the abundance of the R-axions are constrained from the constraint F γ ( E ) < F obs γ as 6 where h ≡ H 0 / (100 km sec -1 Mpc -1 ) and H 0 is the present Hubble parameter.", "pages": [ 17, 18 ] }, { "title": "4. Reionization", "content": "The radiative decay of R-axion after recombination is also constrained from reionization. If the energy of injected photons is relatively small, they are redshifted and interact with intergalactic medium. Then, the intergalactic medium is partially ionized and the R-axion decay is regarded as an additional source of reionization. To be consistent with the observation of the optical depth to the last scattering surface, the R-axion abundance should be small enough. Assuming that the one-third of the energy of photons produced from R-axion decay that leaves the transparency window is converted to the ionization of the intergalactic medium, the R-axion abundance can be constrained from the inequality in Ref. [49, 58], where and x ≡ log 10 (Γ / s -1 ) = -log 10 ( τ a / s). Here Ω b denotes the present density parameter of the baryonic matter. This constraint is complementary to that from the diffuse X-ray and γ -ray background.", "pages": [ 18, 19 ] }, { "title": "5. Dark matter abundance", "content": "If the lifetime of R-axions is longer than the present time t 0 , most of R-axions remain the present Universe and contribute to the dark matter of the Universe. Thus, we can constrain the R-axion abundance in order not to exceed that of the dark matter. In terms of the energy-to-entropy ratio, the R-axion abundance is constrained as [59]", "pages": [ 19 ] }, { "title": "B. Constraints on model parameters", "content": "Now we are ready to show cosmological constraints for spontaneous R-symmetry breaking models. In Fig.3, we show the constraints on the model parameters, m a and f a coming from various conditions argued in the previous subsection. Each colored region is excluded and white region is allowed. As a reference, we also show lines of gravitino mass. Upper dotted lines and lower ones represent m 3 / 2 = 1keV, m 3 / 2 = 1eV, respectively. Here we focus on the region f a > 10 6 GeV since smaller f a is forbidden from laboratory experiments such as rare decays of K + or B 0 [60]. For the higher reheating temperature, T R /greaterorsimilar 10 2 GeV, all the parameter space where R-axions decay at t > 10 6 sec is ruled out regardless of reheating temperature, which comes from the CMB constraint. For m a < 1 MeV, it corresponds to f a /lessorsimilar 10 7 GeV( f a / 1MeV) 3 / 2 , and 1MeV < m a < 4 . 5 MeV, it corresponds to f a < 10 9 . 5 GeV( f a / 1MeV) 1 / 2 . For m a > 4 . 5 MeV, the BBN constraint opens and all the parameter space where the R-axion lifetime is t > 10 4 sec is ruled out, again, regardless of reheating temeperature. For 4 . 5MeV < m a < 200 MeV, it corresponds to f a /lessorsimilar 10 9 GeV( f a / 10MeV) 1 / 2 , and for m a > 200 MeV, it corresponds to f a < 10 12 GeV( f a / 200MeV) 1 / 2 . This is because R-axions are inevitably produced so much that cannot pass any constraints discussed above, especially, the BBN and CMB constraints. Short-lived R-axion is allowed because any entropy production before is not forbidden and there are no cosmological constraints. On the other hand, for the smaller reheating temperature, T R /lessorsimilar 10 2 GeV, several param- eter space where R-axions decay later is allowed. This can be understood from Eqs. (36) and (38). For larger f a , nonthermal production is dominant and the R-axion abundance is expressed as ρ a /s ∝ f 2 a , whereas thermal production, which depends on f a as ρ a ∝ f -2 a , dominates for smaller f a . Thus, the R-axion abundance takes its lower value at f a ∼ 10 11 -12 GeV. As a result, allowed parameter region appears at m a ∼ 10 keV and 1-100 MeV for f a ∼ 10 12 GeV and T R ∼ 10 -2 -1 GeV. One may regard that there is a parameter region where R-axions can be dark matter for smaller reheating temperature, f a ∼ 10 14 GeV. However, in this parameter region, R-saxion mass is considerably small, m s /lessorsimilar 1 MeV for a naive model discussed in Sec. II. For the point of view of vacuum selection and R-string stability [29], this requires unacceptably small messenger mass. Therefore, we conclude that an ingenious model building is necessary for R-axions to be dark matter of the present Universe. Thus far, we did not take into account constraints from R-axinos or gravitinos. Gravitinos are produced from gluino scattering in thermal plasma and their abundance is evaluated as [45, 61], where m ˜ g is the gaugino mass. Since gravitinos are stable in this case, gravitino abundance is constrained from dark matter abundance (Eq. (54)). As a result, depending on the gaugino mass, another stringent constraint is imposed for model parameters in higher reheating temperature case T R /greaterorsimilar 10 GeV: Smaller f a and m a would be forbidden. In summary, we have shown that spontaneous R-symmetry breaking models are severely constrained from cosmological considerations and generally long-lived R-axions are forbidden. In order to avoid that, careful model building and smaller reheating temperature are required.", "pages": [ 19, 20, 21 ] }, { "title": "V. DISCUSSION", "content": "We have studied general cosmological constraints on spontaneous R-symmetry breaking models. We estimated the abundance of R-axion produced firstly via their dynamics such as coherent oscillation and decay of cosmic string/wall system, and secondly via thermal scattering process from gluon-axion interaction. It is interesting that R-axion production from R-string and wall systems are large enough and can be dominant in some parameter region. Basically the models were motivated by gauge mediation, gravitino as well as Raxion are relatively light. Therefore, R-axion tends to be long-lived. The conditions for the R-axion density coming from BBN, X-ray/ γ -ray background, reionization and overclosure severely constrain the scale of R-symmetry breaking. As a result, smaller R-symmetry breaking scale and SUSY-breaking scale are disfavored from cosmological constraints. In the point of view of gauge mediation, this result weakens its motivation, but is consistent with the recent LHC results with 125 GeV Higgs-like boson and without SUSY particles [2]. It would be interesting to study further constraints for R-axion with relatively large mass. When the R-axion mass is larger than 1 GeV, various decay channels to hadronic particle open. We expect that weaker but non-negligible constraints for large decay constant will be imposed, thought analysis would become involved. A phenomenologically viable model with long-lived R-axions can be constructed by introducing a mechanism diluting the R-axion density in the early universe. Although our primary interest was gauge mediation models, it would be easy to apply our analysis the closely related situations such as spontaneous R-symmetry breaking in thermal inflation models [62]. In this paper, we have not explicitly shown a mechanism of vacuum selection of false vacuum. Existence of the R-string and walls are highly depend on the scenario of the early stage of universe. Also, as mentioned in the Introduction, imhomogenious vacuum decay by impurities such as a cosmic string depends on the details of the scenario [29]. So it may be useful to show an explicit example of full scenario and study R-axion cosmology in detail. This is beyond the scope of our study, so we will leave it as a future work.", "pages": [ 21, 22 ] }, { "title": "Acknowledgment", "content": "The authors would like to thank M. Eto for the early stage of collaboration, and T. Hiramatsu, Y. Inoue, K. Nakayama, A. Ogasahara, A. Ringwald, and T. Takahashi for useful comments and discussions. KK would like to thank Kyoto University for their hospitality where this work was at the early stage. TK is supported in part by the Grant-in-Aid for the Global COE Program 'The Next Generation of Physics, Spun from Universality and Emergence' and the JSPS Grant-in-Aid for Scientific Research (A) No. 22244030 from the Ministry of Education, Culture,Sports, Science and Technology of Japan. YO's research is supported by The Hakubi Center for Advanced Research, Kyoto University.", "pages": [ 22, 23 ] }, { "title": "Appendix A: Higgs sector", "content": "In this appendix, we show the Higgs sector and its mixing with the R-axion following Ref. [32]. Here, we concentrate on the neutral components, H 0 u and H 0 d , of the Higgs sector in the minimal supersymmetric standard model. The R-axion appears through the so-called B-term. Then, the relevant terms in their scalar potential are given by where µ is the supersymmetric mass, originated from the µ term, m 2 H u and m 2 H d are soft SUSY breaking masses squared for H u and H d , and Bµ is the SUSY breaking B-term. The B-term has an R-charge, and the axion appears there. At the potential minimum, the Higgs fields develop their vacuum expectation values and the electroweak symmetry is broken. Around the vacuum, we decompose the neutral Higgs fields as where v u and v d are VEVs of H 0 u and H 0 d . We denote v 2 = v 2 u + v 2 d , which is related to the Z -boson mass m Z as v 2 = 4 m 2 Z / ( g 2 + g ' 2 ) = (246) 2 (GeV) 2 . Also we denote their ratio as Furthermore, the stationary conditions, at H 0 u = v u and H 0 d = v d , lead to the following relations: Using them, the mixing mass matrix of the axial parts, ξ u,d and the R-axion is given by Then, the mass eigenstates are given by where G 0 and ˜ a denote the would-be Nambu-Goldstone boson and low-energy R-axion, respectively.", "pages": [ 23, 24 ] }, { "title": "Appendix B: R-axion production from R-saxion decay", "content": "Here we estimate the R-axion abundance from the R-saxion decay and show that it is smaller than those from R-axion dynamics. Since we have assumed that the R-symmetry is restored in the early Universe, there should be homogeneous R-saxion oscillation associated with the spontaneous breaking of R-symmetry. It can take place when the Hubble parameter becomes smaller than the saxion mass. Here we assume that the R-symmetry is broken at H = m s . If R-saxions receive thermal mass, the Hubble parameter at the time of phase transition becomes lower, but it requires large reheating temperature and we do not consider it here. The energy density of R-saxion is given by where the subscript 'so' indicates that the parameter or variable is evaluated at the onset of R-saxion oscillation. The energy density of R-saxion oscillation decreases as ρ s, osc ∝ a -3 due to the Hubble expansion, and gradually R-saxion decays into R-axions at H sd = Γ sax . Here the subscript 'sd' represents the parameter or variable is evaluated at R-saxion decay. The number density of R-axions from saxion decay, then, is evaluated as  where s ( T ) = (2 πg ∗ s ( T ) / 45) T 3 is the entropy density. Such R-axions are relativistic at R-saxion decay and loose their energy due to the cosmic expansion. After some time, they become nonrelativistic. The energy-to-entropy ratio, ρ a, sax /s , is fixed at that time (if reheating is completed.) Therefore, we can estimate the abundance of R-axions from R-saxion decay as  with T so = ( π 2 g ∗ ( T so ) / 90) 1 / 4 ( m s M pl ) 1 / 2 . Here we neglected the interaction of R-axion and assumed that the number of R-axions in a comoving volume is conserved. We can easily show that the abundance of R-axions from R-saxion decay is always smaller than that from R-string and R-string-wall system. Note that if the phase transition is driven by thermal potential, R-axion abundance from R-saxion decay is larger than that is estimated above. However, as noted, it requires high reheating temperature, in which the model has already been constrained by the thermal contribution strictly. As a result, the conclusion does not change. Phys. Lett. B 719 , 148 (2013) [arXiv:1211.4676 [hep-ph]]; M. Endo, K. Hamaguchi, S. Iwamoto and N. Yokozaki, JHEP 1206 , 060 (2012) [arXiv:1202.2751 [hep-ph]].", "pages": [ 24, 25, 26 ] } ]

[ { "title": "Jon Emery, Gianmassimo Tasinato and David Wands", "content": "Institute of Cosmology & Gravitation, University of Portsmouth, Dennis Sciama Building, Portsmouth, PO1 3FX, United Kingdom E-mail: [email protected], [email protected], [email protected] Abstract. We study a model of multiple-field DBI inflation leading to mixed form of primordial non-Gaussianity, including equilateral and local bispectrum shapes. We present a general formalism based on the Hamilton-Jacobi approach, allowing us to go beyond slow-roll, combining the three-point function for the fields at Hubble-exit with the non-linear evolution of super-Hubble scales. We are able to obtain analytic results by taking a separable Ansatz for the Hubble rate. We find general expressions for both the equilateral and local type nonGaussianity parameter f NL . The equilateral non-Gaussianity includes the usual enhancement for small sound speeds, but multiplied by an analytic factor which can lead to a suppression. We illustrate our results with two scenarios. In the first model, previously found to have detectable local non-Gaussianity, we find that the equilateral signal is not sufficiently suppressed to evade current observational bounds. In our second scenario we construct a model which exhibits both a detectable equilateral f NL and a negative local f NL . Keywords: Cosmology, Inflation, Non-Gaussianity", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Inflation is widely believed to be responsible for the specific set of initial conditions on which the hot big bang relies. Whilst a compelling mechanism, a consistent model proves elusive however (see [1-3] for reviews), which is in part due to the limited information available in the two-point statistics of the primordial density perturbations. Potential non-Gaussian signatures have therefore become an increasingly popular observable with which to discern between otherwise degenerate models [4], particularly since impending observations are set to improve by at least an order of magnitude [5]. It is therefore important to try to understand the correspondence between inflationary dynamics and the different forms of non-Gaussianity (see [6, 7] for recent reviews). Non-Gaussianity can be produced by inflation in a number of distinct ways. For example, by converting between entropy and adiabatic modes during 1 multiple-field inflation [15-17], the curvature perturbation ζ can evolve on super-horizon scales [18, 19]. Such nonlinearities can in principle produce local-type non-Gaussianity [20-42], where the effect is associated with a turn in the trajectory and is often enhanced by violations of slow roll. Alternatively, single field models with non-standard kinetic terms provide an alternative source of non-Gaussianity (see [43, 44] and references therein). Often motivated by string theory, the models we are concerned with have a characteristic sound speed c s , where c s = 1 in the canonical case. 2 As a result, equilateral non-Gaussianity can be produced by the interactions of quantum fields on sub-horizon scales. This is the case in Dirac-Born-Infeld (DBI) inflation [46, 47], in which a probe D-brane moving along the radial direction of a warped throat drives inflation. More generally however, it is expected that both equilateral and local contributions will be relevant in models characterised by non-standard kinetic terms and multiple-field dynamics [48-60]. For example, in previous work [59] we studied a multiple-DBI model, akin to that of [56-58], as a concrete example of multi-component inflation with non-standard kinetic terms. Using the δN formalism, we tracked the super-horizon evolution of perturbations using the field fluctuations at horizon exit and the subsequent background trajectory. With the adoption of a sum separable Hubble parameter, as in [29], we were able to treat the two-field case both analytically and beyond slow variation to calculate the local contribution. Moreover, by considering inflation in the tip regions of two warped throats, we illustrated that rapidly varying sound speeds can produce large local type non-Gaussianity during a turn in the trajectory. Our previous work did not include the equilateral contribution produced on sub-horizon scales however, which is what we intend to address in this paper. Whilst this contribution is dominant in the single field case, the introduction of multiple fields can alter this conclusion through the conversion of entropy and adiabatic modes (see, for example, [48-50]). In this paper we again consider the multiple-DBI model as in [59] and compute the full third order action using the Arnowitt-Deser-Misner (ADM) formalism [61]. After considering the leading contributions in slow variation and small sound speeds, we calculate the three point function for the field fluctuations at horizon exit using the path integral approach [62]. Thereafter, we implement the δN formalism and assume a separable Hubble parameter as in our previous work to calculate, fully analytically, the combined local and equilateral contributions to the bispectrum of the curvature perturbation, giving one of the few explicit examples of models characterised by both contributions (see [60, 63] for alternatives). Finally, as a first step towards assessing the viability of such a signal, we apply our results to two specific cases. The outline of the paper is as follows. We begin in section 2 by briefly reviewing the multiple-DBI model and introducing and reformulating the relevant quantities using the δN formalism. Thereafter, we use the path integral method to calculate the three point function of field fluctuations at horizon crossing in section 3, having first derived the third order action for this scenario. We then use the δN formalism to present the corresponding equilateral non-linearity parameter. In section 4 we assume a separable Hubble parameter to combine this result with the local contribution found in our previous work, giving the total three point function for the curvature perturbation. We briefly assess the feasibility of such a signal by studying two specific examples in section 5. Finally, we conclude in section 6. Throughout this paper we use the ( -, + , + , +) metric signature and set M P = c = 1, where M P = 1 √ 8 πG is the reduced Planck mass. Capital latin indices label scalar fields and any summation is explicit. Greek indices label space-time co-ordinates whilst lower case latin indices label spatial co-ordinates only, where the Einstein summation convention is adopted. Finally, commas denote partial derivatives and over-dots represent derivatives with respect to cosmic time.", "pages": [ 2, 3 ] }, { "title": "2 Multiple-DBI inflation and non-Gaussianity", "content": "We begin this section by briefly reviewing the multiple-DBI model and the relevant observational quantities, paying particular attention to non-Gaussianity. Thereafter, we use the δN formalism to re-write these expressions and investigate their evolution on super-horizon scales. This section is intended to be relatively brief, since full details can be found in [59].", "pages": [ 3 ] }, { "title": "2.1 Background evolution in multiple-DBI", "content": "Multiple-DBI inflation is encompassed by the following action where P I is a function of the single scalar field φ I and kinetic function X I = -1 2 g µν φ I,µ φ I,ν , whilst the potential V is a function of the set of scalar fields φ = { φ 1 , φ 2 , ..., φ n } . We model inflation through n probe D3 branes descending n distinct warped throats glued to a compact Calabi-Yau manifold in type IIB string theory. 3 First considered by [56-58] to investigate the effect of multiple sound horizons on equilateral type non-Gaussianity, the corresponding expression for P I is where f ( I ) parameterises the warped brane tension of throat I and is a function of φ I only. The corresponding homogeneous equations of motion are given by Here c ( I ) is defined as the sound speed associated with the field I and we have used (2.3) to eliminate X I such that f ( I ) remains a function of φ I whilst H , c ( I ) , V and ˙ φ I generally depend on the collection of fields φ . Notice that the above equations of motion are written in Hamilton-Jacobi form [65, 66] in which the Hubble parameter is written as a function of the scalar fields, taking precedence over the potential. This is more suited to the case of non-trivial sound speeds and will allow us to consider departures from slow variation. To define slow variation we introduce the following parameters where we require /epsilon1 < 1 for inflation. Slow variation is defined as /epsilon1 ( I ) , η ( I ) , s ( I ) /lessmuch 1 and we shall state explicitly when this additional restriction is required.", "pages": [ 3, 4 ] }, { "title": "2.2 Perturbations and non-Gaussianity", "content": "We characterise the scalar degree of freedom in the primordial density perturbations by introducing the primordial curvature perturbation on uniform density hypersurfaces ζ ( t, x i ) (see [67, 68] for explicit definitions). The two and three-point correlation functions then define the power spectrum P ζ and bispectrum B ζ respectively where ζ k is the Fourier transform of ζ , k i are comoving wavevectors and δ 3 is the three dimensional Dirac delta function. For a Gaussian ζ the two-point function completely defines the statistics of the field. Signatures of non-Gaussianity are encoded in the connected contributions to higher order correlators, such as the three-point function. To parameterise the deviation from Gaussianity we introduce the k -dependent non-linearity parameter 4 f NL , given by the ratio of the bispectrum to a combination of power spectra By assuming a scale invariant dimensionless power spectrum P ζ = k 3 2 π 2 P ζ ( k ) the above can be written as We now adopt the δN formalism [18, 70-73] to evolve ζ on super-horizon scales using only the field fluctuations at horizon exit and the homogeneous field evolution thereafter. To facilitate this we make two restrictions on the background dynamics. First we demand that the sound speeds are comparable whilst observable scales exit during inflation, such that c ( I ) /similarequal c /star for all I during this interval. 5 Horizon exit 6 therefore equates to evaluating a quantity when c /star k = a /star H /star . To simplify the spectrum of field fluctuations we also assume slow variation during horizon exit. Given these restrictions, we use the separate Universe approach [18, 71, 72, 74] to identify the curvature perturbation ζ with the difference in the number of e-folds between the perturbed ( N ) and homogeneous background ( N 0 ) universes, evaluated between an initially flat hypersurface t /star (e.g. shortly after horizon exit) and a final uniform density hypersurface t f (e.g. early in the radiation dominated epoch). This allows us to calculate the relevant quantities (e.g. f NL ) at time t f given the field fluctuations at time t /star and the homogeneous field evolution between these times. For example, the dimensionless power spectrum can be written as [73, 75] where N ,I is with respect to the field I at horizon exit. Here we have defined the dimensionless power spectrum of scalar field fluctuations at horizon exit using the two point function where we use slow variation at horizon exit and δ IJ is the kronecker delta symbol. Similarly the three-point function is given by where in this case /star denotes a convolution and 'perms' denotes cyclic permutations over the momenta. Neglecting the connected part of the four-point function and using Wick's theorem to rewrite the four-point functions as products of two-point functions, the latter term can be written as [73, 75] such that the bispectrum becomes Inspection of f NL ( k 1 , k 2 , k 3 ) = f (3) NL ( k 1 , k 2 , k 3 ) + f (4) NL shows that there are two distinct contributions to the bispectrum. Adopting the notation of [21], the k -independent parameter 7 f (4) NL is due to non-linear behaviour in ζ on super-horizon scales and is referred to as the local contribution, given by The result of our previous work was to provide an analytic expression for this parameter in the subset of cases described by a sum-separable Hubble parameter, in which the derivatives N ,I and N ,IJ can be fully evaluated. We neglected the contribution from the k -dependent parameter f (3) NL ( k 1 , k 2 , k 3 ) however, which is due to the intrinsic non-Gaussianity of the δφ I , produced by quantum field interactions on sub-horizon scales. The aim of this paper then is to explicitly calculate the equilateral contribution and so arrive at the total expression for the bispectrum. Inspection of (2.13) shows that this requires two distinct steps. In the following section we use the path integral method to first calculate the three point function of field fluctuations at horizon crossing. Thereafter we use the δN formalism to find the equilateral non-linearity parameter of the curvature perturbation in this scenario.", "pages": [ 5, 6, 7 ] }, { "title": "3 The equilateral contribution", "content": "In this section we use a standard prescription to calculate the three point function of field fluctuations at horizon crossing, analogous to calculations in [49, 50, 62, 78]. We begin by presenting the third order action of field fluctuations for a more general scenario in the spatially flat gauge, before restricting ourselves to slow variation and small sound speeds around horizon exit in the multiple-DBI case. Thereafter, we use this result and the path integral formalism to find the three point function of field fluctuations at horizon exit and in turn an expression for f (3) NL .", "pages": [ 7 ] }, { "title": "3.1 The third order action", "content": "To calculate the three point function of field fluctuations we begin with the general action (2.1). For a spatially flat Friedman-Robertson-Walker (FRW) Universe, the background equations of motion are given by which here we write in the conventional form, as opposed to the Hamilton-Jacobi form in section 2.1. The Klein-Gordon equation, which is not independent of (3.1) and (3.2), is given by Progressing to perturbations about the homogeneous background, we construct the third order action by recasting (2.1) using the Arnowitt-Deser-Misner (ADM) formalism [61]. This will be useful since the lapse function N and shift vector N i become Lagrange multipliers under variation. This, along with an appropriate choice of gauge, will simplify the task of isolating the physical degrees of freedom when we consider perturbations. Until then, however, we stress that the equations remain exact with no choice of gauge. The ADM metric is given by where h ij is the spatial 3-metric. In terms of this metric, the action (2.1) and kinetic term become where K = K i i , R (3) is the three dimensional Ricci scalar and indices are raised and lowered using the spatial metric. K ij is the extrinsic curvature, given by where | i denotes the covariant derivative with respect to the spatial metric. To derive the energy and momentum constraint equations we vary the ADM action (3.5) with respect to the lapse function N and shift vector N i respectively where for notational convenience we have introduced v I = ˙ φ I -N i φ I,i . To solve the energy and momentum constraints we consider a first order 8 expansion of the inhomogeneous quantities about a spatially flat FRW background where we consider scalar perturbations only and δ ij is the Kronecker delta symbol. This presents n +4 scalar degrees of freedom: δφ I , α, β, ψ and E . We can eliminate two degrees of freedom by adopting the spatially flat gauge, whereby ψ = 0 and E = 0, such that leaving n + 2 scalar degrees of freedom. Note that for notational convenience we drop the overbar on homogeneous quantities for the remainder of this section. To eliminate two further degrees of freedom we substitute the above into the constraint equations (3.8) and (3.9), giving algebraic equations for α and β | i | i The above can be substituted back into the action (3.5) expanded to the desired order and, after removing total derivatives and using the background equations of motion, yields the perturbed action in terms of the n physical degree of freedom δφ I . For simplicity we begin with the second order action where the effective mass matrix is given by: which can be used to evaluate the two point function of field fluctuations, as in (2.12), using the standard prescription. We then find, after some lengthy calculations, the corresponding third order action (see [49, 50, 62] for analogous calculations) where for notational convenience we have introduced ˜ X I = ˙ φ I ( δ ˙ φ I -˙ φ I α ) and we note again that only the first order energy (3.12) and momentum (3.13) constraints are required. The above results are consistent with an analogous calculation by [50], who consider a slightly more general action where P in Eq. (3.15) is a function of the kinetic functions X IJ = -1 2 g µν φ I,µ φ J,ν and the scalar fields φ = { φ 1 , φ 2 , ..., φ n } . Note that where we use a -2 δ ij to raise spatial indices, [50] use δ ij only, which accounts for the additional factors of a -2 and a -4 in the latter's results. The two sets of expressions are identical when the metric is written explicitly. We now consider the leading contributions to the action (3.16) in slow variation and small sound speeds, since the dominant contribution in the following path integrals will be around horizon exit. Neglecting the purely gravitational part of the action and following the arguments regarding the more general versions of slow variation in [49, 50] we find Consider now the multiple-DBI scenario, described by the Lagrangian (2.2), which we repeat for reference It is then straight forward to compute the following derivatives Upon substitution into (3.17) and keeping only terms at leading order small sound speeds, we arrive at This is the third-order action for field fluctuations in the multi-DBI scenario to leading order in slow variation and small sound speeds, which is justified around horizon exit.", "pages": [ 7, 8, 9, 10, 11 ] }, { "title": "3.2 The path integral formalism", "content": "With the action (3.20) we are now in a position to calculate the three point function of field fluctuations at horizon crossing to leading order in slow variation and small sound speeds. To this end we adopt the path integral technique and, for brevity, we refer to [62] for a clear and detailed description of this method. We first require some standard results, the first being the propagator and its time derivative. The following is easily obtained from the second order theory (3.15) assuming slow variation at horizon exit, in exactly the same way as the power spectrum (2.12) where τ represents conformal time. Indeed, by considering equal times τ 1 = τ 2 in the superhorizon limit | c I k 1 τ | /lessmuch 1, the above yields exactly the definition of the dimensionless power spectrum (2.12). In addition we require the following time integrals, which can be obtained by choosing the appropriate contour in the complex plane (analogous integrals appear in [48]) Given the above we are now in a position to use the standard prescription, as in [62], to find the following contributions to 〈 δφ I k 1 δφ J k 2 δφ K k 3 〉 from the first and second terms in (3.20) respectively where terms in the above are to be evaluated at horizon crossing, since this is the dominant contribution to the relevant time integrals. Note that there is a sign ambiguity in the above from using (2.6) to write H ,I in terms of /epsilon1 ( I ) . The above are valid provided we assume ˙ φ I < 0, which will be the case in the scenarios we consider. Finally then, we sum these contributions to arrive at the three point function of field fluctuations at horizon exit, to leading order in slow variation and small sound speeds where the k -dependent parameter Λ is given by It is then trivial to check that the above expressions recover the single field result [79].", "pages": [ 12, 13 ] }, { "title": "3.3 The equilateral non-linearity parameter", "content": "Given the expression for the three point function of the field fluctuations at horizon exit (3.26), it is then straight forward to find the corresponding contribution to the three point function of ζ using the δN formalism. To this end, we substitute the result (3.26) into the first term in (2.13) and consider the two field scenario with fields φ and χ where the subscript 'eq' denotes that we are considering only the equilateral contribution. Comparison of the above with the definition of the bispectrum (2.8) gives, where we have used (2.11) to replace P /star with P ζ . Finally then, the terms in parenthesis can be associated with the non-linearity parameter f (3) NL by inspection of (2.15) The above expression for f (3) NL is the main result of this section and, in the absence of additional dynamical restrictions to evaluate the derivatives of N , cannot be developed further analytically. The k -dependence is unchanged compared to that of the single field scenario [79], such that this contribution does indeed peak in the equilateral limit k 1 = k 2 = k 3 . By considering ˙ χ → 0, such that N ,χ → 0 and N ,φ → H ˙ φ , and using the background equations of motion (2.3), it can be shown that the final terms becomes equal to one, recovering the single field result. We notice then that the above has the form of the single field result, which is precluded observationally by the strong c -2 /star dependence, modulated by an expression dependent on the background evolution after horizon exit. It seems possible then that, in some circumstances, this modulation may suppress the value of f (3) NL to remain within observational bounds. Such modulation has been found in similar multiple-field DBI scenarios. For example, [49] use the adiabatic-entropy perturbation basis to obtain the single field result modulated by a cos 2 Θ term, where Θ depends on the background trajectory after horizon exit. This modulation has then been exploited in concrete examples to suppress the otherwise observationally precluded value of f (3) NL , as in [60]. It is not a priori obvious if this possible in our case by inspection of (3.30) however, since the behaviour of this term is highly model dependent. We address this issue in the following section by considering scenarios in which the derivatives of N can be evaluated analytically, as we did for the local contribution in [59].", "pages": [ 13, 14 ] }, { "title": "4 The total three point correlation function", "content": "The expressions for f (3) NL (3.30) and f (4) NL (2.16) contain field derivatives of the number of efolds N . Given the lack of a unique attractor in multiple-field scenarios however, additional restrictions to the background dynamics are required to further develop such expressions analytically. For example, in our previous work [59] we adopted the method of [29] and demanded a sum separable Hubble parameter 9 . Not only did this allow the violation of slow variation after horizon exit, it also suited the case of non-standard kinetic terms, since the dynamics are better described in the Hamilton-Jacobi formalism. Given this restriction, we exploited the resultant integral of motion to derive analytic expressions for the derivatives of N and in turn f (4) NL in the multiple-DBI case. In this section we apply those results to present an analogous expression for f (3) NL which, to the best of our knowledge, is the first example of the application of the separable technique towards the equilateral contribution. To make analytical progress we now restrict our attention to two-field models, with fields φ and χ , that posses a sum separable Hubble parameter which leads to a number of simplifications. Inspection of (2.5) shows that the sound speed c ( I ) becomes a function of its respective field φ I only, such that c ( φ ) ( φ ) and c ( χ ) ( χ ). Moreover, mixed derivatives of H (i.e. H ,φχ ) become zero. The combination of the above reduces the number of relevant slow variation parameters where /epsilon1 = /epsilon1 ( φ ) + /epsilon1 ( χ ) and we emphasise again that η ( I ) and s ( I ) can become much greater than one after horizon exit. Crucially though, the above assumption enables us to calculate the field derivatives of N analytically by defining an integral of motion. Here we simply quote the results of our previous work [59], where the full details of the calculation can be found. The derivatives of N can be expressed in terms of slow variation parameters where, for brevity, we have introduced the following definitions On substitution of the above into the result (3.30) , we arrive at the following expression for f (3) NL which we emphasise is valid for the two-field DBI scenario assuming comparable small sound speeds and slow variation at horizon exit, in addition to a separable Hubble parameter. By setting ˙ χ → 0, we find Z f → H ( χ ) f , u → 1 and v → 0 such that (4.7) again recovers the single field result [79]. The above is analogous to the result (3.30), in that we find the single field result modulated by a term dependent on the background evolution after horizon exit. The advantage here however, is that the behaviour of the modulation becomes more transparent. For example, we note that this term is approximately O (1) in slow variation at horizon exit. Moreover, since neither of the terms in the numerator are positive definite it is conceivable that, with a sufficient level of cancellation, the modulation term may suppress the otherwise prohibitively large contribution of c -2 /star , providing an observationally viable value of f (3) NL at the end of inflation. It remains to be seen if this is possible in a concrete setup however, which we intend to address in the following section. Alongside the above equilateral contribution (4.7), we must also consider the local signal. Here we can directly quote the analogous expression for f (4) NL from [59] where the parameter A is defined as and we have introduced These additional parameters appear because the expression for f (4) NL (2.16) contains second derivatives of N , whilst f (3) NL (3.30) has only first. Note that, whilst the terms preceding the parenthesis in the expression for A (4.9) are O ( /epsilon1 /star ), η ss f and s ss f can become much larger than unity, producing observable local type non-Gaussianity. We demonstrated this in a concrete model in [59] by considering inflation in the tip regions of two warped throats, in which s ss f becomes enhanced by the abrupt change of c ( φ ) and c ( χ ) at the end of inflation. Equations (4.7) and (4.8) together provide the full expression for f NL , and in turn the bispectrum (2.15), to leading order in slow variation and small, comparable sound speeds at horizon exit, given a sum separable Hubble parameter. We have noted that both contributions can, in principle at least, produce contributions to the bispectrum, rendering this one of the few explicit models capable of doing so (see [60, 63] for alternatives). It remains to be seen if this is possible in practice however. Ideally, a consistency relation between the two contributions would elucidate this point further but, given the considerable freedom within the model, finding a general relation has so far proved difficult. In the absence of a consistency relation therefore, it is useful to consider some specific models, as we did in [59] to study the local contribution produced by inflation in the tip regions of the throats. We leave this for the following section and here simply highlight that this scenario does indeed provide the potential mechanism to produce a mixed non-Gaussian signal. Before proceeding to specific scenarios, we first note that whilst the expressions (4.7) and (4.8) describe the production and evolution of non-Gaussianity during inflation, these are not necessarily the final observed values. For completeness we must track their evolution from the end of inflation until they are imprinted on the cosmic microwave background (CMB) at decoupling. Given our lack of knowledge of the early Universe however, this is not generally feasible. As such, recent work has considered whether such non-Gaussianity produced during inflation can indeed imprint upon the CMB [33-40]. Such study provides valuable clues as to what we can infer about inflationary dynamics from observations of non-Gaussianity and it would be interesting to include such considerations in this scenario. Here however, we simply illustrate the potential production of mixed non-Gaussianity through multiple-field dynamics and small sound speeds during inflation.", "pages": [ 14, 15, 16, 17 ] }, { "title": "5 Illustrative Examples", "content": "In the previous section we demonstrated the possibility of producing mixed local and equilateral non-Gaussianity in the multiple-DBI scenario, through the expressions for f (3) NL (4.7) and f (4) NL (4.8). The aim of this section is to look more closely at the feasibility of producing such a signal by considering some concrete examples. We begin by revisiting the case of inflation in the tip regions of two warped throats, where previously we found enhanced local non-Gaussianity caused by the rapid increase in the sound speeds at the end of inflation. We find that the required modulation is insufficient to suppress the prohibitively large contribution of c -2 /star in (4.7) however, and leave further exploration of parameter space to future work. We then consider a phenomenologically similar but fully analytic model by choosing exponential warp factors, finding a regime in which observationally viable mixed non-Gaussianity is produced. Whilst not derived from string theory, this does provide a first step towards understanding the regimes in which this occurs, before progressing to more realistic setups.", "pages": [ 17 ] }, { "title": "5.1 Inflation in two cut-off throats", "content": "Previously, we studied the case of inflation in the tip regions of two warped throats and found enhanced local non-Gaussianity at the end of inflation, caused by a sudden increase in the sound speeds [59]. Here we revisit exactly this model and include the equilateral contribution, to study whether the background evolution is sufficient to suppress the contribution of c -2 /star in (4.7). We keep our discussion of the model brief, since full details can be found in [59]. We choose a model whereby two probe D3 branes traverse two distinct warped throats glued to a compact Calabi-Yau in type IIB string theory [56-58]. As such, the warp factors are given by where f ( χ ) is given by replacing φ → χ and for simplicity we have assumed the same warping in each throat. A full discussion of the physical significance of the λ i and the infrared singularity at φ = λ 3 e -1 /λ 2 can be found in [59]. We also choose a linear, separable Hubble parameter such that η ( φ ) = η ( χ ) = 0. Using (2.5), we arrive at the following expression for the sound speeds where c ( χ ) is again given by replacing φ → χ . We then solve the background field equations (2.3) with λ 1 = 6 × 10 16 , which is typically required in standard DBI [47], λ 2 = 2 and λ 3 = 1. Furthermore, we choose φ ( t /star ) = χ ( t /star ) = 1 as our initial conditions, where t /star is the time at which observable scales exit the horizon. We require N /similarequal 60 e-folds between t /star and the end of inflation, which we choose to define as /epsilon1 = 1. Finally we introduce a small asymmetry such that H ( φ ) = 1 . 188 × 10 -6 and H ( χ ) = 1 . 192 × 10 -6 . With this choice of parameters c ( φ ) /c ( χ ) ∼ 1+10 -3 at t /star and is therefore consistent with our approximation that the sound horizons are comparable when observable scales exit. Given this, we solve the field equations and plot the inflationary trajectory in figure 1. Whilst approximately straight for most of inflation, the trajectory finally curves in the χ direction as the sound speeds rapidly increase in the tip regions of the throats. The turn in the trajectory is due to our choice of H ( χ ) > H ( φ ) , such that c ( χ ) increases slightly before c ( φ ) towards the end of inflation. Given this trajectory, we study the evolution of the relevant quantities as a function of t f for a fixed t /star . We find that the quantities associated with the two-point statistics are consistent with observations [80], where P ζ = 2 . 44 × 10 -9 and n ζ -1 = -0 . 0108 at the end of inflation (see [59] for the full expressions of these observables in the separable Hubble approach). With respect to the three-point function, figure 2 shows the evolution of f (4) NL and f (3) NL in the equilateral configuration, calculated using (2.16) and (3.30) respectively, as a function of the slow roll parameter /epsilon1 . As discussed in [59], the rapidly increasing sound speeds in the tip of the throats produce f (4) NL /similarequal -20 at the end of inflation. Looking now at the equilateral contribution, we see that the value of c /star ∼ 10 -2 provides f (3) NL ∼ 10 4 at horizon Χ 1.0 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1.0 Φ exit. Thereafter however, the background evolution actually enhances this value between horizon exit and the end of inflation, eventually giving f (3) NL /similarequal 12700. We conclude then that this scenario is inconsistent with current observations [80]. Whilst this need not necessarily be the case for all parameter values, we leave a fuller exploration of the parameter space to future work.", "pages": [ 17, 18, 19 ] }, { "title": "5.2 Exponential warp factors", "content": "In this section we consider a phenomenologically similar, analytic model that is able to produce both equilateral and local non-Gaussianity that satisfies current constraints. Let us consider the following separable Hubble parameter We make the following choice for the warp factors All the parameters in the previous formulae are positive. This choice of warp factors is motivated by the fact that it will lead to an analytically solvable system. The sound speeds read The equation of motion for the scalar φ results where the solution, assuming φ (0) = φ /star , is while an analogous solution can be found for χ ( t ). The solution for the scalar field is a decreasing function of time t . The speeds of sound are increasing functions of time, provided that γ and δ are larger than one. The slow variation parameters read (in the limit in which A φ e -αφ /lessmuch 1) (5.14) and we note the useful relations The number of e-folds is then given by We would like to find an inflationary trajectory for which the amplitude of the nonGaussianity parameter f (3) NL can be tuned sufficiently small to satisfy present constraints. At the same time, we would like that the amplitude of the local non-Gaussianity parameter f (4) NL is sufficiently large to be detectable in the future, but still satisfying present-day constraints. Recall the expression for f (3) NL (4.7) where in the approximation /epsilon1 /star /lessmuch 1. If the speed of sound c /star /lessmuch 1, the amplitude of f (3) NL is prohibitively large, unless u is tuned in such a way that, at the end of inflation, the following inequality is satisfied Recall also the expression for f (4) NL (4.8), where the dominant contribution is given by where In order to obtain a detectable f (4) NL , we have to find situations in which either or both η ss and s ss are large at the end of inflation. Let us start discussing the conditions for being able to tune the value of f (3) NL . We assume the following hierarchy between the slow-roll parameters at horizon exit The function σ ( u ), defined in (5.22), vanishes at the point r 2 3 / ( r 2 3 -1). We then demand that the parameter u satisfies at the end of inflation, where λ small in absolute value. Expanding the function σ ( u ) at first order in λ , one finds We can therefore set σ ( u end ) to be sufficiently small to compensate the enhancement associated with the speed of sound in (4.7). Let us assume that, at the end of inflation, /epsilon1 ( φ ) is much larger than /epsilon1 ( χ ) Moreover, we write the following expressions for the speeds of sound at the end of inflation such that we can write Tuning properly the κ i then, this quantity can assume the desired value of (5.26). We now consider possible inflationary trajectories, with the specific requirements listed above, that lead to observationally viable non-Gaussianities of both local and equilateral type. We would like to determine which conditions we have to impose to the model parameters in order to satisfy all our requirements. For simplicity, we assume that the γ and δ are very large, so that we can safely neglect corrections weighted by inverse powers of these parameters. We proceed by discussing one by one the conditions that fix our parameters. As discussed in the previous sections, in all our analysis we make the hypothesis that, at horizon exit, c ( φ ) /star = c ( χ ) /star = c /star . This implies The quantity φ /star is associated with the speed of sound c /star at horizon exit by The condition (5.25) gives where we neglect corrections that scale as 1 /γ and 1 /δ since, as we assumed above, these are negligible. Condition (5.29) gives Hence, combining with (5.33), we find In the approximation H /similarequal H 0 , the slow-roll parameters at horizon exit are given by The dominant contributions to the number of e-folds depends on the terms evaluated at horizon exit and with the aid of the previous equations we find The dominant contributions to the non-Gaussian parameters in this scenario read as follows (we write only their amplitude, and not the scale dependence) Without providing an exhaustive analysis, let us consider a concrete set-up in which we assign the following numerical values to the parameters κ i to obtain a sufficient number of e-folds. With this choice, the value of u end is implying that (see (5.26)) We demand also that Hence, assuming that λ < 10 -3 (we will see that this assumption is satisfied in our set-up) (5.36) gives The condition (5.46) then provides the relations implying that δ /similarequal 10 γ . Let us consider the local non-Gaussian parameter f (4) NL : using (5.42) we find Since our parameters α and δ are positive, this quantity is always negative. Choosing for definiteness we get Substituting the information of (5.50) into (5.48), we get We are not allowed to choose too large values for α , since condition (5.50) would lead to small values of δ , against our working hypothesis. We set α = 3, that gives Hence, f (3) NL is easily obtained from (5.41) by substituting our values of the parameters: By tuning appropriately the parameter λ to a value of order 10 -7 one finds and its sign depends on the sign of λ . Hence, this semi-quantitative analysis shows that within this concrete model we can obtain relatively large non-Gaussianities both of local and equilateral type. It would be interesting to perform a more complete analysis of this set-up to determine more precisely its predictions. Moreover, since the exponential warp factors were introduced for analytical ease, it would also be prudent to use these results as a starting point for further parameter exploration of the previous example (section 5.1), which whilst more realistic is less analytically tractable.", "pages": [ 20, 21, 22, 23, 24, 25 ] }, { "title": "6 Conclusions", "content": "We analysed a setup of multiple-field DBI inflation leading to mixed form of primordial non-Gaussianity, including equilateral and local bispectrum shapes. Previously, we studied a multiple-DBI model as an example of multi-component inflation with non-standard kinetic terms, showing that rapidly varying sound speeds can produce large local type nonGaussianity during a turn in the trajectory [59]. Here we have included the equilateral contribution produced on sub-horizon scales, and found the possibility of an observationally viable mixed non-Gaussian signal. We used a general formalism based on the Hamilton-Jacobi approach, allowing us to go beyond slow-roll, combining the three-point function for the fields at Hubble-exit with the non-linear evolution of super-Hubble scales calculated using the δN formalism. We were able to obtain analytic results by taking a separable Ansatz for the Hubble rate. We found general expressions for both the equilateral and local type non-Gaussianity parameter f NL . The equilateral non-Gaussianity includes the usual enhancement for small sound speeds, but multiplied by an analytic factor which can lead to a suppression of this quantity. This modulation may suppress the value of f (3) NL to within observational bounds (see [49, 60] for analogous results). We applied our findings in two explicit scenarios. In the first model, previously found to have detectable local non-Gaussianity, we found that the equilateral signal is not sufficiently suppressed to evade current observational bounds. In our second set-up we constructed an observationally viable model which exhibits both equilateral f (3) NL and a negative local f (4) NL , providing a first step towards understanding regimes in which this can occur in more realistic scenarios. Open issues remain however, the most prevalent being a better understanding of the likelihood of such a mixed non-Gaussian signal. For example, it would be interesting to use the results of our analytical example in section 5.2 to more methodically assess the parameter space of the phenomenologically similar but more realistic scenario in section 5.1. Moreover, a consistency relation between the more general formulae for f (3) NL (4.7) and f (4) NL (4.8) would be desirable, since this would be more widely applicable. The use of a sum separable Hubble parameter, whilst providing analytically tractable expressions, also restricts the choice of potential. Thus it would be more general still to consider alternative analytical or numerical methods. Finally, we again note that our approach only treats multiple-field dynamics during inflation and we cannot necessarily conclude that such values are observed in the CMB. It would be prudent therefore to use analogous techniques to [33-40] to further our understanding of inflationary dynamics from observations of non-Gaussianity in the CMB. We conclude by noting two further directions to develop the above, in addition to increased generality. In our previous work we made a first attempt at addressing the next order statistic, the trispectrum, by plotting the evolution of the analogues of the non-linearity parameter, g NL and τ NL , in the example of inflation in two cut-off throats. In light of impending observations by Planck [5] however, it would be interesting to further characterise the model by more rigorously considering its predictions for the trispectrum. Finally, we note again that in the third order action (3.20) we worked to leading order in small sound speeds at horizon exit. It is conceivable however that the next to leading order terms may still provide a significant contribution, with the potential to generate shapes distinct from the local and equilateral types. Since tentative signals for the orthogonal shape have been detected by WMAP [81], and again given the impending results from Planck, it would be interesting to consider such contributions in future work.", "pages": [ 25, 26 ] }, { "title": "Acknowledgments", "content": "The authors would like to thank Taichi Kidani and Guido W. Pettinari for useful discussions. JE is supported by an STFC doctoral training grant ST/F007531/1. GT is supported by an STFC Advanced Fellowship ST/H005498/1. DW is supported by STFC grant ST/H002774/1.", "pages": [ 26 ] }, { "title": "References", "content": "slow-roll inflation , JCAP 0810 (2008) 008, [ arXiv:0807.1101 ].", "pages": [ 28 ] } ]

[ { "title": "Jounghun Lee a and Gerard Lemson b", "content": "E-mail: [email protected] Abstract. We utilize the Millennium-II simulation databases to study the spin bias of dark subhalos in the Local Group-like systems which have two prominent satellites with comparable masses. Selecting the group-size halos with total mass similar to that of the Local Group (LG) from the friends-of-friends halo catalog and locating their subhalos from the substructure catalog, we determine the most massive (main) and second to the most massive (submain) ones among the subhalos hosted by each selected halo. When the dimensionless spin parameter ( λ ) of each subhalo is derived from its specific angular momentum and circular velocity at virial radius, a signal of correlation is detected between the spin parameters of the subhalos and the main-tosubmain mass ratios of their host halos at z = 0: The higher main-to-submain mass ratio a host halo has, the higher mean spin parameter its subhalos have. It is also found that the correlations exist even for the subhalo progenitors at z = 0 . 5 and 1. Our interpretation of this result is that the subhalo spin bias is not a transient effect but an intrinsic property of a LG-like system with higher main-to- submain mass ratio, caused by stronger anisotropic stress in the region. A cosmological implication of our result is also discussed. Keywords: galaxy evolution, galaxy morphology, cosmological simulations", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The Local Group (LG) is a dumbbell-shaped group of galaxies which include the great Andromeda (M31), the Triangulum (M33) and the Magellanic Clouds (MC) as well as our home, the Milky Way (MW) [1]. The two centers of the dumbbell shaped LG are nothing but MW and M31 which are known to have very similar masses of ∼ 10 12 h -1 M /circledot [2, 3]. These two prominent galaxies contribute most of the total mass of LG, M LG , which has been estimated to be log[ M LG /M /circledot ] = 12 . 72 with a 2 σ range of [12 . 26 , 13 . 01] based on the accurate measurements of the distance and pair-wise speed between MW and M31 [4]. The majority of the other LG member galaxies are the satellites of either MW or M31, having orders of magnitude lower masses. It is expected that MW and M31 will eventually form a large central galaxy through a major merger between them when LG completes its virialization. An intriguing question to ask is if and how the presence of two prominent galaxies in the LG and their mutual interaction caused their satellite galaxies to possess any biased or anomalous properties compared with other typical group galaxies. Occurring rarely in the LG-like environment [5], the major merger event is one of those few mechanisms that can have a significant effect on the geometrical and physical properties of the subhalos. For instance, ref. [6] claimed that the major mergers of the M31 progenitors should be responsible for the change of the satellite hosts between M31 and MW. Ref. [7] also attributed the detected vast polar structures of the MW dwarf satellites [8] to the major mergers of the M31 progenitors [see also, 9]. Here, we suggest a scenario that the spin parameters of the LG member galaxies have higher mean value than the that of the typical group galaxies due to the high anisotropic stress in the LG site. According to the recent study of ref. [10], the evolution of the angular momentum of a galaxy in the nonlinear regime is driven primarily by the local vorticity effect. The dumbbell shape of LG and the ongoing gravitational interaction between its two prominent galaxies, MW and M31, reflects the enhanced anisotropic stress in the local region around LG, which must have originated from the external tidal effect [11]. Given that the spin parameter of a galactic halo is strongly correlated not only with its surface stellar density [24, and references therein] but also with the temperature and mass of its gas contents [15], understanding the spin parameter distributions of the LG member galaxies may provide a crucial key to explaining their physical properties as well as their evolution. The goal of this Paper is to numerically test the above scenario by analyzing the data from the high-resolution N-body simulations. The contents of the upcoming sections are outlined as follows. In section 2 we describe the data from high-resolution N-body simulations and explain the numerical analysis of the simulation data. In section 3 we present the main result on the spin bias of the dark subhalos in the LG-like systems and its redshift dependence. In section 4 we discuss a physical interpretation of our result and its cosmological implication as well.", "pages": [ 2, 3 ] }, { "title": "2 Main-to-submain mass ratios of group-size halos", "content": "For the numerical investigation we utilize the catalogs of dark matter halos and their subhalos resolved in the Millennium II simulations [16]. Under the assumption of a flat ΛCDM cosmology with the key parameters of Ω m = 0 . 25 , Ω Λ = 0 . 75 , n s = 1 . 0 , σ 8 = 0 . 9 , h = 0 . 73, the Millennium-II simulations were performed at various epochs in a periodic box of linear size 100 h -1 Mpc with 2156 3 dark matter particles each of which has mass of 6 . 89 × 10 6 h -1 M /circledot . The dark matter halos and their subhalos were identified by applying the friends-of-friends (FoF) and the subfind algorithms [17] to the particle data from the Millennium-II simulations, respectively. The full descriptions of the Millennium-II simulation and how to retrieve information from the halo catalogs are provided in refs. [16, 18], respectively. From the Millennium-II FoF catalogs at z = 0, we first select those group-size halos whose FoF masses, M h , are in the 2 σ mass range of LG [4]: 10 12 . 26 ≤ M h /M /circledot ≤ 10 13 . 01 . A total of 2079 dark halos in the Millennium-II FoF catalog at z = 0 are found to satisfy this mass constraint. For each selected group-size halo, we extract their subhalos from the Millennium-II subhalo catalog at z = 0 but consider only those well-resolved ones consisting of 200 or more dark matter particles ( N p ) for our analysis. Figure 1 plots the mean number of the well-resolved (all) subhalos versus the FoF masses of their host halos at z = 0 as solid (dashed) line. As can be seen, the mean numbers of the well-resolved subhalos with N p ≥ 200 increase monotonically with M h but do not exceed 100 in the whole range of M h . We define the main and the submain subhalos of each halo as the most massive and the second to the most massive subhalos, respectively. Then, we assign each selected halo its unique value of the main-to-submain mass-ratio, M s 2 /M s 1 , where M s 1 and M s 2 denote the masses of the main and submain subhalos, respectively. If some halo has this mass-ratio close to unity, it is similar to the Local Group, having a dumbbell shape with two centers. Figure 2 illustrates the spatial distributions of the subhalos in the projected x -y plane for the two different cases of M s 2 /M s 1 . The left panel corresponds to the case that the main-to-submain mass ratio is close to unity with two prominent subhalos of comparable masses (largest filled circles). Note that most of the other subhalos for this case seem to be the satellites of these two prominent subhalos. The right panel corresponds to the case where the main-to-submain mass ratio is much smaller than unity with one single central dominant subhalo. In each panel, the medium and small-size filled circles represent the projected positions of the subhalos other than the prominent ones with N p ≥ 200 and N p < 200, respectively. To see how rare the dumbbell-shaped systems are among the selected group-size halos, we bin the values of M s 2 /M s 1 and count the numbers of the group-size halos belonging to each bin to determine the probability density distribution of M s 2 /M s 1 , the result of which is shown in Fig. 3. As can be seen, the probability density reaches its maximum value around M s 2 /M s 1 = 0 . 02, dropping rapidly as M s 2 /M s 1 approaches unity. To see if the distances between the main and the submain subhalos depend on their mass-ratios, we also calculate the mean main-between-submain distances averaged over those hosts belonging to each bin of M s 2 /M s 1 , the result of which is plotted in Fig. 4. The horizontal dotted line corresponds to the separation distance between the MW and the M31 [4]. As can be seen, the mean distance between the main and the submain subhalos increases as M s 2 /M s 1 increases. Note also that it matches the separation distance between MW and M31 when M s 2 /M s 1 has the value around 0 . 8, which indicates that those FoF halos with M s 2 /M s 1 ≥ 0 . 8 are indeed similar to the LG. To see if the main-to-submain mass-ratio of a host halo depends on its total mass, we bin the values of M h and calculate the mean value of M s 2 /M s 1 averaged over those hosts belonging to each bin of M h . Figure 5 plots the mean value of the main-tosubmain mass ratios of the selected group-size halos as a function of its FoF mass. The errors represent one standard deviation σ r in the measurement of 〈 M s 2 /M s 1 〉 computed as σ 2 r = [ 〈 ( M s 2 /M s 1 ) 2 〉 - 〈 M s 2 /M s 1 〉 2 ] / ( N h -1) where N h denotes the number of those group-size halos belonging to each bin of M h . As can be seen in Figure 5, the mean value, 〈 M s 2 /M s 1 〉 , does not vary strongly with the total mass, M h . To see if the mass distribution of the subhalos depends on the main-to-submain mass ratios of their host halos, we bin the values of M s 2 /M s 1 and calculate the mean value of the maximum circular velocity, V max , averaged over the subhalos whose host halos belong to each bin of M s 2 /M s 1 . The Millennium-II substructure catalog provides information on V max for each subhalo, which is a good indicator of the subhalo mass. Figure 6 plots the average value of V max as a function of the main-to-submain mass ratios of their host halos. The errors represent again one standard deviation in the measurements of 〈 V max 〉 . As can be seen, there is only very weak, if any, correlation between V max and M s 2 /M s 1 with the mean value of V max around 50 s -1 km , regardless of the value of M s 2 /M s 1 .", "pages": [ 3, 4, 5, 6, 7 ] }, { "title": "3 Spin bias in the LG-like Environments", "content": "Now that the LG-like groups are found atypical in the respect that most of the groupsize halos with masses comparable to that of LG have main-to-submain halo ratios much less than unity, we would like to investigate what environmental effect the atypical LG-like systems have on their subhalos. We are particularly interested in the environmental effect on the subhalo's dimensionless spin parameter λ which is conveniently defined as λ = j/ ( √ 2 V vir R vir ) [19] where R vir represents the virial radius and V vir is the circular velocity measured at R vir , j is the magnitude of the subhalo's specific angular momentum (angular momentum per mass). For the subhalos identified by the SUBFIND algorithm, the virial radius R vir is related to the spherical radius R max at which the subhalo's circular velocity curve reaches its maximum as R vir = R max / 0 . 18 [20], while V vir , can be calculated from the virial mass M vir and the virial radius R vir as V vir = √ GM vir /R vir where G is the Newtonian constant. Using these relations along with information on j , V max and R max provided in the Millennium-II substructure catalog, we compute the dimensionless spin parameter λ of each selected subhalo. Binning the mass ratio M s 2 /M s 1 of each selected group-size halo and calculating the mean value of the spin parameters of those well-resolved subhalos whose host halos belong to each bin of M s 2 /M s 1 , we determine 〈 λ 〉 as a function of M s 2 /M s 1 , which is shown in the top panel of Fig. 7. The errors represent one standard deviation σ λ in the measurement of 〈 λ 〉 computed as σ 2 λ = [ 〈 λ 2 〉 - 〈 λ 〉 2 ] / ( N h -1) where N h denotes the number of those group-size halos belonging to each bin of M s 2 /M s 1 . As can be seen, there exists a clear signal of correlation between λ and M s 2 /M s 1 : The higher main-tosubmain mass ratio a host halo has, the higher mean spin parameters their subhalos have. Recalling that the main-to-submain mass ratio of a host halo has no mass bias (see Fig. 6) and that the subhalo spin parameters are insensitive to the subhalos' mass, we affirm that the correlation detected between λ and M s 2 /M s 1 is not due to any mass bias. The bottom panel of Fig. 7 shows one standard deviation scatter of λ (dotted line), computed as [ 〈 λ 2 〉 - 〈 λ 〉 2 ] 1 / 2 , around its mean value (solid line). Although the width of the scatter of λ is much wider than the range of the detected trend in λ with M s 2 /M s 1 , it does not necessarily mean that the correlation between λ and M s 2 /M s 1 is not meaningful since the spin parameter λ is well known to be widely scattered following the log-normal distribution [19]. The presence of the correlation between λ and M s 2 /M s 1 is important and meaningful because it implies that for the case of higher M s 2 /M s 1 the fraction of λ ≥ λ c in the log-normal tail will be larger where λ c is some threshold of the spin parameter. Now that a signal of correlation between λ and M s 2 /M s 1 is detected, it is inter- sting to examine whether or not λ also depends on M s 3 /M s 2 where M s 3 denotes the mass of the third to the most massive subhalo. We repeat the same calculation to determine 〈 λ 〉 but as a function of M s 3 /M s 2 , the result of which is shown in Fig. 8. As can be seen, the mean spin parameter of the subhalos depends weakly on M s 3 /M s 2 , reaching the maximum value at M s 3 /M s 2 ≈ 0 . 05 and decreasing as M s 3 /M s 2 increases. This result implies that the LG may be the optimal environment for the highest spin parameters: In addition to its high value of M s 2 /M s 1 ≈ 0 . 8 [2], the value of M s 3 /M s 2 of the LG is approximately 0 . 05 since the Triangulum galaxy (the third to the most massive member galaxy in the LG) has mass approximately M s 3 = 5 × 10 10 h -1 M /circledot [23]. Since it is only the central galaxies whose physical properties are known to depend on the spin parameters of their host halos [12-14], we repeat the whole calculations using only the central prominent subhalos, the result of which is shown in Fig. 9. As can be seen, we observe stronger correlation between the spin parameters of the central prominent subhalos and the main-to-submain mass ratios of their host halos. From this results, it can be inferred that the subsequent tidal stripping effect tends to reduce the strength of the correlation between the spin parameters of their subhalos and the main-to-submain mass ratios of their host halos. Given that those host halos with higher M s 2 /M s 1 are likely to be recent merger remnants, it should be worth checking whether or not the observed correlation between λ and M s 1 /M s 2 is a transient effect. Locating the progenitors of the subhalos belonging to each host halo at higher redshifts, z = 0 . 5 and z = 1, in the Millennium Merger Tree catalog, we investigate the correlations between the spin parameters of the subhalo progenitors and the main-to-submain mass ratios of their descendant hosts at z = 0. Figure 10 shows the same as Fig. 7 but for the subhalo progenitors at z = 0 . 5 and z = 1 in the top and bottom panels, respectively. The results are obtained by considering only those well resolved subhalo progenitors with N p ≥ 200. As can be seen, the mean spin parameters of the subhalo progenitors are correlated with the main-to-submain mass ratios of the descendant hosts and the strength of the correlations are similar to that observed at z = 0. Using only the central prominent subhalos, we repeat the whole calculations, the result of which is shown in Fig. 11. As can be seen, we observe stronger correlation between the spin parameters of the progenitors of the central prominent subhalos and the main-to-submain mass ratios of their descendant hosts. Noting the results shown in Fig. 10, we think that the observed trend in λ with M s 2 /M s 1 is not a mere transient phenomena due to the merging but an intrinsic effect of the anisotropic stress which increases with the main-to-submain mass ratios. Previous theoretical studies asserted that the surface stellar density of a disc galaxy is inversely proportional to the spin parameter of its host halo and that a dark galaxy whose surface stellar density falls below 100 pc -2 forms in the critically fast spinning halos whose spin parameters exceeds some threshold of λ c ≈ 0 . 06 [1214]. Very recently, ref. [24] performed a high-resolution hydrodynamic simulation to numerically confirm that the spin parameters are indeed strongly correlated with the surface stellar and gas densities of a disc galaxy. Their result revealed clearly that the halo's higher spin leads to the lower stellar and gas surface densities of its disk galaxy. To see how abundant the dark galaxies are in the LG-like systems, we calculate the number fraction of the prominent subhalos with λ ≥ λ c = 0 . 06 as a function of the main-to-submain mass ratio, the result of which is plotted in Fig. 12. As can be seen, the fraction of the fast-spinning central subhalos with λ ≥ λ c increases almost monotonically with the main-to-submain mass ratio. For the case of M s 2 /M s 1 ≥ 0 . 3, approximately 18% of the central subhalos have λ ≥ λ c while for the case of M s 2 /M s 1 ≤ 0 . 05 only 2% of the central subhalos satisfy the condition.", "pages": [ 7, 8, 9, 10, 11, 12, 13 ] }, { "title": "4 Summary and discussion", "content": "By analyzing the halo and subhalo catalogs from the Millennium-II simulations, we have detected a clear signal of correlations between the main-to-submain mass ratios of group-size halos and the spin parameters of their subhalos at present epoch. We have also found that the central prominent subhalos exhibit stronger correlations and that the correlations with similar strength exist even for the subhalo progenitors at z = 0 . 5 and z = 1. We conclude that the observed high mean spin parameters of the subhalos in the LG-like groups are not transient merger remnants but likely to be intrinsic property of the LG-like systems induced by the high anisotropic stress in the local site. An important implication of our result is that the LG member galaxies are biased toward the high spins and thus likely to have on average lower stellar and gas surface densities than the typical group galaxies [12-14, 24]. When the observed properties of the MW satellites are used to be compared with the predictions of ΛCDM cosmology the spin bias of the MW satellites should be taken into account. For instance, our result may help alleviate the tension between the observed satellite populations of the MW and the predictions of the ΛCDM cosmology [25, 26, and references therein] since the presence of more dark satellite galaxies due to their biased spins in the MW system could explain the lower abundance of the observed satellites of the MW. It is, however, worth mentioning here that the robustness of our result obtained from the Millennium-II simulations against the subhalo-finding algorithm will have to be tested further in the future before connecting it to the observed properties of galaxies. Especially the masses and spin parameters of the subhalos calculated using the number of DM particles can have large variations in their values at the consecutive time steps due to the limitation of the SUBFIND algorithm.", "pages": [ 13, 14 ] }, { "title": "Acknowledgments", "content": "We thank an anonymous referee for providing many useful comments which help us improve the original manuscript. The Millennium-II Simulation databases used in this paper and the web application providing online access to them (http://www.mpagarching.mpg.de/galform/millennium-II/) were constructed as part of the activities of the German Astrophysical Virtual Observatory. The work of JL was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST, No.2012-0004916) and partially by the research grant from the National Research Foundation of Korea to the Center for Galaxy Evolution Research (NO. 2010-0027910). The work of GL was supported by Advanced Grant 246797 'GALFORMOD' from the European Research Council.", "pages": [ 15 ] } ]

[ { "title": "Domain Wall Model in the Galactic Bose-Einstein Condensate Halo", "content": "J. C. C. de Souza ∗ and M. O. C. Pires † Centro de Ciˆencias Naturais e Humanas, Universidade Federal do ABC, Rua Santa Ad'elia 166, 09210-170, Santo Andr'e, SP, Brazil We assume that the galactic dark matter halo, considered composed of an axionlike particles Bose-Einstein condensate [1], can present topological defects, namely domain walls, arising as the dark soliton solution for the Gross-Pitaevskii equation in a self-graviting potential. We investigate the influence that such substructures would have in the gravitational interactions within a galaxy. We find that, for the simple domain wall model proposed, the effects are too small to be identified, either by means of a local measurement of the gradient of the gravitational field or by analysing galaxy rotation curves. In the first case, the gradient of the gravitational field in the vicinity of the domain wall would be 10 -31 ( m/s 2 ) /m . In the second case, the ratio of the tangential velocity correction of a star due to the presence of the domain wall to the velocity in the spherical symmetric case would be 10 -8 . PACS numbers: 98.80.Cq; 98.80.-k; 95.35.+d", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The existence of a mysterious kind of matter, rather different from the usual barionic matter, presents itself as a great challenge for modern Physics. This so-called dark matter corresponds to almost 23% of the energy density of the Universe [2] and can amount to approximately 90% of the total mass in galaxies. Recently, it has been proposed that this type of matter can be composed of some kind of weakly interacting bosons [3]. When these bosons are spinless they can be identified with axions, hypothetical particles proposed in the context of Peccei-Quinn models [4]. On the other hand, in the case of sub-eV spin-1 particles, they are called hidden bosons or hidden photons [5, 6]. Axions and hidden bosons form a class of particles known as WISPs (Weakly Interacting Slim Particles), due to their diminute masses. In the last few years, the possibility that the dark matter content of galaxies is in the form of a self-graviting Bose-Einstein condensate (BEC) has been considered. Using this approach, the authors in [7] were able to relate the mass and the scattering length of an axionlike particle with the radius of the galactic dark matter halo. By proposing a new density profile based in the BEC features they could construct rotation curves that fit well a sample of galaxies. Using the same initial hypothesis, and extending it to spin-1 particles, the authors in [1] showed that the mass of the WISP's is constrained by galaxy radii data to the range 10 -6 -10 -4 eV . The next natural step in the identification of the dark matter halo with a BEC is to study the possible presence of substructures. BEC's can present a number of different substructures, called topological defects, such as vortexes, domain walls, monopoles and textures. The existence of these substructures is verified in laboratory experiments, for ultra-cold alkali atom gases trapped in a magnetic optical potential, and their features are well studied under these circumstances [8-10]. They are observed as a small region (much smaller than the size of the condensate) of null mass density in the gas. Mathematically, these defects have origin in zeros of the condensate wave-function, stressing the quantum nature of these phenomena in the gas. Topological defects have also been studied in the framework of cosmology and gravitation [11], in which they have notable differences from the condensed matter ones. For instance, they can be massive and carry a large amount of energy. Recently, the authors in [12] have suggested an experiment to detect a massive axionic domain wall via magnetic interaction in the context of field theory. To the authors knowledge, topological defects for a self-graviting BEC have never been proposed before. Our goal in the present paper is to explore the possibility that the galactic BEC is endowed with a domain wall, a finite region in space where the density vanishes. We are interested in the effect that such a structure can have on the galactic dynamics (specially on rotation curves), and if it is possible to detect a local domain wall by means of gravitation interaction effects. We restrict ourselves to the axionlike particle case. This paper is organized as follows. In section II we perform the derivation of the density functions for the halo endowed with a domain wall, and estimate its width. In section III, we give an estimate of the gradient of the halo gravitational field in the vicinity of a domain wall. In section IV we derive a correction term for the rotation curves of stars in spiral galaxies taking into account the influence of a domain wall perpendicular to the galactic disk plane (as depicted in figure 1). Section V shows our final remarks. PSfrag replacements galactic disk", "pages": [ 2, 3 ] }, { "title": "II. DARK SOLITONS", "content": "The zero-temperature mean field energy of a weakly interacting BEC confined in a self-graviting potential, V , is given by [1] where m is the mass of the particle composing the condensate, a is the s -wave scattering length and ψ ( /vectorr ) is the condensate wave function, satisfying ∫ d 3 r | ψ | 2 = N with N being the total number of particles. The mean field dynamics of the system is described by the Gross-Pitaevskii (GP) equation where g = 4 π /planckover2pi1 2 a/m . Some of the solutions of equation (2) may be quantized vortexes or dark solitons. These functions are topological defects in scalar BECs, in which the density vanishes due to the topological constraint on the phase of the wave function. In order to investigate the topological defect that corresponds to the domain wall that can appear perpendicularly to the z direction, we coupled the topological defect with the ground state of the galactic condensate in the Thomas-Fermi approximation [7] in the form where ψ TF ( x, y, z ) ≡ ψ TF ( r ) is the Thomas-Fermi solution for the GP equation (with r 2 = x 2 + y 2 + z 2 ), with k = √ Gm 3 / /planckover2pi1 2 a , R = π/k is the condensate radius and ρ 0 is the central number density of the condensate. φ ( z, t ) corresponds to the topological defect solution. We are interested in characterizing the defect by its position z , hence we eliminate the ThomasFermi solution in GP equation, as well as its dependence on x and y coordinates, by multiplying (2) by ψ ∗ TF and integrating it in these coordinates, obtaining We assume that the width of the topological defect is much smaller than the size of the condensate. In this situation we can consider the particle number density and the effective interaction parameter as almost constants in the defect vicinity. As the self-graviting potential obeys the Poisson equation, we can approximate the potential as V ( r ) ≈ V 0 and, therefore, we substitute in equation (5) to obtain the one-dimensional Gross-Pitaevskii equation for u ( z, t ) where g ' is supposed be locally constant. Equation (8) has the solution where µ = g ' = g ( z 0 ). This solution is called a planar dark soliton, describing a domain wall at z = z 0 , since the density vanishes at that point. where η ( z ) = -/planckover2pi1 k 2 m sin( kz ) (1+cos( kz )) is an extremely small factor and the term it couples to can be neglected. The effective interaction parameter, g ( z ) = gρ 0 f ( kz ) / 2, is proportional to the central density and the form factor where x = z/R . The quantity ξ , called healing length, is related to the width of the domain wall. It is possible to show that it is a function of the parameters of the condensate in the form The density function for the domain wall ρ DW = | u ( z, t ) | 2 is shown in figure 3. The density function for the condensate with a domain wall is depicted in figure 4. Local dark matter density measurements indicate a value of 0 . 4 GeV/cm 3 . Using that information and the masses and scattering lengths for an axionic dark matter halo as suggested in [1], we can infer the order of magnitude for the healing length of the domain wall. The results are shown in table I. We can see that the healing length decreases very quickly with the particle's mass, becoming negligible for larger masses. The value for a mass of 10 -4 eV is already beyond any physical significance at galactic scales.", "pages": [ 4, 5, 6, 7 ] }, { "title": "III. GRAVITATIONAL EFFECT IN THE VICINITY OF THE DOMAIN WALL", "content": "We proceed now to the calculation of the effect of a domain wall located on the galaxy disk, more specifically crossing Earth's position, on a test body (e. g., a satellite). In the presence of a domain wall, the total density distribution is not symmetrical, then the gravitational effects are distinct from the case of the halo density without the domain wall. We intent to estimate this difference by analysing the movement of a massive test body crossing the domain wall. The gravitational field on the body is given by the solution of the equation where /rho1 is the mass density which can be related to the wave function by The gravitational effect in the test body will be maximized if this body is moving along the z-axis and between the border and the center of the domain wall. By the symmetry of the spatial configuration, the gravitational field has only a z-direction component. In this case, the equation to be solved is The difference in the gravitational field is given by where x 0 = z 0 /R is the domain wall position relative to the galaxy radius. For the Sun's relative position, x 0 ∼ 0 . 5 and the gravitational effect exerted in the massive body crossing the topological defect is of the order of 10 -28 m/s 2 , for a healing length of the order of 10 3 m and the gradient of the gravitational field is 10 -31 ( m/s 2 ) /m . Because the Earth's movement (along with the Sun) in the galaxy has a velocity of ∼ 10 5 m/s , this effect in the vicinity of our planet could only be detected by an experiment with a precision greater than 10 -32 m , which is far beyond present day technological capability.", "pages": [ 7, 8 ] }, { "title": "IV. TANGENTIAL VELOCITY CORRECTION", "content": "When the domain wall is present, the gravitational field presents tangential components. However, the projection of the gravitational field in the tangential direction is much smaller than the projection in the radial direction even near the domain wall. Then, we can neglect the tangential components and assume that the gravitational field is radial and given by where Using Gauss theorem in the equation (16), we obtain where the mass profile of the dark condensate galactic halo is, with V the volume of a sphere with radius r . Equation (18) allows to represent the tangential velocity v 2 tg ( r ) = rg r ( r ) of a test particle moving in the halo as where is the squared tangential velocity for the spherically symmetric case (already obtained in [7]) and is the correction in the squared velocity due to the presence of the domain wall. Θ( x ) is the Heaviside step function. v 2 corr is proportional to ξ/R /lessmuch 1, then the correction is maximal when the wall is located near the center of the galaxy. As the domain wall width is always many orders of magnitude smaller than the radius of the halo, this correction is also small. In figure 5 both terms of (20) are shown, to stress the difference in magnitude they present. With the addition of a barionic matter term (after choosing an appropriated barionic matter density profile), equation (20) can represent a rotation curve for stars in a spiral galaxy. The term v 2 ss had already been obtained in [7], and it was found to fit observed rotation curves for a number of galaxies. Our correction, v 2 corr , because of its small magnitude, cannot be detected in these types of rotation curves. As the factor ξ/R , for typical galaxies, would amount to about 10 -16 , the ratio between the correction and the tangential velocity for the spherical symmetric case would be v corr /v ss ∼ 10 -8 . The small efect that a domain wall would have in the galactic dynamics can be explained when we take in consideration the mass that could fill a disk the same size as the domain wall in a galaxy similar to the Milky Way ( R ≈ 10 kpc ). This mass would amount roughly to 10 22 kg , or about the mass of the Moon.", "pages": [ 8, 9, 10 ] }, { "title": "V. CONCLUSIONS", "content": "By assuming that the dark matter halo in galaxies is composed of a condensate of bosonic particles (with axionlike properties), as previous works hypothesized, we were able to model one type of substructure in the halo, in the form of a topological defect known as domain wall, derived from a dark soliton solution for the Gross-Pitaevskii equation in a self-graviting potential. Because other types of topological defects (such as vortexes, monopoles, textures and ring solitons) would occupy a smaller volume in the halo, and therefore would have a smaller influence on the total dark matter density, we decided to restrict to the study of domain walls. Even in this case, the magnitude of the effects are too small to be subject to detection by present methods, at least for the choice of parameters (mainly the healing length ξ , which depends on the mass and the scattering length of the axionlike dark matter particle estimated in [1]) and simplifications we have made here. For the local gravitational interaction, we have a gradient in the field of the order of 10 -31 ( m/s 2 ) /m . The correction factor on the velocity rotation curves for stars in spiral galaxies is typically of the order of 10 -8 . However, there may exist some kind of cumulative effect that renders the influence of domain walls considerable in a system with a larger number of topological defects or a greater dark matter density. They also may be important in other phases of galaxy evolution. The main result of this work is the implementation of a methodology for the inclusion of topological defects in a quantum gas dark matter halo. The sequence of this study implies, for example, the determination of the dynamical and thermodynamical stability of the domain wall, the extension of the method for a spin-1 particle condensate (as suggested in [1]), and the manifestation of such defects in a fermionic quantum fluid, among other possibilities. These issues will be the subject of future work.", "pages": [ 10, 11 ] }, { "title": "ACKNOWLEDGMENTS", "content": "J. C. C. S. thanks CAPES (Coordena¸c˜ao de Aperfei¸coamento de Pessoal de N'ıvel Superior) for financial support.", "pages": [ 11 ] } ]

[ { "title": "Chris Orban a,b", "content": "E-mail: [email protected] Abstract. In setting up initial conditions for ensembles of cosmological N-body simulations there are, fundamentally, two choices: either maximizing the correspondence of the initial density field to the assumed fourier-space clustering or, instead, matching to real-space statistics and allowing the DC mode (i.e. overdensity) to vary from box to box as it would in the real universe. As a stringent test of both approaches, I perform ensembles of simulations using power law and a 'powerlaw times a bump' model inspired by baryon acoustic oscillations (BAO), exploiting the self-similarity of these initial conditions to quantify the accuracy of the matter-matter two-point correlation results. The real-space method, which was originally proposed by Pen 1997 [1] and implemented by Sirko 2005 [2], performed well in producing the expected self-similar behavior and corroborated the non-linear evolution of the BAO feature observed in conventional simulations, even in the strongly-clustered regime ( σ 8 /greaterorsimilar 1). In revisiting the real-space method championed by [2], it was also noticed that this earlier study overlooked an important integral constraint correction to the correlation function in results from the conventional approach that can be important in ΛCDM simulations with L box /lessorsimilar 1 h -1 Gpc and on scales r /greaterorsimilar L box / 10. Rectifying this issue shows that the fourier space and real space methods are about equally accurate and efficient for modeling the evolution and growth of the correlation function, contrary to previous claims. An appendix provides a useful independent-of-epoch analytic formula for estimating the importance of the integral constraint bias on correlation function measurements in ΛCDM simulations. Keywords: cosmology: theory - large-scale structure of universe - methods: N-body sim- ulations ArXiv ePrint: 1201.2082", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Next-generation astronomical surveys will demand increasingly precise predictions from theory in order to properly interpret observations and constrain the nature of dark energy. As emphasized by [3, 4], this will be a challenging task: inaccuracies in the predictions of halo abundance and halo bias, for example, can affect cosmological inferences [5, 6], and measurements of the baryon acoustic oscillations (BAO) clustering feature will soon reach the stage where theoretical estimates of the shift of this feature from non-linear dynamics become important [7]. Although current state-of-the-art cosmological N-body simulations, given a specific set of cosmological parameters, are in many ways well-equipped to deliver highly precise predictions of the dark matter two-point correlation function and power spectrum for a relatively wide range of scales [8], the difficult-to-estimate covariances of these statistics are also crucial for placing constraints on cosmological parameters [9, 10]. While much creativity has gone into methods and algorithms that ultimately save substantial computer time in delivering these predictions [11-14] the cosmological N -body simulations that these methods draw upon, with very few exceptions, are conducted without allowing the overdensity in each box to vary as it would if boxes of a cosmologically-relevant size were randomly positioned in the universe. The goal of this paper is twofold: (1) to assess the ramifications of this choice and in doing so explore the predictions of the conventional (or 'standard') method, (2) to explore the predictions of a method that does allow the overdensity to vary from boxto-box or otherwise to document - for lack of existing references - why the field has largely abandoned this approach. Since other authors have adequately described the conventional method [15-17], which seeks to maximize the correspondence between the assumed initial fourier space clustering properties in the simulation and the fourier-space properties of the assumed cosmological model, I here focus the discussion on a method for running ensembles of simulations that is designed to instead maximize the correspondence between simulated real-space clustering statistics (e.g. σ 8 , ξ ( r )) and the real-space properties of the assumed cosmological model. Originally proposed by Pen [1] and implemented by Sirko [2] 1 , as this method allows the socalled DC mode of each simulation (in an ensemble of simulations) to vary self-consistently according to the clustering power on the scale of the box in much the same way that the density within randomly placed boxes in the real universe will fluctuate around the mean density. In the early days of fully cosmological N -body simulations [e.g. 18] this effect was sometimes included, albeit in less-sophisticated ways than in [1] and [2]. 2 In the Sirko [2] framework the initial power spectrum used with the Zeldovich [21] (and by extension 2LPT [22, 23]) approximation is convolved such that the matter correlation function matches exactly the linear theory correlation function for r < L box / 2, while for r > L box / 2 the correlation function is set to zero. With this in mind Sirko refers to this approach as ' ξ -sampled' initial conditions (ICs), while the standard method is referred to as ' P -sampled', since by using an unconvolved linear theory power spectrum with the Zeldovich approximation the initial conditions are instead matched to the fourier space clustering statistics. The ξ -sampled strategy, by matching the correlation function out to r = L box / 2, should avoid biases on all real space statistics, since the rms overdensity in spheres, σ ( R ), is simply related to the correlation function, and the halo mass function to good approximation is only a function of σ ( R ) [24-26]. Without this convolution these real space statistics become biased (e.g. from P ( k ) = 0 for k /lessorsimilar 2 π/L box ), as discussed by [1] and [27]. Sirko [2] presents a set of ΛCDM simulations with 100 h -1 Mpc box sizes that indicate that the conventional, P -sampled method can give strongly biased results for the matter correlation function on scales near 1/4th the size of the box, independently of epoch, while the results of ξ -sampled simulations with the same parameters give much more reasonable matter correlation functions on these scales. This conclusion is revisited in § 3, which argues that if a measurement-bias correction is applied to the P -sampled results, the two methods are consistent. Although a number of groups have published results using the initial conditions code developed by Sirko, which was the among the first include the 2nd order Lagrangian corrections [22, 23] to the Zeldovich [21] displacements, the code is very seldom used to generate ξ -sampled ICs. To my knowledge, only Reid et al. [28] have utilized the code in this mode, citing the success of convergence tests in [29]. In that study they create mock catalogues from a suite of 42 simulations with L box = 558 h -1 Mpc, and N = 512 3 for comparison with SDSS LRG data [30]. They chose the ξ -sampled method for this task, citing the attractive feature of allowing the DC mode of the box to vary, thereby modeling the power spectrum covariance of real surveys more realistically. [29] and Appendix A of [28] present a wide variety of convergence tests that explore the effects of increasing the resolution with either fixed initial conditions (i.e. with a particular randomly sampled value for the DC mode) or for a set of a few initial conditions realizations. More recently, [31] argued that the DC mode should be re-introduced and compared the results of five L box = 20 h -1 Mpc, ΛCDM simulations using the ξ -sampled method to a high-resolution, standard-method simulation with L box = 80 h -1 Mpc, finding good correspondence between the results for the variance of the halo mass function. This study systematically explores the predictions of the two different methods using relatively large ensembles of simulations (20 unless otherwise noted) and a diverse set of initial conditions. Where the results disagree it may be ambiguous which approach is more accurate, therefore I focus on pure powerlaw models which should evolve self-similarly. This allows highly-accurate self-consistency checks of the simulation results, since each output should, in a statistical sense, resemble scaled versions of earlier and later outputs. These kinds of 'self-similar' tests were decisive in confirming the accuracy of the first generation of fully cosmological N-body codes [32]. I also show a few tests where, instead of a pure powerlaw, I simulate BAO-inspired initial conditions consistent with a configuration space powerlaw times a gaussian bump. Investigated in great depth in [33] using the conventional method, this test is self-similar in a different sense - namely that the evolution of the dark matter clustering should only depend on the ratio of the scale of non-linearity to the scale of the BAO. I include these initial conditions as another test of the ξ -sampled method and as a valuable cross-check for the conventional method's predictions for the non-linear evolution of the BAO feature. Importantly, these simulations can explore the shift and broadening of the BAO bump even in the strongly-clustered regime ( σ 8 /greaterorsimilar 1). I test these models extensively, focusing on pure powerlaw models with spectral slopes of n = -1, -1 . 5, and -2, and on the three models explored in [33] which resemble n = -0 . 5, -1, and -1 . 5 powerlaws in fourier space. § 2 gives an overview of the ξ -sampled method. § 3 describes aspects of measuring the correlation function in ξ -sampled and P -sampled simulations, including the importance of the integral constraint measurement bias which led Sirko [2] to believe incorrectly that correlation functions in P -sampled, ΛCDM simulations are suppressed for r /greaterorsimilar L box / 10. § 4 describes powerlaw initial conditions in the ξ -sampled context. I compare predictions from the two methods, showing results for the matter-matter two-point correlation function in § 5. In § 6 I investigate results for the variance of the correlation function, comparing the results from the two methods to each other and to expectations from theory. In § 7 I summarize my main conclusions. Appendix A presents a simple, independent-of-epoch analytic formula that, given the box size, can estimate the importance of the integral constraint in ΛCDM simulations.", "pages": [ 2, 3, 4 ] }, { "title": "2 Overview of the ξ -sampled Method", "content": "In the ξ -sampled method implemented by [2], the (real space) matter correlation function for a given cosmological model is the (usual) fourier transform of the power spectrum To convolve P ( k ) such that the simulated ξ ( r ) is an exact match to Eq. 2.1 for r < L box / 2, but is zero for larger separations, one simply fourier transforms ξ ( r ) while cutting off the integral at L box / 2 since ξ ( r ) = 0 for r > L box / 2, I will refer to this result as P real ( k ) to emphasize that this power spectrum is designed to maintain correspondence with the real space properties of the cosmological density field. Importantly, P real (0) can be non-zero even if P (0) = 0; this term sets the fluctuations in the DC mode. In Appendix A of [2], using the subscript 'uni' to denote variables in the model of interest and 'box' to identify the parameters of the simulated volume, these fluctuations are mapped self-consistently onto fluctuations in cosmological parameters, where ∆ 0 is a gaussian variable with mean zero and variance P real (0) /L 3 box and D (1) is the value of the linear growth function at the present epoch. Note that Eq. 2.3 implies that in h -1 length units the box size of each simulation varies with the value of φ , whereas in length units without the inverse hubble factor (e.g. Mpc) the box size remains fixed. Similarly the box integrated mass, M box = ρ m L 3 box , varies from box-to-box in h -1 M /circledot units, but is fixed in M /circledot units. Of crucial importance in deriving Eqs. 2.3-2.6 is the relationship between the scale factor of interest, a uni , and the corresponding scale factor in a particular realization, a box . In [2] this relationship is set by an approximate formula which determines a box as the epoch where the age of the universe in the box is the same as the age of the unperturbed universe during the epoch of interest, 3 [2] justified this formula by arguing that the ratio of the average density of the universe to the average density of a given box, ¯ ρ uni / ¯ ρ box = a 3 box /a 3 uni , is simply related to the overdensity of the box, which grows according to the linear theory growth function. Eq. 2.7 can also be obtained by Taylor expanding the perturbed H ( a box ) for small φ and equating the age of the universe in the box to the age of the universe at the epoch of interest.", "pages": [ 4, 5 ] }, { "title": "2.1 Integration of Particle Trajectories", "content": "Having set up the initial conditions, determined the perturbed cosmological parameters of a given realization and computed the relevant scale factors, a box , for the epochs of interest, the initial conditions can be evolved using any cosmological N-body code. I use the publiclyavailable Gadget2 code with no modifications [35]. As a hybrid Tree-based code with a PM grid for large scale forces, Gadget2 is a highly scalable N-body code which compares well to other codes used in the literature [e.g. 8]. Unless otherwise noted I show results /negationslash from simulations with 256 3 particles and a 512 3 PM grid. Initial redshifts were set using ∆ 2 ( k Ny ) /lessorsimilar 0 . 001 as a rule of thumb [36], and the force softening was set to 1/20th the initial mean interparticle spacing.", "pages": [ 5, 6 ] }, { "title": "3 Measurements of the Two-Point Correlation Function", "content": "With ensembles of simulations in the conventional method, the measurements of dark matter clustering at a given output, a uni , can typically be combined, and the statistical precision improved, with a simple average. In ξ -sampled simulations this procedure is somewhat more complicated. For clarity, the Sirko 2005 approach for measuring the matter-matter two-point correlation function will be described in § 3.1, and then a conceptual subtlety with this formulation will be highlighted with an alternate derivation in § 3.2. Following these subsections, the integral constraint bias will be discussed in both the P -sampled and ξ -sampled contexts. In what follows ,i subscripts are used to distinguish quantities that change from realization to realization from those without ,i subscripts that stay fixed. Also, it is helpful to remember that the number of simulation particles in each realization is kept fixed and that the box size is fixed in Mpc units, so in any box i , where L uni is the mean box size of the realizations in comoving Mpc units ( L box ,i = a box ,i L uni /a uni ) and accordingly both ¯ n box ,i and ¯ n uni are in Mpc -3 units (instead of h 3 Mpc -3 units). Depending on the context, N is either the total number of simulation particles in the box or the total number of randomly-selected tracer particles being used to compute the correlation function. Both contexts hold N fixed and therefore ¯ n box ,i and ¯ n uni are simply related, This also connects the scale factors to the overdensity, which is very similar to the expression in Eq. 2.7. For brevity, the symbol ∆ i ≡ ( D ( a uni ) /D (1))∆ 0 ,i will frequently be used to denote the overdensity of a given box at a particular epoch.", "pages": [ 6 ] }, { "title": "3.1 Estimation in Sirko 2005", "content": "In the Sirko 2005 approach, the principal subtlety in calculating the mean correlation function from an ensemble of ξ -sampled simulations is simply that the mean number density in each box, ¯ n box ,i , deviates from the mean number density, ¯ n uni . We naturally begin with a correlation function measurement that is totally ignorant of the 'uni' cosmology. Using the overdensity, δ = n/ ¯ n -1, and the well-known formula for the two-point correlation function, this is where the 〈 〉 symbols denote an average over the simulation box; n i ( /vectorx ) and n i ( /vectorx + /vectorr ) are number densities at different positions within the box, i . It is straightforward to show that Eq. 3.4 is equivalent to since 〈 n i ( /vectorx ) 〉 = 〈 n i ( /vectorx + /vectorr ) 〉 = ¯ n box ,i . The goal now is to find the relation between ξ box ,i ( r ) and a correlation function measurement in the 'uni' cosmology, Using Eq. 3.2, Eq. 3.6 can be expanded to become, Combining Eqs. 3.4 & 3.7 we obtain, which is equivalent to Eq. 25 from Sirko 2005. In the final averaging, ξ uni ,i ( r ) in Eq. 3.8 is weighted by w i = ( a box ,i /a uni ) 3 to ensure that boxes with larger volumes receive higher weight. Unless otherwise noted Eq. 3.8 is used with the weighting just mentioned in calculations of the two-point correlation function in ξ -sampled simulation ensembles.", "pages": [ 6, 7 ] }, { "title": "3.2 Subtleties of Eq. 3.8: Survey-like versus 'better informed' estimators", "content": "To highlight the subtleties of Eq. 3.8, let us re-derive the expression in a different way. The two-point correlation function can be equivalently defined as the joint probability, δP , to find a particle in volume, dV 1 , and another particle, at some distance, r , in the volume dV 2 , For a given realization, one of these volume elements integrate to the volume of the simulation box, L 3 box ,i , while the other volume is integrated over a radial shell, V shell . For the correlation function of an individual box, ξ box ,i ( r ), for which ¯ n = ¯ n box ,i , this yields an expression for the total number of pairs in the box within a given radial separation, where the 1 / 2 factor avoids the double counting of pairs. The above expression is useful as an algorithm for measuring ξ box ,i ( r ) from counting the number of pairs at various separations in a given simulation box. There are two ways of converting ξ box ,i ( r ) in Eq. 3.10 into a correlation function measurement in the 'uni' cosmology. Most simply, one can define ξ uni ,i ( r ) according to Eq. 3.9 using the 'uni' number density for ¯ n and appreciating that the correlation function measurement is over the volume of the box for a specific realization, L 3 box ,i , This leads to the conclusion that The remarkable consequence of assuming Eq. 3.12 is that even if the particle distribution in the simulation volume is completely uncorrelated ( ξ box ,i ( r ) → 0), the correlation function in the 'uni' cosmology can still be non-zero since, in that case, Importantly this result remains after volumetric weighting is applied to ξ uni ,i ( r ). From the non-zero result of Eq. 3.13 it is clear that Eq. 3.12 is a survey-like approach to measuring the correlation function in the simulation ensemble in the sense that the measurement knows about the volume of the box but it does not know the true overdensity of the box. This ignorance is transferred to ξ uni ,i ( r ) and it is only in averaging over many simulations that the mean of the ∆ i values will be close to zero and a precise measurement of the mean correlation function can be made. This is very much like surveys where, in principle, one would benefit from perfectly knowing the overdensity of a particular subvolume which would be useful for measuring the correlation function. Perfect knowledge of the overdensity would help determine how much of a measured excess (or decrement) of pairs in a subvolume reflects the the true non-linear correlation function and how much of the excess (or decrement) reflects a difference between the mean density of the subvolume and the mean density of the universe. However, in practice, the overdensity of a particular subvolume in a survey is uncertain at some level and this uncertainty must be taken into account in estimating the errors on the clustering measurement. A more-sophisticated (a.k.a. 'better-informed') approach to connecting ξ box ,i ( r ) and ξ uni ,i ( r ) is therefore to use the overdensity information, as just described, to compare the measured number of pairs, N p,i ( r, ∆ r ), to a 'better-informed' expectation of the number of random pairs for that simulation volume. To do this one can introduce a correlation function offset, denoted by ξ δ,i , that will make this adjustment, At large separations, or in a hypothetical situation where the clustering in each box is totally uncorrelated, then ξ box ,i ( r ) → 0 and we can define ξ uni ,i ( r ) so that by fiat in each box ξ uni ,i ( r ) → 0 and the box-to-box fluctuations in overdensity are entirely captured by ξ δ,i . This implies or just Solving for ξ uni ,i ( r ) in Eq. 3.14, the 'better informed' estimator becomes which is identical to the result Sirko derived (Eq. 3.8 in this work). Sirko's estimator therefore implicitly uses the knowledge of the overdensity in each box to improve the correlation function estimate. Parenthetically, note that as in Eq. 3.8 and in Sirko [2] the above expression for ξ uni ,i ( r ) must be volumetrically weighted by w i = ( a box ,i /a uni ) 3 when averaging over all realizations. Interestingly, this 'better-informed' estimator is not unlike correlation function measurements in conventional, P -sampled simulations. Since the density of finite volumes in the real universe fluctuates around the mean, arguably one should account for this source of uncertainty in the error bars of a given correlation function measurement from a P -sampled simulation. But instead, rather than degrade the error on the mean correlation function, one naturally uses the extra information that the overdensity of a given P -sampled simulation is always zero, regardless of the box size, to inform the expectation for the number of random pairs. Thus for P -sampled simulations ¯ n box ,i is always equal to ¯ n uni (in general and in Eq. 3.14) and consequently it is perfectly known that ∆ i = 0 (i.e. ξ δ,i = 0) for all realizations. In this sense, correlation function measurements in P -sampled simulations are also performed with a 'better informed' estimator without any extra effort.", "pages": [ 7, 8, 9 ] }, { "title": "3.3 Integral-Constraint Bias in P -sampled Simulations", "content": "An important but sometimes neglected measurement bias that affects correlation function estimation is an integral constraint that arises from the fact that summing over the number of pairs in the volume must naturally yield 1 2 N 2 where N is the number of randomly selected tracer particles. This issue has been identified by other authors (e.g. [58]) and it is is entirely orthogonal to the question of which estimator [38, 39, etc.] converges most rapidly to the true ξ ( r ) in the presence of Poisson noise. Orban & Weinberg [33, Appendix B] outline an approach for correcting the correlation function measurement. Appendix A demonstrates that for ΛCDM simulations with large boxes ( L box /greaterorsimilar 2 h -1 Gpc) the integral constraint is a minor issue. For significantly smaller boxes this is an important concern. Notably, [2] presented simulations with L box = 50 -100 h -1 Mpc without any kind of correction for this effect. The present section will discuss the integral constraint bias in P -sampled simulations. This subtlety is also relevant to ξ -sampled simulations. The next subsection will discuss how the ξ -sampled approach using Eq. 3.8 as in Sirko [2], includes a correction for the problem. Since the notation in this section differs slightly from that in Orban & Weinberg [33, Appendix B], a brief re-derivation of that result will help explain the problem. For P -sampled simulations, the number of pairs in a given radial bin is given by 4 If integrated over the entire box this expression becomes where 4 3 πR 3 box ≡ L 3 box , implying that R box = (4 π/ 3) -1 / 3 L box ≈ L box / 1 . 61. Note that ¯ nL 3 box = N , so Eq. 3.19 becomes and since ¯ n 4 3 πR 3 box = ¯ nL 3 box = N , a measurement constraint is imposed on ξ uni ,i ( r ), Our reproduction of the P -sampled ΛCDM simulations presented in Fig. 9 of Sirko 2005 [2], shown here in the left panel of Fig. 1, indicates that this measurement bias is quite important for the L box = 100 h -1 Mpc simulations they present, suppressing the correlation function at 1/4th the scale of the box by almost a factor of two and causing a severe disagreement with the linear theory correlation function for r ∼ 20 -25 h -1 Mpc despite the fact that ξ L ( r ) /lessmuch 1 on these scales. To correct for this measurement bias, following the approach used in Orban & Weinberg [33], one defines where ξ bias is a radially-independent term and ξ uni , true ,i ( r ) is understood to be the correlation function of the box without the integral-constraint bias. Using Eq. 3.21, ξ bias can be solved for, giving where the integral over ξ uni , true ,i ( r ), which is weighted heavily towards large scales, has been well approximated using linear theory. The corrected estimator for the correlation function is therefore This result is identical to the prescription presented in Orban & Weinberg [33]. Results from using the integral-constraint corrected estimator are presented in the right panel of Fig. 1. For separations of r ∼ 20 -25 h -1 Mpc the amplitude of the correlation function is nearly a factor of two higher at all epochs, which agrees much better with the linear theory correlation function on these scales as would be expected. Therefore the conclusion in Sirko 2005 that P -sampled simulations suppress the correlation function for separations approaching the box scale is found to stem from an overlooked integral-constraint correction. Importantly, as is clear from Fig. 1, the integral constraint correction matters for separations as small as r ∼ 10 h -1 Mpc ∼ L box / 10 or perhaps slightly smaller . While most practitioners would regard clustering measurements at separations of r ∼ L box / 4 or r ∼ L box / 5 in a simulation volume to be too large compared to the scale of the box to be trustworthy, it should be received with some amount of surprise that clustering measurements at separations as small as r ∼ L box / 10 are significantly biased, independently-of-epoch in conventional, P -sampled simulations. Thankfully, correlation function measurements at these scales can be corrected using Eq. 3.24 without re-running the simulation and Appendix A provides a useful independent-of-epoch formula for ΛCDM simulations that can estimate this bias at the BAO scale given the size of the simulation box.", "pages": [ 9, 10, 11 ] }, { "title": "3.4 Integral-Constraint Bias in ξ -sampled Simulations", "content": "Returning to the conclusions of Sirko [2] one may ask why the ξ -sampled results in [2] agreed so well with linear theory approaching the box scale if Sirko did not also apply an integral constraint correction to the ξ -sampled correlation function measurements? The answer is that Eq. 3.8 (which is what Sirko used) includes, in its average, a term very much like the integral constraint correction. At large enough separations in the simulation box, the particle distribution will be approximately uncorrelated ( ξ box ,i ( r ) ≈ 0). On these scales Eq. 3.8 gives with ∆ i ≡ D ( a uni ) D (1) ∆ 0 ,i as used elsewhere. Taking the average over many realizations i , While not exact, the term P real (0) /L 3 box is very similar to the ξ bias correction term in Eq. 3.23 & 3.24. It is this term that does the work, so to speak, of correcting for the integral constraint in the ξ -sampled simulations in [2]. This is why Sirko concluded that the ξ -sampled method matched well with linear theory on scales approaching the box without an explicit correction term. If this ∆ 2 i term had not emerged (as it does in Eq. 3.25, essentially by accident) the ξ -sampled correlation function results would have been suppressed much like the P -sampled result shown in the left panel of Fig. 1. But since the term does appear there is no need to explicitly correct for the integral constraint bias in ξ -sampled simulations. Instead the correction is understood to be built into Eq. 3.8, which is the formula employed in all the ξ -sampled correlation function measurements presented here.", "pages": [ 11, 12 ] }, { "title": "4 ξ -sampled ICs with Powerlaw Models", "content": "For powerlaw models, where P ( k ) = Aa 2 k n , the task of computing Eq. 2.2 is made substantially easier because an exact analytic solution for ξ L ( r ) is known in this case, [41]. Eq. 2.2 therefore becomes 5 Eq. 4.2 can be used straightforwardly to express the DC power, Analytic and special-function solutions to Eq. 4.2 exist for certain powerlaws. In this study I am interested in n = -1, -1 . 5 and -2 which can be expressed by where Si(x) is the sine integral and S( x ) is a Fresnel integral. These formulae can be very useful for generating accurate initial conditions, especially for steep power spectra. I show these power spectra in Fig. 2, fixing r 0 /L box = 1 / 16 to set the relative amplitudes. Notice that steeper powerlaws have larger DC power, easily seen on the plot as the asymptotic value of P real ( k ) /L 3 box as k → 0. Noticing that P ( k ) does not go to zero at small k for the box P -sampled powerlaws, one might be concerned that these models are unphysical. However, despite the high levels of large scale clustering power the rms overdensity in spheres and other statistics can remain finite for n > -3. It so happens that a close inspection of Eq. 4.2 reveals that P real ( k ) is finite and positive (or equal to zero) for all k only if n ≥ -2. It is unclear how to circumvent this issue to simulate steeper power spectra. For ΛCDM initial conditions this limitation translates into assuming L box /greaterorsimilar 2 . 5 h -1 Mpc to avoid P real ( k ) < 0 because the effective slope of the ΛCDM correlation function at small scales, using ξ L ( r ) = ( r 0 /r ) n eff ( r )+3 , is n eff /lessorsimilar -2 for r /lessorsimilar 2 . 5 h -1 Mpc.", "pages": [ 12, 13 ] }, { "title": "4.1 Scale free?", "content": "Although pure powerlaw models are often referred to in the literature as 'scale free,' since P ( k ) = Ak n is featureless, the ξ -sampled initial power spectra shown in Fig. 2 clearly depend on the choice of L box . In practice, these oscillatory features die away in simulations and the effect of the box size is merely to change the variance of the DC mode (which is set by P real (0) /L 3 box ). Since dark energy introduces a new scale into the problem (e.g. the age of the universe when ρ m = ρ Λ ), I consider only Ω m, uni = 1 . 0 , Ω Λ , uni = 0 , Ω k, uni = 0 so as to keep the simulations as 'scale free' as possible and allow the self-similar tests discussed in the next section. In the Zeldovich [21] and adhesion [42, 43] approximations (as in linear theory), the effect of dark energy on structure formation is entirely captured by changing the linear theory growth function. [44] and [45] convincingly argue that this approximation is remarkably accurate even in the non-linear regime - the second order effect of dark energy is relatively small. Therefore the results of my Ω m = 1 tests should still be quite relevant to studies that include a dark energy component. As one final comment on the scale-free nature of my simulations, throughout I adopt, as a time variable, where k box ≡ 2 π/L box and k NL is defined by the dimensionless linear theory power spectrum, ∆ 2 L ( k NL ) ≡ 1. The scale-free nature of the simulations demands that only the ratio a/a ∗ is meaningful (e.g. as the square root of the dimensionless power on the scale of the box) and so a ∗ is defined implicitly in the definitions already given. Eq. 4.8 is also simply related to the σ miss formula of [46], which quantifies the missing power on the scale of the box in P -sampled simulations as another choice for a time variable. I adopt Eq. 4.8 for ease of comparison with [47] and because the σ miss formula in [46] would be inappropriately applied to ξ -sampled simulations, which have a turnoff in P real ( k ) near the box scale (Fig. 2).", "pages": [ 13, 14 ] }, { "title": "5.1 Powerlaw Models", "content": "Fig. 3 shows my primary results for the self-similar scaling of the matter correlation function. The x-axis is shown in r/r 0 units where ξ L ( r 0 ) ≡ 1. Insofar as the dark matter clustering is negligibly affected by numerical limitations such as the finite scale of the box or the scale of the force softening, with this scaling the correlation function results from different outputs should all lie upon the same line. To the extent that this is achieved the correlation function can be said to evolve with self-similarity and it is clear from Fig. 3, excluding the first outputs which are severely affected by transients from initial conditions, that over a wide range of scales the results from these relatively modest, N = 256 3 , simulations do fall upon the the same locus as expected. This locus is different for each powerlaw; for steeper power spectra (e.g. n = -2) power is transferred from large scales to small scales and the non-linear growth of ξ ( r ) out paces linear theory whereas for shallower power spectra (e.g. n /greaterorsimilar -1) there is so much small scale power that the process of halo formation and collapse causes the non-linear growth to fall behind linear theory in a process sometimes called 'pre-virialization' [39]. In the language of the halo model [48] this implies that the predicted linear theory clustering on small scales is so high that the amplitude of the 1-halo term is below the linear theory clustering amplitude on those scales. The n = -1 case falls between these two extremes and the amplitude of the correlation function is both above and below linear theory, depending on the regime. (For a bracketing case of an even shallower power spectrum see, e.g., the n = -0 . 5 results in [33, Appendix A].) In Fig. 3, the ξ -sampled and P -sampled methods generally agree well on the shape of the self-similar solution. This is significant for the ξ -sampled results, on some level verifying the method. Alongside the measurements in each case fitting functions for the self-similar correlation function from higher resolution simulations are shown (black lines). For n = -1 and n = -2 this comparison is made by numerically fourier transforming the non-linear power spectrum fitting functions published in [47]; note in the n = -1 case I include subtle but important corrections to their fit at small k/k nl as determined in [33, Appendix A]. For n = -1 . 5, I compare with ξ ( r ) measurements from 10 P -sampled simulations with N = 512 3 [33, Appendix A]. These high-resolution results are used more quantitatively in Fig. 4 where the correlation function results are presented relative to the box size. Overall, the agreement with the high-resolution self-similar results is quite good and excluding the initial and final outputs in each case my simulation set tends to match the self-similar evolution to better than about 5% in most outputs and on most scales. This is similar to the precision on the results from higher-resolution simulations. The last output is excluded from this conclusion since the linear theory clustering level is so high that one expects departures from the true non-linear clustering from the suppression of power on the scale of the box. Also, the correction for the integral constraint, which assumes a linear theory correlation function in ξ bias (Eq. 3.23), likely becomes inaccurate in the highly-clustered regime as well. Another caveat to the overall good agreement is at small r/r o especially for early outputs. Fig. 4 presents the same correlation function measurements in Fig. 3 relative to the scale of the simulation box and shows that these deviations from self-similarity are all below the scale of the initial mean interparticle spacing. This is as expected since at best the initial conditions will only match the self-similar solution down to these separations. Rather than excluding these scales from Fig. 3, they are included to highlight, in Fig. 4, that as structure evolves the self-similar behavior extends further and further below this scale, in some cases approaching the force softening. This result is non-trivial and difficult to anticipate from first principles. It bears mentioning some of the previous work on how non-linear clustering proceeds near or below the scale of the initial mean interparticle spacing. [49], using n = -1 simulations, show that Fourier modes in the non-linear regime are largely determined by the collapse of large-scale modes rather than by evolution of power initially on those scales. This nicely explains the trend in Fig. 4 for later outputs to match better with the self-similar solution on small scales and why the poisson noise in the dark matter density on those small scales does not prevent this from happening. However, Joyce et al. [50] and collaborators have argued that the common practice of setting the force softening significantly smaller than the initial mean interparticle spacing (as in the simulations presented here) introduces errors which arise from the possibility that with this choice the equations of motion for the particles are no longer true to the Vlasov-Poisson fluid equations. Despite this, their results concur with Fig. 4 that ξ ( r ) can reliably be modeled below the scale of the mean interparticle spacing. According to [50] the main effect of aggressive force softening is to cause ∼ 5% disagreement with the true non-linear ξ ( r ) on scales larger than the mean interparticle spacing. The P -sampled results shown in Fig. 4 are in qualitative agreement with the simulations presented in [50] in the sense that accurate non-linear behavior is observed below the mean interparticle spacing and on larger scales the measurements are consistent with the self-similar solution also at the level of ∼ 5%. Although beyond the scope of this paper, it would be interesting to run the P -sampled simulation set with less aggressive force softening (e.g. half the mean interparticle spacing) to test if the measured error on the mean ξ ( r ) is detectably smaller, as predicted in [50]. At any rate, for all three powerlaws the self-similar behavior extends well below the scale of the mean interparticle spacing; it does not significantly depend on whether power is being rapidly 'transferred' to smaller scales as for n = -1 . 5 and n = -2 or whether the non-linear growth proceeds less quickly than the linear theory prediction on small scales (i.e. r < r 0 ), as for n = -1.", "pages": [ 14, 16 ] }, { "title": "5.2 Powerlaw Times a Bump Results", "content": "As discussed in depth in [33], a real-space powerlaw times a bump can be used as a self-similar numerical test in addition to providing insight into the non-linear physics of the evolution of the BAO feature. In this case, and for resemblance to the ΛCDM correlation function I chose A bump = 2 . 75, σ bao /r bao = 0 . 075, and powerlaws of n = -0 . 5, -1, and -1 . 5. Unlike ΛCDM, this setup can be evolved much further than σ 8 ∼ 1 to investigate the non-linear physics of the problem. For each powerlaw, in Fig. 5 I compare results from 20 ξ -sampled simulations with N = 256 3 , r bao / 〈 L box ,i 〉 = 1 / 20 to the results of 7 P -sampled, N = 512 3 , r bao /L box = 1 / 20 simula- tions from [33]. In Fig. 5 these P -sampled results are shown with dot-dashed lines of various colors corresponding to different outputs. Since the first two outputs from the P -sampled n = -0 . 5 simulations are noisy because of the very low clustering amplitude, Fig. 5 presents the best fit gaussians to those results for ease of comparison. All other dot-dashed lines are the mean correlation function results from the P -sampled simulations. Error bars in Fig. 5 show the error on the mean for the ξ -sampled results. Qualitatively, the correlation function results agree well and importantly the non-linear shift in the n = -1 . 5 results and lack of shift in the n = -0 . 5 and -1 results are consistent. This conclusion should be reassuring to the wider effort to characterize the non-linear shift of the BAO peak using standard P -sampled simulations. A quantitative comparison of the results in Fig. 5 is presented in Fig. 6. Here the ξ -sampled results are shown with solid points with thick error bars, and the P -sampled results are shown as open circles with thin error bars, both with colors corresponding to the powerlaw (cyan for n = -0 . 5, green for n = -1, and red for n = -1 . 5). As in Fig. 5 of Orban & Weinberg [33], the error bars for both methods come from jackknife error estimation by sequentially omitting one of the realizations and computing the best fit gaussian and shift of the peak. The ξ -sampled results typically have tighter error bars than the P -sampled results because more ξ -sampled simulations were performed. The upper left panel of Fig. 6 echoes what was said earlier that the non-linear shift of the BAO peak is consistent between the two methods. The n = -0 . 5 results for both methods show some preference for a BAO shift to slightly larger scales, however the shift in this case is degenerate with the broadening (notice that the error bars in the bottom left plot are relatively large at later outputs) and the error bars are consistent with no movement of the BAO peak. A real movement of the peak to larger scales would have been counter-intuitive since the non-linear physics of the shift stems from a small but non-negligible tendency for particle pairs with initial separations of r = r bao to be found in regions with a slight overdensity, causing (on average) a very small movement inward [53]. Fortunately for BAO surveys, the broadening of the BAO feature from the growth of structure is a much larger effect than the non-linear shift. The bottom left panel of Fig. 6 highlights the results for broadening of the gaussian width of the BAO feature as it evolves from its initial value of σ bao /r bao = 0 . 075. The results from the two methods compare well and there are no pairs of points from any particular output or powerlaw that are statistically inconsistent with each other. In Fig. 5 of Orban & Weinberg [33] a very similar plot compared these P -sampled results to a simple diffusion model inspired by [53]. This earlier comparison was reasonably good and remarkably the n = -1 . 5 results agreed with an ab initio prediction of the diffusion model. The agreement between the ξ -sampled and P -sampled results argues that this same physics is properly included in ξ -sampled simulations and, e.g., that including the DC mode fluctuations in overdensity does little to change this result. The top right panel of Fig. 6 presents the results for the amplitude of the BAO feature. At later outputs the two methods agree well, however there is some tension with the first few outputs. This would be concerning except that the P -sampled N = 256 3 results in Fig. 8 of Orban & Weinberg [33] show a similar decrement in bump amplitude compared to N = 512 3 P -sampled simulations at early outputs. This mismatch seems to be some finite-particle numerical effect as opposed to some orthogonal concern relating to box scale cutoffs of large scale power. The bottom right panel shows the results for the normalized area of the bump, A bump × σ bao /r bao which tends to be constant in spite of the non-linear evolution of the BAO feature in agreement with the diffusion model discussed in Orban & Weinberg [33]. In the ξ -sampled simulations at early outputs the bump area falls somewhat below its initial value for both n = -0 . 5 and -1 by about one sigma. This can be attributed to the decrement of A bump since σ bao evolves as expected, but more importantly this tension with the constantbump-area evolution at these early outputs seems to corroborate the conclusion that it is merely an inaccuracy from using N = 256 3 particles instead of N = 512 3 .", "pages": [ 16, 17, 18, 19, 20 ] }, { "title": "6.1 Preliminaries", "content": "Having explored the ensemble-averaged predictions for the mean ξ ( r ), in this section I compare the results for the box-to-box variance of ξ ( r ) from the ξ -sampled and P -sampled methods, focusing on separations approaching the box scale ( r /greaterorsimilar L box / 10). While the variance (or, more generally, covariance) of statistics like ξ ( r ) is important for surveys so as to precisely and accurately infer cosmological constraints from a finite data set [9, 10, 52, 54-56], the primary goal of this section is somewhat more prosaic. Namely, if the box-to-box variance of ξ ( r ) from one or the other method is substantially larger then substantially more simulations must be performed via this method to obtain the same precision on the mean ξ ( r ). This would be the only reason to perform additional simulations since § 3.3 and § 5 show that as long as the integral constraint correction is applied to P -sampled correlation function measurements, the mean ξ ( r ) is consistent between the two methods. The box-to-box variance comes from the (usual) definition, where ξ uni ,i ( r ) is a correlation function measurement from an individual box. For ξ -sampled simulations, since the overdensity of each box is perturbed from the mean density of the true cosmology, one must use Eq. 3.8 to 'convert' ξ box ,i ( r ) (a statistic that assumes incorrectly that the overdensity of the box is zero) to ξ uni ,i ( r ) as discussed in § 3. For P -sampled simulations this step is unnecessary because every realization has zero overdensity by design and consequently ξ uni ,i ( r ) = ξ box ,i ( r ). This section will discuss three different sources of box-to-box variance (or, equivalently, three different considerations that degrade the precision on the mean ξ ( r ) from a finite number of simulations). These sources are: (1) variance from ignorance of the overdensity of the realizations, (2) variance from measuring ξ ( r ) from a finite number of randomly realized density fields, and (3) variance from correlations on weakly to strongly non-linear scales. The first item is mentioned only for completeness. As discussed in § 3, the ξ -sampled correlation function estimator in Eq. 3.8 is implicitly 'informed' of the overdensity and likewise the P -sampled estimator is informed of the overdensity in the sense that the overdensity of each box is identically zero. Were this not the case, then on large scales where the correlation is weak, following the discussion in § 3.2 the correlation function measurement would yield the overdensity of the box at that epoch, ξ uni ,i ( r ) ≈ 2∆ i , and applying Eq. 6.1 one would find In the powerlaw models investigated in this section but certainly also in ΛCDM cosmologies the above result would be an order of magnitude larger than any of the other sources of boxto-box variance just mentioned and many more simulations would need to be performed to measure the mean ξ ( r ) with any kind of precision. This underscores the importance of using a 'better informed' estimator for which ξ uni ,i ( r ) = 0 when the particles are uncorrelated. The cost of using an estimator that is ignorant of the overdensity is severe. One may ask, then, what Eq. 6.1 really means for ξ -sampled simulations using a 'better informed' estimator. The answer is that the definition of the variance is not substantially changed. Eq. 6.1 is the variance (or, equivalently, width of the distribution) of correlation function measurements at a particular separation, r , from finite volumes in a situation where the mean density of the universe is perfectly known and the overdensities of each volume are also perfectly known. In other words, the only unknown is ξ ( r ), which is what we are trying to measure. This is exactly as it would be in a P -sampled ensemble of simulations where all overdensities are perfectly known to be zero and the mean density of the universe is likewise perfectly known. So although the task of measuring ξ ( r ) is somewhat more complicated in ξ -sampled simulations (i.e. because of Eq. 3.8 and the volumetric weighting), the measurement in principle is not qualitatively different from P -sampled ensembles. This being the case, the most important source of variance for both methods comes from the fact that we are trying to measure the mean correlation function of the universe from a finite number of randomlyrealized density fields with known overdensities.", "pages": [ 20, 21 ] }, { "title": "6.2 Expectations from Gaussian Statistics", "content": "Mindful that the correlation function is also the fourier transform of the power spectrum (Eq. 2.1), the statement just made regarding the most important source of variance can also be conveyed by pointing out that finite volumes contain a finite number of fourier modes that can be used to compute statistics like the correlation function. Since the number of modes in the simulation box for each k value is straightforwardly determined this consideration can be used to estimate the variance of ξ ( r ) using a linear theory approximation for P ( k ). Applying this reasoning one arrives at an estimate for the variance, [55]. This is referred to as a 'gaussian' estimate of the variance because in the approximation of gaussian random fields, wherein all higher order statistics (e.g. 3-point and 4-point functions) are assumed to be negligible, Eq. 6.3 perfectly models the variance of ξ ( r ). For pure powerlaw models, P ( k ) = Ak n , it can be shown using Eq. 6.3 that where A ≡ A n r n +3 0 , and Γ(1 + 2 n ) is the usual gamma function. Notice that all of the r 0 dependence has canceled out with the division by ξ pow ( r ) = ( r 0 /r ) n +3 . Unfortunately, Eq. 6.4 is only convergent for the limited range of -1 . 5 < n < -0 . 5. Fig. 7 compares Eq. 6.3 with a lowk cutoff at k box (solid gray lines) to the box-to-box variance measured from the simulation ensembles in detail, showing the separation, r , relative to the scale of the box and normalizing the y -axis by ξ pow ( r ) so that the gaussian expectation of Eq. 6.4 is independent of epoch. For the convergent case of n = -1, Eq. 6.4 without a lowk cutoff (dashed black lines) is also compared to the simulation data. The ξ -sampled results are also compared to another source of variance (black dot-dashed lines, Eq. 6.6) that will be explained in the next section.", "pages": [ 21, 22 ] }, { "title": "6.3 Commentary on Figure 7", "content": "The n = -1 results in Fig. 7 are most instructive since Eq. 6.3 is compared to the measured variance from simulations both with and without the lowk cutoff. In each panel in Fig. 7 the range r/L box /greaterorsimilar 1 / 10 is most important for this comparison because on smaller scales and increasingly for later outputs the box-to-box variance from non-linear correlations, which are not accounted for in Eq. 6.3, become important and greatly exceed the linear theory expectation of Eq. 6.3 6 . But for r/L box /greaterorsimilar 1 / 10, the n = -1 case measurements of the variance generically fall below Eq. 6.3 without the lowk cutoff and are either consistent with or slightly above the expected variance from including the lowk cutoff in Eq. 6.3. That both methods fall below the expectation of Eq. 6.3 without the lowk cutoff is a sensible result, especially for P -sampled simulations because it is an explicit assumption of the method that P ( k ) = 0 for all k -modes from scales larger than the size of the simulation box. As a result, clustering power on these scales do not contribute to the box-to-box variance of ξ ( r ). where Q 3 and Q 4 are constants from HEPT that depend on n . Notice that r o does not cancel out as in Eq. 6.4, so this source of variance grows larger as the simulation progresses. More exactly, Eq. 6.5 predicts that σ ξ /ξ pow ( r ) on non-linear scales will increase in proportion to the linear growth function. This prediction was confirmed with a detailed comparison of Eq. 6.5 to the measurements from simulations in [34] however HEPT was overall consistent with the simulations only at an order-of-magnitude level. Note that the Journal version of [57] contains a typo for Q 4 . The arXiv version is correct or, c.f., [58] (R. Scoccimarro private communication). The ξ -sampled n = -1 results falling below the expectation of Eq. 6.3 is also sensible for two reasons: (1) as argued earlier, the definition of the box-to-box variance is not substantially changed. And (2) despite including the fluctuations in the DC mode of the simulations, one still expects the ξ -sampled method to under-represent clustering on scales larger than the box. Notice, for example, that P real ( k ) is totally insensitive to clustering power in ξ ( r ) from r > L box / 2 (Eq. 2.2) and it is P real ( k ) that is used in the Zel'dovich formulism to generate the initial conditions. Turning to the n = -1 . 5 and -2 results, clearly the measurements from the P -sampled simulations for n = -1 . 5 and -2 also compare well to the expectation from Eq. 6.3 with a cutoff at k box as expected. However, the ξ -sampled results clearly exceed the expectation of Eq. 6.3 with the lowk cutoff. This stems from the fact that, compared to the n = -1 simulations, the fluctuations in the overdensity are larger for the n = -1 . 5 simulations and even larger for the n = -2 simulations (c.f. P real ( k → 0) in Fig. 2). But the key is that in ξ -sampled simulations the correction for the integral constraint bias arises naturally because ξ uni ,i ( r ) → ∆ 2 i on large scales where the particle distribution is approximately uncorrelated (Eq. 3.25). When this is the case, the box-to-box variance (Eq. 6.1) yields Since ∆ i = D ( a uni ) D (1) ∆ 0 ,i and ∆ 0 ,i is a gaussian random variable, then 〈 ∆ 4 i 〉 = 3 〈 ∆ 2 i 〉 2 and Although the above expression is smaller than, e.g., Eq. 6.2 it can be as large or larger than Eq. 6.3. So while the expectation of Eq. 6.3 with a lowk cutoff compared well to the measurements from simulation in all the other panels, this is why the ξ -sampled n = -1 . 5 and -2 results in Fig. 7 so greatly exceed the expectation from Eq. 6.3 with the lowk cutoff even in the r /greaterorsimilar L box / 10 region where non-linear effects are small. In ξ -sampled ΛCDM simulations, such as those in Sirko [2], this issue would likewise artificially increase the box-to-box variance and degrade the error on the mean ξ ( r ). If the box size is small enough then 〈 ∆ 4 i 〉 1 / 2 will be comparable to σ ξ from Eq. 6.3 7 . Indeed this seems to be the case in their Fig. 9 which presents ξ -sampled simulations with L box = 100 h -1 Mpc. The 1-sigma error on the mean ξ ( r ) in that case is noticeably larger than the 1-sigma error on the mean from the P -sampled simulations. Sirko [2] does not comment on this interesting result. The work here suggests that this is just a consequence of ξ uni ,i ( r ) ≈ ∆ 2 i on the scale of the simulation box and how large typical values of ∆ 2 i can be for 100 h -1 Mpc boxes (c.f. their Fig. 4). Perhaps in future investigations a ξ -sampled estimator can be constructed to prevent this extra source of variance from contributing but without removing the compensation for the integral constraint bias ( § 3.4). Viewed another way, this additional complication with ξ -sampled simulations highlights the simplicity and economy of P -sampled simulations which with only a small correction for the integral constraint (Eq. 3.23) yields an unbiased estimate of the mean ξ ( r ) and does so with a large-scale variance that corresponds very closely to the approximation of gaussian random fields as it should.", "pages": [ 22, 23, 24 ] }, { "title": "7 Summary and Conclusions", "content": "This paper explores the predictions from both the conventional method of running ensembles of cosmological simulations and an alternative approach proposed by [1] and implemented by [2]. The conventional method is dubbed the P -sampled approach because it aims to maximize the correspondence between the fourier space properties of the simulation and the fourier space statistics of the assumed cosmological model whereas [1] and [2] outline a ξ -sampled approach which is built from focusing on real-space statistics. Unlike the conventional method, the real-space approach allows the DC mode to vary from box to box. In an investigation comparing the ξ -sampled and P -sampled methods for the growth and evolution of the matter-matter two-point correlation function the following conclusions were drawn: Now that the ensemble-averaged predictions for the correlation function using the ξ -sampled method have been explored and validated in some depth, future investigations with the ξ -sampled method would do well to explore the ensemble-averaged predictions for halo clustering, halo mass functions and the power spectrum. Indeed, there may be a statistic of interest for which including the fluctuations in the DC mode or some other aspect of the ξ -sampled method is of particular importance [1, 31].", "pages": [ 25 ] }, { "title": "Acknowledgements", "content": "The author thanks the Ohio State University Center for Cosmology and AstroParticle Physics for its support, and David Weinberg for guidance. Thanks also goes to Jeremy Tinker for insightful conversations, Ed Sirko for helpful correspondence and an anonymous referee who clarified some conceptual issues. A special thanks to Stelios Kazantzidis (CCAPP) and the OSU astronomy department for making available compute nodes for this project, as well as the Ohio Supercomputer Center which was also a valuable resource. This project has been supported by NSF grant AST-1009505 and AST-0707985.", "pages": [ 26 ] }, { "title": "A A Simple Expression for the Integral Constraint Bias in Λ CDM Simulations", "content": "A simple derivation can be used to estimate the bias introduced by the integral constraint for large boxes assuming a ΛCDM initial power spectrum. In this case, The integral over infinity is equivalent to P ( k → 0) which goes to zero because P ( k ) ∼ k on large scales. The other term within the brackets can be approximated analytically since on scales larger than r ∼ 250 h -1 Mpc, ξ Λ CDM ≈ ξ ∗ ( r ∗ /r ) 4 where r ∗ is a constant and the amplitude, ξ ∗ , is negative. It can be easily shown that Applying this result to estimate the fractional bias in the amplitude of the BAO feature yields where ˆ ξ ( r bao ) is the uncorrected correlation function and I have assumed ( -ξ ∗ ) /ξ ( r bao ) ≈ 2.71e-4 and r ∗ ≈ 1 h -1 Gpc using CAMB [60] and parameters from WMAP7 [61]. Formally, because of a cancellation of the square of the linear theory growth function in the ratio ( -ξ 0 ) /ξ ( r bao ), Eq. A.3 is independent of redshift and, if left unaccounted for, this measurement bias will propagate to change inferences regarding the broadening and shift of the BAO feature in the correlation function as well regardless of epoch. In more detail, redshiftdependent contributions arising from higher-order correlations can also bias the correlation function [58], so in practice Eq. A.3 can be thought of as a lower bound.", "pages": [ 26 ] } ]

[ { "title": "A. Spies, a G. Anton a", "content": "a Erlangen Center for Astroparticle Physics, Department of Physics, Friedrich-AlexanderUniversity of Erlangen-Nuremberg E-mail: [email protected], [email protected] Abstract. In this paper we study a class of supersymmetric models with non-universal gaugino masses that arise from a mixture of SU(5) singlet and non-singlet representations, i.e. a combination of 1 , 24 , 75 and 200 . Based on these models we calculate the expected dark matter signatures within the linear combination 1 ⊕ 24 ⊕ 75 ⊕ 200 . We confront the model predictions with the detected boson as well as current experimental limits from selected indirect and direct dark matter search experiments ANTARES respective IceCube and XENON. We comment on the detection/exclusion capability of the future XENON 1t project. For the investigated parameter span we could not find a SU(5) singlet model that fulfils the Higgs mass and the relic density constraint. In contrary, allowing a mixture of 1 ⊕ 24 ⊕ 75 ⊕ 200 enables a number of models fulfilling these constraints. Keywords: Supersymmetry, SU(5), non-universal, Gaugino, Higgs Boson, direct detection, indirect detection, XENON, Antares, IceCube", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Supersymmetric extensions of the standard model (SM) of particle physics provide an elegant way to solve some major problems of the SM, e.g. by explaining the huge gap between the weak scale and the Planck scale. Moreover, if R-parity is conserved, supersymmetry (SUSY) can provide a dark matter candidate in a natural way. In most SUSY scenarios, the lightest neutralino χ 0 1 is the lightest supersymmetric particle (LSP). Being a massive, electrical neutral, stable and weakly interacting particle makes it an excellent candidate to explain dark matter from the particle physics point of view. Throughout this paper we refer to χ 0 1 as the neutralino χ . Several observations constrain the parameter space of SUSY models. Such constraints are the correct parameter of electro-weak symmetry breaking or the relic dark matter density determined from WMAP data. Recently, the LHC experiments reported the discovery of a particle of 125 GeV/ c 2 mass which very probably is a Higgs boson [1], [2]. This Higgs particle will further constrain the SUSY parameter space. In this paper we study a class of supersymmetric models introduced in [3] where a mixture between the singlet representation 1 of the gauge group SU(5) and its adjoint representation 24 was considered. It was shown, that these models provide a solution to the 'little hierarchy problem' of supersymmetry and a neutralino dark matter candidate. While in [3] only a rather limited parameter range was consistent with the constraints, especially with constraints coming from the mass of the Higgs boson, we try to find more extended parameter regions that simultaneously incorporate a Higgs boson with the correct mass (within theoretical and experimental uncertainties) and an explanation for dark matter. We adapt the parameterization of [3] and extend it to the more general case of 1 ⊕ 24 ⊕ 75 ⊕ 200 . We investigate the phenomenological consequences with respect to dark matter search experiments. Two classes of experiments are considered: direct detection experiments which search for signatures produced by scattering a dark matter particle off atomic nuclei while indirect detection experiments search for self annihilation products of dark matter particles. We refer to the results published by the direct detection experiment XENON and to the indirect detection experiments ANTARES and IceCube. This paper is organized as follows. In Section 2 we introduce the parameterization used in this analysis. We comment on the recent experimental results from the LHC concerning a Higgs particle and its implications for dark matter searches in Section 3. Section 4 is dedicated to the phenomenological implications of the parameterization introduced in Section 2 with respect to the relic density resulting from such models. In Section 5 we comment on indirect detection observables, i.e. the muon neutrino flux and related muon flux from neutralino annihilations in the Sun with respect to indirect detection experiments ANTARES and IceCube with its low energy extension DeepCore (ICDC). The direct detection experiment XENON is part of our discussion in Section 6. The excluded regions of the parameter space from the combined results of IceCube and XENON 100 is presented in Section 7. In Section 8 we present a preliminary study of models with non-universal gaugino masses. We deviate from gaugino mass ratios given by SU(5) group theoretical factors, allowing for variable gaugino mass ratios. Our conclusions are presented in Section 9.", "pages": [ 3 ] }, { "title": "2 Parameterization of SU(5) Singlet and Non-singlet Combination", "content": "Non-universalities in the gaugino sector arise from a chiral function f ab ( φ I ) in the gauge kinetic part of the Lagrangian L gaugekin [4]. φ I is a chiral superfield and L gaugekin is given by [5] where a , b are indices of the gauge generators, φ I 's denote the chiral superfields and λ a is the SU(5) gaugino field. ϕ I is the scalar component of φ I whereas F I is its auxiliary F-component. f ab transforms as the symmetric product of two adjoint representations and is given by [5] In the above equations Einsteins sum convention was used for indices appearing twice. f 0 and ζ Mult are functions of gauge singlets φ singlet . The index 'Mult' labels possible multiplets of Eq. 2.2, that are allowed as a linear term of φ Mult in f ab ( φ I ). Supersymmetry is broken by the F-components F I of the chiral superfields φ I when they acquire non-zero vevs and thus gaugino masses are generated. For the case of a non-singlet these gaugino masses ( M 1 , M 2 , M 3 ) are unequal but related to each other [6]. Their relative magnitude at the scale of grand unification is given by group theoretical factors according to [7] with the coefficients c a listed in Table 1 A mixture of singlet and non-singlet representations can be written in form of three nonuniversality equations for M 1 , M 2 and M 3 where i = 24 , 75 , 200 labels the possible multiplets, ( a 24 , a 75 , a 200 ) = (1 , 5 , 10), ( b 24 , b 75 , b 200 ) = (3 , -3 , 2) and ( c 24 , c 75 , c 200 ) = ( -2 , -1 , 1) . θ 1 reflects the contribution of the singlet. θ i reflects the contribution of the corresponding multiplet to the non-universality of the model. If θ 1 = 0 and all θ i = 0 we obtain the cMSSM scenario (often referred to as mSUGRA) where M 1 = M 2 = M 3 = m 1 / 2 . For θ 1 = π/ 2 and all θ i = π/ 2 we have a pure SU (5) non-singlet contribution reflecting the given mass ratios of Table 1. The above parameterization was adapted from [3]. There, only 1 ⊕ 24 was considered. Instead of coefficients a i , b i and c i only coefficients a 24 , b 24 , and c 24 with ( a 24 , b 24 , c 24 ) = (1 , 3 , -2) occur in Equation 2.5. Moreover, the analysis of [3] imposed θ 1 = θ 24 . We will refer to that model briefly in the next Section. In total we obtain a 9 dimensional parameter space glyph[negationslash] We restricted the 5 parameters m 0 , m 1 / 2 , A 0 , tan β , sign( µ ) to the region which was evaluated in [3] in order to be able to investigate how far a generalised mixing ( θ i = 0) changes the results. Accordingly our simulations use the following parameter range: When we use a different set of parameters it is explicitely mentioned in the text. Our results for the linear combination 1 ⊕ 24 are in good agreement with those given in [3]. We found the combinations 1 ⊕ 75 and 1 ⊕ 200 provide less models fulfilling the constraints of the following sections compared to the parameterization of Younkin and Martin [3]. In principle any of the representations appearing in the symmetric product (Eq. 2.2) must be treated equal and non of them should be preferred. To calculate the supersymmetric particle spectrum we used the public code SuSpect [8]. DarkSUSY [9] was employed for simulating dark matter observables. As such observables we investigated the muon neutrino flux φ ν µ and the resulting muon flux φ µ for indirect dark matter detection. As signal of direct detection we calculated the spin independent WIMP nucleon cross-section, σ nucleon SI .", "pages": [ 4, 5, 6 ] }, { "title": "3 Higgs candidate from LHC results", "content": "Last year, LHC experiments reported the discovery of a new boson with mass of 125 GeV/ c 2 which might turn out to be the Higgs boson [1], [2]. In order to take theoretical uncertainties of the calculated mass of the Higgs boson into account we allow an uncertainty of ± 3 GeV/ c 2 [19] on the mass of the Higgs boson. The implications of such a Higgs boson on the kind of models investigated in this paper is discussed in this Section. Younkin and Martin indicated that only a few models survive the experimental and theoretical constraints from the Higgs boson. In their simulations they kept the gluino mass parameter M 3 and subsequently the bino mass parameter M 1 fixed. Here, we use a slightly different parameterization for 1 ⊕ 24 . Instead of fixing M 3 and M 1 , we fix the overall gaugino mass scale m 1 / 2 = 600 GeV (see Section 2). In contrast to Younkin et al, we simulated the model predictions with respect to the Higgs mass for independently varying the singlet mixing angle, θ 1 , and the 24-plet mixing angle θ 24 . The predicted distribution of the Higgs mass is shown on the left hand side of Figure 1. The simulations were carried out for m 0 = 4 TeV and tan β = 10. From the results of [3], we expected only a few models above our required lower limit of m h > 122 GeV. Indeed, only ∼ 4% of the simulated models achieve m h > 122 GeV. Models that do not provide a Higgs boson mass with 122 < m h < 128 GeV must be rejected. So, the aim of this paper is to find a linear combination whose predictions with respect to the Higgs boson mass are more promising compared to that of Younkin et al. We investigated the linear combination 1 ⊕ 24 ⊕ 75 ⊕ 200 . To compare the results with those of 1 ⊕ 24 we also keep m 0 = 4 TeV and tan β = 10. We independently varied the mixing angles θ 1 , θ 24 , θ 75 and θ 200 in the range given in Section 2. In our model ∼ 36% of the simulated models provide a Higgs boson with m h > 122 GeV. Thus, a significant larger number of models in our parameterization can provide a Higgs boson in agreement with experimental measurements. This result is shown on the right hand side of Figure 1. In order to identify regions in the two (four) dimensional parameter space of mixing angles θ 1 and θ 24 for 1 ⊕ 24 ( θ 1 , θ 24 , θ 75 and θ 200 for 1 ⊕ 24 ⊕ 75 ⊕ 200 ), where 122 < m h < 128 GeV is given, we fixed the remaining input parameters of the model to the values given in the caption of Figure 1 leaving only the two (four) angles as free parameters with -0 . 25 < θ i /π < 0 . 75. For 1 ⊕ 24 the number of models resulting in predictions 122 < m h < 128 GeV is plotted versus the mixing angle θ 1 /π respectively θ 24 /π in Figure 2. In Figure 3 the number of models resulting in predictions 122 < m h < 128 GeV is plotted versus the mixing angles θ 1 /π , θ 24 /π , θ 75 /π and θ 200 /π (for 1 ⊕ 24 ⊕ 75 ⊕ 200 ). Although the relic density is part of the discussion in the next Section, Figures 2 and 3 also show models that simultaneously fulfil 122 < m h < 128 GeV and Ω h 2 < 0 . 13 (red line). In the case of 1 ⊕ 24 only restricted regions for θ 1 /π and θ 24 /π provide a Higgs boson with 122 < m h < 128 GeV. These are the regions with -0 . 16 < θ 1 /π < 0 . 16, 0 . 34 < θ 1 /π < 0 . 42 and 0 . 62 < θ 1 /π < 0 . 68 for the singlet contribution. The 24-plet contribution leads to 122 < m h < 128 for θ 24 glyph[lessorsimilar] -0 . 18 and two small bumps at θ 24 /π ∼ -0 . 05 and 0.03. The situation changes drastically when considering the most general linear combination 1 ⊕ 24 ⊕ 75 ⊕ 200 . Possible models that provide a Higgs boson with the correct mass can be obtained for all angles θ i over the complete range -0 . 25 < θ i /π < 0 . 75 (Figure 3). As mentioned above, the red line in Figure 2 and 3 represents the number of models with 122 < m h < 128 GeV and Ω h 2 < 0 . 13. In the case of 1 ⊕ 24 regions with 122 < m h < 128 GeV and Ω h 2 only partially coincide, such that in total only a few models provide a Higgs boson with the right mass and correct dark matter relic density. In the case of 1 ⊕ 24 ⊕ 75 ⊕ 200 over the complete range of the θ i 's models that simultaneously fulfil Ω h 2 < 0 . 13 and 122 < m h < 128 can be found. The relic density is part of the discussion in the next Section where we combine particle physics with astroparticle physics. There, we investigate our models predictions with respect to the relic density of the neutralino, which is assumed to be the dark matter particle. From now on we concentrate on linear combination 1 ⊕ 24 ⊕ 75 ⊕ 200 . For simplicity and graphical representation purposes we unify the mixing angles θ i , where θ 1 = θ 24 = θ 75 = θ 200 ≡ θ . Of course, reducing the number of free parameters also reduces the allowed regions of the parameter space where the Higgs mass is consistent with measurements. But with this restriction still a factor of ∼ 10 more models in our parameterization survive the Higgs constraints compared to 1 ⊕ 24 .", "pages": [ 7, 8, 9 ] }, { "title": "4 Relic Density for Non-universal Gaugino Masses", "content": "The calculated relic density, Ω h 2 , predicted by the models introduced in Section 2, determines whether the corresponding model can give an explanation for dark matter. The relic density of dark matter, deduced from WMAP data, is restricted to 0 . 106 < Ω h 2 < 0 . 118 given in [10]. We relaxed the previous constraint to Ω h 2 < 0 . 13. The relaxed upper constraint accounts for the possibility that R-Parity may not be conserved, while the relaxed lower bound includes the possibility that dark matter may not be only made up by one single particle. Furthermore, a lower value of Ω h 2 with respect to WMAP data can be accepted if the missing relic density is filled up with dark matter particles produced non-thermally, e.g. the decay of long lived particles or cosmic strings [11], [12], [13]. Throughout the following sections we present our results for tan β = 10 on the left and tan β = 45 on the right hand side. In Figure 4 Ω h 2 is shown for the mixing of representations 1 ⊕ 24 ⊕ 75 ⊕ 200 . All models with Ω h 2 > 0 . 13 are colored red, while models with Ω h 2 < 0 . 13 are colored yellow. Colored faint blue are parameter regions which yield squark masses m squark < 1 . 4 TeV. Such squark masses are already excluded by LHC [14]. Colored green regions refer to models with Ω h 2 < 0 . 13 and a Higgs mass of 122 < m h < 128 GeV. Green colored models are capable to account for dark matter and providing a Higgs boson with mass 122 < m h < 128. The parameters in the white region between 0 . 1 glyph[lessorsimilar] θ/π glyph[lessorsimilar] 0 . 4 for both tan β values do not have a correct electro weak symmetry breaking, i.e. do not have a convergent µ from solving the renormalization group equations (RGE). The region between -0 . 16 glyph[lessorsimilar] θ/π glyph[lessorsimilar] -0 . 12 for both tan β violate the LEP2 bound on the chargino mass [10]. The white region between -0 . 07 glyph[lessorsimilar] θ/π glyph[lessorsimilar] 0 . 04 is forbidden due to tachyonic third generation sfermions (tan β = 45) or has a Higgs potential that is unbound from below or lead to charge and color breaking minima (see e.g. [15] - [18]), for tan β =10. θ/π values greater than ∼ 0.4 for tan β =45 lead to a tachyonic pseudoscalar Higgs boson A and are excluded. For tan β = 10, several coannihilation regions occur. For -0 . 1 glyph[lessorsimilar] θ/π glyph[lessorsimilar] 0 . 08 and m 0 glyph[lessorsimilar] 1 . 5 TeV as well as 0 . 54 < θ/π < 0 . 6 and m 0 < 400 GeV the tau slepton, ˜ τ , and the tau sneutrino, ˜ ν τ , are nearly degenerate in their masses with the neutralino mass. Further, the top squark, ˜ t , coannihilates with the neutralino for 0 . 02 glyph[lessorsimilar] θ/π glyph[lessorsimilar] 0 . 08 and m 0 glyph[lessorsimilar] 2 TeV. Large parts of this coannihilation regions are excluded because the squarks are lighter than 1.4 TeV for -0 . 05 < θ/π < 0 . 12 and m 0 < 2 TeV (indicated in faint blue in Figure 4). Resonant annihilation regions with the pseudoscalar Higgs Boson A and the lightest Higgs boson h similar to h- and A- funnel regions of the cMSSM can be found in the vicinity of the LEP2 bound, for h , and 0 . 02 glyph[lessorsimilar] θ/π glyph[lessorsimilar] 0 . 08 and m 0 < 1 TeV for A . The latter one is excluded by squarks that are too light. For tan β = 45, the above regions coincide, except the ˜ τ and ˜ ν τ coannihilation region for tan β ∼ 0 . 5, because of a tachyonic pseudoscalar Higgs boson. As for tan β = 10, most of the coannihilation regions are excluded by squarks that are too light for -0 . 04 glyph[lessorsimilar] θ/π glyph[lessorsimilar] 0 . 1 and m 0 < 2 TeV. The red region on the left side of Figure 4 has a neutralino entirely made up of the bino similar to the bulk region of the cMSSM. The yellow regions are characterized by a neutralino, that is either wino dominated ( θ/π < -0 . 05 and θ/π > 0 . 58) or higgsino dominated ( θ/π > 0 . 04 and θ/π < 0 . 52) or a mixture between wino and higgsino (0 . 02 glyph[lessorsimilar] θ/π glyph[lessorsimilar] 0 . 06 and 0 . 53 glyph[lessorsimilar] θ/π glyph[lessorsimilar] 0 . 56). The relic density in this regions is smaller than the lower bound deduced from WMAP data, i.e. smaller than 0.106. Moreover, in wino and/or higgsino dominated regions the lightest neutralino is almost degenerate with the lightest chargino and pairs of neutralinos annihilate very efficiently via t-channel chargino exchange into pairs of W bosons and the neutralinos relic population is depleted. Nevertheless, boundary conditions from WMAP correspond to thermally produced dark matter. Wino or higgsino like dark matter could have been produced non-thermally, in a sense that Ω h 2 is decomposed into the sum of thermally plus non-thermally produced dark matter, Ω = Ω therm +Ω nontherm (see e.g. [13]). The total relic density of cold dark matter can than be kept in agreement with observations. That is why we relax WMAP constraints to 0 < Ω h 2 < 0 . 13. The same arguments apply for tan β =45 (right hand side of Figure 4). Wino dominated regions are found for θ/π < -0 . 05. For θ/π glyph[greaterorsimilar] 0 . 04 the neutralino is dominated by the higgsino and for 0 . 02 glyph[lessorsimilar] θ/π glyph[lessorsimilar] 0 . 04 the neutralino is a mixture between wino and higgsino. Again the red region is characterized by a bino dominated neutralino. Models, where µ has a desired small value, so that it can solve the little hierarchy problem coincide partially with higgsino dominated regions. Small µ regions are found as thin contours on top and on the right edge for both tan β = 10 and tan β = 45 of the allowed models. For tan β = 10 they are also found on top and on the left edge of allowed models. For tan β = 45 only a thin arc remains on the right side. After introducing the possible dark matter scenarios from the model setup mentioned in Section 2 we now investigate constraints given by selected indirect and direct detection experiments.", "pages": [ 10, 11 ] }, { "title": "5 Indirect Detection", "content": "Although, there are plenty of indirect detection observables, e.g. photon flux, anti-proton flux, e + -e -flux, that are worth being investigated, this paper is limited to the muon neutrino flux φ µ ν and the muon flux φ µ when talking about indirect detection. We focus on the indirect detection experiments ANTARES and IceCube. These experiments measure muons via the detection of Cerenkov light, which is emitted by the charged muons traveling through water or ice. The muon flux coming from below the detector is due to muon neutrinos which interact in charge current interactions close to the detector. In Figure 5 the predicted integrated muon and anti muon neutrino flux from the Sun is plotted logarithmically versus the mass of the neutralino, m χ . The highest neutrino fluxes and therefore also highest muon fluxes are expected at the 'high higgsino' or 'small µ ' region explained in Section 3. In this parameter region the neutralinos can annihilate into Higgs and weak vector bosons resulting in a high muon neutrino flux. The black, magenta and brown lines in Figure 5 correspond to the ANTARES limit at 90% Confidence Level assuming that all neutralinos annihilate exclusively into either b ¯ b , W + W -or τ + τ -, so that the limit is independent from the choice of the SUSY model. As can be seen in Figure 5 ANTARES is not yet able to exclude the kind of models introduced in Section 2. The published ANTARES limit [20] is based on 282.84 days of data taking that include a correction of 20% for the 5 line detector configuration. More stringent limits are expected from the analysis of further data taken with the 12 line detector configuration. The predicted neutrino induced muon and anti muon fluxes from neutralino annihilations in the Sun are displayed in Figure 6 for combination 1 ⊕ 24 ⊕ 75 ⊕ 200 and both tan β = 10 and tan β = 45. As in the case of the neutrino flux, the limits given in Figure 6 assume a 100% annihilation into W + W -. The gray line in Figure 6 correspond to the limit at 90% C.L. for the IceCube neutrino telescope [21] with 86 strings including the low energy extension DeepCore (ICDC). From Figure 6 it follows that still a reasonable amount of models, consistent with Ω h 2 < 0 . 13 and 122 < m h < 128, are not yet excluded by IceCube The exclusion limit of Figure 6 is projected onto the m 0 -θ plane to visualize which part of the parameter space can be excluded. These regions are shown in Figure 7 for the annihilation channel W + W -. For tan β = 10, large parts of the 'high higgsino' or 'small µ ' regions for 0 glyph[lessorsimilar] θ/π glyph[lessorsimilar] 0 . 1 and on top ( m 0 glyph[greaterorsimilar] 4 TeV, -0 . 2 glyph[lessorsimilar] θ/π glyph[lessorsimilar] 0 and 0 . 64 glyph[lessorsimilar] θ/π < 0 . 75) can be excluded assuming all annihilations go into W + W -. This also applies for tan β = 45, where models with m 0 > 3 . 5 TeV and -0 . 2 < θ/π glyph[lessorsimilar] 0 are excluded as well as models with 0 glyph[lessorsimilar] θ/π < 0 . 1. Also, the thin arc on the right hand side for tan β = 45 is excluded by IceCube. From the indirect detections point of view the limits on the muon and anti muon flux given by the IceCube collaboration have the best power to exclude the kind of models we investigated. After investigation of exclusion capabilities of indirect detection with neutrino telescopes, in the next Section we focus on direct detection methods, i.e. elastic scattering interactions of a WIMP with a nucleon of the target material. We concentrate on results of the XENON 100 experiment [24] and comment on the future extension XENON 1t [25].", "pages": [ 12, 13, 14 ] }, { "title": "6 Direct Detection", "content": "In this Section we focus on direct detection methods of dark matter, i.e. we compare the phenomenology of our models with the latest results given by the XENON collaboration. We concentrate on the spin independent WIMP nucleon cross-section σ nucleon SI . The predicted cross-sections are plotted in Figure 8 versus the mass of the neutralino m χ which is assumed to be the WIMP. Clearly visible is the fact, that only XENON 100 of all direct detection experiments shown here is able to exclude any of the simulated models. Nevertheless, several of the simulated models with a Higgs mass of 122 < m h < 128 and Ω h 2 < 0 . 13, are not yet excluded by direct detection experiments. Even with the predicted sensitivity of the future extension XENON 1t (black line in Figure 8) a reasonable number of models that fulfill our requirements for Ω h 2 and the Higgs mass m h survive. This is not the case for the previously studied models [3], where ∼ 95% of these models for low tan β and all models for high tan β are excludable by XENON 1t. The corresponding excluded (excludable) parameter space for XENON 100 (XENON 1t) in the m 0 -θ plane is shown in Figure 9 and 10. For tan β = 10 (left hand side of Figure 9), XENON 100 excludes many of the 'small µ /high higgsino' models on the right and upper edge of the left 'island'. The faint red band on the right 'island' that is excluded by XENON 100 corresponds to models where the neutralino is a mixture between wino and higgsino. In this region µ is already too high to solve fine-tuning problems, as it is the case for small µ . For tan β = 45 (right hand side of Figure 9), again the right and upper edge of the left 'island' is excluded by XENON 100. As for the case of low tan β models belonging to that region have a small µ and solve the little hierarchy problem of supersymmetry. Complementary to IceCube, no models on the thin arc on the right hand side are excluded by XENON 100. Almost the whole cMSSM like focus point region with a small µ and a large higgsino fraction in the neutralinos composition can be tested by XENON 1t for tan β = 10 and 45 (Figure 10). Only a few models for -0 . 04 glyph[lessorsimilar] θ/π < 0 and m 0 > 4 TeV (tan β = 10) and m 0 > 3 . 5 TeV (tan β = 45) may survive. In the case tan β = 10, all models on the right 'island' that fulfill Ω h 2 < 0 . 13 and 122 < m h < 128 GeV would be excludable by XENON 1t. Only models with θ/π glyph[lessorsimilar] 0 . 18 still provide a consistent Higgs mass with the relic density requirement. For tan β = 45 a triangular shaped region survives the predicted sensitivity of XENON 1t. It can be found for m 0 > 3 . 4 TeV and θ < 0 . 16. When comparing the experimental limits presented in Section 5 and 6 to the model predictions one should keep in mind the uncertainties and approximations which are contained in the models. These uncertainties comprise the WMIP nucleon cross-section, σ nucleon SI , from uncertainties in nuclear matrix elements and uncertainties in the local WIMP density, which affect the capture rate of neutralinos in the Sun and also the predictability with respect to direct detection.", "pages": [ 15, 16 ] }, { "title": "7 Combined Excluded Regions of the Parameter Space", "content": "In this section we summarize the number of models simulated, models that fulfill our relic density requirement and models that can be excluded by ANTARES for W + W -, τ + τ -, IceCube + DeepCore (ICDC) for W + W -and XENON with and without respect to a correct relic density. Numbers can be found in Table 2. The prefix 'DM' in Table 2 means models that have relic density with Ω h 2 < 0 . 13 while the prefix 'Higgs' are those with 122 < m h < 128 GeV. We present excluded regions of the parameter space when combining the most stringent constraints coming from the Higgs boson as well as XENON 100 and IceCube limits. The allowed regions (colored blue) that fulfill the constraints Ω h 2 < 0 . 13 and 122 < m h < 128 GeV are summarized in Figure 11 (for the parameter space of representation 1 ⊕ 24 ⊕ 75 ⊕ 200 ). Colored red are those models, that have a relic density Ω h 2 < 0 . 13 and Higgs boson with 122 < m h < 128 GeV and are excluded by either IceCube or XENON 100 at 90% C.L. Models not satisfying 122 < m h < 128 GeV or Ω h 2 < 0 . 13 are colored gray.", "pages": [ 17 ] }, { "title": "8 Variable Gaugino Mass Ratios", "content": "In the previous Sections we have shown, that a correct Higgs mass can be achieved for models with fixed mixing angles θ i ≡ θ and m 0 glyph[greaterorsimilar] 4 TeV. Thus, the linear combination 1 ⊕ 24 ⊕ 75 ⊕ 200 provides significantly more models that fulfil the constraint 122 < m h < 128 GeV compared to 1 ⊕ 24 . Nevertheless, we had to relax the relic density requirements and include the possibility for dark matter to be produced non-thermally. Otherwise, most of the simulated models do not produce a sufficient amount of thermally produced dark matter. In this Section we present a preliminary study of deviations from the mass ratios of SU(5) representations. We parameterize the gaugino masses M 1 , M 2 and M 3 similar to Equation 2.5, given by where coefficients a, b and c are allowed to vary in the range [-10,10]. We fixed m 0 = 3 TeV, m 1 / 2 = 600 GeV and A 0 = -m 1 / 2 . We simulated models for θ and tan β pairs ( θ, tan β ) = (-45,10), (-45,45), (115,10) and (115,45). The number of models simulated, as well as the number of models that are consistent with 122 < m h < 128 GeV and 0 . 09 < Ω h 2 < 0 . 13 are listed in Table 3. We required a more stringent bound on the relic density 0 . 09 < Ω h 2 < 0 . 13, in contrast to the previous sections. Models with 122 < m h < 128 GeV are labeled ' n models Higgs' in Table 3. Models with relic densities 0 . 09 < Ω h 2 < 0 . 13 are labeled ' n models DM'. The constraint on Ω h 2 slightly deviates from the limits given by the WMAP collaboration. It includes the possibility of a broken R-parity and thus decaying dark matter as well as an additional dark matter component, e.g. axions. To find the optimal triple (a,b,c) for each of the pairs ( θ ,tan β ) from Table 3 that describes m h and Ω h 2 best, we performed a χ 2 analysis according to where we took Ω h 2 observed = 0 . 11 ± 0 . 02. m h, observed = 125 . 3 GeV is the observed mass of the Higgs boson given by the CMS collaboration [1]. σ theo Higgs = 3 GeV and σ observed Higgs = 0 . 4(stat . ) + 0 . 5(syst . ) GeV are the theoretical and experimental uncertainties, respectively on the Higgs boson. The resulting χ 2 -values for (a,b,c) are listed in Table 4 We use the triples (a,b,c) from the four simulated nodes listed in Table 4 to fit linear functions a(x,y), b(x,y) and c(x,y) for the coefficients given in Eq. 8.1, where x = sin( θ ) and y = tan β . These linear functions are given by With these linearized coefficients we simulated ∼ 300 000 models with our benchmark point input parameters m 0 = 3 TeV, m 1 / 2 = 600 GeV and A 0 = -m 1 / 2 . We varied θ from -45 to 135 degree and tan β from 2 to 60. Only for -0 . 1 glyph[lessorsimilar] θ/π glyph[lessorsimilar] 0 . 1 m h is below 122 GeV (Figure 12). As for these small angles sin( x ) ∼ x and cos( x ) ∼ 1, the region -0 . 1 glyph[lessorsimilar] θ/π glyph[lessorsimilar] 0 . 1 is obviously dominated by the singlet and unified gaugino masses like in cMSSM. This 'cMSSM' like Higgs region is already known to be disfavored by the measurements of CMS and ATLAS. Further, small values of tan β , i.e. tan β glyph[lessorsimilar] 5 do not satisfy the lower bound m h > 122 GeV. Further, the curly white band in the middle of the plot is excluded by the LEP2 limit on the chargino mass [10]. Large parts of the parameter space in the θ -tan β plane have Ω h 2 > 0 . 16 (Figure 12). Two extended regions where Ω h 2 glyph[lessorsimilar] 0 . 13 and 122 < m h < 128 GeV is fulfilled can be found for 5 glyph[lessorsimilar] tan β glyph[lessorsimilar] 32 and θ/π < -0 . 16, 8 glyph[lessorsimilar] tan β glyph[lessorsimilar] 54 and 0 . 1 glyph[lessorsimilar] θ/π 0 . 75. These regions are characterized by a neutralino that is a pure wino. Further, the neutralino is nearly degenerate in its mass with the lightest chargino and pairs of neutralinos can annihilate into pairs of Wbosons via t-channel chargino exchange. Similar regions were already found in Section 4. Unfortunately, this annihilation process is very efficient and the relic density is pushed below 0.09 (light green/turky and purple regions in Figure 12). Like in Section 4, where we allowed dark matter to be produced non-thermally, this assumption has to be made to make models in this regions viable dark matter models. Solid black colored regions on the right hand side of Figure 12 fulfil 0 . 09 < Ω h 2 < 0 . 13 and 122 < m h < 128 GeV. There, the wino neutralino is heavy enough ( O (2 TeV)) to produce the right amount of thermally produced dark matter [26]. Summarizing the above results: we simulated approximately 300 000 models. ∼ 73% of them are consistent with 122 < m h < 128 GeV. ∼ 4% of all models have a relic density with 0 . 09 < Ω h 2 < 0 . 13. 91% of models with 0 . 09 < Ω h 2 < 0 . 13 provide a Higgs boson with a mass consistent with measurements. Of course, relaxing the relic density bound to Ω h 2 < 0 . 13 (see Section 4) increases the number of models consistent with Ω h 2 . In that case ∼ 45% of all models fulfil the constraint on Ω h 2 . ∼ 90% of these models have a Higgs boson with mass 122 < m h < 128 GeV. We imposed a linear dependence of coefficients (a,b,c) on the mixing angle θ and tan β . Simulating more nodes for pairs ( θ, tan β ) would allow to introduce coefficients (a,b,c) that depend non-linearly on ( θ, tan β ), allowing broader regions in the θ -tan β plane that fulfil 0 . 09 < Ω h 2 < 0 . 13. Nevertheless, this preliminary analysis indicates, that variable coefficients of Eq. 8.1 easily provide models that describe the Higgs mass as well as the neutralino as dark matter candidate. Further, these models escaped LHC measurements, as squarks and gluinos are too heavy ( > 1.5 TeV for gluinos, > 2.1 TeV for squarks).", "pages": [ 19, 20, 21 ] }, { "title": "9 Conclusion", "content": "We investigated supersymmetric models with non-universality in the gaugino sector. This class of models was first introduced by Younkin and Martin [3], who investigated a mixing of SU (5)'s singlet representation with the 24 representation. We extended this mixture to the more general case of all representations appearing in the symmetric product of 24 ⊗ 24 . We focused on the phenomenological implications with respect to the relic density, to recent experimental results from selected direct and indirect detection measurements and to a possible Higgs boson with a mass of 122 < m h < 128 GeV. The probable detection of the Higgs boson last year puts strongest constraints on the parameter space investigated by Younkin et al. Extending the parameter space by the four mixing angles θ 1 , θ 24 , θ 75 and θ 200 (in contrast to one single mixing angle θ in [3]) extends the phenomenological implications of models with non-universal gaugino masses, i.e. provides a solution to this 'Higgs' problem. We found a factor of ∼ 9 more models provide a candidate model with a Higgs boson mass that is consistent with measurements within experimental and theoretical uncertainties. These regions are not constrained to a certain range of angles θ i but cover the complete simulated range of θ i . Furthermore, models with Ω h 2 < 0 . 13 highly coincide with models where 122 < m h < 128 GeV is respected. Gluino and squark masses are sufficiently high to escape LHC experiments from detection, such that this kind of models are still viable models that can explain a Higgs boson with a mass of 125 GeV/ c 2 and provide the neutralino as a dark matter candidate, given the possibility for dark matter to be produced non-thermally. We performed a detailed study on the dark matter relic density and regions that are excluded by direct and selected indirect detection experiments as well as Higgs boson mass constraints. For simplicity and the sake of clarity we unified the mixing angles θ 1 = θ 24 = θ 75 = θ 200 ≡ θ . We found that the parameter space of the considered model can be classified into four regions with respect to the neutralinos composition. These are a pure wino region ( θ/π glyph[lessorsimilar] -0 . 06 and θ/π glyph[greaterorsimilar] 0 . 58 for tan β = 10, θ/π glyph[lessorsimilar] -0 . 04 for tan β = 45), and second a pure bino region for -0 . 05 glyph[lessorsimilar] θ/π glyph[lessorsimilar] 0 . 03 for both tan β = 10 and 45. In the third region the neutralino is a pure higgsino (0 . 06 glyph[lessorsimilar] θ/π glyph[lessorsimilar] 0 . 52 for tan β = 10 and θ/π glyph[greaterorsimilar] 0 . 06 for tan β = 45). The last region is characterized by a neutralino that is a wino/higgsino mixture. It can be found for 0 . 02 glyph[lessorsimilar] θ/π glyph[lessorsimilar] 0 . 06 and 0 . 52 glyph[lessorsimilar] θ/π glyph[lessorsimilar] 0 . 56 (tan β = 10) and for 0 . 02 glyph[lessorsimilar] θ/π glyph[lessorsimilar] 0 . 06 (tan β = 45). The relic density in the pure bino region is higher than the upper constraint on Ω h 2 (Ω h 2 < 0 . 13), analogue to the bulk region of the cMSSM. Concerning the other regions Ω h 2 drops rapidly below the lower bound deduced from WMAP data. We showed, that relaxing the WMAP constraint to lower values, allowing the possibility for dark matter to be produced non-thermally, provides significantly more models that satisfy the relaxed relic density requirement compared to the model parameterization introduced by Younkin et al. To obtain a mass of the Higgs boson within 122 < m h < 128 GeV, m 0 must be at least 4 TeV and θ/π must be smaller than zero, or bigger than 0.54 for tan β = 10. For tan β = 45, m 0 must exceed ∼ 3.4 TeV and θ/π must be lower than 0 . 02. Nevertheless, more models provide a Higgs boson ( ∼ factor of two for tan β = 10, and ∼ factor of 9 for tan β = 45), that satisfies 122 < m h < 128 GeV compared to the parameterization introduced by [3]. Almost all of these models ( O (95%)) agree with Ω h 2 < 0 . 13. Currently, the best exclusion limit for non-universal models investigated in this work are given by the IceCube and XENON collaborations. IceCube can exclude 8% and 18.5% of all models in the channel W + W -for tan β = 10 and tan β = 45, respectively. This means that approximately 20% (44%) of models that agree with a Higgs of 122 < m h < 128 GeV and Ω h 2 are excluded for tan β = 10 (45). The current XENON 100 direct detection experiment with a life time of 225 days excludes ∼ 11% of all models for tan β = 10 and ∼ 18% for tan β = 45. This corresponds to approximately 21% of excluded models with 122 < m h < 128 GeV and Ω h 2 (tan β = 10) and ∼ 30% for tan β = 45. The future XENON 1t, will be even more restrictive. It can exclude ∼ 83% and ∼ 76% of all models, for tan β = 10 and 45,respectively, which means that ∼ 78% and ∼ 81% of models consistent with Higgs and Ω h 2 can be tested. For the parameter regions tested in this paper we find that a SU(5) singlet is not able to describe a supersymmetric scenario which is in agreement with the dark matter relic density and the observed Higgs mass. Instead, a mixing of other representations into the singlet allows for models consistent with observations. A linear combination including all non-singlet representations of SU(5) that appear in the symmetric product of 24 ⊗ 24 cannot be excluded by current measurements from direct and the considered indirect detection methods. With the newly detected particle at the LHC being the Higgs boson reduces the possible parameter space, but neither existing data from dark matter search nor predicted sensitivities, e.g. XENON 1t, can ultimately exclude the model investigated in this work. Last but not least we did a preliminary analysis on the gaugino mass ratios. We allowed a variable mass ratio between the gaugino mass parameters M 1 , M 2 and M 3 . Therefore, we varied the coefficients (a,b,c) that determine the ratios of M i at a given mixing angle θ at the GUT scale. We found that a Higgs boson with a mass of 122 < m h < 128 GeV is easily achieved, even at lower values of m 0 = 3 TeV compared to m 0 glyph[greaterorsimilar] 4 TeV with respect to the other linear combinations. Furthermore, it is possible to require more stringent constraints on the relic density, e.g. 0 . 09 < Ω h 2 < 0 . 13. A reasonable amount of models with these constraints remain and provides candidate models for dark matter produced thermally while simultaneously satisfying constraints from the Higgs boson.", "pages": [ 22, 23 ] }, { "title": "Acknowledgments", "content": "We would like to thank S.P. Martin for various fruitful discussions. This work was partially funded by the German Ministry of Education and Research (BMBF) contract number 05A11WEA.", "pages": [ 24 ] }, { "title": "References", "content": "http://cds.cern.ch/record/1472710/files/ATLAS-CONF-2012-109.pdf", "pages": [ 24 ] } ]

[ { "title": "G. A. Caliandro 1 , D. F. Torres 1 , 2 , & N. Rea 1", "content": "1 Institute of Space Sciences (IEEC-CSIC), Campus UAB, Fac. de Ci'encies, Torre C5, parell, 2a planta 08193 Barcelona, Spain 2 Instituci'o Catalana de Recerca i Estudis Avanc¸ats (ICREA) Barcelona, Spain Abstract. Here, we present a new method to evaluate the expectation value of the power spectrum of a time series. A statistical approach is adopted to define the method. After its demonstration, it is validated showing that it leads to the known properties of the power spectrum when the time series contains a periodic signal. The approach is also validated in general with numerical simulations. The method puts into evidence the importance that is played by the probability density function of the phases associated to each time stamp for a given frequency, and how this distribution can be perturbed by the uncertainties of the parameters in the pulsar ephemeris. We applied this method to solve the power spectrum in the case the first derivative of the pulsar frequency is unknown and not negligible. We also undertook the study of the most general case of a blind search, in which both the frequency and its first derivative are uncertain. We found the analytical solutions of the above cases invoking the sum of Fresnel's integrals squared. Keywords: stars: neutron, pulsars, gamma-rays: observations, time series, power spectrum", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Pulsar timing from long observations is a current topic of astrophysics. This is especially due to the discovery of an unprecedented population of gamma-ray pulsars by the Large Area Telescope (LAT) on board the Fermi satellite [1]. The low rate of gamma-rays implies the need of very long observations so as to reach a significant detection of the pulsations. The survey mode of Fermi -LAT perfectly matches this need, and a timing consortium of radio and X-ray observers has been formed in order to build long ephemeris to adequately extract pulsations [2]. More recently, adapting the radio techniques, [3] presented a method to build long ephemeris using gamma-ray data. Since Fermi -LAT continuously monitors all the sky, these ephemeris have an accuracy that is even better than the radio ones for bright gamma-ray pulsars. Efforts were also spent in order to find new techniques to make the blind search for gamma-ray pulsations more efficient and computationally feasible. Indeed, due to the long integration times, a full Fourier analysis is computationally too demanding for the search of pulsations in gamma-rays, particularly when lacking knowledge from other wavelengths (a blind search). A sub-population of radio quiet gamma ray pulsars ([4], [5]) has been discovered with LAT data thanks to the 'time differencing technique' [6]. Recently, a new technique dedicated especially to the search of millisecond pulsars has also been developed [7]. The cited works on pulsar timing have greatly improved the analysis of isolated pulsars, and their blind search on gamma-rays. On the other hand, the search of gamma-ray pulsars in binary systems is performed only when their radio ephemeris are known. Indeed, the first pulsar catalog of Fermi -LAT [1] includes only five radio loud millisecond pulsars in binary systems. The reason is mainly that the Doppler shift due to the orbital motion of the pulsar wash out the pulsations at the observer's frame of reference. The correction for the orbital motion (as well as for other effects due to the companion star) need a very precise knowledge of the orbit. Since there are several parameters that define the orbit, uncertainties in one or more of them can vanish the effect of the correction. We have recently presented a systematic study on how the uncertainties of the orbital parameters affect the pulsed signal [8]. In that paper we already made use of a new method to compute the expectation value of the power spectrum at the signal and nearby frequencies, based on a statistical approach, which we will detail and prove here. The rest of the paper is organized as follows: In Section 2 we recall the main approaches used in the literature to compute the expectation value of a time series up to now. In Section 3 we demonstrate our method, which we validate it in Section 4 by means of numerical simulations and by reproducing known features of the power spectrum of a periodic times series. In sections 5 and 6, respectively, we apply the method to analytically solve the power spectrum in the case the first derivative of the frequency is uncertain and not negligible, and in the case of a blind search where both the frequency and its derivative are unknown. Finally, in Section 7 we conclude underlining some interesting aspects of this approach.", "pages": [ 2, 3 ] }, { "title": "2 Context", "content": "In our work we calculate the expectation value of the power spectrum following a statistical approach. Specifically, we treat the phase associated to each time stamp as a random variable, and the power spectrum as a function of it. Therefore, starting from the probability density function of the phases, we make use of the known rules of statistics in order to derive the properties of the power spectrum. We will show that with this approach we reach the same results obtained by other authors in different ways. But the novelty of our approach is that we introduce a different concept of the event phases and a different way to treat them. We found that the approach we introduce can be particularly useful to solve specific problems that can arise in a timing analysis. In order to highlight the difference of our method with respect to what can be found in literature, we first summarize other approaches in what follows.", "pages": [ 3 ] }, { "title": "2.1 Lomb's approach", "content": "In 1975, Lomb [9] undertook the study of the frequency analysis by means of the least-square method, which consists in fitting sine waves to the data ( y i , taken at times t i , i = 1 , 2 , ..., n ) in the time domain, and plotting versus the frequency ( ν ) the reduction ( ∆ R ) in the sum of the residual squared respect to fitting a constant value (see also [10]) Lomb [9] (see his Section 2) found the formula of the least-squared spectrum (a posteriori called Lomb-Scargle periodogram) and demonstrates that the formula of the classical periodogram is an approximation for it. From this base the statistical properties of the power spectrum of random noise were examined, as well as the effect of noise on the spectrum of a sine wave.", "pages": [ 3 ] }, { "title": "2.2 van der Klis' approach", "content": "van der Klis [11] (see his Section 2.5) explained the relation between the continuous Fourier transform and the Discrete Fourier Transform (DFT) introducing the concepts of windowing and sampling. The Fourier transform of an infinitely extended continuous function x ( t ) ( -∞ < t < ∞ ) is defined as When a real instrument is used to detect the phenomena described by the function x ( t ) , we need to take into account two effects. The observation will be finite in time, and the instrument hardware will detect discrete samples of the function x ( t ) . In other words, what we detect is the function x ( t ) multiplied by both the window function, which describe the structure and the duration of the observations, and by the characteristic sampling function of the instrument. The simplest window function is where T is the duration of the observation. The sampling function is where N is the number of bins in the observation time T . The observed function is then Substituting (2.5) into (2.2) we end up with the definition of the DFT. A powerful theorem of the Fourier analysis states that the Fourier transform of the product of two functions is the convolution of the Fourier transforms of these two functions. So, if a ( ν ) is the Fourier transform of x ( t ) , and b ( ν ) that of y ( t ) , then the Fourier transform of x ( t ) y ( t ) is a ( ν ) ∗ b ( ν ) ≡ ∫ ∞ -∞ a ( ν ' ) b ( ν -ν ' ) dν ' . Making use of this theorem applied to Eq. 2.5, van der Klis [11] derived the main properties of the power spectrum calculated for an event data time series.", "pages": [ 3, 4 ] }, { "title": "2.3 Ransom et al.'s approach", "content": "In an alternative way, Ransom et al. [12] (see their Section 2.1) derived the properties of the Fourier power of a time series from a direct study of the Discrete Fourier Transform. The k th element of a DFT of a uniformly-spaced time series n j ( j = 0 , 1 , ...., N -1 ) is defined as where for a time spacing dt , and a total observation length T , the frequency of the k th element is f k = k/T , and the index j indicate the time stamp t = jdt . If the DFT summation is drawn on a complex plane, it appears as a simple vector addition with each element rotated by -2 πk/N from the previous element. Starting from this representation of the DFT summation, [12] derived the Fourier response and its power to full noise time series as well as to periodic signals. To describe the statistical properties of the power spectrum of a time series composed by noise and a periodic signal, both [11], and [12] follow the description given by Groth [13]. He considers the Fourier power of a single frequency bin as a random variable. Invoking the central limit theorem, Groth [13] demonstrates in a few steps that the Fourier power of a noisy time series has a χ 2 distribution with 2 degrees of freedom. As one can notice, none of the authors cited above directly define the phase of the time stamps ( θ = 2 πνt ), nor use it as a random variable, in contrast with what follows.", "pages": [ 4 ] }, { "title": "3.1 First approach", "content": "Following the definition given in [14], the power spectrum of a sample data set { X ( t i ) , i = 1 , 2 , ...., N 0 } calculated at an arbitrary test frequency ω is given by Here, we consider the series of the arrival times of single events on a detector, i.e. the sample data set X is such that X i = 1 for each i . Attributing a phase value to each event θ i = ωt i we have for the power spectrum This definition corresponds to the Rayleigh power, and it differs by just a factor 2 with respect to the Z 2 1 test introduced by [15]. The variable Z 2 1 has a probability density function (pdf) for the null hypothesis equal to that of a χ 2 with 2 degrees of freedom, where null hypothesis is used to signify that there is no periodic signal at frequency ω . In what follows we focus on how to calculate the expectation value of the power spectrum in Eq. (3.2) using the statistical properties of the phase distribution P θ ( θ ) . Here and throughout the paper θ does not refer to phase of the pulsar, but to a phase relative to an arbitrary test frequency, as defined above. P θ ( θ ) refers to the theoretical continuous distribution of the phases. For a large number of events ( N 0 /greaterorsimilar 100 ) in Eq. (3.2), the sums of the trigonometric functions of the phases, cos ( θ i ) and sin ( θ i ) , are well approximated by the expectation value of their continuous distributions times N 0 . Thus, where P cos ( θ ) and P sin ( θ ) are the continuous distributions of cos( θ ) and sin( θ ) expressed as function of θ , respectively. Substituting these in Eq. (3.2), the power spectrum is given by Since we used the mean values of cosine and sine, Eq. (3.5) is the expectation value of the power spectrum at the frequency ω .", "pages": [ 5 ] }, { "title": "3.2 A more rigorous approach", "content": "This can be demonstrated more rigorously in the following way. Take a set of N independent random variables z i , all with the same variance σ and the same average 〈 z 〉 . The expectation value of their squared sum is /negationslash /negationslash /negationslash In the last step of Eq. (3.6) it is shown that 〈 ( ∑ z i ) 2 〉 = N 2 〈 z 〉 2 only if 〈 z 〉 /negationslash = 0 , whereas in the opposite case, the term with the variance ( Nσ 2 ) is dominant. This represents a limitation of our method, which is correct only for frequencies ω which are close to the signal ( ω 0 ). Indeed, for frequencies far from the signal, the distribution of the phases is flat and the power spectrum is dominated by noise. Several works dedicated to the study of the noise in the power spectrum can be found in literature (e.g. [16]). Here, we focus on the power spectrum close to the signal peak, with the aim of using the results for a future development of optimized software dedicated to the search of periodic signals. where we used the commutative properties of the average operator ( 〈〉 ), and we expressed the number of the double products from the square of a sum as ∑ i = j 1 = ( N 2 -∑ N i =1 i ) . Averaging Eq. (3.2), and doing algebra as above on ( ∑ cos( θ i )) 2 and ( ∑ sin( θ i )) 2 , it can be proven that the expectation value of the power spectrum is given by Eq. (3.5).", "pages": [ 5, 6 ] }, { "title": "3.3 Distributions", "content": "In the next paragraphs we will find a formula for P cos ( θ ) and P sin ( θ ) , in order to solve the integrals in Eqs. (3.3) and (3.4), and calculate Eq. (3.5). We shall use that the probability density function (pdf) of a variable z = f ( x ) that is function of a random variable x whose pdf P x ( x ) is known, can be calculated as (Eq. 3.7 is taken from [17]. It can also be found in e.g. [18]). Here, the prime (as in f ' ) represents a derivative with respect to x , and the intervals [ x 1 i , x 2 i ] are those for which for a given z = z 0 , f ( x ) < z 0 . At the first or the last interval (with respect to the validity range of x ), it could happen that the edges, x 1 i and x 2 i , respectively, can not be properly defined. The corresponding term in Eq. (3.7) is null in these cases. In our case, P z in Eq. (3.7) corresponds to P cos or P sin , while P x is the phase distribution P θ . The phases θ = ωt are defined in the full range (0 , ωT obs ) , where T obs is the total length of the observation. We will find useful to express ωT obs as where N is the number of entire cycles of 2 π/ω contained in the observation time T obs and R is the fractional part of the last cycle ( R = fmod( ωT obs , 2 π ) ) expressed in phase. /negationslash", "pages": [ 6 ] }, { "title": "3.4 P cos", "content": "We shall start with the calculation of P cos . Figure 1 shows the cosine function. We marked a generic value z + 0 > 0 and found the edges of the intervals for which cos ( θ ) < z + 0 . They are marked in the figure as θ + 1 i , and θ + 2 i , where the subindex i indicates the cycle number of the cosine function. In the same way we marked a generic value lower than 0 ( z -0 < 0 ), and the respective intervals [ θ -1 i , θ -2 i ] for which cos ( θ ) < z -0 . Since the codomain of the inverse function acos( z ) is [0 , π ] , it is convenient to express the edges of the intervals marked in figure 1 in the following way Substituting these values in Eq. (3.7), the most general expression for the pdf of cos( θ ) is Here we have substituted θ + / -1 0 with θ , which for this expression is constrained in the range [0 , π ] , as is specified in the formula at the feet of the closing of the square bracket. In Eq. (3.10), sin ( θ ) is the derivative of the cosine calculated at (2 πi + θ ) and (2 π ( i + 1) -θ ) . The sum is over the number N of entire cycles contained within the observation (see Eq. 3.8), while the term B ( θ ; R ) takes into account the fractional part of the last cycle. This term is in general negligible, because commonly one would have N /greatermuch 1 . It strongly depends on how long is the fraction of the last cycle, thus, on R , defined by Eq. (3.8). For example, if R < π the interval for which cos ( θ ) < z 0 is [ θ 1 N , R ] . Since the second edge of the interval is not properly defined, but it is set to the end of the observation, the term P x ( x 2 N ) /f ' ( x 2 N ) in Eq. (3.7) has to be set to zero. Finally, B ( θ ; R ) is equal in this case to [ P θ (2 πN + θ ) / sin( θ )] 0 ,R , where the subindices indicate the range of validity of θ only in this last cycle.", "pages": [ 7 ] }, { "title": "3.5 P sin", "content": "The solution for P sin ( θ ) is similarly found. Figure 2 plots the sine function. We marked there a generic value z + 0 > 0 , and the edges of the intervals for which sin ( θ ) < z + 0 ( θ + 1 i and θ + 2 i , where the subindex i indicates the cycle number of the sine function). In the same way we marked a generic value lower than 0 ( z -0 < 0 ), and the respective intervals [ θ -1 i , θ -2 i ] for which sin ( θ ) < z -0 . Since in this work θ = ωt is positively defined, the phase θ 0 also marked in the figure is not a solution of interest for us, but we need it because the inverse function asin( z ) has the codomain [ -π/ 2 , π/ 2] . The intervals defined for z + 0 > 0 are symmetric respect to π/ 2 + 2 πi , while the intervals defined for z -0 < 0 are symmetric respect to 3 2 π +2 πi . For this reason it is not immediate to join the two cases. Indeed, we will split them defining P + sin and P -sin . For positive values of z 0 , the edges of the intervals can be expressed as Substituting in Eq. (3.7), we get where here θ = θ + 1 0 is defined in the range [0 , π/ 2] , as indicated by the subindices of the closing square bracket. For negative z 0 , the edges of the intervals are Substituting in Eq. (3.7), we obtain where here θ = θ 0 is defined in the range [ -π/ 2 , 0] , as indicated by the subindices of the closing square bracket. In Eqs. (3.12) and (3.14) cos ( θ ) at the denominator is the derivative of the sine calculated at (2 πi + θ ) , (2 π ( i +1 / 2) -θ ) , and (2 π ( i +1) + θ ) . Finally, we can sum the Eqs. (3.12) and (3.14) to get the complete solution for P sin , Also here the term B ( θ ; R ) takes into account the fraction of the last cycle of the sine function, and it is in general negligible. We have now all the ingredients to solve the integrals in Eqs. (3.3) and (3.4), and calculate the expectation value of the power spectrum in Eq. (3.5). In both, Eqs. (3.10) and (3.15), a key role is played by the sum of the terms ∑ N -1 i =0 P θ (2 π ( i + k ) ± θ ) , where k = 0 , 1 / 2 , 1 , as well as the sign of θ depend on the cases. This corresponds to nothing more than the distribution of the phases folded by 2 π . This will be commented in the last Section. The solutions we found here are free from any assumption on P θ , and in this sense they are universal.", "pages": [ 7, 8 ] }, { "title": "4 Numerical validation", "content": "In this section we are going to validate in two different ways the method formulated to evaluate the expectation value of the power spectrum. First, the results found for the pdf of the sine and cosine (Eqs. 3.15, 3.10) will be checked with simulations. Secondly, we shall show that using our approach we can derive all the already known features of the power spectrum.", "pages": [ 8 ] }, { "title": "4.1 Simulations", "content": "In order to check the formulae we have found in the previous Section, we wrote a code to simulate the arrival time series from a sinusoidal signal of frequency ω 0 (Eq. 4.1), rate r , observed for a time T obs and having a distribution given by /negationslash The arrival time series is simulated in two steps. First, all the time stamps are simulated as random numbers in the range 0 , 2 π/ω 0 , following the distribution of Eq. (4.1). Then, in order to cover the full duration of the observation, each time stamp has been randomly delayed adding a value (2 π/ω 0 ) n , where n is a random integer number uniformly distributed in 0 , ω 0 T obs / 2 π . Figure 3 shows the distributions of the time stamps in the first step for ω 0 = 1 . 0 s -1 (left panel), and the distribution of n for the second step, for T obs = 1 . 0 E+4 s (right panel). This procedure requires that the observation time T obs is exactly an integer multiple of the signal period. This implies that for ω = ω 0 the term R in equation 3.8 is null, while it is in general not true for ω = ω 0 . Once the arrival time series has been fully simulated, we calculate the phases θ i = ωt i , their cosine, and sine value, and finally the power at the frequency ω from Eq. (3.2). Figure 4 shows the distributions of cos( θ ) and sin( θ ) (left and right panel, respectively) obtained for ω = ω 0 . In this case P θ ( θ ) = 1 + sin( θ ) , and it allows for a simplification of the terms 2 π ( i + k ) in Eqs. (3.10, 3.15). The pdf P cos and P sin , depicted in red on the plots of Figure 4 are equal to More generally, for ω = ω 0 , the phase distribution is equal to /negationslash and Eqs. (3.10, 3.15) can not be simplified. Specifically, when the terms B ( θ ; R ) are not negligible it is particularly hard to analytically reproduce the pdf P cos and P sin , because they have different points of discontinuity. In Figure 5, we choose these cases that most severely test our formulae. The phase distribution folded in 2 π is plotted in the left panels of the figure. Middle and right panels plot the pdf of cos ( θ ) and sin ( θ ) , respectively. Each row in the figure corresponds to a different simulation with observation time T obs chosen such that N from Eq. (3.8) is equal to zero or one, and the phases are calculated for different ω = ω 0 , as specified in the caption. The red line in each plot correspond to the analytical solutions. The perfect agreement with the simulations also in reproducing the discontinuities validates our computations in the previous Section. In the search of pulsations, the condition N /greatermuch 1 is always satisfied, even for only few hours of observations. In this most common case, then, the distributions we plotted appear smoother (as shown in Figure 6), and the term B ( θ ; R ) is completely negligible, because its weight is 1 /N ∼ 0 .", "pages": [ 9, 10 ] }, { "title": "4.2 Power spectrum features", "content": "Now we are going to demonstrate that the power spectrum of a sinusoidal signal observed for a finite time has a sinc squared shape ( [sin( x ) /x ] 2 ) centered on its proper frequency. With this aim, the power spectrum calculated by Eq. (3.5) needs the evaluation of the average values of sin ( θ ) and cos ( θ ) , /negationslash The pdf P cos ( θ ) and P sin ( θ ) are given by Eqs. (3.10) and (3.15), respectively. As we already underlined, they are composed by sums of the terms P θ (2 π ( i + k ) ± θ ) . For a sinusoidal signal, which general distribution of the phases is given by Eq. (4.4), these terms are equal to The power will be not null at frequencies close enough to the signal, such that ω 0 /ω ∼ 1 . This approximation can be adopted in Eq. (4.7), but just for the terms that do not contain the integer i , which is the index of the sums in Eqs. (3.10) and (3.15). Indeed, for large values of i , the term 2 πiω 0 /ω can significantly differ by an integer multiple of 2 π , and so can not be simplified. Eq. (4.7) is then approximately equal to To calculate 〈 sin( θ ) 〉 , only the term containing sin ( θ ) in Eq. (4.8) will lead to a not null quantity when substituted in Eq. (3.15), and this one into Eq. (4.5). The average value of sin ( θ ) is then equal θ to where 1 /N come from the normalization of P sin ( θ ) . For a random value of the ratio ω 0 /ω the former sum is negligible, but when the ratio is close to 1 we can write it as and all the values in the sum will be positive until 2 πN | ∆ ω | /ω ≤ π/ 2 , which neglecting R in Eq. (3.8) becomes where ω T = 2 π/T obs . With good approximation, the term in square brackets in Eq. (4.9) is equal to the following integral expression with ε = ∆ ω/ω . In the same way, the average value of cos ( θ ) is evaluated substituting Eq. (4.8) in Eq. (3.10), and this one into Eq. (4.6), leading to In this case, all the terms in the sum have the same sign when 2 πN | ∆ ω | /ω ≤ π . This condition is less constraining with respect to Eq. (4.11), and can be adopted to define the half peak width ( HPW ) in the power spectrum around ω 0 The integral expression for the term in square brackets in Eq. (4.13) is From Eq. (3.5), the power spectrum is calculated adding the squares of the sine and cosine averages. Correspondingly, from Eqs. (4.12) and (4.15) we have where πNε = π ∆ ω/ω T . Figure 7 shows the power spectrum (black curve) calculated from Eq. (3.2) for a sinusoidal signal, centered at its proper frequency ( ω 0 ), and in units of ω T . The contribution of the sine sum is shown in blue ( ∑ sin( θ i )) 2 and that of the cosine sum is shown in green ( ∑ cos( θ i )) 2 , which are equal to the first and second term on the left hand of Eq. (4.16), respectively. The right hand term of Eq. (4.16) is a squared sinc function centered on the signal frequency, and with width inversely proportional to the observation time. In Figure 7 the power spectrum is normalized so that the peak is equal to 1. We have considered in this demonstration a 100% pulsed sinusoidal signal (see Eq. (4.1)). In contrast, a signal partially pulsed can be represented by the following distribution of the arrival times where 0 ≤ a ≤ 1 determines the fraction of the signal that is pulsed. Then, in Eqs. (4.9) and (4.13) the term multiplying the square brackets is a/ 2 , which substituting in Eq. (3.5) results in the power spectrum being proportional to a 2 . The peak power in Figure 7 would be equal to a 2 . On the other hand, the mean power at frequencies far away from ω 0 remains unchanged. Then, the signal to noise ratio in the power spectrum is proportional to a 2 . Specifically, it is P ( ω 0 ) / 〈 P ( ω = ω 0 ) 〉 = N 0 a 2 / 4 . /negationslash", "pages": [ 10, 11, 12, 13, 14 ] }, { "title": "5 Pulsar frequency derivative", "content": "Weare going to apply the method developed in this paper to the practical case of observations so long that the first derivative of the pulsar frequency can not be neglected. The time series of the emitted photons ( t e ) by an isolated pulsar can be corrected for the first frequency derivative as: where t c is the corrected time series, and ˙ ω 0 is the frequency derivative. If the frequency derivative is un-known, one should try different values of ˙ ω , which will affect the correction of time series Here t w stays for generally corrected time series, while t c is the properly corrected time series. Anyway, once corrected the phase assigned to each photon is In order to apply our method, we need to evaluate the distribution of the phases P θ . With this aim we have first to find the relationship between t c and t w . From Eqs. 5.1 and 5.2 where δ ˙ ω = ˙ ω -˙ ω 0 . Solving Eq. 5.2 for t e we have where we choose the solution with the negative sign of the square root because this satisfies the condition that t e = 0 when t w = 0 . Squaring Eq. 5.5, expanding the root square in the Taylor series until the third term ( ˙ ω/ω 0 t w /lessmuch 1 for all the pulsars), and substituting in Eq. 5.4 we have and its inverse The distribution of the corrected time series P t w can be calculated applying the formula in Eq. 3.7. Since t w is a monotonic function of t c , the evaluation of P t w is simplified as where here -and hereafterU indicates a normalisation factor. In the same way, the distribution of the phase assigned to each photon can be caculated considering Eq. 5.3. Since in Eq. 5.3 ω 0 acts like a constant, P θ has the same form as P t w We assume that the properly corrected times have a sinusoidal distribution Substituting in Eq. 5.9 The square root at the denominator can be approximated to one, since 2 δ ˙ ω ω 0 t c /lessmuch 1 for all the pulsars even for observations as long as some years. Then, substituting t c with Eq. 5.6 in the argument of the sine we have Finally, the distribution P θ as function of θ is obtained substituting t w = θ/ω 0 We should substitute Eq. 5.13 in Eqs. 3.10 and 3.15 to calculate P cos ( θ ) and P sin ( θ ) , which are composed by sums of the terms P θ (2 π ( i + k ) ± θ ) . In this case these terms are equal to Since here 0 /lessorequalslant θ < 2 π while 2 πi can be as large as ω 0 T obs (see Eq. 3.8), then in the squared term ± θ can be neglected. Thus P (2 π ( i + k ) ± ∼ ± - δ ˙ ω ω 2 0 To calculate 〈 sin θ 〉 , only the term containing sin ( θ ) in Eq. (5.15) will lead to a not null quantity when substituted in Eq. (3.15), and this one into Eq. (4.5). The average value of sin ( θ ) is then equal to k = 0 , 1 / 2 , 1 can be neglected. With good approximation, the term in square brackets in Eq. (5.16) is equal to the following integral expression where Then where C ( x ) = ∫ x 0 cos( t 2 ) dt is the cosine Fresnel integral. In the same way, the average value of cos ( θ ) is evaluated substituting Eq. (5.15) in Eq. (3.10), and this one into Eq. (4.6), leading to All the terms in the sum are positive until 2 π 2 δ ˙ ω ω 2 0 N 2 ≤ π . This condition can be adopted to define the width of the peak in the power spectrum at variance of δ ˙ ω θ ) 1 + sin(2 πk θ 2 π 2 ( i + k ) 2 ) = θ The integral expression for the term in square brackets in Eq. (5.20) is Then where S ( x ) = ∫ x 0 sin( t 2 ) dt is the sine Fresnel integral. From Eq. (3.5), the power spectrum is calculated adding the squares of the sine and cosine averages. Correspondingly, from Eqs. (5.19) and (5.23) we have where U is a normalization factor, which in Figure 8 is choosen so that the power peak is equal to 1. Figure 8 shows the shape of the power spectrum at variance of δ ˙ ω as function of the variable y = 2 πδ ˙ ω/ω 2 T . In these units the width of the peak is equal to y = 1 , and the first minimum is at y ∼ 1 . 8 . Figure 8 and Eq. (5.21) show that in a pulsation search the first frequency derivative can not be neglected when ˙ ω 0 is of the order of magnitude of 1 /T 2 obs , or greater.", "pages": [ 14, 15, 16, 17 ] }, { "title": "6 Blind search", "content": "Amore general case happens when both the pulsar frequency and its first derivative are unknown. Of course, this happen every time one search for new pulsars, but there are at least two situations where the first frequency derivative can not be neglected in the search. On the one hand, the search for radio quiet γ -ray pulsars with γ -ray data needs integration times of few weeks or more, so that ˙ ω 0 is not negligible. On the other hand, in radio searches for pulsars with fast spin down ˙ ω 0 , it is important even for observations of few hours. The general form of Eqs. (5.1), and (5.2) is: Following the same steps and approximations from Eq. (5.4) to Eq. (5.3), and from Eq. (5.8) to Eq. (5.13) we get for the general case where To calculate P cos ( θ ) , and P sin ( θ ) we substitute in Eq. (6.4) θ with 2 π ( i + k ) ± θ , and we apply the same approximations as in Eq. (5.15) (neglecting k and θ when possible): The first term within the sine is equal to Substituting in Eq. (6.5) we have Only the term multiplying sin ( θ ) in Eq. (6.8) gives a not null contribution to 〈 sin( θ ) 〉 Setting z = 2 πi/ω , the term within the square brackets can be approximated with where S ( x ) and C ( x ) are the sine and cosine Fresnel integrals, respectively. Similarly, the average value of cos ( θ ) is and the term within the square brackets can be approximated with Before writing the formula of the expectation value of the power spectrum is useful to make the following simplifications. where from Eq. (3.8) N = ω/ω T , δR is given by Eq. (6.5), and we set With this notation the arguments of the Fresnel integrals in Eqs. (6.10), and (6.12) are Finally, the power spectrum given by the sum of the squares of Eqs.(6.10), and (6.12) is Figure 9 shows the power spectrum at variance of both ∆ ω and δ ˙ ω . The shape of the power spectrum follow an oblique structure, which is the typical one observed in plots produced for example by the program PRESTO when a blind search is performed. The diagonal axis of the structure has a weak dependence by the parameter K of Eq. (6.18) when it is lower than 1. For K → 0 the diagonal axis has the equation y = -2 x , that means", "pages": [ 17, 18, 19, 20 ] }, { "title": "7 Conclusions", "content": "In this paper we describe and validate a method to calculate the expectation value of the power spectrum. Adopting the definition given by [14] (see Eq. 3.2) we calculate the expectation value making use of the statistical properties of the arrival time series, and consequently of the phases attributed to each event. Our results are summarized by Eqs. (3.5), (3.10), and (3.15). We validate the method focusing on the simple case of a sinusoidal signal assumed to come from an isolated pulsar. But since the solutions in Eq. (3.10) and (3.15) are free from any assumption on the event phase distribution, the method can be generalized to any situation. As noticed at the end of Section 2, a key ingredient of our method is the sum of the terms ∑ N -1 i =0 P θ (2 π ( i + k ) ± θ ) , which corresponds to the distribution of the event phases folded by 2 π . At the proper frequency ω 0 , the folded distribution is equivalent to the pulse profile, but this is not /negationslash true anymore at a different frequency ω = ω 0 , as shown for example by Eq. (4.4) in the case of a sinusoidal signal. There are several factors that can modify the folded distribution of the phases. In this paper we applied our method to the case in which the folded distribution is perturbed by the first derivative of the pulsar frequency. Also, we considered the power spectrum expected in a blind search, in which both the frequency and its first derivative are uncertain. The analytical descriptions of the power spectra in these cases are given by Eq. (5.24) and Eq. (6.19), respectively. These are novel results in the field of timing. In a separate paper we make direct use of the method developed here to evaluate the effects of the uncertainties of orbital parameters in the timing of pulsars in binary systems [8].", "pages": [ 20, 21 ] }, { "title": "Acknowledgments", "content": "This work was supported by the grants AYA2012-39303, SGR2009-811, and iLINK2011-0303. DFT was additionally supported by a Friedrich Wilhelm Bessel Award of the Alexander von Humboldt Foundation.", "pages": [ 21 ] } ]

[ { "title": "Cheng Cheng 1 , 2 ∗ , Qing-Guo Huang 1 † , Yin-Zhe Ma 3 , 4 ‡", "content": "Abstract: We constrain models of single field inflation with the prePlanck CMB data. The data used here is the 9-year Wilkinson Microwave Anisotropy Probe ( WMAP ) data, South Pole Telescope (SPT) data and Atacama Cosmology Telescope (ACT) data. By adding in running of spectral index parameter, we find that the χ 2 is improved by a factor of ∆ χ 2 = 8 . 44, which strongly indicates the preference of this parameter from current data. In addition, we find that the running of spectral index α s does not change very much even if we switch to different pivot scales, which suggests that the power law expansion of power spectrum is accurate enough till the 1st order term. Furthermore, we find that the joint constraints on r -n s give very tight constraints on single-field inflation models, and the models with power law potential φ p can only survive if 0 . 9 /lessorsimilar p /lessorsimilar 2 . 1, so a large class of inflation models have already been ruled out before Planck data. Finally, we use the f NL data to constrain the non-trivial sound speed c s . We find that the current constraint is dominated by the power spectrum constraints which have some inconsistency with the constraints from f NL . This poses important questions of consistency between power spectrum and bispectrum of WMAP data. Keywords: CMB, inflation.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The inflationary model [1, 2, 3] has achieved a great success in modern cosmology, and it has been confirmed by many high precision CMB and Large scale structure experiments [4, 5, 6]. It provides a good explanation to a series problems such as flatness problem, horizon problem, and monopole problem in the standard cosmology scenario. In addition, inflation paradigm provides a natural explanation for the origin of primordial perturbations which constitute the seeds for the large scale structure we can see today. Therefore, identifying the realistic inflation model becomes an important task in observational cosmology. /negationslash Astronomical observations provide a large mount of data to constrain the cosmological parameters, especially inflation models. The default cosmology model people always use is the 'six-parameter' ΛCDM cosmology model, in which the canonical single-field slow-roll inflation (sound speed c s = 1) is assumed in the model. However, the class of slow-roll inflation models already have some weak tension with the observational data. In Ref. [4], it is shown that the generic φ p inflation model cannot provide consistent r -n s values within reasonable range of number of e -folds. In addition, WMAP 9-year data [7] suggests that the local non-Gaussianity has a large positive value, while the orthogonal non-Gaussianity is a large negative value, and these values are hardly to be produced in the canonical single-field slow-roll inflation models. Given these interesting tension between the canonical single-field slow-roll inflation model and the current observational data, we would like to explore the possibilities of non-trivial sound speed c s = 1 as well as non-zero running of spectral index dn s /d ln k to test their consistency with current combination of WMAP 9-year data [4], ACT data [6] and SPT data [5]. We intend to finish this work right before Planck data release (expected on 21st March, 2013) in order to make an immediate comparison before and after the Planck data. We hope that our work will motivate theorists to explore more phenomena in the general single-field slow-roll inflation model given the tight constraints on single field inflation models. This paper is organized as follows. In Sec. 2, we will discuss the model we are focusing on, and the data we will use to constrain the models. In Sec. 3, we will present our results of fitting. The concluding remarks will be presented in the last section.", "pages": [ 2, 3 ] }, { "title": "2.1 The Model", "content": "We will use standard 6 -parameter ΛCDM model as our basic model 1 . We then allow r (tensor-to-scalar ratio), dn s /d ln k (running of spectral index), c s (sound speed for the curvature perturbation modes) to be varied since we want to explore the level of constraints from these parameters. The sound speed is related to the tilt of tensor power spectrum in the general single-field inflation model through [8, 9] 2 where the tensor power spectrum is parameterized as Here k 0 is the pivot scale. Thus the tensor to scalar ratio is defined as Since tensor power spectrum also contribute to CMB angular power spectrum C TT l on very large scales, we will use the CMB temperature angular power spectrum to constrain r and c s . For more discussion on how the sound speed c s changes the data fitting is given in [10]. In addition, we add the 'running of running' parameter which characterizes the running of running of spectral index, i.e. Thus the scalar power spectrum is parameterized as Note that once the 'running of running' ( β s ) is introduced into the model, the running of spectral index α s becomes a scale-dependent quantity. To remove any ambiguity, we need to specify the pivot scale in the power law expansion (Eq. (2.5)), this is why the α s is related to k 0 . However, if α s turns out to be less dependent on k 0 , it means that the truncation till α s is enough (1st order), and there is no need to introduce a higher order truncation ( β s ). The reason we want to release β s as the running of running parameter is that SPT data [5] gives a detection of a negative value of the running of spectral index α s = dn s /d ln k at k 0 = 0 . 025Mpc -1 . So we would like to add this parameter as a higher order effect to monitor any possible 'running of running'. Even though it has not been detected, it is expected to be significantly constrained and is useful for the reconstruction of canonical single-field slow-roll inflation [11, 12].", "pages": [ 3, 4 ] }, { "title": "2.2 The data", "content": "We will use the most precise class of CMB data up-to-date, which is the combination of WMAP 9-year data [4], SPT data [5] and ACT data [6]. The temperature angular power spectrum from three data sets is shown in Fig. 1. The combined data is named as ' CMB data' in the following discussion. We set the maximum l -range of scalar model to be 7000 ( l s max = 7000), and maximal tensor l -range to be 3000 ( l t max = 3000) in the running of MCMC chains. In addition, we add Baryon Acoustic Oscillation data [13] as well as H 0 prior from HST (Hubble-Space-Telescope) project [14] into our data source. In order to explore the variation of sound speed, we add constrained f NL data provided by WMAP 9-year bispectrum into our likelihood. The sound speed c s is related to the equilateral and orthogonal type of non-Gaussianity f NL through Eq.(57) in [7]. So according to [7] we assign f i = ( f eq NL , f orth NL ) as the data vector, which is Then we use the χ 2 function (Eq.(58) in[7]) to calculate the best-fit value of c s 3 , i.e. where F ij is the lower right four elements of the Fisher Matrix and ˆ f i = (51 , -245). For the extended model of ΛCDM, we will release r and α s in the CAMB code [15] and further modify the code to incorporate running of running parameter ( β s ). We run CosmoMC [16, 17] to generate MCMC samples. We will express our results in term of best-fit value of marginalized likelihood, as well as 1 σ and 2 σ confidence level (CL) (68 . 3% and 95 . 4% CL).", "pages": [ 5 ] }, { "title": "3.1.1 Λ CDM cosmology model", "content": "We first fix c s = 1 and investigate the constraints on parameters r and α s . The data sets we use here are CMB data, BAO and H 0. Here we consider '6 -parameter model', '6 -parameter+ r model', '6 -parameter+ α s ' model and '6 -parameter+ r + α s ' model which are expressed as 'ΛCDM', 'ΛCDM+ r ', 'ΛCDM+ α s ' and 'ΛCDM+ r + α s ' models respectively. /s32 /s32 In Fig. 2, we can see that the likelihood of r shifts a little if we switch α s on and off. The solid line is ΛCDM+ r model, and the dotted line is ΛCDM+ r + α s model. In addition, the likelihood becomes broader in ΛCDM+ r + α s model, and the upper limit is also higher. This indicates that without the direct polarization power spectrum, it is hard to draw concrete upper limit on the amplitude of tensor mode r , since adding a single extra-parameter can greatly broaden the constraint on r . Similar thing exists in Fig. 3. The solid line is the ΛCDM+ α s , and the dotted one is ΛCDM+ α s + r . One can see that the likelihood of α s is broader if r is released as a free /s32 /s32 parameter. This means that the two parameters have some level of degeneracy, which is potentially able to be broken if the future polarization data is added. The Fig. 4 shows the likelihoods of n s for three models. Here we consider all of the three models, i.e. ΛCDM+ r , ΛCDM+ α s , ΛCDM+ r + α s . We can see that not only the peak of distribution shift, but also the range of confidence level of n s changes quite a lot in three different model: if we add r , the spectral index still prefers a 'red' spectrum as n s < 1, but such situation does not exist anymore in the case of ΛCDM+ α s and ΛCDM+ α s + r . /s32 /s32 Fig. 5 shows the contours of joint constraints on α s -n s in ΛCDM+ α s model(the blue solid curves) and ΛCDM+ r + α s model(the red dashed curves). We can see adding r leads to the shift of n s towards bluer region, and the constraints become broader. /s32 /s32 In Table 1, we list the results of fitting by fixing c s = 1. One can see that by introducing α s parameter, the χ 2 really improve significantly (∆ χ 2 = 4 . 22), indicating that the current data prefer the inflation model with running of the spectral index. In the left panel of Fig. 6, we compare our joint constraints on r and n s with the results from WMAP 9-year paper [4]. WMAP 9-year results used WMAP 9-year data, combined with old SPT data, old ACT data, BAO data and H 0 data and obtain the black contours (1 σ and 2 σ CL). We used the similar combination, except that our SPT and ACT data are the corresponding new data sets [5, 6]. By updating the new data of ACT and SPT, one can see that the constraints are tightened up to some extent. This suggests that the new SPT and ACT data really provide a large level arm for WMAP 9 data, which offer more constraining power on small scales CMB angular power spectrum. We use our results of joint constraints on plane of r -n s to discuss its implication for inflation models (Right panel of Fig. 6). /s32 /s110 /s115 /s114 /s32 between two dashed lines in Fig. 6 indicates the prediction of chaotic inflation. One can see that the models with p = 2 [18] and p = 2 / 3 [19] are disfavored at around 2 σ level, and only the models with p ∈ [0 . 9 , 1 . 8] for N = 50 or p ∈ [1 . 5 , 2 . 1] for N = 60 are still consistent with data within 95% CL.", "pages": [ 5, 6, 7, 8, 9 ] }, { "title": "3.1.2 Comparison of different pivot scale and the influence of running of runing of spectral index ( β s )", "content": "In the former sections, all the fittings are done at pivot scale k 0 = 0 . 002 Mpc -1 and the running of spectral index is preferred at more than 2 σ level. In this section, we investigate the distributions of α s at different pivot scales. We use the model ΛCDM+ r + α s . The solid line is k = 0 . 002 Mpc -1 , and the dotted line is k = 0 . 025 Mpc -1 in Fig. 7. It shows that when the pivot scale change, the distribution of α s almost does not change at all. This means that the constraints on α s is not sensitive to the pivot scale you choose, which indicates that the truncation of power index expansion (Eq. (2.5)) is accurate enough till 1st order. /s32 Considering the higher order power effect of the primordial power spectrum, we introduce a new parameter β s to characterize the 'running of running' (Eqs. (2.4) and (2.5)). Left panel of Fig. 8 shows the joint constraint on α s and β s , and the right panel shows the marginalized distribution of β s with a flat prior. We can see that the peak of β s slightly deviates from 0, but is perfectly consistent with zero within 1 σ CL. This means that the current data do not support the 'running of running of spectral index', and therefore the power law expansion of the scalar power spectrum (Eq. (2.5)) is accurate enough till the α s term. This is consistent with what we find in Sec. 3.1.2. The fitting results are shown in Table 1. /s32 /s32 /s32 /s32 /s32", "pages": [ 9, 10 ] }, { "title": "3.2 General single-field inflation Model ( c s free)", "content": "/s32 In this section, we release c s as a free parameter which is constrained by the f NL data from WMAP 9-year results [7] 4 . Table. 2 shows the best fit (-log(Like)) and confidence level of n s , r and α s of two models. It can be seen that adding α s significantly reduces the best fit -log(Like), but enlarge both the confidence interval of r and n s . Fig. 9 shows the distribution of c s . The dotted line is the likelihood from f NL data while the solid line is marginalized probabilities from the CMB+BAO+ H 0 + f NL data. We can see that the f NL prefers a very low value of c s while the CMB data sets prefer a larger value, which indicates some tension between each other. In addition, the combined constraints are dominated by CMB power spectrum simply because the number of CMB power spectrum data is far greater than the f NL data. The tension between the f NL data and the CMB power spectrum data may have a variety of indications: In any case, we need to develop a method which can direct relate c s with power spectrum and bispectrum (not just f NL data) and globally fit this parameter by using full spectrum of CMB data. Such work is in progress.", "pages": [ 11 ] }, { "title": "4. Conclusion", "content": "In this paper, we combine the most recent prePlanck CMB data to constrain the inflation /s32 model parameters. Our data consists of WMAP 9-year data [4], ACT data [6], SPT data [5], Baryon Acoustic Oscillation data [13] as well as H 0 prior [14]. We mainly find four interesting results from our numerical fitting: In conclusion, we find that prePlanck data have already been able to set tight constraints on single field inflation model. But current observational data still leave many open questions to be solved. We hope such issues will be resolved when the Planck data becomes available in a few days. AcknowledgementWe would like to thank Mark Halpern to share his figure with us. This work is supported by the project of Knowledge Innovation Program of Chinese Academy of Science and a grant from NSFC (grant NO. 10821504).", "pages": [ 11, 12, 13 ] } ]

[ { "title": "Lingfei Wang, Ernestas Pukartas and Anupam Mazumdar", "content": "Consortium for Physics, Lancaster University, Lancaster LA1 4YB, UK Abstract. Inflation creates perturbations for the large scale structures in the universe, but it also dilutes everything. Therefore it is pertinent that the end of inflation must explain how to excite the Standard Model dof along with the dark matter. In this paper we will briefly discuss the role of visible sector inflaton candidates which are embedded within the Minimal Supersymmetric Standard Model (MSSM) and discuss their merit on how well they match the current data from the Planck. Since the inflaton carries the Standard Model charges their decay naturally produces all the relevant dof with no dark/hidden sector radiation and no isocurvature fluctuations. We will first discuss a single supersymmetric flat direction model of inflation and demonstrate what parameter space is allowed by the Planck and the LHC. We will also consider where the perturbations are created by another light field which decays after inflation, known as a curvaton . The late decay of the curvaton can create observable nonGaussianity. In the end we will discuss the role of a spectator field whose origin may not lie within the visible sector physics, but its sheer presence during inflation can still create all the perturbations responsible for the large scale structures including possible non-Gaussianity, while the inflaton is embedded within the visible sector which creates all the relevant matter including dark matter, but no dark radiation.", "pages": [ 1 ] }, { "title": "1 Introduction and motivation for a visible sector inflation", "content": "The primordial inflation [1] is the simplest dynamical mechanism which explains the seed perturbations for the cosmic microwave background (CMB) radiation with almost Gaussian perturbations as suggested by the recent Planck data [2-4]. Since inflation dilutes everything other than stretching the initial vacuum fluctuations, after the end of inflation the coherent oscillations of the inflation must excite the Standard Model (SM) quarks and leptons at temperatures sufficiently high to realize SM baryons and dark matter in the current universe [5, 6]. In principle, inflation could have occurred in many many phases [7], and perhaps even future-eternal [8, 9], but it must end in our Hubble patch with the right thermal history and three light neutrino species [2]. In principle inflaton whose potential drives inflation could be an arbitrary hidden sector field 1 , whose properties can be constructed solely to match the observational data from Planck [3, 4]. However note that the CMB observables merely probe the gravitational aspect of the problem, it is not sensitive to the inflation's couplings to the SM matter and neither its origin. Typically hidden sector inflatons are SM gauge singlets 2 , whose mass and couplings can never be probed directly. In order to explain the universe filled with the SM quarks and leptons such an inflaton should primarily couple only to the SM sector [13], which is an ad-hoc assumption. A gauge singlet could in principle couple to other sectors, i.e. hidden or visible, there is no symmetry which can completely forbid their couplings to the hidden sector. Especially, string theory provides many viable SM gauge singlet inflaton candidates, for a review see [14]. Inflation is typically driven either by close string moduli or open string 2 There could be more than one inflaton fields and perhaps even of the order of O (10 2 -10 3 ) as in the case of assisted inflation [10]. Although there are some embeddings of such models of inflation within large SU(N) gauge theories [11], and in string theory [12], but it is highly unlikely that nature would prefer such a route since none of these fields can be embedded within a visible sector physics. moduli. In either case they are SM gauge singlets - therefore it is not at all clear why and how such an inflaton would decay solely into the SM dof . Typically string compactification yields many moduli and hidden sectors [15]. A high scale inflation, i.e. scale which is higher than the mass of the moduli, could in principle excite all the moduli and dump all its entropy in the hidden sectors fields [16]. The reason for this is kinematical , the inflaton can decay into hidden sectors due to typically large branching ratio, i.e. there are more hidden sectors and only one visible SM sector. Furthermore, one might as well worry whether the inflaton could excite dark radiation, provided some of the dof become extremely light, such as in the case of string axions [17], or dark matter, such as in case of Kaluza-Klein dark matter [18]. In this respect, it is vital that the last phase of primordial inflation, i.e. last 50 -60 efoldings of inflation must end in a vacuum of BSM physics which can solely excite the relevant SM dof required for the success of Big Bang Nucleosynthesis (BBN), see for a review [19]. In this regard, Minimal Supersymmetric Standard Model (MSSM) [20] provides a perfect setup where all the matter content is known and can be probed at the LHC [21, 22] 3 . SUSY also helps inflation model building, since inflation needs a potential which remains sufficiently flat along which the slow-roll inflation can take place in order to generate the observed temperature anisotropy in the CMB. SUSY at any scale guarantees the flatness of such flat directions at a perturbative and a non-perturbative level (for a review see [24]), besides providing a falsifiable framework at low scales. Furthermore, the lightest SUSY particle can be absolutely stable under R-parity, and thus provides an ideal cold dark matter candidate [25]. The minimalistic realization would be to embed inflation, dark matter within MSSM which are all determined by the known SM couplings which provides credibility not only to particle physics but also to cosmological predictions. Our aim of this paper will be to show this within three paradigms for the inflationary cosmology - in all the cases inflation happens in the visible sector of MSSM. The origin of perturbations could be sourced from the visible sector physics or it might as well arise from the hidden sector, we will discuss the role of hidden sector here which might be responsible for creating mild non-Gaussianity. We will also discuss their merits in conjunction with the release of the Planck data along with the constraints arising from the LHC.", "pages": [ 2, 3 ] }, { "title": "2 Three paradigms within visible sector inflation", "content": "One can envisage three realistic scenarios. A simple single field model of inflation and a scenario with multi-fields. In the latter case we can capture all the essence by mimicking just two fields - one which is inflaton and the other could be either curvaton [26-28], or a spectator field [29, 30] as the simplest examples.", "pages": [ 3 ] }, { "title": "· A single field model of inflation:", "content": "It is well known that a single field model of inflation with a canonical kinetic term will yield almost Gaussian perturbations. Of course, one can depart from the simplest assumptions to generate non-Gaissianity, such as sudden change in the potential, modifying the initial vacuum from Bunch-Davis, or introducing non-canonical kinetic term, for a review on non-Gaussianity see [31]. All these have interesting consequences for the primordial non-Gaussianity, but most of them are severely constrained by the current observations [3] 4 . In this paper we will revisit the parameter space of a visible sector single field model of inflation embedded within MSSM with canonical kinetic term and with the BunchDavis initial vacuum condition [33-35]. In all these models inflation happens below the Planck scale and generate small non-Gaussianities. Furthermore, these models produce the right thermal history of the universe without any dark radiation.", "pages": [ 3, 4 ] }, { "title": "· Curvaton scenario:", "content": "A light subdominant field during inflation can also seed the perturbations for the CMB. In the simplest scenarios it is assumed that the inflaton fluctuations are sub-dominant. The light field known as a curvaton [26-28] can slow roll after the end of inflation, and decays later on once the inflaton has completely decayed. While decaying the curvaton converts its initial isocurvature fluctuations into curvature perturbations. This conversion leads to a pure adiabatic fluctuations if the curvaton dominates while decaying, on the other hand if the curvaton decay products are sub-dominant compared to the energy density of the inflaton decay products, then there is a residual isocurvature fluctuations. Furthermore since the conversion itself is non-adiabatic, there is a generation of non-Gaussian perturbations of the local configuration. In order not to generate residual isocurvature fluctuations, the inflaton decay products must thermalize with that of the curvaton decay products. A priori this is a non-trivial condition. The only way it could be satisfied provided both inflaton and curvaton can be embedded within the visible sector, i.e. MSSM, then this problem could be addressed amicably since both the fields would decay into the MSSM dof [36].", "pages": [ 4 ] }, { "title": "· Spectator scenario:", "content": "This is a completely new paradigm where a light subdominant field like curvaton is present during inflation, but it decays into radiation much before the end of inflation [29, 30]. The sheer presence of such a light field can create perturbations for the CMB, but since the field decays during inflation, its decay products need not be that of the SM or MSSM dof . In principle if inflation is occurring within a visible sector the perturbations can be seeded by the hidden sector field, which is advantageous for many theories of BSM including string theory. We will illustrate this for the first time with an example of inflation occurring within MSSM, while the spectator field is made up of arbitrary gauge singlet arising from the hidden sector physics. There could be two possibilities for the observed tensor to scalar ratio being negligible ( r < 0 . 11 with 95% CL) [4]. The scale of inflation could be genuinely below the GUT scale, which is the case we will be considering in all the examples below, or the second option could be that the gravity is purely classical and so is the vacuum [37], while matter component is treated quantum mechanically; for a review on cosmological perturbation, see [38]. A linearized Einstein gravity has no source term, therefore for a classical gravity without any source for exciting gravity waves in a homogeneous and isotropic universe, the resultant primordial gravitational waves will be absolutely zero [37]. Any positive detection of primordial gravitational waves will indeed shed an important light on whether gravity should be treated classically or not.", "pages": [ 4 ] }, { "title": "3 Inflection point potential for MSSM flat directions", "content": "The MSSM provides nearly 300 gauge-invariant F -and D -flat directions [39, 40], which are all charged under the SM gauge group. Out of these flat directions, there are particularly 2 D -flat directions: ˜ u ˜ d ˜ d and ˜ L ˜ L ˜ e , which carry the SM charges and can be the ideal inflaton candidates [33-35], where ˜ u, ˜ d correspond to the right handed squarks, ˜ L corresponds to the left handed slepton, and ˜ e corresponds to the right handed selectron. Both the inflaton candidates provide inflection point in their respective potentials where inflation can be driven for sufficiently large e-foldings of inflation to explain the current universe and explain the seed perturbations for the temperature anisotropy in the CMB [33-35], see also [41]. Since both ˜ u ˜ d ˜ d and ˜ L ˜ L ˜ e flat directions are lifted by higher order superpotential terms of the following form, which would provide non-vanishing A -term in the potential even at large VEVs, but below the cut-off scale: where λ ∼ O (1) 5 , and M p = 2 . 4 × 10 18 GeV is the reduced Planck mass. The scalar component of Φ superfield, denoted by φ , is given by 6 for the ˜ u ˜ d ˜ d and ˜ L ˜ L ˜ e flat directions respectively. After minimizing the potential along the angular direction θ (Φ = φe iθ ), we can situate the real part of φ by rotating it to the corresponding angles θ min . The scalar potential is then found to be [33, 34] where m φ and A are the soft breaking mass and the A -term respectively ( A is a positive quantity since its phase is absorbed by a redefinition of θ during the process). The masses for ˜ L ˜ L ˜ e and ˜ u ˜ d ˜ d are given by: glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] Note that the masses are now VEV dependent, i.e. m 2 ( φ ). The inflationary perturbations will be able to constrain the inflaton mass only at the scale of inflation, i.e. φ 0 , while LHC will be able to constrain the masses at the LHC scale. However both the physical quantities are related to each other via RGE as we will discuss below. For where α 2 glyph[lessmuch] 1, there exists a point of inflection ( φ 0 ) in V ( φ ), where From now on we only keep the leading order terms in all expressions. Note that inflation occurs within an interval 7 in the vicinity of the point of inflection, within which the slow roll parameters glyph[epsilon1] ≡ ( M 2 p / 2)( V ' /V ) 2 and η ≡ M 2 p ( V '' /V ) are smaller than 1. The Hubble expansion rate during inflation is given by In order to obtain the flat potential, it is crucial that the A ( φ 0 )-term ought to be close to m φ ( φ 0 ) in the above potential Eq. (3.3). This can be obtained within two particular scenarios - (1) Gravity Mediation: in gravity-mediated SUSY breaking, the A -term and the soft SUSY breaking mass are of the same order of magnitude as the gravitino mass, i.e. m φ ∼ A ∼ m 3 / 2 [45], and (2) Spilt SUSY: in Split SUSY scenario the scale of SUSY is high and sfermions are very heavy, the A -term is typically protected by R-symmetry, see Refs. [46, 47], as a result the A -term could be very small compared to the soft masses. However, if the Yukawa hierarchy arises from the Froggatt-Nielsen mechanism, then the A -term can be made as large as that of the soft mass, i.e. m φ ∼ A , as in the case of Ref. [48]. at which The above potential Eq. (3.3) has been studied extensively in Refs. [34, 49, 50]. The amplitude of density perturbations δ H and the scalar spectral index n s are given by: and respectively, where In the above, N COBE is the number of e-foldings between the time when the observationally relevant perturbations are generated till the end of inflation and follows: N COBE glyph[similarequal] 66 . 9 + (1 / 4)ln( V ( φ 0 ) /M 4 p ) ∼ 50. The running of the spectral tilt is negligible [34, 49, 50] within the current bound of the Planck observations [4]. The perturbations are due to single canonical field, therefore one would not expect large non-Gaussianity from this model. The observed non-Gaussianity parameter denoted by f NL ≤ 1 is bounded by the slow roll parameters, see Ref. [31], and is consistent with Planck [3]. The scale of inflation is low enough that one would not expect any observed tensor perturbations in any future CMB experiments 8 . Instant reheating and thermalization [53] occurs when a single MSSM flat direction is responsible for inflation. This is due to the gauge couplings of the inflaton to gauge/gaugino fields. Within 10 -20 inflaton oscillations radiation-dominated universe prevails, as shown in Ref. [54]. The resultant reheat temperature at which all the MSSM dof are in thermal equilibrium (kinetic and chemical equilibrium) is given by [54] Since the temperature of the universe is so high, it immediately thermalizes the LSP provided it has gauge interactions. The LSP relic density is then given by the Standard (thermal) Freeze-out mechanism. In particular, if the neutralino is the LSP, its relic density is determined by its annihilation and coannihilation rates [42, 55]. The advantage of realizing inflation in the visible sector is that it is possible to nail down the thermal history of the universe precisely. At temperatures below 10 -100 GeV there will be no extra degrees of freedom in the thermal bath except that of the SM, therefore BBN can proceed without any trouble within low scale SUSY scenario. This reheat temperature is marginally compatible with the BBN bound for the gravitino mass m 3 / 2 ≥ O (TeV). It is also sufficiently high that various mechanisms of baryogenesis may be invoked to generate the observed baryon asymmetry of the universe. In Fig. 1 we have explored a wide range of the inflaton mass, m φ , where inflation can explain the observed temperature anisotropy in the CMB with the right amplitude, P ζ = 2 . 196 × 10 -9 , and the tilt in the power spectrum, n s = 0 . 9603 ± 0 . 0073 [2]. The observables P ζ and n s have been shown by blue region. We have restricted ourselves to VEV below the GUT scale. Within the current parameter range the model provides negligible running in the tilt which is well within the observed limit. We have allowed a wide range for m φ and φ 0 (the inflection point) to show that inflation can indeed happen within SUSY from low scales to high scale SUSY breaking soft-masses. High scale soft masses could be made compatible within split-SUSY scenario [48]. Using renormalization group equations the mass of the inflaton can be evaluated at any energy scales, thus providing connection between physics at the very high energies in early universe and experimentally probed scales at LHC. For the ˜ u ˜ d ˜ d flat direction RGE is [42, 55]: where ˆ µ = ˆ µ 0 = φ 0 is the VEV at which inflation occurs. For ˜ L ˜ L ˜ e : where M 1 , M 2 , M 3 are U (1), SU (2) and SU (3) gaugino masses, which all, assuming SUSY models which obey universality conditions like constrained MSSM (CMSSM) [56], equate to m 1 / 2 at the unification scale, and g 1 , g 2 and g 3 are the associated couplings. To solve these equations, one needs to take into account of the running of the gaugino masses and coupling constants which are given by, see [45]: with α 1 = 11 / 16 π 2 , α 2 = 1 / 16 π 2 and α 1 = -3 / 16 π 2 . Within CMSSM one can try to constrain the inflaton mass for ˜ u ˜ d ˜ d and ˜ L ˜ L ˜ e similar to the analysis of Ref. [55]. The current LHC searches for SUSY particles put a stringent limit on squarks and sleptons, see [57, 58], and as a result on the inflaton mass as shown in Fig. 2. 9", "pages": [ 5, 6, 7, 8, 9 ] }, { "title": "4 Inflation and curvaton both embedded within MSSM", "content": "The curvaton scenario [26-28] is an alternative mechanism for creating perturbations. In this scenario, the density perturbations are sourced by the quantum fluctuations of a light scalar field σ , known as the curvaton, which makes a negligible contribution to the energy density during inflation and decays after the decay of the inflaton field φ . The advantage of the curvaton mechanism is that it can generate non-Gaussianity [26, 60] in the primordial density perturbations and also significant residual isocurvature perturbations, neither of which are possible in the usual single-field inflation models. Both signatures are now well constrained by the current Planck data [2, 3]. If the curvaton does not completely dominate the energy density at the time of its decay, the process of conversion of initial isocurvature perturbations into adiabatic curvature perturbations can enhance the local form of non-Gaussian fluctuations by where is the curvaton's energy density ratio at the time the curvaton decays [26]. Here ρ γ is the energy density of the radiation as the decay products of the inflaton. However, if either the curvaton or the inflaton belongs to a hidden sector of BSM, they may decay into other fields beyond the SM dof . There is no guarantee that the hidden and visible sector dof would reach thermal equilibrium before the BBN takes place. In this case, residual anti-correlated isocurvature perturbations are expected to be in conflict with the CMB data, which constrain them to be glyph[lessorsimilar] 5% [4]. If the curvaton belongs to the visible sector but the inflaton does not, a value of r ∼ 1 would avoid this conflict [61], but would render any non-Gaussianity undetectable. Note that if r ∼ 1 the curvaton is solely responsible for exciting all the SM dof , so it must carry the SM charges in order to avoid dark radiation for instance [61, 62]. The curvaton scenario lends strong support to a visible sector dark matter such as neutralino in the case of the LSP, because either from the decay of the inflaton or from the curvaton, the neutralino would thermalize with the rest of the plasma soon after its decay, and its final abundance will be determined by its annihilation and co-annihilation rates. Keeping all these constraints in mind we need to embed both inflaton and curvaton within a visible sector of BSM physics where they both decay into the SM dof . Let us consider the case where the inflaton, φ , and the curvaton, σ , both originate from different saddle point directions which are orthogonal to each other at least at the lowest orders in an effective field theory 10 . The total potential is Let us first discuss the origin of the curvaton, which we take to be an R -parity conserving D -flat direction of the MSSM. For the purpose of illustration we consider that to be ˜ L ˜ L ˜ e , which is lifted by the non-renormalizable operator: where λ is a non-renormalizable coupling induced by integrating out the heavy fields at the intermediate scale, M ∗ , which could be close to the GUT scale, i.e. M ∗ ∼ M GUT . The scalar component of the Σ superfield and its mass are given by: where at the lowest order the potential along the σ direction is given by similar to Eq. (3.3) 11 : where A ∼ m σ ∼ O (1 -10) TeV, are the soft SUSY-breaking terms. We will assume that the curvaton rolls on a saddle point of the potential, i.e. A = √ 40 m σ , so the saddle point lies at We now turn to the origin of V ( φ ) within the MSSM. Let us consider a flat-direction orthogonal to the curvaton. If the curvaton is ˜ L ˜ L ˜ e , the inflaton could be ˜ u ˜ d ˜ d direction. In which case both inflaton and curvaton are embedded within MSSM. We take the inflaton direction to be squarks, typically they are expected to be heavier than the sleptons: Note that ˜ u ˜ d ˜ d and ˜ L ˜ L ˜ e remain two independent directions for the entire range of VEVs. This flat direction will also be lifted by the non-renormalizable operators. However, at larger VEVs the potential energy density stored in the ˜ u ˜ d ˜ d direction will be larger than that of ˜ L ˜ L ˜ e , so it would be lifted by higher order terms: The potential at lowest order would be: where · · · contains the higher order terms. Note that the λ m in Eq. (4.9) are all nonrenormalizable couplings induced by integrating out the heavy fields at the intermediate scale. At energies below the cut-off scale these coefficients need not necessarily be of O (1). Potentials like Eq. (4.10) were studied in Refs. [63, 64]. For λ 2 glyph[lessmuch] λ 3 glyph[lessmuch] λ 4 glyph[lessmuch] λ n ≤ O (1), they provide a unique solution for which the first and second order derivatives of the potential vanish along both radial and angular direction in the complex plane: ∂V/∂φ = ∂V/∂φ ∗ = ∂ 2 V/∂φ 2 = ∂ 2 V/∂φ ∗ 2 = 0 (a saddle point condition). For the first three terms in Eq. (4.10), it is possible to show that this happens when at the VEVs: φ = φ 0 exp[ iπ/ 3 , iπ, i 5 π/ 3], where Concentrating on the real direction, the potential energy density stored in the inflaton sector is given by: where φ 0 glyph[lessmuch] M ∗ . With the inflaton and curvaton potentials, the universe would then undergo the evolution as indicated in Fig. 3. The curvaton σ only ends slow roll some time after inflation. In our case both φ and σ will decay within one Hubble time of the evolution. Both carry the SM gauge charges for which thermalization is similar to the case of instant preheating as shown in Refs. [53, 54]. As is also shown in Fig. 3, we will use subscripts ' ∗ ', ' e ', and ' c ' to indicate the Hubble exit of the relevant perturbations, the end of inflation, and the time when σ decays, respectively. During inflation, both fields are slowly rolling. The curvaton σ remains subdominant so inflation is totally determined by the inflaton φ , where the e-folds from the Hubble exit of relevant modes to the end of inflation is N 1 . Since inflation is held near the saddle point, we can approximate the inflaton energy to be nearly constant, V ( φ 0 ). The curvaton's motion can be solved by integrating out its slow roll equation of motion where After inflation, the inflaton decays into radiation, whose energy density then satisfies where N 2 is the number of e-folds of universe expansion after inflation, till the curvaton σ decays. From the violation of the second order slow roll condition, i.e. η σc = -1, we find the curvaton slowly rolls after inflation for the number of e-folds N 2 , which satisfies where characterizes the typical 'width' of the slow roll region of the curvaton field near the saddle point. The slow roll equation of motion for the curvaton can also be integrated out after inflation, although now the universe is dominated by radiation. Similar to Eq. (4.14), here we have Therefore Eq. (4.14) and Eq. (4.19) fully describe the motion of the curvaton, before it ends slow roll. Since, m σ glyph[greatermuch] H ∗ after the end of slow roll, σ decays into radiation and the universe then evolves adiabatically 12 . When the curvaton receives the quantum fluctuations δσ ∗ at the Hubble exits, its initial perturbation will be maintained during inflation. Its perturbation evolves according to the perturbed Eq. (4.14), which is After inflation, the curvaton's perturbations are converted into the curvature perturbations when the curvaton decays 13 . This can be specified by the perturbation in the number of e-folds of the curvaton's slow roll after inflation, δN 2 , and the perturbation in the curvaton field at the end of slow roll, δσ c . Together they should comply the same end-of-slow-roll condition for the curvaton, i.e. δη σc = 0, which gives The curvaton perturbation, δσ , also evolves after the end of inflation, according to the perturbed Eq. (4.19), with the relation Combining Eq. (4.20), Eq. (4.21) and Eq. (4.22), we are able to solve the perturbation in the number of e-folds before the curvaton ends slow roll After the curvaton ends slow roll, the σ field instantly decays into relativistic dof within one Hubble time and we can ignore its evolution. The e-folds of the radiation dominated era, N 3 , can be written as a constant plus a quarter of the logarithmic of the total energy density, i.e. When the initial perturbation δσ ∗ is present, it also changes the energy density at the time the curvaton ends slow roll. The perturbation in σ 's energy density, in this case, would be small compared to that of the radiation. This is both because the curvaton is subdominant, and because its potential is relatively flat around the saddle point. Therefore, the major contribution to δN 3 comes from the perturbation in the radiation energy density, which comes from perturbing Eq. (4.24) as (according to Eq. (4.16)) where is the energy density ratio of the curvaton when it decays. After taking into account the energy density perturbation at the curvaton's end of slow roll, the universe then enters an adiabatic evolution, and no super Hubble perturbations will be generated. Therefore the total perturbation in the e-folds of expansion is where Therefore the power spectrum of curvature perturbation becomes The spectral index of the curvature perturbation can then be calculated as From the above equations, we find that the observed spectral index n s constrains the initial condition of the curvaton σ ∗ by The local bispectrum can be calculated, according to Eq. (4.23) and Eq. (4.26), as We can scan the parameter space in m σ and V ( φ 0 ), and calculate the possible local bispectrum, as shown in Fig. 4. In Fig. 4, we require the coupling λ < 1, and the curvaton remains subdominant ( r < 0 . 1). These two constraints narrow the allowed parameter space to the green band. The local bispectrum typically acquires f NL ∼ O (20). As a specific example, we pick the inflaton as having the parameters λ 2 = 10 -9 and λ 4 = 10 -3 , with an inflation energy scale relatively low, at V ( φ 0 ) = (2 . 4 × 10 9 GeV) 4 , with negligible curvature perturbations. A low scale inflation is helpful in obtaining the right tilt in the power spectrum for the curvature perturbations. For the curvaton mass m σ = 7 . 4 TeV, we can acquire the observed power spectrum of curvaton perturbation by taking the coupling constant λ = 0 . 012. The energy density ratio when curvaton decays is r = 0 . 086. This gives the local bispectrum f NL = 8 . 3.", "pages": [ 9, 10, 11, 12, 13, 14, 15 ] }, { "title": "5 Spectator mechanism with a visible sector inflation", "content": "The spectator mechanism is a new mechanism, which has been proposed recently in Refs. [29, 30]. The perturbations are created by a sub-dominant field which decays during inflation, known as a spectator. The spectator field cannot modify the dominant inflaton dynamics, but it can leave its imprint in the cosmological perturbations. The decay of the spectator field can create non-Gaussianity due to the conversion of the entropy perturbations into the curvature perturbations. The spectator field decays completely into relativistic species during inflation, thus leaving no residual isocurvature perturbations. All the matter is created by the decay of the inflaton field after inflation, therefore it is important that the inflaton sector must be embedded within a well motivated visible sector. For all practical purposes the inflaton field's perturbations could be assumed to be sub dominant as compared to that of the spectator's. In order to create the right thermal history, we provide a simple example of embedding inflation within MSSM × U (1) B -L gauge group, where the latter is also gauged. A simple D -gauge invariant flat direction which can be the inflaton candidate in our case is given by [35, 65]: where h is the Yukawa coupling and, N , H u , L are corresponding right handed neutrino, Higgs and slepton superfields. Note that the above superpotential can generate Dirac mass for the light neutrinos if the scale of U (1) B -L breaking is of order O (TeV) and the Yukawa is h ∼ 10 -11 -10 -12 . The inflaton field φ corresponds to the superpotential: So its potential can be written as where A is the trilinear A-term and the soft SUSY breaking mass term for the flat direction is given by: Note that for A = 4 m φ , there exists a saddle point for which V ' ( φ 0 ) = V '' ( φ 0 ) = 0. The saddle point and the potential are given by: Here m ν denotes the neutrino mass which is given by m ν = h 〈 H u 〉 , with 〈 H u 〉 glyph[similarequal] 174 GeV. For neutrino masses with a hierarchical pattern, the largest neutrino mass is m ν glyph[similarequal] 0 . 05 eV in order to explain the atmospheric neutrino oscillations [66]. In our case we will be investigating a range of the inflaton masses which can accommodate the right handed sneutrino mass close to the low scale supersymmetry, and also yield the correct neutrino masses: The above range of h also guarantees the curvature perturbation contribution by the inflaton is negligible, see [30]. r After inflation the inflaton NH u L will start coherent oscillations and will dump all its energy into the light relativistic species of MSSM and the lightest of the right handed sneutrinos. In fact the lightest right handed sneutrino could be the dark matter candidate if it is the lightest SUSY particle. Dark matter analysis with the lightest sneutrino has been performed in Ref. [65]. Since U (1) B -L is gauged, it will lead to a quick thermalization of all the relativistic dof with a reheat temperature very similar to the analysis of Ref. [54]. The reheat temperature will be roughly given by T rh ∼ (30 /π 2 g ∗ ) 1 / 4 V ( φ 0 ) 1 / 4 ∼ 10 8 GeV for m φ ∼ 1 TeV [54, 65]. Let us now imagine that the spectator field has a simple potential arising from some hidden sector physics. For the sake of illustration we consider this to have a flat potential with a hyperbolic tangent profile: where U 0 and σ 0 are constant parameters whose values we will scan to show how this simple potential can explain the amplitude of the perturbations and also the observable nonGaussianity of the local form. Note that U 0 glyph[lessmuch] V ( φ 0 ). A typical timeline for the spectator scenario in our case is summarized in Fig. 5. We will use the subscripts ' ∗ ', ' c ' and ' e ' to indicate the respective slices as the relevant perturbations from the spectator field leave the Hubble patch during inflation, the spectator ends slow roll, and the end of inflation. We can solve the background evolution of the spectator field σ , typically σ ends slow roll when 10 From this we can solve σ c as for σ c > σ 0 . Here the subscript ' c ' indicates at the time when σ ends slow roll. We can also solve the background motion of the spectator field σ before it ends slow roll, since the inflaton is dominating with a constant energy density. By integrating the slow roll equation of motion for σ , we obtain σ , as a function of N , the remaining e-folds of inflation, as where N c is number of e-folds of inflation from σ ends slow roll to the end of inflation. The second and third order slow roll parameters for σ then simplify to For the pivot scale N = N ∗ , the spectral index n s , the local bispectrum f NL , and the local trispectrum g NL are determined by η σ ∗ in this case, giving the leading order terms, see for details [29, 30] is the energy density ratio of the spectator σ at its end-of-slow-roll boundary. The power spectrum of curvature perturbations then, becomes Therefore to achieve the observed amplitude for P ζ , this requires σ 0 to take the value When the inflaton model is given, the parameters N ∗ and V 0 are fixed. The spectral index n s , the local bispectrum f NL , the local trispectrum g NL , and the relative value σ 0 /H ∗ then only depend on the r , the energy density ratio, and N c , the number of e-folds from the spectator ends slow roll to the end of inflation. We can then parametrically plot their dependences on N c and r in Fig. 6. One can see that the model predicts the spectral tilt, (as shown in green shade which depicts the 2 σ range,) the negligible running of the spectral tilt and the local bispectrum in the range observed by the current Planck data [2-4]. We have also shown the value of g NL where in Fig. 6. Since σ decays into radiation there is no residual isocurvature fluctuations, which matches the data perfectly well. Since the origin of σ field belongs to the hidden sector, it could arise from billions of hidden sectors of string theory, all the inflationary models which claim a successful inflationary cosmology could potentially act like a spectator field. One of the common feature for the spectator field is the flat potential and this can be achieved in many string motivated models. Now the advantage is that these stringy origins need not have to explain the matter content of the universe. The latter could be obtained from the visible sector model of inflation.", "pages": [ 15, 16, 17, 18, 19, 20 ] }, { "title": "6 Nonlocal bispectra from curvaton and spectator mechanisms", "content": "So far we have discussed the non-Gaussianity of local type or in the squeezed limit. In this section we briefly discuss how to obtain the equilateral and orthogonal types of the bispectrum. The bispectrum for any shape f NL ( k 1 , k 2 ) is in general defined as [31] Here B and P indicate the strengths of the two-and three-point correlation functions of the gauge invariant curvature perturbation ζ . They are defined as In Eq. (6.3), the delta function indicates k 1 , k 2 , and k 3 form a closed momentum triangle, because of this multi-point correlation function expansion. Depending on the shape of the triangle, the bispectrum f NL ( k 1 , k 2 ) can take different values, corresponding to different types of the bispectrum. The most considered and also best observationally constrained type of bispectrum, the 'local' type which we have discussed above, corresponds to the squeezed limit k 1 ≈ k 2 glyph[greatermuch] k 3 . For any single field slow roll inflation, the local bispectrum of curvature perturbations is constrained to be small by the almost scale invariant spectrum [67]. In principle, every other shape of the momentum triangle corresponds to a unique type of bispectrum, and is considered 'nonlocal'. Among them, however, what raises people's most interests are the 'equilateral' and 'orthogonal' types. The equilateral type comes from taking the equilateral momentum triangle where k 1 = k 2 = k 3 , and the excess equilateral bispectrum (compared to the local one) typically can be generated by models with noncanonical kinetic terms [68, 69]. The orthogonal type is constructed in [69], to account for the bispectrum contribution that is orthogonal/uncorrelated to the local and equilateral shapes, for the general single field inflation. For this reason, the orthogonal bispectrum is in general a linear combination of various shapes, and does not originate from a single-shape definition. It has been shown in the past that the non-canonical inflatons can generate nonlocal bispectra with various patterns [68]. If we embed such a non-canonical field in the de Sitter universe, this non-canonical field can still contribute to the nonlocal bispectra as a curvaton or a spectator [70]. The inflaton field dominating the de Sitter universe can still come from the visible sector, as demonstrated in Section 4 and Section 5, to be responsible for the matter production. The generic nonlocal bispectrum for the curvature perturbation ζ for any nonlocal shape, written as f (nloc) NL ( k 1 , k 2 ), can be estimated as follows. In both the curvaton and the spectator cases, the universe evolves almost adiabatically till the boundary where the curvaton or spectator changes its equation of state significantly. Before this boundary, the gauge invariant perturbation of the curvaton or spectator, which is defined as does not evolve after the Hubble exit of the relevant modes. Here σ is the curvaton or the spectator field we are concerned about, ρ σ is its energy density, ψ ( k ) is the scalar perturbation in the metric, and here the prime means derivative w.r.t the conformal time. We have omitted the time dependence. Here we can define the nonlocal bispectrum for the gauge invariant perturbation ζ σ ( k ), as f (nloc) NL( σ ) ( k 1 , k 2 ), similarly with Eq. (6.1) to Eq. (6.3) 14 . In particular, we have At the spectator or the curvaton boundary, the quantum fluctuations in σ then transfer to the curvature perturbations. In the simplest setup where we can assume that the inflaton contribution to the curvature perturbation is negligible, the total curvature perturbation becomes where r is the energy density ratio of σ compared to the total energy density when it reaches the boundary, i.e. at the time when the perturbations are converted. After the boundary, the spectator is quickly redshifted away and the curvaton decays into radiation which has the same equation of state with the inflaton's decay products. The universe then becomes adiabatic and ζ ( k ) does not evolve afterwards. As the observation confirms, we can assume the local bispectrum, determined by Eq. (4.32) or Eq. (5.15), is much smaller than the nonlocal bispectrum we are interested in. Therefore the nonlocal bispectrum of the curvature perturbations can then be calculated, according to Eq. (6.8), as The above equation also works for both the equilateral and the orthogonal types of bispectra. The multi-point correlation functions has been well studied for a noncanonical slow roll field in a de Sitter universe, showing possible large equilateral and/or orthogonal types of bispectrum [68]. The Eq. (6.9) then gives a simple enhancing relation for the nonlocal bispectrum of curvature perturbations, showing that the same mechanism also works in the curvaton or the spectator setup, in order to generate large nonlocal bispectra. Typically non-canonical kinetic terms such those as appearing in the DBI-inflation occur in the hidden sector rather than the visible sector particle physics. It is simple to hide the uncertainties of the hidden sector physics in the spectator scenario as compared to the visible sector curvaton. One can imagine a spectator field with a non-canonical kinetic term slowly rolling its potential and decaying well before the end of visible sector inflation. The decay products of the spectator field would be diluted away anyway during the remaining inflation, rendering the universe solely with the visible sector inflaton which produces all the relevant matter for the BBN and without any trace of dark radiation.", "pages": [ 20, 21, 22 ] }, { "title": "7 Summary", "content": "In this paper we have emphasized only the visible sector models of inflation, curvaton and spectator scenarios in light of Planck data [2-4]. There are plethora of hidden sector gauge singlet models of inflation and curvaton with ad-hoc couplings and mass parameters, but these models are unable to explain why such a gauge singlet inflaton or curvaton would decay solely into the visible sector dof . We summarize our findings in Table-1. Although we have restricted ourselves to the local form of f NL but one advantage of a spectator scenario is that it can generate nonlocal forms of f NL compatible with the Planck data. The spectator mechanism is amazing since it can fit all the observed parameters, P ζ , n s , dn s /d ln k, f local NL , f oath NL , f equil NL of the CMB without generating any isocurvature perturbations, besides providing all the SM dof , which means no dark radiation by virtue of the inflaton being within the visible sector physics.", "pages": [ 22 ] }, { "title": "Acknowledgments", "content": "The research of AM is supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant ST/J000418/1. EP is supported by STFC ST/J501074.", "pages": [ 22 ] }, { "title": "References", "content": "[arXiv:1205.2815 [hep-ph]].", "pages": [ 25 ] } ]

[ { "title": "ISO(4,1) Symmetry in the EFT of Inflation", "content": "Paolo Creminelli, 1 Razieh Emami, 2, 1 Marko Simonovi´c, 3, 4 and Gabriele Trevisan 3 1 Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151, Trieste, Italy 2 School of Physics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531, Teheran, Iran 3 SISSA, via Bonomea 265, 34136, Trieste, Italy 4 Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, I-34136, Trieste, Italy In DBI inflation the cubic action is a particular linear combination of the two, otherwise independent, cubic operators ˙ π 3 and ˙ π ( ∂ i π ) 2 . We show that in the Effective Field Theory (EFT) of inflation this is a consequence of an approximate 5D Poincar'e symmetry, ISO(4,1), non-linearly realized by the Goldstone π . This symmetry uniquely fixes, at lowest order in derivatives, all correlation functions in terms of the speed of sound c s . In the limit c s → 1, the ISO(4,1) symmetry reduces to the Galilean symmetry acting on π . On the other hand, we point out that the non-linear realization of SO(4,2), the isometry group of 5D AdS space, does not fix the cubic action in terms of c s . Motivations. The study of non-linearly realized symmetries in the context of inflation has proven to be a powerful tool to make model-independent predictions. A spontaneously broken symmetry is manifested in relations among operators with different number of fields: for example, in the framework of the EFT of inflation [1] one finds a relation between the kinetic term and the cubic operators, as a consequence of the non-linear realization of time diffeomorphisms. This implies that in any model with small speed of sound c s glyph[lessmuch] 1, one has parametrically large non-Gaussianities ∝ c -2 s . This regime is still allowed by observations, although severely constrained by the beautiful Planck data [2]. In this note we study the consequences of the nonlinear realization of ISO(4,1), the 5D Poincar'e symmetry, in the EFT of inflation. The motivation is twofold. On one hand this symmetry is typical of inflationary models based on brane constructions, where the position of a brane moving in an extra dimension plays the role of the inflaton. Although the inflationary solution (spontaneously) breaks ISO(4,1), the dynamics of perturbations is constrained by the non-linearly realized symmetries. On the other hand, observations are only sensitive to small perturbations around the inflating solution and their dynamics is encoded in the EFT of inflation. It is then of interest to study the possible symmetries that can be imposed in this theory. In this respect ISO(4,1) naturally stands out, since it contains both the 4D Poincar'e group and the shift symmetry of the inflaton, which is usually imposed to justify slow-roll and the consequent approximate scale-invariance of the spectrum. We will show, for example, that the relation between the cubic operators ˙ π 3 and ˙ π ( ∂ i π ) 2 which occurs in DBI inflation [3] does not require any UV input, but it is just a consequence of the ISO(4,1) symmetry at the level of the EFT of inflation. Nonlinear realization of ISO(4,1). In general, the homogeneous inflaton background φ 0 ( t ) breaks the 4D Poincar'e symmetry to translations and rotations: ISO(3,1) → ISO(3). (We here concentrate on scales much shorter than the Hubble scale H , where spacetime can be considered flat; we will consider gravity later on.) At leading order in slow-roll, the inflaton φ is also endowed with an approximate shift symmetry φ → φ + c and a solution φ 0 ( t ) = vt preserves a combination of this shift symmetry and time translations. Perturbations around this background can be parametrized by the Goldstone mode π and the most general action compatible with the symmetries reads All the constants are time independent as a consequence of the residual shift symmetry 1 . Let us now impose the extra symmetry. We want to enlarge ISO(3,1) × shift (11 generators) to a 15-dimensional group, ISO(4,1). The additional four transformations act as 2 These are rotations and boosts in the 5th dimension, if we interpret φ as a coordinate in the extra dimension, for example describing the position of a brane. The shift symmetry of φ is interpreted as translation in the 5th dimension to complete the isometry group of 5D flat space. However, the geometric interpretation is not mandatory and we may remain agnostic about the origin of this symmetry. These transformations act on the Goldstone π as 3 where in the last equality we have reabsorbed 1 /v into the definition of ω µ . Demanding that the action (2) is invariant under these additional transformations imposes some conditions on the coefficients a 0 , a 1 , a 2 , . . . ( 4 ). If we focus on the variation of the action quadratic in π , we get the following relations The first equation says that the speed of propagation of π excitations, the 'speed of sound' c s is related to v as From the 5D geometrical point of view, this is a consequence of the relativistic sum of velocities. Here it is simply a consequence of the ISO(4,1) symmetry in the EFT of inflation. The cubic action is fixed by the second and third relation, so that up to cubic order the action (up to an overall coefficient) reads This is exactly the same result one gets in DBI inflation [3], but here we see that one does not need any UV input: this action follows from the ISO(4,1) symmetry in the EFT of inflation. As we are going to discuss later, these results will not change when gravity is taken into account. In the notation of [5] the coefficient ˜ c 3 (that is in general free), is fixed by ISO(4,1): ˜ c 3 = 3 2 (1 -c 2 s ). In terms of the relative coefficient between the two operators A ≡ -( c 2 s + 2 3 ˜ c 3 ), the symmetry fixes A = -1. The Planck limits [2] on these parameters are shown in Fig. 1. We can go to higher order and set to zero the cubic variation of the action (2). We get a simple system of algebraic equations whose solution is Again all the coefficients are completely fixed in terms of a single parameter, the speed of sound c s . This does not come as a surprise: the only operator with one derivative per field, that linearly realizes the 4D Poincar'e group and non-linearly realizes ISO(4,1) is the brane tension operator so it is not surprising that everything is fixed for operators with one derivative per field. One can check that expanding (10) around φ 0 = vt one gets operators which satisfy (5) and (9). Still it is nice to see the constraints directly at the level of the EFT of inflation, without assuming to be able to extrapolate far from the inflationary solution. One can also explore the consequences of ISO(4,1) for operators with more derivatives. If we look at operators with two derivatives on one of the π 's then the effective action starts with cubic terms (quadratic terms are total derivatives) and reads Using the transformation (4) we can easily find the relations among λ 1 , λ 2 , µ 1 and µ 2 As a check, one can start from the brane picture and consider an operator with one extra derivative on π compared to the brane tension: there is only one, the extrinsic curvature of the brane. This gives the following operator which non-linearly realizes ISO(4,1) [6] Indeed, expanding (13) around φ 0 = vt we find that the cubic action for the Goldstone is which satisfy the constraints (12). The limit of Galilean symmetry and the coupling with gravity. The ISO(4,1) transformation (4) contains a dimensionless parameter v , which can be interpreted in a 5D picture as the brane velocity in the bulk. As we discussed, this parameter fixes the speed of sound of perturbations, eq. (6). One can consistently take the limit v → 0 of the symmetry 5 . This is a group contraction and in this limit the symmetry does not act on coordinates anymore and it thus commutes with the 4D Poincar'e group. It reduces to an internal symmetry acting on π only This is the Galilean symmetry studied in [7], whose implications for the EFT of inflation have been discussed in [8] (see also [9]). This symmetry requires c s = 1 and forbids all interactions with a single derivative per field. All interactions come from higher derivative terms. For example in eq. (14), for c s = 1 we have only the second operator which can be written as ( ∂π ) 2 glyph[square] π , i.e. the cubic Galileon. So far we discussed the ISO(4,1) symmetry in Minkowski space, without including gravity. Ultimately we are interested in calculating correlation functions during inflation, so that the coupling with gravity cannot be neglected. Similarly to what happens in the case of the Galilean symmetry discussed above, gravity breaks the ISO(4,1) symmetry 6 . This implies that the symmetry is not a good one for the background evolution, since in general the Hubble friction plays an important role. This is an additional motivation to formulate the symmetry directly in the EFT of inflation as a non-linearly realized symmetry for π on scales much shorter than Hubble, without reference to the background solution. Another point to address is whether the actions for π derived above can be used, once minimally coupled to gravity, to calculate observables during inflation or gravity will completely change the picture. The breaking of the symmetry due to gravity will manifest in two ways. First of all, graviton radiative corrections will induce operators which do not respect the symmetry. This effect is arguably small, as suppressed by powers of M Pl . Second, in calculating π loops on a gravitational background, non-invariant terms will also be generated. These operators will be invariant under a shift of π , as the shift symmetry is compatible with the coupling with gravity, but not fully ISO(4,1) invariant. As these terms arise only on a curved background they will contain powers of the Riemann tensor, schematically On a quasi de Sitter background R glyph[similarequal] H 2 , so we expect these terms to be suppressed with respect to the ones we This does not depend on v and is still non-linearly realized for v → 0. considered above by powers of ( H/ Λ) 2 glyph[lessmuch] 1, where Λ is the UV cut-off of the theory. These corrections can become relevant if the coefficient of some operator is unnaturally large. For example, the effect of the induced gravity term on a brane is studied in [10, 11] and the conclusion is that the cubic action is in general not uniquely fixed in terms of c s : a different linear combination of the operators ˙ π 3 and ˙ π ( ∂ i π ) 2 is possible, giving in particular an orthogonal shape of non-Gaussianity. This is at first surprising as the model respect the ISO(4,1) symmetry we are discussing. However, the deviations are indeed due to cubic operators with more than three derivatives in the EFT of inflation [11]: in curved space some of these derivatives can be traded for the curvature scale H and one is left with only three derivatives on π . However a basic tenet of the EFT approach is that operators of higher dimension give small corrections: if they induce O (1) changes, it is not clear why one can neglect all the other higher dimensional terms. ISO(4,1) or SO(4,2)? In DBI inflation [3] a probe brane lives in an AdS throat and non-linearly realizes the SO(4,2) group, so that one may wonder why we did not consider this group instead of ISO(4,1). One simple answer is that during inflation the brane does not move much in units of the AdS radius L , so that the difference between flat and curved bulk is immaterial. It is still interesting to understand whether SO(4,2) would give the same predictions. The answer is no. It is straightforward to check, for example supplementing the DBI action with other SO(4,2)invariant operators like the AdS conformal Galileons [6], that the nice predictions of ISO(4,1) are lost. In particular the speed of sound is not fixed in terms of the velocity v in the bulk and the cubic operators ˙ π 3 and ˙ π ( ∂ i π ) 2 can appear in a general linear combination. The fact that c 2 s is not fixed in terms of v may come as a surprise: after all it simply comes from the relativistic sum of velocities and this should apply locally also in AdS. This intuition however requires that higher derivative operators are suppressed by a cutoff scale Λ glyph[greatermuch] L -1 : in this case only the tension of the brane is important and we get back to the DBI inflation case. When, on the other hand, Λ ∼ L -1 higher derivative operators are unsuppressed, the brane is a thick object in comparison with the AdS radius: it will not follow geodesics and we do not expect the same predictions as for DBI inflation, though the SO(4,2) symmetry is preserved. All this can also be seen at the level of the EFT. The most general action allowed by the symmetry up to quadratic order is where all the coefficients are now time dependent. As in the ISO(4,1) case, a background solution with constant velocity is not in general a solution, therefore we have to keep a 0 that will be cancelled by additional terms which are not SO(4,2) symmetric. The non-linear transformation of π that realizes SO(4,2) is Again, requiring the invariance of the action under this transformation leads to a set of three constraints on the coefficients It is straightforward to verify that, for a constant ˙ φ 0 , these constraints do not fix the relation among the coefficients of the kinetic term and therefore c s . Aparticular case in which these constraints actually fix the form of the speed of sound, is the one in which the background preserves an SO(4,1) subgroup of SO(4,2) [12]. This happens for a background solution φ 0 ( t ) = α t -1 . In this case the tadpole term is absent and π is massless so that the constraints in (20) greatly simplify. The speed of sound is then fixed by the last constraint and the residual dilation symmetry [12]: c 2 s = 1 -α -2 . Conclusions . Imposing an ISO(4,1) symmetry in the EFT of inflation leaves (at leading order in derivative) a single free parameter, the speed of sound c s . It should be straightforward to study the consequences of ISO(4,1) in the EFT of multi-field inflation [13]: this symmetry is indeed at play in multi-field DBI models [14]. Acknowledgements. It is a pleasure to thank M. Serone for useful discussions, N. Bartolo and M. Liguori for correspondence about Planck data.", "pages": [ 1, 2, 3, 4 ] } ]

[ { "title": "Reheating temperature in non-minimal derivative coupling model", "content": "H. Mohseni Sadjadi ∗ and Parviz Goodarzi Department of Physics, University of Tehran, P. O. B. 14395-547, Tehran 14399-55961, Iran November 1, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "We consider the inflaton as a scalar field described by a non-minimal derivative coupling model with a power law potential. We study the slow roll inflation, the rapid oscillation phase, the radiation dominated and the recombination eras respectively, and estimate e-folds numbers during these epochs. Using these results and recent astrophysical data we determine the reheating temperature in terms of the spectral index and the amplitude of the power spectrum of scalar perturbations.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "To solve some dilemmas in the standard model of cosmology such as the flatness, the horizon, the monopoles problems and so on, inflation as an accelerated expansion era in the early universe was introduced by [1]. This scenario is now dubbed as old model of inflation, in which the universe underwent a de-Sitter expansion in a supercooled unstable false vacuum. Afterwards, by proposing a scalar field (inflaton) as the source of inflation, a new inflationary model was introduced in [2]. In this context, inflation was driven by the inflaton which slowly rolled down towards the minimum of its effective potential. To provide enough e-folds number, the potential must be nearly flat near its minimum. The nature of this scalar field has not yet been identified, but a simple possible candidate might be the Higgs boson [3]. To adapt the inflaton to the Higgs boson, a non-minimally derivative coupling model in which the kinetic term of the inflaton is coupled to the Einstein tensor, was proposed in [4]. This model does not suffer from unitary violation problem and is safe from quantum corrections. Besides, slow roll inflation can be described by steep potentials in this framework[5, 6]. More general non-minimal derivative coupling model has also been considered in the literature to study the accelerated expansion of the universe in the early universe as well as in the late time [7]. After the end of the inflation, the universe was cold and dominated by the inflaton scalar field energy. This energy had to be converted to relativistic particles to reheat the universe via a procedure called the reheating process [8]. A proposal for reheating, is the decay of the inflaton to ultra-relativistic particles (radiation) during a rapid coherent oscillation phase about the minimum of the potential. These particles interacted rapidly to become in thermal equilibrium characterized by the reheating temperature T reh , and the universe entered on the radiation dominated era. Although the exact value of T reh has not yet been known, but some upper and lower bounds for this temperature have been obtained in the literature. By considering that the reheating process occurred before the big bang nucleosynthesis (BBN), and by combining constraints on light elements abundance and data obtained from large scale structure and cosmic microwave background (CMB), one can find a lower bound for T reh , 4 MeV /lessorsimilar T reh [9]. An upper bound may be taken as the energy scale at the end of inflation which is around the GUT scale T reh /lessorsimilar 10 16 GeV . These assumptions give a wide range for the reheating temperature. In [10] by involving supersymmetry and considering the gravitino production, and on the base of cosmic microwave background (CMB) radiation data, this range was tightened to 6 TeV /lessorsimilar T reh /lessorsimilar 10 4 TeV . A more accurate method to determine T reh in terms of CMB data was introduced in [11]. This method is based on determining the number of efolds during the evolution of the universe from the inflation until the present time. Although in this context T reh may be determined in terms of spectral index and amplitude of power spectrum of scalar perturbations, but due to uncertainties of theses quantities in WMAP7 data [12], a large relative uncertainty for T reh is arisen: σ ( T reh ) T ≈ 53, where T reh = 3 . 5 × 10 6 GeV . reh In this paper we assume that the inflaton is a scalar field described by non-minimal derivative coupling model introduced in [4]. We study the inflationary era, the rapid oscillation phase of the inflaton, the radiation dominated epoch respectively and employ the method proposed in [11] to determine the reheating temperature in terms of the spectral index and power spectrum of scalar perturbations. Finally, the value of T reh and its relative uncertainty are computed from recent astrophysical data such as WMAP9 and Planck 2013 results. We use units /planckover2pi1 = c = 1 through the paper.", "pages": [ 1, 2 ] }, { "title": "2 Evolution of the universe and the reheating temperature", "content": "Weconsider the spatially flat Friedmann-Lemaˆıtre- Robertson-Walker (FLRW) space-time and choose an arbitrary length scale, λ 0 , crossing the Hubble radius R H := 1 H = a ˙ a at some time, denoted by t ∗ , during the inflation [13]. By using the red-shift relation where subscript '0' denotes the present time and by taking a 0 = 1, we obtain This reference time will be used in division of the evolution of the universe into four parts as follows: II- From t e until the reheating or beginning of the radiation dominated epoch, denoted by t reh . The number of e-folds from t ∗ until t 0 is then given by where the subscripts denote the value of the parameter at their corresponding times. In the following we will try to use eq.(4) to determine the reheating temperature in a non-minimal derivative coupling model in which the inflaton kinetic term is non-minimally coupled to Einstein tensor. This inflationary model is described by the action [4] where G µν = R µν -1 2 g µν R is the Einstein tensor, R is the scalar curvature, M P is the reduced planck mass given by M P = √ 1 8 πG = 2 . 4 × 10 18 GeV , and M is a scale with mass dimension.", "pages": [ 3 ] }, { "title": "2.1 Slow roll", "content": "In the era (I), the universe is dominated by the inflaton scalar field. The Friedmann equation is where is the energy and the upper dot is ' . ' = d dt . The pressure is obtained as With the help of continuity equation one can derive the equation of motion of the inflaton as In the continue, we restrict ourselves to high friction regime [4] This choice, as we will see, by enhancing the slow roll procedure, enables us to consider more general steep potentials. During the slow roll, we have H 2 M 2 ˙ φ 2 /lessmuch V ( φ ), ¨ φ /lessmuch H ˙ φ , and therefore the Friedmann equation and inflaton equation of motion reduce to respectively. The slow roll parameters /epsilon1 , δ satisfy δ can be written as where η is defined with Eqs.(13-15), imply that by choosing an appropriate M in high friction regime (11), slow roll conditions do not oblige us to adopt approximately flat potentials. Hereafter we will restrict ourselves to power law potential where v is a real number, and n is an even positive integer to guarantee that the potential has a minimum, about which the rapid oscillation of the inflaton occurs after slow roll. The number of e-folds in the era (I) is To obtain the above equation we have used (16), and φ ∗ /greatermuch φ e . To estimate φ ∗ , we consider the spectral index n s [5], For the power law potential, this equation reduces to The number od e-folds in the time interval I is then obtained by substituting φ ∗ from (19) into (17):", "pages": [ 4, 5 ] }, { "title": "2.2 Reheating era", "content": "At the end of slow roll we have /epsilon1 ( φ e ) /similarequal 1, which yields At this time the energy density is approximately given by and the scalar field commences a rapid coherent oscillation around the bottom of the potential (see fig.(1), depicted for the quadratic potential V ( φ ) = 1 2 m 2 φ 2 via numerical methods). To obtain the equation of state of the universe in this era, we follow the steps used in [14]. In this high frequency regime, the behavior of the scalar field is opposite to the slow roll, and its quasi-periodic evolution may be described as [14, 15] W ( t ) is some function of time and the time dependent amplitude , Φ( t ), is given by The rapid oscillation of the scalar field φ occurs after the slow roll. This high frequency regime is characterized by (for more details, see the first reference in [14] and also [15]) ∣ ∣ ∣ ∣ ∣ ∣ ∣ implying that the energy density and the Hubble parameter decrease very slowly (insignificantly) in one period of oscillation of the scalar field [14]. This can be seen numerically in fig .(1) which shows that although φ oscillates rapidly, but δ Φ( t ) Φ( t ) /lessmuch 1, where δ Φ( t ) is the change of Φ( t ) during one period of φ oscillation. With the help of (7), (8), and (11), the continuity equation of the scalar field can be written as Now let us take the time average of both sides of the above equation over one oscillation of the scalar field( < ... > = ∫ t + T t ...dt ' T is the time average over an oscillation whose period is T ). The left hand side of (26) gives This approximation is valid only on time scales large with respect to the short period of high frequency quasi-oscillation. Converting time integration to φ integration and using and (24), we obtain the time average over an oscillation of the right hand side of (26) as To obtain the above relation we have used the evenness of V , (24), and also (25) which implies that the variables in the integral except φ may be replaced by their averaged values in one oscillation of φ in rapid oscillation phase. So finally we get In (29) and (30), ρ φ ( t ) and H are the averaged values in the sense explained above and (30) holds for time scales large with respect to T ( t /greatermuch T ). Therefore on time scales much larger than the period of rapid oscillation, the effective equation of state parameter of the scalar field is w = γ -1 where If the non-minimal derivative coupling was absent we would have γ = 2 n n +2 [14]. In this regime, following our analysis and by using (30), the Hubble parameter can be approximated by [15] To confirm and justify our results, based on eqs.(6), (7), and (10), the behavior of H ( ∝ √ ρ φ ) is depicted via numerical methods in fig.(2). This shows a good agreement between numerical result and (32) for large time scales. We assume that in this epoch the inflaton decays to ultra-relativistic particles (whose the energy density is denoted by ρ r ), to reheat the universe. From the beginning of rapid oscillation, i.e. ρ r = 0, until ρ r = ρ reh /similarequal ρ φ , which is the beginning of radiation dominated era, the universe is approximately dominated by the rapidly oscillating scalar field. Therefore, in this era the Hubble parameter can be approximated by (32) [15], and the scale factor during this era scales as At t reh we can estimate the energy density as where T reh is the temperature of ultra relativistic particles at t reh , and g reh is the number of (massless) degrees of freedom corresponding to the ultrarelativistic particles present in the model at t reh [16]. Collecting all together, we can estimate the number of e-folds during rapid oscillation To be more specific we must determine v . To do so, consider the power spectrum of the scalar perturbation which is computed at the horizon crossing k = k 0 = a ∗ H ∗ (see (2) and (3)), where k 0 = 1 λ 0 is a pivot scale. Thus Using this equation together with (19) and H 2 ∗ /similarequal 1 3 M 2 P V ( φ ∗ ), and after some computations we find that Substituting (38) into (35) yields So far, in our computations we have implicitly assumed that H 2 M 2 /greatermuch 1 holds during inflation and reheating. But as the inflaton oscillation amplitude and consequently H 2 (see eq.(24)) decrease, the validity of this assumption must be investigated. At the end of the reheating era (beginning of radiation dominated era) ρ φ /similarequal ρ reh , we have H 2 reh /similarequal 1 3 M 2 P ρ reh . As H decreases, H reh is the the minimum of the Hubble parameter in the era I and II; so if then the assumption H 2 M 2 /greatermuch 1 is safe in our computations. In terms of the reheating temperature, (40) may be written as", "pages": [ 5, 6, 7, 8, 9 ] }, { "title": "2.3 Radiation dominated and recombination eras", "content": "In the radiation dominated era the universe is constituted of ultra-relativistic particles which are in thermal equilibrium, and undergoes an adiabatic expansion during which the entropy per comoving volume is conserved: dS = 0 [16]. In this era the entropy density, defined by s = Sa -3 , is obtained as [16] So we have In the recombination era, g rec corresponds to photons degrees of freedom and consequently g rec = 2. Hence The temperature decreases by the expansion of the universe via T ( z ) = T ( z = 0)(1+ z ), where z is the redshift parameter. Therefore we can express T rec in terms of T CMB as We have also Thus", "pages": [ 9, 10 ] }, { "title": "2.4 Reheating temperature", "content": "So far, we have determined the number of e-folds in the the right hand side of (4). To obtain the reheating temperature we need also to determine N in (4). Taking a 0 = 1, the number of e-folds from the horizon crossing until the present time is derived as N = exp(∆), where To obtain the above relation we have made use of (37) and which is derived from (13-15) and (19). Using (4), (20), (39), (47), and (48) we finally arrive at where α is defined through α , up to this order of approximation, is independent of v and M . Note that, following (41), validity of the high friction assumption (11) requires that M satisfies For the quartic potential, n = 4, (50) reduces to and for quadratic potential, n = 2, it reduces to In the minimal model, for the quadratic potential, the reheating temperature was obtained in [11]: Note that in contrast to the non-minimal derivative coupling model, T reh in (55) does not depend on relativistic degrees of freedom g reh . By taking g reh = 106 . 75, which is the ultrarelativistic degrees of freedom at the electroweak energy scale, the reheating temperature in minimal model for quadratic potential was computed in [11] as T reh = 3 . 5 × 10 6 GeV . This result was based on WMAP7 data [12] which, for the pivot scale k 0 = 0 . 002 Mpc -1 , imply (for %68 CL, or 1 σ error) The relative uncertainty, σ ( T reh ) T reh up to first order Taylor expansion, where was derived as σ ( T reh ) T reh ≈ 53 [11]. In our nonminimal model, for quadratic potential, and by using (56), we find T reh = 6 . 53 × 10 12 GeV which is much larger than what was obtained in the minimal case. The relative uncertainty is now Due to exponential dependence of T reh on n s , the uncertainty in determining the spectral index has a large effect on the reheating temperature uncertainty. Fortunately recent results from WMAP9, ACT, SPT and Planck 2013, may be employed to obtain more exact value for reheating temperature with less relative uncertainty. These results are quoted in the table1, for %68 CL, or 1 σ error . The adopted pivot scale for the two first column is k 0 = 0 . 002 Mpc -1 , while for the two last columns k 0 = 0 . 05 Mpc -1 . Note that in the last column the relative uncertainty is less than one. This could occur in the context of WMAP9 results provided that σ ( n s ) ≤ 0 . 003. To compute the above uncertainties, a first order Taylor expansion was employed (see (57)) which, because of exponential dependence of temperature on 1 1 -n s , is insufficient. To obtain more accurate bounds for T reh one can insert n s and P s directly in (55). For example for quadratic potential and using Planck+WP+highL+BAO data, at 2 sigma error (%95 CL ), we obtain", "pages": [ 10, 11, 12 ] }, { "title": "3 Conclusion", "content": "Non-minimal derivative coupling model with a power law potential was employed to describe the inflation (see (5)). In this context, the slow roll inflationary phase of the inflaton was discussed in high friction regime (see (11)). The rapid oscillation phase after the slow roll, during which the inflaton decays to ultra-relativistic particles, was studied. From the beginning of this rapid oscillation until the radiation dominated epoch, the equation of state parameter of the universe can be approximated by a constant (see (31)), which enabled us to compute the number of e-folds in this era (see (35)). We also estimated the number of e-folds number in radiation dominated, and recombination era, in the same way as the minimal model (see (47)). By gathering all these results together, we obtained the reheating temperature in terms of T CMB , spectral index and the amplitude of the power spectrum of scalar perturbations (see (50)). Finally according to recent astrophysical data, we determined the value of T reh which is much bigger than the reheating temperature obtained in minimal model. This is due to high friction regime adopted in this paper, which enhances the decay of the inflaton. We also showed that the uncertainty in our result is very smaller with respect to the minimal model. Due to the transparency of the universe to gravitational wave the detection of primordial gravitational wave may also be used as a powerful tool to study the history of the universe evolution. Expansion rate of the universe and his thermal history after the inflationary phase have direct imprints on the gravitational wave spectrum and its detectability [18] . In [19], depending on the tensor-to-scalar ratio r , and the reheating temperature, the required sensitivity of some experiments to detect gravitational wave was discussed. Indeed there was shown that r and the reheating temperature, T reh , are the main parameters for determining the gravitational wave spectrum. As an outlook, in future studies, these results may be extended to our non-minimal derivative coupling model for which r was computed in [20].", "pages": [ 12, 13 ] } ]

[ { "title": "Chiaki Hikage 1 and Kazuhiro Yamamoto 2 , 3", "content": "2 Department of Physical Sciences, Hiroshima University, Higashi-hiroshima, Kagamiyama 1-3-1, 739-8526, Japan Kagamiyama 1-3-1, 739-8526, Japan E-mail: [email protected], [email protected] Abstract. We study the impacts of the satellite galaxies on the redshift-space distortions. In our multipole power spectrum analysis of the luminous red galaxies (LRGs) samples of the Sloan digital sky survey (SDSS), we have clearly detected the non-zero signature of the hexadecapole and tetrahexadecapole spectrum, which almost disappears in the power spectrum with the sample of the brightest LRGs only. We thus demonstrate that the satellite LRGs in multiple systems make a significant contribution to the multipole power spectrum though its fraction is small. The behavior can be understood by a simple halo model, in which the one-halo term, describing the Finger of God (FoG) effect from the satellite galaxies, makes the dominant contribution to the higher multipole spectra. We demonstrate that the smallscale information of higher multipole spectrum is useful for calibrating the satellite FoG effect and improves the measurement of the cosmic growth rate dramatically. We further demonstrate that the fiber collision in the galaxy survey influences the one-halo term and the higher multipole spectra, because the number of satellite galaxies in the halo occupation distribution (HOD) is changed. We also discuss about the impact of satellite galaxies on future high-redshift surveys targeting the H-alpha emitters. Keywords: power spectrum, redshift surveys, cosmic flows, modified gravity ArXiv ePrint: 1303.3380", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The luminous red galaxies (LRGs) in the Sloan digital sky survey (SDSS) demonstrated the usefulness of a large redshift-survey of galaxies. Especially, it proved that a precise measurement of the statistical features in their spatial distribution provides us with the very useful methodology not only for the cosmology but also for the fundamental physics. For example, the baryon acoustic oscillation signature in the large scale structure is now recognized as a promising way for exploring the origin of the accelerated expansion of the universe [1, 2]. A stringent constraint on the neutrino mass is also obtained [3, 4]. Furthermore, the LRG sample showed that a measurement of the redshift-space distortions gives us a unique chance of testing the theory of gravity (e.g., [5, 6], cf. [7]). The LRGs in SDSS are massive early-type galaxies, and most part of them are considered to be residing in the center of halos. However, it is clarified that the some fraction of the LRGs consists of multiple galaxies system. The halo occupation distribution (HOD) of the LRGs was clarified by Reid and Spergel ([8], cf. [9]). The correspondence between the LRGs and halos has been investigated and has illuminated the importance of the Finger of God (FoG) effect [3, 10-12], which is non-linear redshift distortion due to the internal motion of galaxies within halos [13]. Recent investigations with N-body simulations have discovered that halos at the redshift 2 could be the origin of the LRG host halos [14]. In the present paper, we investigate the contribution of the satellite LRGs in multiple systems to the redshift-space distortions. A related topic has been investigated in the literature [3, 10-12], but the previous works investigated the contribution to the monopole spectrum. We here focus our investigation on the redshift-space distortions described by the higher multipole power spectrum. The redshift-space distortions are measured in terms of the anisotropic correlation function or the anisotropic power spectrum, e.g., [7, 15, 16]. The anisotropic correlation function of the SDSS LRG sample has been measured in the literature, e.g., [2, 17, 18]. The anisotropic power spectrum P ( k, µ ), where µ denotes the directional cosine between the line of sight direction and the wave number vector, is the Fourier transform of the anisotropic correlation function. They are equivalent to each other. The multipole power spectrum P /lscript ( k ) is defined as the coefficient of the multipole expansion, 1 2 or where L /lscript ( µ ) is the Legendre polynomial, which is normalized as In refs. [5, 21], the multipole power spectrum of the SDSS LRG sample was measured. In the present paper, we demonstrate that the satellite galaxies make a significant contribution to the higher multipole power spectrum, though its fraction is small. The primary purpose of the present paper is to understand the contribution of satellite galaxies to the multipole power spectrum. To this end, we measure the multipole power spectrum of the SDSS LRG samples, and compare the results with the predictions of a simple halo model with the HOD of the SDSS LRG catalog. Then, we show the importance of the one halo term in the higher multipole power spectrum. We demonstrate that the information of multipole power spectra such as hexadecapole P 4 ( k ) and tetrahexadecapole P 6 ( k ) are useful for calibrating the satellite properties and significantly improve the measurement of the growth rate. We also investigate the influence of satellite galaxies in a future redshift survey targeting H-alpha emitters on the multipole power spectrum in a measurement of the redshift-space distortions, because their contamination could give rise to a systematic error when comparing with theoretical models. An assessment of the systematic error is also the purpose of the present paper. This paper is organized as follows. In section 2, we show the multipole power spectrum of the satellite LRGs and their contribution to the total LRG sample and the impact on the growth rate measurement. In section 3, we introduce a simple halo model for a system consisting of central galaxies and satellite galaxies, then we show that the halo model with the HOD of LRGs explains the behavior of the LRG multipole spectra. 4, we demonstrate a constraint from the LRG samples by comparing the observational results and the theoretical model. In section 5, we also discuss about the impact of satellite galaxies in a future survey targeting H-alpha emitters at high redshifts. Section 6 is devoted to summary and conclusions. Appendix outlines the derivation of our theoretical expression for the multipole power spectrum in redshift space in the halo model. Throughout the present paper, we adopt the Hubble constant H 0 = 100 h km/s/Mpc with h = 0 . 7 unless otherwise stated.", "pages": [ 2, 3 ] }, { "title": "2 Impacts of satellite galaxies on the redshift-space distortions", "content": "In this section, we demonstrate the contribution of the satellite LRGs to the multipole power spectrum. We here use the halo sample described in [12] using observed SDSS DR7-Full LRG sample in Northern sky (publicly available catalog prepared by [22]). The sample consists of 96762 LRGs with the magnitude -23 . 2 < M g < -21 . 2 in the redshift range 0 . 16 < z < 0 . 47 (the mean redshift is 0.32) covering 1.44(Gpc/h) 3 comoving volume. Halo is identified with the counts-in-cylinders techniques developed by [8]: two galaxies are considered neighbors when the transverse separation ∆ r ⊥ ≤ 0 . 8Mpc/h and the redshift difference ∆ z/ (1 + z ) ≤ 0 . 006 corresponding to the velocity difference δv p = 1800km/s. The total number of halos is 92046. When the missing galaxies due to fiber collisions are taken into account, the total number of LRGs become 98991. If all of them are hosted by the same halos of the observed LRGs, the actual number of satellite LRGs becomes 6945 (7%). Most of halos (95.5%) occupy single LRG (hereafter we call them 'single LRG systems') and the rest of them contain multiple LRGs ('multiple LRG systems'). The multiplicity distribution of LRGs in halos is listed in Table 1 of [12]. For the multiple LRG systems, we choose the brightest LRG (BLRG) in each group as the central LRG and the rest of them are the nonbrightest LRGs (NBLRGs), which we regard as satellite LRGs. Strictly speaking, BLRGs are not always central LRGs as suggested by several observations (e.g., [12]), and our satellite sample contains central LRGs to some extent. We have used different samples described in Table 1 to see the impact of satellite galaxies on the redshift-space power spectrum. Figure 1 shows the histogram of the number density of the galaxy samples as a function of the redshift z .", "pages": [ 4, 5 ] }, { "title": "2.1 Multipole power spectrum", "content": "We adopt the method to measure the multipole power spectrum developed in [23]. For simplicity, we adopt the weight factor ψ = 1, and the parameter α = 0 . 1 for the random catalog (see [23] for details). The method doesn't take the window effect of the survey region into account, but it is demonstrated that the window effect in our method is negligible by comparing with other method incorporating it explicitly [21]. We perform the multipole power spectrum analysis for each sample, whose results are shown in Figure 2 and 3. Figure 2 compares the multipole power spectrum of the All LRG sample (black curve) and that of the NBLRGs (red diamond with large error bars). The squares with small error bars are the results in a previous work in [6], which are obtained from the LRG sample with 7150 square degrees sky coverage with the total number 100157 in the range of redshift 0 . 16 ≤ z ≤ 0 . 47. Thus the previous sample is almost same as the 'All' LRG sample in the present paper. This figure shows that the amplitude of the correlation of the NBLRGs is quite large compared with the dominant component.", "pages": [ 5, 6, 7 ] }, { "title": "2.2 Impact on parameter estimation", "content": "Here let us demonstrate the impact of the contamination from the satellite galaxies (NBLRGs) in an estimation of cosmological parameters. For simplicity, let us consider the simple model of the anisotropic power spectrum P ( k, µ ) = ( b ( k ) + fµ 2 ) 2 P NL m ( k ) D [ kµ ˜ σ v /H 0 ] , (2.1) where P NL m ( k ) denotes a nonlinear matter power spectrum, D [ kµ ˜ σ v /H 0 ] is the damping factor due to the FoG effect and ˜ σ 2 v is the velocity dispersion parameter, for which we adopt the function Here we determined the bias b ( k ) so that the observational and the theoretical monopole spectra match. Then computed the chi-squared using the quadrupole spectrum by χ 2 = ∑ i [ P obs . 2 ( k i ) -P theo . 2 ( k i )] 2 / [∆ P obs . 2 ( k i )] 2 , where P obs . 2 ( k i ) and ∆ P obs . 2 ( k i ), are the observed values and errors, and P theo . 2 ( k i ) is the corresponding theoretical value. See reference [5] for details. Figure 4 shows the 1 sigma and 2 sigma contours of ∆ χ 2 on the parameter plane ˜ σ v and γ , where the growth factor and the growth rate are parametrized as where Ω m ( a ) is the matter density parameter at the scale factor a . Here we fixed the other parameters n s = 0 . 97, Ω m = 0 . 28, Ω b = 0 . 046, σ 8 = 0 . 8 and assumed the cold dark matter model with a cosmological constant (ΛCDM model) as the background universe model. In each panel, the dotted curve, solid curve, and the dashed curve are the Single, Brightest, and All LRG sample, respectively. The left (right) panel used the data with k ≤ 0 . 2 h Mpc -1 ( k ≤ 0 . 3 h Mpc -1 ). The value γ = 0 . 55 is the prediction of the model on the basis of the general relativity [24]. Though our theoretical model is very simple, the results clearly show that the contamination of the satellite galaxies (NBLRGs) significantly biases the parameter estimation. This figure also indicates that the results are influenced by including the brightest LRGs consisting of the multiple systems.", "pages": [ 7, 8 ] }, { "title": "3 Halo model description of satellite Finger-of-God", "content": "In this section, we consider the FoG effect of satellite galaxies based on the halo model picture [25-27]. In the halo model, the power spectrum of LRGs are decomposed into 1-halo and 2-halo terms. Then we write the anisotropic power spectrum in the redshift-space consisting of the 1-halo and 2-halo terms, We here consider the sample which consists of the central galaxies and the satellite galaxies, and adopt the following expressions (3.2) and (3.11) for P 1h ( k, µ ) and P 2h ( k, µ ), respectively. A brief summary of the derivation for a general case is described in the appendix (See also below for details). One-halo term is given by where we adopt the halo mass function dn/dM given by [28] and ¯ n is the mean number density of LRGs given by ¯ n = ∫ dM ( dn/dM ) N HOD ( M ) and N HOD ( M ) is the halo occupation distribution (i.e., the average number of galaxies inside the halo with mass M ). We use the following form of the HOD of central LRGs and satellite LRGs [29] where erf( x ) is the error function. We adopt M min = 5 . 7 × 10 13 M /circledot /h , σ log M = 0 . 7, M cut = 3 . 5 × 10 13 M /circledot /h , M 1 = 3 . 5 × 10 14 M /circledot /h , and α = 1 to match the HOD of SDSS DR7 LRG catalog [8] as shown in Figure 5. Assuming the number of groups with N sat satellites is Poisson distributed [30], the averaged satellite-satellite pair number 〈 N sat ( N sat -1) 〉 per halo goes to 〈 N cen 〉〈 N sat 〉 2 . We also take into account the missing galaxies due to the fiber collision by multiplying the satellite HOD with a following mass-dependent factor where 1 -f col ( M ) represent the fraction of missing satellite LRGs due to the fiber collision effect for the host halo mass of M . The factor A col and A col + B col corresponds to f col ( M ) for M = M cut and M = M 1 where the averaged number of satellites is 0 and 1 respectively. Here we set A col = 0 . 7 and B col = -0 . 05 to match the number fraction of NBLRGs and the number of NBLRG pairs in groups. We do not consider the fiber collision effect on central HOD, for simplicity. Central LRGs locate near the halo center and thus their velocity difference relative to the host halo should be small. Note that it is difficult to verify that each central LRG is located at the center of each halo in observational data. However, 20-40% of brightest LRGs are found to be off-centered (satellite) galaxies using lensing and cross-correlation analysis [12]. Therefore, large part of the NBLRGs are off-centered and their velocity should be the main source of the FoG effect. The functions ˜ p cs ( k, µ ; M ) and ˜ p ss ( k, µ ; M ) are the Fourier transform of central-satellite and satellite-satellite distribution inside the halo with the mass of M , and the internal motion of satellite LRGs elongate the distributions in the line-ofsight direction. We assume that the internal velocity of the satellite LRGs has a Gaussian distribution determined by virial velocity as σ v, off ( M ) = ( GM/ 2 R vir ) 1 / 2 , in which the virial radius of the halo with mass of M is R vir = (3 M/ 4 π ¯ ρ m ( z )∆ vir ( z )) 1 / 3 with ∆ vir = 265 at z = 0 . 32. When the satellite motion is uncorrelated with each other, ˜ p cs ( k, µ, M ) and ˜ p ss ( k, µ, M ) are given by We assume that the distribution of the satellite galaxies follows the NFW profile [31] and ˜ u NFW ( k ) denotes the Fourier transform of truncated NFW profile, equation (A.4), (see also [32]). In order to test the validity of Gaussian assumption of satellite velocity distribution, equations (3.7) and (3.8), we compare the distribution functions of pairwise velocity for central-satellite pairs and satellite-satellite pairs based on the halo model, as shown in Figure 6. We compute the pairwise velocity between NBLRGs and BLRGs within the same group from their redshift difference as ∆ v = c ∆ z/ (1 + z ). We find that the distributions are well explained by the mass integral of the Gaussian velocity distribution with the Virial velocity dispersion of each mass σ v, off ( M ) = ( GM/ 2 R vir ) 1 / 2 , where the normalization of P (∆ v ) is determined so that the integral over ∆ v is unity. With the velocity probability distribution functions, we compute σ 2 ∆ v = ∫ ∆ v 2 P (∆ v ) d ∆ v for the theoretical value of pairwise velocity dispersions. The model predictions become 663km/s for central-satellite pairs and 937km/s for satellite-satellite pairs. These values well agree with the observed pairwise velocity dispersions: 653km/s for BLRG-NBLRG pairs and 909km/s for NBLRG-NBLRG pairs. The good agreement validates our models of central-satellite and satellite-satellite distributions in redshift space (equations (3.7) and (3.8)). For comparison, we also plot the exponential profile, which also well describes the behavior of the observed pairwise velocity distribution. The 2-halo term is given by where P NL m ( k ) is the real-space non-linear matter power spectrum. Here we use the non-linear matter power spectrum to describe the non-linear power spectrum of velocity divergence for simplicity, while the matter and velocity power spectra are actually different (c.f.,[36, 37]). Here we add the volume exclusion effect of halos ˜ u vol ( k ; M ) in addition to the satellite distribution in order to include that two different halos cannot approach each other closer than a halo size. We use a Gaussian form ˜ u vol ( k ; M ) = exp( -( akR vir ( M )) 2 / 2) and we choose the width parameter a = 2 to fit the observed power spectrum. Again we consider only the velocity distribution of satellite LRGs. We simply use the linear Kaiser formula [33] given by the term of ( b ( M ) + fµ 2 ) with the growth rate f ≡ d ln D/d ln a and the linear halo bias b ( M ) [27, 32, 34]. Without the FoG effect, that is σ v, off = 0, we have P 2h /similarequal ( b eff + fµ 2 ) 2 P NL and P 1h /similarequal N 1h , where b eff is the effective bias of LRGs given by b eff = ∫ dM ( dn/dM ) b ( M ) N HOD ( M ) / ¯ n and N 1h is defined by N 1h = ∫ dM ( dn/dM ) [ 2 〈 N cen 〉〈 N sat 〉 + 〈 N sat ( N sat -1) 〉 ] / ¯ n 2 . In this case, N 1h is a constant and we have P 1 h /lscript ( k ) = 0 for /lscript ≥ 2. Figure 7 compares the halo model predictions of multipole power spectra for All LRG sample with the observations. Our model qualitatively well explain the observations although we simply adopted the linear Kaiser redshift distortion and the linear halo bias. The halo model well explains the differences between ALL and BLRG samples as shown in the below of this section. The satellite FoG effect in 1-halo term becomes significantly important at larger k and dominant in the multipole spectra for /lscript ≥ 4 even though the satellite fraction is only 5%. The 1-halo term contribution causes a systematic bias in the measurement of the growth rate as shown in Figure 4 because the FoG effects from the 1 and 2-halo terms have different feature and the simple form of equation (2.1) is not enough to describe both of the FoG effects very well. In the following section, we show how the constraints on the growth rate changes by taking into account the 1-halo term. The behavior of higher-order multipole spectrum is sensitive to the satellite FoG effect in one-halo term. In other words, the higher multipole spectra can be a good probe of the satellite fraction and the satellite velocity distribution. The one halo term making contribution to the multipole power spectrum of (3.2) can be written as follows: where we defined and q = σ v, off ( M ) k/ √ 2 aH ( z ). Specifically, we have The error function has the asymptotic form erf( q ) → 1 for q /greatermuch 1. Therefore, Q /lscript ( q ) is in proportion to ( -1) /lscript/ 2 q -1 in the limit q /greatermuch 1, which explains the asymptotic behavior of the multipole power spectrum at the large wave numbers. The central-satellite contribution is dominant in the 1-halo term of LRG samples and thus kP 1h /lscript ∼ 2 kQ /lscript ( q ) f sat / ¯ n where f sat is the satellite fraction. When we use the values of ¯ σ v, off = 663km/s, aH ( z ) = 88 h km/s at z = 0 . 32, f sat = 0 . 07 and ¯ n /similarequal 10 -4 (Mpc / h) -3 for the LRG sample, the large-scale limit of kP 1h /lscript goes to 230 for /lscript = 0, -120 for /lscript = 2, 85 for /lscript = 4, and -74 for /lscript = 6. Figure 7 shows that 1-halo term contribution approaches these values roughly. Figure 8 compares the halo model predictions of the multipole power spectra for the 'NBLRG' sample and the observed spectra. As 40% of BLRGs in multiple LRG systems are satellites (the number of multiple LRG systems N mul is 4157), the same number of central galaxies are mixed in the NBLRG sample. Here we consider that 35% (=0 . 4 N mul / N sat ) of the NBLRG sample are central galaxies and the rest of them are satellites. Based on this assumption we write the one-halo and two-halo terms of the power spectrum of the NBLRG sample as where q (sat) cen is the fraction of central galaxies in the NBLRG sample. Here we set q (sat) cen = Min(0 . 35 , 1 / 〈 N sat 〉 ) so that q (sat) cen 〈 N sat 〉 does not exceed unity. The one-halo and two-halo terms for NBLRGs are plotted with the dotted curve and the dashed curve, respectively. The halo model explains the observed multipole spectral very well. One-halo term (dotted curve) is a dominant contribution to the multipole spectra and reaches kP NBLRG /lscript ( k ) ∼ O (10 3 ). The brightest LRG power spectrum does not have one-halo term because each halo contains one LRG at most. Then, the BLRG power spectrum is written only with the twohalo term. This can be clearly seen in that the BLRG multipole power spectra with /lscript = 4 and 6 are almost zero in Figure 9, which indicates that the 1-halo term from satellite galaxies becomes significantly smaller by removing NBLRGs. This means that the halo reconstruction method we use well succeeds in removing the one-halo term. This is important for the study of precision cosmology because the uncertainty of the satellite HOD and its FoG effect becomes significantly small. In a strict sense, however, P 4 ( k ) has slightly positive signature compared to the halo model predictions. This may come from that the reconstruction method is incomplete and some of satellite LRGs are included in the BLRG sample. Multipole spectra such as P 4 ( k ) dominated by the 1-halo term is useful for estimating the residual 1-halo term effect. However, several observations indicate that some fraction of BLRGs are satellite or off-centered galaxies (e.g., [35]), which causes the FoG effect [11]. When the fraction that BLRGs locate on the mass center of their host halos is q (BLRG) cen , the BLRG power spectrum is given As shown in [12], the lensing and cross-correlation measurements indicate 20% fraction of BLRGs are off-centered. Figure 9 shows the model predictions of the BLRG power spectra with q (BLRG) cen = 1 (all of BLRGs are centrals) and q (BLRG) cen = 0 . 8 (20% of BLRGs are satellite). The figure shows that their agreement is better at high k when the satellite FoG effect is included, which indicates that even BLRG sample may have significant FoG effect on the multipole power spectra. Note that the result may change if we take into account the nonlinearity in the galaxy biasing. The linear Kaiser formula is the simplest model, then more careful analysis will be necessary using the sophisticated perturbation theories as well as numerical simulations ([36-39]), though such analysis is beyond the scope of the present paper. Figure 10 shows the differences of multipole power spectra P 2 and P 4 between ALL and BLRG samples. The curves in each panel are the theoretical prediction of our model using the satellite HOD parameters of the NBLRG samples. The theoretical curves much better fit the observational results, compared with those in Figure 7. This agreement indicates the contamination of the FoG effect of the off-centered velocities in the BLRGs.", "pages": [ 8, 9, 10, 11, 12, 13, 14, 15, 16 ] }, { "title": "4 Constraints on the growth rate and the properties of satellite galaxies", "content": "In this section, we consider a constraint by comparing the observed multipole power spectrum of the LRG samples and our theoretical model including the one-halo term. This demonstrates how the one-halo term influences a cosmological constraint. We define χ 2 by where P obs . /lscript ( k i ) and ∆ P /lscript ( k i ) are the observed power spectrum and the error, respectively, and P model /lscript ( k i ) is the theoretical model, described in the below. Based on the halo model developed in previous section, we fit the observed power spectra with the following form of the power spectra averaged over halo mass where ¯ b ( k ) is the averaged bias of LRGs and linearly fitted as b 0 + b 1 k . Here the growth rate is f = Ω m ( z ) γ , assuming the ΛCDM model as background universe, and the other cosmological parameters are fixed as n s = 0 . 96, Ω m = 0 . 28, Ω b = 0 . 044, σ 8 = 0 . 8. We consider the FoG of satellite LRGs and parametrize it with the satellite fraction f sat and the averaged velocity dispersion ¯ σ v, off . We use a Lorentzian form of FoG damping function of (2.2), which well approximates the observed satellite velocity distribution as shown in Figure 6. In the limit of small k , the FoG damping function D ( x ), eq. (2.2), becomes 1 -x 2 / 2. In this lowest-order approximation, ˜ σ 2 v in equation (2.1) corresponds to 2 f sat (¯ σ v, off H 0 /aH ( z )) 2 in equation (4.4). For the 1-halo term, we only take the dominant contribution of the central-satellite pairs into account, and neglect that from the satellite-satellite pairs. We do not introduce additional parameter of central fraction (i.e., q cen ), for simplicity, while it is still controversial issue whether BLRGs are off-centered or not. Instead, we leave f sat as a free parameter because the observed multipole power spectrum, P 4 ( k ) in Figure 9, systematically deviates from zero even at small k , which suggests the residual 1-halo terms. In summary, the number of the fitting parameters is 5 in total: b 0 , b 1 , γ, f sat , and ¯ σ v, off . First, let's see how adding the 1-halo term in the theoretical model changes the fitting results. Table 2 compares the constraints on the parameters of γ, f sat , ¯ σ v, off without 1-halo term and those with 1-halo term for All, BLRG, and Single LRG samples, respectively. The fitting range is up to k = 0 . 2 h/ Mpc for all of P l ( l = 0 , 2 , 4 , 6), and the bias parameters are marginalized over. In the fitting (I) without 1-halo term, the value of γ for All sample is overestimated (or growth rate f is underestimated) compared to that of the BLRG or Single LRG sample, which is also shown in Figure 4: the difference of best-fit values of γ between All and Single is 0.16 and that between BLRG and Single is 0.07. The deviation is mildly alleviated by including the 1-halo term in the fitting (II): 0.12 between ALL and Single and 0.04 between BLRG and Single. However, the 1-halo term effect is highly degenerated with the growth rate. Next we add the information of the small-scale measurements of P 4 ( k ) and P 6 ( k ) of the range of wavenumbers up to k = 0 . 6 h /Mpc in the fitting. As shown in the previous section, P 4 ( k ) and P 6 ( k ) at large k is dominated by the FoG effect of 1-halo term, then the information can be used to calibrate the uncertainty of the satellite FoG. We find that the information of P 4 ( k ) and P 6 ( k ) on small scales (at large k ) significantly improves the ± ± ± error of satellite fraction by a factor 3 ∼ 4 and the error of γ by a factor 2. Here P 4 ( k ) plays an important role, especially. Our constraints on γ from the 3 different LRG samples becomes consistent with each other by including the higher multipole spectra at large k . This indicates that our fitting formula including 1-halo term well describes the behavior of three different LRG samples. Figure 11 shows the contour of the joint constraints on γ and f sat when the small-scale information of P 4 ( k ) and P 6 ( k ) is included (red) and not included (blue). It is clearly seen that the measurements of P 4 ( k ) and P 6 ( k ) on small-scales break the degeneracy between γ and f sat and improves their errors dramatically. Figure 12 compares the observations (black filled circles with error bars) of the multipole power spectra and the corresponding best-fitted curve (red solid curves). Our model well describes the observations of the three samples including P 4 ( k ) and P 6 ( k ) at large k . Actually the satellite fraction for BLRG and Single LRG samples significantly decreases, as described in Table 2. However, it still remains ∼ 2% fraction of central-satellite pair, accordingly the satellite fraction for All sample becomes ∼ 7 %, which is higher than the expected value including the fiber collision ∼ 5%. This may indicate that the halo reconstruction is incomplete and some of satellite galaxies are still included in BLRG and Single LRG samples. Multipole power spectra such as P 4 ( k ) and P 6 ( k ) are a good indicator for the residual 1-halo term and may be useful for finding a better grouping method. Our constraint on the velocity dispersion of ¯ σ v, off is ∼ 800km/s, which is roughly consistent with the observed pairwise velocity dispersion between BLRG and NBLRGs, that is 653km/s as shown in Figure 6. Our method using the measurements of higher multipole spectrum P 4 ( k ) provides a promising way to calibrate the satellite FoG effect and improve the error of the growth rate measurement. The measurements of the satellite fraction and the velocity dispersion can be translated to the constraints on the satellite HOD and/or the velocity bias between LRGs and halos. However, our theoretical model is still very simple and uses various approximations such as Kaiser approximation. In order to obtain more robust estimates on the growth rate and satellite properties, we need more precise theoretical models of halo clustering and velocity probability distribution function by comparing with simulated mock samples. Such detailed analysis is beyond the scope of this paper and is left as future work.", "pages": [ 16, 17, 18, 19, 20 ] }, { "title": "5 Forecast on multipole power spectra for H α emitters", "content": "Main targets of high-redshift ( z = 1 ∼ 2) galaxy surveys, planed in such as Subaru/PFS [40] and Euclid [41], are H α emitters (HAE). In this section, we perform Fisher analysis to estimate the impact of satellite galaxies for such future surveys targeting HAEs.", "pages": [ 20 ] }, { "title": "5.1 HOD of H α emitters", "content": "The relation of HAEs to halos are less known observationally, and will be more complicated than that of LRGs. We use the following form of HOD based on the sample of 370 HAEs at z=2.23 detected in Hi-Z Emission Line Surveys (HiZELs) [42] Here the central HAE distribution is described with Gaussian and smoothed step-like components with their amplitudes determined by the normalization factors F a and F b . The typical mass and the dispersion are parametrized with M min , c and σ log M, c , respectively. Satellite HOD is described with a smoothed step-like component multiplied by power-law with scaling of α , the typical satellite mass M min , s and the amplitude F s . The values of HOD parameters for different luminosity samples are listed in Table . The plots of HODs are shown in Figure 13. Figure 14 shows the comparison of multipole power spectra for central HAEs with those for all HAEs including satellites at z = 2 . 23. The FoG effect on the HAE power spectrum from satellite in a halo is much smaller than that on LRGs power spectrum because the typical halo mass of HAEs is much smaller than that of LRGs. In our halo model, averaged virial velocity of halos hosting HAEs is 170km/s, while those hosting LRGs are 660km/s. For the faint HAE sample, the contamination of the satellite changes the higher multipole spectrum at a few percent or 10 percent level depending on the wave number, while the effect becomes smaller for the luminous HAE sample because the satellite fraction decreases. FoG effect for HAEs are expected to be much smaller than LRGs, however, upcoming galaxy surveys are expected to measure the growth rate measurement at the percent-level accuracy and thus it is still important to estimate the systematic errors of the FoG effect.", "pages": [ 20, 21 ] }, { "title": "5.2 Fisher matrix", "content": "We here discuss about systematic errors from uncertainties of the satellite galaxies in future redshift survey at a quantitative level. To this end, we adopt the Fisher matrix technique to estimate the systematic errors from the one halo term (see, e.g., [11, 43, 44]). The bias in a parameter is estimated by where F θθ ij is the Fisher matrix, whose inverse matrix is [ F θθ ] -1 ik , and F θψ kj δψ j is a vector which describes the systematic bias caused by ignoring the one-halo term. In case A , we adopt the with In the above expressions, we consider the power spectrum that is the combination of the one-halo term (3.2) and the two-halo term (3.11), with the growth rate f = Ω m ( z ) γ , expressions where V is a survey volume, and we set k min = 0 . 01 h Mpc -1 and k max = 0 . 3 h Mpc -1 . In case B , we use [45] where γ , b 0 , b 1 , α are parameters, and b halo ( M ) is the halo bias, fixed as [46] with ν = δ c /σ ( M,z ), δ c = 1 . 686, a = 0 . 132, b = 1 . 5 and c = 2 . 4. We adopt the 4 parameters, γ , b 0 , b 1 , α for the Fisher matrix analysis, where the target parameter is γ = 0 . 55, b 0 = 1, b 1 = 0 . 2, α = 1. The background cosmology is fixed to be the ΛCDM model with Ω m = 0 . 3, Ω b = 0 . 044, and σ 8 = 0 . 8. In the present paper, we focus on the systematic bias in γ , which is considered to be useful for testing gravity. Figure 15 shows the systematic bias ∆ γ as a function of the redshift. In each panel, the solid curve (dashed curve) adopts the HOD with L > 10 43 erg/s ( L > 10 42 erg/s), and the thin (thick) curve is the case A ( case B ) for the estimation of the Fisher matrix, respectively. The left panel assumes no Fiber collision, while the right panel take the fiber collision into account by assuming f col = 0 . 5. Figure 15 means that the fiber collision reduces the systematic bias because the satellite fraction, which causes the systematic error, is reduced. Furthermore, brighter H α emitters do not generally contain satellite, which also reduces the systematic bias. The mean number density is ¯ n /similarequal (2 ∼ 3) × 10 -4 ( h/ Mpc) 3 for the H α emitter with L > 10 43 erg/s (solid curve), while ¯ n /similarequal (2 ∼ 3) × 10 -3 ( h/ Mpc) 3 for H α emitter with L > 10 42 erg/s (dashed curve). The number density of galaxies of a optimized redshift survey would be ¯ n /similarequal (2 ∼ 3) × 10 -4 ( h/ Mpc) 3 . In this case, the sample with L > 10 43 erg/s (solid curve) will be a realistic sample, whose systematic bias in γ is not large. It might be worthy to note that an analysis with the multipole power spectrum ( case B : thick curve) makes a larger systematic bias compared with an analysis with the full anisotropic power spectrum ( case A : thin curve). In general, the amplitude of the one-halo term becomes smaller at higher redshift because the halo mass becomes smaller. However, the power spectrum is less sensitive to the cosmological parameter at higher redshift, which reduces the Fisher matrix elements at higher redshift. This is one of the reason why the systematic bias becomes larger at higher redshift. In the present paper, we omitted the random velocity dispersion between halos. For the H α emitters, however, the halo random velocity could be large. This effect will be included in the two halo term, but not in the one halo term. Then, this might not be included as an uncertainty of the one halo term, but is related with the modeling of the two halo term. A more precise theoretical model for the H α emitters will be necessary including the HOD model and the fiber collision, depending on observational strategy. Our results here are obtained by extensively using the HOD model, which was originally obtained at z > 2.", "pages": [ 21, 23, 24 ] }, { "title": "6 Summary and Conclusions", "content": "In the present paper, we have investigated the influence of the satellite galaxies on the redshift-space distortions. We have found the following points, for the first time. First, the satellite galaxies significantly contribute to the higher-order multipole power spectrum though the fraction is small. Second, the contribution of the satellite galaxies to the higherorder multipole power spectrum is explained by a simple halo model, and the one halo term makes the dominant contribution. We have also demonstrated that the contribution from satellite galaxies depends on the HOD of galaxy samples and the effect of the fiber collision. These findings are based on the SDSS LRG sample, but generally means that an uncertainty of the HOD might give rise to a systematic error in measuring redshift-space distortion when satellite galaxies are contaminated. We have also demonstrated that the small-scale information of higher multipole spectra P 4 ( k ) and P 6 ( k ) at large wavenumbers help calibrate the satellite FoG effect and improve the measurement of growth rate dramatically. For the H α emitters, which are the target galaxies of the PFS redshift survey and the Euclid redshift survey, we have shown that the satellite's contribution to the redshift-space distortion is much smaller than the case of LRGs, because the host halo mass is small. The results are based on the HOD of the H α emitters at the redshift z > 2, it would be interesting to investigate how the results change depending on the redshift especially in the lower redshift regions. Combination with weak lensing survey might help to resolve the uncertainty in HOD [47-49]. A simple Fisher matrix analysis shows that the systematic error from the HOD uncertainty in the parameter γ is not large for H α emitters with L > 10 43 erg/s. But this conclusion is based on the simple model with the HOD model, which was originally obtained at z > 2. Then further check will be necessary, including a modeling of the peculiar velocity of halos. The one-halo term makes the significant contribution to the higher multipole power spectrum of the LRG sample. It is expected that the same situation happens in the CMASS sample of the BOSS survey. The one-halo term reflects the HOD as well as the random velocities of satellite galaxies in a halo. This fact might provide us with an additional cosmological information on the scales of cluster of galaxies. For example, in a class of modified gravity model, the effective gravitational constant in a halo could be larger than that of the solar system. This enhances the velocity of satellite galaxies, which might be detected a signature of modified gravity theories (c.f. [50]). Such a signature might be constrained from the observation of higher multipole power spectrum. But we have also demonstrated that such a gravity-test requires the precise information of the velocity probability distribution function of satellite galaxies as well as the HOD, plus the fiber collision effect. This subject is also left as a future problem.", "pages": [ 24, 25 ] }, { "title": "Acknowledgments", "content": "We thank M. Takada and S. Masaki for useful discussions at the early stage of this work. We also thank A. Oka, S. Saito, T. Nishimichi, A. Taruya, T. Matsubara, T. Okumura, T. Kanemaru, and A. Terukina for useful communications related to the topic of the present paper. We acknowledge anonymous referree for useful and constructive comments. The research by K.Y. and C.H. is supported in part by Grant-in-Aid for Scientific researcher of Japanese Ministry of Education, Culture, Sports, Science and Technology (No. 21540270 and No. 21244033 for K.Y. and No. 24740160 for C.H.). K.Y. is also supported by exchange visitor program between JSPS and DFG.", "pages": [ 25 ] }, { "title": "A Derivation of power spectrum", "content": "In this appendix, we derive a general expression of the multipole power spectrum in the halo model, which gives the grounds to adopt the expressions in section 3. Following the halo model approach, the correlation function is written as the sum of the 1-halo term and the 2-halo term. The power spectrum is the Fourier transform of the correlation function, then the power spectrum is also written as the combination of the 1-halo term and the 2-halo term. We start with the real-space power spectrum in a halo model presented in reference [51], where we defined and ˜ u NFW ( k ; M ) is the Fourier transform of the density profile of galaxy distribution. We assume that the galaxy density profile is the same as the dark matter density profile. For the NFW density profile, we have [27, 32] where The redshift-space power spectrum of the halo model may be evaluated as follows. Tinker investigated the formulation for the redshift-space correlation function in a halo model [52], in which the redshift-space correlation function is obtained by [36, 53] where ξ R ( r ) is the real-space correlation function, s ⊥ is the projected separation, s ‖ is the line of sight separation, r 2 = s 2 ⊥ + z 2 and v z = H ( s ‖ -z ), P ( v z ) is the probability distribution function of the galaxy pairwise velocity, and H is the Hubble parameter. P ( v z ) maps the pairs at separation in the line-of-sight direction z to s ‖ with the probability P ( v z ) [36, 53]. This gives the prescription to include the random velocity of galaxies in a halo in redshift-space power spectrum. Then, we may write the redshift-space power spectrum in the form where and where we defined when the pair wise velocity between the central galaxy and the satellite galaxy obeys the Gaussian probability distribution function P ( v z ) = ( √ 2 πσ v ) -1 e -v 2 z / 2 σ 2 v . Here we assume that the random velocity of the central galaxies can be neglected, then we may write ˜ p ss ( k, µ ; M ) = ˜ p 2 cs ( k, µ ; M ) for the satellite-satellite galaxy pair. In the case of the exponential velocity distribution function, P ( v z ) = ( √ 2 σ v ) -1 e -√ 2 | v z | /σ , we have As is shown in section 3, the one-halo term dominates the higher multipole power spectrum of the All LRG sample. It is useful to present the analytic formula, as is given by equation (3.12) with (3.14)-(3.17) for the case of the Gaussian velocity distribution function. In the case of the exponential velocity distribution function, (3.14)-(3.17) are replaced with", "pages": [ 25, 26, 27 ] } ]

[ { "title": "Microscopic Picture of Non-Relativistic Classicalons", "content": "Felix Berkhahn, a Sophia Müller, a,b Florian Niedermann a,c and Robert Schneider a,c E-mail: [email protected], [email protected], [email protected], [email protected] Abstract. A theory of a non-relativistic, complex scalar field with derivatively coupled interaction terms is investigated. This toy model is considered as a prototype of a classicalizing theory and in particular of general relativity, for which the black hole constitutes a prominent example of a classicalon. Accordingly, the theory allows for a non-trivial solution of the stationary Gross-Pitaevskii equation corresponding to a black hole in the case of GR. Quantum fluctuations on this classical background are investigated within the Bogoliubov approximation. It turns out that the perturbative approach is invalidated by a high occupation of the Bogoliubov modes. Recently, it was proposed that a black hole is a Bose-Einstein condensate of gravitons that dynamically ensures to stay at the verge of a quantum phase transition. Our result is understood as an indication for that claim. Furthermore, it motivates a non-linear numerical analysis of the model.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Recently, Dvali and Gomez proposed a microscopic picture of black holes [1-3]. According to them, black holes can be understood as Bose-Einstein condensates of gravitons. In this picture, the Schwarzschild geometry would effectively emerge from the interaction of a test particle with the condensate of gravitons. In [4, 12] this picture was further elaborated and the authors concluded that the black hole is at the point of quantum phase transition. Within the Schwarzschild radius, the graviton theory is strongly coupled. This necessitates to sum up a large number of equally important terms in the perturbation series. This fact and the relativistic nature of the graviton theory makes it hard to obtain any quantitive predictions along the lines of [1-4, 12] within the theory of general relativity. Therefore, in this paper we propose a non-relativistic, derivatively coupled toy model that allows to quantitatively compute properties expected for black holes according to [1-4, 12]. Our model is constructed such that it contains a ground state corresponding to the black hole of general relativity, which is nothing else but a non-relativistic classicalon state. For a description of the concept of classicalization in the case of gravity see [5, 6] and for its generalisation to other derivatively coupled theories compare to [7-10]. We perform a quantum perturbation theory around a highly occupied classical state (so called 'Bogoliubov approximation') which is supposed to make up the classicalon. Our results indicate that the perturbative approach is not applicable, which is exactly what we expect to see if the system indeed manages to stay at the point of quantum phase transition. Therefore, we see indications for the claims of [4, 12], even though only a subsequent numerical and non-linear analysis will clearly decide about the status of our model. Our paper is organized as follows: Section 2 summarizes the main ideas of [1]. Section 3 contains our model and results. Future prospects of our theory are discussed in Section 4.", "pages": [ 2 ] }, { "title": "2.1 Quantum Portrait", "content": "The starting point in the approach of [1-4] is the observation that the graviton interaction strength α gr is momentum dependent due to the derivatively coupled nature of interaction terms of the metric fluctuation field with itself: where G N is Newtons constant and λ is the typical graviton wavelength involved in a given scattering process. For the case of black holes, the characteristic wavelength is set by the Schwarzschild radius r g = 2 G N M ∼ λ , where M is the mass of the black hole. Accordingly, each graviton contributes an energy ∼ h/ (2 G N M ) . The total number N of gravitons constituting a black hole is thus where we have introduced the planck length L P = √ hG N . Equation (2.2) is also true for the number of gravitons contained in the gravitational field of other objects such as planets since it can be obtained from summing up the Fourier modes of any Newtonian gravitational field φ = -r g /r . Inserting (2.2) in (2.1) yields the dependence of the coupling with N The occupation number N can be understood as the parameter measuring the classicality of a given object composed out of gravitons, in this case black holes. Intrinsic quantum processes such as the decay into a two particle state are exponentially suppressed 〈 Out | exp( -S ) | In 〉 ∼ exp( -N ) . Additionally, the number of gravitons produced in the gravitational field of any elementary particle is negligibly small, for example for an electron we get N = 2 G n m 2 e /h ≈ 10 -44 . This shows why elementary particles cannot be considered as a classical gravitating object (even though they contribute a standard Newton law at large distances), and in particular it becomes clear why a single elementary particle does not collapse into a black hole. Let us contrast black holes with the gravitational field of other objects such as planets. Assuming that the characteristic wavelength of the gravitons is in any case given by the characteristic size R of the object, we obtain as the gravitational part of the energy This shows that for objects not being a black hole (i.e., for R > r g ) a substantial part of the energy is carried by other constituents than gravitons. This is why the gravitational field of other objects than black holes cannot exist without an external source, for example a planet. However, once the extension of the gravitational object reaches R = r g , the whole energy M of our object is stored in the gravitational field, so that an external source is not required to balance the energy budget. It is exactly at this point where the interaction of an individual graviton with the collective potential generated by the other gravitons becomes significant. This can most easily be seen by appreciating that the classical perturbation series in the metric fluctuation field h about a Minkowski background breaks down at the horizon r g . However, the interaction of two individual gravitons is still small as long as we consider regions r > L P . Given that the dominant interaction is gravity itself, the authors of [1] concluded that black holes are self-sustained bound states of gravitons. Moreover, black holes are maximally packed in the sense that the only characteristic of a black hole in the semi-classical limit is the number of gravitons N composing it, and any further increase of this number results inevitably in an increase of the size and mass of the black hole. This becomes clear since by default the extension of the black hole is no free parameter but given by r g , and accordingly all physical black hole quantities (mass, size, entropy, etc.) can be quantified by N . This is nothing else but the famous no-hair theorem translated in the language of gravitons. An important consequence of this picture is that black holes always balance on the verge of self-sustainability, since the kinetic energy h/r g of a single graviton is just as large as the collective binding potential -α gr Nh/r g produced by the remaining N -1 gravitons. Thus, if you give a graviton just a slight amount of extra energy, its kinetic energy will be above the escape energy of the bound state. In [1] it was therefore concluded that black holes are leaky condensates . The above reasoning strictly applies only in the (semi-)classical limit N →∞ . This is important, because we might wonder how a quantum effect like Hawking radiation can be understood in our picture of highly occupied graviton states, since usually we expect quantum effects to be exponentially suppressed. Actually, to explain this, the authors in [4] conjectured that the black hole is at a point of quantum phase transition . Thus, quantum excitations are always significant and cannot be ignored. In particular, given that black holes are leaky condensates, every quantum excitation will lead to the escape of the corresponding graviton. These escaped particles are interpreted as the Hawking radiation of the black hole. Moreover, due to the quantum phase transition, the leading corrections to the above (semi-)classical ( N → ∞ ) picture are not exponentially but only 1 /N suppressed. This makes it possible for any finite N to retrieve information from the black hole (for instance the Hawking spectrum contains 1 /N corrections, making it for example possible to read out the amount of Baryons originally stored in the black hole). The famous information paradox is thus just a relict of working in the strict (semi-)classical N → ∞ approach in which the hair of the black hole is negligible compared to the N graviton state. In the next section we discuss the well known physics of quantum phase transition for the example of a non-relativistic condensed matter system. Assuming that black holes behave similar to this model, we will qualitatively discuss the implications for black hole physics, as it was done in [4].", "pages": [ 2, 3, 4 ] }, { "title": "2.2 On the Verge of Quantum Phase Transition", "content": "The discussion of this section closely follows [11], where the properties of a quantum phase transition are studied. We want to describe a system of N bosons of mass m with an attractive interaction in one dimension of size V at zero temperature. The second quantized field ˆ Ψ( x, t ) in the Heisenberg representation is measuring the particle density at position x . The corresponding Hamiltonian reads where U is a positive parameter of dimension [energy] × [length] controlling the interaction strength. The dynamics of ˆ Ψ( x, t ) are given by the Heisenberg equation where the equal time commutation relations have been used. Applying the mean-field approximation amounts to replacing the operator ˆ Ψ( x, t ) by a classical field Ψ 0 ( x, t ) . This replacement is justified when the quantum ground state is highly occupied. In this case the non-commutativity of the field operator is a negligible effect. Since we are looking for stationary solutions, the time dependence is separated in the usual way where µ is the chemical potential. Inserting this ansatz in (2.6), yields the stationary GrossPitaevskii equation. A trivial solution that fulfils the periodic boundary conditions Ψ 0 (0) = Ψ 0 ( V ) is given by This solution corresponds to the homogenous Bose-Einstein condensate. However, this solution is the minimal energy configuration only for U < U c . The critical value has been be derived in [11] to be: U c = glyph[planckover2pi1] 2 π 2 / ( mVN ) . For U > U c the ground state is given by an inhomogenous solution Ψ (sol) 0 ( x ) describing a soliton. By increasing the parameter U , i.e. the interaction strength, the ground state of the system undergoes a phase transition from the Bose-Einstein phase to the soliton phase once the critical point U c is reached. As the authors in [11] have shown, this point of phase transition is characterized by a cusp in the chemical potential µ ( U ) , the kinetic energy glyph[epsilon1] kin ( U ) and the interaction energy glyph[epsilon1] int ( U ) per particle as functions of U . The main result of [11] was to show that at the point of phase transition quantum corrections to Ψ 0 become important and a purely classical description is no longer possible, therefrom the name 'quantum phase transition'. A suitable way to investigate this effect is provided by the Bogoliubov approximation in which the classical field Ψ 0 is furnished with small quantum corrections δ ˆ Ψ . A proper quantum mechanical treatment, of which the details are given in the next section, allows to derive the famous energy spectrum of the Bogoliubov excitations Due to the periodic boundary conditions, the momentum δk of the Bogoliubov modes is quantized in steps of 2 π/V . From (2.11) it is clear that once the interaction strength approaches the value U c , the energy of the first Bogoliubov mode ( δk = 2 π/V ) vanishes. Consequently, the excitation of the first mode becomes energetically favourable and the condensate is depleting very efficiently. This is the characteristic property of a quantum phase transition. This picture is further substantiated by calculating the occupation number of excited Bogoliubov modes which shows that the vanishing of glyph[epsilon1] ( δk ) is accompanied by an extensive occupation of the corresponding quantum states. This means that the Bogoliubov approximation is no longer applicable and quantum corrections are significant. For values U > U c the energy becomes imaginary, which signals the formation of a new ground state that is given by the soliton solution Ψ (sol) 0 ( x ) , compare to the discussion in [11]. Moreover, the work of [13, 14] shows that the system becomes drastically quantum entangled at the critical point, which is yet another characterization of quantum phase transition. By making the N dependence of U c explicit and introducing the new dimensionless coupling parameter α = UmV/ ( glyph[planckover2pi1] 2 π 2 ) , the condition for the breakdown of the Bogoliubov approximation becomes This is exactly the condition for self-sustainability in the case of a black hole (2.3). These considerations closely follow [4], where the authors wanted to illustrate the relation between black hole physics and Bose-Einstein condensation at the critical point. Of course, in this toy model the relation (2.13) is not generically realized, but has to be imposed by adjusting the model parameters by hand. (For a given value of N , the interaction strength U has to be chosen appropriately.) In the case of GR the left hand side of equation (2.13) is k -dependant which in principal could allow for a generic cancelation between the two terms in the squared bracket in the last line of (2.11). This cancelation is assumed to take place up to 1 /N -corrections. The aim of our work is to present a non-relativistic scalar model that is in principle able to account for this cancelation and thus generically stays at the point of quantum phase transition independent of the chosen parameters. It is not possible to derive this result within the Bogoliubov approximation since a high occupation of quantum states is the defining property of a quantum phase transition. However, the breakdown of the perturbative approach is a necessary condition and therefore provides an indication for it.", "pages": [ 4, 5, 6 ] }, { "title": "3.1 The Model", "content": "Non-relativistic classicalizing theories have the advantage of being computable without a resummation of infinitely many equally important terms as it would be the case for example in GR. In the following, we will consider a special non-relativistic, classicalizing theory that was constructed to mimic general relativity. As in [11], we choose to confine our theory in a 1-dimensional box of size V . To be concrete, we consider the following Hamiltonian for the second quantized field ˆ Ψ( x ) measuring the particle density at position x : where : : denotes the normal ordering. We are looking for homogenous solutions of the Heisenberg equation in which the field operator is again replaced by a classical field Ψ 0 ( x ) . (The subscript 0 will be suppressed throughout the rest of this work.) We try to generalize the known homogenous BEC solution (2.10). We can separate the time dependence as in (2.9). Since Ψ( x ) is a complex field, (3.2) has in general the following class of solutions where the momentum k is quantized in steps of 2 π/V by implementing periodic boundary conditions. The number of particles is denoted by N . Inserting (3.3) in the Hamiltonian (3.1) results in the polynomial where z = N V k 2 . However, not every solution (3.3) is a local minimum of the energy (3.4). For sure, one minimum is given by k = 0 (since the kinetic energy contributes positively), which would exactly correspond to the Minkowski vacuum in the case of general relativity given that this is the global energetic minimum of the theory (3.1). Moreover, by appropriately choosing the coefficients λ and κ , we can construct a second minimum of (3.4) at z 0 = Nk 2 0 /V with positive energy, denoted with Ψ k 0 , where k 0 > 0 . It is easy to show that the corresponding solution not only minimizes (3.4) (that is, minimizing the energy within the sub-class of homogenous solutions (3.3)) but is also given as a minimum in complete field space (that is, it is a minimum for general fluctuations Ψ = Ψ k 0 + δ Ψ ). It is this solution that will turn into the classicalon which corresponds to the black hole solution of general relativity. Furthermore, it should be noted that the chemical potential is zero due to the relation µ ∝ ∂H (0) /∂z | z 0 .", "pages": [ 6, 7 ] }, { "title": "3.2 Bogoliubov Theory", "content": "Wewill study the leading quantum perturbations δ ˆ Ψ( x ) about the classical condensate Ψ k 0 ( x ) . To this end, we write glyph[negationslash] where ˆ a ( k ) is the annihilation operator of the momentum mode k . The Bogoliubov approximation consists in treating the first term in (3.5) classically due to the large occupation of the state with momentum k 0 . The second term presents a small quantum correction. On account of this, the replacement glyph[negationslash] is introduced, which allows to identify Ψ k 0 ( x ) with the first term in (3.5). The second term is simply the Fourier representation of the quantum perturbation δ ˆ Ψ( x ) . We want to calculate the perturbation series up to second order in δ ˆ Ψ( x ) or ˆ a ( k = k 0 ) . Note that once we allow for an occupation of the momentum states with k = k 0 , we have to distinguish between N 0 , glyph[negationslash] the number of particles in the ground state, and N , the total number of particles. Since we want to express everything in terms of N , the normalisation condition glyph[negationslash] has to be employed. This means that the zeroth order H (0) terms contribute to the second order H (2) when we express N 0 in terms of N . Inserting (3.5) and (3.7) into the Hamiltonian (3.1), results in the following quadratic order expression: glyph[negationslash] where the decomposition k = k 0 + δk has been used and the (re-)definitions apply. Here, the polynomials P 0 , P 1 and Λ 0 are functions of the combination z 0 and the coefficients m , λ and κ : Note that when using the minimal energy condition ∂H (0) /∂z | z 0 = 0 , see equation (3.4), we obtain P 0 = P 1 and Λ 0 = 0 due to the relations 2 V ( P 0 -P 1 ) = ∂H (0) /∂z | z 0 and 2 V Λ 0 /k 2 0 = -∂H (0) /∂z | z 0 , respectively. Furthermore, it can be checked that P 0 > 0 if z 0 corresponds to the minimum of (3.4) because 2 V P 1 = z 0 ∂ 2 H (0) /∂z 2 | z 0 . The Hamiltonian (3.8) is almost of the Bogoliubov form and can be diagonolised by means of the transformation where u, v ∈ R . Setting the off-diagonal terms to zero and requiring standard commutation relations for ˆ α and ˆ β implies as well as as well as These two equations are solved by where Note that glyph[epsilon1] (1) 0 and glyph[epsilon1] (2) 0 are strictly positive, whereas the sign of glyph[epsilon1] 1 depends on the value of δk . Thus in order to fulfill (3.18), we have to choose u and v in (3.20) both positive when δk < k 0 and one of both has to be chosen negative when δk > k 0 . In both cases the diagonalized version of (3.8) reads glyph[negationslash] Using the definitions (3.11), (3.12) and (3.13), we find glyph[epsilon1] = 2 P 0 k 0 | δk | and ( glyph[epsilon1] (1) 0 -glyph[epsilon1] (2) 0 ) / 2 = 2 P 0 k 0 δk . Note that glyph[epsilon1] is strictly positive. By employing the relation ˆ α ( δk ) = ˆ β ( -δk ) we find glyph[negationslash] Accordingly, the vacuum | 0 〉 of the Fock space is defined as It follows from the Hamiltonian (3.23) that the combination is the energy of the quasi particles created by ˆ α † ( δk ) with momentum k 0 + δk . Since the vacuum of our theory is defined with respect to ˆ α , it contains a non-vanishing amount of excited real particles associated with ˆ a (and ˆ b equivalently). This effect goes under the name quantum depletion and occurs physically due to the interactions amongst the particles which necessarily pushes some of them to excited states. Their precise number is given by This allows to rewrite the energy of the quasi particles associated with ˆ α as and the number of depleted real particles with momentum k 0 + δk as The above results can easily be generalized to a derivatively coupled theory with an arbitrary number of higher order terms Note that the coefficients c r have dimension [energy][length] 3r -1 . The standard kinetic term corresponds to r = 1 for which the coefficient is c 1 = glyph[planckover2pi1] 2 / (2 m ) . The energy of the quasi particles and the number of depleted particles are given by (3.27) and (3.28) where P 0 now is given by the generalized expression and k 0 is determined as a minimum of the generalized version of (3.4) The coefficients c r have again to be chosen such that there is a non trivial minimum.", "pages": [ 7, 8, 9, 10 ] }, { "title": "3.3 Discussion", "content": "Our results incorporate the vanishing of the energy gap for δk < 0 . This (at least partly) vanishing energy gap can be considered as an indication for the occurrence of a quantum phase transition, as we discussed in section 2.2. Moreover, we see that the Bogoliubov modes become highly occupied for δk glyph[greatermuch] k 0 . This in fact signals a breakdown of the Bogoliubov theory anyways, as two succeeding terms in the quantum perturbation theory compare as where δN denotes the number of excited particles in the momentum state k 0 + δk . Equation (3.32) clearly shows that the number of excited particles should at least be suppressed as δN ∼ N 0 k 2 0 /δk 2 . The result for the number of depleted particles (3.28) is, however, completely the opposite, as it is not suppressed but enhanced for large δk . Therefore, we can safely conclude that the perturbative approximation has broken down anyways. Again, this is in accordance with the expectation of being at the quantum critical point because at this point the system behaves purely quantum and cannot even approximately be described classically. Therefore, the breakdown of the Bogoliubov theory was expected, since it amounts to calculate the perturbative quantum corrections around a classical ground state. Note that the breakdown is also intuitive from the viewpoint of a vanishing energy gap for the quasi particles with δk < 0 . Of course, neither ˆ a or ˆ b particles can directly be related with the direction of ˆ α or ˆ β particles in phase space. But the vanishing of the energy gap should somehow be transferred into the sector of physical ˆ a and ˆ b particles. Since a vanishing energy gap means that it is indefinitely easy to excite the quasi particles, we seem to recover this behavior in the high momentum sector of ˆ a and ˆ b particles. We can also perform the Bogoliubov approximation around the global minimum of (3.4) at k = 0 . Due to the derivatively coupled nature of the interaction terms, the higher order terms in (3.1) do not contribute, which in turn implies that the Hamiltonian (3.8) is already diagonal. Therefore, there is no depletion of the vacuum which allows us to further extend the GR analogy: This state would simply correspond to the Minkowski vacuum in the case of GR.", "pages": [ 10 ] }, { "title": "4 Future Prospects", "content": "Contrary to model (2.5), where the critical point is actually reached and crossed by sufficiently increasing the interaction strength U , in our model there is some indication that the system stays at the point of quantum phase transition and does not organize itself in a new classical ground state. However, this indication is only inferred from the observation of the breakdown of the Bogoliubov theory. To get some solid measures, we need to go beyond the Bogoliubov approximation in the next step [15]. This can be achieved by a full quantum mechanical treatment of the theory (3.1). The diagonalization of the Hamiltonian can be performed under the assumption that only the lowest l momentum eigenstates are significantly occupied (given that we are supposed to sit in a local minimum, this seems to be a good assumption). Therefore, it suffices to diagonalize the Hamiltonian within a Hilbert subspace containing only a finite number of states describing N bosons occupying l different momentum eigenstates. For l chosen appropriately small the calculation is numerically feasible and has been performed in the case of the non-derivativly coupled model in [11]. By means of this calculation we would be able to address quantitative questions, such as the size of the energy gap, the number and spectrum of depleted particles or the amount of quantum entanglement in the system. The generalization of our results to a relativistic classicalon theory offers another promising prospect of future research. This necessitates to apply the ideas of the Bogoliubov approach to a relativistic theory and would be a significant step towards a more quantitative treatment of the black hole condensate in general relativity.", "pages": [ 11 ] }, { "title": "Acknowledgements", "content": "The authors would like to thank Gia Dvali, Daniel Flassig, Stefan Hofmann, Michael Kopp, Florian Kühnel, Alexander Pritzel and Nico Wintergerst for inspiring discussions. The work of FB was supported by TRR 33 'The Dark Universe'. The work of SM was supported by a research grant of the Max Planck Society. The work of FN and RS was supported by the DFG cluster of excellence 'Origin and Structure of the Universe'.", "pages": [ 11 ] } ]

[ { "title": "Large Scale Anisotropic Bias from Primordial non-Gaussianity", "content": "Shant Baghram, 1, ∗ Mohammad Hossein Namjoo, 2, † and Hassan Firouzjahi 1, ‡ 1 School of Astronomy, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran 2 School of Physics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran In this work we study the large scale structure bias in models of anisotropic inflation. We use the Peak Background Splitting method in Excursion Set Theory to find the scale-dependent bias. We show that the amplitude of the bias is modified by a direction-dependent factor. In the specific anisotropic inflation model which we study, the scale-dependent bias vanishes at leading order when the long wavelength mode in squeezed limit is aligned with the anisotropic direction in the sky. We also extend the scale-dependent bias formulation to the general situations with primordial anisotropy. We find some selection rules indicating that some specific parts of a generic anisotropic bispectrum is picked up by the bias parameter. We argue that the anisotropic bias is mainly sourced by the angle between the anisotropic direction and the long wavelength mode in the squeezed limit.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Inflation [1] has emerged as the leading paradigm for the theory of early Universe and structure formation. Basic predictions of inflation indicate that the curvature perturbations are nearly scale-invariant, nearly adiabatic and nearly Gaussian which are in very good agreements with cosmological observations such as WMAP [2] and PLANCK [3]. The simplest models of inflation are based on a scalar field rolling slowly on a flat potential. Any detection of primordial non-Gaussianity (NG) will have significant implications for inflationary model buildings, for a review see [4-6]. For example, many models of single field inflation predict a very small amount of local non-Gaussianity in the squeezed limit f NL ∼ (1 -n s ) [7], in which n s is the curvature perturbation power spectrum spectral index and f NL parametrizes the amplitude of local NG. With n s /similarequal 0 . 96 from PLANCK [3], one expects f NL ∼ O (10 -2 ) for conventional models of single field inflation. However, this expectation is violated if the system has not reached the attractor regime [8, 9] or if one allows for a non Bunch-Davies initial condition [10-13]. Furthermore, inflationary models with large NG predicts different shapes for bispectrum. Therefore, any detection or otherwise of large primordial NG with different shapes will go a long way to rule out many inflationary scenarios or put constraints on model parameters. Having this said, the recent PLANCK collaboration data [14] showed no significant deviation from Gaussian initial conditions. PLANCK constrained the amplitude of NG for different known shapes and accordingly the Gaussian initial conditions are consistent with the picture. The most suitable cosmological observation to constrain the primordial NG is CMB. This is because the perturbations in the last scattering surface are in the linear regime and the fingerprints of non-Gaussianity are mainly preserved [15, 16]. However, recently the interests in Large Scale Structure (LSS) observations and their implications for non-Gaussianity are boosted due to the theoretical findings of scale-dependent bias [17]. In general, the distribution of baryonic matter in the Universe, mainly clustered in galaxies and the clusters of galaxies, is the fundamental observable of LSS [18, 19]. The distribution of galaxies and clusters of galaxies can be studied by a): the mass function of the structures (i.e. galaxies and cluster of galaxies) and b): the correlation functions, power spectrum and even higher moments of distribution. In the standard theories of structure formation, the Gaussianity assumption plays a crucial role in finding the distribution of structures via the primordial density contrast distribution [20]. Accordingly, changing the initial condition from a Gaussian to non-Gaussian primordial density perturbations will change the mass function of structures. This change mainly shows itself in the tail of distribution function. Consequently, this effect manifests itself mainly in the statistics of the clusters of galaxies in high mass and high redshifts distributions [21-25]. The bispectrum of LSS observations is also affected by the primordial NG and by the secondary NG induced by the non-linear growth of structures [26]. The NG introduced by non-linear growth of structures has its own signatures on the bispectrum of structures, but its shape changes in deep non-linear regime. One can find the bispectrum of galaxies and clusters of galaxies to detect the primordial non-Gaussianity and distinguish its effect from the effects of gravitational instability. Recent works have shown that the galaxy bispectrum can be a very promising way to constrain primordial NG [18, 27-29]. On the other hand, the primordial NG may have an effect on the clustering of halos. As an intuitive example we can consider the local NG where the long wavelength mode changes the background linear density fluctuations. This change, in the picture of linear bias theory, will have an influence on the density peaks where structures are formed. In other words, the non-Gaussian long mode changes the threshold which structure goes from linear to non-linear regime. The primordial NG has unique feature in LSS by introducing a scale-dependent bias [17, 30, 31]. This fingerprint of primordial NG provides the opportunity to constrain primordial NG using the power spectrum of galaxies. Many works are done in numerical simulations to check the scale-dependence of the bias parameter introduced by primordial NG and also to study the change in the statistics of structures in Universe [25, 32, 33]. From the observational side different groups used the LSS probes to constrain the primordial NG [34-36]. Furthermore, there are other LSS observables such as the Integrated Sachs Wolfe cross correlation with the galaxy power spectrum [36], the 3D bispectrum of Ly-alpha forest, the redshifted 21-cm signal from the post re-ionization epoch [37, 38], the statistics of voids [39], the cosmological weak lensing [40] etc. which can also be used to study the primordial NG. In a word, the LSS observations will become a very important tool, complementary to CMB observations, to constrain the properties of primordial NG. There have been some indications of statistical anisotropies on CMB power spectrum. Although The statistical significance of the violation of statistical isotropy is not high in WMAP data [41, 42], but nonetheless the possibility of having statistically anisotropic seed perturbations are intriguing. Recently the data from PLANCK collaboration also confirmed the anomalies observed by WMAP, including the anisotropy in CMB sky [43]. This observation triggers the interests in anisotropic models both theoretically and observationally. The statistical anisotropy is usually parameterized via [44] P ζ = P ζ 0 (1 + g ∗ ( ˆ k. ˆ n ) 2 ) in which ˆ n is the preferred direction in sky, P ζ is the curvature perturbation power spectrum of the Fourier mode /vector k with the direction along the unit vector ˆ k and P ζ 0 represents the isotropic power spectrum. Constraints from CMB and large scale structure indicate that | g ∗ | /lessorsimilar 0 . 4 [14, 45, 46]. Recently the bispectrum and the trispectrum in a model of anisotropic inflation [47] have been calculated in [48-51]. It has been shown that large non-Gaussianities with non-trivial shapes are generated. Considering our motivation in using non-Gaussianity fingerprints in LSS as a probe of inflationary universe, we would like to study the effects of large scale-dependent and orientation-dependent bispectrum on bias in these models. The rest of the paper is organized as follows: In Section II we present the anisotropic inflation model and the corresponding bispectrum which is used in subsequent analysis. In Section III we review the basics of halo bias with non-Gaussian initial conditions. In Section IV, we present a general mathematical formulation for the anisotropic bispectrum in terms of spherical harmonics which can be used to calculate the halo bias in models with generic anisotropic bispectrum. In Section V we present our results of halo bias in the models of anisotropic inflation with the anisotropic bispectrum obtained in Section II. We leave some technical issues of halo bias analysis into the Appendix. In this paper we work with the natural unit in which c = /planckover2pi1 = 1.", "pages": [ 1, 2 ] }, { "title": "II. ANISOTROPIC INFLATIONARY MODEL", "content": "In this section we review the anisotropic inflation model and its anisotropic bispectrum which will be used in subsequent analysis. The best method to introduce statistical anisotropies is to incorporate U (1) gauge fields or vector fields in models of inflation. However, due to conformal invariance of the standard Maxwell theory in an expanding background, the background gauge field and its quantum fluctuations are diluted during inflation. Therefore, in order to produce an almost scale-invariant power spectrum of gauge field fluctuations, one has to consider a time-dependent gauge kinetic coupling. This prescription was originally used in [52, 53] in the context of primordial magnetic field. An interesting model of anisotropic inflation is introduced in [47] in which it is shown that, with a suitably chosen gauge kinetic coupling, the inflationary system has an attractor solution in which the gauge field energy density reaches a small but observationally detectable fraction of the total energy density. As a result, the anisotropy produced are at the order of slow-roll parameters. The Lagrangian of the system is in which φ is the inflaton field and F µν = ∂ µ A ν -∂ ν A µ is the field strength associated with the U (1) gauge field A µ . The background is in the form of Bianchi I universe with the metric Here H ≡ ˙ α is interpreted as the average Hubble expansion rate, H a ≡ ˙ a/a and H b ≡ ˙ b/b are the expansion rates along the spatial directions x and y and ˙ σ/H ≡ ( H b -H a ) /H is a measure of anisotropic expansion. We note that this metric enjoys only a two-dimensional rotational symmetry in y -z plane. The details of the dynamics of the system are given in [47, 50]. For the simple chaotic potential V = m 2 φ 2 / 2, the conformal coupling (the time-dependent gauge kinetic coupling) is chosen to be where c is a constant. With c /similarequal 1, one can check that the system admits the attractor solution in which the ratio of the gauge field energy density in the form of electric field energy density is a small and constant fraction of the total energy density. Defining the fraction of electric field energy density to the potential energy as during the attractor regime one obtains in which /epsilon1 H ≡ -˙ H/H 2 is the slow-roll parameter and I ≡ c -1 c . The power spectrum of the curvature perturbation is defined via For the particular anisotropic inflation model described above the power spectrum was calculated in [48-51, 54-59] which has the form where the anisotropy parameter g ∗ is given by Here N ( k i ) represents the number of e-folding that the mode of interest k leaves the horizon. In our notation, the number of e-folding is counted backwards from the time of the end of inflation by a ( N ) = a f exp( N ) so N ≤ 0. For example, N = N CMB = -60 to solve the flatness and the horizon problem. As a result, N ( k ) is calculated to be in which k CMB represents the comoving mode which leaves the horizon at N CMB = -60 e-folds before the end of inflation. To satisfy the observational constraints from CMB and large scale structure we require | g ∗ | < 0 . 3 [14, 45, 46] corresponding to I /lessorsimilar 10 -5 [55-58]. The bispectrum of curvature perturbations, B ζ ( /vector k 1 , /vector k 2 , /vector k 3 ), is defined via The Bispectrum for the model of anisotropic inflation was calculated using in-in formalism in [48, 49] and using δN formalism in [50] with the result Here the anisotropic shape function C ( /vector k 1 , /vector k 2 ) is defined as: where ˆ n is the specific anisotropic direction in the sky. Note that in Eq. (11), P 0 ( k i ) represents the isotropic power spectrum so all anisotropies are encoded in shape function C ( /vector k 1 , /vector k 2 ) (and the appropriate permutations) with the amplitude 288 IN ( k 1 ) N ( k 2 ) N ( k 3 ). It is instructive to look into the bispectrum in the squeezed limit in which one mode is much longer than the other two, say k 3 /lessmuch k 1 /similarequal k 2 , so from the condition ∑ i /vector k i = 0 we also conclude that /vector k 1 /similarequal -/vector k 2 . In this limit, one obtains in which to obtain the above result, Eq. (8) has been used to express the parameter IN ( k 1 ) N ( k 2 ) in terms of g ∗ . The non-Gaussianity parameter f is defined in the squeezed limit k k k via [4, 5] NL 3 /lessmuch 1 /similarequal 2 In general, f NL is an orientation-dependent and scale-dependent quantity. As an order of magnitude estimation, and neglecting the logarithmic scale-dependence in N ( k i ), we can define an orientation-dependent effective f eff NL via keeping in mind that N ( k ) = N CMB +ln( k/k CMB ). Setting N CMB = 60, we can easily get f eff NL ∼ 60 with g ∗ ∼ 0 . 1, compatible with observational constraints. A very interesting observation from Eq.(13) is that when the long wavelength mode /vector k 3 is in the direction of anisotropy, (i.e. /vector k 3 ‖ ˆ n ), then the term inside the big-bracket in Eq.(13) vanishes. Consequently, in this configuration, we do not expect to see the NG effects in LSS. We discuss this feature in more details in Sec.III. For the subsequent analysis we adopt the coordinate system such as the anisotropic direction ˆ n coincides with the ˆ z direction in the spherical coordinates so the other momentum vectors are described by ˆ k 1 = ( θ 1 , ψ 1 ) and ˆ k 2 = ( θ 2 , ψ 2 ), where θ and ψ are the polar and azimuthal angles in spherical coordinates, respectively. For the future reference, we also need the angles between two arbitrary unit vectors ˆ q i = ( θ q i , ψ q i ) defined via cos γ = ˆ q 1 . ˆ q 2 , which is The power spectrum and the bi-spectrum presented in Eqs. (7) and (11) are for the particular model of anisotropic inflation as studied in [48-50]. For a generic anisotropic model the most general power spectrum can be written as [60] where P 0 is the isotropic power spectrum, Y LM ( ˆ k ) (with L ≥ 2) are spherical harmonics and g LM ( k ) quantify the departure from statistical isotropy as a function of wavenumber k . Since each Fourier mode /vector k is related to -/vector k , in the case of real g LM ( k ), the multipole moment L must be even, and in the limit of k → 0 we recover the isotropic power spectrum P 0 ( k ). However, in the general case, (real and imaginary g LM ), we have This condition is imposed by the fact that the matter power spectrum is a real quantity. Comparing Eq. (17) with Eq. (7) for our particular anisotropic inflation model we have g 20 ∝ g ∗ while the rest of g LM are zero. In Section IV we extend the general definition of Eq. (17) for the power spectrum to the bispectrum and look into its implications in halo bias analysis. In this section we review the concept of bias, a parameter that shows the dependence of dark matter halo abundance to the background dark matter density perturbations. The reader who is familiar with these analysis can directly jump to the next Sections in which we present our results of halo bias for anisotropic primordial power spectrum and bispectrum. It is worth to mention that in this work we are not interested in galaxy bias, which is the weighted integral of the halo bias, corresponding to the mechanism of halo occupation distribution (HOD). In order to find an expression for the bias parameter we follow the work by Scoccimarro et al [61]. However, there are other studies which use Excursion Set Theory (EST) to calculate the halo bias. In Adshead et al [62] the authors solved the more complicated problem of non-spherical halos for which the collapse threshold becomes scale-dependent. In D'Aloisio et al. [63] EST is extended to path integral approach taking into account the non-Markovianity effects of random walks in EST. The halo bias relates the halo abundance to the dark matter over-density. In Excursion Set Theory (EST) [64] it is defined as where δ h is the halo over-density and δ m is the matter density perturbation. The EST framework is a very useful tool to calculate the abundance of structures. It is based on the concept of threshold crossing when we go from larger scales to smaller scales with the exclusion of the cloud-in-cloud effect which is present in Press-Schechter formalism[65]. At large scales EST is known to reproduce the initial condition while in small scales it determines the local bias parameter with linear and nonlinear terms [66, 67] which are in reasonably good agreements with numerical simulations [68-70]. It is worth to mention that the halo bias is a function of redshift and scales. This scale-dependance is introduced by applying the initial non-Gaussian condition. According to Appendix A the large scale halo bias can be treated in peak-background splitting [72]. The idea of splitting of the density contrast to short and long wavelength can be translated to a similar splitting of Bardeen potential (correspondingly the matter density contrast) due to Poisson equation which depends on cosmological parameters. The matter density in PBS can be written as The number density of formed structures with mass m can be expressed as a function of small scale statistics, (i.e. smalls scale power spectrum P s ( k )) and the background long wave-length perturbation δ l (i.e. n = n [ δ l ( /vectorx ) , P s ( k s ); m ]) [74]. The bias parameter in the context of peak-background splitting (PBS) is described by the fact that the background large scale over-density changes the critical threshold of spherical collapse [71]. Therefor, the criteria for collapse becomes where δ s is the matter density contrast of the structure, (the subscript 's' stands for the short wavelength); δ l is the background (long-wavelength) density contrast and δ c /similarequal 1 . 68 is the critical density contrast in spherical collapse formalism [72, 73] (for a review of PBS see Appendix A). Now in order to find the bias parameter we have to compare the dark matter halo abundance, in cases with and without the presence of long wavelength (background) over-density. For this task we use the EST approach. In Appendix A, we review the concept of EST in more details and we will derive the bias parameter using the PBS in EST context. The primordial potential in Fourier space can be translated into the late time potential as Here Φ ini represents the initial Bardeen potential sourced by the inflaton field quantum fluctuations which is related to the curvature perturbation in radiation dominated via Φ ini = 2 / 3 R , T ( k ) is the transfer function and D ( z ) is the growth function normalized to scale factor at early times. An important point here is that we use the usual formalism of the isotropic linear perturbation theory when we calculate the effects of anisotropic NG on LSS observables. This is reasonable to first order because after inflation ends we recover the isotropic FRW Universe at the background level. The anisotropies are inherited only in seed perturbations which show themselves only through power-spectrum and bispectrum. Now one can relate the initial non-Gaussian potential to δ l via Poisson equation in sub-horizon scale and linearregime where Here, Ω 0 and H 0 are the matter fraction energy density and Hubble parameter at present time, respectively. In order to calculate the halo-bias term, we should calculate the effect of large scale perturbation, δ l , on the Probability Distribution Function (PDF) density fluctuations . This yields the following relation between the Lagrangian halo number density and the PDF of density fluctuations [61] where Π( δ s , σ 2 m , δ c ; δ l , σ 2 l ) is the conditional PDF of density fluctuations for δ s with corresponding variance σ m , when there is a background perturbation of δ l and variance σ l . The notation used in the conditional PDF, means that the variance σ m at large scales converges to the value σ m = σ l , in contrast to the unconditional PDF of density fluctuations Π 0 ( δ s , σ 2 m , δ c ) in which the variance vanishes ( σ m → 0) at large scales. It will be relevant to define a quantity that shows the probability of first up-crossing in the time interval between σ 2 m and σ 2 m + dσ 2 m in EST language as From the above formalism, we can see the effect of primordial NG on LSS. Assuming that the Bardeen Potential has a local-type NG we have where φ is the Gaussian field. Now we can use the splitting idea on φ by applying φ = φ l + φ s , where φ l is the long wavelength mode of potential and φ s is short wavelength corresponding to the scale of structure. In Appendix A we discuss how the above non-linear form can be generalized to a model with arbitrary shape of non-Gaussianity, in which case, the non-linear term generalizes into a kernel. Now in the presence of primordial NG, the modes are not independent and the conditional PDF of density fluctuations is modified by the non-Gaussian long-wavelength mode. In this case the PDF of density fluctuations will be a function of φ l through the variance and also higher order cumulants ( c p ≡ 〈 δ p s 〉 c ) as The non-Gaussian initial conditions introduces a dependence on higher-order cumulants which does not exist in the Gaussian case. These higher order cumulants depend on the long wavelength mode. Under the assumption that all these effects are small, using the EST formalism we can Taylor expand the conditional PDF of density fluctuations, Π, around unconditional one, Π 0 . The EST formalism with a sharp k -space filter and with the assumption of Gaussian initial conditions leads to a Markovian random walk condition for the density contrast value when changing the mass scale/radius in each step. This means that in the case of Markovianity we neglect the environmental dependence in halo formation process. Recently Maggiore and Riotto [75-77] showed how to extend the EST with the pathintegral method to include the non-Markovian condition. Also there are many follow up works where this effect on non-Gaussian halo bias is studied [78-82] (for more details, see appendix A). In this work we study the effects on linear bias from the anisotropic primordial NG and include only the first derivative contribution in Taylor expansion, Eq.(A17). As it was shown in Scoccimarro et al. [61], the bias parameter calculated in first order of f NL is not sensitive to the Markovianity/non Markovianity condition. In this work we concentrate on NG at the order of f NL . It is worth mentioning that in higher order NG, such as in trispectrum analysis yielding the g NL parameter, non-Markovianity is induced which results in to a new scale-dependence in bias. The analysis in [78-82] show that the departure from Markovian condition in bias parameter is more significant for low mass ranges. As we showed in appendix A, up to where the linear bias is A very important point is that in above equation we have omitted the subscript of F , which means that the nonGaussianity changes the mass function of the structures so the first linear term will have a contribution from primordial non-Gaussianity. Consequently, we have and where F = R NG F 0 , and R NG comes from the deviation of PDF density fluctuations from the Gaussian case [24, 87]. In other words, the effects of non-Gaussianity appeared both in the mass function and in the power spectrum via scale-dependent bias parameter. Since in this work we are interested in the scale-dependence features of bias, the contribution of R NG is not much of interest. For the Gaussian case we use the Sheth-Tormen [88] Gaussian mass function. For the non-Gaussian mass function effect we use the results of [21] in which the non-Gaussian mass function is expanded in the Press-Schechter framework [20] such that first order in f NL , only the first two terms in Taylor expansion of PDF density fluctuations appear. These terms are derivatives of PDF density fluctuations with respect to the long wavelength mode δ l and the variance σ l . The higher order terms, corresponding to derivatives with respect to c p ( p ≥ 3), contribute to O ( f 2 NL ) and O ( g NL ) bias. As mentioned above, in this work we consider only NG at the order of f NL . The p = 1 contribution, the first term in Taylor expansion, Eq.(A18), is the usual scale-independent linear bias from Gaussian perturbations. Keeping in mind b ≡ δ h /δ l , for the first order linear bias ( b 1 L ) we have which can be written as: In the presence of primordial non-Gaussianity, there are new contributions from higher order cumulants p ≥ 2. As a result, the next to leading order term gives which in general is a scale-dependent correction to the leading order, scale-independent bias, Eq. (A20). The key quantity here is I 21 which is the derivative of second cumulant σ 2 m , ( p = 2), with respect to the long wavelength mode φ l which is obtained as [61] where B ˆ δ ˆ δφ is the cross bispectrum of small-scale smoothed density ˆ δ and φ . As a result, I 21 is the quantity which we are looking for in the case of non-Gaussian initial condition which introduces scale-dependent bias at the order of O ( f NL ). So far only the Lagrangian bias appeared because peaks are those of the initial density field (linearly extrapolated). Making the standard assumptions that halos move coherently with the underlying dark matter, and using the techniques outlined in [83-86], one can obtain the final Eulerian bias in linear order as where x ≡ δ c /σ M and δ c = 1 . 68 is the critical density and s 3 is the reduced skewness defined as The skewness is related to the matter bispectrum as where /vectorq 12 = -( /vector q 1 + /vector q 2 ), W is the window function in Fourier space and R is the smoothing scale. For more details see Appendix A. On the other hand, the scale-dependent bias can be rewritten as So the total Eulerian bias, up to first order in f NL , can be split into the scale-independent term, b si , and scaledependent term, b sd , as where b si and b sd are with b G ≡ 1 + b 1 L ( G ) and In next section we find the scale-independent and the scale-dependent bias for general anisotropic initial power spectrum and discuss the effects of the primordial anisotropies on the bias parameter.", "pages": [ 2, 3, 4, 5, 6, 7, 8 ] }, { "title": "IV. GENERAL FORMULATION OF ANISOTROPIC BIAS", "content": "In this section we calculate the bias parameter, assuming a general model independent primordial anisotropy. For this task we write the power spectrum and bispectrum in their most general anisotropic forms. Then we find the possible configurations that give the scale-dependent bias. As mentioned in section II, the general anisotropic power spectrum can be written as [60] which is a generalization of our anisotropic power spectrum defined in Eq.(7), where all the k -dependence is buried in the coefficient g LM . In this work we assume that the mechanism of spherical collapse for structure formation is still applicable in our model with primordial anisotropic perturbations. This is motivated from the fact that after inflation the background becomes isotropic and anisotropies are encoded only on primordial perturbations. Secondly, the collapse mechanism is a local process so up to first order it is not affected by large scale anisotropies. As a result, we assume that the transfer function at leading order is not affected by the primordial anisotropies. Having this said, it would be interesting to perform the analysis with the above assumptions being relaxed, but this is beyond the scope of this work. First of all the variance is defined through the linear matter power spectrum where P L and W ( kR ) are the linear matter power spectrum and window function in Fourier space which are discussed in details in the Appendix A. The anisotropic variance reads as The spherical harmonics are orthonormal via As a result, noting that Y 00 = 1 / √ 4 π , we have so the variance, Eq.(46), can be simplified to Ignoring the scale-dependence of g 00 , we have with Similarly, we can generalize the bispectrum in squeezed limit, where k 3 /lessmuch k 1 ≈ k 2 , as follows where P L is the Legendre function, Y l,m ( θ 1 , φ 1 ) are the spherical harmonics, by which the whole direction-dependence of bispectrum is encoded and the c Llm are the k -dependent coefficients. This decomposition is possible because we set the z coordinate along the anisotropy direction and rotate the x -y plane in Fourier (k-space) such that the azimuthal angle of k 3 is set to zero. We can do this rotation because of the symmetry in the x -y plane. This kind of decomposition, i.e. decomposing into a k -dependent sector ( c Llm ) and the angular-dependent part is especially useful to find the corresponding I 21 term in the case of generic anisotropic bispectrum and the corresponding scale-dependent bias. Since in the real space the bispectrum is a real quantity, and noting that the bispectrum above is defined in Fourier space, the following condition holds Now, using Eq.(52), we can write I 21 as which, again using the orthonormality of spherical harmonics, simplifies to /negationslash This is an interesting result showing that the anisotropic bispectrum induces an anisotropic, scale-dependent bias through the Legendre function of the angle between the anisotropic direction ˆ n and the long wavelength mode. It is also interesting to note that there is a Selection Rule for scale-dependent bias at O ( f NL ), where the harmonic numbers of the short wavelength must vanish ( l = 0, m = 0). All the other parts of anisotropic bispectrum with l, m = 0 do not contribute to the leading order scale-dependent bias. It is worth to mention that the reality assumption of the bispectrum implies where in the case of real c L 00 , L is even. Now, if we define a direction-independent parameter I di ( L )21 as then I 21 will be re-expressed as where θ k is the angle between ˆ n and /vector k . Now the scale-dependent bias becomes where in this case σ 2 A is defined as in Eq.(49). In next Section, as a specific example of the general formulation presented above, we study our model of anisotropic inflation introduced in Sec. (II), corresponding to L = 2 , l = 0 , m = 0.", "pages": [ 8, 9, 10 ] }, { "title": "V. ANISOTROPIC BIAS FOR GAUGE FIELD INFLATIONARY MODELS", "content": "In this Section we study the LSS bias in the model of anisotropic inflation as a special example of general formulation developed in previous Section. For a related work with a phenomenological modeling of bispectrum and its implication for bias see [89]. As in the previous Section, we continue with our simplifying assumptions of spherical collapse and take the transfer function to be that of the isotropic background. We investigate the change in PDF of density perturbations and the corresponding cumulants, and the effect of these changes on bias. In our model the variance is modified due to the fact that we use anisotropic power spectrum. However, we show below that it is not direction-dependent (as it was shown in general case). Without loss of generality, we can assume that the anisotropy is pointed along the z -direction in spherical coordinates (ˆ n = ˆ z ). Then, starting from Eq. (7) for the primordial anisotropic power spectrum, we have where θ and ψ are spherical coordinate angles and g ∗ ( k ) = -24 IN ( k 1 ) N ( k 2 ). Note that σ 2 A is the variance obtained from the full anisotropic power spectrum, where σ 2 m is the variance corresponding to the isotropic part. Since the scale-dependence of g ∗ is logarithmic through N ( k i ), as given in Eq. (9), as a first approximation we can ignore its scale-dependence so we have the last approximate equality in Eq. (60). Note that, as we mentioned before, the variance does not have any direction-dependence, which is somewhat an obvious observation, since one should integrate over the full 3D Fourier space to obtain the variance, eliminating any direction present in power spectrum. However, it is interesting to note that depending on the sign of anisotropy parameter g ∗ the correction to the variance due to anisotropy enhances or suppresses the leading order term. From the above variance, we can obtain the leading order scale-independent bias b 1 L . As for the next step, we obtain the bias from the anisotropic bispectrum which is both scale-dependent and directiondependent. This would be the main effect of anisotropy in the bias parameter. In order to obtain b 2 L , we need I 21 where the information from primordial bispectrum is encoded. We use the squeezed limit ( k 3 /lessmuch k 2 /similarequal k 1 ) bispectrum predicted by the model, Eq.(13). Because of the symmetry in x -y plane we rotate the long wavelength mode such that ψ ˆ k 3 = 0. Inserting the bispectrum to I 21 , Eq.(32) results in where k and q correspond to long and short wavelength ( k 3 and k 2 in Eq. (13)), and θ , ψ are polar and azimuthal angles in spherical coordinates, defined by the angle between the anisotropy direction ˆ n and wavenumbers ˆ q and ˆ k respectively Now we can integrate the angular dependence γ q which yields It is interesting to note that the orientation-dependence appears in the form of sin 2 θ k . As a result, the bias vanishes when sin θ k = 0, i.e. when the long wavelength mode is aligned with the anisotropic direction. This result originates from the fact that in this specific direction the bispectrum vanishes in squeezed limit as one can check from Eq. (13). Furthermore, since in the model under consideration g ∗ has a mild scale-dependence via logarithmic correction in N ( k ), we observe that there is an extra but mild k-dependent factor in I 21 in comparison with the standard local non-Gaussian shape [61, 90]. Besides that, I 21 linearly depends on I which is the free parameter of the anisotropic inflationary model. Now, by using the variance and I 21 parameter obtained above we can find the bias parameter in the anisotropic model by where, for the first order linear bias in the case of anisotropy, we have in which the subscript ( A ) has been added to point out that the parameters are obtained in the presence of anisotropy. Here R ( A ) NG is the anisotropic non-Gaussian correction to the PDF of density fluctuations defined by where x A ≡ δ c /σ A and δ c = 1 . 68 is the critical density and s ( A )3 is the reduced skewness defined as Note that, in the above formula, we have to use the full variance σ A including the correction due to anisotropy. The anisotropic skewness is also defined as where B A is the anisotropic bispectrum. The next order term in bias will give the scale-dependent and direction-dependent effect So the total Eulerian anisotropic bias is defined as The first three terms above gives the scale-independent bias, whereas the last term is scale-dependent and directiondependent bias due to the anisotropic bispectrum. Since the model we consider has a bispectrum shape very close to the standard local non-Gaussian shape and since b 1 L ( NG ) is small in comparison with b sd in local shape, [90], we can ignore b 1 L ( NG ) term in our analysis. In Fig. 1 we plot the relative magnitude of the scale-dependent bias Eq. (70) to Gaussian bias, b ( A ) ( sd ) /b G , versus the wavenumber. As expected we have approximately k -2 scale-dependence similar to the local non-Gaussian shape. However, this k -dependence is slightly different because of mild dependence of g ∗ to wavenumber. A completely new feature is the direction-dependence of bias originated form primordial anisotropy, proportional to sin 2 θ k 3 . In the case of θ k 3 = π/ 2 we have the maximum scale-dependent bias, whereas at angles θ k 3 = 0 , π , as the anisotropic bispectrum vanishes, the scale-dependent bias also vanishes accordingly. In order to compare our results with the conventional local non-Gaussian models, we set θ k 3 = 33 · to have f eff NL /similarequal 100 and then compare the bias in this specific direction with the bias in local shape with amplitude f NL = 100. In Fig 2 and Fig. 3, for both long and short wavelengths, we compare the fraction of scale-dependent bias, b ( sd ) /b G , from local non-Gaussianity compared to our anisotropic model. In Fig. 4 we compare their relative magnitudes. Since g 1 / 2 ∗ ∝ N ( k ) = N CMB +ln( k/k CMB ) the bias of our model is slightly higher than local non-Gaussian case for long wavelength modes while it is slightly lower for short wavelength modes. The free parameter of our model is I . In Fig. 5 we plot b ( A ) ( sd ) /b G versus wave number for different values of I . The I -dependence of bias is linear according to Eq. (64). In Fig. 6 we plot the redshift-dependence of b ( A ) ( sd ) /b G . As can be seen, at higher redshifts we have higher bias. This is the standard expectation, since at earlier times the non-linearity of local, small scale perturbations was weaker and thus they are more sensitive to large scale perturbations, resulting in larger bias.", "pages": [ 10, 11, 12, 13, 14 ] }, { "title": "VI. CONCLUSION AND DISCUSSIONS", "content": "The large scale structure observations, like the statistics of rare objects and the scale-dependence of bias parameter, can be used as complementary cosmological observations to CMB data to constrain the inflationary models. In this work we obtained the scale-dependent and the direction-dependent bias of dark matter halos in anisotropic inflationary models. The anisotropic model studied in this work has a bispectrum shape very close to the local non-Gaussian shape with an extra mild k -dependence and also a direction-dependence. We showed that the bias parameter is mainly influenced by the angle ( θ k 3 ) between the anisotropy direction and the long wavelength mode /vector k 3 in the squeezed limit by a factor of sin 2 θ k 3 . An interesting observation is that the scale-dependent bias vanishes at linear order along the direction θ k 3 = 0 , π . This can be explained by the fact that the bispectrum vanishes in squeezed limit when the long wavelength mode is aligned with the anisotropy direction. This means that, at the level of bispectrum, the model reduces to the Gaussian model along this direction. As it is clear from the PBS formalism, the bias parameter is mainly affected by the long wavelength mode. We can see this explicitly in our work where the angle between the anisotropy direction and the long mode appears in the bias formula. The free parameter of our model is I . Constraints from the CMB and LSS observations require I /lessorsimilar 10 -5 [55-58]. Roughly speaking, our bias parameter analysis also imply the same order of magnitude for I . Considering the fact that the observations are now in good agreement with a Gaussian bias we expect that in quasi-linear regime (where we have the strict constraint on bias) the ratio b ( A ) sd /b G is at the order of one so from our results we also find I /lessorsimilar 10 -5 . In this work we also formulated the general, model-independent anisotropic scale-dependent bias at linear order. For this purpose, by assuming the rotational symmetry in x -y plane of Fourier space, we modeled the bispectrum as a function of spherical harmonics, Y lm , and the Legendre functions, P L , of the angles between the short/long wavelength modes and the anisotropy direction. Interestingly, we find a selection rule for scale-dependent anisotropic bias, which shows that the bias parameter only responses to the Y 00 part of the anisotropic bispectrum. We also show that b sd is independent of short wavelength mode and its direction. This is an obvious check for the formalism, since the bias should be only a function of the long wavelength mode. In this work we assumed that the mechanism of spherical collapse and the conventional form of the transfer function are still applicable. This is motivated from the fact that the collapse of structures is a local mechanism which is less affected by the large scale anisotropies. Having this said in principle it is an interesting question to see how one can generalize the spherical collapse mechanism in the presence of primordial anisotropic fluctuations. On the other hand, the modification of the transfer function due to Boltzmann and perturbed Einstein equations in the presence of an anisotropic cosmic fluid is an interesting question which is beyond the scope of this work. A very interesting question to ask is whether the ideas presented in this work can be used observationally to find the fingerprints of NG and anisotropies in LSS data. The future LSS surveys such as the Large Synoptic Survey Telescope (LSST) is designed to obtain photometric redshift for almost 4 billion galaxies. The galaxies are distributed in redshift space with the distribution peaking around z = 1. This survey enables us to determine the galaxy bias with high accuracy. The galaxy cluster count with combination of other cosmological observations, such as the weak gravitational lensing and CMB data, can measure the full sky bias parameter in the redshift range between 0 to 1, with a precision as good as 2% accuracy [97]. However, the challenge from observational side is that if we want to determine the amplitude of the direction-dependent bias we need almost a full sky survey with enough statistics in each patch. This is the most important obstacle in determining the bias via galaxy cluster counting. There are other promising observations, such as the 21cm hydrogen intensity power-spectrum, which can be used to detect the fingerprints of NG and anisotropy. It is easier to have a full map data on the hydrogen intensity map although the astrophysical uncertainties will interfere in determining the bias parameter [98-101]. The statistical analysis and future observational forecasts are potential extensions of this work to quantify the chance of the detection. The other point to mention is that the strength of the anisotropy signal depends on f NL , the anisotropy parameter I and the angle θ k . Now we have a strong constrain on the local NG from PLANCK data [14] f NL = 2 . 7 ± 5 . 8 making the signal small. However there are two crucial points here. First, there is a degeneracy between the parameters f NL , I and θ k and second, the local NG measured by PLANCK collaboration is on large scales (CMB scale) and there is a room for running of f NL towards the smaller scales (sub CMB/LSS scales) making f NL large enough for our analysis.", "pages": [ 14 ] }, { "title": "Acknowledgments", "content": "We would like to thank Razieh Emami for many insightful discussions on anisotropic inflation. We also thank the anonymous referee for the careful and insightful comments on the draft which were very helpful to improve the presentations.", "pages": [ 15 ] }, { "title": "Appendix A: Halos bias in the Peak Background Splitting in the context of Excursion Set Theory", "content": "In this appendix first we review the Excursion Set theory (EST) then we apply the Peak Background Splitting (PBS) in EST context and finally we argue that how we can find the halo bias. In the context of EST halo formation can be described as the random walk of matter density contrast as the smoothing radius goes from very large radius, corresponding to infinitesimal variance σ 2 and small δ , to the scales crossing the linear threshold for collapse δ c at some finite smoothing radius. This radius is related to the scale in which the halos form. Within the EST formulation the number density of collapsed objects (dark matter halos of mass m ) per unit mass is given by where σ 2 m is the variance of the small scale density field smoothed with filter (window function) at spatial scale R and ¯ ρ is the background energy density, relating the halo mass to smoothing radius by m = 4 π ¯ ρR 3 / 3. Furthermore, Π 0 ( δ s , σ 2 m , δ c ) is the unconditional probability distribution function (PDF of density fluctuations) of small scale perturbations reaching δ s (short wavelength density contrast) at variance σ 2 . By the subscript 0 as well as the unconditional assumption for PDF of density fluctuations, we mean that the initial condition (first step in random walk) is δ s = 0 where σ 2 m = 0, and it satisfies the absorbing barrier condition Π 0 ( δ c , σ 2 m , δ c ) = 0. The smoothing procedure will be done by top-hat window function in Fourier space [91] where x = kR , k is the wavenumber and R is Lagrangian radius of collapsed objects, related to the mass via in which m is the mass of the structure and Ω m = Ω 0 m (1 + z ) 3 . Since the variance σ m is a monotonic function of mass scale due to matter power spectrum we use σ ( M ) as the 1-D variable in random walk. So it will be relevant to define a quantity that shows the probability of first up-crossing in the time σ 2 m and σ 2 m + dσ 2 m as Now the number density of collapsed objects will be It is worth to mention that the number density of dark matter halos obeys the normalization condition The key parameter here is the matter density variance which is related to the linear matter power spectrum as where the density contrast power spectrum is [91] Here n s is the spectral index, A is the linear matter primordial power spectrum amplitude in k = 0 . 002 h -1 Mpc , T ( k ) is the transfer function and D ( z ) is the growth factor normalized to scale factor at early times. The evolution of density contrast is imprinted in growth function D ( z ) and the transfer function T ( k ), showing the scale-dependence of gravitational potential during cosmic evolution. In this work we use the growth function of standard Λ CDM cosmology and the transfer function of Bardeen, Bond, Kaiser and Szalay (BBKS) [73] respectively as below and where Ω m = Ω 0 m a -3 / (Ω 0 m a -3 +Ω Λ ) and Ω Λ = Ω 0 Λ / (Ω 0 m a -3 +Ω Λ ). We define the growth function, in a way that is normalized to scale factor in deep matter dominated era. An important point to note is that the variance is linearly dependent to growth function due to Eq. (A8) where σ 2 ( M,z ) = σ 2 ( M,z = 0)[ D ( z ) /D ( z = 0)] 2 , where σ 2 ( M,z = 0) is the present value of variance. This redshift-dependance is important in the sense that the statistics of structure and even the bias parameter will be redshift-dependent. In Fig. (6) we showed the redshift-dependence of bias parameter for our specific anisotropic inflation model. In the matter dominated era D ( z ) scales like 1 / (1 + z ) which is a decreasing function with respect to redshift. Now that we set the matter density variance and large scale statistics of matter distribution, by setting the initial condition of perturbations we can find the mass function of structures in the Universe. For the Gaussian initial condition the probability distribution function (PDF of density fluctuations) in Fourier-space top-hat filter is [64] where P G is the Gaussian PDF of density fluctuations. The assumption of the universality of mass-function (like Press-Schechter) yields which leads us to write σ 2 m F 0 just as a function of threshold quantity ν = δ c /σ m where f ( ν ) is the usual Gaussian factor in Press-Schechter theory [20]. Now in order to find the halo bias in Gaussian and non-Gaussian inflationary models, first we discuss the PBS. We will show how this method is applicable for non-Gaussian fields. For inflationary models with primordial non-Gaussianity, the gravitational potential is usually written as [92 ? , 93] where φ is a Gaussian random field. The above relation is the simplest extension of Bardeen potential to include nonlinearity, called local non-Gaussianity. We can generalize the local type non-Gaussianity to arbitrary shape, simply by replacing the non-linear term by a kernel [61] In order to obtain the bias parameter, we can use the PBS idea on the Gaussian potential φ = φ s + φ l , where φ l is the long-wavelength mode of the potential and φ s is the short-wavelength mode. Since the large scale perturbations change the background for small scale modes, the PDF of density fluctuations for small scale perturbations is modified to the conditional PDF of density fluctuations, Π( δ s , σ 2 m , δ c ; δ l , σ 2 l ). In conditional PDF of density fluctuations, the initial condition is σ m → σ l in large scale limit (first step in random walk), in contrast with the unconditional case in which the variance vanishes on large scales. One of the important assumptions in PBS method is the Markovianity where we can write where Pi is the PDF of density perturbations. The Markovianity condition let us to change the threshold of critical density and density fluctuation of the structure by δ l . In this work we are in he regime where we can neglect the non-Markovianity induced from the primordial NG (for an extended discussion refer to Scoccimarro et al. [61]). On the other hand, the existence of any type of NG modifies the PDF of density fluctuations in a way that it is no longer independent of higher cumulants, while in Gaussian case the PDF of density fluctuations is completely determined by zeroth order ( δ s ) and first order ( σ 2 m ) cumulants. By the splitting procedure explained above, and knowing that the PDF of density fluctuations is now a function of all cumulants, we can expand the Lagrangian halo over-density as an expansion over large-scale φ l modes [61], where ( ... ) 0 means that the corresponding quantity is evaluated at φ l = 0. Note that because the background is now modified by large scale perturbation φ l , the cumulants are now functions of φ l . The first derivative in Eq. (A17), to all orders in primordial NG, is where the cumulants are defined by in which the subscript m for cumulants show that they are evaluated in the absence of the background φ l (independent of φ l ). It is worth to indicate that to first order in f NL , only the first two terms in the Taylor expansion above ( p = 1 , 2) contributes to the bias, while p = 3 contributes to O ( f 2 NL ) and O ( g NL ). Now we can obtain the bias parameter, using the above formulation. The p = 1 contribution is the usual scaleindependent bias presented in initially Gaussian case. Keeping in mind b ≡ δ h /δ l , we have which can be simplifies as The p = 2 contribution is the scale-dependent correction to the leading order bias coming from primordial NG The quantity I 21 includes the information about the primordial NG which is the derivative of second cumulant σ 2 m , (p=2), with respect to long wavelength mode φ l which is obtained to be [61]: where B ˆ δ ˆ δφ is the cross bispectrum of small-scale smoothed density ˆ δ and φ . So far only the Lagrangian bias appeared in our analysis because the peaks are those of the initial density field (linearly extrapolated). By the standard assumptions that halos move coherently with the underlying dark matter, and using the techniques outlined in [83-86], one can obtain the final Eulerian bias as Note that, due to the existence of primordial NG, the leading order scale-independent bias also modifies as where in Eq. (A25) we have omitted the subscript of F , which means that the NG will change the mass function, resulting in a modification of b 1 L . As a result we have and as described in Section (III). where x ≡ δ c /σ M , δ c = 1 . 68 is the critical density and s 3 is the reduced skewness defined as The skewness is related to the matter bispectrum as where /vectorq 12 = -( /vector q 1 + /vector q 2 ) and W ( kR ) is the window function in Fourier space, smoothing perturbations up to scale R . In obtaining the mass function of non-Gaussianity model in this approximation, we have assumed that all the deviation is imprinted in the skewness which may not be entirely true. In order to improve the results, numerical simulations are done [25, 75, 94]. Consequently, a scaling parameter κ defined by R NG ( x ) → R NG ( κx ) are introduced where, in the work of [33], from simulation of [95], it is obtained to be κ = 0 . 91. (For a similar correction from simulation see [96]). On the other hand, the scale-dependent bias can be rewritten as So the total Eulerian bias up to first order in f NL can be split to scale-independent b si and scale-dependent b sd terms as where b si and b sd are with b G ≡ 1 + b 1 L ( G ) and [45] N. E. Groeneboom, L. Ackerman, I. K. Wehus and H. K. Eriksen, 'Bayesian analysis of an anisotropic universe model: systematics and polarization,' Astrophys. J. 722 , 452 (2010) [arXiv:0911.0150 [astro-ph.CO]]. [55] R. Emami and H. Firouzjahi, 'Curvature Perturbations in Anisotropic Inflation with Symmetry Breaking,' arXiv:1301.1219 [hep-th]. [80] M. Musso and A. Paranjape, 'Non-Gaussian halo abundances in the excursion set approach with correlated steps,' Mon. Not. Roy. Astron. Soc. 420 , 369 (2012) [arXiv:1108.0565 [astro-ph.CO]]. [astro-ph.CO].", "pages": [ 15, 16, 17, 18, 20, 21, 22 ] } ]

[ { "title": "Fluid Inflation", "content": "Xingang Chen 1 , 2 , Hassan Firouzjahi 3 , Mohammad Hossein Namjoo 4 , and Misao Sasaki 5 1 Centre for Theoretical Cosmology, DAMTP, University of Cambridge, Cambridge CB3 0WA, UK 2 Department of Physics, The University of Texas at Dallas, Richardson, TX 75083, USA 3 School of Astronomy, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran 4 School of Physics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran and 5 Yukawa Institute for theoretical Physics, Kyoto University, Kyoto 606-8502, Japan (Dated: November 1, 2018) In this work we present an inflationary mechanism based on fluid dynamics. Starting with the action for a single barotropic perfect fluid, we outline the procedure to calculate the power spectrum and the bispectrum of the curvature perturbation. It is shown that a perfect barotropic fluid naturally gives rise to a non-attractor inflationary universe in which the curvature perturbation is not frozen on super-horizon scales. We show that a scale-invariant power spectrum can be obtained with the local non-Gaussianity parameter f NL = 5 / 2. PACS numbers:", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Cosmic inflation has emerged as a very successful paradigm for the early universe and structure formations. The basic predictions of simple models of inflation for the curvature perturbation power spectrum and bispectrum are in very good agreement with recent cosmological observations such as WMAP [1] and PLANCK [2, 3]. Without fully addressing the UV completion aspects of inflation, at the low energy effective field theory level, one can explore a variety of possibilities in the inflationary model building. In fact, many models of inflation based on scalar fields are constructed purely phenomenologically. Furthermore, one may add various features to such models, for example, by introducing extra phenomena such as particle creation and field annihilation, or local departures from inflation such as steps in the potential, turning trajectories and waterfall mechanisms. These additions have been used to explain the local features or glitches seen in CMB observations [4-40]. In this work we consider a different type of low energy effective field theory model for inflation. Namely, we present a formalism to obtain inflation from a fluid. Our starting point is the Lagrangian formalism for a perfect fluid in Einstein gravity, which enables us to calculate the power spectrum and bispectrum of the curvature perturbation. Depending on the equation of state and whether it is an isentropic (barotropic) or non-isentropic fluid, different inflationary scenarios are possible. As a first step, we concentrate on an isentropic fluid in which the pressure is a given function of the energy density. In principle, one should be able to extend this formalism to a non-isentropic fluid. This paper is organized as follows. In Section II we present a Lagrangian formalism for fluid inflation and the background equations. In Section III we present the cosmological perturbation theory in our setup and calculate the power spectrum and bispectrum of the curvature perturbation. In Section IV we present a simple scalar field model that mimics our fluid model. We then coclude the paper with a short discussion.", "pages": [ 1 ] }, { "title": "II. THE FORMALISM", "content": "To calculate the power spectrum and the bispectrum we need to have a Lagrangian formalism of fluid dynamics coupled with Einstein gravity. Here we use the Lagrangian for the perfect fluid in the presence of gravity proposed by Ray [41, 42] where ρ is the rest mass density, e ( ρ ) is the specific internal energy, U µ is the 4-velocity and λ 1 and λ 2 are Lagrange multipliers for the two constraints; the first is the normalization of the 4-velocity and the second is the conservation of the rest mass density. Note that the total energy density, E , is given by Below we show that the above Lagrangian gives the correct equations of motion for an isentropic perfect fluid minimally coupled to gravity. In this work we concentrate on an isentropic fluid for which e = e ( ρ ). In principle one can consider more general situations in which e is als a function of other thermodynamic variables such as entropy. Varying the action with respect to the Lagrange multipliers λ 1 and λ 2 yields the normalization condition for U µ and the energy conservation equations, respectively, and Varying the action with respect to ρ and U µ yields, respectively, where the constraint Eq. (3) have been used to obtain the latter equation. Finally, varying the action with respect to g µν yields the Einstein equation, where G µν is the Einstein tensor and the energy momentum tensor T µν is given by In addition to the above Euler-Lagrange equations, using the second law of thermodynamics, one has where s is the entropy density and P is the pressure. For an isentropic fluid we have ds = 0, hence Knowing that e = e ( ρ ), the above equation also implies that P is a function of ρ . Alternatively, in terms of the energy density E , using Eq. (2) we obtain Equations (10) and (11) imply that P is a function of E , P = P ( E ), which is expected for an isentropic or barotropic fluid. Plugging Eq. (11) into the definition of T µν yields Thus we recover the standard form for the energy momentum tensor of a perfect fluid.", "pages": [ 1, 2 ] }, { "title": "A. The background equations", "content": "Here we provide the background equations. As for the background, we assume a flat FLRW universe, Noting that at the background level U µ = (1 , 0 , 0 , 0), from Eqs. (5) and (6) one obtains the equations for the Lagrange multipliers as Furthermore, the rest mass conservation equation (4) yields where H ≡ ˙ a/a is the Hubble expansion rate. The background Einstein equations are Combining the above Einstein equations, one can easily recover the energy conservation equation in an expanding background, Now we consider the inflationary background. First, let us look at the slow-roll parameter /epsilon1 ≡ -˙ H/H 2 . Using the background Friedmann equation (16) and the energy conservation equation (18), one has The second slow-roll parameter η is given by Here the speed of sound c s for our isentropic fluid is given by For an infinitesimal perturbation, this implies Note that the definition (21) is relevant since we consider an isentropic fluid. It is important to note that for stable perturbations with c 2 s > 0, the magnitute of the η parameter is never smaller than unity as clear from Eq. (20). Indeed, taking /epsilon1 /lessmuch 1 to sustain a long enough period of inflation, one obtains η /similarequal -3(1 + c 2 s ). As we shall see below, to have an almost scale-invariant power spectrum, we must require c s /similarequal 1. So we conclude η /similarequal -6. This signals that our fluid inflationary system is within the domain of 'ultra slow-roll inflation' scenarios [43-47]. For a nearly constant η , one obtains where /epsilon1 i is the value of /epsilon1 at an initial/reference time t = t i . The fact that η /similarequal -6 as explained above implies that /epsilon1 decays during the ultra slow-roll inflation like a -6 . It is also instructive to look at the equation of state parameter w ≡ P/E . Using the relation ˙ P = c 2 s ˙ E and the background Friedmann and the energy conservation equations, one can easily check that We are interested in model in which the fluid has a (nearly) constant sound speed. With a constant c s , the above equation can be integrated, yielding where w i is the initial value of w . As inflation proceeds, F rapidly decays and one has This means that w approaches -1 exponentially rapidly. As mentioned before, this means we are within the domain of ultra slow-roll inflation. Finally, with the assumption that w /similarequal -1 and /epsilon1 is rapidly decaying, the background can be approximated by a pure de Sitter solution to a high accuracy, Here τ is the conformal time, dτ = dt/a ( t ), H = a ' /a is the conformal Hubble parameter, and the subscript e denotes the value of a quantity at the end of ultra slow-roll inflation.", "pages": [ 2, 3, 4 ] }, { "title": "III. THE PERTURBATION", "content": "Now we consider the perturbation in our fluid coupled to gravity. For relevant studies in different context see [48, 49]. For this purpose, we employ the ADM formalism in which the metric components are expressed as Plugging the above into the action yields where L G is the gravitational part of the Langrangian, K ij is the extrinsic curvature of the t = constant hypersurface, in which (3) ∇ represents the covariant derivative with respect to the three-dimensional metric h ij and K is the trace of K ij . The matter Lagrangian L m is given by Note that, by integration by parts, the above Lagrangian density is equivalent to Lagrangian density ˆ L m The lapse function N and the shift vector N i are Lagrange multipliers. Varying the action with respect to them gives the Hamiltonian and momentum constraint equations,", "pages": [ 4 ] }, { "title": "A. Linear perturbation", "content": "Now we consider linear perturbations. To proceed further, we have to choose a gauge. Since our system is based on the fluid dynamics, it is convenient to choose the comoving gauge in which 1 Here u represents the velocity scalar potential to all order in perturbations and R denotes the curvature perturbations in the comoving gauge. As usual we decompose the lapse and the shift functions into its scalar degrees of freedom, Similarly, we perturb the Lagrange multipliers λ i and the density field ρ as In the above decompositions, we have focused on the scalar perturbations and neglected the tensor and vector perturbations. From now on we omit the superscript 0 from the background quantities. Now we obtain the perturbed field equations. Perturbing the normalization condition (3) and the rest mass conservation equation (4) yields Perturbing the expressions for the Lagrange multipliers λ i in Eqs. (5) and (6) yields Furthermore, perturbing the constraint equations (33) and (34) results in Alternatively, one can perturb the Einstein equations. In particular, the (0 i )-component of the Eistein equations gives Comparing this equation with (42) yields δλ 2 = 0. The other components of the Einstein equations are not necessary thanks to the contracted Bianchi identities, or the energy momentum conservation law T µ ν ; µ = 0. From the momentum conservation equation, T µ i ; µ = 0, one has Again this is consistent with the constraint (41) if δλ 2 = 0. Perturbing the energy conservation equation, T µ 0 ; µ = 0, gives where we have and Eliminating the lagrange multipliers and the other fields in favor of R , we obtain the Lagrangian for R as where /epsilon1 ≡ -˙ H/H 2 is the slow-roll parameter as defined before. Note, however, that the action (54) is obtained without any slow-roll assumptions. Also one can check that the above quadratic action results in the same linear equation for R as given in Eq. (47). Now let us quantize the system. Changing the time variable t to the conformal time τ , the quadratic action (54) becomes This equation can be obtained using the constraint equations as well as the relation between ρ , E and P , mentioned before. By setting δλ 2 = 0 in the constraint equations and solving for all the variables but R , one obtains an equation of motion for R which represents the unique propagating degree of freedom, where we recall that the sound speed c s is defined in Eq. (21), and it appears in the perturbed relations (22), namely,", "pages": [ 5, 6 ] }, { "title": "B. Power spectrum", "content": "To calculate the power spectrum we need to expand the action to second order. Let us firt recapitulate the action given by Eq. (29), where L G = N √ hL G . Accordingly the second order action is given in the form, where the prime denotes a derivative with respect to the conformal time and The momentum conjugate to the field R is They satisfy the canonical commutation relation, The quantized field can be expressed in the Fock representation, where R k is a positive frequency mode function that satisfies the equation of motion, and the normalization condition, The annihilation and creation operators, a k and a † k , satisfy Assuming that R k should approach a conventional positive frequencty function at high frequencies, R k ∝ e -ic s kτ for τ →-∞ , the solution is uniquely determined as where H (1) ν is the Hankel function of the first kind, and Here again the subscript i denotes an initial/reference time τ = τ i . One might suspect that the absolute value of C k would depend on the choice of the initial time τ i . However, for a nearly constant η , one can show that it is independent of τ i because one has /epsilon1 a 2 ∝ a η +2 and x 1 -2 ν ∝ a 2 ν -1 = a η +2 . One of the important properties of our model is that the curvature perturbation is not conserved after horizon crossing. Expanding the Hankel function at x /lessmuch 1 gives As a result, the final curvature perturbation at the end of ultra slow-roll inflation τ = τ e is given by The power spectrum of curvature perturbation at the end of ultra slow-roll inflation is given by By using Eq. (65) the above reduces to which, using the approximation η /similarequal -3(1 + c 2 s ), further reduces to The spectral index is easily read off as Interestingly, the sound speed explicilty appears in the spectral index in this model, in contrast to the standard inflationary scenarios in which only ˙ c s plays a role in the spectral index. In order to have a scale-invariant perturbations we require c s = 1. The amplitude of the spectrum in this case is given by A red tilted power spectrum can be achieved by a slightly superluminal sound speed. With c s = 1, from Eq. (20) we obtain η /similarequal -6 and from Eq. (64) ν /similarequal -3 / 2. This yields /epsilon1 ∝ a -6 as mentioned before. Of course, recent cosmological observations by WMAP and PLANCK strongly favor a red-tilted power spectrum [3]. We see that in our model a subluminal sound speed implies n s > 1. This is a direct consequence of the starting assumption of our considering an isentropic fluid. To obtain a red spectral index for a subluminal sound speed, perhaps one should consider a more general, non-isentropic fluid. As for the tensor to scalar ratio, since the tensor spectrum is exactly the same as the standard case, one finds Since /epsilon1 decreases exponentially during ultra slow-roll inflation, we conclude that the amplitude of the tensor perturbation is exponentially suppressed in this model. The above simple model is not complete by itself, since there is no mechanism to terminate inflation. In principle, one can match the non-attractor phase of inflation to an attractor phase of conventional slow-roll inflation or of a hot Friedmann stage at which /epsilon1 is not decaying exponentially. At such a second stage, R becomes frozen on super-horizon scales as usual. This implies that one can read off the final value of R by computing its value at τ = τ e when the transition from the non-attractor phase to an attractor phase starts. This picture was employed in the context of a single scalar field theory in [44]. The second phase of inflation is necessary also because the non-attractor inflationary phase we considered here cannot last long enough to solve the horizon problem. Because the slow-roll parameter is decreasing exponentially with time, to get P R ∼ 6 × 10 -9 , we need a low-scale H [46]. For example, if we assume the lower-bound reheating energy to be ∼ 1 GeV, we have /epsilon1 min ∼ 10 -66 . This means that the upper bound of the inflationary efold for this non-attractor phase is 25.", "pages": [ 6, 7, 8 ] }, { "title": "C. Cubic Action and non-Gaussianity", "content": "Here we expand the action to third order which will be suitable to calculate the bispectrum. Starting with the action given in Eq. (49), one has where L (3) G represents the cubic order gravitational Lagrangian density, while L ( i ) m stands for the i -th order matter Lagrangian. With the expansion, one can check that where we have introduced (Not to be confused with the entropy density.) Using the constraint equations to remove non-dynamical variables, the full cubic action from the matter sector is where As demonstrated in Appendix A, similar to [52] and [50], one can check that the above cubic matter Lagrangian is equivalent to that for the theory of a scalar field with the action L m = P ( X ) where X = -g µν X µ X ν / 2, ˜ Σ = XP X +2 X 2 P XX and ˜ λ = X 2 P XX + 2 3 X 3 P XXX . Since the gravitational part of the action is the same by construction, this conclusion enables us to cast the cubic action for our model to the well-studied cubic action for a general P ( X,φ ) theory for k-inflation [51, 52] or DBI inflation [53] with [54, 55] where the field χ is defined by and f ( R ) and δL/δ R| 1 , respectively, by and is obtained to be This value of f NL is consistent with the recent Planck constraints on primordial non-Gaussianity [56].", "pages": [ 8, 9, 10 ] }, { "title": "IV. A MODEL", "content": "Here we present a single field model which shows the behavior similar to what we pointed out in the previous sections. Consider a canonically normalized field, so c s = 1, with the potential, During the first stage, the system approaches rapidly towards a de Sitter universe since /epsilon1 ∝ a -6 . This model was originally studied in [43] as 'ultra slow-roll' (USR) and was further studied in [44] as a toy single field model which can produce non-negligible local non-Gaussianity. During this phase, the curvature perturbation is not frozen on super-horizon scales, exhibitin the non-attractor nature of the system. As studied in [44], the background dynamics during the non-attractor phase is Thus ˙ φ ∝ a -3 and hence The power spectrum and bispectrum were computed in [44], and the local-type non-Gaussianity with f NL = 5 / 2 was obtained. It is instructive to look at the bispectrum in the squeezed limit using the δN method. One has where N is the number of e -folds counted backward from the end of ultra slow-roll inflation at which φ = φ c (not to be confused with the lapse function). It is important to note that N is a function not only of φ but also of ˙ φ , in contrast to the conventional slow-roll inflation for which ˙ φ is not independent but a function of φ . Taking the variations of φ and ˙ φ yields So far our analysis of the cubic action was general and no assumption on the value of c s has been made. However, from our power spectrum analysis, Eq. (71), we see that to obtain a scale-invariant power spectrum we need c s = 1. Therefore, from now on we concentrate on the case c s = 1. In this limit, all the interaction terms in the cubic action becomes small except for the last term involving f ( R ). It is known that this last term can be eliminated by the field redefinition R → R n + f ( R n ). This means that the leading contribution to non-Gaussianity comes only from the field redefinition. As emphasized in [44] both of the first two terms in f ( R ) in Eq. (86) contribute to non-Gaussianity. This is in contrast to the usual attractor situation in which ˙ R vanishes on the super-horizon scales and only the first term in f ( R ) contributes to non-Gaussianity. Following the same steps as in [44], the amplitude of local type non-Gaussianity, f NL , defined in the squeezed limit, k 1 /lessmuch k 2 = k 3 , as On super-horizon scales, δφ follows the evolution of background φ , and one can check that δ ˙ φ /similarequal 0 on super-horizon scales. As a result This automatically yields f NL = 5 / 2 in agreement with the result obtained from the in-in formalism. As mentioned before, inflation never ends unless there is a mechanism to terminate the non-attractor phase. In the current example, we have introduced a non-trivial potential for φ > φ c . At the second phase, inflation proceeds as in the conventional slow-roll inflation and R freezes out on super-horizon scales. Therefore, the physical parameters such as f NL and n s can be read off by calculating these quantities at τ = τ c when the non-attractor phase is matched to the attractor phase. In summary, in this work we have presented a fluid description of inflation. To be specific, we have considered the action of a single barotropic perfect fluid with appropriate Lagrange multipliers. After eliminating the Lagrange multipliers and the other non-dynamical variables we have obtained the quadratic and cubic actions for R . We have shown that this barotropic fluid naturally gives rise to a non-attractor inflationary phase in which R is not frozen on super-horizon scales. An interesting prediction of this model is that the curvature perturbation power spectrum is scale-invariant with the value of local type non-Gaussianity given by f NL = 5 / 2. We have also shown that at the level of cosmological perturbation theory this fluid model is equivalent to a scalar field theory with the Lagrangian P ( X ). The natural question which arises is how one can extend this formalism to a non-barotropic fluid for which the pressure is not uniquely determined by the energy density. This may help to keep n s as a free parameter to obtain a slightly red-tilted power spectrum as suggested by the PLANCK data [3] without appealing to a superluminal fluid. However, this may also result in generating entropy perturbations which are under strong observational constraints by the PLANCK data [3]. We would like to come back to this issue elsewhere. Also in this work we have considered a model with constant c s . In principle one may relax this assumption and consider the case in which c s is time-dependent. As a result, this will add the new contribution ˙ c s /c s (in the limit where c s is changing slowly with time) into n s . It is an interesting question to see if this can help to obtain a red-tilted power spectrum.", "pages": [ 10, 11 ] }, { "title": "Acknowledgments", "content": "We would like to thank Nima Khosravi, Javad Taghizadeh Firouzjaee, Alberto Nicolis and Jonathan White for useful discussions. This work was supported in part by the JSPS Grant-in-Aid for Scientific Research (A) No. 21244033.", "pages": [ 11 ] }, { "title": "Appendix A: The action for P ( X,φ ) theory", "content": "In this appendix we prove the equivalence between the perturbation theory in our isentropic fluid and a scalar field theory with the matter action, similar to k-inflation models [51, 52]. This equivalence will be used to map the bispectrum in our model to that of a well-studied P ( X,φ ) theory, such as in [55]. Our aim here is to expand the matter Lagrangian up to third order of perturbations. It is convenient to adopt the comoving gauge in which δφ = 0 and Noting that α = ˙ R /H , up to third order in comoving gauge we have Gathering all cubic order terms we obtain 2 2 where E = 2 XP X -P is the total energy density that appears in the Friedmann equation, 3 M P H = E . Comparison between Eq. (A6) and Eq. (81) demonstrates the equivalence between the above theory and the matter sector of our fluid theory with the identifications ˜ Σ ↔ Σ and ˜ λ ↔ λ . [5] S. M. Leach, M. Sasaki, D. Wands and A. R. Liddle, 'Enhancement of superhorizon scale inflationary curvature pertur- bations,' Phys. Rev. D 64 , 023512 (2001) [astro-ph/0101406]. [9] M. Joy, V. Sahni, A. A. Starobinsky, 'A New Universal Local Feature in the Inflationary Perturbation Spectrum,' Phys. Rev. D77 , 023514 (2008). [arXiv:0711.1585 [astro-ph]]. [13] F. Arroja, A. E. Romano and M. Sasaki, 'Large and strong scale dependent bispectrum in single field inflation from a sharp feature in the mass,' Phys. Rev. D 84 , 123503 (2011) [arXiv:1106.5384 [astro-ph.CO]]. [15] V. Miranda, W. Hu and P. Adshead, 'Warp Features in DBI Inflation,' Phys. Rev. D 86 , 063529 (2012) [arXiv:1207.2186 [astro-ph.CO]].", "pages": [ 11, 12 ] } ]

[ { "title": "Joyce Byun and Rachel Bean", "content": "Department of Astronomy, Cornell University, Ithaca, NY 14853, USA. E-mail: [email protected], [email protected] Abstract. A detection of primordial non-Gaussianity could transform our understanding of the fundamental theory of inflation. The precision promised by upcoming cosmic microwave background (CMB) and large-scale structure (LSS) surveys raises a natural question: if a detection given a particular template is made, what does this truly tell us about the underlying theory? Even in the case of non-detections and upper bounds on deviations from Gaussianity, what can we then infer about the viable theories that remain? In this paper we present a systematic way to constrain a wide range of non-Gaussian shapes, including general single and multi-field models and models with excited initial states. We present a separable, divergent basis able to recreate many shapes in the literature to high accuracy with between three and seven basis functions. The basis allows shapes to be grouped into broad 'template classes', satisfying theoretically-relevant priors on their divergence properties in the squeezed limit. We forecast how well a Planck-like CMB survey could not only detect a general nonGaussian signal but discern more about its shape, using existing templates and new ones we propose. This approach offers an opportunity to tie together minimal theoretical priors with observational constraints on the shape in general, and in the squeezed limit, to gain a deeper insight into what drove inflation.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "An early period of accelerated expansion, perhaps a trillionth of a second after the Big Bang, is proposed to solve a number of problems unresolved by the Big Bang scenario, such as the flatness and horizon problems. The paradigm of single-field slow-roll inflation is the simplest model to describe this acceleration, and makes broad predictions of adiabatic and Gaussiandistributed primordial density (scalar) perturbations, described by a nearly scale-invariant 2-point correlation, and smaller gravitational metric (tensor) perturbations. In this model, the scalar and tensor amplitudes and scale dependence are related through a 'consistency relationship'. The precision of astrophysical measurements has dramatically improved over the last decade. Cosmic Microwave Background (CMB) measurements, such as from the Wilkinson Microwave Anisotropy Probe [1, 2], small-scale CMB measurements from the Atacama Cosmology Telescope and South Pole Telescope [3, 4], and large-scale structure (LSS) observations such as from the Sloan Digital Sky Survey [5, 6], are all entirely consistent with single-field slow-roll predictions, placing strong constraints on the scalar power spectrum and upper limits on the degree of deviations from Gaussianity and amplitude of tensor modes. The agreement between single-field inflationary predictions and observations is a profound success for cosmology, but it is as yet, insufficient to inform us about the underlying theory from which inflation derives. Rapid theoretical progress in high energy effective field theory has led to a wide range of possible Lagrangians for inflation [7-10]. These often go beyond making distinct predictions for the form of the single-field inflationary potential and can include multiple dynamical fields and non-canonical derivative (kinetic) terms in the action. These alternative mechanisms can generate new observational signatures, including different consistency relationships relating scalar and tensor perturbations [11, 12], the addition of non-adiabatic (isocurvature) modes [13, 14], and the possibility of observationally measurable non-Gaussian correlations [15]. In particular, the possibility that primordial nonGaussianity may be detectable as a non-zero 3-point correlation function, or bispectrum, has been a major development in the search for observational signatures of the underlying inflationary theory. What is most exciting is that different theories can give rise to bispectra with distinct scale dependencies, such that measuring not only the amplitude but also the scaledependence, or 'shape', of the bispectrum could provide a direct insight into the inflationary mechanism. Much work has focused on potentially measuring the amplitude of commonly predicted shapes, such as the local [15-17], equilateral [18], and orthogonal [19] templates. Recent theoretical developments have also led to a wider population of bispectra, including those from fast-roll inflation [20-24], quasi-single field inflation [25, 26], warm inflation [27-29], and non-Bunch-Davies or excited initial states [20, 30-32]. There are also hybrids of multi-field and non-slow-roll models [33-35], and the inclusion of isocurvature modes in the non-Gaussian correlations [36-38]. These bispectra can have very different shapes, meaning their signal is weighted towards different configurations of the 3 wavenumbers in (Fourier) k -space. How divergent shapes are in the 'squeezed' k -configuration, when one of the three length scales contributing to the 3-point function becomes much larger than the other two, in particular can signal whether inflation is derived from a single-field or multi-field model. The divergence in the squeezed limit could also be constrained by its effect on large-scale structure. A non-Gaussian signal peaking in the squeezed limit would directly couple large scale modes to small scales, on which non-linear halos are forming [39]. This gives rise to an additional contribution to the halo bias, determining how the number density of halos of a given mass are related to the underlying linear power spectrum. In theory, wide field large-scale structure surveys could provide a sensitive constraint on the divergence properties of non-Gaussianity [40-46]. Given the diversity of theoretically motivated shapes, an intriguing question is how well one might actually be able to determine the shape of primordial non-Gaussianity, rather than purely assuming a shape template is the true shape a priori. To what extent can the shape of non-Gaussianity be reconstructed using the CMB and LSS 3-point correlations? If a positive detection is made assuming a template, how well would such a detection really constrain the underlying shape and the theoretical model that generated it? This 'reconstruction' approach has been widely considered in the context of the inflationary power spectrum, both in terms of P ( k ) reconstruction (e.g. [47-49]), and measuring the hierarchy of slow-roll parameters (e.g. [50-53] and references therein), instead of assuming a nearly scale-invariant spectrum parametrized by a constant tilt n s and a constant running dn s /d ln k . Unfortunately, calculating theoretical predictions for CMB bispectra is computationally cumbersome in its exact form, requiring 4-dimensional integrals to be performed. A formalism to make the calculation tractable for general bispectra was introduced in [54]. The authors proposed a technique to create templates for shapes by expanding non-separable shapes on a basis set of bispectra that are explicitly separable functions of the three wavenumbers. The separability reduces the 4-dimensional integral to a tractable computation without a significant reduction in the accuracy of the computed CMB bispectra. This approach has been used to forecast bispectrum constraints for a variety of fundamental shapes [2] and adapted to other basis sets to describe oscillatory, rather than monotonic, shapes [55]. Furthermore, the method of modal expansions on a separable basis has been shown to be advantageous and applicable in a variety of contexts, for example in studying CMB 3-point correlations with wavelets [56], CMB trispectra [57, 58], and matter density bispectra in LSS [59-61]. In this work we present an alternative separable basis to efficiently describe and investigate the broad class of nearly scale-invariant general bispectra in terms of their squeezed limit properties. We discuss a way to expand a general shape in the basis, which is specifically tuned to enable us to systematically increase the complexity of the template in a theoretically motivated way. We forecast the potential for determining the underlying non-Gaussian shape given upcoming CMB temperature and E-mode polarization data modeled on the Planck survey. The format of the paper is as follows. In section 2, we review the formalism used to calculate CMB bispectra. We introduce a separable basis to describe general shapes that are scale-invariant and potentially divergent, and discuss how this basis can be applied to describe a wide variety of shapes in the literature. Using the basis, we develop an expansion that allows us to incrementally investigate classes of bispectra motivated by theories. In section 3, we present a Fisher analysis quantifying how well a Planck-like survey will be able to distinguish between and constrain individual bispectrum shapes. Using a principal component analysis, we find the best to worst measured uncorrelated shapes, and compute the overall uncertainties in the bispectrum measurement as a function of k -space configuration under different theoretical priors. We use these results to establish how much we can learn about the bispectrum shape, and hence with what confidence we might be able to narrow down the underlying inflationary theory. In section 4, we summarize our findings and discuss implications for future work.", "pages": [ 2, 3, 4 ] }, { "title": "2 Efficient calculation of a general non-Gaussian shape", "content": "In this section we lay out the formalism to describe and compute general bispectra. In subsections 2.1 and 2.2, we respectively review the calculation of the CMB bispectrum given the primordial 3-point function and the definitions of covariances in wavenumber and multipole space that roughly quantify the theoretical similarity of two bispectra. In subsection 2.3 we introduce a separable basis set to describe general bispectra and develop computationally tractable templates. Subsection 2.4 discusses the application of the basis set to a variety of theoretical bispectra and templates in the literature. How bispectra can be classified and presented pictorially is reviewed in subsection 2.5.", "pages": [ 4 ] }, { "title": "2.1 The CMB bispectrum", "content": "While Gaussian fluctuations are wholly described by a 2-point correlation function, a full description of non-Gaussian fluctuations requires higher order correlations that are not trivially related to the 2-point function. The simplest higher order correlation is the 3-point function, where the 3-point Fourier space statistic analogous to the 2-point power spectrum is the bispectrum, B Φ , defined by Φ( k ) is the primordial gravitational potential, related to the curvature perturbation by Φ = 3 5 R . Under the assumptions of statistical isotropy and homogeneity, the bispectrum is dependent on only the magnitudes of the wavenumbers, k 1 , k 2 , and k 3 . The bispectrum is often parameterized by a shape, S ( k 1 , k 2 , k 3 ) , and an amplitude, f NL , at an arbitrary configuration in k -space which together determine the bispectrum at all scales, The typical convention is to choose N = 6[2 π 2 ( 3 5 ) 2 ∆ 2 R ( k 0 )] 2 , where ∆ 2 R ( k 0 ) is the amplitude of the primordial power spectrum of the curvature perturbations at a pivot scale k 0 . Shapes are typically normalized such that S ( k 0 , k 0 , k 0 ) = 1 . CMB statistics are commonly described by correlations between angular moments on the sky, a glyph[lscript]m , calculated through a spherical harmonic decomposition of the photon transfer functions, ∆ glyph[lscript] ( k ) , integrated along the line of sight and sourced by the primordial perturbations, The CMB 3-point correlation function is given by To perform the integrals over k and x , we use the CAMB 1 code [62], which uses the line of sight approximation [63] to calculate the photon transfer functions ∆ glyph[lscript] . We consider purely isotropic bispectra for which the integral over ∫ d Ω ˆ x is a separable geometrical factor called the Gaunt integral. The properties of the Gaunt integral require that the non-zero correlations have glyph[lscript] 1 , glyph[lscript] 2 , and glyph[lscript] 3 satisfying an even sum glyph[lscript] 1 + glyph[lscript] 2 + glyph[lscript] 3 and | glyph[lscript] 1 -glyph[lscript] 2 | ≤ glyph[lscript] 3 ≤ glyph[lscript] 1 + glyph[lscript] 2 for glyph[lscript] 1 , glyph[lscript] 2 ≤ glyph[lscript] 3 . Under the assumption of isotropy, the angle-averaged angular bispectrum is the 3-point analogue to the C glyph[lscript] , where the bracketed term is the Wigner-3j symbol. To further separate out a purely geometrical factor from the angular-averaged bispectrum, it is convenient to work with the reduced bispectrum b glyph[lscript] 1 glyph[lscript] 2 glyph[lscript] 3 , so that Here x is a dummy variable which should be integrated between zero and infinity. We note that in an analogous evaluation of C glyph[lscript] , x has a physical interpretation as the comoving distance to the surface of last scattering. One might assume therefore that in the 3-point integral the upper limit x max could be set to ( τ 0 -τ rec ) . However, as others have previously also commented [19], the integral over x arises out of rewriting the delta function in (2.1) as an integral over a product of Bessel functions. Numerically, for a general bispectrum we find the value of x max ensuring the required degree of convergence is glyph[lscript] -dependent, and typically needs to be greater than ( τ 0 -τ rec ) .", "pages": [ 4, 5, 6 ] }, { "title": "2.2 Shape similarity", "content": "The degree to which a bispectrum B is theoretically similar to another, B ' , can be quantified by a k -space correlation coefficient, or ' k -space cosine', corr k , integrated over the k -space tetrapyd volume V with weight w [54, 64], An analogous statistic describing the similarity of two bispectra in multipole space can be quantified by their glyph[lscript] -space correlation coefficient, or ' glyph[lscript] -space cosine', corr glyph[lscript] [54], These correlation statistics are frequently used within non-Gaussian shape studies to quantify how well a template matches a given shape. A template can be obtained, given a set of n basis shapes, in our case {K n } , by first applying the Gram-Schmidt algorithm to give a set of orthonormal basis functions {R n } (in either k or glyph[lscript] space), where the last line defines the matrix λ . The new basis can then be used to create a matched template for a specific non-separable shape S , We note that in general the classical Gram-Schmidt algorithm can be numerically unstable, resulting in {R n } that are not exactly orthogonal. This issue can be abated by implementing the well-known modified Gram-Schmidt algorithm, and any numerical issues that may remain can be checked by verifying that all of the {R n } are orthogonal to each other, i.e. 〈R i , R j 〉 = δ ij . For our case this was true to within a few × 10 -6 at worst, across the first 7 modes. In addition, a faster check can be conducted for those shapes for which the coefficients on the original basis are known, since computing αλ should return the input coefficients on the {K n } basis. We verified that this was the case for the equilateral, orthogonal, and enfolded templates, with the worst coefficients being off by a fractional error of a few × 10 -5 . We find this accuracy is more than sufficient for the forecasting analyses to constrain shape measurements with upcoming surveys as we discuss in section 3. The efficacy of the template can be quantified by the cumulative cosine, corr ( S, S ( n ) template ) as in (2.9) or (2.11). A high correlation coefficient signals a good fit. If this cosine is close to one then the two shapes are sufficiently alike that one might expect constraints on the amplitude of B can be taken as constraints on B ' as well, without having to do a separate analysis of the data. If on the other hand, the cosine is low, then it is likely that separate analyses of the data are needed for B and B ', because a template for B will not be able to pick out a non-zero signal for B ', and vice versa. We note though that while a template may have a large cosine with the shape, this does not automatically mean that the template will be able to accurately model a correlation between the true shape and a third shape, with which it is not similar. The extreme example of this would be if the third shape were exactly proportional to the discrepancy between the template and shape. In constructing a template, if this was a concern, one might want to tailor it to the purpose by altering the weight in the Gram-Schmidt decomposition to ensure a minimization of the covariance between the shape and template over a given region of ( k or glyph[lscript] ) configuration space in which a third shape was relevant. While it is extremely useful to establish the similarity of shapes, it is the converse of this, how well two shapes can be distinguished from one another, using data, that is the main focus of this work. This provides a motivation to consider an efficient way to generate glyph[lscript] -space bispectra explicitly by creating templates described by basis functions separable in k 1 , k 2 and k 3 as we discuss below. To do this, in sections 2.3 and 2.4 we develop a framework to describe the possible degrees of freedom that a general shape might have under a specific theoretical prior. We note that corr k and corr glyph[lscript] represent simplified correlation statistics that purely take into account the cosmic variance limitations. Neither statistic, as they are written above, takes into account the noise, sky coverage or resolution characteristics of a particular survey. As described in [54], corr glyph[lscript] can be modified to include these experimental effects by changing the weighted sum over glyph[lscript] 1 , glyph[lscript] 2 , glyph[lscript] 3 to reflect the measurement covariance matrix. The modified corr glyph[lscript] is then a refined, survey-dependent extension of (2.11) that tailors the correlation statistic to reflect the observational, rather than intrinsic, distinguishability of shapes. Distinguishability between several shapes can be done by conducting a Fisher or χ 2 analysis that includes the measurement covariance matrix [15]. In section 3, we perform a Fisher analysis and use the correlation statistics, including experimental effects such as instrument noise, beam size, and incomplete sky coverage, to quantify how well upcoming surveys might distinguish one shape from another.", "pages": [ 6, 7 ] }, { "title": "2.3 A new separable basis, K n", "content": "The 4-dimensional integral over the product of highly oscillatory functions given in (2.7) is computationally intensive. This has been a barrier to efficiently calculating observational predictions for the CMB bispectrum. As a result many studies have focused on models for which the primordial bispectrum can be written as, or well-approximated by, a separable (symmetric) function of ( k 1 , k 2 , k 3 ) , such that the 3-dimensional integral over k 1 , k 2 , and k 3 in (2.7) is reduced to a product of three 1-dimensional integrals. In [54] the authors proposed a way to reduce the computation time for general models by expanding the shape in terms of a separable basis, Each Q i is constructed from symmetrized products of three 1-dimensional polynomials of k 1 , k 2 , and k 3 , where n maps onto a combination of { p, r, s } ≥ 0 , and c pi , c rj , c sk are constants. Using the Q n basis, the equilateral template can be reconstructed to 98% accuracy (according to the cumulative cosine) using 6 basis functions [54], while other shapes motivated by single-field inflation models can require 20 or more mode functions to get > 95% convergence [65]. An analysis of data to constrain the bispectrum depends not only on the uncertainties inherent in the data itself, but also the theoretical priors determining the model being compared with the data. The choice of a separable basis set to describe the theory is therefore also influenced by this prior. An analysis allowing the primordial bispectrum to take any form (i.e. with no shape prior on the forms of the separable functions f i , g i , h i ) would use a discrete set of k -space bins to describe the uncorrelated amplitudes at each scale and configuration. In such a scenario, no theoretical prior is applied and the constraints on the bispectrum are simply those determined by the data. In studying theoretically motivated models of inflation, however, there can be broad or specific characteristics of the bispectra that it would be reasonable to impose in conjunction with the data that suggest a form for the separable basis functions. For example, a Fourier basis may be more efficient than a polynomial one for describing bispectra with oscillatory features [55]. The two minimal assumptions we consider here as theoretical priors are that the bispectrum i) has a roughly monotonically changing amplitude as a function of scale, and ii) like the power spectrum, it is nearly scale-invariant. The polynomial basis of [54] does not naturally confine shapes to these two common theoretical properties of bispectra. Firstly, the polynomial basis does not naturally restrict itself to scale-invariant shapes, because i + j + k ≥ 0 in (2.18); resulting sums of the basis functions are thus scale-dependent in general. Most theoretically-motivated bispectra in the literature, however, are nearly scale-invariant, with i + j + k ≈ 0 (see [66] for a review). Such shapes can be reduced to functions of two variables, k 1 /k 3 and k 2 /k 3 . There are exceptions to this, of course, such as non-Gaussianity from particle production [67] or from features in the inflationary potential [68, 69]. These models can strongly deviate from scale-invariance because modes leaving the horizon at a specific moment, when particle production is occurring or a feature in the potential is important, are preferentially populated. Secondly, different types of theoretical mechanisms generating bispectra predict different divergence properties in the squeezed limit, where k 1 glyph[lessmuch] k 2 ≈ k 3 . We consider the squeezed limit as k 1 = k glyph[lscript] , the long wavelength mode, and k 2 = k 3 = k s , the short wavelength modes, so that for scale-invariant shapes the squeezed limits purely dependent on x sq ≡ k glyph[lscript] /k s . Single-field inflation models, through a consistency relation [70] predict the bispectrum will vanish in this limit. Local bispectra, typically arising in multi-field models, have a x -1 sq divergence, while excited states can have x -2 sq divergence. Since the powers i , j , and k in (2.18) are ≥ 0 , the {Q n } all tend towards a constant value in the squeezed limit, and thus cannot effectively describe shapes diverging in the squeezed limit. As a result, this basis is not immediately suited to reconstructing templates that display specific divergence behaviors in this limit, without a further prior being imposed. For example, the compelling and well-studied local template cannot easily be recovered using the {Q n } basis without either using many more basis functions or ignoring the divergent part of the shape that makes the local template distinct from others [65]. In this paper we introduce a set of separable basis functions, {K n } , that efficiently describe nearly scale-invariant and potentially divergent shapes, and explicitly consider the forms of the shapes generated using this basis under various divergence constraints. Explicitly, we consider Here f n NL are expansion coefficients and n again denotes a combination of powers { p, r, s } of the wavenumbers ( k 1 , k 2 , k 3 ) . K n is defined as where N n is the number of distinct permutations of { p, r, s } . p ' is defined as and similarly for r ' and s ' . Equations (2.20)-(2.21) ensure that each K n is normalized in the conventional way, with K n ( k 1 , k 2 , k 3 ) = 1 at k 1 = k 2 = k 3 = k 0 . In the scale-invariant case where n s = 1 , K n only depends on k 1 /k 3 and k 2 /k 3 , and K n ( k 1 , k 2 , k 3 ) = 1 for all k 1 = k 2 = k 3 . To allow for potentially divergent shapes, we allow the powers { p, r, s } to be negative as well as positive, and to make each K n nearly scale-invariant, we require the powers to satisfy p + r + s = 0 . Each shape has a well-determined behavior in the squeezed limit, The set of {K n } with the allowed combinations of { p, r, s } is given in Table 1 along with their divergence properties. The basis modes are also written in the equivalent short-hand notation used by [71]. Since we will use this notation to describe non-separable shapes in the next section we summarize it here: glyph[negationslash] glyph[negationslash] glyph[negationslash] with", "pages": [ 7, 8, 9, 10 ] }, { "title": "2.4 Application of the basis to shapes arising in inflationary theory", "content": "In this subsection, we illustrate the efficiency and accuracy allowed by our basis in describing shapes in the literature. First we discuss cases involving shapes and templates which are exactly expressed in terms of, or well-approximated by, linear combinations of the first 3 modes of the basis. Then we extend the basis to include more divergent modes, and present the basis of shapes we will use under different divergence priors.", "pages": [ 10 ] }, { "title": "2.4.1 Shapes exactly expressed in terms of {K 0 -K 2 }", "content": "Some commonly considered templates can be exactly expressed in terms of the first three modes of the basis {K 0 -K 2 } . The local shape, S local = K 2 , can be derived from a simple ansatz for describing the nonlinear contribution to the primordial curvature perturbation in real space as a local effect [15], Local shapes arise out of single-field slow-roll models, though the amplitude of the bispectrum in this case is predicted to be undetectably small [70, 72]. Large, local non-Gaussianity is predicted by a wide variety of other physically-motivated models, such as multifield inflation (e.g. curvaton scenario) [73, 74], (p)reheating mechanisms [75], and ekpyrotic inflation [74, 76]. The constant shape, S const = 1 = K 0 , was originally studied for its very simple form [71]. More recently the shape has been studied in the context of shapes arising from quasi-single field inflation (QSFI) models [25, 26, 44, 77]. The more general shape of QSFI is discussed in more detail below. Models with higher-derivative kinetic terms and/or non-trivial speeds of sound in the inflationary Lagrangian generally produce non-separable shapes, sensitive to the sum k t = k 1 + k 2 + k 3 in the denominator, and thus cannot be exactly written in terms of a separable basis. The equilateral template [18], S equil = -2 K 0 +6 K 1 -3 K 2 , is widely used as a template to detect evidence of such shapes. Examples include generalized single-field models [7, 20, 78, 79], k -inflation [11, 80, 81], ghost inflation [82], DBI inflation [83, 84], single-field non-slow roll and bimetric theories [22, 23, 85]. A general, effective field theory of inflation is dominated by contributions from two shapes [20], While each can typically be well-described by the equilateral template, a linear combination of these picking out the differences between them can yield a very different shape. This realization led to the generation of the 'orthogonal' template, S orth = -8 K 0 + 18 K 1 -9 K 2 [19]. While inflation derived from a Bunch-Davies vacuum can be written in terms of a plane wave basis with positive k modes, excited states that are not in the Bunch Davies-vacuum can have initial states with both positive and negative k . Models motivated by non-trivial vacuum states can produce shapes with denominators containing k 1 + k 2 -k 3 (and its permutations), rather than k t [20, 30-32]. Unlike the equilateral and local templates, these shapes peak in the flattened configuration, when k 3 = k 1 + k 2 . While this shape again cannot be reconstructed perfectly using separable basis functions, an ansatz proposed as a proxy to this shape can be given by S enf = -3 K 0 + 6 K 1 -3 K 2 [31]. The shape has zero amplitude at k 1 = k 2 = k 3 , making the conventional normalization at this configuration unsuitable for this template. Though flattened shapes such as this one are usually associated with generalized initial states, it is in some cases possible to obtain flattened shapes through single-field inflation [10].", "pages": [ 10, 11 ] }, { "title": "2.4.2 Shapes well-approximated by {K 0 -K 2 }", "content": "Non-Gaussian templates to describe single-field theories are not limited to equilateral and orthogonal shapes. Fast-roll single-field non-Gaussian models [21, 22] retain the scale-invariant spectra but relax the condition for slow-roll inflation. [65] showed these can be written in terms of seven constituents, four of which are S local , S const , K 1 , and S single . The remaining three constituent shapes are 2 all of which have significant cosines with the local template. Other shapes exist in the literature that, while not separable, to some degree interpolate between the templates discussed above and hence can be reasonably-well described by linear combinations of {K 0 -K 2 } . For example, non-Bunch-Davies vacua generate shapes that can be equilateral, local, or enfolded [32]. Quasi-single field (QFSI) models [25, 26, 44, 77] motivated by string theory and supergravity inspired inflation contain multiple fields, but the extra fields have masses comparable to the Hubble scale. These models can be well described by a family of bispectrum templates dependent on a single parameter, ν , where N ν is the Neumann function of order ν . This shape interpolates between the constant and local templates. Another set of models that combine multiple fields and higher-derivative terms [33-35] also generate configurations that interpolate between standard shapes, spanning the local and equilateral templates. We use the basis modes, {K 0 -K 2 } to create templates for these non-separable shapes, S 3 -5 , S DBI , S single , and S QFSI ( ν ) . To demonstrate this, we generate an orthonormal basis {R n } using the Gram-Schmidt algorithm in k -space, taking R 0 = K 0 , and create a template S template = ∑ n i =0 α n R n as in (2.15) that reduces the covariance between the shape and template. The effectiveness of the template's fit can be quantified by the cumulative cosine. In Figure 1 we show how the shapes discussed above can be well modeled by templates using linear combinations of the {K 0 -K 2 } templates. In each case the cumulative cosine for the template and shape exceeds 0.98.", "pages": [ 11, 12 ] }, { "title": "2.4.3 Shapes well-approximated by more divergent {K 0 -K n }", "content": "There are two strong motivations to extend template design beyond these three core templates. Firstly, expansions using the first three templates do not necessarily ensure that theoretical priors on the divergence properties are satisfied by the template. An example of this is the consistency relation that requires shapes of single-field inflation to vanish in the squeezed limit [70]. However, the orthogonal and enfolded templates constructed to describe singlefield shapes tend toward a constant value in the squeezed limit. [19] proposed an orthogonal template, S ortho (2) , and [86] an enfolded template, S enf (2) , that are somewhat more complex, using linear combinations of shapes that diverge as x sq -2 , but they have the benefit of showing the correct divergence properties and more accurately reproducing the original non-separable shape. They can be written in terms of the K n modes as where p is a variable chosen to maximize the template's fit to the physical shape. Using our basis we can generalize this approach and write down classes of templates, denoted S [ R,r ] , constructed from basis modes with maximal divergence R that in the squeezed limit diverge as x r sq , where r ≥ R . In general, a shape written in terms of the basis will have a squeezed limit behavior given by with d nm summarized in Table 1. We find S [ R,r ] can be written in terms of an irreducible set of shapes given in Table 2 for which α n d nm = 0 for R ≤ m S local and S equil are the only shapes constructed from R = -1 modes that respectively have -1 and vanishing divergence. There are an infinite set of shapes, however, with constant divergence described by β K 0 +(1 -β )(2 K 1 -K 2 ) where β is free parameter which could take any value except β = -2 , for which the equilateral template is recovered. Instead of varying the parameter β , we could instead select a value of β to generate a template from the set. β = -8 corresponds to the orthogonal template chosen by [19] to maximize the resulting shape's orthogonality with S local and S equil . We could then choose to write general shapes in terms of linear combinations of { S equil , S ortho , S local } , rather than α n K n , S [ -1 , 0] = α E S equil + α O S ortho , (2.33) S [ -1 , -1] = α E S equil + α O S ortho + α L S local . (2.34) If these are the only shapes being used, the normalization constraint S [ R, -r ] ( k 0 , k 0 , k 0 ) = 1 fixes one α coefficient. We can extend this approach to include basis modes that diverge as x sq -2 , S [ -2 , 1] = α E S equil + α O ( S ortho +6 K 4 -6 K 3 ) + α L ( S local +2 K 3 -2 K 5 ) +(1 -α E -α O -α L )(2 K 3 -K 6 ) , (2.35) S [ -2 , 0] = α E S equil + α O S ortho + α L ( S local +2 K 3 -2 K 5 ) + β 3 (2 K 3 -K 6 ) +(1 -β 3 -α L -α E -α O )(2 K 4 -K 6 ) (2.36) S [ -2 , -1] = α E S equil + α O S ortho + α L S local + β 3 (2 K 3 -K 6 ) + β 4 (2 K 4 -K 6 ) +(1 -β 3 -β 4 -α L -α E -α O )(2 K 5 -K 6 ) , (2.37) S [ -2 , -2] = α E S equil + α O S ortho + α L S local + β 3 (2 K 3 -K 6 ) + β 4 (2 K 4 -K 6 ) + β 5 (2 K 5 -K 6 ) + (1 -β 3 -β 4 -β 5 -α L -α E -α O ) K 6 . (2.38) To tie this general approach to specific shapes in the literature, S ortho (2) and S enf (2) can be written in this form by the following choice of coefficients: S ortho (2) = (1 + p ) S equil -pS [ -2 , 1] [ α E = -19 9 , α O = 5 9 , α L = 2 3 ] (2.39) S enf (2) = (1 + p ) S equil -pS [ -2 , 1] [ α E = 9 5 , α O = -3 5 , α L = 0 ] . (2.40) The inclusion of extra basis shapes can be particularly important when the shape has undulations and is not just a smooth monotonic function. Shapes arising out of Galileon inflation are a good example of this. Imposing a Galilean symmetry on a single-field inflation model [86-89] gives rise to a non-Gaussian shape generated by three cubic interaction terms in the inflaton Lagrangian. While the shapes associated with each of these three operators, individually, are well-approximated by S equil and S enf (2) , there exist combinations of them for which the resulting Galileon shape has little overlap with any of the shapes we have mentioned so far. Non-separable templates for Galileon inflation have been developed in [86] and [89] which have high cosines both with the underlying shape and each other. For illustrative purposes, we consider the shape presented in [86], based on equations (26)-(28) of this reference. When we use the Gram-Schmidt decomposition to construct a template with only the first three modes, we find a poor fit with a cumulative cosine of only 0.13. The Galileon shape derives from a single-field action and a Bunch-Davies vacuum so theoretical consistency requires that it vanishes in the squeezed limit. Motivated by this, if we fit the Galileon model using the 4 shapes in S [ -2 , 1] , we obtain a template with a cosine of 0.93. This reconstruction is not improved if we allow an unconstrained combination of the seven K 0 -K 6 modes. We can extend our approach to R = -3 modes, and for example consider the following general shape that vanishes in the squeezed limit: S [ -3 , 1] = α E S equil + α O ( S ortho +6 K 4 -6 K 3 ) + α L ( S local -2 K 5 +2 K 3 ) + β 3 (2 K 3 -K 6 ) + β 7 (2 K 7 -K 11 ) + β 8 ( S ortho +6 K 8 -6 K 7 ) + β 9 ( S local -2 K 9 +2 K 7 ) +(1 -α E -α O -α L -β 3 -β 4 -β 7 -β 8 -β 9 )( K 6 -2 K 10 +2 K 7 ) . (2.41) Fitting these eight distinct shapes in S [ -3 , 1] to the Galileon shape, we obtain an improved template with cosine of 0.99. The second reason to consider a basis including more divergent terms is that some inflationary scenarios, such as excited initial states and warm inflation, in which inflation occurs in a warm radiation bath [27-29] (see [90] for a review), can give rise to shapes that are more divergent than the local shape, with an overall divergence of x sq -2 . This would suggest using an unconstrained combination of K 0 -K 6 modes, or using constrained combinations of the R = -3 modes for which the x -3 sq divergent term vanishes. One such example of this is a template for warm inflation proposed by [65], S warm = K 2 + K 7 -K 9 . (2.42) The realization that the differences between similar shapes can be important and provide an additional insight into the underlying model, implies that we should not just compare a small number of templates to the data. It is reasonable to extend beyond this and create more refined templates, sensitive to more than just properties that models have in common with the equilateral, orthogonal, and local templates. 2.5 Shape classification and depiction The models discussed in the previous section reflect only a sample of the wide range of non-Gaussian inflationary shapes arising in the literature. Putting a coarse filter on their properties, one might characterize them using three descriptors: i) their divergence in the squeezed limit, ii) how many modes it takes to accurately describe them, and iii) the 'family' to which they belong. Many of the physical shapes tend to be grouped in terms of a 'family' resemblance to an existing template, reflecting the type of configurations of triangles with side lengths k 1 , k 2 , and k 3 where the shapes have most of their power [71, 91]. For scale invariant shapes this is equivalent to studying the distribution of power over the space { k 1 k 3 , k 2 k 3 } for a fixed k 3 > k 1 , k 2 . This space can be pictorially represented by a triangle with sides 0 ≤ k 1 k 3 ≤ 1 and 1 / 2 ≤ k 2 k 3 ≤ 1 . We introduce it here in the context of the shapes already discussed, because we use this format to present some of our forecasting results. In Figure 2 we show examples of the shapes discussed in the previous section. S const = K 0 is the archetypal component of a family with similar power over all scales, homogeneous over the whole triangular region plotted. 'Squeezed' shapes have a bispectrum amplitude that is peaked in the top left-hand corner of the plot where k 1 k 3 glyph[lessmuch] 1 and k 2 k 3 = 1 , while 'equilateral' type shapes peak in the top right-hand corner where k 1 k 3 = k 2 k 3 = 1 . 'Flattened' shapes peak along the left edge, where k 1 k 3 + k 2 k 3 = 1 . Of the shapes we've discussed so far, some clearly fall within these family categories: S local , S warm , S 4 and S 5 are 'squeezed' shapes, while S equil , S DBI , S single are 'equilateral' and S enf is 'flattened'. 3 k GLYPH<144> 2 k 3 k GLYPH<144> 2 k 1.0 0.9 0.8 0.7 0.6 0.5 1.0 0.9 0.8 0.7 0.6 0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 ortho (g) S 0.2 0.4 0.6 0.8 1.0 (a) S Local k 1 GLYPH<144> k 3 local Figure 2 : Plots showing the comparative spatial distribution of non-Gaussian shape, S ( k 1 , k 2 , k 3 ) , as a function of k 1 /k 3 and k 2 /k 3 . From left to right we show [top] the local, equilateral, and enfolded separable templates, [middle] the orthogonal(2) template, and non-separable shapes derived from a QSFI model with ν = 1 . 3 and Galileon inflation, and [bottom] shapes contributing to S [ -2 , 1] . All shapes are normalized to unity at the equilateral configuration ( k 1 k 3 = k 2 k 3 = 1 ). The color scales for all but the local and QSFI shapes are the same to aid comparison. Figure 2 : Plots showing the comparative spatial distribution of non-Gaussian shape, S ( k 1 , k 2 , k 3 ) , as a function of k 1 /k 3 and k 2 /k 3 . From left to right we show [top] the local, equilateral, and enfolded separable templates, [middle] the orthogonal(2) template, and non-separable shapes derived from a QSFI model with ν = 1 . 3 and Galileon inflation, and [bottom] shapes contributing to S [ -2 , 1] . All shapes are normalized to unity at the equilateral configuration ( k 1 k 3 = k 2 k 3 = 1 ). The color scales for all but the local and QSFI shapes are the same to aid comparison. S orth + 6 K k 1 GLYPH<144> k +6 K 4 - 6 K 3 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 (h) S local (b) S Quasi - single H n= 1.3 L k 1 QSFI S loc - 2 K k 1 - 2 K 5 GLYPH<144> k ( ν = 1 . 3) 5 + 2 K 3 3 +2 K 3 (f) S (i) 2 K 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 (c) S Folded k 1 GLYPH<144> k 3 enf Galilean k 1 GLYPH<144> k 3 Galileon 2 K k 3 1 3 - K GLYPH<144> k 3 -K 6 6 GLYPH<144> k 3 (e) S 4 - 6 K 3 3 10 0 3 - 3 3 k GLYPH<144> 2 k 3 k GLYPH<144> 2 k 3 k GLYPH<144> 2 k 1.0 0.9 0.8 0.7 0.6 0.5 1.0 0.9 0.8 0.7 0.6 0.5 1.0 0.9 0.8 0.7 0.6 0.5 3 - 3 10 0 3 - 3 Equilateral k 1 GLYPH<144> k 3 equil 3 k GLYPH<144> 2 k 3 k GLYPH<144> 2 k 3 k GLYPH<144> 2 k 1.0 0.9 0.8 0.7 0.6 0.5 1.0 0.9 0.8 0.7 0.6 0.5 1.0 0.9 0.8 0.7 0.6 0.5 3 - 3 3 - 3 3 - 3 There exist other additional shapes generated by modes K 3 through K 7 . For example, Figure 2 also includes three shapes that contribute to S [ -2 , 1] that could describe a general single-field model with Bunch Davies vacuum. While each vanishes in the squeezed limit by construction, we find they differ from the equilateral shape in still having a component of their signal focused along the flattened configuration. The comparative size of this component correlates with the divergence of the shapes from which they are created, S ortho , S local , and K 3 . There are shapes that do not fall clearly into any of these families: S ortho (2) peaks in both the flattened and equilateral configurations, excited states can peak in squeezed and flattened configurations, and S QSFI shapes interpolate between constant and local properties. Beyond this there are shapes with distinct undulating forms, the S Galileon shape for example, that do not peak at either edges or corners. Moreover, not all shapes within each family are alike. For example, the local and warm shapes both peak in squeezed configurations, but their divergence properties in this region are different, leading to a low cosine between them. Given the breadth of bispectrum shapes that could be created, and the comparatively loose characteristics on which 'families' are formed, there is strong motivation to ask how much information we can discern observationally about bispectra. This will help quantify how well we might determine the underlying non-Gaussian shape, if a detection of non-Gaussianity is made. 3 Forecasting constraints on general shapes In the following analysis, we apply the separable, divergent basis and template classes from the previous section to assess how we can constrain the shape of primordial bispectra with upcoming CMB data. Our goal is to quantify what properties of shapes are measurable, and the respective roles of the experimental uncertainties and theoretical priors on determining distinguishability. Motivated by a broad cross-section of models in the literature, we will focus on shapes described by basis functions {K 0 -K 6 } that are nearly scale-invariant and contain terms that are potentially as divergent as x -2 sq in the squeezed limit. We describe the Fisher matrix approach we use assuming a Planck-like CMB experiment in section 3.1. In section 3.2, we present the results of a principal component analysis for the set of shapes S [ -2 ,r ] with different divergence criteria in the squeezed limit imposed. Doing so generates the experiment's preferred orthogonal basis of best to worst measured bispectrum configurations, the principal components (PCs) and their corresponding uncertainties, subject to the theoretical prior. We consider the implications for shape normalization and the bestmeasured k -configuration in sections 3.3 and 3.4, respectively, and finish in section 3.5 by quantifying our potential ability to determine and distinguish shapes. 3.1 Fisher matrix approach We compute the 7 × 7 Fisher matrix for the amplitudes of the basis modes, K n , { f n NL , n = 0 , ..., 6 } defined in Eq. (2.19) as F ( f i NL , f j NL ) = f sky ∑ abc,pqr ∑ glyph[lscript] 1 glyph[lscript] 2 glyph[lscript] 3 ∂B abc ( i ) glyph[lscript] 1 glyph[lscript] 2 glyph[lscript] 3 ∂f i NL ( Cov -1 ) abc,xyz glyph[lscript] 1 glyph[lscript] 2 glyph[lscript] 3 ∂B xyz ( j ) glyph[lscript] 1 glyph[lscript] 2 glyph[lscript] 3 ∂f j NL , (3.1) where { abc } and { xyz } each sum over the 8 possible temperature ( T ) and polarization ( E ) combinations of bispectra: TTT,TTE,TET,ETT,TEE,ETE,EET,EEE . Given a general primordial shape expanded on the {K n } basis as in (2.19), the corresponding CMB reduced bispectrum is b abc glyph[lscript] 1 glyph[lscript] 2 glyph[lscript] 3 = N ∑ n f n NL K abc ( n ) l 1 l 2 l 3 , (3.2) where K ( n ) glyph[lscript] 1 glyph[lscript] 2 glyph[lscript] 3 denotes the reduced bispectrum of the basis function K n in (2.20)-(2.21). Here we compute it as K abc ( n ) glyph[lscript] 1 glyph[lscript] 2 glyph[lscript] 3 = 1 N ( n ) perm k 2( n s -1) 0 ∫ x 2 dx [ I ap glyph[lscript] 1 ( x ) I br glyph[lscript] 2 ( x ) I cs glyph[lscript] 3 ( x ) + { prs } perms ] (3.3) I ap glyph[lscript] ( x ) ≡ 2 π ∫ k max k min dkk p ' ∆ a glyph[lscript] ( k ) j glyph[lscript] ( kx ) (3.4) where p ' is defined as in (2.21). We have modified the CAMB 3 code [62] to numerically evaluate the values of I ap glyph[lscript] 1 and then written code to appropriately combine them to form each K ( n ) glyph[lscript] 1 glyph[lscript] 2 glyph[lscript] 3 . Specifically, we take k min = 6 . 6 × 10 -6 Mpc -1 , k max = 0 . 56 Mpc -1 , and x max = 16 . 5 × 10 3 Mpc. We include a note of caution that since the integrals over k and x for cases where p is very negative (positive) depend on having accurate transfer functions at small (large) values of k , numerical results for these integrations should be carefully checked for robustness. To verify the numerical robustness of our results, we have checked that I ap glyph[lscript] ( x ) obtained numerically for p < 0 match the expected analytic result in the Sachs-Wolfe limit. We have also quantified how the Fisher matrix results quoted in the next section are robust or exhibit instabilities to changes in the accuracy boost parameter in CAMB, which allows for fine resolution in the k and x integrals. In particular, for the most divergent K 6 mode, which is a combination of the most extreme integrals (with p = -2 and 4) and thus we would expect to have the greatest amount of numerical error, we find that the Fisher results quoted in the next section changed by less than 0 . 01% when the accuracy boost was increased from 1.5 to 2. However, we find that the worst measured eigenmode, in the PCA, is far more sensitive to the integral resolution. We find with an accuracy boost of around 2 we get convergence of a few percent in all but the worst measured mode. This last mode oscillates with a variation of around 15% in the standard deviation. This sensitivity in the worst measured mode (which we will find is the least divergent shape in the squeezed limit), can affect the constraints for shapes which have a component described by this mode. In the following sections, we present our results with these cautions attached when appropriate. The covariance matrix we use from [92] is ( Cov -1 ) abc,xyz glyph[lscript] 1 glyph[lscript] 2 glyph[lscript] 3 = ( C -1 ) ax glyph[lscript] 1 [ ( C -1 ) by glyph[lscript] 2 ( C -1 ) cz glyph[lscript] 3 +( C -1 ) bz glyph[lscript] 2 ( C -1 ) cy glyph[lscript] 3 ] +( C -1 ) ay glyph[lscript] 1 [ ( C -1 ) bz glyph[lscript] 2 ( C -1 ) cx glyph[lscript] 3 +( C -1 ) bx glyph[lscript] 2 ( C -1 ) cz glyph[lscript] 3 ] +( C -1 ) az glyph[lscript] 1 [ ( C -1 ) bx glyph[lscript] 2 ( C -1 ) cy glyph[lscript] 3 +( C -1 ) by glyph[lscript] 2 ( C -1 ) cx glyph[lscript] 3 ] , (3.5) with ( C -1 ) ax glyph[lscript] = ( ˆ C TT glyph[lscript] ˆ C TE glyph[lscript] ˆ C TE glyph[lscript] ˆ C EE glyph[lscript] ) -1 (3.6) ˆ C ax glyph[lscript] = C ax glyph[lscript] + N ax glyph[lscript] . (3.7) Here f sky is the overall fraction of the sky observed, and we assume f sky = 0 . 8 . N ax glyph[lscript] is the instrument noise for a correlation between observables a and x . We model CMB noise by considering the three lowest frequency bands of the Planck HFI instrument for temperature and E-mode polarization, as described in the Planck Bluebook [93]. We assume each frequency channel has Gaussian beam profile of width θ FWHM and isotropic noise with error in X = T, E of σ X . The noise in each frequency channel c is then given by N ax glyph[lscript],c = ( σ x,c θ fwhm ) 2 e glyph[lscript] ( glyph[lscript] +1) θ 2 fwhm,c / 8 ln 2 δ ax (3.8) N ax glyph[lscript] = [ ∑ c ( N ax glyph[lscript],c ) -1 ] -1 . (3.9) Our fiducial flat Λ CDM cosmology is described by the following parameters, which are consistent with the latest WMAP 9-year constraints [2]: Ω b h 2 = 0 . 02258 , Ω c h 2 = 0 . 1109 , ∆ 2 R ( k 0 ) = 2 . 43 × 10 -9 , n s = 0 . 963 , and τ = 0 . 088 . As has been done in other recent Fisher forecasts on non-Gaussianity parameters, such as [94], we consider the uncertainties on the non-Gaussian amplitudes independent of the uncertainties in the fundamental cosmological parameters that also affect the power spectrum, as these are comparatively small relative to the uncertainties from the bispectrum shape functions [95]. For this initial analysis, we neglect the effect of imperfect measurements of the lensing signal [96, 97], secondary anisotropies [15], and inhomogeneous sky coverage/noise on the constraints (e.g. [54, 98]). 3.2 Fisher matrix results A general bispectrum can be expanded in terms of either K n or the component shapes, { S X } , in S [ R,r ] , given in (2.33)-(2.38), B Φ ( k 1 , k 2 , k 3 )( k 1 k 2 k 3 ) 2 N = f NL S = ∑ n f n NL K n ( k 1 , k 2 , k 3 ) = ∑ X f X NL S X ( k 1 , k 2 , k 3 ) (3.10) While the Fisher matrix we used based on S [ R,r ] automatically includes the additional priors to constrain the divergence properties, these could also be introduced into the K n Fisher analysis by using Lagrange multipliers to systematically impose each divergence constraint. The latter makes no assumption a priori about what linear combinations of the shapes given in Table 2 should have their amplitudes constrained. While we use the shape expansion in our discussion below, we investigated both approaches and found they led to consistent conclusions. We use the Fisher matrix in terms of K n to construct Fisher matrices for the component shapes in S [ -2 ,r ] for r = -2 , -1 , 0 , 1 . In Table 3 we give the glyph[lscript] -space correlation coefficients based on (2.11), but here weighted by the data covariance between pairs of the component shapes, S X and S Y , Corr glyph[lscript] ( S X , S Y ) = F XY √ F XX F Y Y . (3.11) This gives a measure of the similarity of the component shapes based on how they are measured by the survey, integrated over all glyph[lscript] combinations. We find the similarity between pairs of the four basis shapes in S [ -2 , 1] , each of which vanishes in the squeezed limit, are primarily related to the divergence of the shapes from which they are derived. S equil and S ortho +6 K 4 -6 K 3 are very similar to each other, while Table 3 : Correlation coefficients between shapes that diverge as x n sq . These shapes are components in the general template classes, S [ -2 ,r ] , for r ≤ n . Table 3 : Correlation coefficients between shapes that diverge as x n sq . These shapes are components in the general template classes, S [ -2 ,r ] , for r ≤ n . S local -2 K 5 +2 K 3 and 2 K 3 -K 6 also have a high degree of overlap. Interestingly the S local -2 K 5 +2 K 3 and 2 K 3 -K 6 shapes also have significant similarities with the shapes that diverge as x 0 sq . This is derived from their strong signal along the configurations between squeezed and flattened configurations, as discussed in section 2.5. The shape with x -1 sq divergence constructed from the R = -2 modes, 2 K 5 -K 6 , is highly degenerate with the local template; essentially this implies the two are indistinguishable from one another using the CMB data. √ The unmarginalized errors, σ ( f X NL ) = 1 / F XX , give the uncertainty in the measurement of a specific template if the underlying theory is known to be wholly described by that template. We find these are comparatively insensitive to the integral resolution discussed in section 3.1. The covariance matrices obtained from inverting the Fisher matrices give the uncertainties on the amplitudes of the component shapes, σ ( f NL ) , marginalized over the freedom allowed by each model. The marginalization does make the results precision dependent in the worst measured mode, i.e. the results are accurate to better than 15%. We summarize the results in Table 4. The covariance matrix in each case can be diagonalized to obtain the orthonormal eigenvectors, ˆ e i = ∑ X c iX S X , (3.12) and associated eigenvalues, which give the variances σ 2 ( b i ) in the amplitudes of the eigenvectors. These then provide a way to rank the best to worst measured bispectra. Given this orthonormal basis, any general bispectrum may be expanded as f NL S = ∑ i b i ˆ e i . (3.13) The principal components obtained by diagonalizing the covariance matrix are not immediately 'shapes' in the way we considered so far. They have unit norm with respect to the Table 4 : The uncertainties on the amplitudes of the component shapes, in the general template classes S [ -2 ,r ] , that diverge as x r sq in the squeezed limit. We give both the unmarginalized errors, assuming the underlying shape is exactly described by the component shape, and the marginalized errors if we allow the shape to be a general linear combinations of components consistent with the prior on the divergence properties. Table 4 : The uncertainties on the amplitudes of the component shapes, in the general template classes S [ -2 ,r ] , that diverge as x r sq in the squeezed limit. We give both the unmarginalized errors, assuming the underlying shape is exactly described by the component shape, and the marginalized errors if we allow the shape to be a general linear combinations of components consistent with the prior on the divergence properties. component shape basis, ∑ X | c iX | 2 = 1 , rather than being normalized at the equilateral configuration, ∑ X c iX S X ( k 0 , k 0 , k 0 ) = 1 . If we restrict the shapes to those described by the first three modes, marginalization does not significantly alter the constraints from the unmarginalized errors, i.e. the three common templates are essentially the principal components (PC) of the covariance matrix, with the eigenvalues showing that the more divergent the shape, the better it is measured. In contrast, when extended to general shapes, constructed of all seven modes, we find marginalized errors for individual shapes are far larger because of observational similarities between shapes of similar divergence, or similar properties in the flattened limit. It seems that only K 6 is well constrained if any shape from the S [ -2 ,r ] type is allowed. When extended to shapes constructed of seven modes, the correspondence between the PC's and divergence remains. We find that, in general, divergence in the squeezed limit, followed by a second divergence measure, corresponding to the signal near the flattened configurations, can be used as coarse indicators of comparative constraining power with the CMB. For the general shape without any additional divergence constraints, the best measured PC is almost completely composed of the most divergent shape, K 6 . The second best measured PC has dominant contributions from S local and 2 K 5 -K 6 with which it is very degenerate. If the general shape is restricted to have vanishing divergence in the squeezed limit, then the best measured PC is very similar to a shape like 2 K 3 -K 6 which has large signal in the flattened configurations despite vanishing in the squeezed limit. The next best measured PC is then similar to shapes like equilateral or the orthogonal-derived shape S ortho +6 K 4 -6 K 3 , which has less power on flattened configurations. In both cases, none of the templates look like the two worst measured modes, which exhibit large oscillatory features along flattened configurations. Table 5 : Properties of the principal components for each template class S [ -2 ,r ] in terms of their component shapes. The properties in the squeezed limit is determined by the value of r . The table provides uncertainties, for a unit norm eigenvector, σ ( b i ) , and an effective σ ( f NL (ˆ e i )) , when the eigenvector is normalized consistently at the equilateral configuration. Table 5 : Properties of the principal components for each template class S [ -2 ,r ] in terms of their component shapes. The properties in the squeezed limit is determined by the value of r . The table provides uncertainties, for a unit norm eigenvector, σ ( b i ) , and an effective σ ( f NL (ˆ e i )) , when the eigenvector is normalized consistently at the equilateral configuration. 3.3 Drawbacks of normalization at the equilateral configuration As stated earlier, the PC's as they are originally generated, are not shapes in the usual sense because they are not bispectra normalized at k 1 = k 2 = k 3 = k 0 . They have a unit norm in terms of the basis shapes. With this normalization, as usual in PCA, their eigenvalues quantify which combinations of the basis shapes are best and worst constrained by data, and the eigenvectors can be combined to create general shapes. We can convert σ ( b i ) to an effective σ ( f NL (ˆ e i )) , corresponding to the amplitude of each eigenvector shape normalized in the conventional way, σ ( f NL (ˆ e i )) = | σ ( b i )ˆ e i ( k 0 , k 0 , k 0 ) | . Table 5 gives the values of σ ( b i ) and σ ( f NL (ˆ e i )) . We quote the results when both temperature and polarization data are included. We find that the exclusion of the E-mode polarization from the Fisher analysis does not noticeably change the shape of the principal components, but does increase the eigenvalues by about a factor of ∼ 1 -3 across all eigenvectors. The constraints on all but the last eigenvalue under each divergence constraint shown in Table 5 are accurate to a few percent. The worst measured eigenmode is measured to ∼ 15% accuracy. Normalizing our PC's at the arbitrarily chosen equilateral configuration allows us to compare them to other shapes consistently at one point in k -space. σ ( f NL ) does not in itself, however, quantify a shape's overall variance across all k . An analogous situation arises in quoting uncertainties on the power spectrum amplitude from two different surveys, say a large-scale CMB survey and a galaxy survey. Both surveys could quote uncertainties at a common arbitrary scale, say k 0 = 0 . 05 h/Mpc , but while this uncertainty might represent the best measured scale for the galaxy survey, it would grossly overstate the minimum uncertainty in the CMB survey, which is best measured at a much larger scale. It is entirely possible for a well measured mode to have a significant part of its small variance located in the equilateral configuration, while a poorly measured mode could have its lowest variance in the equilateral configuration but be poorly measured over other regions of k -space. Indeed we find this to be the case, given that the best measured shapes have signal peaked near the squeezed, rather than equilateral, configuration. This means that σ ( f NL ) is not a useful measure in itself to assess how well a shape can be measured. This shortcoming of the conventional normalization has been discussed previously in other studies, e.g. [18] and [54], where alternative normalization schemes based on an integrated total amount of Figure 3 : Configurations of the principal components for S [ -2 , -2] , a general shape that can be as divergent as x -2 sq in the squeezed limit. The plots show the amplitude of the eigenvectors for the best ˆ e 1 to worst ˆ e 7 measured modes as a function of k 1 k 3 versus k 2 k 3 . The principal components are each normalized to be unity at the equilateral configuration. Figure 3 : Configurations of the principal components for S [ -2 , -2] , a general shape that can be as divergent as x -2 sq in the squeezed limit. The plots show the amplitude of the eigenvectors for the best ˆ e 1 to worst ˆ e 7 measured modes as a function of k 1 k 3 versus k 2 k 3 . The principal components are each normalized to be unity at the equilateral configuration. non-Gaussianity have been proposed. The overall spread in uncertainties from the best to worst eigenvector is much reduced when normalized at the equilateral configuration and can in some cases produce a switch in the ordering of the modes for σ ( f NL ) relative to that of σ ( b i ) . This does not present an inconsistency in the analysis, but simply demonstrates the perils of considering a normalization at an arbitrary scale. Figure 3 shows the variety of profiles in the 2-dimensional ( k 1 k 3 , k 2 k 3 ) space shown in the triangle plots. Given that the power spectrum we consider is not perfectly scale invariant, there is some small dependency of the bispectrum amplitude on the value of k 3 , described by p ' in (2.21). The spatial profiles, however, in terms of k 1 k 3 and k 2 k 3 are k 3 -independent. The gradients in the PC configurations reflect the rough ordering from squeezed to flattened to equilateral as the modes span from best to worst. The complementarity of the eigenvectors, reflected by the different directions of gradients of the signals in the configuration space, has implications for the location of the best measured configuration, as we discuss in section 3.4. 3.4 Best measured k -configurations In the analysis that follows, we avoid splitting up bispectra into shapes and amplitudes, normalized at an arbitrary configuration. Instead we consider the overall constraints on the bispectrum, B ( k 1 , k 2 , k 3 ) , itself up to the constant normalization, given in (2.2), f NL S = k 2 1 k 2 2 k 2 3 B ( k 1 , k 2 , k 3 ) /N . The eigenmodes and eigenvalues from the PCA provide a way to compute an error on a general k -space bispectrum. We can calculate the posterior distribution of the uncertainties on f NL S given the data, D , with a theoretical prior given by the eigenvectors { ˆ e i } , p ( f NL S | D ) = ∫ n ∏ i =0 db i p ( f NL S | b i ) p ( b i | D ) , (3.14) p ( f NL S | b i ) = δ ( f NL S -n ∑ i =0 b i ˆ e i ( k 1 , k 2 , k 3 )) , (3.15) p ( b i | D ) = 1 √ 2 πσ ( b i ) exp ( -b 2 i 2 σ 2 ( b i ) ) . (3.16) Under this assumption of Gaussian errors this gives the commonly used result, σ 2 ( f NL S ( k 1 , k 2 , k 3 )) = ∑ i σ 2 ( b i )ˆ e i ( k 1 , k 2 , k 3 ) 2 . (3.17) This equation for computing the error can be applied to each set of PC's generated for each divergence scenario in the previous section. The errors in the ( k 1 k 3 , k 2 k 3 ) configuration space can be plotted and the best measured k -configuration, and the associated uncertainty, calculated for each scenario. σ ( f NL S ) varies only very weakly across slices in k 3 ; its functional form can be divided into a dependence on ( k 1 k 3 , k 2 k 3 ) and a weak dependence on k 3 , going as k 2( n s -1) 3 , for fixed k 1 k 3 and k 2 k 3 . For our choice of theoretical priors on the model, σ ( f NL S ) decreases with increasing k 3 . This is because the noise scales as the signal for the near scale-invariant theoretical prior we impose. An alternative prior would give very different dependencies on k 3 . For example if we were to remove the theoretical prior all together and model the bispectrum amplitude as bins in k , the only constraints on the model come from the observational uncertainties, and the noise would diverge exponentially on small scales. 3 k GLYPH<144> 2 k 1.0 0.9 0.8 0.7 0.6 0.5 0.0 3 - 3 0.2 0.4 0.6 0.8 1.0 PC 1 k (a) ˆ 1 GLYPH<144> k e 1 Figure 4 : As in Figure 3, but showing the configurations of the principal components for S [ -2 , 1] , a general shape that vanishes in the squeezed limit. Figure 4 : As in Figure 3, but showing the configurations of the principal components for S [ -2 , 1] , a general shape that vanishes in the squeezed limit. PC 2 k (b) ˆ 1 GLYPH<144> k e 2 PC 4 k (d) ˆ 1 GLYPH<144> k e 4 The weak k 3 dependence implies that the uncertainties at one k 3 reasonably reflect the overall uncertainties if one were to marginalize over k 3 . Figure 5 shows the error on the k 3 = 0 . 01 Mpc -1 slice for three different divergence cases. The location of minimum σ ( f NL S ) comes from the sum of the eigenmodes that is weighted by each mode's error, which arises out of the complementarity of the degeneracy directions of the PC's. We find the location of the best measured configuration is consistent for the scenarios that diverge as x -2 sq through to 3 3 3 3 k GLYPH<144> 2 k 3 k GLYPH<144> 2 k 1.0 0.9 0.8 0.7 0.6 0.5 1.0 0.9 0.8 0.7 0.6 0.5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 3 - 3 3 - 3 a constant in the squeezed limit for R = -2 . This location is not situated in any one of the corners of the triangle plot associated with squeezed, equilateral, and flattened configurations. Instead it is somewhat centrally located adjacent to the flattened edge. The best-measured configurations are located at k 1 k 3 ≈ 0 . 32 and k 2 k 3 ≈ 0 . 80 , with minimum σ ( f NL S ) ≈ 37 . For the vanishing divergence prior, the best-measured configuration approaches the squeezed limit, as we have required the noise to scale as the shapes, which go to zero there. We also find that for shapes constructed from the local, equilateral, and orthogonal templates ( R = -1 ), the best measured location spans a degeneracy direction also along the flattened edge, with minimum σ ( f NL S ) ≈ 20 . While the best measured region, in which the error is a minimum, is useful in the absence of knowledge about the theory, the signal-to-noise ratio, for a given underlying model can also help determine a survey's ability to distinguish between shapes. We consider this in the following subsection. 3.5 Shape determination and distinguishability Wenow turn to discussing a central question of the paper: given a detection of non-Gaussianity using a specific template, what can be confidently inferred about the true underlying shape? We have already considered this from one perspective in section 3.2 by considering the uncertainties in ascribing a detection using a template to the template's shape itself. If we allow for the possibility that a detection using a specific template could be detecting the component of another shape allowed by the theoretical prior we are considering, then the errors on shape determination can increase significantly, especially for shapes that do not peak in the squeezed or flattened configurations. In this section we approach the question of shape distinguishability from a second direction, considering the range of possible general shapes, under a divergence prior, that could create the detected template signal and fit the bispectrum data within some confidence range. Such analyses have already been considered in the context of specific models, for example, how well we might disentangle a QSFI model (e.g. in (2.29)) from S equil or S local as a function of ν [44, 77]. Here we extend this approach to a more general shape, and consider what implications a detection with one of the common templates has for general models. For specificity we consider a subset of general shapes consistent with S [ -2 , 1] , S gen = (1 -α X -α Y -α Z ) S equil + α X S X + α Y S Y + α Z S Z , (3.18) where S X,Y,Z can be { S ortho +6 K 4 -6 K 3 , S local -2 K 5 +2 K 3 , 2 K 3 -K 6 } . This is investigating a general set of single-field inflation models from which S ortho (2) in (2.39) and S enf (2) in (2.40) are drawn. How large must a template signal f T NL be to be confident that the signal is not from a different, more general shape S gen ? We set this distinguishable detection threshold to be σ ( f T NL ) , the error on f T NL for the template, marginalized over f S NL , the amplitude of the general shape. The marginalized constraint is computed by inverting the 2 × 2 Fisher matrix for ( f T NL , f S NL ) . Thus we are comparing two shapes, where one is a template, and the other is a general shape, in which α X , α Y , and α Z parametrize the deviation from S equil . In the simplest case, we allow only α X to be non-zero, such that S gen is a linear combination of two shapes, S equil and S X , that varies with one parameter. In Figure 6, we show σ ( f T NL ) for the local, equilateral, and orthogonal templates when S X takes different forms. The minimum value of σ ( f T NL ) for each template across all values of α recovers the Figure 5 : Example contour slices of σ ( f NL S ) under different divergence constraints, [top left] S [ -2 , -2] , [top right] S [ -2 , 1] , and [bottom] S [ -1 , -1] . Figure 5 : Example contour slices of σ ( f NL S ) under different divergence constraints, [top left] S [ -2 , -2] , [top right] S [ -2 , 1] , and [bottom] S [ -1 , -1] . (a) Diverges as 1 /k 2 in the squeezed limit. (b) Vanishing divergence in the squeezed limit. (c) Diverges as 1 /k in the squeezed limit. glyph[negationslash] unmarginalized errors of each. A detected value of f equil NL must be larger to produce a 1 σ detection of the equilateral shape, as opposed to a more general shape with α = 0 , while f local NL never has to be much larger than the unmarginalized σ ( f local NL ) to favor the local model over this general shape, because S X and S local are weakly correlated. To illustrate the use of Figure 6, for example, a detection of f orth NL = 40 , while greater than the unmarginalized error of 19, would only be sufficient to rule out a false 1 σ detection of S gen with S X = 2 K 3 -K 6 for -5 glyph[lessorsimilar] α glyph[lessorsimilar] 0 . 9 . On the other hand, if f orth NL is detected to be larger than 46, then S gen of the 1-parameter form would be disfavored, as σ ( f orth NL ) is smaller than this over all values of α . Models with S X = 2 K 3 -K 6 are most easily differentiated from S equil because they have the lowest correlation with S equil . An application of comparing constraints on α X from two distinct templates is to test whether a given model is consistent with or disfavored by the data. If two template measurements each individually remain consistent with two non-overlapping regions of α -space, then it would be clear that modeling the underlying shape with α alone is not able to produce a viable model. This would be true for dual measurements of { f equil NL = 60 , f ortho NL = 45 } for S X = 2 K 3 -K 6 , since they would imply non-overlapping ranges of α , -0 . 7 ≤ α ≤ 0 . 3 versus 1 . 3 ≤ α ≤ 3 . 8 to each be consistent with the data. We can extend the same analysis to a comparison between templates and a 2-parameter general shape by allowing both α X and α Y to vary simultaneously, while α Z is fixed to zero. For example, S enf (2) is a specific template for which this is true. In Figure 7 we show σ ( f equil NL ) and σ ( f orth NL ) over different choices of the 2-dimensional space and find that there exist degeneracy directions that are not fully captured by the 1-dimensional projections in Figure 6. We find that σ ( f local NL ) remains close to the unmarginalized value in this case as well. In the most general 3-parameter model, we can ask the question of whether there is any area of this space corresponding to a general model that vanishes in the squeezed limit, with a significant enough overlap with the local template to require that a potentially detected f local NL be much greater than the unmarginalized value of 3. If this were the case, then it may be that a local template detection cannot definitively rule out a general shape that satisfies the single-field consistency relation. However, we find that nowhere in the parameter space does the σ ( f loc NL ) become greater than 4.2, showing that a detection of the local template above this threshold would effectively rule out a general shape, vanishing in the squeezed limit, subject to the assumption that it can be written in terms of our basis in S [ -2 , 1] . The same distinguishing power is not present for S local if we allow a weaker prior given by S [ -2 , -1] . In this case the significant cosine between S local and 2 K 5 -K 6 , means we may never be able to confidently attribute a detection with S local to be definitive evidence that the diverging signal is unambiguously S local . A long shot could be to additionally look at the correlation of the bispectrum signal with 2 K 4 -K 6 which is mildly negatively correlated with S local and essentially uncorrelated with 2 K 5 -K 6 . This last point raises an interesting application of our study: to ask if there are distinct, new templates that we might use to learn about the origins of a detected non-Gaussian signal. In the context of models described by the first three modes, K 0 to K 2 , the local, equilateral, and orthogonal templates are almost perfectly aligned with the principal components. If we extend the templates to include K 3 through K 6 , however, we find these no longer represent the PC's. For example, what might be the best way to extend the template pool to search for signatures of single-field inflation models with Bunch-Davies vacua? In the context of r = 1 shapes, 2 K 3 -K 6 is well-aligned with the best measured PC and is only mildly correlated with the existing templates which would make it a reasonable candidate to add as an additional template. We show the resulting constraints on general shapes in Figures 6 and 7. The figures show that this template probes regions of the allowed α -space which the equilateral and orthogonal templates do not constrain in the same way. Thus it may be possible to combine constraints from the common templates and motivated choices of a small number of new templates, like 2 K 3 -K 6 , to probe the underlying shape of non-Gaussianity. Figure 6 : Detection thresholds on the amplitude of templates, f T NL , for distinguishing between the template and a general shape, S = (1 -α ) S equil + αS X , at the 1 σ confidence level. [Top] S X = S ortho +6 K 4 -6 K 3 , [bottom] S X = 2 K 3 -K 6 . Blue, orange, red, and black curves denote f equil NL , f orth NL , f local NL , and f 2 K 3 -K 6 NL , respectively. Since S local -2 K 5 +2 K 3 is very similar to 2 K 3 -K 6 , the case where S X = S local -2 K 5 +2 K 3 is not shown. Figure 6 : Detection thresholds on the amplitude of templates, f T NL , for distinguishing between the template and a general shape, S = (1 -α ) S equil + αS X , at the 1 σ confidence level. [Top] S X = S ortho +6 K 4 -6 K 3 , [bottom] S X = 2 K 3 -K 6 . Blue, orange, red, and black curves denote f equil NL , f orth NL , f local NL , and f 2 K 3 -K 6 NL , respectively. Since S local -2 K 5 +2 K 3 is very similar to 2 K 3 -K 6 , the case where S X = S local -2 K 5 +2 K 3 is not shown. 4 Conclusion At the heart of this work is the discussion about how uncertainties quoted on shape detection are inherently dependent on the underlying assumptions made about the shape. While a detection of non-Gaussianity with any template will be extraordinarily transformative in our field, its interpretation, in what it tells us about the underlying shape, has to be considered carefully in terms of our underlying theoretical prior we impose. Even if no detection of nonGaussianity is made, upper bounds on the deviations from Gaussianity according to templates will have broader impacts for constraints on general shapes. We have presented an approach for quantifying how well upcoming CMB temperature and E -mode polarization data can determine the shape of primordial non-Gaussianity under minimal assumptions. We proposed a set of polynomial divergent basis functions, { K n } , that are well-tuned to describing many nearly scale-invariant, smoothly varying, but potentially divergent shapes discussed in the literature. We find we need only three to seven modes to generate matched templates to describe a wide range of physically motivated shapes. In this sense, the divergent basis is more efficient than the polynomial basis used in previous studies (e.g. [54]). Each K n in our basis is generally divergent, but linear combinations of the K n can be constructed to have cancellations in the squeezed limit, thus creating templates that are less divergent (e.g. equilateral shape). For example, S equil and (2 K 3 -K 6 ) both vanish in the squeezed limit, but still have a low correlation because the latter has more power near the flattened and squeezed configurations. Using the {K n } it is straightforward to form template classes, S [ R,r ] , that have specific, common divergence properties in the squeezed limit. Each class is constructed from an irreducible set of shapes, that while constructed out of a basis sets with maximum divergence x R sq ( R < 0 ), through cancellations of divergent terms, have squeezed limit x r sq ( r > R ) . The choice of R controls how many basis modes are used to develop the templates, e.g. R = -1 includes K 0 through K 2 , while R = -2 uses K 0 through K 6 . As R becomes more negative it allows templates to be refined and shapes with a broader set of features across configuration space to be modeled. The classes allow templates to be developed with priors that are well-motivated by theories: S [ R, 1] represents the class of all single-field models derived from a Bunch-Davies vacuum, S [ R, -1] in addition includes all multi-field models that diverge like the local shape in the squeezed limit, and S [ R, -2] is the most general class of shapes which includes models from non-Bunch-Davies vacuum initial states. While the constituent shapes making up each class have the same divergence properties in the squeezed limit, away from this limit they have power weighted differently in the configuration space. For example, we discuss a new shape, 2 K 3 - K 6 , used in S [ R, 1] , that has the same squeezed limit behavior as the equilateral shape but has glyph[lscript] -space cosines with the standard equilateral, orthogonal, and local templates of 0.07, 0.80 and -0.29 respectively. While the divergent terms cancel in the squeezed limit, 2 K 3 -K 6 has significant power just away from the squeezed and flattened configurations that differentiates it from the equilateral shape, and leads to it being most similar (though only mildly) with the orthogonal template. An added benefit of using the divergent basis and template classes to consider general shapes is that it ties together the methods we use to search for evidence of shapes with CMB data to LSS constraints from a scale-dependent halo bias, which probes the squeezed limits of shapes. It is well-known that templates for physical shapes which work for generating CMB predictions can fail when used for LSS predictions [45], because while CMB constraints represent a weighted average over all k -space configurations, the halo bias traces the squeezed limit region of k -space only. Thus our approach provides a way of generating templates that can potentially be used consistently for both CMB and LSS studies. We adopt a Fisher matrix approach modeled on a Planck-like survey to estimate uncertainties on the amplitudes of shapes within each shape class, r . As summarized in Table 4, we computed the uncertainties on shape attribution under each prior and how these uncertainties on confidently being able to determine that a template is the true shape can change substantially dependent upon the type of prior we impose. We find that the best measured shapes are those with the strongest divergence and with principal power near squeezed and flattened k -configurations. Though the conventional approach is to quote constraints at the equilateral configuration, k 1 = k 2 = k 3 , we show, as summarized in Table 5, that this convention can mask how well or badly a shape is measured, as doing so has the effect of re-normalizing constraints such that badly measured modes can appear to have constraints similar to the best measured mode. Using the PCA results, we map out the k -dependence of the constraints for a general shape given a prior, and show its dependence on the prior. For all but the r = 1 case, the best measured location is not in the equilateral configuration where shapes and constraints on f NL are typically normalized, but in a configuration that is neither squeezed, flattened, or equilateral, but somewhere in between. This best measured location at roughly k 1 /k 3 ≈ 0 . 32 and k 2 /k 3 ≈ 0 . 80 arises out of the complementary gradients of the power in the PC's. For the r = 1 case, the best measured location is weighted more strongly towards the squeezed configuration, reflecting that the signal and the noise, with which it is correlated, both go to zero in this limit. Given our parametrization of a general shape under a divergence prior, we then ask how well it could be constrained using measurements of amplitudes of common templates, like the local, equilateral, and orthogonal templates. We focus on the class of general shapes that can represent the possible range of single-field models that vanish in the squeezed limit ( r = 1 ). We calculate bounds on the subset of shapes that can remain consistent with constraints on the local, equilateral, and orthogonal templates, and find again-consistent with what we found earlier in the analysis-that templates with more power in the squeezed and flattened configurations provide more stringent constraints on this class of general shapes. Thus, the local, equilateral, and orthogonal templates serve different roles in constraining general shapes; the local template, if detected with sufficient amplitude, will rule out any shape of this type, while the equilateral and orthogonal templates serve to put constraints around different regions of the parameter space. In this sense, constraints from different templates can be complementary. Furthermore, a general (unknown) shape, will have different overlaps with the templates, creating a possibility that by combining constraints on templates, the overall constraint will shed more light on the underlying theory than any one constraint alone. We find it can also be advantageous to look for signals with a new, distinct template, beyond the three standard ones, that could help constrain models more efficiently; we explored the potential for using 2 K 3 -K 6 in this context. In this initial study we use somewhat idealized assumptions focusing on the effects of cosmic variance and Gaussian noise from a homogeneous sky coverage. We recognize the rich potential for further study to other basis sets, that better characterize sharp or oscillatory features in bispectra, the presence of isocurvature modes, and stronger deviations from scale-invariance. To confidently attribute a primordial source to any measured nonGaussianity one would also want to fully account for contributions from astrophysical and instrumental sources, including gravitational lensing, inhomogeneous sky coverage, and secondary anisotropies from astrophysical foregrounds. There is also the substantial question of how large-scale structure measurements, with sensitivity to the squeezed limit, can complement the CMB data in constraining these general shapes, as well as whether 4-point statistics and checks of non-Gaussian consistency ansatzes can play a role. We are tackling some of these intriguing issues in work in preparation. 5 Acknowledgements We would like to thank Nishant Agarwal, Xingang Chen, Tom Loredo, Liam McAllister, Sarah Shandera, and the anonymous referee for useful discussions during the preparation of this paper. JB and RB's research was supported by NSF CAREER grant AST0844825, NASA Astrophysics Theory Program grants NNX08AH27G and NNX11AI95G and by Research Corporation. References [1] C. Bennett, D. Larson, J. Weiland, N. Jarosik, G. Hinshaw, et al., Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Final Maps and Results , arXiv:1212.5225 . [2] G. Hinshaw, D. Larson, E. Komatsu, D. Spergel, C. Bennett, et al., Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Parameter Results , arXiv:1212.5226 . [3] K. Story, C. Reichardt, Z. Hou, R. Keisler, K. Aird, et al., A Measurement of the Cosmic Microwave Background Damping Tail from the 2500-square-degree SPT-SZ survey , arXiv:1210.7231 . [4] J. L. Sievers, R. A. Hlozek, M. R. Nolta, V. Acquaviva, G. E. Addison, et al., The Atacama Cosmology Telescope: Cosmological parameters from three seasons of data , arXiv:1301.0824 . [5] A. G. Sanchez, C. Scoccola, A. Ross, W. Percival, M. Manera, et al., The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological implications of the large-scale two-point correlation function , arXiv:1203.6616 . [6] A. J. Ross, W. J. Percival, A. Carnero, G.-b. Zhao, M. Manera, et al., The Clustering of Galaxies in SDSS-III DR9 Baryon Oscillation Spectroscopic Survey: Constraints on Primordial Non-Gaussianity , arXiv:1208.1491 . [7] C. Cheung, P. Creminelli, A. L. Fitzpatrick, J. Kaplan, and L. Senatore, The Effective Field Theory of Inflation , JHEP 0803 (2008) 014, [ arXiv:0709.0293 ]. [8] S. Weinberg, Effective Field Theory for Inflation , Phys.Rev. D77 (2008) 123541, [ arXiv:0804.4291 ]. [9] L. Senatore and M. Zaldarriaga, The Effective Field Theory of Multifield Inflation , JHEP 1204 (2012) 024, [ arXiv:1009.2093 ]. [10] N. Bartolo, M. Fasiello, S. Matarrese, and A. Riotto, Large non-Gaussianities in the Effective Field Theory Approach to Single-Field Inflation: the Bispectrum , JCAP 1008 (2010) 008, [ arXiv:1004.0893 ]. [11] J. Garriga and V. F. Mukhanov, Perturbations in k-inflation , Phys.Lett. B458 (1999) 219-225, [ hep-th/9904176 ]. [12] R. Bean, S. E. Shandera, S. Henry Tye, and J. Xu, Comparing brane inflation to WMAP , JCAP 0705 (2007) 004, [ hep-th/0702107 ]. [13] D. Polarski and A. A. Starobinsky, Isocurvature perturbations in multiple inflationary models , Phys.Rev. D50 (1994) 6123-6129, [ astro-ph/9404061 ]. [14] J. Garcia-Bellido and D. Wands, Metric perturbations in two field inflation , Phys.Rev. D53 (1996) 5437-5445, [ astro-ph/9511029 ]. [15] E. Komatsu and D. N. Spergel, Acoustic signatures in the primary microwave background bispectrum , Phys.Rev. D63 (2001) 063002, [ astro-ph/0005036 ]. [16] A. Gangui, F. Lucchin, S. Matarrese, and S. Mollerach, The Three point correlation function of the cosmic microwave background in inflationary models , Astrophys.J. 430 (1994) 447-457, [ astro-ph/9312033 ]. [17] L. Verde, L.-M. Wang, A. Heavens, and M. Kamionkowski, Large scale structure, the cosmic microwave background, and primordial non-gaussianity , Mon.Not.Roy.Astron.Soc. 313 (2000) L141-L147, [ astro-ph/9906301 ]. [18] P. Creminelli, A. Nicolis, L. Senatore, M. Tegmark, and M. Zaldarriaga, Limits on non-gaussianities from wmap data , JCAP 0605 (2006) 004, [ astro-ph/0509029 ]. [19] L. Senatore, K. M. Smith, and M. Zaldarriaga, Non-Gaussianities in Single Field Inflation and their Optimal Limits from the WMAP 5-year Data , JCAP 1001 (2010) 028, [ arXiv:0905.3746 ]. [20] X. Chen, M.-x. Huang, S. Kachru, and G. Shiu, Observational signatures and non-Gaussianities of general single field inflation , JCAP 0701 (2007) 002, [ hep-th/0605045 ]. [21] J. Khoury and F. Piazza, Rapidly-Varying Speed of Sound, Scale Invariance and Non-Gaussian Signatures , JCAP 0907 (2009) 026, [ arXiv:0811.3633 ]. [22] J. Noller and J. Magueijo, Non-Gaussianity in single field models without slow-roll , Phys.Rev. D83 (2011) 103511, [ arXiv:1102.0275 ]. [23] R. H. Ribeiro, Inflationary signatures of single-field models beyond slow-roll , JCAP 1205 (2012) 037, [ arXiv:1202.4453 ]. [24] J. Noller, Constraining fast-roll inflation , arXiv:1205.5796 . [25] X. Chen and Y. Wang, Large non-Gaussianities with Intermediate Shapes from Quasi-Single Field Inflation , Phys.Rev. D81 (2010) 063511, [ arXiv:0909.0496 ]. [26] X. Chen and Y. Wang, Quasi-Single Field Inflation and Non-Gaussianities , JCAP 1004 (2010) 027, [ arXiv:0911.3380 ]. [27] S. Gupta, A. Berera, A. Heavens, and S. Matarrese, Non-Gaussian signatures in the cosmic background radiation from warm inflation , Phys.Rev. D66 (2002) 043510, [ astro-ph/0205152 ]. [28] I. G. Moss and C. Xiong, Non-Gaussianity in fluctuations from warm inflation , JCAP 0704 (2007) 007, [ astro-ph/0701302 ]. [29] D. Lopez Nacir, R. A. Porto, L. Senatore, and M. Zaldarriaga, Dissipative effects in the Effective Field Theory of Inflation , JHEP 1201 (2012) 075, [ arXiv:1109.4192 ]. [30] R. Holman and A. J. Tolley, Enhanced Non-Gaussianity from Excited Initial States , JCAP 0805 (2008) 001, [ arXiv:0710.1302 ]. [31] P. D. Meerburg, J. P. van der Schaar, and P. S. Corasaniti, Signatures of Initial State Modifications on Bispectrum Statistics , JCAP 0905 (2009) 018, [ arXiv:0901.4044 ]. [32] N. Agarwal, R. Holman, A. J. Tolley, and J. Lin, Effective field theory and non-Gaussianity from general inflationary states , arXiv:1212.1172 . [33] D. Langlois, S. Renaux-Petel, D. A. Steer, and T. Tanaka, Primordial perturbations and non-Gaussianities in DBI and general multi-field inflation , Phys.Rev. D78 (2008) 063523, [ arXiv:0806.0336 ]. [34] F. Arroja, S. Mizuno, and K. Koyama, Non-gaussianity from the bispectrum in general multiple field inflation , JCAP 0808 (2008) 015, [ arXiv:0806.0619 ]. [35] S. Renaux-Petel, Combined local and equilateral non-Gaussianities from multifield DBI inflation , JCAP 0910 (2009) 012, [ arXiv:0907.2476 ]. [36] D. Langlois and A. Lepidi, General treatment of isocurvature perturbations and non-Gaussianities , JCAP 1101 (2011) 008, [ arXiv:1007.5498 ]. [37] D. Langlois and B. van Tent, Hunting for Isocurvature Modes in the CMB non-Gaussianities , Class.Quant.Grav. 28 (2011) 222001, [ arXiv:1104.2567 ]. [38] D. Langlois and B. van Tent, Isocurvature modes in the CMB bispectrum , JCAP 1207 (2012) 040, [ arXiv:1204.5042 ]. [39] N. Dalal, O. Dore, D. Huterer, and A. Shirokov, The imprints of primordial non-gaussianities on large-scale structure: scale dependent bias and abundance of virialized objects , Phys.Rev. D77 (2008) 123514, [ arXiv:0710.4560 ]. [40] M. LoVerde, A. Miller, S. Shandera, and L. Verde, Effects of Scale-Dependent Non-Gaussianity on Cosmological Structures , JCAP 0804 (2008) 014, [ arXiv:0711.4126 ]. [41] L. Verde and S. Matarrese, Detectability of the effect of Inflationary non-Gaussianity on halo bias , Astrophys.J. 706 (2009) L91-L95, [ arXiv:0909.3224 ]. [42] A. Becker, D. Huterer, and K. Kadota, Scale-Dependent Non-Gaussianity as a Generalization of the Local Model , JCAP 1101 (2011) 006, [ arXiv:1009.4189 ]. [43] A. Becker, D. Huterer, and K. Kadota, Constraining Scale-Dependent Non-Gaussianity with Future Large-Scale Structure and the CMB , JCAP 1212 (2012) 034, [ arXiv:1206.6165 ]. [44] J. Norena, L. Verde, G. Barenboim, and C. Bosch, Prospects for constraining the shape of non-Gaussianity with the scale-dependent bias , JCAP 1208 (2012) 019, [ arXiv:1204.6324 ]. [45] C. Wagner and L. Verde, N-body simulations with generic non-Gaussian initial conditions II: Halo bias , JCAP 1203 (2012) 002, [ arXiv:1102.3229 ]. [46] I. Agullo and S. Shandera, Large non-Gaussian Halo Bias from Single Field Inflation , JCAP 1209 (2012) 007, [ arXiv:1204.4409 ]. [47] WMAP Collaboration Collaboration, D. Spergel et al., Wilkinson Microwave Anisotropy Probe (WMAP) three year results: implications for cosmology , Astrophys.J.Suppl. 170 (2007) 377, [ astro-ph/0603449 ]. [48] R. Hlozek, J. Dunkley, G. Addison, J. W. Appel, J. R. Bond, et al., The Atacama Cosmology Telescope: a measurement of the primordial power spectrum , Astrophys.J. 749 (2012) 90, [ arXiv:1105.4887 ]. [49] S. Bridle, A. Lewis, J. Weller, and G. Efstathiou, Reconstructing the primordial power spectrum , Mon.Not.Roy.Astron.Soc. 342 (2003) L72, [ astro-ph/0302306 ]. [50] E. J. Copeland, E. W. Kolb, A. R. Liddle, and J. E. Lidsey, Reconstructing the inflaton potential: Perturbative reconstruction to second order , Phys.Rev. D49 (1994) 1840-1844, [ astro-ph/9308044 ]. [51] W. H. Kinney, E. W. Kolb, A. Melchiorri, and A. Riotto, WMAPping inflationary physics , Phys.Rev. D69 (2004) 103516, [ hep-ph/0305130 ]. [52] R. Bean, D. J. Chung, and G. Geshnizjani, Reconstructing a general inflationary action , Phys.Rev. D78 (2008) 023517, [ arXiv:0801.0742 ]. [53] P. Adshead and R. Easther, Constraining Inflation , JCAP 0810 (2008) 047, [ arXiv:0802.3898 ]. [54] J. Fergusson, M. Liguori, and E. Shellard, General CMB and Primordial Bispectrum Estimation I: Mode Expansion, Map-Making and Measures of f NL , Phys.Rev. D82 (2010) 023502, [ arXiv:0912.5516 ]. [55] P. D. Meerburg, Oscillations in the Primordial Bispectrum I: Mode Expansion , Phys.Rev. D82 (2010) 063517, [ arXiv:1006.2771 ]. [56] D. Regan, P. Mukherjee, and D. Seery, General CMB bispectrum analysis using wavelets and separable modes , arXiv:1302.5631 . [57] D. Regan, E. Shellard, and J. Fergusson, General CMB and Primordial Trispectrum Estimation , Phys.Rev. D82 (2010) 023520, [ arXiv:1004.2915 ]. [58] J. Fergusson, D. Regan, and E. Shellard, Optimal Trispectrum Estimators and WMAP Constraints , arXiv:1012.6039 . [59] J. Fergusson, D. Regan, and E. Shellard, Rapid Separable Analysis of Higher Order Correlators in Large Scale Structure , Phys.Rev. D86 (2012) 063511, [ arXiv:1008.1730 ]. [60] D. Regan, M. Schmittfull, E. Shellard, and J. Fergusson, Universal Non-Gaussian Initial Conditions for N-body Simulations , Phys.Rev. D86 (2012) 123524, [ arXiv:1108.3813 ]. [61] M. Schmittfull, D. Regan, and E. Shellard, Fast Estimation of Gravitational and Primordial Bispectra in Large Scale Structures , arXiv:1207.5678 . [62] A. Lewis, A. Challinor, and A. Lasenby, Efficient computation of CMB anisotropies in closed FRW models , Astrophys. J. 538 (2000) 473-476, [ astro-ph/9911177 ]. [63] U. Seljak and M. Zaldarriaga, A Line of sight integration approach to cosmic microwave background anisotropies , Astrophys.J. 469 (1996) 437-444, [ astro-ph/9603033 ]. [64] D. Babich, P. Creminelli, and M. Zaldarriaga, The Shape of non-Gaussianities , JCAP 0408 (2004) 009, [ astro-ph/0405356 ]. [65] T. Battefeld and J. Grieb, Anatomy of bispectra in general single-field inflation - modal expansions , JCAP 1112 (2011) 003, [ arXiv:1110.1369 ]. [66] X. Chen, Primordial Non-Gaussianities from Inflation Models , Adv.Astron. 2010 (2010) 638979, [ arXiv:1002.1416 ]. [67] N. Barnaby, On Features and Nongaussianity from Inflationary Particle Production , Phys.Rev. D82 (2010) 106009, [ arXiv:1006.4615 ]. [68] X. Chen, R. Easther, and E. A. Lim, Large Non-Gaussianities in Single Field Inflation , JCAP 0706 (2007) 023, [ astro-ph/0611645 ]. [69] X. Chen, R. Easther, and E. A. Lim, Generation and Characterization of Large Non-Gaussianities in Single Field Inflation , JCAP 0804 (2008) 010, [ arXiv:0801.3295 ]. [70] J. M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models , JHEP 0305 (2003) 013, [ astro-ph/0210603 ]. [71] J. Fergusson and E. Shellard, The shape of primordial non-Gaussianity and the CMB bispectrum , Phys.Rev. D80 (2009) 043510, [ arXiv:0812.3413 ]. [72] V. Acquaviva, N. Bartolo, S. Matarrese, and A. Riotto, Second order cosmological perturbations from inflation , Nucl.Phys. B667 (2003) 119-148, [ astro-ph/0209156 ]. [73] D. Wands, Multiple field inflation , Lect.Notes Phys. 738 (2008) 275-304, [ astro-ph/0702187 ]. [74] C. T. Byrnes and K.-Y. Choi, Review of local non-Gaussianity from multi-field inflation , Adv.Astron. 2010 (2010) 724525, [ arXiv:1002.3110 ]. [75] K. Enqvist, A. Jokinen, A. Mazumdar, T. Multamaki, and A. Vaihkonen, Non-Gaussianity from preheating , Phys.Rev.Lett. 94 (2005) 161301, [ astro-ph/0411394 ]. [76] D. Wands, Local non-Gaussianity from inflation , Class.Quant.Grav. 27 (2010) 124002, [ arXiv:1004.0818 ]. [77] E. Sefusatti, J. R. Fergusson, X. Chen, and E. Shellard, Effects and Detectability of Quasi-Single Field Inflation in the Large-Scale Structure and Cosmic Microwave Background , JCAP 1208 (2012) 033, [ arXiv:1204.6318 ]. [78] P. Creminelli, On non-Gaussianities in single-field inflation , JCAP 0310 (2003) 003, [ astro-ph/0306122 ]. [79] D. Seery and J. E. Lidsey, Primordial non-Gaussianities in single field inflation , JCAP 0506 (2005) 003, [ astro-ph/0503692 ]. [80] C. Armendariz-Picon, T. Damour, and V. F. Mukhanov, k - inflation , Phys.Lett. B458 (1999) 209-218, [ hep-th/9904075 ]. [81] M. Li, T. Wang, and Y. Wang, General Single Field Inflation with Large Positive Non-Gaussianity , JCAP 0803 (2008) 028, [ arXiv:0801.0040 ]. [82] N. Arkani-Hamed, P. Creminelli, S. Mukohyama, and M. Zaldarriaga, Ghost inflation , JCAP 0404 (2004) 001, [ hep-th/0312100 ]. [83] E. Silverstein and D. Tong, Scalar speed limits and cosmology: Acceleration from D-cceleration , Phys.Rev. D70 (2004) 103505, [ hep-th/0310221 ]. [84] M. Alishahiha, E. Silverstein, and D. Tong, DBI in the sky , Phys.Rev. D70 (2004) 123505, [ hep-th/0404084 ]. [85] J. Magueijo, J. Noller, and F. Piazza, Bimetric structure formation: non-Gaussian predictions , Phys.Rev. D82 (2010) 043521, [ arXiv:1006.3216 ]. [86] P. Creminelli, G. D'Amico, M. Musso, J. Norena, and E. Trincherini, Galilean symmetry in the effective theory of inflation: new shapes of non-Gaussianity , JCAP 1102 (2011) 006, [ arXiv:1011.3004 ]. [87] A. Nicolis, R. Rattazzi, and E. Trincherini, The Galileon as a local modification of gravity , Phys.Rev. D79 (2009) 064036, [ arXiv:0811.2197 ]. [88] C. Burrage, C. de Rham, D. Seery, and A. J. Tolley, Galileon inflation , JCAP 1101 (2011) 014, [ arXiv:1009.2497 ]. [89] R. H. Ribeiro and D. Seery, Decoding the bispectrum of single-field inflation , JCAP 1110 (2011) 027, [ arXiv:1108.3839 ]. [90] A. Berera, I. G. Moss, and R. O. Ramos, Warm Inflation and its Microphysical Basis , Rept.Prog.Phys. 72 (2009) 026901, [ arXiv:0808.1855 ]. [91] M. Liguori, E. Sefusatti, J. Fergusson, and E. Shellard, Primordial non-Gaussianity and Bispectrum Measurements in the Cosmic Microwave Background and Large-Scale Structure , Adv.Astron. 2010 (2010) 980523, [ arXiv:1001.4707 ]. [92] D. Babich and M. Zaldarriaga, Primordial bispectrum information from CMB polarization , Phys.Rev. D70 (2004) 083005, [ astro-ph/0408455 ]. [93] P. collaboration, Planck Collaboration Blue Book , http://www.sciops.esa.int/SA/PLANCK/docs/Bluebook-ESA-SCI%282005%291_V2.pdf . [94] E. Sefusatti, M. Liguori, A. P. Yadav, M. G. Jackson, and E. Pajer, Constraining Running Non-Gaussianity , JCAP 0912 (2009) 022, [ arXiv:0906.0232 ]. [95] M. Liguori and A. Riotto, Impact of Uncertainties in the Cosmological Parameters on the Measurement of Primordial non-Gaussianity , Phys.Rev. D78 (2008) 123004, [ arXiv:0808.3255 ]. [96] A. Lewis, A. Challinor, and D. Hanson, The shape of the CMB lensing bispectrum , JCAP 1103 (2011) 018, [ arXiv:1101.2234 ]. [97] R. Pearson, A. Lewis, and D. Regan, CMB lensing and primordial squeezed non-Gaussianity , JCAP 1203 (2012) 011, [ arXiv:1201.1010 ]. [98] K. M. Smith and M. Zaldarriaga, Algorithms for bispectra: Forecasting, optimal analysis, and simulation , Mon.Not.Roy.Astron.Soc. 417 (2011) 2-19, [ astro-ph/0612571 ]. y a y a equil s H f NL orth L S = H 1 -a x -a y L S equil + a X H S orth + 6 K 4 -6 K 3 L + a Y H S local -2 K 5 + 2 K 3 L s H f NL orth L S = H 1 -a x -a y L S equil + a X H S orth + 6 K 4 -6 K 3 L + a Y H S local -2 K 5 + 2 K 3 L 4 2 0 - 2 - 4 - 2 - 4 50 0 - 4 - 2 0 2 4 a x s H f NL 2 K 3 -K 6 L S = H 1 -a x -a y L S equil + a X H S orth + 6 K 4 -6 K 3 L + a Y H S local -2 K 5 + 2 K 3 L 4 2 0 200 0 - 4 - 2 0 2 4 a x S = H 1 -a 200 0 - 4 - 2 0 2 4 a x s H f NL orth L S = H 1 -a x -a y L S equil + a X H S local -2 K 5 + 2 K 3 L + a Y H 2 K 3 -K 6 L 50 0 - 4 - 2 0 2 4 a x s H f NL 2 K 3 -K 6 L S = H 1 -a x -a y L S equil + a X H S local -2 K 5 + 2 K 3 L + a Y H 2 K 3 -K 6 L 200 0 - 4 - 2 0 2 4 a x Figure 7 : Detection thresholds on the amplitude of templates, [top] f equil NL , [center] f orth NL , and [bottom] f 2 K 3 -K 6 NL , for distinguishing between each template and two forms of a general shape S at the 1 σ confidence level. The general shapes considered are [left panels] S gen = (1 -α X -α Y ) S equil + α X ( S ortho + 6 K 4 -6 K 3 ) + α Y ( S local -2 K 5 + 2 K 3 ) or [right panels] S gen = (1 -α X -α Y ) S equil + α X ( S local -2 K 5 +2 K 3 ) + α Y (2 K 3 -K 6 ) . Contours for f loc NL are not pictured, because the marginalized σ ( f loc NL ) remains close to its unmarginalized value over these 2-dimensional spaces. The case where S X = S ortho +6 K 4 -6 K 3 and S Y = 2 K 3 -K 6 is not pictured because S X and S Y are nearly uncorrelated, thus no additional information is revealed beyond that in Figure 6. y a y a y a - 2 - 4 - 2 - 4 - 2 - 4 4 2 0 4 2 0 4 2 0 x -a y L S equil + a X H S s H f local - 2 K equil 5 L NL + 2 K 3 L + a Y H 2 K 3 - K 6 L S local and S equil are the only shapes constructed from R = -1 modes that respectively have -1 and vanishing divergence. There are an infinite set of shapes, however, with constant divergence described by β K 0 +(1 -β )(2 K 1 -K 2 ) where β is free parameter which could take any value except β = -2 , for which the equilateral template is recovered. Instead of varying the parameter β , we could instead select a value of β to generate a template from the set. β = -8 corresponds to the orthogonal template chosen by [19] to maximize the resulting shape's orthogonality with S local and S equil . We could then choose to write general shapes in terms of linear combinations of { S equil , S ortho , S local } , rather than α n K n , If these are the only shapes being used, the normalization constraint S [ R, -r ] ( k 0 , k 0 , k 0 ) = 1 fixes one α coefficient. We can extend this approach to include basis modes that diverge as x sq -2 , To tie this general approach to specific shapes in the literature, S ortho (2) and S enf (2) can be written in this form by the following choice of coefficients: The inclusion of extra basis shapes can be particularly important when the shape has undulations and is not just a smooth monotonic function. Shapes arising out of Galileon inflation are a good example of this. Imposing a Galilean symmetry on a single-field inflation model [86-89] gives rise to a non-Gaussian shape generated by three cubic interaction terms in the inflaton Lagrangian. While the shapes associated with each of these three operators, individually, are well-approximated by S equil and S enf (2) , there exist combinations of them for which the resulting Galileon shape has little overlap with any of the shapes we have mentioned so far. Non-separable templates for Galileon inflation have been developed in [86] and [89] which have high cosines both with the underlying shape and each other. For illustrative purposes, we consider the shape presented in [86], based on equations (26)-(28) of this reference. When we use the Gram-Schmidt decomposition to construct a template with only the first three modes, we find a poor fit with a cumulative cosine of only 0.13. The Galileon shape derives from a single-field action and a Bunch-Davies vacuum so theoretical consistency requires that it vanishes in the squeezed limit. Motivated by this, if we fit the Galileon model using the 4 shapes in S [ -2 , 1] , we obtain a template with a cosine of 0.93. This reconstruction is not improved if we allow an unconstrained combination of the seven K 0 -K 6 modes. We can extend our approach to R = -3 modes, and for example consider the following general shape that vanishes in the squeezed limit: Fitting these eight distinct shapes in S [ -3 , 1] to the Galileon shape, we obtain an improved template with cosine of 0.99. The second reason to consider a basis including more divergent terms is that some inflationary scenarios, such as excited initial states and warm inflation, in which inflation occurs in a warm radiation bath [27-29] (see [90] for a review), can give rise to shapes that are more divergent than the local shape, with an overall divergence of x sq -2 . This would suggest using an unconstrained combination of K 0 -K 6 modes, or using constrained combinations of the R = -3 modes for which the x -3 sq divergent term vanishes. One such example of this is a template for warm inflation proposed by [65], The realization that the differences between similar shapes can be important and provide an additional insight into the underlying model, implies that we should not just compare a small number of templates to the data. It is reasonable to extend beyond this and create more refined templates, sensitive to more than just properties that models have in common with the equilateral, orthogonal, and local templates.", "pages": [ 12, 13, 14, 15 ] }, { "title": "2.5 Shape classification and depiction", "content": "The models discussed in the previous section reflect only a sample of the wide range of non-Gaussian inflationary shapes arising in the literature. Putting a coarse filter on their properties, one might characterize them using three descriptors: i) their divergence in the squeezed limit, ii) how many modes it takes to accurately describe them, and iii) the 'family' to which they belong. Many of the physical shapes tend to be grouped in terms of a 'family' resemblance to an existing template, reflecting the type of configurations of triangles with side lengths k 1 , k 2 , and k 3 where the shapes have most of their power [71, 91]. For scale invariant shapes this is equivalent to studying the distribution of power over the space { k 1 k 3 , k 2 k 3 } for a fixed k 3 > k 1 , k 2 . This space can be pictorially represented by a triangle with sides 0 ≤ k 1 k 3 ≤ 1 and 1 / 2 ≤ k 2 k 3 ≤ 1 . We introduce it here in the context of the shapes already discussed, because we use this format to present some of our forecasting results. In Figure 2 we show examples of the shapes discussed in the previous section. S const = K 0 is the archetypal component of a family with similar power over all scales, homogeneous over the whole triangular region plotted. 'Squeezed' shapes have a bispectrum amplitude that is peaked in the top left-hand corner of the plot where k 1 k 3 glyph[lessmuch] 1 and k 2 k 3 = 1 , while 'equilateral' type shapes peak in the top right-hand corner where k 1 k 3 = k 2 k 3 = 1 . 'Flattened' shapes peak along the left edge, where k 1 k 3 + k 2 k 3 = 1 . Of the shapes we've discussed so far, some clearly fall within these family categories: S local , S warm , S 4 and S 5 are 'squeezed' shapes, while S equil , S DBI , S single are 'equilateral' and S enf is 'flattened'. 3 k GLYPH<144> 2 k 3 k GLYPH<144> 2 k 1.0 0.9 0.8 0.7 0.6 0.5 1.0 0.9 0.8 0.7 0.6 0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 ortho (g) S 0.2 0.4 0.6 0.8 1.0 (a) S Local k 1 GLYPH<144> k 3 local S orth + 6 K k 1 GLYPH<144> k +6 K 4 - 6 K 3 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 (h) S local (b) S Quasi - single H n= 1.3 L k 1 QSFI S loc - 2 K k 1 - 2 K 5 GLYPH<144> k ( ν = 1 . 3) 5 + 2 K 3 3 +2 K 3 (f) S (i) 2 K 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 (c) S Folded k 1 GLYPH<144> k 3 enf Galilean k 1 GLYPH<144> k 3 Galileon 2 K k 3 1 3 - K GLYPH<144> k 3 -K 6 6 GLYPH<144> k 3 (e) S 4 - 6 K 3 3 10 0 3 - 3 3 k GLYPH<144> 2 k 3 k GLYPH<144> 2 k 3 k GLYPH<144> 2 k 1.0 0.9 0.8 0.7 0.6 0.5 1.0 0.9 0.8 0.7 0.6 0.5 1.0 0.9 0.8 0.7 0.6 0.5 3 - 3 10 0 3 - 3 Equilateral k 1 GLYPH<144> k 3 equil 3 k GLYPH<144> 2 k 3 k GLYPH<144> 2 k 3 k GLYPH<144> 2 k 1.0 0.9 0.8 0.7 0.6 0.5 1.0 0.9 0.8 0.7 0.6 0.5 1.0 0.9 0.8 0.7 0.6 0.5 3 - 3 3 - 3 3 - 3 There exist other additional shapes generated by modes K 3 through K 7 . For example, Figure 2 also includes three shapes that contribute to S [ -2 , 1] that could describe a general single-field model with Bunch Davies vacuum. While each vanishes in the squeezed limit by construction, we find they differ from the equilateral shape in still having a component of their signal focused along the flattened configuration. The comparative size of this component correlates with the divergence of the shapes from which they are created, S ortho , S local , and K 3 . There are shapes that do not fall clearly into any of these families: S ortho (2) peaks in both the flattened and equilateral configurations, excited states can peak in squeezed and flattened configurations, and S QSFI shapes interpolate between constant and local properties. Beyond this there are shapes with distinct undulating forms, the S Galileon shape for example, that do not peak at either edges or corners. Moreover, not all shapes within each family are alike. For example, the local and warm shapes both peak in squeezed configurations, but their divergence properties in this region are different, leading to a low cosine between them. Given the breadth of bispectrum shapes that could be created, and the comparatively loose characteristics on which 'families' are formed, there is strong motivation to ask how much information we can discern observationally about bispectra. This will help quantify how well we might determine the underlying non-Gaussian shape, if a detection of non-Gaussianity is made.", "pages": [ 15, 16, 17 ] }, { "title": "3 Forecasting constraints on general shapes", "content": "In the following analysis, we apply the separable, divergent basis and template classes from the previous section to assess how we can constrain the shape of primordial bispectra with upcoming CMB data. Our goal is to quantify what properties of shapes are measurable, and the respective roles of the experimental uncertainties and theoretical priors on determining distinguishability. Motivated by a broad cross-section of models in the literature, we will focus on shapes described by basis functions {K 0 -K 6 } that are nearly scale-invariant and contain terms that are potentially as divergent as x -2 sq in the squeezed limit. We describe the Fisher matrix approach we use assuming a Planck-like CMB experiment in section 3.1. In section 3.2, we present the results of a principal component analysis for the set of shapes S [ -2 ,r ] with different divergence criteria in the squeezed limit imposed. Doing so generates the experiment's preferred orthogonal basis of best to worst measured bispectrum configurations, the principal components (PCs) and their corresponding uncertainties, subject to the theoretical prior. We consider the implications for shape normalization and the bestmeasured k -configuration in sections 3.3 and 3.4, respectively, and finish in section 3.5 by quantifying our potential ability to determine and distinguish shapes.", "pages": [ 17 ] }, { "title": "3.1 Fisher matrix approach", "content": "We compute the 7 × 7 Fisher matrix for the amplitudes of the basis modes, K n , { f n NL , n = 0 , ..., 6 } defined in Eq. (2.19) as where { abc } and { xyz } each sum over the 8 possible temperature ( T ) and polarization ( E ) combinations of bispectra: TTT,TTE,TET,ETT,TEE,ETE,EET,EEE . Given a general primordial shape expanded on the {K n } basis as in (2.19), the corresponding CMB reduced bispectrum is where K ( n ) glyph[lscript] 1 glyph[lscript] 2 glyph[lscript] 3 denotes the reduced bispectrum of the basis function K n in (2.20)-(2.21). Here we compute it as where p ' is defined as in (2.21). We have modified the CAMB 3 code [62] to numerically evaluate the values of I ap glyph[lscript] 1 and then written code to appropriately combine them to form each K ( n ) glyph[lscript] 1 glyph[lscript] 2 glyph[lscript] 3 . Specifically, we take k min = 6 . 6 × 10 -6 Mpc -1 , k max = 0 . 56 Mpc -1 , and x max = 16 . 5 × 10 3 Mpc. We include a note of caution that since the integrals over k and x for cases where p is very negative (positive) depend on having accurate transfer functions at small (large) values of k , numerical results for these integrations should be carefully checked for robustness. To verify the numerical robustness of our results, we have checked that I ap glyph[lscript] ( x ) obtained numerically for p < 0 match the expected analytic result in the Sachs-Wolfe limit. We have also quantified how the Fisher matrix results quoted in the next section are robust or exhibit instabilities to changes in the accuracy boost parameter in CAMB, which allows for fine resolution in the k and x integrals. In particular, for the most divergent K 6 mode, which is a combination of the most extreme integrals (with p = -2 and 4) and thus we would expect to have the greatest amount of numerical error, we find that the Fisher results quoted in the next section changed by less than 0 . 01% when the accuracy boost was increased from 1.5 to 2. However, we find that the worst measured eigenmode, in the PCA, is far more sensitive to the integral resolution. We find with an accuracy boost of around 2 we get convergence of a few percent in all but the worst measured mode. This last mode oscillates with a variation of around 15% in the standard deviation. This sensitivity in the worst measured mode (which we will find is the least divergent shape in the squeezed limit), can affect the constraints for shapes which have a component described by this mode. In the following sections, we present our results with these cautions attached when appropriate. The covariance matrix we use from [92] is with Here f sky is the overall fraction of the sky observed, and we assume f sky = 0 . 8 . N ax glyph[lscript] is the instrument noise for a correlation between observables a and x . We model CMB noise by considering the three lowest frequency bands of the Planck HFI instrument for temperature and E-mode polarization, as described in the Planck Bluebook [93]. We assume each frequency channel has Gaussian beam profile of width θ FWHM and isotropic noise with error in X = T, E of σ X . The noise in each frequency channel c is then given by Our fiducial flat Λ CDM cosmology is described by the following parameters, which are consistent with the latest WMAP 9-year constraints [2]: Ω b h 2 = 0 . 02258 , Ω c h 2 = 0 . 1109 , ∆ 2 R ( k 0 ) = 2 . 43 × 10 -9 , n s = 0 . 963 , and τ = 0 . 088 . As has been done in other recent Fisher forecasts on non-Gaussianity parameters, such as [94], we consider the uncertainties on the non-Gaussian amplitudes independent of the uncertainties in the fundamental cosmological parameters that also affect the power spectrum, as these are comparatively small relative to the uncertainties from the bispectrum shape functions [95]. For this initial analysis, we neglect the effect of imperfect measurements of the lensing signal [96, 97], secondary anisotropies [15], and inhomogeneous sky coverage/noise on the constraints (e.g. [54, 98]).", "pages": [ 17, 18, 19 ] }, { "title": "3.2 Fisher matrix results", "content": "A general bispectrum can be expanded in terms of either K n or the component shapes, { S X } , in S [ R,r ] , given in (2.33)-(2.38), While the Fisher matrix we used based on S [ R,r ] automatically includes the additional priors to constrain the divergence properties, these could also be introduced into the K n Fisher analysis by using Lagrange multipliers to systematically impose each divergence constraint. The latter makes no assumption a priori about what linear combinations of the shapes given in Table 2 should have their amplitudes constrained. While we use the shape expansion in our discussion below, we investigated both approaches and found they led to consistent conclusions. We use the Fisher matrix in terms of K n to construct Fisher matrices for the component shapes in S [ -2 ,r ] for r = -2 , -1 , 0 , 1 . In Table 3 we give the glyph[lscript] -space correlation coefficients based on (2.11), but here weighted by the data covariance between pairs of the component shapes, S X and S Y , This gives a measure of the similarity of the component shapes based on how they are measured by the survey, integrated over all glyph[lscript] combinations. We find the similarity between pairs of the four basis shapes in S [ -2 , 1] , each of which vanishes in the squeezed limit, are primarily related to the divergence of the shapes from which they are derived. S equil and S ortho +6 K 4 -6 K 3 are very similar to each other, while S local -2 K 5 +2 K 3 and 2 K 3 -K 6 also have a high degree of overlap. Interestingly the S local -2 K 5 +2 K 3 and 2 K 3 -K 6 shapes also have significant similarities with the shapes that diverge as x 0 sq . This is derived from their strong signal along the configurations between squeezed and flattened configurations, as discussed in section 2.5. The shape with x -1 sq divergence constructed from the R = -2 modes, 2 K 5 -K 6 , is highly degenerate with the local template; essentially this implies the two are indistinguishable from one another using the CMB data. √ The unmarginalized errors, σ ( f X NL ) = 1 / F XX , give the uncertainty in the measurement of a specific template if the underlying theory is known to be wholly described by that template. We find these are comparatively insensitive to the integral resolution discussed in section 3.1. The covariance matrices obtained from inverting the Fisher matrices give the uncertainties on the amplitudes of the component shapes, σ ( f NL ) , marginalized over the freedom allowed by each model. The marginalization does make the results precision dependent in the worst measured mode, i.e. the results are accurate to better than 15%. We summarize the results in Table 4. The covariance matrix in each case can be diagonalized to obtain the orthonormal eigenvectors, and associated eigenvalues, which give the variances σ 2 ( b i ) in the amplitudes of the eigenvectors. These then provide a way to rank the best to worst measured bispectra. Given this orthonormal basis, any general bispectrum may be expanded as The principal components obtained by diagonalizing the covariance matrix are not immediately 'shapes' in the way we considered so far. They have unit norm with respect to the component shape basis, ∑ X | c iX | 2 = 1 , rather than being normalized at the equilateral configuration, ∑ X c iX S X ( k 0 , k 0 , k 0 ) = 1 . If we restrict the shapes to those described by the first three modes, marginalization does not significantly alter the constraints from the unmarginalized errors, i.e. the three common templates are essentially the principal components (PC) of the covariance matrix, with the eigenvalues showing that the more divergent the shape, the better it is measured. In contrast, when extended to general shapes, constructed of all seven modes, we find marginalized errors for individual shapes are far larger because of observational similarities between shapes of similar divergence, or similar properties in the flattened limit. It seems that only K 6 is well constrained if any shape from the S [ -2 ,r ] type is allowed. When extended to shapes constructed of seven modes, the correspondence between the PC's and divergence remains. We find that, in general, divergence in the squeezed limit, followed by a second divergence measure, corresponding to the signal near the flattened configurations, can be used as coarse indicators of comparative constraining power with the CMB. For the general shape without any additional divergence constraints, the best measured PC is almost completely composed of the most divergent shape, K 6 . The second best measured PC has dominant contributions from S local and 2 K 5 -K 6 with which it is very degenerate. If the general shape is restricted to have vanishing divergence in the squeezed limit, then the best measured PC is very similar to a shape like 2 K 3 -K 6 which has large signal in the flattened configurations despite vanishing in the squeezed limit. The next best measured PC is then similar to shapes like equilateral or the orthogonal-derived shape S ortho +6 K 4 -6 K 3 , which has less power on flattened configurations. In both cases, none of the templates look like the two worst measured modes, which exhibit large oscillatory features along flattened configurations.", "pages": [ 19, 20, 21 ] }, { "title": "3.3 Drawbacks of normalization at the equilateral configuration", "content": "As stated earlier, the PC's as they are originally generated, are not shapes in the usual sense because they are not bispectra normalized at k 1 = k 2 = k 3 = k 0 . They have a unit norm in terms of the basis shapes. With this normalization, as usual in PCA, their eigenvalues quantify which combinations of the basis shapes are best and worst constrained by data, and the eigenvectors can be combined to create general shapes. We can convert σ ( b i ) to an effective σ ( f NL (ˆ e i )) , corresponding to the amplitude of each eigenvector shape normalized in the conventional way, σ ( f NL (ˆ e i )) = | σ ( b i )ˆ e i ( k 0 , k 0 , k 0 ) | . Table 5 gives the values of σ ( b i ) and σ ( f NL (ˆ e i )) . We quote the results when both temperature and polarization data are included. We find that the exclusion of the E-mode polarization from the Fisher analysis does not noticeably change the shape of the principal components, but does increase the eigenvalues by about a factor of ∼ 1 -3 across all eigenvectors. The constraints on all but the last eigenvalue under each divergence constraint shown in Table 5 are accurate to a few percent. The worst measured eigenmode is measured to ∼ 15% accuracy. Normalizing our PC's at the arbitrarily chosen equilateral configuration allows us to compare them to other shapes consistently at one point in k -space. σ ( f NL ) does not in itself, however, quantify a shape's overall variance across all k . An analogous situation arises in quoting uncertainties on the power spectrum amplitude from two different surveys, say a large-scale CMB survey and a galaxy survey. Both surveys could quote uncertainties at a common arbitrary scale, say k 0 = 0 . 05 h/Mpc , but while this uncertainty might represent the best measured scale for the galaxy survey, it would grossly overstate the minimum uncertainty in the CMB survey, which is best measured at a much larger scale. It is entirely possible for a well measured mode to have a significant part of its small variance located in the equilateral configuration, while a poorly measured mode could have its lowest variance in the equilateral configuration but be poorly measured over other regions of k -space. Indeed we find this to be the case, given that the best measured shapes have signal peaked near the squeezed, rather than equilateral, configuration. This means that σ ( f NL ) is not a useful measure in itself to assess how well a shape can be measured. This shortcoming of the conventional normalization has been discussed previously in other studies, e.g. [18] and [54], where alternative normalization schemes based on an integrated total amount of non-Gaussianity have been proposed. The overall spread in uncertainties from the best to worst eigenvector is much reduced when normalized at the equilateral configuration and can in some cases produce a switch in the ordering of the modes for σ ( f NL ) relative to that of σ ( b i ) . This does not present an inconsistency in the analysis, but simply demonstrates the perils of considering a normalization at an arbitrary scale. Figure 3 shows the variety of profiles in the 2-dimensional ( k 1 k 3 , k 2 k 3 ) space shown in the triangle plots. Given that the power spectrum we consider is not perfectly scale invariant, there is some small dependency of the bispectrum amplitude on the value of k 3 , described by p ' in (2.21). The spatial profiles, however, in terms of k 1 k 3 and k 2 k 3 are k 3 -independent. The gradients in the PC configurations reflect the rough ordering from squeezed to flattened to equilateral as the modes span from best to worst. The complementarity of the eigenvectors, reflected by the different directions of gradients of the signals in the configuration space, has implications for the location of the best measured configuration, as we discuss in section 3.4.", "pages": [ 22, 23, 24 ] }, { "title": "3.4 Best measured k -configurations", "content": "In the analysis that follows, we avoid splitting up bispectra into shapes and amplitudes, normalized at an arbitrary configuration. Instead we consider the overall constraints on the bispectrum, B ( k 1 , k 2 , k 3 ) , itself up to the constant normalization, given in (2.2), f NL S = k 2 1 k 2 2 k 2 3 B ( k 1 , k 2 , k 3 ) /N . The eigenmodes and eigenvalues from the PCA provide a way to compute an error on a general k -space bispectrum. We can calculate the posterior distribution of the uncertainties on f NL S given the data, D , with a theoretical prior given by the eigenvectors { ˆ e i } , Under this assumption of Gaussian errors this gives the commonly used result, This equation for computing the error can be applied to each set of PC's generated for each divergence scenario in the previous section. The errors in the ( k 1 k 3 , k 2 k 3 ) configuration space can be plotted and the best measured k -configuration, and the associated uncertainty, calculated for each scenario. σ ( f NL S ) varies only very weakly across slices in k 3 ; its functional form can be divided into a dependence on ( k 1 k 3 , k 2 k 3 ) and a weak dependence on k 3 , going as k 2( n s -1) 3 , for fixed k 1 k 3 and k 2 k 3 . For our choice of theoretical priors on the model, σ ( f NL S ) decreases with increasing k 3 . This is because the noise scales as the signal for the near scale-invariant theoretical prior we impose. An alternative prior would give very different dependencies on k 3 . For example if we were to remove the theoretical prior all together and model the bispectrum amplitude as bins in k , the only constraints on the model come from the observational uncertainties, and the noise would diverge exponentially on small scales. 3 k GLYPH<144> 2 k 1.0 0.9 0.8 0.7 0.6 0.5 0.0 3 - 3 0.2 0.4 0.6 0.8 1.0 PC 1 k (a) ˆ 1 GLYPH<144> k e 1 PC 2 k (b) ˆ 1 GLYPH<144> k e 2 PC 4 k (d) ˆ 1 GLYPH<144> k e 4 The weak k 3 dependence implies that the uncertainties at one k 3 reasonably reflect the overall uncertainties if one were to marginalize over k 3 . Figure 5 shows the error on the k 3 = 0 . 01 Mpc -1 slice for three different divergence cases. The location of minimum σ ( f NL S ) comes from the sum of the eigenmodes that is weighted by each mode's error, which arises out of the complementarity of the degeneracy directions of the PC's. We find the location of the best measured configuration is consistent for the scenarios that diverge as x -2 sq through to 3 3 3 3 k GLYPH<144> 2 k 3 k GLYPH<144> 2 k 1.0 0.9 0.8 0.7 0.6 0.5 1.0 0.9 0.8 0.7 0.6 0.5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 3 - 3 3 - 3 a constant in the squeezed limit for R = -2 . This location is not situated in any one of the corners of the triangle plot associated with squeezed, equilateral, and flattened configurations. Instead it is somewhat centrally located adjacent to the flattened edge. The best-measured configurations are located at k 1 k 3 ≈ 0 . 32 and k 2 k 3 ≈ 0 . 80 , with minimum σ ( f NL S ) ≈ 37 . For the vanishing divergence prior, the best-measured configuration approaches the squeezed limit, as we have required the noise to scale as the shapes, which go to zero there. We also find that for shapes constructed from the local, equilateral, and orthogonal templates ( R = -1 ), the best measured location spans a degeneracy direction also along the flattened edge, with minimum σ ( f NL S ) ≈ 20 . While the best measured region, in which the error is a minimum, is useful in the absence of knowledge about the theory, the signal-to-noise ratio, for a given underlying model can also help determine a survey's ability to distinguish between shapes. We consider this in the following subsection.", "pages": [ 24, 25, 26 ] }, { "title": "3.5 Shape determination and distinguishability", "content": "Wenow turn to discussing a central question of the paper: given a detection of non-Gaussianity using a specific template, what can be confidently inferred about the true underlying shape? We have already considered this from one perspective in section 3.2 by considering the uncertainties in ascribing a detection using a template to the template's shape itself. If we allow for the possibility that a detection using a specific template could be detecting the component of another shape allowed by the theoretical prior we are considering, then the errors on shape determination can increase significantly, especially for shapes that do not peak in the squeezed or flattened configurations. In this section we approach the question of shape distinguishability from a second direction, considering the range of possible general shapes, under a divergence prior, that could create the detected template signal and fit the bispectrum data within some confidence range. Such analyses have already been considered in the context of specific models, for example, how well we might disentangle a QSFI model (e.g. in (2.29)) from S equil or S local as a function of ν [44, 77]. Here we extend this approach to a more general shape, and consider what implications a detection with one of the common templates has for general models. For specificity we consider a subset of general shapes consistent with S [ -2 , 1] , where S X,Y,Z can be { S ortho +6 K 4 -6 K 3 , S local -2 K 5 +2 K 3 , 2 K 3 -K 6 } . This is investigating a general set of single-field inflation models from which S ortho (2) in (2.39) and S enf (2) in (2.40) are drawn. How large must a template signal f T NL be to be confident that the signal is not from a different, more general shape S gen ? We set this distinguishable detection threshold to be σ ( f T NL ) , the error on f T NL for the template, marginalized over f S NL , the amplitude of the general shape. The marginalized constraint is computed by inverting the 2 × 2 Fisher matrix for ( f T NL , f S NL ) . Thus we are comparing two shapes, where one is a template, and the other is a general shape, in which α X , α Y , and α Z parametrize the deviation from S equil . In the simplest case, we allow only α X to be non-zero, such that S gen is a linear combination of two shapes, S equil and S X , that varies with one parameter. In Figure 6, we show σ ( f T NL ) for the local, equilateral, and orthogonal templates when S X takes different forms. The minimum value of σ ( f T NL ) for each template across all values of α recovers the (a) Diverges as 1 /k 2 in the squeezed limit. (b) Vanishing divergence in the squeezed limit. (c) Diverges as 1 /k in the squeezed limit. glyph[negationslash] unmarginalized errors of each. A detected value of f equil NL must be larger to produce a 1 σ detection of the equilateral shape, as opposed to a more general shape with α = 0 , while f local NL never has to be much larger than the unmarginalized σ ( f local NL ) to favor the local model over this general shape, because S X and S local are weakly correlated. To illustrate the use of Figure 6, for example, a detection of f orth NL = 40 , while greater than the unmarginalized error of 19, would only be sufficient to rule out a false 1 σ detection of S gen with S X = 2 K 3 -K 6 for -5 glyph[lessorsimilar] α glyph[lessorsimilar] 0 . 9 . On the other hand, if f orth NL is detected to be larger than 46, then S gen of the 1-parameter form would be disfavored, as σ ( f orth NL ) is smaller than this over all values of α . Models with S X = 2 K 3 -K 6 are most easily differentiated from S equil because they have the lowest correlation with S equil . An application of comparing constraints on α X from two distinct templates is to test whether a given model is consistent with or disfavored by the data. If two template measurements each individually remain consistent with two non-overlapping regions of α -space, then it would be clear that modeling the underlying shape with α alone is not able to produce a viable model. This would be true for dual measurements of { f equil NL = 60 , f ortho NL = 45 } for S X = 2 K 3 -K 6 , since they would imply non-overlapping ranges of α , -0 . 7 ≤ α ≤ 0 . 3 versus 1 . 3 ≤ α ≤ 3 . 8 to each be consistent with the data. We can extend the same analysis to a comparison between templates and a 2-parameter general shape by allowing both α X and α Y to vary simultaneously, while α Z is fixed to zero. For example, S enf (2) is a specific template for which this is true. In Figure 7 we show σ ( f equil NL ) and σ ( f orth NL ) over different choices of the 2-dimensional space and find that there exist degeneracy directions that are not fully captured by the 1-dimensional projections in Figure 6. We find that σ ( f local NL ) remains close to the unmarginalized value in this case as well. In the most general 3-parameter model, we can ask the question of whether there is any area of this space corresponding to a general model that vanishes in the squeezed limit, with a significant enough overlap with the local template to require that a potentially detected f local NL be much greater than the unmarginalized value of 3. If this were the case, then it may be that a local template detection cannot definitively rule out a general shape that satisfies the single-field consistency relation. However, we find that nowhere in the parameter space does the σ ( f loc NL ) become greater than 4.2, showing that a detection of the local template above this threshold would effectively rule out a general shape, vanishing in the squeezed limit, subject to the assumption that it can be written in terms of our basis in S [ -2 , 1] . The same distinguishing power is not present for S local if we allow a weaker prior given by S [ -2 , -1] . In this case the significant cosine between S local and 2 K 5 -K 6 , means we may never be able to confidently attribute a detection with S local to be definitive evidence that the diverging signal is unambiguously S local . A long shot could be to additionally look at the correlation of the bispectrum signal with 2 K 4 -K 6 which is mildly negatively correlated with S local and essentially uncorrelated with 2 K 5 -K 6 . This last point raises an interesting application of our study: to ask if there are distinct, new templates that we might use to learn about the origins of a detected non-Gaussian signal. In the context of models described by the first three modes, K 0 to K 2 , the local, equilateral, and orthogonal templates are almost perfectly aligned with the principal components. If we extend the templates to include K 3 through K 6 , however, we find these no longer represent the PC's. For example, what might be the best way to extend the template pool to search for signatures of single-field inflation models with Bunch-Davies vacua? In the context of r = 1 shapes, 2 K 3 -K 6 is well-aligned with the best measured PC and is only mildly correlated with the existing templates which would make it a reasonable candidate to add as an additional template. We show the resulting constraints on general shapes in Figures 6 and 7. The figures show that this template probes regions of the allowed α -space which the equilateral and orthogonal templates do not constrain in the same way. Thus it may be possible to combine constraints from the common templates and motivated choices of a small number of new templates, like 2 K 3 -K 6 , to probe the underlying shape of non-Gaussianity.", "pages": [ 26, 27, 28 ] }, { "title": "4 Conclusion", "content": "At the heart of this work is the discussion about how uncertainties quoted on shape detection are inherently dependent on the underlying assumptions made about the shape. While a detection of non-Gaussianity with any template will be extraordinarily transformative in our field, its interpretation, in what it tells us about the underlying shape, has to be considered carefully in terms of our underlying theoretical prior we impose. Even if no detection of nonGaussianity is made, upper bounds on the deviations from Gaussianity according to templates will have broader impacts for constraints on general shapes. We have presented an approach for quantifying how well upcoming CMB temperature and E -mode polarization data can determine the shape of primordial non-Gaussianity under minimal assumptions. We proposed a set of polynomial divergent basis functions, { K n } , that are well-tuned to describing many nearly scale-invariant, smoothly varying, but potentially divergent shapes discussed in the literature. We find we need only three to seven modes to generate matched templates to describe a wide range of physically motivated shapes. In this sense, the divergent basis is more efficient than the polynomial basis used in previous studies (e.g. [54]). Each K n in our basis is generally divergent, but linear combinations of the K n can be constructed to have cancellations in the squeezed limit, thus creating templates that are less divergent (e.g. equilateral shape). For example, S equil and (2 K 3 -K 6 ) both vanish in the squeezed limit, but still have a low correlation because the latter has more power near the flattened and squeezed configurations. Using the {K n } it is straightforward to form template classes, S [ R,r ] , that have specific, common divergence properties in the squeezed limit. Each class is constructed from an irreducible set of shapes, that while constructed out of a basis sets with maximum divergence x R sq ( R < 0 ), through cancellations of divergent terms, have squeezed limit x r sq ( r > R ) . The choice of R controls how many basis modes are used to develop the templates, e.g. R = -1 includes K 0 through K 2 , while R = -2 uses K 0 through K 6 . As R becomes more negative it allows templates to be refined and shapes with a broader set of features across configuration space to be modeled. The classes allow templates to be developed with priors that are well-motivated by theories: S [ R, 1] represents the class of all single-field models derived from a Bunch-Davies vacuum, S [ R, -1] in addition includes all multi-field models that diverge like the local shape in the squeezed limit, and S [ R, -2] is the most general class of shapes which includes models from non-Bunch-Davies vacuum initial states. While the constituent shapes making up each class have the same divergence properties in the squeezed limit, away from this limit they have power weighted differently in the configuration space. For example, we discuss a new shape, 2 K 3 - K 6 , used in S [ R, 1] , that has the same squeezed limit behavior as the equilateral shape but has glyph[lscript] -space cosines with the standard equilateral, orthogonal, and local templates of 0.07, 0.80 and -0.29 respectively. While the divergent terms cancel in the squeezed limit, 2 K 3 -K 6 has significant power just away from the squeezed and flattened configurations that differentiates it from the equilateral shape, and leads to it being most similar (though only mildly) with the orthogonal template. An added benefit of using the divergent basis and template classes to consider general shapes is that it ties together the methods we use to search for evidence of shapes with CMB data to LSS constraints from a scale-dependent halo bias, which probes the squeezed limits of shapes. It is well-known that templates for physical shapes which work for generating CMB predictions can fail when used for LSS predictions [45], because while CMB constraints represent a weighted average over all k -space configurations, the halo bias traces the squeezed limit region of k -space only. Thus our approach provides a way of generating templates that can potentially be used consistently for both CMB and LSS studies. We adopt a Fisher matrix approach modeled on a Planck-like survey to estimate uncertainties on the amplitudes of shapes within each shape class, r . As summarized in Table 4, we computed the uncertainties on shape attribution under each prior and how these uncertainties on confidently being able to determine that a template is the true shape can change substantially dependent upon the type of prior we impose. We find that the best measured shapes are those with the strongest divergence and with principal power near squeezed and flattened k -configurations. Though the conventional approach is to quote constraints at the equilateral configuration, k 1 = k 2 = k 3 , we show, as summarized in Table 5, that this convention can mask how well or badly a shape is measured, as doing so has the effect of re-normalizing constraints such that badly measured modes can appear to have constraints similar to the best measured mode. Using the PCA results, we map out the k -dependence of the constraints for a general shape given a prior, and show its dependence on the prior. For all but the r = 1 case, the best measured location is not in the equilateral configuration where shapes and constraints on f NL are typically normalized, but in a configuration that is neither squeezed, flattened, or equilateral, but somewhere in between. This best measured location at roughly k 1 /k 3 ≈ 0 . 32 and k 2 /k 3 ≈ 0 . 80 arises out of the complementary gradients of the power in the PC's. For the r = 1 case, the best measured location is weighted more strongly towards the squeezed configuration, reflecting that the signal and the noise, with which it is correlated, both go to zero in this limit. Given our parametrization of a general shape under a divergence prior, we then ask how well it could be constrained using measurements of amplitudes of common templates, like the local, equilateral, and orthogonal templates. We focus on the class of general shapes that can represent the possible range of single-field models that vanish in the squeezed limit ( r = 1 ). We calculate bounds on the subset of shapes that can remain consistent with constraints on the local, equilateral, and orthogonal templates, and find again-consistent with what we found earlier in the analysis-that templates with more power in the squeezed and flattened configurations provide more stringent constraints on this class of general shapes. Thus, the local, equilateral, and orthogonal templates serve different roles in constraining general shapes; the local template, if detected with sufficient amplitude, will rule out any shape of this type, while the equilateral and orthogonal templates serve to put constraints around different regions of the parameter space. In this sense, constraints from different templates can be complementary. Furthermore, a general (unknown) shape, will have different overlaps with the templates, creating a possibility that by combining constraints on templates, the overall constraint will shed more light on the underlying theory than any one constraint alone. We find it can also be advantageous to look for signals with a new, distinct template, beyond the three standard ones, that could help constrain models more efficiently; we explored the potential for using 2 K 3 -K 6 in this context. In this initial study we use somewhat idealized assumptions focusing on the effects of cosmic variance and Gaussian noise from a homogeneous sky coverage. We recognize the rich potential for further study to other basis sets, that better characterize sharp or oscillatory features in bispectra, the presence of isocurvature modes, and stronger deviations from scale-invariance. To confidently attribute a primordial source to any measured nonGaussianity one would also want to fully account for contributions from astrophysical and instrumental sources, including gravitational lensing, inhomogeneous sky coverage, and secondary anisotropies from astrophysical foregrounds. There is also the substantial question of how large-scale structure measurements, with sensitivity to the squeezed limit, can complement the CMB data in constraining these general shapes, as well as whether 4-point statistics and checks of non-Gaussian consistency ansatzes can play a role. We are tackling some of these intriguing issues in work in preparation.", "pages": [ 29, 30, 31 ] }, { "title": "5 Acknowledgements", "content": "We would like to thank Nishant Agarwal, Xingang Chen, Tom Loredo, Liam McAllister, Sarah Shandera, and the anonymous referee for useful discussions during the preparation of this paper. JB and RB's research was supported by NSF CAREER grant AST0844825, NASA Astrophysics Theory Program grants NNX08AH27G and NNX11AI95G and by Research Corporation.", "pages": [ 31, 32 ] }, { "title": "References", "content": "y a y a equil 4 2 0 - 2 - 4 - 2 - 4 50 0 - 4 - 2 0 2 4 a x 4 2 0 200 0 - 4 - 2 0 2 4 a x S = H 1 -a 200 0 - 4 - 2 0 2 4 a x 50 0 - 4 - 2 0 2 4 a x 200 0 - 4 - 2 0 2 4 a x y a y a y a - 2 - 4 - 2 - 4 - 2 - 4 4 2 0 4 2 0 4 2 0 x -a y L S equil + a X H S s H f local - 2 K equil 5 L NL + 2 K 3 L + a Y H 2 K 3 - K 6 L", "pages": [ 37 ] } ]

[ { "title": "M.M. Hedman", "content": "Center for Radiophysics and Space Research, Cornell University, Ithaca NY 14853 Department of Physics, University of Idaho, Moscow ID 83844-0903 E-mail: [email protected] Abstract. Detecting the extremely low-energy neutrinos that form the Cosmic Neutrino Background (CNB) presents many experimental challenges, but pursuing this elusive goal is still worthwhile because these weakly-interacting particles could provide a new window into the structure and composition of the early universe. This report examines whether cosmological neutrinos can deposit sufficient energy into a target system to be detectable with plausible extensions of current bolometric technologies. While the macroscopic wavelengths of cosmological neutrinos can greatly enhance their cross sections with dense targets, such interactions can only be detectable if they transfer a significant fraction of each neutrino's kinetic energy into the detector system. We find that a large array of dense target masses coupled to suitable motion-sensitive circuits could potentially satisfy both of these conditions and thus might be able to serve as the basis for a more practical cosmological neutrino detector. Keywords: cosmological neutrinos, neutrino detectors", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "As the universe cooled down from its hot and dense initial state, it underwent a series of phase transitions where particle species that were initially in thermal equilibrium decoupled from each other. The most relativistic of these particles have been free-streaming through the universe until the present day, and therefore can carry information about the state of the early universe to detectors currently operating on Earth. The most famous of these relic particle populations is the Cosmic Microwave Background, or CMB, which is one of today's most powerful cosmological probes (see e.g [1-4]) However, the CMB is not the only cosmological particle background in the universe. In particular, there should also be a Cosmic Neutrino Background, or CNB. Standard cosmological models predict the present universe is filled with a nearly homogeneous population of each of the three neutrino species and three antineutrino species. These neutrino populations should follow a thermal Fermi-Dirac distribution with an effective temperature of 1.95 K, such that the mean number density of each species of neutrino or anti-neutrino is ∼ 56 / cm 3 [5]. Recent cosmological observations, including measurements of the CMB power spectrum, are roughly consistent with this basic model [2, 3, 6], and thus give us confidence that such a background does indeed exist. If these cosmological neutrinos could be observed, they could provide invaluable information about the structure, composition and early history of the universe [7, 8]. Directly detecting these extremely low-energy and weakly-interacting particles poses major experimental challenges, and while a variety of detection techniques have been considered over the last four decades, none has proven to be practical with reasonable extensions of current technologies. For example, multiple authors have explored the possibility of detecting the mechanical forces cosmological neutrinos can apply to various laboratory systems [9-18]. These scenarios often rely heavily on enhancement factors in the interaction cross sections that arise when the low-energy neutrinos coherently scatter from all the particles within their de Broglie wavelength. However, even with the resulting enhancements in the interaction rates, the resulting accelerations are still orders of magnitude below the detection threshold of any laboratory system conceived thus far [10, 11, 14-16, 18]. More recently, there has been interest in the possibility of detecting cosmological neutrinos in beta-decay experiments such as KATRIN and MARE [19, 20]. In these cases the observational signature is a distortion in the energy spectra of the electrons emitted by unstable nuclei that arises because some of these nuclei capture a cosmological neutrino instead of undergoing normal beta decay. Compared with detecting mechanical forces, these techniques have the advantage that each neutrino interaction produces a distinct change in the target system. Unfortunately, the neutrino cannot be coherently captured by many nuclei, so the relevant reaction cross sections are not enhanced by the neutrinos' macroscopic wavelengths. Indeed, the rates of such relic neutrino captures are so low that planned experiments like KATRIN could only detect a signal from the CNB if it was ∼ 10 9 times larger than that expected in standard cosmological models [19]. Thus, while these machines could detect nonstandard neutrino backgrounds, they may not yet provide a clear path towards a practical method of detecting the CNB expected from standard cosmological models. Despite these discouraging findings, one must remember that mechanical forces and nuclear transitions are only two ends on the spectrum of detectable signals that could be produced by cosmological neutrinos. In between these limiting cases are a wide range of detection schemes where the neutrinos excite transitions between internal states of the target system. These sorts of detection scenarios are worth considering because the relevant signal does not need to be a force or displacement, but could instead be a transfer of energy from the CNB into the detector. Advances in bolometer technology have led to devices capable of measuring extremely tiny energy fluxes [21-24], so such energy transfers could potentially be easier to detect than tiny accelerations or rare nuclear changes. The magnitude of the relevant energy fluxes has only been computed for a few specific cases (e.g. when the neutrino excites phonons in a solid medium [14, 15]), and so it is worth exploring whether other systems could experience larger and more detectable energy changes due to their interactions with the CNB. The report examines how efficiently energy can be transferred from cosmological neutrinos into several different potential detection systems. We begin with a review of low-energy neutrino scattering and develop a formalism for quantifying the energy flux carried into a target system by the CNB. These calculations suggest that under ideal circumstances the energy carried by cosmological neutrinos could in principle be detectable with reasonable extensions of current bolometric technologies. Next, we consider specific systems that extract energy from the CNB with various efficiencies. Consistent with previous analyses, we find that an isolated mass at rest or a mass in a static potential well cannot extract a detectable amount of energy from low-energy neutrinos. However we also find that other systems, such as a moving mass, a mass attached to a charge embedded in a magnetic field, or a mass attached to a motion-sensitive circuit, are much more efficient at extracting energy from cosmological neutrinos. Indeed, a mass coupled to a motion-sensitive circuit may even be sufficiently efficient at extracting energy from cosmological neutrinos to serve as a basis for a viable CNB detector.", "pages": [ 2, 3 ] }, { "title": "2 Low-energy neutrino scattering", "content": "A cosmological neutrino can interact with an arbitrary laboratory system in a large number of ways, so a completely generic analysis of low-energy neutrino scattering is next to impossible. Hence this study will only consider potential neutrino detection systems where (1) the neutrino is not captured by the target system, (2) the neutrino interacts only with a dense object (here called the 'antenna mass' of the target system) that is much smaller than the neutrino's de Broglie wavelength, and (3) the interaction only induces motion in that object's center-of-mass, and does not excite any internal modes of the object. The first condition is not a major constraint, since the extremely low-energy cosmological neutrinos are very inefficient at exciting nuclear reactions, and the most promising capture reactions are those already examined in the context of the beta-decay experiments mentioned above [19]. The second and third conditions, by contrast, are more restrictive. For example, they exclude scenarios where the neutrino directly excites phonons in a solid [14, 15]. Still, the above class of detection schemes includes not only the well-studied case of an isolated antenna mass, but also more complex (and less explored) systems where the mass is trapped in a potential well or coupled to electromagnetic fields. Pursuant to criterion (2), the neutrino is assumed not to interact directly with the components of the target system responsible for producing these fields or forces. This can easily be achieved if the antenna mass is much denser than any other part of the system (e.g. the antenna mass is made of lead while the other components are made of aluminum). Previous investigations focusing on the recoil motion of an isolated mass have demonstrated that the neutrino-induced acceleration and kinetic energy are too small to be detectable [10, 11, 14-16, 18]. However, for the broader class of systems considered here the energy transferred into the target system by the neutrino interaction does not necessarily equal the kinetic energy of the object's recoil motion. Hence we require more general expressions for how often cosmological neutrinos will interact with the target system and how much energy the neutrinos will impart into the detector. From standard time-dependent perturbation theory, the interaction rate between the neutrinos and the target system can be expressed as: where E i and E f are the total energies of the initial and final energy eigenstates of the combined system (neutrino plus target), | I 〉 and | F 〉 are the spatial parts of the same eigenstates, and H ' is the interaction Hamiltonian between the neutrino and the target system. Furthermore, if the neutrinos are not captured by the target (cf. condition 1 above) and the initial and final states of the neutrinos have energies E i and E f , respectively, then the rate at which energy is transferred into the target system by the neutrinos is: These expressions for R and P can now be evaluated assuming that the neutrinos only interact with a mass containing N fermions packed into a region that is much smaller than the neutrinos' de Broglie wavelength (cf. condition 2 above) and that the internal modes of the mass are not excited by the interaction (cf. condition 3 above). In this case, the interaction Hamiltonian can be approximated by the following simple form: where G F = 10 -5 GeV -2 is the Fermi constant, c V is the vector amplitude factor that depends on the fermion content in the antenna, ξ is a numerical constant that ranges between √ 2 for relativistic neutrinos and 1 / √ 2 for non-relativistic Dirac neutrinos [15], and x ν and x A are the spatial coordinates of the neutrino and the antenna mass, respectively. Regardless of the target-system's structure, the initial and final states of the neutrino will correspond to those of free particles. For the sake of clarity, the following calculations assume a monochromatic, unidirectional flux of neutrinos onto the target. Hence we can write the initial and final states of the combined system as | I 〉 = ψ ν i | i 〉 and | F 〉 = ψ ν f | f 〉 , where | i 〉 and | f 〉 are the spatial components of the initial and final eigenstates of the target system, while ψ ν i and ψ ν f are the initial and final states of the neutrinos: where p i,f are the incoming and outgoing neutrino momenta and the normalization constant N ν is set such that |N ν | 2 = 1 / V , where V is a quantization volume that can be expressed in terms of either the number density of incident neutrinos n i or a differential element of the outgoing momenta p f : Inserting these expressions into the above equations for R and P , and integrating over x ν to eliminate the delta function in the Hamiltonian gives the following expressions for the interaction and energy transfer rates: where δ p = p i -p f is the change in the neutrino's momentum during the interaction. The above rates are for an interaction that yields a neutrino with a specific momentum p f . The total interaction and energy transfer rates are obtained by integrating over all possible outgoing momenta, yielding the following expressions: These expressions may be simplified slightly by recognizing that the differential element: where p f = | p f | and the second equality follows from the standard relation E 2 f = p 2 f + m 2 ν . We may therefore integrate over E f to eliminate the energy-conserving delta function, leaving only the angular integral: Finally, these expressions can be re-written in the following forms: Where F R and F P are the dimensionless efficiency factors: The above formulation of the interaction and the energy transfer rates has the useful property that the target-dependent factors have now been isolated into the efficiency factors. Furthermore, provided E f < E i , neither efficiency factor can exceed unity, so F P = F R = 1 represents an ideal cosmological neutrino detector. Such a detector not only has the highest possible neutrino interaction rate, but also extracts all of the kinetic energy avialible from each scattered neutrino. Thus we can use this ideal case to establish whether any system of this type could ever yield a detectable signal. Real cosmological neutrinos have a range of energies and approach the target from all directions, so a precise estimate of the relevant rates would require integrating the above expressions for R and P over the appropriate distribution functions. While we will consider the variability in the neutrinos' approach directions as appropriate below, accounting for the finite range of neutrino energies would just complicate the expressions and this level of precision is not needed for the order-of-magnitude calculations presented in this initial study. Instead we will simply insert 'typical' values for cosmological neutrinos into Equations 2.13 and 2.14. For the sake of simplicity, we assume massless cosmological neutrinos, so p i = E i glyph[similarequal] 10 -4 eV, and the local number density n i glyph[similarequal] 100/cm 3 . Also, since the antenna mass would best be constructed of a dense metal like lead (with ∼ 10 25 fermions per cubic centimeter), and it must also be smaller than the ∼ 1 cm de Broglie wavelength of these neutrinos, a reasonable value for the total number of fermions in the antenna mass is N = 10 24 . Inserting these numbers into Equations 2.13 and 2.14 gives: Thus a system with F R glyph[similarequal] 1 would interact with cosmological neutrinos once every few hours, and if F P glyph[similarequal] 1, the target system could extract 10 -27 Watts from the Cosmic Neutrino Background. While 10 -27 Watts is not much power, it could be within the reach of current technologies. Modern bolometric detectors are now approaching sensitivities of order a few times 10 -19 W/ √ Hz [21-24]. If such devices could be coupled to targets with F P glyph[similarequal] 1, then the above power flux could be detected in ∼ 10 8 detector-years, or in a single year with ∼ 10 8 detectors. This number, while large, does not necessarily correspond to an impossibly large instrument. If each individual antenna mass is less than ∼ 1 cm 3 in size, the entire threedimensional array of 10 8 detectors could in principle fit within a region 10 meters across. Hence it may be possible to construct an array with sufficient raw sensitivity to detect cosmological neutrinos if we can find an 'efficient' target system with F P of order unity.", "pages": [ 3, 4, 5, 6 ] }, { "title": "3 Efficiencies of specific detector systems", "content": "The calculations in the previous section reveal that a system with F P ∼ 1 could be able to extract a detectable amount of energy from cosmological neutrinos. This might appear to contradict previous analyses which demonstrated that the mechanical forces produced by cosmological neutrinos are undetectable [10, 11, 14-16, 18]. However, as demonstrated below, the scenarios considered in these earlier works yield a F P ∼ E i /M << 1, which means these systems can only extract an undetectably small amount of energy from cosmological neutrinos, consistent with the published calculations. Fortunately, a careful consideration of these detectors' limitations allows us to identify detection schemes that could extract energy much more efficiently from the Cosmic Neutrino Background. The following sections consider a series of model detector systems. First, we examine the case of free antenna mass initially at rest, and recover the well-known result that the mechanical forces generated by cosmological neutrinos are too small to detect. Next, we consider a mass trapped in a static potential well, and show that such a system is not significantly more efficient at extracting energy from cosmological neutrinos than a free mass. We then consider systems where the mass is moving with respect to the lab frame or coupled to a charge embedded in a magnetic field, and demonstrate that such systems can be much more efficient neutrino detector than a free mass. Finally, we describe a system composed of a dense mass coupled to a motion-sensitive circuit that may be able to achieve the desired F P ∼ 1. Note that in all these discussions the neutrino momenta p i and p f are always measured relative to the laboratory frame.", "pages": [ 7 ] }, { "title": "3.1 Free mass at rest", "content": "If the target mass is entirely free, then the initial and final states of the target system are those of free particles: where P i,f are the initial and final momenta of the target mass in the laboratory frame, and the normalization factor N A can again either be expressed in terms of a quantization volume V or as a differential momentum element. In this case the matrix element reduces to a momentum-conserving delta function, which is eliminated by integrating over all possible outgoing antenna momenta, leaving the following efficiency factors: If we further stipulate that the mass is initially at rest in the lab frame (i.e. P i = 0), then conservation of energy and momentum requires that E i -E f = δp 2 /M , where δp 2 = | δ p | 2 = p 2 i + p 2 f -2 p i p f cos θ and θ is the scattering angle in the center-of-mass frame. Thus we can re-express the energy-transfer efficiency factor as: For any reasonable antenna mass M will be much larger than E i , p i or δp . In this limit, E i -E f glyph[similarequal] ( p 2 i /M )(1 -cosθ ), so E i -E f << E i and p i -p f << p i , and the above integrals become: Since F R is unity, free masses can interact with cosmological neutrinos once every few hours, so the interaction rates themselves are not necessarily a major obstacle for detecting cosmological neutrinos. Instead, the primary issue is that F P = E i /M is very small. The mass of the target can be written as M = Nµ , where µ is the mass of the relevant fermions. Even in the best (and least realistic) case of a pure electron target with µ = 511 keV, the ratio E i /M ∼ 2 ∗ 10 -34 ( N/ 10 24 ) -1 is extremely small, and the power transfer rate is correspondingly feeble: This power is far too low to be detected with any reasonable technology, consistent with previous analyses [10, 14, 15]. Thus a system which could yield a detectable signal from cosmological neutrinos would need to extract energy from the neutrinos much more efficiently than a free mass at rest.", "pages": [ 7, 8 ] }, { "title": "3.2 Mass trapped in a static potential", "content": "Since free masses cannot efficiently extract energy from the cosmological neutrinos, we must consider more complex target systems where the motion of the antenna mass has a nontrivial spectrum of excited states. One simple example of such a system consists of an antenna mass trapped in a potential well. In this scenario, the neutrino's interaction with the mass excites transitions between eigenstates of the potential, enabling the target system to capture a significant fraction of the incoming neutrino's energy. Unfortunately, it turns out that masses trapped in fixed potentials are not significantly more efficient neutrino detectors than free masses. Consider a mass initially in the ground state of the potential well | g 〉 with energy glyph[epsilon1] g , which the neutrino will excite into a state | e 〉 with energy glyph[epsilon1] e = glyph[epsilon1] g + δglyph[epsilon1] . Note that E i -E f = glyph[epsilon1] e -glyph[epsilon1] g . The energy transfer rate efficiency factor is therefore: So long as δglyph[epsilon1] > 0, this expression can be re-written in the following form: The states | g 〉 and | e 〉 are eigenstates of the Hamiltonian for the bound mass: where ˆ P A is the momentum operator for the antenna mass and V ( x A ) is the trapping potential. Thus H b can replace glyph[epsilon1] g and glyph[epsilon1] e in the above expression. So long as the potential is only a function of the mass position x A , it will commute with e -iδ p · x A and thus cancel out of the expression, leaving: Using the standard commutation rules, this expression reduces to: Now define δ k = δ p /δp and ˆ K A = ˆ P A / √ Mδglyph[epsilon1] . The parameter δ k is just the unit vector pointing along the direction of δ p , while the unitless operator ˆ K A is a linear combination of raising and lowering operators where the coefficients on these operators are of order unity. In terms of these parameters, the expression becomes: Rearranging terms and pulling out a leading factor of p i /M , gives: In order for the efficiency factor to be of order unity, we need the term in brackets to be of order M/p i >> 1, but this is impossible. The factors of p f E f /p 2 i and δp/ 2 p i must both be less than unity if the neutrino is to donate energy to the target system. Furthermore, the δglyph[epsilon1] for a bound particle cannot be less than its value for a free particle δp 2 / 2 M , so δp 2 /Mδglyph[epsilon1] also cannot exceed √ 2. The first matrix element cannot possibly exceed unity because of how the states are normalized. Finally, the above definition of ˆ K A should prevent the second term from being much larger than 1 (This is certainly true for simple potentials such as square wells and harmonic oscillators, but a formal proof that it also applies to more complex potentials is beyond the scope of this report). Thus F P fixed can never be much larger than p i /M , and one cannot construct an efficient neutrino detector from isolated masses trapped in static potential wells.", "pages": [ 8, 9 ] }, { "title": "3.3 Free mass in motion", "content": "At first, the above calculations would appear to suggest that F P must always be of order E i /M . However, much more efficient systems are possible if we critically examine the assumptions behind these computations. For example, let us return to the case of a free mass, but instead of assuming the mass starts at rest, have the mass initially moving at a finite velocity relative to the laboratory frame, so the initial state of the mass has a finite initial momentum P i . In this case, the relevant matrix element still corresponds to a momentum-conserving delta function, but if P i >> p i , then conservation of energy and momentum requires that E i -E f = δ p · P i /M . Thus the momentum impulse δp can potentially produce much larger changes in the kinetic energy of a moving mass than it can for a stationary one. For any realistic detector system P i /M << 1, so E i -E f << E i and the relevant efficiency factors become: where k i and K i are unit vectors aligned with the vectors p i and P i , respectively. Note that P i /M = v i /c , where v i is the initial speed of the target mass If the neutrinos were truly unidirectional, then we could make ( k i · K i ) = 1 by ensuring the mass moves in the same direction as the incident neutrinos. Of course, real cosmological neutrinos will approach the target from all directions. If the incident neutrino flux were perfectly isotropic, then k i · K i would average to precisely zero. In practice, the incident neutrino flux is not exactly isotropic because the solar-system's peculiar velocity v p relative to the mean Hubble flow produces a detectable dipole variation in both the CMB and the CNB. In this situation the average ( k i · K i ) in the laboratory (solar system) frame will be of order v p /c . Based on observations of the CMB, v p /c glyph[similarequal] 0 . 001 [25], and for the sake of argument, we may imagine that the mass is initially moving at a speed of a few centimeters per second towards the peak of the CMB dipole. In that case, the relevant efficiency factor becomes: Since F P is still much less than unity, moving masses are unlikely to serve as practical cosmological neutrino detectors. However, this calculation also shows that a moving mass is many orders of magnitude more efficient than the previous two systems. Thus systems with F P >> E i /M do exist, which offers hope that efficient cosmological neutrino detectors may be possible.", "pages": [ 9, 10 ] }, { "title": "3.4 Mass attached to a charged object", "content": "Systems with initially moving masses are not the only ones that can achieve F P >> E i /M , and additional potential neutrino detection systems can be found by examining the assumptions behind the above calculations for the mass trapped in the static potential well. Specifically, that prior analysis assumed that the Hamiltonian of the target system could be expressed in terms of a fixed potential that depends only on the mass' spatial coordinates (see Equation 3.10). However, there are also systems with Hamiltonians that are more complex functions of momentum. For example, the Hamiltonian of a charged particle coupled to an electromagnetic field is glyph[negationslash] where q is the particle's charge, while Φ and A are the scalar and vector potentials of the electromagnetic field. So long as A = 0 , this Hamiltonian will have momentum-dependent terms that are qualitatively distinct from those associated with a mass trapped in a fixed potential well. These terms do not commute with e -iδ p · x A and therefore could generate larger values for F P . For example, consider the system illustrated in Figure 1, where the antenna mass is attached to a rigid ring of radius r with negligible mass and carrying an electric charge Q . This ring is embedded in a constant magnetic field B , and is held such that it can only rotate along an axis aligned with that field. In this scenario, the constraints on the system's motion allow the Hamiltonian to be written in the following form: + where ˆ L φ is the angular momentum of the mass (and ring) moving in a circular path around the ring's axis. Furthermore, the initial and final states of the target mass are given by the following expression: where φ is the angular coordinate of the mass. For the sake of simplicity, let us assume that the radius r is much larger than the neutrino's de Broglie wavelength and the scale of the neutrino's wavepacket. In this limit, we may approximate the above states as the initial and final states of a particle that is constrained to move in one spatial direction (i.e. the spacing between rotational energy levels is much less than the energy imparted by the neutrino collision). The matrix element |〈 i | e -1 δ p · x A | f 〉| 2 can then be reduced to a momentumconserving delta-function that is eliminated by integrating over all outgoing antenna mass momenta. If we also assume that initially the mass is nearly at rest, then conservation of energy and angular momentum requires that E i -E f = δp ( δp/M -sign( L f -L i ) QBr/Mc ). Note that since δp > 0, the sign of the second term in this equation depends on the direction in which the mass moves in response to the neutrino collision. The relevant efficiency factors B can then be written in the following forms: The above expression for the rate efficiency factor is basically the same as that for the free mass (see Equation 3.2 above), so the interaction between the charge and magnetic field does not directly influence the rate of neutrino interactions. By contrast, the energy transfer factor has a new term proportional to QBr/Mc that did not appear in Equation 3.4 above. This term arises because transferring momentum to the target mass does not just increase the mass' kinetic energy, it also changes the electromagnetic energy associated with the charges' motion through the magnetic field. It is not hard to construct a system where the second term in F P charge is much larger than the first one. First, realize that the charge Q can be expressed as the product C Q Φ o , where C Q is the self-capacitance of the charged ring and Φ o is its electrostatic potential. For a ring of radius r , the self-capacitance will be of order 4 πglyph[epsilon1] o r , where glyph[epsilon1] o is the permittivity of free space. Hence C Q glyph[similarequal] 10 -11 F ( r/ 0 . 1 m ). If we further assume that Φ o ∼ 10V and B ∼ 1 T, and say M ∼ 1g (consistent with 10 24 nucleons) then the relevant ratio becomes: This ratio is orders of magnitude larger than δp/M ∼ E i /M , but is still much less than unity, so E i -E f << E i . Furthermore, because the energy change depends on whether the mass' angular momentum change is positive or negative, the energy transfer efficiency depends on the approach direction of the incident neutrinos. Thus, just like the moving mass described above, the average energy transfer efficiency factor for the nearly-isotropic CNB is reduced by a factor of v p /c . Hence the above expressions for the efficiency factors are: The latter equation shows that F P is still much less than unity (and the moving mass described above), so this system will not yield a truly efficient neutrino detector for any plausible value of Φ o or B . However, this calculation does demonstrate that systems do not need to have a moving mass to achieve F P >> E i /M . Furthermore, these calculations confirm that a coherently-scattered neutrino does not have to deposit the vanishingly small fraction of its initial energy into a target. While the above efficiency factors were computed using a quantum-mechanical formalism, most aspects of this calculation can be reproduced using classical electrodynamics. A ring of radius r carrying a charge Q moving at a speed v = δp/M can be approximated as a current loop which has a magnetic dipole moment µ ∝ Qr ( δp/M ). If this loop is embedded in a magnetic field, it has an energy carried by that magnetic moment is E loop = µB ∼ QBr ( δp/M ). This is the same energy change that forms the dominant term in the above above expression for E i -E f . This concordance arises because the extra terms v in the Hamiltonian do not explicitly depend on the position of the target mass. These terms therefore do not influence the relevant matrix elements, but only affect the energy of the target system's initial and final eigenstates. Hence we should be able to estimate the efficiencies of certain more complex electromagnetically-coupled systems from how their energy should change when the relevant part of the system moves.", "pages": [ 10, 11, 12, 13 ] }, { "title": "3.5 Mass coupled to a motion-sensitive circuit", "content": "A simple charged system like the one described in the previous section is unlikely to yield an efficient neutrino detector because of practical limitations on the magnitudes of the object's charge and the applied magnetic field. In particular, the small self-capacitance of the ring limits the amount of charge that can be applied to it, and thus limits the energy changes associated with the system's motion. Fortunately, other electromechanical systems can have much larger energy changes induced by a moving antenna mass. For example, consider the system illustrated in Figure 2, where the antenna mass is constrained to move along a single linear direction, and is attached to a rigid wire of length glyph[lscript] that is held perpendicular to the mass' direction of motion. The wire in turn is embedded in a uniform magnetic field B that is oriented perpendicular both to the mass' direction of motion and the wire's axis. Thus, if the mass moves, the wire will be drawn across the magnetic field lines, producing an electromotive force across the wire. This wire is also part of a circuit containing a battery that produces a DC voltage Φ o and a capacitor with capacitance C . So long as the mass and wire segment are not moving relative to the magnetic field, the potential across the capacitor will be Φ o , and the energy stored by the capacitor will be E c, 0 = (1 / 2) C Φ 2 o . However, if the mass and wire move at a speed v , then the electromotive force due to the wire's motion through the magnetic field will produce a potential difference of δ Φ = vBglyph[lscript] across the two ends of the moving wire. In general, this potential drop could be either positive or negative depending on the wire's direction of motion. As in the previous two cases, this would reduce the efficiency of the detector to the nearly isotropic CNB by a factor of v p /c . However, in this case one can insure the δ Φ transmitted to the capacitor has the same sign as Φ o with an appropriate rectifier circuit (in the figure we illustrate the rectifier as a single diode that ensures δ Φ > 0). Hence the potential difference across the capacitor changes from Φ o to Φ o + δ Φ and the energy stored in the capacitor increases by δE C = C Φ o δ Φ = C Φ o Bglyph[lscript]v . As with the simple charged system described in the previous section, any state of the above system where the antenna mass has a well-defined momentum will also have a welldefined energy. Hence the relevant matrix elements are again the same as those for a mass free to move in one direction. Furthermore, assuming the target mass is initially at rest, then E i -E f = δE C = C Φ o Bglyph[lscript]δ p/M . The efficiency factor of this system is therefore given by the following expression: If we assume that B is of order 1 T, Φ o is of order 10 Volts, the wire is 1 meter long and the C is of order 1 Farad, then we find that the above efficiency factor will be of order: The efficiency factor of this system is therefore not much less than one. Furthermore, the efficiency of this system can easily be increased by (1) increasing the capacitance of the circuit using multiple capacitors wired in parallel, (2) increasing the battery's voltage, and (3) replacing the straight wire with a coil that has multiple wire segments passing through the magnetic field. These improvements could each increase F P by over an order of magnitude, hence a system similar to the one described above could act as an efficient neutrino detector, with F P ∼ 1.", "pages": [ 13, 14 ] }, { "title": "4 Conclusions", "content": "The above analysis indicates that a compact, dense mass coupled to a suitable motionsensitive circuit should be able to both interact with neutrinos every few hours, and extract a significant fraction of the neutrino's kinetic energy from each interaction. One of these detectors could therefore absorb of order 10 -27 Watts of power from the Cosmic Neutrino Background. An array of 10 8 such detectors could therefore absorb enough energy to be detectable with current state-of-the art bolometers. Of course, it is not yet certain that a large enough array of detectors could be constructed or made sufficiently sensitive to detect cosmological neutrinos. Furthermore, a practical neutrino detector must not just be able to detect the energy from the CNB, but also isolate that signal from other sources of excitation. Such challenges are beyond the scope of this report, but the calculations presented above indicate that a cosmological neutrino detector may not be entirely beyond the reach of nearfuture technologies.", "pages": [ 14, 15 ] }, { "title": "Acknowledgments", "content": "I wish to thank I. Wasserman and D. Chernoff for many useful conversations. I would also like to thank the anonymous reviewers whose comments substantially improved this manuscript.", "pages": [ 15 ] } ]

[ { "title": "Xian Gao a,b,c", "content": "E-mail: [email protected] Abstract. We investigate the coupling relations among perturbations in general multi-field models. We derived the equations of motion for both background and perturbations in a general basis. Within this formalism, we revisit the construction of kinematic orthogonal normal vectors using the successive time derivatives of the background field velocity. We show that the coupling relations among modes in this kinematic basis can be reduced, by employing the background equations of motion for the scalar fields and their high order time derivatives. There are two typical features in the field space: inflationary trajectory and geometry of the potential. Correspondingly, the couplings among modes fall into two categories: one is controlled only by the kinematic quantities, the other involves high order derivatives of the potential. Remarkably, the couplings of the first category, i.e. controlled by the kinematic quantities only, show a 'chain' structure. That is, each mode is only coupled to its two neighbour modes.", "pages": [ 1 ] }, { "title": "1 Introduction and motivation", "content": "The latest observations on the Cosmic Microwave Background (CMB) [1, 2] are compatible with statistically Gaussian primordial perturbation [3], which has a nearly flat spectrum with negligible running spectral tilt. In particular, the data are also compatible with the adiabaticity at 95% CL, which implies there is no evidence for the isocurvature modes and there is only one relevant degree of freedom responsible to the primordial perturbations. In spite of this, there are good reasons to consider models where inflation is driven by multiple scalar fields. On the theoretical side, many inflationary models based on grand unification, supersymmetry and supergravity from string theory involve multiple scalar fields. Models with spectator field(s) other than inflaton such as the curvaton mechanism [4] also introduce additional light field(s), leading to correlations among the adiabatic and isocurvature modes. On the observational side, the asymmetries in the CMB reported in the WMAP data [5] and recently confirmed by Planck [6] indicate nontrivial modifications of our understandings of the primordial Universe, to which multi-field scenarios may supply one possibility [7]. Moreover, there is hint for oscillatory features in the power spectrum [1, 8] (see also [9]), which may also be a signal for the existence of multi-field effects. Despite intensive efforts to understand multi-field inflation over the past decade, most analyses concentrate on specific models with two fields (see e.g [10-15]), although perturbation theories within general multifield/component scenarios have also been developed [16-20]. In this work, instead of investigate concrete models one by one, we would like to examine general features in the presence of additional degree(s) of freedom. To this end, a detailed investigation of the full coupled system of perturbations is needed. In the framework of inflation, multi-field effects manifest themselves as long as the background trajectories are bending in multidimensional field space [21, 22]. When the turning rate is relatively small, the correlations between adiabatic and isocurbature modes have been studied perturbatively [23-25]. Moreover, the impact of possibly existent heavy (with respect to the Hubble scale) modes on the primordial spectrum during inflation has attracted a lot of attention recently [26-30]. The effect from the heavy mode(s) depend on the details of the background trajectories [31, 32]. In particular, features on the power spectrum of curvature perturbation arise when there are nontrivial trajectories in multi-dimensional field space [33-38]. At the level of equations of motion, the coupling relations among perturbations manifest themselves as a set of 'sourcing' relations, i.e. which mode appears as source term in the equation of motion for another mode. Contrary to the two-field cases where the coupling between the adiabatic and isocurvature mode has been investigated in depth, the sourcing relations among the modes in a general multi-field inflation has received less attention. In [39], the authors studied the sourcing relations in kinematic basis under the slow-roll approximation and in the large-scale limit, which can be viewed as a first step attempt in this direction. In this work, we will examine the coupling relations in a general multi-field model in details. We first develop the perturbation theory in a general basis, which is the generalization of the kinematic basis introduced firstly in [21] and developed further in [22, 40-43]. Going beyond the kinematic basis is inspired by the study of two-field models with heavy perturbation mode, i.e. there might be more convenient basis other than the popular kinematic basis when dealing with specific problems [32, 38]. Essentially, this is similar to the idea of treating the scalar field perturbations as vectors in field space [44], which enables us to have a manifestly covariant formalism (see also [45, 46] for a recent investigation). Without specifying any concrete form for the potential, we revisit the kinematic basis. The background trajectory and thus the kinematic basis are characterized by kinematic quantities: the field velocity ˙ φ I and its time derivatives. In practice and as being adopted in this work, these kinematic quantities can be reparameterized in terms of an effective inflaton velocity ˙ σ as well as ( N -1) 'angular velocities' ˙ θ i 's, which generalize the rotation rate ˙ θ in the two-field case [21]. One of the findings in this work is that, the couplings among modes can be reduced by using the background equations of motion as well as their high order time derivatives. This can be done due to the fact that, certain components of the high derivatives of the potential can be re-expressed in terms of the kinematic quantities. Finally, the couplings fall into two categories: one is controlled by only the kinematic quantities associated with the trajectory, the other is controlled by high order derivatives of the potential (together with kinematic quantities). Remarkably, the first category, i.e. couplings controlled by the kinematic quantities show a simple 'chain' structure - each mode is only coupled to its two neighbour modes. By clarifying the dependence of the couplings on these different features in field space - in the background trajectory and in the geometry of the potential, we are able to choose more appropriate basis or to make approximations, and ultimately, to relate these features with observables. This is the main motivation of this work. The paper is organized as following. In Sec.2, we write the equations of motion for both background and perturbations in a general basis. In Sec.3, we first revisit the construction of the set of kinematic basis vectors, then decompose the background equations of motion and their high order time derivatives in the kinematic basis. Finally in Sec.4, we reduce the couplings in the kinematic basis, by using the background equations of motion. Throughout this paper, we work in units such that M Pl := 1 / √ 8 πG N ≡ 1 , and choose the signature of the spacetime metric as {-, + , + , + } .", "pages": [ 2, 3 ] }, { "title": "2 Dynamics in a general basis", "content": "We concentrate ourselves on the simplest model of N scalar fields with the action where g is the determinant of the spacetime metric g µν , X ≡ -1 2 δ IJ ∂ µ φ I ∂ µ φ J and V ( φ I ) is the potential of the scalar fields with I = 1 , · · · , N . The corresponding background evolution equations are well-known: H 2 = 1 3 ( X + V ) and ˙ H = -X ≡ -H 2 /epsilon1 , where an overdot denotes derivative with respect to the cosmic time t . The equations of motion for the scalar fields are where and in the following V ,I ≡ ∂V/∂φ I , V ,IJ ≡ ∂ 2 V/∂φ I ∂φ J etc. When working in the spatially-flat gauge, the scalar degrees of freedom of perturbations are the perturbations of the scalar fields δφ I . The quadratic action for the canonically-normalized variables u I = aδφ I with a the scale factor is (in matrix notation) where a ' ' ' denotes the derivative with respect to the comoving time η defined through dη = dt/a and the mass matrix is given by (see e.g. [47]) In the above, all expressions are written in the primitive basis, with indices I, J etc. However, since the fields can be viewed as coordinates parameterizing the multi-dimensional field space, it is natural to consider other basis, which may be more compatible with features in the field space. There are two natural features in the field space: These two basis differ from each other in general. When the inflationary potential has explicit heavy and light direction, the features in the potential dominate over the features of inflationary trajectory and thus it is more convenient to work in the potential basis [32]. This is also confirmed in [38] that, features in the potential give the main contributions to the resulting power spectrum instead of those in the trajectory. As we have emphasized in the Introduction, the goal of this work is just to clarify the dependence of the couplings on these different features. /negationslash We consider a general basis transformation in field space: e I → e a = e I a e I , where subscripts a, b etc denote indices associated with the general basis. The vielbein e I a satisfy normalization and orthogonal conditions δ IJ e I a e J b = δ ab and δ ab e I a e J b = δ IJ . Quantities carrying indices are supposed to transform 'covariantly': q I = e I a q a , q IJ = e I a e J b q ab etc. This property does not hold after taking ordinary time derivatives, since the basis transformation is time-dependent in general, de I a /dη = 0 . This can be solved by introducing a 'covariant' time derivative D η associated with the given basis: where the 'connection' Z ab is defined as Obviously, the 'covariant' time derivative D η is different from basis to basis. Note Z ab = -Z ba due to the normalization e I a e I b = δ ab . One may check explicitly that ( d/dη ) n q I = e I a D n η q a , i.e. D η is indeed transformed 'covariantly'. D η satisfies all the property of a linear differential operator, and can be naturally generalized for quantities with multiple indices, e.g. D η q ab = q ' ab + Z ac q cb + Z bc q ac . In the following, all quantities are written in the general basis with indices a, b etc. The background velocity for the scalar fields picks a specific direction in field space and is a crucial quantity in our formalism, which transforms as where σ ' := ( | φ ' I φ ' I | ) 1 / 2 = ( | φ ' a φ ' a | ) 1 / 2 which denotes the amplitude of the effective inflaton velocity, and n a satisfies n a n a = 1 which denotes the direction of the inflationary trajectory. We emphasize that both σ ' and n a are abstract notations, especially, σ ' should not be understood as the derivative of any quantity. In terms of σ ' and n a , the background equations of motion for the scalar fields (2.2) can be recast as The quadratic action for the perturbation modes u a = e I a u I takes almost the same form as in (2.3) but with covariant time derivatives: where and in what follows we denote σ '' := dσ ' /dη and V ,a := e I a V ,I etc. where u and M denote u a and M ab ≡ e I a e J b M IJ respectively. The equations of motion for u a can be got by varying (2.9) with mass matrix given by The main purpose of this work is to clarify the coupling relations among modes. From (2.10), the mixing among different modes has two origins: one comes from the 'covariant' time derivatives D 2 η which is associated with a given basis, the other comes from the mass matrix M ab . In the rest of this work, we revisit the kinematic basis, in which the coupling relations can get reduced.", "pages": [ 3, 4, 5 ] }, { "title": "3 Background dynamic in kinematic basis", "content": "The crucial observation in [21] is that, the background inflationary trajectory picks a special direction in field space called the 'adiabatic direction'. In the space of perturbations 1 , component in the perturbations parallel to the background trajectory is the adiabatic mode, which corresponds to the curvature perturbation; while perturbations perpendicular to the background velocity correspond to the entropic modes, which do not contribute to the curvature perturbation directly 2 .", "pages": [ 5 ] }, { "title": "3.1 Kinematic basis revisited", "content": "The adibatic/entropic decomposition was introduced in [21] in the two-field case. In [22], a full set of kinematic basis vectors was constructed through the Gram-Schimdt orthogonalization of the successive high order time derivatives of the background field velocity (see also [39, 48] for a recent discussion). In this work, we use similar procedure to construct the basis vectors e ( i ) 's with i = 1 , · · · , N . The first basis vector of the kinematic basis is defined as the direction of the background trajectory where n a is defined in (2.7). Here and in the following, the indices of the basis vectors are written a general manner. We wish to emphasize that, the kinematic (adiabatic/entropic) decomposition can be made in an arbitrary basis, not necessarily in the kinematic basis. For example, all equations of motion were derived in the mass basis in [32], including the identification of adiabatic and entropic modes. The second basis vector (i.e. the first entropic vector) is chosen to be proportional to the changing rate of the direction of the trajectory with a normalization factor θ ' 1 . With this definition e (2) a is automatically orthogonal to e (1) a since e (1) a is already normalized. The normalization of e (2) a requires | θ ' 1 | ≡ | D η e (1) a | . It is worth emphasizing again that we do not specify the basis associated with the 'covariant' time derivative D η as well as the indices a, b etc. This allows us to freely choose any convenient basis according to the concrete physical situations. For example, in the primitive basis with indices a, b → I, J etc, (3.2) reads since Z IJ ≡ 0 in the primitive basis. On the other hand, if we work in the kinematic basis with indices a, b → i, j etc, by definition e ( i ) ≡ const in kinematic basis, (3.2) now becomes Since in kinematic basis e (1) i = δ 1 i and e (2) i = δ 2 i , (3.4) implies nothing but Z i 1 = θ ' 1 δ 2 i , which is consistent with (3.8) (see the following). In the 'mass basis'with indices a, b → m,n etc, (3.2) is In the two-field case considered in [32, 38] , e (1) m = { cos ψ, sin ψ } and e (2) m = {-sin ψ, cos ψ } where ψ is the angle of the trajectory relative to the mass basis (e.g. approximately the light direction of the potential) and θ 1 ≡ θ = ψ + θ m where θ m is the angle of the mass basis relative to the field manifold. (3.5) implies Z mn → θ ' m ( 0 -1 1 0 ) , which is nothing but the defintion of the connection Z ab in mass basis. Then the idea is to use D η e (2) a to generate e (3) a . Since e (2) a is already normalized, D η e (2) a is orthogonal to e (2) a itself, while simple calculation yields e (1) a D η e (2) a = -θ ' 1 . Thus we may define e (3) a through where θ ' 2 is a new normalization factor independent of θ ' 1 . The sign of θ ' 1 , θ ' 2 etc should be chosen such that the orientation of the set of basis vectors is fixed through the whole evolution. A consistent orientation guarantees both basis vectors and θ ' i 's are smooth functions of time, which is important especially when the trajectory is oscillating [32]. The above procedures can be repeated order by order. In general we have a recurrence relation from which a full set of basis vectors { e ( i ) a } with i = 1 , · · · , N can be constructed. (3.7) implies the entries in the 'connection' matrix Z ij := e ( i ) a D η e ( j ) a in kinematic basis are non-vanishing if and only if | i -j | = 1 [22, 39]. That is, in matrix form, This peculiar structure of Z ij plays a key role in determining the dynamics of the multiple field perturbations. By using (3.7) iteratively, a full set of kinematic basis vectors can be constructed in terms of linear combinations of high order (covariant) time derivatives of the inflaton velocity or precisely e (1) a ≡ n a . For example, etc. The ( N -1) parameters θ ' i 's are determined by the normalization of the basis vectors, which are functions of the background velocity and its high order time derivatives. They are generalizations of the popular θ ' in two-field models [21], which has a simple geometric explanation as the changing rate of the direction of the background trajectory. While in multiple filed cases, there is no intuitive geometric meaning associated with θ ' i 's. Note although θ ' i 's are defined in terms of 'covariant' time derivatives D η which differs from basis to basis, θ ' i 's are basis independent, which characterize the intrinsic geometric properties of the trajectory. Together with the amplitude of background velocity σ ' , { σ ' , θ i } form a complete set of kinematic quantities, in terms of which the equations of motion for both the background and the perturbations can be written more conveniently.", "pages": [ 5, 6, 7 ] }, { "title": "3.2 kinematic decomposition of the background equations", "content": "Having a set of kinematic basis vectors, we are able to decompose the background equations of motion for the scalar fields in this basis. As we will show, this decomposition can be viewed as the linear algebraic equations for the basis vectors { e ( i ) a } . It is convenient to work with comoving time η , in terms of which the background equation for the scalar field (2.2) or (2.8) can be written as In terms of kinematic vectors, (3.11) can be recast as which is an algebraic equation among the kinematic vectors and the derivatives of the potential. (3.12) implies that V ,a completely lies on the plane spanned by e (1) a and e (2) a , which is a 2-dimensional subspace of the space of perturbations. The projection of (3.12) onto e (1) and e (2) yields respectively the well-known adiabatic background equation and [21] where V , 1 ≡ e (1) a V ,a etc. (3.12) set up the relation between e (1) and e (2) . In order to show further relations among e ( i ) 's, the idea is to take time derivatives of (2.7) or (3.11), order by order. This will generate a hierarchy of equations involving the kinematic basis vectors.", "pages": [ 7 ] }, { "title": "3.2.1 D η ¯ E a", "content": "Taking time derivative on (3.11) yields which is a second order equation for the velocity vector φ ' a with an analogue of 'mass matrix': Comparing (3.15) with the equation of motion for the perturbations (2.10), besides the absence of k 2 in (3.15), the only difference is in the 'mass matrices': where we have used (3.12) to replace V ,a in terms of e (1) a and e (2) a . The right-hand-side of (3.17) only depends on the kinematic quantities and has a simple and definite structure in kinematic basis. The term 3 δ ab is a universal self-coupling for all modes due to the expansion of the universe, while term proportional to e (1) a e (1) b induces a self-coupling of the adiabatic mode u (1) , term proportional to e (1) a e (2) b + e (1) b e (2) a implies a mixing between the adiabatic mode u (1) and the first entropic mode u (2) . This fact that M ab and W ab coincide for mixing among different modes other than u (1) and u (2) implies that, the field velocity φ ' a and the perturbation modes u a have essentially the same coupling relations . Thus, to study the coupling relations among different perturbation modes u a in the kinematic basis is essentially equivalent to the investigate the mixing among background kinematic quantities φ ' a , φ '' a , etc, or more conveniently, the kinematic basis vectors e ( i ) a 's. Using φ ' a = σ ' e (1) a and (3.7), after some manipulations, (3.15) can be rewritten in terms e ( i ) a 's as which can also be derived by taking derivative of (3.12) directly. On the left-hand-side of (3.18) we do not expand D 2 η e (1) a , instead, we deliberately group terms into a particular combination V ab e (1) b with As we will see in the next section, this 'operator' V ab also appears in the mixing among different perturbation modes. In particular, (3.18) implies that the action of V ab on e (1) will map e (1) to a linear combination of e (1) and e (2) , which is essentially the reason that the adiabatic mode only couples to the first entropic mode (see Sec.4.1.1). For later convenience, we define which is nothing but the components of the operator V ab in kinematic basis. For later convenience, it is interesting to note V ij is neither symmetric nor antisymmetric, instead where Z ij is given in (3.8). (3.18) is a 'vector' equation, of which the projection onto e (1) yields the adiabatic component with where we have used e (1) a D 2 η e (1) a = -θ ' 2 1 and C 1 is given in (3.19). After some manipulations, (3.24) can be recast as which can be viewed as an equation of motion for /epsilon1 with 'source terms' θ ' 2 1 -a 2 V , 11 . Projecting (3.18) onto e (2) a yields V 21 ≡ a 2 V , 21 + θ ' 1 = C 2 , i.e. [32] which is a propagating equation for θ 1 . (3.25) and (3.26) form a set of coupled equation for /epsilon1 and θ 1 , based on which appropriate approximations can be more easily made than solving (3.12) and (3.14) directly 3 . Since the right-hand-side in (3.18) only contains e (1) and e (2) , projecting (3.18) onto other basis vectors gives which implies (using (3.8)) a 2 V , 31 = -θ ' 1 θ ' 2 and V ,i 1 = 0 for i ≥ 4 .", "pages": [ 7, 8, 9 ] }, { "title": "3.2.2 D 2 η ¯ E a", "content": "When going to higher order, it is convenient to use (3.18) as the starting point. Taking time derivative on (3.18) straightforwardly yields with V ,a 11 ≡ V ,abc e (1) b e (1) c and where we have plugged C 2 and C 1 is given in (3.19). (3.28) is the analogue of (3.12) and (3.18) on the next order, of which the left-hand-side is just V ab e (2) b ≡ V a 2 . The right-hand-side of (3.28) contains two types of terms: one is a summation of kinematic basis vectors e ( i ) a with i = 1 , 2 , 3 , the other is proportional to V ,a 11 which is a high order derivative of the potential. At this point, we denote the coefficient of e (1) a on the right-hand-side of (3.28) as ˜ C (2) 1 , since the component of e (1) can be further reduced, as we show below. For a consistency check, projecting (3.28) onto e (1) a yields where in the last equality we used σ ' V , 111 ≡ e (1) a ( D η V ,ab ) e (1) b ≡ D η ( e (1) a V ,ab e (1) b ) -2 e (1) a V ,ab D η e (1) b . After plugging the expressions for V , 11 and V , 21 (3.24)-(3.26) into (3.32), one finds which is indeed related to V 21 through (3.23). This is not surprising since projecting (3.28) onto e (1) a will not bring any new information. Since V 12 is complete determined by the kinematic quantities, finally (3.28) can be recast as with a redefined (untilded) coefficient As we will see in the next section, for our purpose we are interested in V 32 , which is given by the projection of (3.34) onto e (3) : More over, since the right-hand-side of (3.34) only involves e ( i ) up to i = 3 , it immediately follows", "pages": [ 9, 10 ] }, { "title": "3.2.3 D 3 η ¯ E a", "content": "Taking a further time derivative on (3.28) yields with the 'potential' term and coefficients of the kinematic terms where in the last equality in (3.43) we have plugged (3.31). At the first glance, the right-hand-side of (3.38) involves e (1) a . However, one can show that the projection of the right-hand-side of (3.38) onto e (1) a identically vanishes. In fact, Using (3.32) and plugging (3.30) into (3.40), it immediately follows that ˜ C (3) 1 -a 2 e (1) a ˜ P (3) a ≡ 0 . This fact implies the right-hand-sdie of (3.38) actually contains no component along e (1) a . This is consisitent with the fact that V 31 = V 31 = 0 . Following the same logic, the e (2) component on the right-hand-side of (3.38) can also be reduced. Indeed, the projection of (3.38) onto e (2) a yields V 23 = ˜ C (3) 2 e ( j ) a -a 2 ˜ P (3) 2 , which is related with V 32 through V 23 = V 32 +2 Z ' 23 , while V 32 has already been given in (3.34), i.e. V 32 = C (2) 3 -a 2 σ ' θ ' 1 V , 311 . Combining the above together, finally (3.38) can be recast as with a redefined coefficient and a redefined potential term where ˜ P (3) i ≡ e ( i ) a ˜ P (3) a . Note the e (1) a components of P (3) a and thus of V a 3 given in (3.44) have already been removed.", "pages": [ 10, 11 ] }, { "title": "3.2.4 Higher orders", "content": "The above procedure can be applied to higher orders. Although the expressions become more and more involved, they obey a general structure (see Appendix A for the derivation): where P ( i ) a denotes terms proportional to the higher order derivatives of the potential, which satisfies an iterative relation given in (A.15). The coefficients in front of the kinematic basis vectors satisfy iterative relations with C (1) 1 ≡ C 1 and C (1) 2 ≡ C 2 . After some manipulations, we find from which C ( i ) i can also be evaluated [51]. On the right-hand-side of (3.47), besides the 'potential term' P ( i ) a which are composed of higher order derivative of the potential, the action of V ab on e ( i ) a will generate a linear combination of terms proportional to e ( i ) a and e ( i ± 1) a , which is crucial for our analysis.", "pages": [ 11 ] }, { "title": "4 Couplings among perturbations", "content": "Considering a general set of orthogonal normal vectors { e (1) a , e (2) a , · · · , e ( i ) a , · · · } , the perturbation modes are decomposed as u a = ∑ i u ( i ) e ( i ) a . The action for the projection of perturbation modes in this basis is (with implicit summation over i, j indices) where M ij is the mass matrix (2.11) in this given basis and Z 2 ij stands for Z ik Z kj . The corresponding equations of motion for the i -th mode u ( i ) are: with 'source term' /negationslash /negationslash (4.1)-(4.2) can also be read from (2.9)-(2.10). In kinematic basis with e ( i ) 's defind in Sec.3.1, the mass matrix (2.11) takes the form where we have used (3.12) to replace V ,a in terms of kinematic quantities. Plugging (4.4) into (4.3), the source term get reduced to /negationslash where V ij is defined in (3.22). In (4.5), the first term in the square bracket depends on the background inflationary velocity, which only couples the adiabatic mode u (1) to the first entropic mode u (2) , with coupling depending on the changing rate of the direction of the trajectory. The second term in (4.5) comes from the rotation of the basis, which introduces couplings between u ( i ) and u ( i ± 1) due to the specific structure of Z ij (3.8) in kinematic basis. The third term in (4.5) is the combination of V ,ab and the effect from the rotation of the basis. As we have seen, we can use the background equations of motion as well as their time derivatives investigated in the previous section to reduce the structure of V ij and thus of the source term S ( i ) .", "pages": [ 12 ] }, { "title": "4.1.1 Adiabatic mode", "content": "For the adiabatic mode u (1) , its source term is /negationslash where we have used the fact that V i 1 = 0 for i ≥ 3 (see Sec.3.2.1). At this point, we have seen that, the adiabatic mode is only coupled to the first entropic mode u (2) . From (3.18), and plugging Z 21 = θ ' 1 , we have The source term S (1) has an overall factor θ ' 1 , which reveals the well-known fact that the adiabatic mode is sourced only when the background trajectory is bending with respect to the field manifold [21, 22]. Moreover, (4.8) implies that the adiabatic mode u (1) is only coupled to the first entropic mode u (2) , while none of the other entropic modes can source the adiabatic mode. In [39], the same conclusion was made based on slow-roll/slow-turn approximation and by neglecting second derivative terms in the equations of motion (see also [28] for a concrete formulation in a three field model), while we have shown that it is true exactly. In particular, the couplings are controlled only by kinematic quantities z and θ ' 1 , which satisfy the coupled equations of motion (3.25) and (3.26). For completeness, we also evaluate the left-hand-side of (4.2) for the adiabatic mode. Since a 2 M 11 + Z 2 11 = -z '' /z , the full equation of motion for the adiabatic mode takes the form which is well-known for a long time in two-field cases [21, 22]. In this work we show that (4.9) is exactly valid in general multi-field models.", "pages": [ 12, 13 ] }, { "title": "4.1.2 The first entropic mode", "content": "The source term for the first entropic mode u (2) takes the form /negationslash where we have plugged Z 12 = -θ ' 1 , Z 32 = θ ' 2 and V 12 is given in (3.32) or (3.33). For i ≥ 3 , V i 2 's are given in (3.36)-(3.37). Combine the above together, we can write with where a superscript ' -' (or ' + ') denotes sourcing term from the previous (or next) neighbour mode, i.e u (1) (or u (3) ). (4.11) implies besides terms proportional to high order derivatives of the potential V ,j 11 , the first entropic mode u (2) is only coupled to the adiabatic mode u (1) and the second entropic mode u (3) , with couplings controlled by kinematic quantities: σ ' , z , θ ' 1 and θ ' 2 .", "pages": [ 13 ] }, { "title": "4.1.3 Higher orders", "content": "This procedure can be generalized to higher orders. For i ≥ 3 , the source terms (4.3) take the form /negationslash /negationslash The first term involves Z ji , which simply couples u ( i ) to u ( i ± 1) . The crucial quantity is V ji , of which the components of can be read from (3.47): where we denote P ( i ) j := e ( j ) a P ( i ) a for short. From (A.15), it follows P ( i ) j = P ( j ) i with the iterative relations (3.18) and (3.34) imply P (1) j ≡ 0 and P (2) j = σ ' θ ' 1 V ,j 11 ( j ≥ 2 ) respectively, starting from which the explicit expressions for P ( i ) j can be derived using (4.16) order by order [51]. Plugging (4.15) into (4.14), the source term can be rewritten as /negationslash Then using the expressions for C ( i ) i -1 , C ( i ) i +1 etc (see (3.51)-(3.52)), after some manipulations, finally we have /negationslash", "pages": [ 13, 14 ] }, { "title": "4.2 A 'chain' structure in the coupling relations", "content": "The last term on the right-hand-side in (4.18), which is proportional to P ( i ) j , involves high order derivatives of the potential. On the other hand, the first two terms S ± ( i ) involve only the kinematic quantities, which couple u ( i ) to its two neighbour modes u ( i ± 1) respectively - a 'chain' relationship 4 . This can be seen more explicitly at the level of action. From (4.1), there are two types of couplings among different modes in the Lagrangian, up to total derivatives: a 'friction term' and a 'mass term' with The first type of couplings in L A has already shown a 'chain' structure due to our deliberate construction of the kinematic basis. For L B , one of the main results in this paper is that, we can use the background equations of motion to reduce the components of the combination a 2 M ij + Z 2 ij + Z ' ij , such that the couplings in L B can be divides into two types: one involves the high order derivatives of the potential, which cannot be solved as kinematic quantities; the other is controlled only by the kinematic quantities. Remarkably, the later also shows a 'chain' structure. Precisely, L B can be recast as where γ i can be read from (4.8), (4.11) and (4.18): and P ( i ) j is given in (4.16). (4.23) in one of the main results in this work. In (4.23), the second term containing P ( i ) j , which involves high order derivatives of the potential, introduces complicated couplings among u ( i ) and other modes, whose explicit expressions can be found in [51]. In the present work, we concentrate on the couplings controlled by kinematic quantities, i.e L A and the first term in L B . Such kind of couplings manifest themselves in a 'chain' structure among different modes, which is illustrated in Fig.4.2. As a specific example, for models with triple fields 5 , the interaction terms are which presents an exact chain relation. While for models with more than three fields, there are additional mixings involving high order derivatives of the potential. This fact may be employed to extract characteristic quantities, which can be used to distinguish among models with less or more than three fields.", "pages": [ 14, 15 ] }, { "title": "5 Conclusion", "content": "One of the central topics of the cosmological studies in the recent years has been the interactions during inflation, including interactions at nonlinear order and among multiple perturbation modes. Contrary to the extensive studies of non-Gaussianities, however, the details of interactions in general multi-field models have less been investigated, even at the linear order. In this work, we make a first step in clarifying this issue. First we introduce the equations of motion for the both the background and the perturbations in a general basis, in which couplings from different origins manifest themselves more transparently. In this work, without specifying the concrete form of the potential, we revisit the kinematic basis. In Sec.3.2, we studied the background equations of motion ¯ E a = 0 and their high order (covariant) time derivatives D η ¯ E a = 0 , D 2 η ¯ E a = 0 etc, order by order. Their decomposition in the kinematic basis forms a 'hierarchy' of algebraic equations (3.47), which are nothing but the action of the 'operator' V ab ≡ δ ab D 2 η + a 2 V ,ab on the basis vectors e ( i ) a at each order. In Sec.4.1, we use this hierarchy of equations (3.47) to reduce the form of couplings among different modes order by order. This can be done relies on the fact that, the perturbation modes u a and the field velocity φ ' a have essentially the same coupling relations (see (3.17) and discussions below). Finally, the couplings among modes can be classified into two categories, one is controlled only by the kinematic quantities σ ' and θ ' i 's associated with the inflationary trajectory, the other is controlled by the high order derivatives of the potential, which cannot be solved in terms of kinematic quantities. This can be seen more transparently at the level of action. Remarkably, couplings in the first class manifest a simple 'chain' structure. That is, in kinematic basis, each mode u ( i ) is only coupled to its two neighbour modes u ( i ± 1) . The formalism and results presented in this work have several possible applications. First, the explicit clarification of the dependence of couplings among modes on features of the trajectory and potential will enable us to find basis other than the kinematic/potential basis according to the specific models, in which analysis and approximations can be more easily made. More importantly, the mode interactions serve as the critical bridge between the inflationary models and the observables. In the current work, although we have not solved the coupled system explicitly, the resulting spectra will be functions of both kinematic quantities { σ ' , θ ' 1 , θ ' 2 , · · · } and derivatives of the potentials. The former is controlled by the trajectory as well as the initial conditions, while the later is sensitive to the geometry of the potential. One thus can extract characteristic quantities to distinguish multi-field models from single-field models as well as among multi-field models themselves, or to constraint the inflationary models quantitatively. In particular, the 'chain' structure in the couplings controlled by the kinematic features is sensitive to the effective number of fields active during inflation. For example, in [39] a multi-field observable β 2 was introduced in order to distinguish two-field models from models involving more than three effective fields. In [49, 50], attempts have also been made in extracting information on the number of fields. Last but not least, we emphasize that although the coupling structure revealed in this work manifest themselves transparently in the kinematic basis, they do not essentially rely on the kinematic basis, as everything can be rewritten in a basis-independent manner.", "pages": [ 15, 16 ] }, { "title": "Acknowledgments", "content": "I appreciate C. Peterson and M. Tegmark for helpful correspondence. I was supported by ANR (Agence Nationale de la Recherche) grant 'STR-COSMO' ANR-09-BLAN-0157-01.", "pages": [ 16 ] }, { "title": "A Details in deriving (3.47)", "content": "Weuse the iterative method to prove (3.47). We have shown in Sec.3.2 that for i = 1 , 2 , 3 , the decomposition obeys (3.47). We assume (3.47) is valid at i -th order with i ≥ 3 . Taking time derivative on both sides of (3.47) and after tedious manipulations, we get the ( i +1) -th equation with the 'potential term' and coefficients of the 'kinematic terms' Apparently, the summation over j runs from i -2 to i +2 in (A.1), which does not take the form as (3.47). However, the form of (A.1) can be further reduced. Essentially this is because the components of the ( i + 1) -th order equation (3.47) along e ( j ) a 's with j ≤ i are not independent, which can be given in terms of lower order equations. Projecting (A.1) onto e ( j ) a with j ≤ i -1 yields Inversely, according to our ansatz (3.47), projecting the j -th equation (with j ≤ i -1 ) onto e ( i +1) a yields which is composed of only terms proportional to the high order derivatives of the potential. From (3.23), V j,i +1 ≡ V i +1 ,j (this can also be verified explicitly by comparing terms on the right-hand-sides in (A.8) and (A.9) [51]), which implies the 'kinematic term' ˜ C ( i +1) i -2 e ( i -2) a and ˜ C ( i +1) i -1 e ( i -1) a on the right-hand-side of (A.1) are not necessary and can be absorbed into a redefined 'potential terms'. Similarly, projecting (A.1) onto e ( i ) a yields while inversely, projecting the i -th equation onto e ( i +1) a yields In this case, (3.23) gives V i,i +1 = V i +1 ,i + Z ' i,i +1 ≡ V i +1 ,i -θ '' i . Thus the i -th component of (A.1) can be rewritten as Finally, put all the above together, (A.1) can be recast as with a redefined coefficient (A.13) exactly takes the form as (3.47). In particular, the summation over j runs from i to i +2 . The subtlety is in the redefined 'potential term' P ( i +1) a . From the above analysis, it can be written in a unified form with ˜ P ( i +1) j := e ( j ) a ˜ P ( i +1) a and ˜ P ( i +1) a given in (A.2).", "pages": [ 16, 17, 18 ] } ]

[ { "title": "Joseph Bramante, a Jason Kumar, a Elliot Nelson, b Sarah Shandera b,c", "content": "E-mail: [email protected], [email protected], [email protected], [email protected] Abstract. We demonstrate that local, scale-dependent non-Gaussianity can generate cosmic variance uncertainty in the observed spectral index of primordial curvature perturbations. In a universe much larger than our current Hubble volume, locally unobservable long wavelength modes can induce a scale-dependence in the power spectrum of typical subvolumes, so that the observed spectral index varies at a cosmologically significant level ( | ∆ n s | ∼ O (0 . 04)). Similarly, we show that the observed bispectrum can have an induced scale dependence that varies about the global shape. If tensor modes are coupled to long wavelength modes of a second field, the locally observed tensor power and spectral index can also vary. All of these effects, which can be introduced in models where the observed non-Gaussianity is consistent with bounds from the Planck satellite, loosen the constraints that observations place on the parameters of theories of inflation with mode coupling. We suggest observational constraints that future measurements could aim for to close this window of cosmic variance uncertainty. Keywords: Cosmology, Inflation, Non-Gaussianity", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The temperature fluctuations of the Cosmic Microwave Background (CMB) have been measured to a remarkable precision by the Planck satellite [1-3]. Two of the inferred properties of the primordial scalar curvature fluctuations have particularly important implications for theories of the very early universe: the strong evidence for a red tilt in the primordial power spectrum and the limits on the amplitude of any primordial non-Gaussianity. This evidence from the power spectrum and the bispectrum supports the simplest models of inflation with a single degree of freedom and no significant interactions, but does not yet rule out other possibilities. Future constraints or measurements of non-Gaussianity will continue to provide significant avenues to differentiate models of inflation. Interestingly, while non-Gaussian signatures offer a way to distinguish inflation models with identical power spectra, mode coupling also introduces a new and significant uncertainty in matching observations to theory [4-8]. In a universe much larger than our current Hubble scale, our local background may not agree with the global background used to define homogeneous and isotropic perturbations on a much larger region generated from inflation. If modes are coupled, the observed properties of the statistics in our Hubble volume will depend on the long wavelength background which is not independently observable to us. That is, our local statistics may be biased. This cosmic variance due to mode coupling was discussed for curvaton models in [4, 5] and was recently explored more generally in [6-9] for non-Gaussianity generated by arbitrary non-linear but local transformations of a Gaussian field. This non-Gaussian family is described by the local ansatz [10, 11] for the curvature perturbation ζ : where ζ G ( x ) is Gaussian and f NL , g NL , etc. are constants. For curvature perturbations of this type the amplitude of fluctuations and the amplitude of non-Gaussianity (the observed f NL , g NL , . . . ) vary significantly throughout the entire inflationary region. This is true even with globally small fluctuations, weak non-Gaussianity, and of order 10-100 extra e-folds of inflation. In this paper we explore further implications of mode coupling in the primordial fluctuations. We focus on a generalization of the local ansatz above, allowing f NL , g NL , etc. to be scale-dependent. In that case, the curvature fluctuations measured in subvolumes do not all have the same spectral index and bispectral indices as the parent theory. In other words, the possibility of mode-coupling, even at a level consistent with Planck bounds on non-Gaussianity, relaxes the restrictions that the precisely measured red tilt places on the theory of the primordial fluctuations. This paper and [6-8] work out the observational consequences of mode coupling in the post-inflation curvature fluctuations without asking which dynamics generated the fluctuations. We work with curvature perturbations that are assumed to be output by some inflationary model, and the mode-coupling effects we discuss are only significant if modes of sufficiently different wavelengths are physically coupled. Purely single field models of inflation do not generate such a coupling [12-14], and in Section 4 we demonstrate how to see directly from the shape of the bispectrum that single field type bispectra do not lead to cosmic variance from subsampling. So, although the curvature perturbations in Eq.(1.1) have a single source, the inflationary scenario they come from must be multi-field. For example, the distribution of locally observed non-Gaussian parameters f NL in the curvaton scenario was studied early on in [4, 5]. From the point of view of using observations to constrain inflationary scenarios that generate local non-Gaussianity, the variation allowed in local statistics means that the parameters in an inflaton/curvaton Lagrangian with local type mode coupling are not exactly fixed by observations. Given a Lagrangian and a restriction on the maximum size of a post-inflation region with small fluctuations, there is a probability for a set of observations (e.g., characterized by the power spectrum and f NL , g NL , etc.) in a patch of the universe the size we see today. Put the other way around, the parameters in the 'correct' Lagrangian need only fall within the range that is sufficiently likely to generate a patch with the properties we observe. Although the Planck bounds on local type non-Gaussianity are quite restrictive, they are not restrictive enough to eliminate the possibility of this effect: the data are also consistent with an application of the cosmological principle to a wider range of non-Gaussian scenarios with more than the minimum number of e-folds. In the rest of this section, we review the role of unobservable infrared modes coupled to observables in cosmology and particle physics. We also set up our notation and briefly review previous results for the local ansatz. In Section 2 we introduce a scale-dependent two-source ansatz, which changes the momentum dependence of the correlation functions. We compute features of the power spectrum and bispectrum observed in subvolumes and show that the locally observed spectral index of primordial scalar perturbations ( n obs s ) can be shifted by scale dependent coupling to modes that are observationally inaccessible. In Section 3 we illustrate how cosmic variance from mode-coupling affects the relationship between observation and theory for the spectral index and the amplitude of the power spectrum and bispectrum. The reader interested only in the consequences and not the detailed derivations can skip to those results. There we illustrate, for example, that a spectrum which is scaleinvariant on observable scales may look locally red or blue, and a red spectrum may look locally redder, scale-invariant, or blue when scale-dependent non-Gaussianity is present. It is unlikely to find subvolumes with an observed red tilt inside of a large volume with nearly Gaussian fluctuations with a blue power spectrum. However, a large volume with a blue power spectrum on observable scales due to a significant non-Gaussian contribution may have subvolumes with power spectra that are nearly Gaussian and red. We similarly show that the scale dependence of the bispectrum and higher order spectra in our Hubble volume can be shifted by non-Gaussian correlations with modes that are observationally inaccessible. Section 4 calculates the effect of a generic factorizable bispectrum on the amplitude and scale dependence of the power spectrum in subvolumes and verifies that not all bispectra lead to a variation in the locally observed statistics. The effects of mode coupling on the power spectrum and spectral index of tensor modes is considered in Section 5. We summarize our results in Section 6 and suggest future observational limits that could rule out the need to consider these statistical uncertainties in using observations to constrain (the slow-roll part of) inflation theory.", "pages": [ 2, 3, 4 ] }, { "title": "1.1 Long wavelength modes in cosmology and particle physics", "content": "There has been a great deal of recent literature on mode coupling in the primordial fluctuations, and an infrared scale appearing in loop corrections. In some cases, there may appear to be a naive infrared divergence as this scale is taken to be infinitely large. However, in calculating quantities observable within our own universe, such divergences clearly cannot be physical. For an example of a treatment of the infrared scale from an astrophysical perspective, see [15]. See [16] for a treatment in the simulation literature. Previous discussions in the context of single field inflation, but with a flavor similar to our work here, can be found in [17, 18]. The meaning of the infrared scale appears in computing n -point functions averaged locally in a given Hubble volume . Any long-wavelength modes come outside the expectation value and contribute a constant depending on the particular realization of the long wavelength modes in the local patch. These local n -point functions may differ from the global n -point functions due to the influence of long-wavelength background fluctuations, and the precise relationship between the local and global statistics depends on the local background. Of course, if one averages over all the individual subvolumes, the statistics must recover those initially defined in the large volume regardless of the scale chosen for the subvolume [19]. The relevance of unmeasurable infrared modes is well studied in non-cosmological contexts. A well known example in particle physics is the appearance of Sudakov log factors in the cross section for electron scattering in quantum electrodynamics. One finds infra-red divergences in both the one-loop correction to the cross-section for exclusive e -e -→ e -e -scattering, and in the tree-level cross-section for the process e -e -→ e -e -γ where electrons scatter while emitting a photon. In both cases, the divergence is associated with very long wavelength modes of the electromagnetic field, and can be regulated by introducing an infrared cutoff (analogous to the infrared scale that appears in cosmological calculations). But neither of the scattering processes is observable in and of itself, because one cannot distinguish events in which a photon is emitted from events in which it is not if the wavelength of the photon is much larger than the size of the detector. Instead, the physically measurable quantity is the cross section for electrons to scatter while emitting no photons with energy larger than the energy resolution of the detector, E res . For this quantity, the infra-red divergences (and dependence on the infra-red cutoff) cancel, but a logarithmic dependence on E res is introduced (the Sudakov log factor). This dependence is physically meaningful and represents the fact that, if the energy resolution of the detector is degraded, then events in which a soft photon is emitted may appear to be exclusive e -e -→ e -e -scatters because the long-wavelength photon can no longer be resolved by the detector. In the context of primordial curvature fluctuations, the energy resolution is equivalent to the scale of the subvolume and is ultimately limited by the size of the observable universe; a mode with a wavelength much longer than the observable universe cannot be distinguished from a zero-mode. Furthermore, we only have one universe - we are stuck with one particular set of unmeasurable long wavelength modes. Predictions for observable consequences of inflation models with modecoupling should account for the fact that observations cannot access information about larger scales. In this work, we are assuming that we have been given a set of non-Gaussian inhomogeneities as output from a dynamical model for generating them. We limit ourselves to considering curvature perturbations on a spatial slice, defined with a notion of time appropriate for observations made in our universe after reheating, on which there are small perturbations on a homogeneous and isotropic background. We suppose that the spatial volume over which this description holds is unknown, but that it may be much larger than what we can currently observe. However, we do not consider a spatial slice that is significantly inhomogeneous on large scales so our results should be adjusted to apply to scenarios that enter the eternal inflation regime. Although our calculations contain integrals over momenta, there are no corresponding time integrals. Our momenta integrals are not the 'loops' from dynamics, but merely add up the effects of all the modes coupled to a mode of interest at a particular time. Our analysis will focus on how the fact that long wavelength modes are unobservable affects our ability to compare a particular set of local observations to a model prediction for the larger, statistically homogeneous slice.", "pages": [ 4, 5 ] }, { "title": "1.2 Statistics of ζ in a subsample volume with local type non-Gaussianity", "content": "The curvature perturbation in either a large volume ( L ) or a subsample volume ( M ) is defined as the fractional fluctuation in the scale factor a , where 〈 〉 L,M refers to the value of a field averaged over the volume L or M , respectively. We assume | ζ | , | ζ obs | glyph[lessmuch] 1, and thus keep only the linear term. Throughout the paper, we will denote with an 'obs' superscript quantities as defined within a subsample volume such as the observable universe, which do not correspond (except perhaps by a coincidence of values) to quantities in the larger volume. Since 〈 a 〉 M = 〈 a 〉 L (1 + 〈 ζ 〉 M ), we see that ζ and ζ obs are related by Dividing ζ into long and short-wavelength parts compared to the scale M , ζ ≡ ζ l + ζ s , and considering one particular subvolume we have [6] We have replaced the mean value 〈 ζ 〉 M with the field smoothed on scale M , ζ l (the only difference being the real space vs. Fourier space top-hat window functions). ζ l takes a particular constant value for the subsample in question. For the remainder of the paper, averages 〈 〉 are taken over the large volume L , and averages over the small volume M are represented by a subscript ' l '. In either volume, the two-point function defines the power spectrum, We will consider homogeneous and isotropic correlations, so the bispectrum ( B ζ ( k 1 , k 2 , k 3 )) is defined by From Eq. (1.5), we see that the power spectra in the two volumes are related by The amplitude of linearized fluctuations is thus rescaled by a factor of 1 + ζ l due to the shift in the local background by the same factor: fluctuations appear smaller in overdense regions and larger in underdense regions. In general, an n -point function averaged in the small volume differs from the corresponding n -point function averaged in the large volume by a factor (1 + ζ l ) -n . However, this shift does not affect the level of non-Gaussianity in the small volume, as quantified for example by the dimensionless connected moments M n ≡ 〈 ζ ( x ) n 〉 c / 〈 ζ ( x ) 2 〉 n/ 2 , nor does it affect the shapes of the n -point functions, which will be our focus, but only reflects the rescaling of ζ . In what follows, we will therefore drop factors of 1 + ζ l in expressions for spectral indices, and also in expressions for n -point functions, to which these factors yield corrections smaller than our level of approximation. Suppose the curvature perturbation in the large volume is given by the local ansatz 1 where ζ G is a Gaussian field. Splitting the Gaussian field into long- and short-wavelength modes in comparison to the scale of the subvolume, ζ G = ζ Gl + ζ Gs , the long-wavelength pieces of higher order terms can be recollected in the coefficients of lower order terms. The curvature perturbation observed in a subvolume, ζ obs , is The coefficients f obs NL , g obs NL , etc., and the power spectrum now depend on the particular realization of long-wavelength modes for the subvolume [8]: where we have defined the power spectrum P G of the Gaussian field ζ G , and the bias for a given subvolume (in a fixed size large volume) as The bias is larger for more rare fluctuations and increases as the size of the subvolume considered is decreased. Leading contributions from superhorizon modes in Eqs. (1.11), (1.12) then go like f NL 〈 ζ 2 G 〉 1 / 2 B , where f NL 〈 ζ 2 G 〉 1 / 2 is the level of non-Gaussianity in the large volume. As the size of the subsample approaches the smallest measurable scale (which means there are no measurable modes within the subsample), the bias for average volumes asymptotes to 1. To compare different large volume theories, with different spectral indices and different sizes for the large volume, notice that the degree of bias in any subvolume is also sensitive to the IR behavior of the power spectrum since where N = ln( L/M ) is the number of superhorizon e-folds, P G ( k ) ≡ ( k 3 / 2 π 2 ) P G ( k ) is the dimensionless power spectrum, and we have assumed a constant spectral index n ζ ≡ d ln P G d ln k in evaluating the integral. This is the running if the Gaussian field ζ G , which is distinguished from the total running n s ≡ d ln P ζ d ln k . For a red tilt, n ζ < 1, with enough superhorizon modes, N glyph[greaterorsimilar] | n ζ -1 | -1 , the cumulative power of long-wavelength modes makes the degree of bias much greater than in the scale-invariant case [8]. The relationship between the parameters describing the fluctuations (e.g., P , f NL ) measured in a single small volume and those in the large volume depends in an unobservable way on the unknown IR behavior of the power spectrum. In Figure 1 we show the dependence of 〈 ζ 2 Gl 〉 on the number of superhorizon e-folds for different values of the spectral index. As detailed in [6-8], the coupling of long and short wavelength modes always present in an arbitrary member of the family in Eq.(1.9) (that is, arbitrary values of the coefficients) means that small sub-volumes tend to look like the local ansatz with 'natural' coefficients, f obs NL ( P obs ζ ) 1 / 2 glyph[lessmuch] 1 and higher order terms falling off by the same small ratio. This effect was illustrated several years ago for the case of a purely quadratic term in [20]. In addition, although the shapes of the correlation functions from arbitrary members of the local family are not identical they all have a sizable amplitude in squeezed configurations (e.g, k 1 glyph[lessmuch] k 2 ∼ k 3 for the bispectrum).", "pages": [ 5, 6, 7, 8 ] }, { "title": "2.1 The power spectrum", "content": "Next, we consider mode coupling effects for a generalized local ansatz for the real space curvature that has multiple sources with scale-dependent coefficients: where the dots also contain terms that ensure 〈 ζ ( x ) 〉 = 0. We have defined the fields φ G and σ G to absorb any coefficient of the linear terms, which typically appear, for example, in relating the inflaton fluctuations to the curvature. The Fourier space field, which we will use to do the long and short wavelength split, is For most of the paper, we will take f NL , g NL , etc. to be weakly scale-dependent functions: To determine the mapping between statistics in subsamples to those in the large volume, we follow the same procedure as in Section 1.2, splitting the Gaussian part of the curvature perturbation into long and short wavemode parts: φ G ≡ φ Gl + φ Gs , σ G ≡ σ Gl + σ Gs . The division happens at the intermediate scale M , which we take to be roughly the largest subhorizon wavemode today. Splitting into short and long wavemode parts results in a splitting of the convolution integrals. The non-Gaussian curvature perturbation is also split into ζ l and ζ s . For scales well within the subvolume, the locally observed random field is ζ s ; ζ l is the scalar curvature perturbation smoothed over a scale M , so it is a background to fluctuations observed in a subsample of size M . As described in Section 1.2, the local background ζ l shifts the amplitude of the fluctuations as defined in the small volume, ζ obs = ζ s / (1 + ζ l ), but for our purposes it is safe to neglect this shift, so ζ obs glyph[similarequal] ζ s . Carrying out the long- and short-wavelength split, we find 2 where we have neglected corrections from quartic and higher terms. We see that the scaledependent coefficients are corrected by long-wavelength pieces from higher terms and may scale differently in the small volume. With the assumptions that the two fields are not correlated, 〈 φ k 1 σ k 2 〉 = 0, and neglecting g NL and higher order terms, the total power spectrum in the large volume is where P φ and P σ are the power spectra for the Gaussian fields φ G and σ G , and we identify 〈 σ 2 Gs ( k ) 〉 = ∫ k M -1 d 3 p (2 π ) 3 P σ ( p ), 〈 σ 2 Gl 〉 = ∫ M -1 L -1 d 3 p (2 π ) 3 P σ ( p ), and for future use we define 〈 σ 2 G ( k ) 〉 = 〈 σ 2 Gl 〉 + 〈 σ 2 Gs ( k ) 〉 . We also define n σ ≡ d ln P σ d ln k and n φ ≡ d ln P φ d ln k for future reference, where P σ ( k ) ≡ k 3 2 π 2 P σ ( k ) and P φ ( k ) ≡ k 3 2 π 2 P φ ( k ). In the second line of (2.5) we have split the integral of the first line at the scale M -1 after using the approximation [21-23], The fractional contribution of the non-Gaussian source to the total power is where, P σ,NG ≡ P σ ( k )+ 18 25 f 2 NL ∫ k max L -1 d 3 p (2 π ) 3 P σ ( p ) P σ ( | k -p | ) includes all contributions from the σ sector of the perturbations. In the weakly non-Gaussian regime, ξ m ( k ) ≈ P σ ( k ) /P ζ,G ( k ) = P σ ( k ) / ( P σ ( k ) + P φ ( k )). This ratio is a weakly scale-dependent function if the power spectra are not too different, so we parametrize it as Splitting off the long-wavelength background in Eq. (2.4), the curvature observed in a subvolume is This expression shows that the local power on scale k in typical subvolumes may be nearly Gaussian even if the global power on that scale is not. In other words, consider Eq. (2.5) in the case that 36 25 f 2 NL ( k ) 〈 σ 2 Gl 〉 > 1 and 〈 σ 2 Gl 〉 glyph[greatermuch] 〈 σ 2 Gs ( k ) 〉 . The field σ on scale k is strongly non-Gaussian. However, in subvolumes with σ 2 Gl glyph[similarequal] 〈 σ 2 Gl 〉 the contribution to the local power spectrum quadratic in σ Gl (the last term in the first line of Eq. (2.9)) will give the dominant contribution to the Gaussian power while the local f 2 NL term (the term in the second line of Eq. (2.9)) can be dropped. The locally observed σ field on scale k is weakly non-Gaussian. When the locally observed field is nearly Gaussian (although the global field need not be, as described in the previous paragraph) the observed relative power of the two sources will vary in small volumes and is given by Notice that ξ obs m ( k ) | σ Gl =0 = ξ m ( k ), and that ξ m ( k ) = 1 implies ξ obs m ( k ) = 1.", "pages": [ 8, 9, 10 ] }, { "title": "2.2 The bispectrum and the level of non-Gaussianity", "content": "From the generalized local ansatz in Eq. (2.1), the large volume bispectrum is where the total Gaussian power, P G,ζ , comes from ζ G ≡ φ G + σ G . The terms proportional to three or more powers of f NL (evaluated at various scales) come both from the contribution from three copies of the quadratic σ G term from Eq. (2.1) and from the conversion between P G,σ and P G,ζ . Those terms may dominate the bispectrum if the model is sufficiently nonGaussian over a wide enough range of scales. The same quantity as observed in a weakly non-Gaussian local subvolume is where the . . . again denote terms proportional to more power of f NL . Comparing this expression to the previous equation in the weakly non-Gaussian regime, the observed bispectrum is again a product of functions of k 1 , k 2 , and k 3 . However, those functions are no longer equivalent to the Gaussian power and ratio of power in the two fields that would be measured from the two-point correlation. In other words, the coupling to the background not only shifts the amplitude of non-Gaussianity, but can also introduce new k -dependence which alters the shape of the small-volume bispectrum. Although a full analysis of a generic local type non-Gaussianity would be very useful, for the rest of this section we set g NL and all higher terms to zero for simplicity. These bispectra now have a more complicated shape than in the standard local ansatz, but for weak scale-dependence they are not still not too different. In practice one defines an f NL -like quantity from the squeezed limit of the bispectrum: where P obs ζ and B obs ζ are defined in terms of ζ obs . The definition of f eff NL in Eq. (2.13) is imperfect in any finite volume, since we cannot take the exact limit k l → 0. Instead, we must choose the long and short wavelength modes ( k l glyph[lessmuch] k s ) from within some range of observable scales. Since the best observational constraints over the widest range of scales currently come from the CMB, we will fix k l and k s in terms of the range of angular scales probed by Planck and define The observed non-Gaussianity in a subvolume can then be expressed in terms of the large volume quantities as where ( k s → k l ) indicates the same term as the preceding, except with k s replaced by k l , and this expression is evaluated with k l , k s equal to the limiting wavemodes observed in the CMB. As in the discussion below Eq. (2.9), this expression is valid even for f NL ( k ) σ Gl glyph[greaterorsimilar] 1 as long as we can neglect the 1-loop contribution to P obs ζ ; this must always be the case for our observed universe with very nearly Gaussian statistics. Keep in mind that f CMB NL is not a small volume version of the parameter f NL ( k ) defined in Eqs. (2.1), (2.2), which is a function of a single scale. Rather, f CMB NL corresponds to the observed amplitude of nearly local type non-Gaussianity over CMB scales for given values of f NL , ξ m , k s , k l , σ Gl . In the discussions above, we have considered the case when modes of a particular scale k may be strongly coupled (the term quadratic in f NL ( k ) dominates in P ( k )). However, it is also useful to have a measure of total non-Gaussianity that integrates the non-Gaussian contributions on all scales. For this we use the dimensionless skewness There are two important things to notice about this quantity compared to f NL ( k ) in the local ansatz itself and f CMB NL ( k s , k l ) as defined in Eq. (2.15). First, f NL ( k ) 〈 σ 2 Gl 〉 1 / 2 glyph[greaterorsimilar] 1 does not necessarily imply M 3 > 1, even in the single source case. The behavior of the power spectrum and bispectrum over the entire span of superhorizon and subhorizon e-folds enter M 3 . Evaluating M 3 ≡ 〈 ζ 3 ( x ) 〉 〈 ζ 2 ( x ) 〉 3 / 2 in the single-source case ( σ G → ζ G , n σ → n ζ , as defined in Section 1.2) for a scale-dependent scalar power spectrum, n ζ = 1, yields glyph[negationslash] where we have used the approximation shown in Eq. ((2.6)), and neglected 1-loop and higher contributions to 〈 ζ 3 〉 and 〈 ζ 2 〉 . Splitting the total e-folds on a scale appropriate for our cosmology, the number of subhorizon e-folds is N sub = 60. For a scale-invariant spectrum, n ζ = 1, the expression above becomes For the scale-independent case, n f = 0, this reduces to Notice that for n ζ = 1 and n f < 0 (the case of increasing non-Gaussianity in the IR), M 3 grows rapidly with N . On the other hand, if the power spectrum has a red tilt, n ζ < 1, M 3 will stay small for a wider range of n f values. The second thing to keep in mind about the M n is that the series gives a more accurate characterization of the total level of non-Gaussianity than f CMB NL or M 3 alone would. The level of non-Gaussianity as determined by M n +1 / M n is also what controls the size of the shift small volume quantities can have due to mode coupling. For example, in the two-field case the quantity controlling the level of non-Gaussianity of ζ is ξ m f NL P 1 / 2 ζ , where ξ m ( k ) is the fraction of power coming from σ G in the weakly non-Gaussian case. This quantity determines the scaling of the dimensionless non-Gaussian cumulants, (We specify the scale-dependent functions at some pivot scale as the cumulants involve integrals over these functions at all scales.) The quantity ξ m f NL P 1 / 2 ζ as defined in a subvolume differs from the large-volume quantity due to coupling to background modes [8]: where we have suppressed the scale-dependence, and the bias is now defined as so that it is larger when σ , which biases the subsamples, is a larger fraction of the curvature perturbation.", "pages": [ 10, 11, 12 ] }, { "title": "3 Observational consequences", "content": "In this section we illustrate the range of large-volume statistics that can give rise to locally observed fluctuations consistent with our observations. In considering the relationship between Planck CMB data and inflation theory, we set the scale of the subvolume to be M ≈ H -1 0 .", "pages": [ 12 ] }, { "title": "3.1 The shift to the power spectrum", "content": "Expressed in terms of the large volume power spectrum Eq. ((2.5)), the small volume power spectrum Eq. (2.9) is In the single field, scale-independent, weakly non-Gaussian limit, ξ m = 1 and f NL = const. , and Eq. (3.1) reduces to Eq. (1.11). The shift to the local power spectrum is proportional to the level of non-Gaussianity ξ m ( k ) f NL ( k ) 〈 ζ 2 G 〉 1 / 2 coupling subhorizon modes to long-wavelength modes. We will see in Section 3.2 below that if mode coupling is weaker on superhorizon scales, ξ m ( k ) f NL ( k ) 〈 ζ 2 G 〉 1 / 2 glyph[greaterorsimilar] 1 can be consistent with weak global nonGaussianity. Depending on the value of ξ m ( k ) and on the biasing quantity 6 5 f NL ( k ) σ Gl on the scale k , this shift is approximately In the 6 5 f NL ( k ) σ Gl glyph[lessmuch] 1 limit, the shift to the observed power comes from the O ( f NL σ Gl ) term, which increases or decreases the power from the field σ . In addition, the spectral index can change if the non-Gaussianity is scale-dependent (note the additional k -dependence from the f NL ( k ) σ Gl term in Eq. (2.9) as compared to Eq. (2.5)). New scale dependence can also be introduced if there are two sources contributing to ζ and one is non-Gaussian. In the 6 5 f NL ( k ) σ Gl glyph[greatermuch] 1 limit, where the global power P σ ( k ) on subhorizon scales is dominated by the 1-loop contribution, the O ( f 2 NL σ 2 Gl ) term dominates. If the size of the background fluctuation is larger (smaller) than 1 σ , the power from the field σ will be increased (decreased) relative to the global average, 3 but with the same scale-dependence. Consequently, a shift in n s comes from the difference in running between the two fields: the observed running n obs s will be shifted by the running of the fields φ G or σ , depending on whether the power from the field σ is increased or decreased (see Eq. (3.4) below). Alternatively, if f NL ( k ) σ Gl = O (1) on or near observable scales, n s can be shifted due to the relative change in power of the linear and quadratic pieces of σ ; this scenario is shown below in Figure 6.", "pages": [ 12, 13 ] }, { "title": "3.2 The shift to the spectral index, ∆ n s", "content": "Eq. (3.1) shows that the presence of a superhorizon mode background causes the spectral index d ln P ζ d ln k ≡ n s -1 to vary between subvolumes. 4 Taking the logarithmic derivative of Eq. (3.1) with respect to k , we find where from Eq. (2.3) and Eq. (2.8), n f ≡ d ln f NL d ln k , n ( m ) f ≡ d ln ξ m d ln k , and we have mostly suppressed the k -dependence. From either (3.2) or (3.3) we see that depending on the value of ξ m ( k ) and on the level of non-Gaussianity 6 5 f NL ( k ) σ Gl on the scale k , this shift is approximately glyph[negationslash] where these expressions are approximate, and in particular the single-source limit 6 5 f NL ( k ) σ Gl glyph[greatermuch] 1 cannot be taken simply as the n ( m ) f → 0 , ξ m → 1 limit of Eq. (3.4). That limit requires the full expression, (3.3), from which we find that in the single source case when 6 5 f NL ( k ) σ Gl glyph[greatermuch] 1 and σ 2 Gl glyph[greatermuch] 〈 σ 2 Gs ( k ) 〉 , the correction to the power spectrum vanishes, ∆ n s glyph[similarequal] -n f / ( 3 5 f NL ( k ) σ Gl ) → 0. This equation indicates that these scenarios will also in general have a non-constant spectral index. Although we have not done a complete analysis, Eq.(3.3) shows that α obs s ( k ) ≡ d ln n obs s /d ln k = α s ( k ) should generically be of order slow-roll parameters squared, which is consistent with Planck results [2]. The shift to the spectral index is thus determined by runnings in the large-volume bispectrum, the level of non-Gaussianity on scale k (the strength of mode coupling between this scale and larger scales), and the amount of bias for the subvolume, which will depend on the number of superhorizon e-folds along with the size and running of the power spectrum outside the horizon. We stress that this shift depends on the non-Gaussianity and nonGaussian running of the statistics at the scale being measured , and does not depend directly on the superhorizon behavior of the bispectrum parameters f NL ( k ), ξ m ( k ). We will see below that even if f NL ( k ) or ξ m ( k ) fall swiftly to zero outside the observable volume, the shift ∆ n s will be significant if subhorizon modes k > H -1 0 are strongly coupled to superhorizon modes. Note also that the bias from a given background mode does not depend on the scale of the mode (except through the scale-dependence of P σ ) as σ Gl simply adds up all the background modes equally. We will see in Section 4 that this is not true for nonlocal mode coupling: infrared modes of different wavelength can be weighted differently. For the purpose of model building, it should be pointed out that when 6 5 f NL σ Gl < 0, equations (2.10) and (3.3) can diverge. For instance for a single source model with a hundred superhorizon e-folds ( 〈 ζ 2 Gs 〉 ∼ 0), equation (3.3) is inversely proportional to factors of (1 + 6 5 f NL σ Gl ) 2 . This would be cause for concern - naively it implies extremely large corrections to the spectral index when 6 5 f NL σ Gl ∼ -1. However because of the same proportionality, Eq. (2.15) will also diverge, indicating that the subsamples in this phase space would observe extremely non-Gaussian statistics ( f obs NL glyph[greatermuch] 10). Hence the Planck satellite's bound on nonGaussianity has already excluded the worst-behaved phase space for a negative combination of parameters and background fluctuation, 6 5 f NL σ Gl = 6 5 f NL 〈 σ 2 Gl 〉 1 / 2 B < 0. It was shown in [6-8] that strong non-Gaussianity in a large volume can be consistent with weak non-Gaussianity measured in typical subvolumes. Furthermore, for scaledependent non-Gaussianity, large f NL ( k ) on a given scale can be consistent with weak total non-Gaussianity (adding over all scales). In light of this, we would like to better understand for what values of the parameters, and in particular the global spectral index and bispectral indices, it is possible for a shift | ∆ n s | ∼ 0 . 04 to be typical in Hubble-sized subvolumes, while satisfying the following theoretical and observational conditions: A further possible criteria might be to require | ∆ n s | glyph[lessorsimilar] 0 . 1; for larger values the observed near scale-invariance n obs s glyph[similarequal] 1 might be an unlikely accident given the large variation in scale-dependence among subvolumes. However, Eq. (3.4) shows that this condition is satisfied even for large f NL ( k ) 〈 σ 2 Gl 〉 1 / 2 as long as the non-Gaussian runnings n f , n ( m ) f are not too large, which is also necessary to preserve conditions 1 and 4 above.", "pages": [ 13, 14, 15 ] }, { "title": "Example I: Single source perturbations with constant f NL .", "content": "To understand how the conditions above affect the parameter space, consider first the simple case of single-source, scale-invariant non-Gaussianity with only f NL non-zero: where f NL is constant. In Figure 2, we show the parameter space ( 〈 ζ 2 Gl 〉 , f NL ) consistent with ζ glyph[lessmuch] 1, the observed power spectrum and observed bounds on non-Gaussianity. The dashed black line divides the parameter space where the entire volume is on average weakly or strongly non-Gaussian by setting M 3 glyph[similarequal] 18 5 f NL 〈 ζ 2 G 〉 1 / 2 = 1. This dashed line levels off in parameter space with very small superhorizon contributions to M 3 , 〈 ζ 2 Gl 〉 glyph[lessmuch] 〈 ζ 2 Gs 〉 (meaning subhorizon fluctuations dominate the cumulative skewness), which for a nearly scale-invariant power spectrum implies N glyph[lessmuch] N sub . (Here and in the rest of this Section, we set the number of subhorizon e-folds N sub = 60.) When M 3 glyph[greaterorsimilar] O (1), the dominant contribution to bispectrum in the large volume is given by the higher order terms not explicitly written in Eq. (2.11). The shaded region to the right of the thin gray solid lines shows where ζ is no longer a small perturbation, either due to a large linear or quadratic term. The shaded region on the Gl left shows where f CMB NL in typical subvolumes is inconsistent with constraints from Planck . We see that in the weakly non-Gaussian regime, consistency with Planck reduces to f NL < 10, whereas in the strongly non-Gaussian regime the amplitude of fluctuations must be large enough, 〈 ζ 2 l 〉 1 / 2 ∼ f NL 〈 ζ 2 Gl 〉 glyph[greaterorsimilar] 1 10 , to sufficiently bias Hubble-sized subvolumes so that weak non-Gaussianity is typical. In this regime there is only a small window where 1 σ fluctuations give subvolumes consistent with observation, and requiring f CMB NL to be a factor of 10 smaller would essentially remove this small window. This is because f CMB NL ∼ 1 /f NL ζ 2 Gl ∼ 1 /ζ l , so if f CMB NL is constrained to O (1), ζ is forced to be nonperturbative. Thus, for strongly non-Gaussian, scale-invariant superhorizon perturbations on a homogeneous background geometry to be consistent with observation, the degree of non-Gaussianity in our subvolume would have to exceed the observed degree of inhomogeneity, 1 part in 10 5 . The remaining lines denote curves of constant N = 350 for different values of n ζ (which we take to be constant), fixed by the requirement that the observed amplitude of fluctuations be typical of subvolumes: P obs ζ ( k p ) = (1 + 6 5 f NL ζ Gl ) 2 〈 ζ 2 Gl 〉 n ζ -1 1 -e -N ( n ζ -1) = 2 . 2 × 10 -9 for a typical +1 σ background fluctuation ( ζ Gl = 〈 ζ 2 Gl 〉 1 / 2 ). The entire unshaded parameter space is consistent with the observed amplitude of fluctuations, once we impose this relationship be- rameters plotted. For f NL 〈 ζ 2 Gl 〉 1 / 2 glyph[lessmuch] 1, P obs ζ ≈ P G = 〈 ζ 2 Gl 〉 ( n ζ -1 1 -e -N ( n ζ -1) ) is fixed by the observed power spectrum, so curves of constant N approach a fixed value of 〈 ζ 2 Gl 〉 . On the other hand, for f NL 〈 ζ 2 Gl 〉 1 / 2 glyph[greatermuch] 1, P obs ζ ∝ f 2 NL ζ 2 Gl P G and curves of constant N approach lines of constant f NL 〈 ζ 2 Gl 〉 . The variation with n ζ shows that for a red (blue) tilt, a given number of superhorizon e-folds corresponds to a much larger (smaller) amplitude of fluctuations 〈 ζ 2 Gl 〉 . For a red tilt or flat spectrum, there is a maximum number of e-folds consistent with 〈 ζ 2 Gl 〉 < 1, whereas for a blue tilt as small as n ζ -1 ∼ P obs ζ ∼ 10 -9 , 〈 ζ 2 Gl 〉 will remain perturbatively small for an arbitrarily large number of e-folds. For instance, for n ζ = 1 . 04, having more than 50 superhorizon e-folds does not appreciably change the value of 〈 ζ 2 Gl 〉 in the region where f NL 〈 ζ 2 Gl 〉 1 / 2 glyph[lessmuch] 1 (the vertical part of the blue dashed line in Figure 2 will not shift right with the addition of more superhorizon e-folds). Note that, in the case where f NL is scale-invariant, M 3 is a function of superhorizon e-folds N (Eq. (2.19)). In order to calculate the dashed line for fixed M 3 in Figure 2 we assume n ζ = 1, which along with Eq. (1.15) fixes the number of superhorizon e-folds in terms of f NL and 〈 ζ 2 Gl 〉 . In the strongly non-Gaussian regime, moving along the allowed window in parameter space (along curves of constant N ) does not change the amplitude of fluctuations 〈 ζ 2 〉 or statistics ζ ∝ ζ 2 G , but only gives a relative rescaling to f NL and 〈 ζ 2 Gl 〉 . That is, requiring weak non-Gaussianity of a given size in typical subvolumes from a strongly non-Gaussian large volume singles out (in the scale-invariant case) a particular amplitude of fluctuations in the large volume, and as described above, this amplitude becomes nonperturbative when f obs NL ∼ 1 is typical. In the following section we will see how this condition can be removed in the case of scale-dependent non-Gaussianity: a blue running of f NL implies the level of non-Gaussianity attenuates at large scales.", "pages": [ 15, 16, 17 ] }, { "title": "Example II: Single source with running f NL ( k ) .", "content": "Next, consider a single source local ansatz with scale-dependent non-Gaussianity parameterized by n f ≡ d ln f NL d ln k . The parameter spaces for large volume statistics with n f = ± 0 . 1 and a red or scale-invariant power spectrum P G are shown in Figure 3. All plots here assume an overdense subsample with a +0 . 5 σ background fluctuation. Remarkably, the upper left panel shows that the super-Hubble universe could have a flat spectral index n ζ = 1, while still being consistent with Planck 's observations at the Hubble scale. Conversely, the right panels demonstrate that models with running non-Gaussianity which predict n ζ = 0 . 96 over a super-Hubble volume will typically yield a range of values for n obs s on observable scales in Hubble-sized subsamples. (The spectral index n s ( k p ) on observable scales is only well approximated by n ζ if 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 is sufficiently small; we will see in Figure 5 below that this is still consistent with a sizeable shift | ∆ n s | .) In these plots we require f CMB NL < 10 for a typical background fluctuation ζ Gl = 0 . 5 〈 ζ 2 Gl 〉 1 / 2 . Due to the dependence of f CMB NL on n f this condition is slightly stronger for positive n f , which can be seen by comparing the upper and lower diagrams in Figure 3. On the other hand, for larger background fluctuations, | ζ Gl | > 0 . 5 〈 ζ 2 Gl 〉 1 / 2 , the condition f CMB NL < 10 excludes less parameter space. In Figure 3 we compare only two types of spectral indices, n ζ < 1 and n ζ = 1. While the spectral index n ζ does not directly affect the parameter space constrained by f CMB NL and ζ G < 1, it does have the following two effects: ζ n n ζ = 1 , n = 1 , n f Strongly NG > 10 Weakly NG Nonperturbative N = 10, 100, - 8 10 - 6 10 - 4 10 2 > = 0 . 1 n = 0 . 1 n ζ CMB > 10 Weakly NG Nonperturbative N = 10, 50, 100, - 8 10 - 6 10 - 4 10 2 > - 2 10 - f ζ f L H 5 10 4 10 3 10 2 L p k H NL f f 10 1 10 0 10 - 1 10 - 2 10 NL CMB Gl - 2 10 0 10 5 10 4 10 3 10 2 L p k H NL f 1 10 0 10 - 1 10 - 2 5 10 10 4 10 3 10 2 L p k H NL f 1 10 0 10 - 1 10 - 2 10 10 10 Strongly NG Weakly NG = 0 . 1 D n s ns > 1.16 D D n n s s > > e v i t a b r u t r e p n o N -0.04 -0.04 N = 10, 50, 100, 200, 350 - 8 10 f NL - 6 10 = 0 . 96 , n 10 2 - 4 > f Strongly NG = - 2 10 0 . 1 > 0 = 0 . 96 , n f CMB > 10 NL Gl Gl - 0 10 0 10 to a smaller (larger) number of e-folds in the case of a red tilt (blue tilt), as shown in Figure 2, so it is easier to realize a large shift to a global red tilt than to a global blue tilt. In fact, as previously noted in the discussion of Figure 2, imposing the requirement that P obs ζ be typical of subvolumes for scenarios with a blue tilt n ζ -1 causes 〈 ζ 2 Gl 〉 to converge to a particular value as N is increased. 2. A red tilt in the power spectrum can relax the constraint from requiring weak global non-Gaussianity, as seen by comparing the right panels in Figure 3 to the left panels. For example, when n f > 0, a red tilt in the power spectrum gives more relative weight in M 3 to the more weakly coupled superhorizon modes and damps the power of strongly coupled subhorizon modes. Note that the bottom two panels in Figure 3 permit about the same number of super-horizon e-folds of weakly non-Gaussian parameter space. In the right panel the power removed from subhorizon e-folds by n ζ < 1 is balanced by power added to superhorizon e-folds leading to a larger background 〈 ζ 2 Gl 〉 per e-fold permitted for perturbative statistics as compared to the bottom left panel. For these reasons single-source scenarios with a red power tilt in the large volume have the most significant range of cosmic variance due to subsampling. The solid black lines in Figure 3 show ∆ n s ( k p ) = -0 . 04 in subvolumes with a +0 . 5 σ background fluctuation ( ζ Gl = +0 . 5 〈 ζ 2 Gl 〉 1 / 2 ), and thus show part of the parameter space where | ∆ n s ( k p ) | can be observationally significant. Here we have neglected the subhorizon one-loop correction 〈 ζ 2 Gs ( k p ) 〉 glyph[lessmuch] ζ 2 Gl ; this breaks down for small N but is valid outside of the region of parameter space excluded by the requirement f CMB NL < 10. Rewriting Eq. (3.3) for a single source scenario ( ξ m = 1), ∆ n single source s glyph[similarequal] n f ( 12 5 f NL ζ Gl ( 1 -36 25 f 2 NL 〈 ζ 2 Gl 〉 ) + 72 25 f 2 NL ( ζ 2 Gl -〈 ζ 2 Gl 〉 ) ) ( 1 + 6 5 f NL ζ Gl ) 2 ( 1 + 36 25 f 2 NL 〈 ζ 2 Gl 〉 ) . (3.6) Assuming n f = 0 . 1 and ζ Gl = 0 . 5 〈 ζ 2 Gl 〉 1 / 2 , we can solve this equation to show that ∆ n single source s = -0 . 04 when f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 = 0 . 94 or 5.9, which are the equations of the two black lines plotted in Figure 3. These lines assume positive f NL in the large volume, f NL > 0, but they remain the same for f NL < 0 and a -0 . 5 σ background fluctuation. For values of | n f | larger or smaller than 0 . 1, the distance between these lines grows or shrinks in parameter space. Of course, for the full expression of ∆ n s and a different set of parameter choices, there can be more than two solutions of | ∆ n s | = 0 . 04. For positive f NL ( k p ) ζ Gl (see below), the typical size of ∆ n s is largest in the region between these lines ( 6 5 f NL ( k p ) ζ Gl ∼ 1) and falls towards zero on either side. The upper dotted-dashed lines mark where 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 is large ( O (10)). When that quantity is large, ∆ n s glyph[similarequal] -n f 3 5 f NL 〈 ζ 2 Gl 〉 1 / 2 and thus approaches zero as indicated in Figure 3. Note that in this region the observed spectral index is n obs s glyph[similarequal] n s glyph[similarequal] n ζ +2 n f , so for the parameter choices in Figure 3 the Planck satellite excludes the region above the dotted-dashed lines. All lines and contours in Figure 3 assume that 6 5 f NL ( k p ) ζ Gl > 0 (eg, overdense fluctuations with positive f NL ). If this figure assumed 6 5 f NL ( k p ) ζ Gl < 0 (eg, overdense fluctuations with negative f NL ), the area in parameter space near the line 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 = 1 would be excluded. For further discussion of parameter space with 6 5 f NL ζ Gl < 0, see the discussion after Eq. (3.3). Figure 3 shows that, under the conditions we have imposed and the spectral indices considered, only scenarios where the bispectral tilt is not very red have typical subvolumes where the observed spectral index varies by an amount that is cosmologically interesting for us, | ∆ n s | glyph[greaterorsimilar] 0 . 01. A blue bispectral index may avoid the current observational constraints, which do not probe particularly small scales, and easily remain globally perturbative and weakly non-Gaussian (see paragraph below). In contrast, the bottom panels of Figure 3 illustrate that for either spectral index, a scenario with n f < 0 will be nonperturbative in the interesting part of parameter space where | ∆ n s | ∼ 0 . 04. (In addition, there is only a small window with strongly non-Gaussian but perturbative global statistics.) If both the power spectrum and non-Gaussianity increase in the IR, as in the lower right panel of Figure 3, the statistics will be strongly non-Gaussian across parameter space for a small number of superhorizon e-folds. The upper panels of Figure 3 illustrate a feature discussed in Section 2.2: 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 glyph[greaterorsimilar] 1 does not necessarily imply a large cumulative skewness, M 3 glyph[greaterorsimilar] 1. The dashed curves fix M 3 = 1 as a function of superhorizon e-folds, which are determined at each point in parameter space by the observed level of the power spectrum along with n f , f NL and 〈 ζ 2 Gl 〉 . In regions where M 3 < 1 but f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 glyph[greaterorsimilar] 1, there are a sufficient number of superhorizon modes with weaker coupling ( n f > 0) damp the total non-Gaussianity. To elaborate, in the limit n f ( N + N sub ) glyph[greatermuch] 1, Eq. (2.18) gives M 3 ∝ [ 〈 ζ 2 Gl 〉 /N ( N + N sub )] 1 / 2 . For f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 glyph[lessmuch] 1, N = 〈 ζ 2 Gl 〉 / P obs ζ and so M 3 becomes independent of 〈 ζ 2 Gl 〉 in the limit N glyph[lessmuch] N sub . For f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 glyph[greatermuch] 1, on the other hand, M 3 ∝ 1 /f NL ( k p ) 〈 ζ 2 Gl 〉 3 / 2 , so large f NL ( k p ) and sufficiently large 〈 ζ 2 Gl 〉 are needed to keep the total non-Gaussianity small, and P obs ζ ∼ 2 × 10 -9 typical in subvolumes, as seen in the upper left panel of Figure 3. Note that throughout this analysis, we have assumed n f is constant for all N sub = 60 subhorizon e-folds, so that for blue n f non-Gaussianity continues to grow on subhorizon scales where nonlinear evolution has taken over. If this condition is relaxed, the conditions from weak non-Gaussianity are less restrictive. Figure 4 shows the probability distribution for the shift ∆ n s for the parameters in part of the range of interest for the blue bispectral index shown in the top panels of Figure 3. Both panels show examples that (for appropriate choices of large volume parameters) give local power spectra amplitude and f CMB NL consistent with our observations. Notice that the distribution on the right is substantially less Gaussian than the distribution on the left. This trend continues if one considers larger 〈 ζ 2 Gl 〉 while keeping all other parameters fixed. In Figure 5 we show regions of parameter space in the ( 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 , n f ) plane that are consistent with the Planck measurement n obs s = 0 . 9603 ± 0 . 0073. Assuming that the scalar power spectrum in the full volume of the mode-coupled universe is completely flat, n ζ = 1, we see that 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 must be at least O (10 -1 ) and for weakly non-Gaussian statistics, more than a hundred superhorizon e-folds are required. It is interesting to note that in the case of a blue-tilted f NL , a larger running non-Gaussianity n f loosens parameter constraints coming from requiring perturbative statistics 〈 [ f NL ( k ) ζ 2 Gl ] 2 〉 glyph[lessorsimilar] 0 . 1. Although the dotted lines in Figure 5 will shift to the left with more superhorizon e-folds, these curves exclude less parameter space as n f becomes larger. This is because we have assumed n f is blue and constant so f NL is driven to smaller values in the IR and 〈 [ f NL ( k ) ζ 2 Gl ] 2 〉 becomes smaller for larger n f . Notice the shift in the non-perturbative line in the right panel that occurs at n f > | n ζ -1 | : if the running of the power spectrum is larger than the running of f NL ( k ), then the running of the power spectrum will dominate the variance of the local quadratic term over superhorizon modes, because f 2 NL ( k ) P G ( k ) 2 ∝ k 2( n f + n ζ -1) . Lastly, the Figure 4 . The probability of finding a shift in the spectral index in subvolumes. Left panel: The variance plotted here corresponds to about 195 extra e-folds in a model with n ζ = 0 . 96 or 4 × 10 4 extra e-folds for a scale-invariant spectrum. Right panel: The variance here is consistent with about 240 extra e-folds in a model with n ζ = 0 . 96 or 5 × 10 5 extra e-folds for a scale-invariant spectrum. In both panels the solid black lines show a bispectral index of n f = 0 . 05 while the dotted blue lines show n f = 0 . 1. In the right panel about 24% (6%) of subvolumes in the n f = 0 . 1 ( n f = 0 . 05) have ∆ n s ≥ 0 . 02 and 17% (5%) have ∆ n s ≤ -0 . 04. The points in parameter space that correspond to the dotted lines ( n f = 0 . 1) are shown with black squares in Figure 3. Figure 4 . The probability of finding a shift in the spectral index in subvolumes. Left panel: The variance plotted here corresponds to about 195 extra e-folds in a model with n ζ = 0 . 96 or 4 × 10 4 extra e-folds for a scale-invariant spectrum. Right panel: The variance here is consistent with about 240 extra e-folds in a model with n ζ = 0 . 96 or 5 × 10 5 extra e-folds for a scale-invariant spectrum. In both panels the solid black lines show a bispectral index of n f = 0 . 05 while the dotted blue lines show n f = 0 . 1. In the right panel about 24% (6%) of subvolumes in the n f = 0 . 1 ( n f = 0 . 05) have ∆ n s ≥ 0 . 02 and 17% (5%) have ∆ n s ≤ -0 . 04. The points in parameter space that correspond to the dotted lines ( n f = 0 . 1) are shown with black squares in Figure 3. right panel of Figure 5 shows once again that for a blue tilted f NL , the weakly non-Gaussian parameter space enlarges with the number of superhorizon e-folds, because f NL is driven to very small values over more superhorizon e-folds, decreasing the value of M 3 . To conclude this section, Figure 6 illustrates a single-source scenario in which a power spectrum which appears blue-tilted in the large volume on short scales can appear red on the same scales in a subvolume. On scales where P ζ ( k ) glyph[similarequal] P G ( k ), n s ( k ) glyph[similarequal] n ζ , whereas on scales where the 1-loop contribution dominates P 1-loop ζ ( k ) glyph[similarequal] 36 25 f 2 NL ( k ) 〈 ζ 2 Gl 〉P G ( k ) and the spectral index will be n s ( k ) glyph[similarequal] n ζ + 2 n f . If the transition of power takes place on a scale near the observable range of scales ( f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 = O (1)), the observed spectral index can be shifted. For example, if ζ 2 Gl < 〈 ζ 2 Gl 〉 , the blue-tilted f 2 NL 〈 ζ 2 Gl 〉 contribution loses power in the subvolume, and if f NL ( k p ) ζ Gl > 0, the red-tilted piece gains power (compare Eqs. (2.5), (2.9)). This scenario is shown in Figure 6. Note that as long as f NL ( k p ) is not extremely large (which would violate the constraint on f CMB NL for the value of f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 chosen here), ζ Gl glyph[greatermuch] 〈 ζ 2 Gs ( k ) 〉 1 / 2 and the 1-loop contribution to P obs ζ is very small, suppressed by a factor of 〈 ζ 2 Gs ( k ) 〉 /ζ 2 Gl . Example III: Multiple sources with running ξ m ( k ) . In the single-source case, a large shift to the observed spectral index could only occur if the 1-loop contribution to the power spectrum dominated on small scales. With two sources, a significant shift to n s can be consistent with weak non-Gaussianity ξ m ( k ) f NL ( k ) 〈 σ Gl 〉 1 / 2 < 1 on all scales. If the running of the 1-loop contribution lies between the runnings n σ ≡ d ln P σ ( k ) d ln( k ) and n φ ≡ d ln P φ ( k ) d ln( k ) of the Gaussian contributions to the total power, then it will be subdominant on large and small scales. The transition of power between σ G and φ G takes place over a finite range of scales, over H GLYPH<144> L H L Figure 5 . Left panel: a model with a globally flat power spectrum, but which contains subvolumes where a red tilt would be observed. Right panel: a model with global parameters naively matched to observations that nonetheless contains a significant number of subvolumes with a spectral index at odds with observations. Both cases show single-source perturbations with the running of f NL , n f , plotted against the parameters controlling the size of the bias, 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 . This figure assumes positive f NL and a blue running of f NL . The running of the power spectrum is flat ( n s ( k p ) glyph[similarequal] n ζ = 1) and red ( n s ( k p ) glyph[similarequal] n ζ = 0 . 96) to within ∼ 0 . 01 below the dotted-dashed lines in the left and right panels, respectively. Above the dotted-dashed lines the loop correction to the running of the power spectrum becomes large ( n s ( k p ) -n ζ > 0 . 01). Dashed lines indicate regions where the non-Gaussian cumulant M 3 > 1 for the number of superhorizon e-folds indicated. The dotted line indicates the nonperturbative region ( 〈 [ 3 5 f NL glyph[star] ( ζ 2 Gl -〈 ζ 2 Gl 〉 )] 2 〉 glyph[greaterorsimilar] 0 . 1) for N > 10 3 and N > 100 in the left and right panels, respectively. The grey space shows what region is excluded at 99% confidence by the Planck measurement n obs s = 0 . 9603 ± 0 . 0073, assuming an underdense subsample with a -1 σ background fluctuation. Figure 5 . Left panel: a model with a globally flat power spectrum, but which contains subvolumes where a red tilt would be observed. Right panel: a model with global parameters naively matched to observations that nonetheless contains a significant number of subvolumes with a spectral index at odds with observations. Both cases show single-source perturbations with the running of f NL , n f , plotted against the parameters controlling the size of the bias, 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 . This figure assumes positive f NL and a blue running of f NL . The running of the power spectrum is flat ( n s ( k p ) glyph[similarequal] n ζ = 1) and red ( n s ( k p ) glyph[similarequal] n ζ = 0 . 96) to within ∼ 0 . 01 below the dotted-dashed lines in the left and right panels, respectively. Above the dotted-dashed lines the loop correction to the running of the power spectrum becomes large ( n s ( k p ) -n ζ > 0 . 01). Dashed lines indicate regions where the non-Gaussian cumulant M 3 > 1 for the number of superhorizon e-folds indicated. The dotted line indicates the nonperturbative region ( 〈 [ 3 5 f NL glyph[star] ( ζ 2 Gl -〈 ζ 2 Gl 〉 )] 2 〉 glyph[greaterorsimilar] 0 . 1) for N > 10 3 and N > 100 in the left and right panels, respectively. The grey space shows what region is excluded at 99% confidence by the Planck measurement n obs s = 0 . 9603 ± 0 . 0073, assuming an underdense subsample with a -1 σ background fluctuation. glyph[negationslash] H GLYPH<144> L H GLYPH<144> L H L H GLYPH<144> L which n s changes from n σ to n φ . If the power spectrum of φ G is blue and dominates on small scales ( ξ m ( k glyph[greaterorsimilar] H 0 ) glyph[lessmuch] 1), and the Gaussian contribution from σ is red and dominates on large scales ( ξ m ( k << H 0 ) glyph[similarequal] 1), then the background ζ l glyph[similarequal] σ l for any subvolume couples to and biases the local statistics. For example, a globally flat or blue spectral index n s ( k > H 0 ) > 1 can again appear red, n obs s < 1, in a subvolume. The shift to n s can come only from the modulation of power in σ relative to φ G , and need not rely on running non-Gaussianity n f = 0. That is, a large running of the difference in power of the fields can be achieved without a large level of running non-Gaussianity. This becomes apparent upon inspecting the running of ξ m , n ( m ) f ( k ) ≡ d ln ξ m ( k ) d ln k = (1 -ξ m ( k )) [ n σ -n φ + 2 n f 36 25 f 2 NL ( k ) 〈 σ 2 G ( k ) 〉 1 + 36 25 f 2 NL ( k ) 〈 σ 2 G ( k ) 〉 ] . (3.7) If φ G is more red-tilted than σ G , the background is uncorrelated with short-wavelength modes because φ G dominates on large scales, ζ l glyph[similarequal] φ Gl , so local statistics are not biased. Thus, both n σ ≤ 1 and n φ > n σ are needed for a significant bias. In Figure 7 we show the parameter ln L H n ζ = 0 . 95 n f = 0 . 05 n ζ = 0 . 95 n f = 0 . 05 Figure 6 . Top panel: The contributions to the power spectrum P G ( k ) and P 1-loop ζ ( k ) glyph[similarequal] 36 25 f 2 NL ( k ) 〈 ζ 2 Gl 〉P G ( k ) are shown, for the following parameter choices: n ζ = 0 . 95, n f = 0 . 05, f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 = 3. The total power spectrum is shown with a thin black line, and the corresponding shifted power spectra for a subvolume with a +0 . 1 σ background fluctuation is shown with a thick black line. The vertical scale can be fixed so P obs ζ matches the observed value. Bottom panel: Parameter space for single source non-Gaussianity with n ζ = 0 . 95 and n f = 0 . 05 is shown. The dotted-dashed line indicates f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 = 10, both black lines indicate ∆ n s = -0 . 065 for a +0 . 1 σ background fluctuation, and the red circle indicates the parameter space congruent with the top panel. Dotted lines show the indicated number of superhorizon e-folds for a +0 . 1 σ bias. The exclusion regions are marked the same as those in Figure 3, but these assume a +0 . 1 σ bias. Figure 6 . Top panel: The contributions to the power spectrum P G ( k ) and P 1-loop ζ ( k ) glyph[similarequal] 36 25 f 2 NL ( k ) 〈 ζ 2 Gl 〉P G ( k ) are shown, for the following parameter choices: n ζ = 0 . 95, n f = 0 . 05, f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 = 3. The total power spectrum is shown with a thin black line, and the corresponding shifted power spectra for a subvolume with a +0 . 1 σ background fluctuation is shown with a thick black line. The vertical scale can be fixed so P obs ζ matches the observed value. Bottom panel: Parameter space for single source non-Gaussianity with n ζ = 0 . 95 and n f = 0 . 05 is shown. The dotted-dashed line indicates f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 = 10, both black lines indicate ∆ n s = -0 . 065 for a +0 . 1 σ background fluctuation, and the red circle indicates the parameter space congruent with the top panel. Dotted lines show the indicated number of superhorizon e-folds for a +0 . 1 σ bias. The exclusion regions are marked the same as those in Figure 3, but these assume a +0 . 1 σ bias. space for the two-source scenario described above, with n σ ( k p ) = 0 . 93, n φ ( k p ) = 1 . 005, and ξ m ( k p ) = 0 . 1. We also fix n f = 0 . 001 so that mode coupling is weaker on superhorizon scales. As before, the upper left region shows where f obs NL glyph[greaterorsimilar] 10 in typical subvolumes. We see that adding the second source relaxes the constraint on f NL in the f NL 〈 σ 2 Gl 〉 1 / 2 glyph[lessmuch] 1 regime. This makes it possible to achieve a large shift ∆ n s for smaller values of 〈 σ 2 Gl 〉 and thus fewer superhorizon e-folds. ln k L H Figure 7 . Multifield parameter space for ξ m ( k p ) = 0 . 1, n σ = 0 . 93, n φ = 1 . 005, n f = 0 . 001. The black lines show ∆ n s glyph[similarequal] -0 . 03 for a +3 σ background fluctuation. The dotted-dashed line shows f NL ( k p ) 〈 σ 2 Gl 〉 1 / 2 = 10. The upper left region shows the Planck constraint on f CMB NL for a +3 σ background. Figure 7 . Multifield parameter space for ξ m ( k p ) = 0 . 1, n σ = 0 . 93, n φ = 1 . 005, n f = 0 . 001. The black lines show ∆ n s glyph[similarequal] -0 . 03 for a +3 σ background fluctuation. The dotted-dashed line shows f NL ( k p ) 〈 σ 2 Gl 〉 1 / 2 = 10. The upper left region shows the Planck constraint on f CMB NL for a +3 σ background. The condition ξ m ( k p ) = 0 . 1 makes the field φ G dominant on Planck scales, so from the perspective of the large volume, the power spectrum has a blue tilt n s ( k p ) glyph[similarequal] n φ = 1 . 005 on scale k p . However, for significant biasing (3 σ ) and a small (or zero) non-Gaussian running of the coupled field n f = 0 . 001, the black lines in Figure 7 denote where ∆ n s = -0 . 03, which would be consistent with Planck observations. Here the shift in ∆ n s is coming not from n f but from the difference in running of P σ,NG and P φ , n ( m ) f , as the red-tilted P σ,NG is amplified due to the strong background overdensity. It is also interesting to note that a cursory survey of background fluctuations reveals that biases less than | 3 σ | yield no ∆ n s corrections smaller than -0 . 03, which would seem to partly exclude these parameters for typical Hubble-sized subsamples. In the limit of very small ξ m ( k p ), φ G dominates the power and scale-dependence on observable scales, so unless the bias is extremely strong, any shift in the power and scale-dependence from the σ field will be too small to affect n obs s . Summary. In summary, a significant shift to the observed spectral index from correlations with long-wavelength background modes is possible under the following conditions: 1. A red tilt for the field with mode coupling , n σ ≤ 1 ( n ζ ≤ 1 in the single-source case), is necessary for the cumulative power 〈 σ 2 Gl 〉 on superhorizon scales to be large enough to significantly bias local statistics. 2. A blue bispectral index n f ≥ 0 for f NL ( k ) (assuming constant n f ) is needed to remove the power from the non-Gaussian term on large scales so that strong coupling of shortscale modes to background modes is consistent with weak global non-Gaussianity and ζ being perturbative, while having enough background modes to give a large bias. 3. In a two-source scenario, the ratio of power in the non-Gaussian field to total power should have a red spectrum ( n ( m ) f ( k p ) ≤ 0) so that the non-Gaussian field σ G grows relative to φ G on large scales, causing the background ζ l to be sufficiently correlated with local statistics. If φ G contributes on observable scales ( ξ m ( k p ) < 1), larger values of f NL ( k p ) are consistent with observational constraints on non-Gaussianity, so a smaller background σ Gl is needed to give the same shift to n obs s . Introducing scale-dependence into the spectral indices would relax the conditions for large | ∆ n s | . Although the scenario becomes more complicated in this case, the qualitative features remain valid: scale-dependence of power spectra and non-Gaussian parameters must allow for sufficient cumulative superhorizon power that a large background σ Gl from the source with mode coupling is typical. We note that for given large-volume statistics, the observed red tilt may not be equally consistent with a local overdensity or underdensity in σ G . In the single-source case with n f > 0, for example, an overdensity (underdensity) corresponds to an increase (decrease) of power on small scales. Thus, for a scale-invariant power spectrum in the large volume, the observed red tilt n obs s glyph[similarequal] 0 . 96 could be accounted for in terms of a blue-tilted global bispectrum and local underdensity. However, without information about the global power spectrum, it would be difficult to infer whether we sit on a local underdensity or overdensity. 3.3 The shift to the scale dependence of the bispectrum The bispectrum may also be shifted by mode coupling coming from the soft limits of the large-volume trispectrum and from any non-Gaussian shifts to power spectrum. We can define a spectral index for the squeezed limit of the bispectrum within any particular volume as n sq . ≡ d ln B ζ ( k L , k S , k S ) d ln k L -( n s -1) (3.8) where k L and k S are long wavelength and short wavelength modes, respectively. The small volume quantity, n obs sq . , should be calculated using the observed bispectrum and the observed spectral index. For a single source, scale-invariant local ansatz, n sq . = -3. For the single source, weakly non-Gaussian, scale-dependent scenario with g NL absent, the shift in this bispectral index between the large volume and what is observed in the small volume is Single Source : ∆ n sq . ( k ) ≡ n obs sq . ( k ) -n LargeVol . sq . ( k ) (3.9) ≈ -6 5 f NL ( k L ) σ Gl n f 1 + 6 5 f NL ( k L ) σ Gl . If 6 5 f NL ( k L ) σ Gl = 6 5 f NL ( k L ) 〈 ζ 2 G 〉 1 / 2 B glyph[lessmuch] 1, then ∆ n sq . ( k ) ≈ -6 5 f NL ( k L ) 〈 ζ 2 G 〉 1 / 2 Bn f . This shift is less than one in magnitude, but still relevant for interpreting bispectral indices of order slow-roll parameters. In the two source case, there can be additional scale dependence coming from the ratio of power of the two fields. Considering only the weak coupling case, 6 5 f NL ( k ) σ Gl glyph[lessmuch] 1 (and again setting g NL = 0 for simplicity), Two Source : ∆ n sq . ( k ) = 12 5 f NL ( k ) σ Gl 1 + 12 5 f NL ( k ) σ Gl n f -6 5 f NL ( k ) σ Gl 1 + 6 5 f NL ( k ) σ Gl n f (3.10) -12 5 ξ m ( k ) f NL ( k ) σ Gl 1 + 12 5 ξ m ( k ) f NL ( k ) σ Gl ( n f + n ( m ) f ) ≈ 6 5 f NL ( k ) σ Gl n f -12 5 ξ m ( k ) f NL ( k ) σ Gl ( n f + n ( m ) f ) . Reintroducing g NL and higher terms would lead to additional terms, introducing scaledependence even if f NL in the large volume is a constant. 3.4 Generalized local ansatz and single source vs. multi source effects The two source, weakly scale dependent local ansatz in Eq. (2.1) is representative of the properties of inflation models that generate local type non-Gaussianity. For example, the scale-dependence f NL ( k ) can come from curvaton models with self-interactions [24, 25]. The function ξ m ( k ) comes from the difference in power spectrum of two fields (eg, the inflaton and the curvaton) contributing to the curvature fluctuations. In typical multi-field models, the bispectral indices n f , n ( m ) f are of order slow-roll parameters (like the scale dependence of the power spectrum), and are often not constant. Generic expressions for the squeezed limit behavior of a multi-field bispectrum are given in [26]. The scale-dependent functions f NL ( k ) and ξ m ( k ) are observationally relevant for tests for primordial non-Gaussianity using the bias of dark matter halos and their luminous tracers (eg. quasars or luminous red galaxies). The power law dependence of the squeezed limit on the long wavelength, small momentum mode ( n sq . from Eq. (3.8)) generates the scale-dependence of the non-Gaussian term in the bias. The dependence on the short wavelength modes generates a dependence of the non-Gaussian bias on the mass of the tracer (which is absent in the usual local ansatz). In principle, if local non-Gaussianity is ever detected, it may be within the power of future large scale structure surveys to detect some amplitude of running [27]. However, as demonstrated above, the same shape of bispectrum can be generated locally by a single source for the curvature perturbations, so the presence of the non-trivial function ξ m in the observed bispectrum does not necessarily indicate that two fundamental fields contributed to the primordial curvature perturbations. On the other hand, the presence of one Gaussian source and one non-Gaussian source for the local curvature perturbations is in principle detectable by comparing power spectra that are sensitive in different ways to the total curvature field and to just the non-Gaussian part [28]. Eq. (2.12) shows that in a single source scenario the local background σ Gl can act as a second field to generate the full, multi-source shaped bispectrum, but σ Gl is constant within a single volume. This 'second field' does not have fluctuations on all scales, but its variations are relevant for considering a collection of subvolumes of a particular size. 4 Mode coupling effects from a non-local factorizable bispectrum We have considered the effect of superhorizon modes only for the case of nearly local nonGaussianity, but inflationary theory has generated an expanding space of models exhibiting different types of mode coupling. Intuitively, any scenario that does not couple modes of sufficiently different wavelengths should not lead to correlation functions whose amplitudes or shapes change under subsampling. As a first step towards considering the observational consequences of subsampling general non-Gaussian scenarios, it is straighforward to find corrections from the background to small-volume quantities in the case of a factorizable quadratic kernel in Fourier space with power-law dependence. Consider a curvature perturbation in the large volume given by ζ k = φ G, k + σ G, k + ∫ L -1 d 3 p 1 (2 π ) 3 d 3 p 2 (2 π ) 3 (2 π ) 3 δ 3 ( p 1 + p 2 -k ) F ( p 1 , p 2 , k ) σ G, p 1 σ G, p 2 + ..., (4.1) where F ( k 1 , k 2 , k 3 ) = ∑ j a NL ,j ( k p ) ( k 1 k p ) m 1 ,j ( k 2 k p ) m 2 ,j ( k 3 k p ) m 3 ,j (4.2) is a sum of factorizable terms with power law dependence on the momenta. On the right hand side the a j are amplitudes defined at a pivot scale k p . When ∑ i m i,j glyph[similarequal] 0 for every term j , the bispectrum is approximately scale-invariant. The kernel F ( k 1 , k 2 , k 3 ) can be chosen to generate a desired bispectrum with well behaved one-loop corrections to the power spectrum [29]. Splitting the modes into long and short, the locally defined short wavelength modes with shifts induced from coupling to long wavelength modes from one term in the series above are ζ k s = φ G, k s + σ G, k s + σ G, k s a NL ( k p ) [ ( k k p ) m 1 + m 3 σ ( m 2 ) Gl + ( k k p ) m 2 + m 3 σ ( m 1 ) Gl ] (4.3) + a NL ( k p ) ∫ M -1 d 3 p (2 π ) 3 σ G ( p ) σ G ( | k -p | ) ( k k p ) m 3 ( | k -p | k p ) m 2 ( p k p ) m 1 where σ ( m L ) Gl ≡ ∫ M -1 L -1 d 3 p (2 π ) 3 σ p ( p k p ) m L . (4.4) When the local field is weakly non-Gaussian, the second line is small and we can rewrite the first line as ζ k s ≈ φ G, k s + σ G, k s [1 + ∆ σ ( k )] (4.5) ∆ σ ( k ) = a NL ( k p ) [ ( k k p ) m 1 + m 3 σ ( m 2 ) Gl + ( k k p ) m 2 + m 3 σ ( m 1 ) Gl ] . The leading shift to the power spectrum P obs ζ in a subvolume from unobservable infrared modes in one term of the series above (and assuming weak non-Gaussianity) is: P obs ζ ( k ) = P ζ ( k ) { 1 + ξ m ( k ) [ 2∆ σ ( k ) + ∆ σ ( k ) 2 ]} ; (4.6) where ξ m ( k ) is still the ratio of power in the non-Gaussian source to the total power, defined in Eq.(2.7). In the two-field case with weak non-Gaussianity on all scales, the observed ratio of power in the two fields is related to the same ratio in the large volume by ξ obs m ( k ) = ξ m ( k ) [1 + ∆ σ ( k )] 2 1 + ξ m ( k )[2∆ σ ( k ) + ∆ σ ( k ) 2 ] . (4.7) The induced shift to the spectral index has two terms, but assuming that, say, the first term in the square brackets in ∆ σ is dominant and defining m S = m 1 + m 3 , m 2 = m L , and a NL ( k ) = a NL ( k p )( k/k p ) m S it is ∆ n s ( k ) ≈ 2 aξ m ( k ) a NL ( k ) σ ( m L ) Gl ( n ( m ) f + m S ) . (4.8) The bispectrum in the large volume is B ζ ( k 1 , k 2 , k 3 ) = a NL ( k p ) ( k 3 k p ) m 3 P ζ ( k 1 ) ξ m ( k 1 ) P ζ ( k 2 ) ξ m ( k 2 ) (4.9) × [( k 1 k p ) m 1 ( k 2 k p ) m 2 + ( k 1 k p ) m 2 ( k 2 k p ) m 1 ] +2perm . while the observed bispectrum is B obs ζ ( k 1 , k 2 , k 3 ) = a NL ( k p ) ( k 3 k p ) m 3 [ P obs ζ ( k 1 ) ξ obs m ( k 1 ) 1 + ∆ σ ( k 1 ) ][ P obs ζ ( k 2 ) ξ obs m ( k 2 ) 1 + ∆ σ ( k 2 ) ] (4.10) × [( k 1 k p ) m 1 ( k 2 k p ) m 2 + ( k 1 k p ) m 2 ( k 2 k p ) m 1 ] +2perm . Consider k 1 = k L glyph[lessmuch] k 2 ≈ k 3 . If m 2 < m 1 (so the second term in the second line of the equation above dominates), and m S ≡ m 1 + m 3 , then in the squeezed limit the large volume bispectrum has n sq . ( k ) = -3 + n ( m ) f + m 2 . (4.11) The shift to the observed running of the squeezed-limit bispectrum is ∆ n sq . ( k ) = ∆ n ( m ) f ( k ) -∆ σ ( k ) 1 + ∆ σ ( k ) m S ≈ ∆ σ m S -2 ξ m ∆ σ ( n ( m ) f + m S ) . (4.12) In the case of the generalized, two source local ansatz considered in Sections 2 and 3.3, a NL ( k ) = 3 5 f NL ( k ), m 3 = n f , and m 1 = m 2 = 0 so m S = n f , and both terms in the square brackets of ∆ σ , Eq.(4.5) contribute equally, so we recover the weakly non-Gaussian limits of Eqs. (3.1), (3.3), and Eq. (3.10). As a second example, consider single field inflation (with a Bunch-Davies vacuum and inflation proceeding along the attractor solution). In this case, the squeezed limit of the bispectrum diverges with the long wavelength mode no more strongly than [12-14, 30], B ζ ( k L , k S , k S ) ∝ O ( k L k S ) 2 P ζ ( k L ) P ζ ( k S ) . (4.13) A bispectrum with this squeezed limit can be obtained by using the equilateral template [31] to generate a kernel F ( p 1 , p 2 , k ) ∝ -3 -2 p 1 p 2 /k 2 +2( p 1 + p 2 ) /k +( p 2 1 + p 2 2 ) /k 2 [29]. This yields a squeezed-limit bispectrum with n sq . = -1 and m L = 2 in Eq.(4.4). That is, this bispectrum generates a bias B ∝ ∇ 2 ζ Gl , so there is no sensitivity of locally measured quantities to long wavelength, nearly constant modes. In single field inflation, there is a direct map between local observables and the parameters of the inflationary Lagrangian. Finally, suppose modes are coupled through a bispectrum with a very strong squeezedlimit (eg, n sq . = -4 and m L = -1). Then the biasing of local statistics may come predominantly from background modes farthest in the infrared, which are shared by many neighboring subvolumes. In other words, the dependence of the global bispectrum on the long wavelength mode is related to the average spatial gradient of the bias in the large volume. 5 Tensor mode running as a test of inflation? If the scale dependence of the tensor power spectrum, n t ≡ d ln P t d ln k , can someday be measured, a red tilt would be (nearly) definitive evidence for inflation and against a contracting or ekpyrotic scenario (an interesting special case is 'solid inflation' [32]). Would it be possible to induce a blue tilt n t > 0 in a subvolume the size of the observable universe when the larger volume exhibits a more typical red tilt? If so, a measurement of n t > 0 would not necessarily rule out standard scalar field models of inflation. Conversely, if a red tilt n t < 0 can be induced in a large fraction of subvolumes from non-Gaussianity in a contracting universe scenario, a measurement of n t < 0 may not be a smoking gun for inflation . Consider a three-point interaction 〈 χ k 1 γ s 1 k 2 γ s 2 k 3 〉 ≡ (2 π ) 3 δ 3 ( ∑ k i ) B ( k 1 , k 2 , k 3 ) δ s 1 s 2 (5.1) between two tensor modes γ k i with polarizations s i and one mode from a field χ (here, a scalar field for example). In the squeezed limit, this three-point function will induce a dependence of the local tensor power spectrum on superhorizon χ modes. Any choice of the Fourier space kernel that gives the correct squeezed limit of the bispectrum should show the correct shift to the local power spectrum. So, with a simple choice we find that the tensor power spectrum is shifted by the correlation with long wavelength modes p as γ s i k = γ s i G, k + ∫ L -1 d 3 p 1 (2 π ) 3 d 3 p 2 (2 π ) 3 (2 π ) 3 δ 3 ( p 1 + p 2 -k ) F ( p 1 , p 2 , k )( γ s i G, p 1 χ G, p 2 + γ s i G, p 2 χ G, p 1 ) + ..., (5.2) where we take F ( k 1 , k 2 , k 3 ) = f eff γγχ ( k p ) ∑ j a j ( k 1 k p ) m 1 ,j ( k 2 k p ) m 2 ,j ( k 3 k p ) m 3 ,j . (5.3) For long wavelength modes of the χ field, the tensor power spectrum is shifted by P obs γ = P γ [1 + ∆ χ ( k )] 2 (5.4) ∆ χ ( k ) = af eff γγχ ( k p ) [ ( k k p ) m 1 + m 3 χ ( m 2 ) Gl + ( k k p ) m 2 + m 3 χ ( m 1 ) Gl ] χ ( m L ) Gl ≡ ∫ M -1 L -1 d 3 p (2 π ) 3 χ p ( p k p ) m L . With this parameterization, long wavelength modes of the χ field can shift the locally observed tilt of the tensor power spectrum. In the case that the first term in ∆ χ dominates, we can again define m S = m 1 + m 3 , m L = m 2 and then the shift is approximately ∆ n t ( k ) ≈ 2∆ χ ( k ) m S . (5.5) The quantity m S is zero for an exactly scale-invariant, local type model and more generally cannot be too large if we want to require weak non-Gaussianity for all fields. Depending on the coupling of χ to the scalar curvature, this physics may also introduce a shift in the locally observed scalar power spectrum, the tensor-to-scalar ratio, and a 'fossil' signature in the off-diagonal part of the scalar power spectrum [33], which would be an interesting complementary observable. From these expressions, it looks possible to find scenarios where the locally observed tensor power spectrum would be shifted from red to blue and vice-versa, but a full analysis along the lines of Section 3 should be performed to check consistency with all observables. 6 Discussion and conclusions Non-Gaussianity that couples the statistics of fluctuations on observable scales to wavemodes spanning super-Hubble scales can bias cosmological statistics measured by an observer in a glyph[negationslash] local Hubble volume. Previous work showed that the relative amplitudes of the power spectrum and non-Gaussianity ( f local NL ) can vary in observable subvolumes. In this work we have shown that the spectral index can also vary by enough to be interesting, | ∆ n s | glyph[similarequal] 0 . 04. The scaling of the squeezed limit of the bispectrum can also be shifted, which is relevant for constraints on non-Gaussianity from galaxy bias. These results show that in spite of the excellent precision of the measurements from the Planck satellite (especially n obs s = 0 . 9603 ± 0 . 0073 [2] and constraints on non-Gaussianity), the door is open for a significant cosmic variance uncertainty in comparing our observed patch of the universe to any particular inflation theory even leaving aside issues with eternal inflation. Moreover, rather than just presenting a new source of uncertainty from the super-Hubble background, the correlation between bispectral running in a super-Hubble volume and subvolume power spectrum measurements reopens the door for inflationary models with flat or bluer super-Hubble spectral indices, n s = 0 . 96, provided they also have scale-dependent local non-Gaussianity. This may be particularly useful for hybrid inflation. The numbers measured by the Planck satellite are consistent with a range of levels of non-Gaussianity in a post-inflationary volume, given a model for the statistics in that volume. For example, we recover the observed power spectrum and spectral index, and satisfy current constraints on f CMB NL for a post-inflationary volume with · No local type non-Gaussianity, an arbitrary number of extra e-folds, and any behavior of the power spectrum on superhorizon scales. · Constant f local NL = 5. We observe f local NL = 8 if, for example, the spectral index is a constant n s = 0 . 96 over about 200 extra e-folds of inflation and our Hubble patch sits on top of a 2-sigma under density. · Local non-Gaussianity with constant f local NL = 15. We observe f local NL = 11 if, for example, the spectral index is a constant n s = 0 . 96 over about 150 extra e-folds of inflation and our Hubble patch sits on top of a 2-sigma over-density. · Scale-dependent non-Gaussianity with f NL ( k p ) = -2, n f = 0 . 04, n ζ = 0 . 93, and n s = 0 . 935. We would observe f NL ( k p ) = -1 and n obs s = 0 . 956 if our Hubble patch sits on top of a 2-sigma under density in a volume with about 190 extra e-folds. · Scale-dependent non-Gaussianity with f NL ( k p ) = 20, n f = 0 . 03, n ζ = 0 . 95, and n s = 1 . 005. We would observe f NL ( k p ) = 2 . 5 and n obs s = 0 . 975 if our Hubble patch sits on top of a 0.2-sigma over density in a volume with about 280 extra e-folds. In contrast, we could design an inflation model to have parameters roughly consistent with Planck data, say f NL ( k p ) = 5, n f = 0 . 1, n ζ = 0 . 98, and n s = 0 . 982. However, if the model allows about 400 extra e-folds of inflation, and our Hubble patch were to sit on a 2-sigma over density, we would observe f NL ( k p ) = 4 and n obs s = 1 . 013. These results demonstrate that predictions for our observations in any scenarios with local type non-Gaussianity must be given statistically. To turn the picture around, they also suggest a new route to understanding whether observations can give us any hints about the size of the universe beyond what is directly observable. Previous ideas focused on topologically finite universes (also significantly constrained by Planck [34]) or on evidence for or against a nonperturbatively connected multiverse from bubble collisions [35-37] or curvature [38, 39]. While observations will probably never tell us how long inflation lasted, our work suggests they may at least tell us if that uncertainty is relevant to our interpretation of the data we do have. From a cosmic variance point of view, we are fortunate that there is so far no detection of local type non-Gaussianity. We have shown that future observations could push the mode-coupling uncertainties we have considered here into irrelevance 5 if primordial local non-Gaussianity can be constrained to be | f NL | < 1. Even if | f NL | > 1 is observed, tests for the running of the spectral index, any scale-dependence of | f NL | , and any evidence for extra fields through isocurvature modes or 'fossil' relics hiding in the off-diagonal power spectrum could still limit the size of any subsampling uncertainty. For example, if a blue tilt to f NL is ruled out, biasing of the spectral index is unlikely for single-source models with n ζ and n f constant on all scales. Of course, making these observations statistically well-defined depends on comparing particular competing models. It would be particularly interesting if those models had other cosmological implications related to the size of the universe 6 [40]. It would also be worthwhile to investigate the generic behavior of the local ansatz beyond f NL alone with scale-dependent coefficients, along the lines of the analysis in [6]. It may be that there are statistically natural values for the spectral index in typical small subvolumes. Then, stronger conclusions about generic cosmic variance of the spectral index might be possible. However, it is already clear that if improved limits on the amplitude and scale-dependence of non-Gaussianity can be reached, we could close the window of observational access to a perturbatively connected larger universe. Acknowledgements We thank Chris Byrnes, Bhaskar Dutta, Louis Leblond and Marilena LoVerde for useful suggestions and discussions about this work. The work of J. B. is supported in part by Department of Energy grant DE-FG02-04ER41291. The work of J. K. is supported in part by Department of Energy grants DE-FG02-04ER41291 and DE-FG02-13ER41913. The work of S. S. is supported in part by the National Aeronautics and Space Administration under Grant No. NNX12AC99G issued through the Astrophysics Theory Program. In addition, S. S. thanks the organizers of the Primordial Cosmology Program at KITP for hospitality while this work was being completed and for support by the National Science Foundation under Grant No. NSF PHY11-25915. J. K. thanks the Center for Theoretical Underground Physics and Related Areas (CETUP* 2013) in South Dakota for its support and hospitality while this work was being completed. E. N. is supported by the Eberly Research Funds of The Pennsylvania State University. The Institute for Gravitation and the Cosmos is supported by the Eberly College of Science and the Office of the Senior Vice President for Research at the Pennsylvania State University. References [1] Planck Collaboration Collaboration, P. Ade et. al. , Planck 2013 results. I. Overview of products and scientific results , 1303.5062 . [2] Planck Collaboration Collaboration, P. Ade et. al. , Planck 2013 results. XXII. Constraints on inflation , 1303.5082 . [3] Planck Collaboration, P. Ade et. al. , Planck 2013 Results. XXIV. Constraints on primordial non-Gaussianity , 1303.5084 . [4] A. D. Linde and V. Mukhanov, The curvaton web , JCAP 0604 (2006) 009, [ astro-ph/0511736 ]. [5] V. Demozzi, A. Linde, and V. Mukhanov, Supercurvaton , JCAP 1104 (2011) 013, [ 1012.0549 ]. [6] E. Nelson and S. Shandera, Statistical Naturalness and non-Gaussianity in a Finite Universe , accepted for publication in PRL (2012) [ 1212.4550 ]. [7] S. Nurmi, C. T. Byrnes, and G. Tasinato, A non-Gaussian landscape , 1301.3128 . [8] M. LoVerde, E. Nelson, and S. Shandera, Non-Gaussian Mode Coupling and the Statistical Cosmological Principle , 1303.3549 . [9] C. T. Byrnes, S. Nurmi, G. Tasinato, and D. Wands, Implications of the Planck bispectrum constraints for the primordial trispectrum , 1306.2370 . [10] D. S. Salopek and J. R. Bond, Nonlinear evolution of long wavelength metric fluctuations in inflationary models , Phys. Rev. D42 (1990) 3936-3962. [11] E. Komatsu and D. N. Spergel, Acoustic signatures in the primary microwave background bispectrum , Phys.Rev. D63 (2001) 063002, [ astro-ph/0005036 ]. [12] J. M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models , JHEP 0305 (2003) 013, [ astro-ph/0210603 ]. [13] P. Creminelli and M. Zaldarriaga, Single field consistency relation for the 3-point function , JCAP 0410 (2004) 006, [ astro-ph/0407059 ]. [14] E. Pajer, F. Schmidt, and M. Zaldarriaga, The Observed Squeezed Limit of Cosmological Three-Point Functions , 1305.0824 . [15] P. McDonald, Primordial non-Gaussianity: large-scale structure signature in the perturbative bias model , Phys.Rev. D78 (2008) 123519, [ 0806.1061 ]. [16] M. LoVerde and K. M. Smith, The Non-Gaussian Halo Mass Function with fNL, gNL and tauNL , JCAP 1108 (2011) 003, [ 1102.1439 ]. [17] M. Gerstenlauer, A. Hebecker, and G. Tasinato, Inflationary Correlation Functions without Infrared Divergences , JCAP 1106 (2011) 021, [ 1102.0560 ]. [18] S. B. Giddings and M. S. Sloth, Cosmological observables, IR growth of fluctuations, and scale-dependent anisotropies , Phys.Rev. D84 (2011) 063528, [ 1104.0002 ]. [19] D. H. Lyth, The curvature perturbation in a box , JCAP 0712 (2007) 016, [ 0707.0361 ]. [20] L. Boubekeur and D. Lyth, Detecting a small perturbation through its non-Gaussianity , Phys.Rev. D73 (2006) 021301, [ astro-ph/0504046 ]. [21] J. Kumar, L. Leblond, and A. Rajaraman, Scale Dependent Local Non-Gaussianity from Loops , JCAP 1004 (2010) 024, [ 0909.2040 ]. [22] J. Bramante and J. Kumar, Local Scale-Dependent Non-Gaussian Curvature Perturbations at Cubic Order , JCAP 1109 (2011) 036, [ 1107.5362 ]. [23] N. S. Sugiyama, Consistency Relation for multifield inflation scenario with all loop contributions , JCAP 1205 (2012) 032, [ 1201.4048 ]. [24] C. T. Byrnes, K. Enqvist, and T. Takahashi, Scale-dependence of Non-Gaussianity in the Curvaton Model , JCAP 1009 (2010) 026, [ 1007.5148 ]. [25] Q.-G. Huang, Negative spectral index of f NL in the axion-type curvaton model , JCAP 1011 (2010) 026, [ 1008.2641 ]. [26] M. Dias, R. H. Ribeiro, and D. Seery, Scale-dependent bias from multiple-field inflation , 1303.6000 . [27] S. Shandera, N. Dalal, and D. Huterer, A generalized local ansatz and its effect on halo bias , JCAP 1103 (2011) 017, [ 1010.3722 ]. [28] D. Tseliakhovich, C. Hirata, and A. Slosar, Non-Gaussianity and large-scale structure in a two-field inflationary model , Phys.Rev. D82 (2010) 043531, [ 1004.3302 ]. [29] R. Scoccimarro, L. Hui, M. Manera, and K. C. Chan, Large-scale Bias and Efficient Generation of Initial Conditions for Non-Local Primordial Non-Gaussianity , Phys.Rev. D85 (2012) 083002, [ 1108.5512 ]. [30] T. Tanaka and Y. Urakawa, Dominance of gauge artifact in the consistency relation for the primordial bispectrum , JCAP 1105 (2011) 014, [ 1103.1251 ]. [31] P. Creminelli, A. Nicolis, L. Senatore, M. Tegmark, and M. Zaldarriaga, Limits on non-gaussianities from wmap data , JCAP 0605 (2006) 004, [ astro-ph/0509029 ]. [32] S. Endlich, A. Nicolis, and J. Wang, Solid Inflation , 1210.0569 . [33] D. Jeong and M. Kamionkowski, Clustering Fossils from the Early Universe , Phys.Rev.Lett. 108 (2012) 251301, [ 1203.0302 ]. [34] Planck Collaboration Collaboration, P. Ade et. al. , Planck 2013 results. XXVI. Background geometry and topology of the Universe , 1303.5086 . [35] S. Chang, M. Kleban, and T. S. Levi, Watching Worlds Collide: Effects on the CMB from Cosmological Bubble Collisions , JCAP 0904 (2009) 025, [ 0810.5128 ]. [36] S. M. Feeney, M. C. Johnson, J. D. McEwen, D. J. Mortlock, and H. V. Peiris, Hierarchical Bayesian Detection Algorithm for Early-Universe Relics in the Cosmic Microwave Background , 1210.2725 . [37] S. Osborne, L. Senatore, and K. Smith, Collisions with other Universes: the Optimal Analysis of the WMAP data , 1305.1964 . [38] M. Kleban and M. Schillo, Spatial Curvature Falsifies Eternal Inflation , JCAP 1206 (2012) 029, [ 1202.5037 ]. [39] A. H. Guth and Y. Nomura, What can the observation of nonzero curvature tell us? , Phys.Rev. D86 (2012) 023534, [ 1203.6876 ]. [40] D. B. Kaplan and A. E. Nelson, Inflationary Axion Cosmology Beyond Our Horizon , 0809.1206 . Gl - 2 10 0 10 5 10 4 10 3 10 2 L p k H NL f 1 10 0 10 - 1 10 - 2 5 10 10 4 10 3 10 2 L p k H NL f 1 10 0 10 - 1 10 - 2 10 10 10 Strongly NG Weakly NG = 0 . 1 D n s ns > 1.16 D D n n s s > > e v i t a b r u t r e p n o N -0.04 -0.04 N = 10, 50, 100, 200, 350 - 8 10 f NL - 6 10 = 0 . 96 , n 10 2 - 4 > f Strongly NG = - 2 10 0 . 1 > 0 = 0 . 96 , n f CMB > 10 NL Gl Gl - 0 10 0 10 to a smaller (larger) number of e-folds in the case of a red tilt (blue tilt), as shown in Figure 2, so it is easier to realize a large shift to a global red tilt than to a global blue tilt. In fact, as previously noted in the discussion of Figure 2, imposing the requirement that P obs ζ be typical of subvolumes for scenarios with a blue tilt n ζ -1 causes 〈 ζ 2 Gl 〉 to converge to a particular value as N is increased. 2. A red tilt in the power spectrum can relax the constraint from requiring weak global non-Gaussianity, as seen by comparing the right panels in Figure 3 to the left panels. For example, when n f > 0, a red tilt in the power spectrum gives more relative weight in M 3 to the more weakly coupled superhorizon modes and damps the power of strongly coupled subhorizon modes. Note that the bottom two panels in Figure 3 permit about the same number of super-horizon e-folds of weakly non-Gaussian parameter space. In the right panel the power removed from subhorizon e-folds by n ζ < 1 is balanced by power added to superhorizon e-folds leading to a larger background 〈 ζ 2 Gl 〉 per e-fold permitted for perturbative statistics as compared to the bottom left panel. For these reasons single-source scenarios with a red power tilt in the large volume have the most significant range of cosmic variance due to subsampling. The solid black lines in Figure 3 show ∆ n s ( k p ) = -0 . 04 in subvolumes with a +0 . 5 σ background fluctuation ( ζ Gl = +0 . 5 〈 ζ 2 Gl 〉 1 / 2 ), and thus show part of the parameter space where | ∆ n s ( k p ) | can be observationally significant. Here we have neglected the subhorizon one-loop correction 〈 ζ 2 Gs ( k p ) 〉 glyph[lessmuch] ζ 2 Gl ; this breaks down for small N but is valid outside of the region of parameter space excluded by the requirement f CMB NL < 10. Rewriting Eq. (3.3) for a single source scenario ( ξ m = 1), ∆ n single source s glyph[similarequal] n f ( 12 5 f NL ζ Gl ( 1 -36 25 f 2 NL 〈 ζ 2 Gl 〉 ) + 72 25 f 2 NL ( ζ 2 Gl -〈 ζ 2 Gl 〉 ) ) ( 1 + 6 5 f NL ζ Gl ) 2 ( 1 + 36 25 f 2 NL 〈 ζ 2 Gl 〉 ) . (3.6) Assuming n f = 0 . 1 and ζ Gl = 0 . 5 〈 ζ 2 Gl 〉 1 / 2 , we can solve this equation to show that ∆ n single source s = -0 . 04 when f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 = 0 . 94 or 5.9, which are the equations of the two black lines plotted in Figure 3. These lines assume positive f NL in the large volume, f NL > 0, but they remain the same for f NL < 0 and a -0 . 5 σ background fluctuation. For values of | n f | larger or smaller than 0 . 1, the distance between these lines grows or shrinks in parameter space. Of course, for the full expression of ∆ n s and a different set of parameter choices, there can be more than two solutions of | ∆ n s | = 0 . 04. For positive f NL ( k p ) ζ Gl (see below), the typical size of ∆ n s is largest in the region between these lines ( 6 5 f NL ( k p ) ζ Gl ∼ 1) and falls towards zero on either side. The upper dotted-dashed lines mark where 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 is large ( O (10)). When that quantity is large, ∆ n s glyph[similarequal] -n f 3 5 f NL 〈 ζ 2 Gl 〉 1 / 2 and thus approaches zero as indicated in Figure 3. Note that in this region the observed spectral index is n obs s glyph[similarequal] n s glyph[similarequal] n ζ +2 n f , so for the parameter choices in Figure 3 the Planck satellite excludes the region above the dotted-dashed lines. All lines and contours in Figure 3 assume that 6 5 f NL ( k p ) ζ Gl > 0 (eg, overdense fluctuations with positive f NL ). If this figure assumed 6 5 f NL ( k p ) ζ Gl < 0 (eg, overdense fluctuations with negative f NL ), the area in parameter space near the line 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 = 1 would be excluded. For further discussion of parameter space with 6 5 f NL ζ Gl < 0, see the discussion after Eq. (3.3). Figure 3 shows that, under the conditions we have imposed and the spectral indices considered, only scenarios where the bispectral tilt is not very red have typical subvolumes where the observed spectral index varies by an amount that is cosmologically interesting for us, | ∆ n s | glyph[greaterorsimilar] 0 . 01. A blue bispectral index may avoid the current observational constraints, which do not probe particularly small scales, and easily remain globally perturbative and weakly non-Gaussian (see paragraph below). In contrast, the bottom panels of Figure 3 illustrate that for either spectral index, a scenario with n f < 0 will be nonperturbative in the interesting part of parameter space where | ∆ n s | ∼ 0 . 04. (In addition, there is only a small window with strongly non-Gaussian but perturbative global statistics.) If both the power spectrum and non-Gaussianity increase in the IR, as in the lower right panel of Figure 3, the statistics will be strongly non-Gaussian across parameter space for a small number of superhorizon e-folds. The upper panels of Figure 3 illustrate a feature discussed in Section 2.2: 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 glyph[greaterorsimilar] 1 does not necessarily imply a large cumulative skewness, M 3 glyph[greaterorsimilar] 1. The dashed curves fix M 3 = 1 as a function of superhorizon e-folds, which are determined at each point in parameter space by the observed level of the power spectrum along with n f , f NL and 〈 ζ 2 Gl 〉 . In regions where M 3 < 1 but f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 glyph[greaterorsimilar] 1, there are a sufficient number of superhorizon modes with weaker coupling ( n f > 0) damp the total non-Gaussianity. To elaborate, in the limit n f ( N + N sub ) glyph[greatermuch] 1, Eq. (2.18) gives M 3 ∝ [ 〈 ζ 2 Gl 〉 /N ( N + N sub )] 1 / 2 . For f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 glyph[lessmuch] 1, N = 〈 ζ 2 Gl 〉 / P obs ζ and so M 3 becomes independent of 〈 ζ 2 Gl 〉 in the limit N glyph[lessmuch] N sub . For f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 glyph[greatermuch] 1, on the other hand, M 3 ∝ 1 /f NL ( k p ) 〈 ζ 2 Gl 〉 3 / 2 , so large f NL ( k p ) and sufficiently large 〈 ζ 2 Gl 〉 are needed to keep the total non-Gaussianity small, and P obs ζ ∼ 2 × 10 -9 typical in subvolumes, as seen in the upper left panel of Figure 3. Note that throughout this analysis, we have assumed n f is constant for all N sub = 60 subhorizon e-folds, so that for blue n f non-Gaussianity continues to grow on subhorizon scales where nonlinear evolution has taken over. If this condition is relaxed, the conditions from weak non-Gaussianity are less restrictive. Figure 4 shows the probability distribution for the shift ∆ n s for the parameters in part of the range of interest for the blue bispectral index shown in the top panels of Figure 3. Both panels show examples that (for appropriate choices of large volume parameters) give local power spectra amplitude and f CMB NL consistent with our observations. Notice that the distribution on the right is substantially less Gaussian than the distribution on the left. This trend continues if one considers larger 〈 ζ 2 Gl 〉 while keeping all other parameters fixed. In Figure 5 we show regions of parameter space in the ( 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 , n f ) plane that are consistent with the Planck measurement n obs s = 0 . 9603 ± 0 . 0073. Assuming that the scalar power spectrum in the full volume of the mode-coupled universe is completely flat, n ζ = 1, we see that 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 must be at least O (10 -1 ) and for weakly non-Gaussian statistics, more than a hundred superhorizon e-folds are required. It is interesting to note that in the case of a blue-tilted f NL , a larger running non-Gaussianity n f loosens parameter constraints coming from requiring perturbative statistics 〈 [ f NL ( k ) ζ 2 Gl ] 2 〉 glyph[lessorsimilar] 0 . 1. Although the dotted lines in Figure 5 will shift to the left with more superhorizon e-folds, these curves exclude less parameter space as n f becomes larger. This is because we have assumed n f is blue and constant so f NL is driven to smaller values in the IR and 〈 [ f NL ( k ) ζ 2 Gl ] 2 〉 becomes smaller for larger n f . Notice the shift in the non-perturbative line in the right panel that occurs at n f > | n ζ -1 | : if the running of the power spectrum is larger than the running of f NL ( k ), then the running of the power spectrum will dominate the variance of the local quadratic term over superhorizon modes, because f 2 NL ( k ) P G ( k ) 2 ∝ k 2( n f + n ζ -1) . Lastly, the Figure 4 . The probability of finding a shift in the spectral index in subvolumes. Left panel: The variance plotted here corresponds to about 195 extra e-folds in a model with n ζ = 0 . 96 or 4 × 10 4 extra e-folds for a scale-invariant spectrum. Right panel: The variance here is consistent with about 240 extra e-folds in a model with n ζ = 0 . 96 or 5 × 10 5 extra e-folds for a scale-invariant spectrum. In both panels the solid black lines show a bispectral index of n f = 0 . 05 while the dotted blue lines show n f = 0 . 1. In the right panel about 24% (6%) of subvolumes in the n f = 0 . 1 ( n f = 0 . 05) have ∆ n s ≥ 0 . 02 and 17% (5%) have ∆ n s ≤ -0 . 04. The points in parameter space that correspond to the dotted lines ( n f = 0 . 1) are shown with black squares in Figure 3. Figure 4 . The probability of finding a shift in the spectral index in subvolumes. Left panel: The variance plotted here corresponds to about 195 extra e-folds in a model with n ζ = 0 . 96 or 4 × 10 4 extra e-folds for a scale-invariant spectrum. Right panel: The variance here is consistent with about 240 extra e-folds in a model with n ζ = 0 . 96 or 5 × 10 5 extra e-folds for a scale-invariant spectrum. In both panels the solid black lines show a bispectral index of n f = 0 . 05 while the dotted blue lines show n f = 0 . 1. In the right panel about 24% (6%) of subvolumes in the n f = 0 . 1 ( n f = 0 . 05) have ∆ n s ≥ 0 . 02 and 17% (5%) have ∆ n s ≤ -0 . 04. The points in parameter space that correspond to the dotted lines ( n f = 0 . 1) are shown with black squares in Figure 3. right panel of Figure 5 shows once again that for a blue tilted f NL , the weakly non-Gaussian parameter space enlarges with the number of superhorizon e-folds, because f NL is driven to very small values over more superhorizon e-folds, decreasing the value of M 3 . To conclude this section, Figure 6 illustrates a single-source scenario in which a power spectrum which appears blue-tilted in the large volume on short scales can appear red on the same scales in a subvolume. On scales where P ζ ( k ) glyph[similarequal] P G ( k ), n s ( k ) glyph[similarequal] n ζ , whereas on scales where the 1-loop contribution dominates P 1-loop ζ ( k ) glyph[similarequal] 36 25 f 2 NL ( k ) 〈 ζ 2 Gl 〉P G ( k ) and the spectral index will be n s ( k ) glyph[similarequal] n ζ + 2 n f . If the transition of power takes place on a scale near the observable range of scales ( f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 = O (1)), the observed spectral index can be shifted. For example, if ζ 2 Gl < 〈 ζ 2 Gl 〉 , the blue-tilted f 2 NL 〈 ζ 2 Gl 〉 contribution loses power in the subvolume, and if f NL ( k p ) ζ Gl > 0, the red-tilted piece gains power (compare Eqs. (2.5), (2.9)). This scenario is shown in Figure 6. Note that as long as f NL ( k p ) is not extremely large (which would violate the constraint on f CMB NL for the value of f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 chosen here), ζ Gl glyph[greatermuch] 〈 ζ 2 Gs ( k ) 〉 1 / 2 and the 1-loop contribution to P obs ζ is very small, suppressed by a factor of 〈 ζ 2 Gs ( k ) 〉 /ζ 2 Gl . Example III: Multiple sources with running ξ m ( k ) . In the single-source case, a large shift to the observed spectral index could only occur if the 1-loop contribution to the power spectrum dominated on small scales. With two sources, a significant shift to n s can be consistent with weak non-Gaussianity ξ m ( k ) f NL ( k ) 〈 σ Gl 〉 1 / 2 < 1 on all scales. If the running of the 1-loop contribution lies between the runnings n σ ≡ d ln P σ ( k ) d ln( k ) and n φ ≡ d ln P φ ( k ) d ln( k ) of the Gaussian contributions to the total power, then it will be subdominant on large and small scales. The transition of power between σ G and φ G takes place over a finite range of scales, over H GLYPH<144> L H L Figure 5 . Left panel: a model with a globally flat power spectrum, but which contains subvolumes where a red tilt would be observed. Right panel: a model with global parameters naively matched to observations that nonetheless contains a significant number of subvolumes with a spectral index at odds with observations. Both cases show single-source perturbations with the running of f NL , n f , plotted against the parameters controlling the size of the bias, 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 . This figure assumes positive f NL and a blue running of f NL . The running of the power spectrum is flat ( n s ( k p ) glyph[similarequal] n ζ = 1) and red ( n s ( k p ) glyph[similarequal] n ζ = 0 . 96) to within ∼ 0 . 01 below the dotted-dashed lines in the left and right panels, respectively. Above the dotted-dashed lines the loop correction to the running of the power spectrum becomes large ( n s ( k p ) -n ζ > 0 . 01). Dashed lines indicate regions where the non-Gaussian cumulant M 3 > 1 for the number of superhorizon e-folds indicated. The dotted line indicates the nonperturbative region ( 〈 [ 3 5 f NL glyph[star] ( ζ 2 Gl -〈 ζ 2 Gl 〉 )] 2 〉 glyph[greaterorsimilar] 0 . 1) for N > 10 3 and N > 100 in the left and right panels, respectively. The grey space shows what region is excluded at 99% confidence by the Planck measurement n obs s = 0 . 9603 ± 0 . 0073, assuming an underdense subsample with a -1 σ background fluctuation. Figure 5 . Left panel: a model with a globally flat power spectrum, but which contains subvolumes where a red tilt would be observed. Right panel: a model with global parameters naively matched to observations that nonetheless contains a significant number of subvolumes with a spectral index at odds with observations. Both cases show single-source perturbations with the running of f NL , n f , plotted against the parameters controlling the size of the bias, 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 . This figure assumes positive f NL and a blue running of f NL . The running of the power spectrum is flat ( n s ( k p ) glyph[similarequal] n ζ = 1) and red ( n s ( k p ) glyph[similarequal] n ζ = 0 . 96) to within ∼ 0 . 01 below the dotted-dashed lines in the left and right panels, respectively. Above the dotted-dashed lines the loop correction to the running of the power spectrum becomes large ( n s ( k p ) -n ζ > 0 . 01). Dashed lines indicate regions where the non-Gaussian cumulant M 3 > 1 for the number of superhorizon e-folds indicated. The dotted line indicates the nonperturbative region ( 〈 [ 3 5 f NL glyph[star] ( ζ 2 Gl -〈 ζ 2 Gl 〉 )] 2 〉 glyph[greaterorsimilar] 0 . 1) for N > 10 3 and N > 100 in the left and right panels, respectively. The grey space shows what region is excluded at 99% confidence by the Planck measurement n obs s = 0 . 9603 ± 0 . 0073, assuming an underdense subsample with a -1 σ background fluctuation. glyph[negationslash] H GLYPH<144> L H GLYPH<144> L H L H GLYPH<144> L which n s changes from n σ to n φ . If the power spectrum of φ G is blue and dominates on small scales ( ξ m ( k glyph[greaterorsimilar] H 0 ) glyph[lessmuch] 1), and the Gaussian contribution from σ is red and dominates on large scales ( ξ m ( k << H 0 ) glyph[similarequal] 1), then the background ζ l glyph[similarequal] σ l for any subvolume couples to and biases the local statistics. For example, a globally flat or blue spectral index n s ( k > H 0 ) > 1 can again appear red, n obs s < 1, in a subvolume. The shift to n s can come only from the modulation of power in σ relative to φ G , and need not rely on running non-Gaussianity n f = 0. That is, a large running of the difference in power of the fields can be achieved without a large level of running non-Gaussianity. This becomes apparent upon inspecting the running of ξ m , n ( m ) f ( k ) ≡ d ln ξ m ( k ) d ln k = (1 -ξ m ( k )) [ n σ -n φ + 2 n f 36 25 f 2 NL ( k ) 〈 σ 2 G ( k ) 〉 1 + 36 25 f 2 NL ( k ) 〈 σ 2 G ( k ) 〉 ] . (3.7) If φ G is more red-tilted than σ G , the background is uncorrelated with short-wavelength modes because φ G dominates on large scales, ζ l glyph[similarequal] φ Gl , so local statistics are not biased. Thus, both n σ ≤ 1 and n φ > n σ are needed for a significant bias. In Figure 7 we show the parameter ln L H n ζ = 0 . 95 n f = 0 . 05 n ζ = 0 . 95 n f = 0 . 05 Figure 6 . Top panel: The contributions to the power spectrum P G ( k ) and P 1-loop ζ ( k ) glyph[similarequal] 36 25 f 2 NL ( k ) 〈 ζ 2 Gl 〉P G ( k ) are shown, for the following parameter choices: n ζ = 0 . 95, n f = 0 . 05, f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 = 3. The total power spectrum is shown with a thin black line, and the corresponding shifted power spectra for a subvolume with a +0 . 1 σ background fluctuation is shown with a thick black line. The vertical scale can be fixed so P obs ζ matches the observed value. Bottom panel: Parameter space for single source non-Gaussianity with n ζ = 0 . 95 and n f = 0 . 05 is shown. The dotted-dashed line indicates f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 = 10, both black lines indicate ∆ n s = -0 . 065 for a +0 . 1 σ background fluctuation, and the red circle indicates the parameter space congruent with the top panel. Dotted lines show the indicated number of superhorizon e-folds for a +0 . 1 σ bias. The exclusion regions are marked the same as those in Figure 3, but these assume a +0 . 1 σ bias. Figure 6 . Top panel: The contributions to the power spectrum P G ( k ) and P 1-loop ζ ( k ) glyph[similarequal] 36 25 f 2 NL ( k ) 〈 ζ 2 Gl 〉P G ( k ) are shown, for the following parameter choices: n ζ = 0 . 95, n f = 0 . 05, f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 = 3. The total power spectrum is shown with a thin black line, and the corresponding shifted power spectra for a subvolume with a +0 . 1 σ background fluctuation is shown with a thick black line. The vertical scale can be fixed so P obs ζ matches the observed value. Bottom panel: Parameter space for single source non-Gaussianity with n ζ = 0 . 95 and n f = 0 . 05 is shown. The dotted-dashed line indicates f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 = 10, both black lines indicate ∆ n s = -0 . 065 for a +0 . 1 σ background fluctuation, and the red circle indicates the parameter space congruent with the top panel. Dotted lines show the indicated number of superhorizon e-folds for a +0 . 1 σ bias. The exclusion regions are marked the same as those in Figure 3, but these assume a +0 . 1 σ bias. space for the two-source scenario described above, with n σ ( k p ) = 0 . 93, n φ ( k p ) = 1 . 005, and ξ m ( k p ) = 0 . 1. We also fix n f = 0 . 001 so that mode coupling is weaker on superhorizon scales. As before, the upper left region shows where f obs NL glyph[greaterorsimilar] 10 in typical subvolumes. We see that adding the second source relaxes the constraint on f NL in the f NL 〈 σ 2 Gl 〉 1 / 2 glyph[lessmuch] 1 regime. This makes it possible to achieve a large shift ∆ n s for smaller values of 〈 σ 2 Gl 〉 and thus fewer superhorizon e-folds. ln k L H Figure 7 . Multifield parameter space for ξ m ( k p ) = 0 . 1, n σ = 0 . 93, n φ = 1 . 005, n f = 0 . 001. The black lines show ∆ n s glyph[similarequal] -0 . 03 for a +3 σ background fluctuation. The dotted-dashed line shows f NL ( k p ) 〈 σ 2 Gl 〉 1 / 2 = 10. The upper left region shows the Planck constraint on f CMB NL for a +3 σ background. Figure 7 . Multifield parameter space for ξ m ( k p ) = 0 . 1, n σ = 0 . 93, n φ = 1 . 005, n f = 0 . 001. The black lines show ∆ n s glyph[similarequal] -0 . 03 for a +3 σ background fluctuation. The dotted-dashed line shows f NL ( k p ) 〈 σ 2 Gl 〉 1 / 2 = 10. The upper left region shows the Planck constraint on f CMB NL for a +3 σ background. The condition ξ m ( k p ) = 0 . 1 makes the field φ G dominant on Planck scales, so from the perspective of the large volume, the power spectrum has a blue tilt n s ( k p ) glyph[similarequal] n φ = 1 . 005 on scale k p . However, for significant biasing (3 σ ) and a small (or zero) non-Gaussian running of the coupled field n f = 0 . 001, the black lines in Figure 7 denote where ∆ n s = -0 . 03, which would be consistent with Planck observations. Here the shift in ∆ n s is coming not from n f but from the difference in running of P σ,NG and P φ , n ( m ) f , as the red-tilted P σ,NG is amplified due to the strong background overdensity. It is also interesting to note that a cursory survey of background fluctuations reveals that biases less than | 3 σ | yield no ∆ n s corrections smaller than -0 . 03, which would seem to partly exclude these parameters for typical Hubble-sized subsamples. In the limit of very small ξ m ( k p ), φ G dominates the power and scale-dependence on observable scales, so unless the bias is extremely strong, any shift in the power and scale-dependence from the σ field will be too small to affect n obs s . Summary. In summary, a significant shift to the observed spectral index from correlations with long-wavelength background modes is possible under the following conditions: 1. A red tilt for the field with mode coupling , n σ ≤ 1 ( n ζ ≤ 1 in the single-source case), is necessary for the cumulative power 〈 σ 2 Gl 〉 on superhorizon scales to be large enough to significantly bias local statistics. 2. A blue bispectral index n f ≥ 0 for f NL ( k ) (assuming constant n f ) is needed to remove the power from the non-Gaussian term on large scales so that strong coupling of shortscale modes to background modes is consistent with weak global non-Gaussianity and ζ being perturbative, while having enough background modes to give a large bias. 3. In a two-source scenario, the ratio of power in the non-Gaussian field to total power should have a red spectrum ( n ( m ) f ( k p ) ≤ 0) so that the non-Gaussian field σ G grows relative to φ G on large scales, causing the background ζ l to be sufficiently correlated with local statistics. If φ G contributes on observable scales ( ξ m ( k p ) < 1), larger values of f NL ( k p ) are consistent with observational constraints on non-Gaussianity, so a smaller background σ Gl is needed to give the same shift to n obs s . Introducing scale-dependence into the spectral indices would relax the conditions for large | ∆ n s | . Although the scenario becomes more complicated in this case, the qualitative features remain valid: scale-dependence of power spectra and non-Gaussian parameters must allow for sufficient cumulative superhorizon power that a large background σ Gl from the source with mode coupling is typical. We note that for given large-volume statistics, the observed red tilt may not be equally consistent with a local overdensity or underdensity in σ G . In the single-source case with n f > 0, for example, an overdensity (underdensity) corresponds to an increase (decrease) of power on small scales. Thus, for a scale-invariant power spectrum in the large volume, the observed red tilt n obs s glyph[similarequal] 0 . 96 could be accounted for in terms of a blue-tilted global bispectrum and local underdensity. However, without information about the global power spectrum, it would be difficult to infer whether we sit on a local underdensity or overdensity. 3.3 The shift to the scale dependence of the bispectrum The bispectrum may also be shifted by mode coupling coming from the soft limits of the large-volume trispectrum and from any non-Gaussian shifts to power spectrum. We can define a spectral index for the squeezed limit of the bispectrum within any particular volume as n sq . ≡ d ln B ζ ( k L , k S , k S ) d ln k L -( n s -1) (3.8) where k L and k S are long wavelength and short wavelength modes, respectively. The small volume quantity, n obs sq . , should be calculated using the observed bispectrum and the observed spectral index. For a single source, scale-invariant local ansatz, n sq . = -3. For the single source, weakly non-Gaussian, scale-dependent scenario with g NL absent, the shift in this bispectral index between the large volume and what is observed in the small volume is Single Source : ∆ n sq . ( k ) ≡ n obs sq . ( k ) -n LargeVol . sq . ( k ) (3.9) ≈ -6 5 f NL ( k L ) σ Gl n f 1 + 6 5 f NL ( k L ) σ Gl . If 6 5 f NL ( k L ) σ Gl = 6 5 f NL ( k L ) 〈 ζ 2 G 〉 1 / 2 B glyph[lessmuch] 1, then ∆ n sq . ( k ) ≈ -6 5 f NL ( k L ) 〈 ζ 2 G 〉 1 / 2 Bn f . This shift is less than one in magnitude, but still relevant for interpreting bispectral indices of order slow-roll parameters. In the two source case, there can be additional scale dependence coming from the ratio of power of the two fields. Considering only the weak coupling case, 6 5 f NL ( k ) σ Gl glyph[lessmuch] 1 (and again setting g NL = 0 for simplicity), Two Source : ∆ n sq . ( k ) = 12 5 f NL ( k ) σ Gl 1 + 12 5 f NL ( k ) σ Gl n f -6 5 f NL ( k ) σ Gl 1 + 6 5 f NL ( k ) σ Gl n f (3.10) -12 5 ξ m ( k ) f NL ( k ) σ Gl 1 + 12 5 ξ m ( k ) f NL ( k ) σ Gl ( n f + n ( m ) f ) ≈ 6 5 f NL ( k ) σ Gl n f -12 5 ξ m ( k ) f NL ( k ) σ Gl ( n f + n ( m ) f ) . Reintroducing g NL and higher terms would lead to additional terms, introducing scaledependence even if f NL in the large volume is a constant. 3.4 Generalized local ansatz and single source vs. multi source effects The two source, weakly scale dependent local ansatz in Eq. (2.1) is representative of the properties of inflation models that generate local type non-Gaussianity. For example, the scale-dependence f NL ( k ) can come from curvaton models with self-interactions [24, 25]. The function ξ m ( k ) comes from the difference in power spectrum of two fields (eg, the inflaton and the curvaton) contributing to the curvature fluctuations. In typical multi-field models, the bispectral indices n f , n ( m ) f are of order slow-roll parameters (like the scale dependence of the power spectrum), and are often not constant. Generic expressions for the squeezed limit behavior of a multi-field bispectrum are given in [26]. The scale-dependent functions f NL ( k ) and ξ m ( k ) are observationally relevant for tests for primordial non-Gaussianity using the bias of dark matter halos and their luminous tracers (eg. quasars or luminous red galaxies). The power law dependence of the squeezed limit on the long wavelength, small momentum mode ( n sq . from Eq. (3.8)) generates the scale-dependence of the non-Gaussian term in the bias. The dependence on the short wavelength modes generates a dependence of the non-Gaussian bias on the mass of the tracer (which is absent in the usual local ansatz). In principle, if local non-Gaussianity is ever detected, it may be within the power of future large scale structure surveys to detect some amplitude of running [27]. However, as demonstrated above, the same shape of bispectrum can be generated locally by a single source for the curvature perturbations, so the presence of the non-trivial function ξ m in the observed bispectrum does not necessarily indicate that two fundamental fields contributed to the primordial curvature perturbations. On the other hand, the presence of one Gaussian source and one non-Gaussian source for the local curvature perturbations is in principle detectable by comparing power spectra that are sensitive in different ways to the total curvature field and to just the non-Gaussian part [28]. Eq. (2.12) shows that in a single source scenario the local background σ Gl can act as a second field to generate the full, multi-source shaped bispectrum, but σ Gl is constant within a single volume. This 'second field' does not have fluctuations on all scales, but its variations are relevant for considering a collection of subvolumes of a particular size. 4 Mode coupling effects from a non-local factorizable bispectrum We have considered the effect of superhorizon modes only for the case of nearly local nonGaussianity, but inflationary theory has generated an expanding space of models exhibiting different types of mode coupling. Intuitively, any scenario that does not couple modes of sufficiently different wavelengths should not lead to correlation functions whose amplitudes or shapes change under subsampling. As a first step towards considering the observational consequences of subsampling general non-Gaussian scenarios, it is straighforward to find corrections from the background to small-volume quantities in the case of a factorizable quadratic kernel in Fourier space with power-law dependence. Consider a curvature perturbation in the large volume given by ζ k = φ G, k + σ G, k + ∫ L -1 d 3 p 1 (2 π ) 3 d 3 p 2 (2 π ) 3 (2 π ) 3 δ 3 ( p 1 + p 2 -k ) F ( p 1 , p 2 , k ) σ G, p 1 σ G, p 2 + ..., (4.1) where F ( k 1 , k 2 , k 3 ) = ∑ j a NL ,j ( k p ) ( k 1 k p ) m 1 ,j ( k 2 k p ) m 2 ,j ( k 3 k p ) m 3 ,j (4.2) is a sum of factorizable terms with power law dependence on the momenta. On the right hand side the a j are amplitudes defined at a pivot scale k p . When ∑ i m i,j glyph[similarequal] 0 for every term j , the bispectrum is approximately scale-invariant. The kernel F ( k 1 , k 2 , k 3 ) can be chosen to generate a desired bispectrum with well behaved one-loop corrections to the power spectrum [29]. Splitting the modes into long and short, the locally defined short wavelength modes with shifts induced from coupling to long wavelength modes from one term in the series above are ζ k s = φ G, k s + σ G, k s + σ G, k s a NL ( k p ) [ ( k k p ) m 1 + m 3 σ ( m 2 ) Gl + ( k k p ) m 2 + m 3 σ ( m 1 ) Gl ] (4.3) + a NL ( k p ) ∫ M -1 d 3 p (2 π ) 3 σ G ( p ) σ G ( | k -p | ) ( k k p ) m 3 ( | k -p | k p ) m 2 ( p k p ) m 1 where σ ( m L ) Gl ≡ ∫ M -1 L -1 d 3 p (2 π ) 3 σ p ( p k p ) m L . (4.4) When the local field is weakly non-Gaussian, the second line is small and we can rewrite the first line as ζ k s ≈ φ G, k s + σ G, k s [1 + ∆ σ ( k )] (4.5) ∆ σ ( k ) = a NL ( k p ) [ ( k k p ) m 1 + m 3 σ ( m 2 ) Gl + ( k k p ) m 2 + m 3 σ ( m 1 ) Gl ] . The leading shift to the power spectrum P obs ζ in a subvolume from unobservable infrared modes in one term of the series above (and assuming weak non-Gaussianity) is: P obs ζ ( k ) = P ζ ( k ) { 1 + ξ m ( k ) [ 2∆ σ ( k ) + ∆ σ ( k ) 2 ]} ; (4.6) where ξ m ( k ) is still the ratio of power in the non-Gaussian source to the total power, defined in Eq.(2.7). In the two-field case with weak non-Gaussianity on all scales, the observed ratio of power in the two fields is related to the same ratio in the large volume by ξ obs m ( k ) = ξ m ( k ) [1 + ∆ σ ( k )] 2 1 + ξ m ( k )[2∆ σ ( k ) + ∆ σ ( k ) 2 ] . (4.7) The induced shift to the spectral index has two terms, but assuming that, say, the first term in the square brackets in ∆ σ is dominant and defining m S = m 1 + m 3 , m 2 = m L , and a NL ( k ) = a NL ( k p )( k/k p ) m S it is ∆ n s ( k ) ≈ 2 aξ m ( k ) a NL ( k ) σ ( m L ) Gl ( n ( m ) f + m S ) . (4.8) The bispectrum in the large volume is B ζ ( k 1 , k 2 , k 3 ) = a NL ( k p ) ( k 3 k p ) m 3 P ζ ( k 1 ) ξ m ( k 1 ) P ζ ( k 2 ) ξ m ( k 2 ) (4.9) × [( k 1 k p ) m 1 ( k 2 k p ) m 2 + ( k 1 k p ) m 2 ( k 2 k p ) m 1 ] +2perm . while the observed bispectrum is B obs ζ ( k 1 , k 2 , k 3 ) = a NL ( k p ) ( k 3 k p ) m 3 [ P obs ζ ( k 1 ) ξ obs m ( k 1 ) 1 + ∆ σ ( k 1 ) ][ P obs ζ ( k 2 ) ξ obs m ( k 2 ) 1 + ∆ σ ( k 2 ) ] (4.10) × [( k 1 k p ) m 1 ( k 2 k p ) m 2 + ( k 1 k p ) m 2 ( k 2 k p ) m 1 ] +2perm . Consider k 1 = k L glyph[lessmuch] k 2 ≈ k 3 . If m 2 < m 1 (so the second term in the second line of the equation above dominates), and m S ≡ m 1 + m 3 , then in the squeezed limit the large volume bispectrum has n sq . ( k ) = -3 + n ( m ) f + m 2 . (4.11) The shift to the observed running of the squeezed-limit bispectrum is ∆ n sq . ( k ) = ∆ n ( m ) f ( k ) -∆ σ ( k ) 1 + ∆ σ ( k ) m S ≈ ∆ σ m S -2 ξ m ∆ σ ( n ( m ) f + m S ) . (4.12) In the case of the generalized, two source local ansatz considered in Sections 2 and 3.3, a NL ( k ) = 3 5 f NL ( k ), m 3 = n f , and m 1 = m 2 = 0 so m S = n f , and both terms in the square brackets of ∆ σ , Eq.(4.5) contribute equally, so we recover the weakly non-Gaussian limits of Eqs. (3.1), (3.3), and Eq. (3.10). As a second example, consider single field inflation (with a Bunch-Davies vacuum and inflation proceeding along the attractor solution). In this case, the squeezed limit of the bispectrum diverges with the long wavelength mode no more strongly than [12-14, 30], B ζ ( k L , k S , k S ) ∝ O ( k L k S ) 2 P ζ ( k L ) P ζ ( k S ) . (4.13) A bispectrum with this squeezed limit can be obtained by using the equilateral template [31] to generate a kernel F ( p 1 , p 2 , k ) ∝ -3 -2 p 1 p 2 /k 2 +2( p 1 + p 2 ) /k +( p 2 1 + p 2 2 ) /k 2 [29]. This yields a squeezed-limit bispectrum with n sq . = -1 and m L = 2 in Eq.(4.4). That is, this bispectrum generates a bias B ∝ ∇ 2 ζ Gl , so there is no sensitivity of locally measured quantities to long wavelength, nearly constant modes. In single field inflation, there is a direct map between local observables and the parameters of the inflationary Lagrangian. Finally, suppose modes are coupled through a bispectrum with a very strong squeezedlimit (eg, n sq . = -4 and m L = -1). Then the biasing of local statistics may come predominantly from background modes farthest in the infrared, which are shared by many neighboring subvolumes. In other words, the dependence of the global bispectrum on the long wavelength mode is related to the average spatial gradient of the bias in the large volume. 5 Tensor mode running as a test of inflation? If the scale dependence of the tensor power spectrum, n t ≡ d ln P t d ln k , can someday be measured, a red tilt would be (nearly) definitive evidence for inflation and against a contracting or ekpyrotic scenario (an interesting special case is 'solid inflation' [32]). Would it be possible to induce a blue tilt n t > 0 in a subvolume the size of the observable universe when the larger volume exhibits a more typical red tilt? If so, a measurement of n t > 0 would not necessarily rule out standard scalar field models of inflation. Conversely, if a red tilt n t < 0 can be induced in a large fraction of subvolumes from non-Gaussianity in a contracting universe scenario, a measurement of n t < 0 may not be a smoking gun for inflation . Consider a three-point interaction 〈 χ k 1 γ s 1 k 2 γ s 2 k 3 〉 ≡ (2 π ) 3 δ 3 ( ∑ k i ) B ( k 1 , k 2 , k 3 ) δ s 1 s 2 (5.1) between two tensor modes γ k i with polarizations s i and one mode from a field χ (here, a scalar field for example). In the squeezed limit, this three-point function will induce a dependence of the local tensor power spectrum on superhorizon χ modes. Any choice of the Fourier space kernel that gives the correct squeezed limit of the bispectrum should show the correct shift to the local power spectrum. So, with a simple choice we find that the tensor power spectrum is shifted by the correlation with long wavelength modes p as γ s i k = γ s i G, k + ∫ L -1 d 3 p 1 (2 π ) 3 d 3 p 2 (2 π ) 3 (2 π ) 3 δ 3 ( p 1 + p 2 -k ) F ( p 1 , p 2 , k )( γ s i G, p 1 χ G, p 2 + γ s i G, p 2 χ G, p 1 ) + ..., (5.2) where we take F ( k 1 , k 2 , k 3 ) = f eff γγχ ( k p ) ∑ j a j ( k 1 k p ) m 1 ,j ( k 2 k p ) m 2 ,j ( k 3 k p ) m 3 ,j . (5.3) For long wavelength modes of the χ field, the tensor power spectrum is shifted by P obs γ = P γ [1 + ∆ χ ( k )] 2 (5.4) ∆ χ ( k ) = af eff γγχ ( k p ) [ ( k k p ) m 1 + m 3 χ ( m 2 ) Gl + ( k k p ) m 2 + m 3 χ ( m 1 ) Gl ] χ ( m L ) Gl ≡ ∫ M -1 L -1 d 3 p (2 π ) 3 χ p ( p k p ) m L . With this parameterization, long wavelength modes of the χ field can shift the locally observed tilt of the tensor power spectrum. In the case that the first term in ∆ χ dominates, we can again define m S = m 1 + m 3 , m L = m 2 and then the shift is approximately ∆ n t ( k ) ≈ 2∆ χ ( k ) m S . (5.5) The quantity m S is zero for an exactly scale-invariant, local type model and more generally cannot be too large if we want to require weak non-Gaussianity for all fields. Depending on the coupling of χ to the scalar curvature, this physics may also introduce a shift in the locally observed scalar power spectrum, the tensor-to-scalar ratio, and a 'fossil' signature in the off-diagonal part of the scalar power spectrum [33], which would be an interesting complementary observable. From these expressions, it looks possible to find scenarios where the locally observed tensor power spectrum would be shifted from red to blue and vice-versa, but a full analysis along the lines of Section 3 should be performed to check consistency with all observables. 6 Discussion and conclusions Non-Gaussianity that couples the statistics of fluctuations on observable scales to wavemodes spanning super-Hubble scales can bias cosmological statistics measured by an observer in a glyph[negationslash] local Hubble volume. Previous work showed that the relative amplitudes of the power spectrum and non-Gaussianity ( f local NL ) can vary in observable subvolumes. In this work we have shown that the spectral index can also vary by enough to be interesting, | ∆ n s | glyph[similarequal] 0 . 04. The scaling of the squeezed limit of the bispectrum can also be shifted, which is relevant for constraints on non-Gaussianity from galaxy bias. These results show that in spite of the excellent precision of the measurements from the Planck satellite (especially n obs s = 0 . 9603 ± 0 . 0073 [2] and constraints on non-Gaussianity), the door is open for a significant cosmic variance uncertainty in comparing our observed patch of the universe to any particular inflation theory even leaving aside issues with eternal inflation. Moreover, rather than just presenting a new source of uncertainty from the super-Hubble background, the correlation between bispectral running in a super-Hubble volume and subvolume power spectrum measurements reopens the door for inflationary models with flat or bluer super-Hubble spectral indices, n s = 0 . 96, provided they also have scale-dependent local non-Gaussianity. This may be particularly useful for hybrid inflation. The numbers measured by the Planck satellite are consistent with a range of levels of non-Gaussianity in a post-inflationary volume, given a model for the statistics in that volume. For example, we recover the observed power spectrum and spectral index, and satisfy current constraints on f CMB NL for a post-inflationary volume with · No local type non-Gaussianity, an arbitrary number of extra e-folds, and any behavior of the power spectrum on superhorizon scales. · Constant f local NL = 5. We observe f local NL = 8 if, for example, the spectral index is a constant n s = 0 . 96 over about 200 extra e-folds of inflation and our Hubble patch sits on top of a 2-sigma under density. · Local non-Gaussianity with constant f local NL = 15. We observe f local NL = 11 if, for example, the spectral index is a constant n s = 0 . 96 over about 150 extra e-folds of inflation and our Hubble patch sits on top of a 2-sigma over-density. · Scale-dependent non-Gaussianity with f NL ( k p ) = -2, n f = 0 . 04, n ζ = 0 . 93, and n s = 0 . 935. We would observe f NL ( k p ) = -1 and n obs s = 0 . 956 if our Hubble patch sits on top of a 2-sigma under density in a volume with about 190 extra e-folds. · Scale-dependent non-Gaussianity with f NL ( k p ) = 20, n f = 0 . 03, n ζ = 0 . 95, and n s = 1 . 005. We would observe f NL ( k p ) = 2 . 5 and n obs s = 0 . 975 if our Hubble patch sits on top of a 0.2-sigma over density in a volume with about 280 extra e-folds. In contrast, we could design an inflation model to have parameters roughly consistent with Planck data, say f NL ( k p ) = 5, n f = 0 . 1, n ζ = 0 . 98, and n s = 0 . 982. However, if the model allows about 400 extra e-folds of inflation, and our Hubble patch were to sit on a 2-sigma over density, we would observe f NL ( k p ) = 4 and n obs s = 1 . 013. These results demonstrate that predictions for our observations in any scenarios with local type non-Gaussianity must be given statistically. To turn the picture around, they also suggest a new route to understanding whether observations can give us any hints about the size of the universe beyond what is directly observable. Previous ideas focused on topologically finite universes (also significantly constrained by Planck [34]) or on evidence for or against a nonperturbatively connected multiverse from bubble collisions [35-37] or curvature [38, 39]. While observations will probably never tell us how long inflation lasted, our work suggests they may at least tell us if that uncertainty is relevant to our interpretation of the data we do have. From a cosmic variance point of view, we are fortunate that there is so far no detection of local type non-Gaussianity. We have shown that future observations could push the mode-coupling uncertainties we have considered here into irrelevance 5 if primordial local non-Gaussianity can be constrained to be | f NL | < 1. Even if | f NL | > 1 is observed, tests for the running of the spectral index, any scale-dependence of | f NL | , and any evidence for extra fields through isocurvature modes or 'fossil' relics hiding in the off-diagonal power spectrum could still limit the size of any subsampling uncertainty. For example, if a blue tilt to f NL is ruled out, biasing of the spectral index is unlikely for single-source models with n ζ and n f constant on all scales. Of course, making these observations statistically well-defined depends on comparing particular competing models. It would be particularly interesting if those models had other cosmological implications related to the size of the universe 6 [40]. It would also be worthwhile to investigate the generic behavior of the local ansatz beyond f NL alone with scale-dependent coefficients, along the lines of the analysis in [6]. It may be that there are statistically natural values for the spectral index in typical small subvolumes. Then, stronger conclusions about generic cosmic variance of the spectral index might be possible. However, it is already clear that if improved limits on the amplitude and scale-dependence of non-Gaussianity can be reached, we could close the window of observational access to a perturbatively connected larger universe. Acknowledgements We thank Chris Byrnes, Bhaskar Dutta, Louis Leblond and Marilena LoVerde for useful suggestions and discussions about this work. The work of J. B. is supported in part by Department of Energy grant DE-FG02-04ER41291. The work of J. K. is supported in part by Department of Energy grants DE-FG02-04ER41291 and DE-FG02-13ER41913. The work of S. S. is supported in part by the National Aeronautics and Space Administration under Grant No. NNX12AC99G issued through the Astrophysics Theory Program. In addition, S. S. thanks the organizers of the Primordial Cosmology Program at KITP for hospitality while this work was being completed and for support by the National Science Foundation under Grant No. NSF PHY11-25915. J. K. thanks the Center for Theoretical Underground Physics and Related Areas (CETUP* 2013) in South Dakota for its support and hospitality while this work was being completed. E. N. is supported by the Eberly Research Funds of The Pennsylvania State University. The Institute for Gravitation and the Cosmos is supported by the Eberly College of Science and the Office of the Senior Vice President for Research at the Pennsylvania State University. References [1] Planck Collaboration Collaboration, P. Ade et. al. , Planck 2013 results. I. Overview of products and scientific results , 1303.5062 . [2] Planck Collaboration Collaboration, P. Ade et. al. , Planck 2013 results. XXII. Constraints on inflation , 1303.5082 . [3] Planck Collaboration, P. Ade et. al. , Planck 2013 Results. XXIV. Constraints on primordial non-Gaussianity , 1303.5084 . [4] A. D. Linde and V. Mukhanov, The curvaton web , JCAP 0604 (2006) 009, [ astro-ph/0511736 ]. [5] V. Demozzi, A. Linde, and V. Mukhanov, Supercurvaton , JCAP 1104 (2011) 013, [ 1012.0549 ]. [6] E. Nelson and S. Shandera, Statistical Naturalness and non-Gaussianity in a Finite Universe , accepted for publication in PRL (2012) [ 1212.4550 ]. [7] S. Nurmi, C. T. Byrnes, and G. Tasinato, A non-Gaussian landscape , 1301.3128 . [8] M. LoVerde, E. Nelson, and S. Shandera, Non-Gaussian Mode Coupling and the Statistical Cosmological Principle , 1303.3549 . [9] C. T. Byrnes, S. Nurmi, G. Tasinato, and D. Wands, Implications of the Planck bispectrum constraints for the primordial trispectrum , 1306.2370 . [10] D. S. Salopek and J. R. Bond, Nonlinear evolution of long wavelength metric fluctuations in inflationary models , Phys. Rev. D42 (1990) 3936-3962. [11] E. Komatsu and D. N. Spergel, Acoustic signatures in the primary microwave background bispectrum , Phys.Rev. D63 (2001) 063002, [ astro-ph/0005036 ]. [12] J. M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models , JHEP 0305 (2003) 013, [ astro-ph/0210603 ]. [13] P. Creminelli and M. Zaldarriaga, Single field consistency relation for the 3-point function , JCAP 0410 (2004) 006, [ astro-ph/0407059 ]. [14] E. Pajer, F. Schmidt, and M. Zaldarriaga, The Observed Squeezed Limit of Cosmological Three-Point Functions , 1305.0824 . [15] P. McDonald, Primordial non-Gaussianity: large-scale structure signature in the perturbative bias model , Phys.Rev. D78 (2008) 123519, [ 0806.1061 ]. [16] M. LoVerde and K. M. Smith, The Non-Gaussian Halo Mass Function with fNL, gNL and tauNL , JCAP 1108 (2011) 003, [ 1102.1439 ]. [17] M. Gerstenlauer, A. Hebecker, and G. Tasinato, Inflationary Correlation Functions without Infrared Divergences , JCAP 1106 (2011) 021, [ 1102.0560 ]. [18] S. B. Giddings and M. S. Sloth, Cosmological observables, IR growth of fluctuations, and scale-dependent anisotropies , Phys.Rev. D84 (2011) 063528, [ 1104.0002 ]. [19] D. H. Lyth, The curvature perturbation in a box , JCAP 0712 (2007) 016, [ 0707.0361 ]. [20] L. Boubekeur and D. Lyth, Detecting a small perturbation through its non-Gaussianity , Phys.Rev. D73 (2006) 021301, [ astro-ph/0504046 ]. [21] J. Kumar, L. Leblond, and A. Rajaraman, Scale Dependent Local Non-Gaussianity from Loops , JCAP 1004 (2010) 024, [ 0909.2040 ]. [22] J. Bramante and J. Kumar, Local Scale-Dependent Non-Gaussian Curvature Perturbations at Cubic Order , JCAP 1109 (2011) 036, [ 1107.5362 ]. [23] N. S. Sugiyama, Consistency Relation for multifield inflation scenario with all loop contributions , JCAP 1205 (2012) 032, [ 1201.4048 ]. [24] C. T. Byrnes, K. Enqvist, and T. Takahashi, Scale-dependence of Non-Gaussianity in the Curvaton Model , JCAP 1009 (2010) 026, [ 1007.5148 ]. [25] Q.-G. Huang, Negative spectral index of f NL in the axion-type curvaton model , JCAP 1011 (2010) 026, [ 1008.2641 ]. [26] M. Dias, R. H. Ribeiro, and D. Seery, Scale-dependent bias from multiple-field inflation , 1303.6000 . [27] S. Shandera, N. Dalal, and D. Huterer, A generalized local ansatz and its effect on halo bias , JCAP 1103 (2011) 017, [ 1010.3722 ]. [28] D. Tseliakhovich, C. Hirata, and A. Slosar, Non-Gaussianity and large-scale structure in a two-field inflationary model , Phys.Rev. D82 (2010) 043531, [ 1004.3302 ]. [29] R. Scoccimarro, L. Hui, M. Manera, and K. C. Chan, Large-scale Bias and Efficient Generation of Initial Conditions for Non-Local Primordial Non-Gaussianity , Phys.Rev. D85 (2012) 083002, [ 1108.5512 ]. [30] T. Tanaka and Y. Urakawa, Dominance of gauge artifact in the consistency relation for the primordial bispectrum , JCAP 1105 (2011) 014, [ 1103.1251 ]. [31] P. Creminelli, A. Nicolis, L. Senatore, M. Tegmark, and M. Zaldarriaga, Limits on non-gaussianities from wmap data , JCAP 0605 (2006) 004, [ astro-ph/0509029 ]. [32] S. Endlich, A. Nicolis, and J. Wang, Solid Inflation , 1210.0569 . [33] D. Jeong and M. Kamionkowski, Clustering Fossils from the Early Universe , Phys.Rev.Lett. 108 (2012) 251301, [ 1203.0302 ]. [34] Planck Collaboration Collaboration, P. Ade et. al. , Planck 2013 results. XXVI. Background geometry and topology of the Universe , 1303.5086 . [35] S. Chang, M. Kleban, and T. S. Levi, Watching Worlds Collide: Effects on the CMB from Cosmological Bubble Collisions , JCAP 0904 (2009) 025, [ 0810.5128 ]. [36] S. M. Feeney, M. C. Johnson, J. D. McEwen, D. J. Mortlock, and H. V. Peiris, Hierarchical Bayesian Detection Algorithm for Early-Universe Relics in the Cosmic Microwave Background , 1210.2725 . [37] S. Osborne, L. Senatore, and K. Smith, Collisions with other Universes: the Optimal Analysis of the WMAP data , 1305.1964 . [38] M. Kleban and M. Schillo, Spatial Curvature Falsifies Eternal Inflation , JCAP 1206 (2012) 029, [ 1202.5037 ]. [39] A. H. Guth and Y. Nomura, What can the observation of nonzero curvature tell us? , Phys.Rev. D86 (2012) 023534, [ 1203.6876 ]. [40] D. B. Kaplan and A. E. Nelson, Inflationary Axion Cosmology Beyond Our Horizon , 0809.1206 . Gl Gl - 0 10 0 10 to a smaller (larger) number of e-folds in the case of a red tilt (blue tilt), as shown in Figure 2, so it is easier to realize a large shift to a global red tilt than to a global blue tilt. In fact, as previously noted in the discussion of Figure 2, imposing the requirement that P obs ζ be typical of subvolumes for scenarios with a blue tilt n ζ -1 causes 〈 ζ 2 Gl 〉 to converge to a particular value as N is increased. 2. A red tilt in the power spectrum can relax the constraint from requiring weak global non-Gaussianity, as seen by comparing the right panels in Figure 3 to the left panels. For example, when n f > 0, a red tilt in the power spectrum gives more relative weight in M 3 to the more weakly coupled superhorizon modes and damps the power of strongly coupled subhorizon modes. Note that the bottom two panels in Figure 3 permit about the same number of super-horizon e-folds of weakly non-Gaussian parameter space. In the right panel the power removed from subhorizon e-folds by n ζ < 1 is balanced by power added to superhorizon e-folds leading to a larger background 〈 ζ 2 Gl 〉 per e-fold permitted for perturbative statistics as compared to the bottom left panel. For these reasons single-source scenarios with a red power tilt in the large volume have the most significant range of cosmic variance due to subsampling. The solid black lines in Figure 3 show ∆ n s ( k p ) = -0 . 04 in subvolumes with a +0 . 5 σ background fluctuation ( ζ Gl = +0 . 5 〈 ζ 2 Gl 〉 1 / 2 ), and thus show part of the parameter space where | ∆ n s ( k p ) | can be observationally significant. Here we have neglected the subhorizon one-loop correction 〈 ζ 2 Gs ( k p ) 〉 glyph[lessmuch] ζ 2 Gl ; this breaks down for small N but is valid outside of the region of parameter space excluded by the requirement f CMB NL < 10. Rewriting Eq. (3.3) for a single source scenario ( ξ m = 1), ∆ n single source s glyph[similarequal] n f ( 12 5 f NL ζ Gl ( 1 -36 25 f 2 NL 〈 ζ 2 Gl 〉 ) + 72 25 f 2 NL ( ζ 2 Gl -〈 ζ 2 Gl 〉 ) ) ( 1 + 6 5 f NL ζ Gl ) 2 ( 1 + 36 25 f 2 NL 〈 ζ 2 Gl 〉 ) . (3.6) Assuming n f = 0 . 1 and ζ Gl = 0 . 5 〈 ζ 2 Gl 〉 1 / 2 , we can solve this equation to show that ∆ n single source s = -0 . 04 when f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 = 0 . 94 or 5.9, which are the equations of the two black lines plotted in Figure 3. These lines assume positive f NL in the large volume, f NL > 0, but they remain the same for f NL < 0 and a -0 . 5 σ background fluctuation. For values of | n f | larger or smaller than 0 . 1, the distance between these lines grows or shrinks in parameter space. Of course, for the full expression of ∆ n s and a different set of parameter choices, there can be more than two solutions of | ∆ n s | = 0 . 04. For positive f NL ( k p ) ζ Gl (see below), the typical size of ∆ n s is largest in the region between these lines ( 6 5 f NL ( k p ) ζ Gl ∼ 1) and falls towards zero on either side. The upper dotted-dashed lines mark where 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 is large ( O (10)). When that quantity is large, ∆ n s glyph[similarequal] -n f 3 5 f NL 〈 ζ 2 Gl 〉 1 / 2 and thus approaches zero as indicated in Figure 3. Note that in this region the observed spectral index is n obs s glyph[similarequal] n s glyph[similarequal] n ζ +2 n f , so for the parameter choices in Figure 3 the Planck satellite excludes the region above the dotted-dashed lines. All lines and contours in Figure 3 assume that 6 5 f NL ( k p ) ζ Gl > 0 (eg, overdense fluctuations with positive f NL ). If this figure assumed 6 5 f NL ( k p ) ζ Gl < 0 (eg, overdense fluctuations with negative f NL ), the area in parameter space near the line 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 = 1 would be excluded. For further discussion of parameter space with 6 5 f NL ζ Gl < 0, see the discussion after Eq. (3.3). Figure 3 shows that, under the conditions we have imposed and the spectral indices considered, only scenarios where the bispectral tilt is not very red have typical subvolumes where the observed spectral index varies by an amount that is cosmologically interesting for us, | ∆ n s | glyph[greaterorsimilar] 0 . 01. A blue bispectral index may avoid the current observational constraints, which do not probe particularly small scales, and easily remain globally perturbative and weakly non-Gaussian (see paragraph below). In contrast, the bottom panels of Figure 3 illustrate that for either spectral index, a scenario with n f < 0 will be nonperturbative in the interesting part of parameter space where | ∆ n s | ∼ 0 . 04. (In addition, there is only a small window with strongly non-Gaussian but perturbative global statistics.) If both the power spectrum and non-Gaussianity increase in the IR, as in the lower right panel of Figure 3, the statistics will be strongly non-Gaussian across parameter space for a small number of superhorizon e-folds. The upper panels of Figure 3 illustrate a feature discussed in Section 2.2: 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 glyph[greaterorsimilar] 1 does not necessarily imply a large cumulative skewness, M 3 glyph[greaterorsimilar] 1. The dashed curves fix M 3 = 1 as a function of superhorizon e-folds, which are determined at each point in parameter space by the observed level of the power spectrum along with n f , f NL and 〈 ζ 2 Gl 〉 . In regions where M 3 < 1 but f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 glyph[greaterorsimilar] 1, there are a sufficient number of superhorizon modes with weaker coupling ( n f > 0) damp the total non-Gaussianity. To elaborate, in the limit n f ( N + N sub ) glyph[greatermuch] 1, Eq. (2.18) gives M 3 ∝ [ 〈 ζ 2 Gl 〉 /N ( N + N sub )] 1 / 2 . For f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 glyph[lessmuch] 1, N = 〈 ζ 2 Gl 〉 / P obs ζ and so M 3 becomes independent of 〈 ζ 2 Gl 〉 in the limit N glyph[lessmuch] N sub . For f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 glyph[greatermuch] 1, on the other hand, M 3 ∝ 1 /f NL ( k p ) 〈 ζ 2 Gl 〉 3 / 2 , so large f NL ( k p ) and sufficiently large 〈 ζ 2 Gl 〉 are needed to keep the total non-Gaussianity small, and P obs ζ ∼ 2 × 10 -9 typical in subvolumes, as seen in the upper left panel of Figure 3. Note that throughout this analysis, we have assumed n f is constant for all N sub = 60 subhorizon e-folds, so that for blue n f non-Gaussianity continues to grow on subhorizon scales where nonlinear evolution has taken over. If this condition is relaxed, the conditions from weak non-Gaussianity are less restrictive. Figure 4 shows the probability distribution for the shift ∆ n s for the parameters in part of the range of interest for the blue bispectral index shown in the top panels of Figure 3. Both panels show examples that (for appropriate choices of large volume parameters) give local power spectra amplitude and f CMB NL consistent with our observations. Notice that the distribution on the right is substantially less Gaussian than the distribution on the left. This trend continues if one considers larger 〈 ζ 2 Gl 〉 while keeping all other parameters fixed. In Figure 5 we show regions of parameter space in the ( 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 , n f ) plane that are consistent with the Planck measurement n obs s = 0 . 9603 ± 0 . 0073. Assuming that the scalar power spectrum in the full volume of the mode-coupled universe is completely flat, n ζ = 1, we see that 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 must be at least O (10 -1 ) and for weakly non-Gaussian statistics, more than a hundred superhorizon e-folds are required. It is interesting to note that in the case of a blue-tilted f NL , a larger running non-Gaussianity n f loosens parameter constraints coming from requiring perturbative statistics 〈 [ f NL ( k ) ζ 2 Gl ] 2 〉 glyph[lessorsimilar] 0 . 1. Although the dotted lines in Figure 5 will shift to the left with more superhorizon e-folds, these curves exclude less parameter space as n f becomes larger. This is because we have assumed n f is blue and constant so f NL is driven to smaller values in the IR and 〈 [ f NL ( k ) ζ 2 Gl ] 2 〉 becomes smaller for larger n f . Notice the shift in the non-perturbative line in the right panel that occurs at n f > | n ζ -1 | : if the running of the power spectrum is larger than the running of f NL ( k ), then the running of the power spectrum will dominate the variance of the local quadratic term over superhorizon modes, because f 2 NL ( k ) P G ( k ) 2 ∝ k 2( n f + n ζ -1) . Lastly, the Figure 4 . The probability of finding a shift in the spectral index in subvolumes. Left panel: The variance plotted here corresponds to about 195 extra e-folds in a model with n ζ = 0 . 96 or 4 × 10 4 extra e-folds for a scale-invariant spectrum. Right panel: The variance here is consistent with about 240 extra e-folds in a model with n ζ = 0 . 96 or 5 × 10 5 extra e-folds for a scale-invariant spectrum. In both panels the solid black lines show a bispectral index of n f = 0 . 05 while the dotted blue lines show n f = 0 . 1. In the right panel about 24% (6%) of subvolumes in the n f = 0 . 1 ( n f = 0 . 05) have ∆ n s ≥ 0 . 02 and 17% (5%) have ∆ n s ≤ -0 . 04. The points in parameter space that correspond to the dotted lines ( n f = 0 . 1) are shown with black squares in Figure 3. Figure 4 . The probability of finding a shift in the spectral index in subvolumes. Left panel: The variance plotted here corresponds to about 195 extra e-folds in a model with n ζ = 0 . 96 or 4 × 10 4 extra e-folds for a scale-invariant spectrum. Right panel: The variance here is consistent with about 240 extra e-folds in a model with n ζ = 0 . 96 or 5 × 10 5 extra e-folds for a scale-invariant spectrum. In both panels the solid black lines show a bispectral index of n f = 0 . 05 while the dotted blue lines show n f = 0 . 1. In the right panel about 24% (6%) of subvolumes in the n f = 0 . 1 ( n f = 0 . 05) have ∆ n s ≥ 0 . 02 and 17% (5%) have ∆ n s ≤ -0 . 04. The points in parameter space that correspond to the dotted lines ( n f = 0 . 1) are shown with black squares in Figure 3. right panel of Figure 5 shows once again that for a blue tilted f NL , the weakly non-Gaussian parameter space enlarges with the number of superhorizon e-folds, because f NL is driven to very small values over more superhorizon e-folds, decreasing the value of M 3 . To conclude this section, Figure 6 illustrates a single-source scenario in which a power spectrum which appears blue-tilted in the large volume on short scales can appear red on the same scales in a subvolume. On scales where P ζ ( k ) glyph[similarequal] P G ( k ), n s ( k ) glyph[similarequal] n ζ , whereas on scales where the 1-loop contribution dominates P 1-loop ζ ( k ) glyph[similarequal] 36 25 f 2 NL ( k ) 〈 ζ 2 Gl 〉P G ( k ) and the spectral index will be n s ( k ) glyph[similarequal] n ζ + 2 n f . If the transition of power takes place on a scale near the observable range of scales ( f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 = O (1)), the observed spectral index can be shifted. For example, if ζ 2 Gl < 〈 ζ 2 Gl 〉 , the blue-tilted f 2 NL 〈 ζ 2 Gl 〉 contribution loses power in the subvolume, and if f NL ( k p ) ζ Gl > 0, the red-tilted piece gains power (compare Eqs. (2.5), (2.9)). This scenario is shown in Figure 6. Note that as long as f NL ( k p ) is not extremely large (which would violate the constraint on f CMB NL for the value of f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 chosen here), ζ Gl glyph[greatermuch] 〈 ζ 2 Gs ( k ) 〉 1 / 2 and the 1-loop contribution to P obs ζ is very small, suppressed by a factor of 〈 ζ 2 Gs ( k ) 〉 /ζ 2 Gl . Example III: Multiple sources with running ξ m ( k ) . In the single-source case, a large shift to the observed spectral index could only occur if the 1-loop contribution to the power spectrum dominated on small scales. With two sources, a significant shift to n s can be consistent with weak non-Gaussianity ξ m ( k ) f NL ( k ) 〈 σ Gl 〉 1 / 2 < 1 on all scales. If the running of the 1-loop contribution lies between the runnings n σ ≡ d ln P σ ( k ) d ln( k ) and n φ ≡ d ln P φ ( k ) d ln( k ) of the Gaussian contributions to the total power, then it will be subdominant on large and small scales. The transition of power between σ G and φ G takes place over a finite range of scales, over H GLYPH<144> L H L Figure 5 . Left panel: a model with a globally flat power spectrum, but which contains subvolumes where a red tilt would be observed. Right panel: a model with global parameters naively matched to observations that nonetheless contains a significant number of subvolumes with a spectral index at odds with observations. Both cases show single-source perturbations with the running of f NL , n f , plotted against the parameters controlling the size of the bias, 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 . This figure assumes positive f NL and a blue running of f NL . The running of the power spectrum is flat ( n s ( k p ) glyph[similarequal] n ζ = 1) and red ( n s ( k p ) glyph[similarequal] n ζ = 0 . 96) to within ∼ 0 . 01 below the dotted-dashed lines in the left and right panels, respectively. Above the dotted-dashed lines the loop correction to the running of the power spectrum becomes large ( n s ( k p ) -n ζ > 0 . 01). Dashed lines indicate regions where the non-Gaussian cumulant M 3 > 1 for the number of superhorizon e-folds indicated. The dotted line indicates the nonperturbative region ( 〈 [ 3 5 f NL glyph[star] ( ζ 2 Gl -〈 ζ 2 Gl 〉 )] 2 〉 glyph[greaterorsimilar] 0 . 1) for N > 10 3 and N > 100 in the left and right panels, respectively. The grey space shows what region is excluded at 99% confidence by the Planck measurement n obs s = 0 . 9603 ± 0 . 0073, assuming an underdense subsample with a -1 σ background fluctuation. Figure 5 . Left panel: a model with a globally flat power spectrum, but which contains subvolumes where a red tilt would be observed. Right panel: a model with global parameters naively matched to observations that nonetheless contains a significant number of subvolumes with a spectral index at odds with observations. Both cases show single-source perturbations with the running of f NL , n f , plotted against the parameters controlling the size of the bias, 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 . This figure assumes positive f NL and a blue running of f NL . The running of the power spectrum is flat ( n s ( k p ) glyph[similarequal] n ζ = 1) and red ( n s ( k p ) glyph[similarequal] n ζ = 0 . 96) to within ∼ 0 . 01 below the dotted-dashed lines in the left and right panels, respectively. Above the dotted-dashed lines the loop correction to the running of the power spectrum becomes large ( n s ( k p ) -n ζ > 0 . 01). Dashed lines indicate regions where the non-Gaussian cumulant M 3 > 1 for the number of superhorizon e-folds indicated. The dotted line indicates the nonperturbative region ( 〈 [ 3 5 f NL glyph[star] ( ζ 2 Gl -〈 ζ 2 Gl 〉 )] 2 〉 glyph[greaterorsimilar] 0 . 1) for N > 10 3 and N > 100 in the left and right panels, respectively. The grey space shows what region is excluded at 99% confidence by the Planck measurement n obs s = 0 . 9603 ± 0 . 0073, assuming an underdense subsample with a -1 σ background fluctuation. glyph[negationslash] H GLYPH<144> L H GLYPH<144> L H L H GLYPH<144> L which n s changes from n σ to n φ . If the power spectrum of φ G is blue and dominates on small scales ( ξ m ( k glyph[greaterorsimilar] H 0 ) glyph[lessmuch] 1), and the Gaussian contribution from σ is red and dominates on large scales ( ξ m ( k << H 0 ) glyph[similarequal] 1), then the background ζ l glyph[similarequal] σ l for any subvolume couples to and biases the local statistics. For example, a globally flat or blue spectral index n s ( k > H 0 ) > 1 can again appear red, n obs s < 1, in a subvolume. The shift to n s can come only from the modulation of power in σ relative to φ G , and need not rely on running non-Gaussianity n f = 0. That is, a large running of the difference in power of the fields can be achieved without a large level of running non-Gaussianity. This becomes apparent upon inspecting the running of ξ m , n ( m ) f ( k ) ≡ d ln ξ m ( k ) d ln k = (1 -ξ m ( k )) [ n σ -n φ + 2 n f 36 25 f 2 NL ( k ) 〈 σ 2 G ( k ) 〉 1 + 36 25 f 2 NL ( k ) 〈 σ 2 G ( k ) 〉 ] . (3.7) If φ G is more red-tilted than σ G , the background is uncorrelated with short-wavelength modes because φ G dominates on large scales, ζ l glyph[similarequal] φ Gl , so local statistics are not biased. Thus, both n σ ≤ 1 and n φ > n σ are needed for a significant bias. In Figure 7 we show the parameter ln L H n ζ = 0 . 95 n f = 0 . 05 n ζ = 0 . 95 n f = 0 . 05 Figure 6 . Top panel: The contributions to the power spectrum P G ( k ) and P 1-loop ζ ( k ) glyph[similarequal] 36 25 f 2 NL ( k ) 〈 ζ 2 Gl 〉P G ( k ) are shown, for the following parameter choices: n ζ = 0 . 95, n f = 0 . 05, f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 = 3. The total power spectrum is shown with a thin black line, and the corresponding shifted power spectra for a subvolume with a +0 . 1 σ background fluctuation is shown with a thick black line. The vertical scale can be fixed so P obs ζ matches the observed value. Bottom panel: Parameter space for single source non-Gaussianity with n ζ = 0 . 95 and n f = 0 . 05 is shown. The dotted-dashed line indicates f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 = 10, both black lines indicate ∆ n s = -0 . 065 for a +0 . 1 σ background fluctuation, and the red circle indicates the parameter space congruent with the top panel. Dotted lines show the indicated number of superhorizon e-folds for a +0 . 1 σ bias. The exclusion regions are marked the same as those in Figure 3, but these assume a +0 . 1 σ bias. Figure 6 . Top panel: The contributions to the power spectrum P G ( k ) and P 1-loop ζ ( k ) glyph[similarequal] 36 25 f 2 NL ( k ) 〈 ζ 2 Gl 〉P G ( k ) are shown, for the following parameter choices: n ζ = 0 . 95, n f = 0 . 05, f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 = 3. The total power spectrum is shown with a thin black line, and the corresponding shifted power spectra for a subvolume with a +0 . 1 σ background fluctuation is shown with a thick black line. The vertical scale can be fixed so P obs ζ matches the observed value. Bottom panel: Parameter space for single source non-Gaussianity with n ζ = 0 . 95 and n f = 0 . 05 is shown. The dotted-dashed line indicates f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 = 10, both black lines indicate ∆ n s = -0 . 065 for a +0 . 1 σ background fluctuation, and the red circle indicates the parameter space congruent with the top panel. Dotted lines show the indicated number of superhorizon e-folds for a +0 . 1 σ bias. The exclusion regions are marked the same as those in Figure 3, but these assume a +0 . 1 σ bias. space for the two-source scenario described above, with n σ ( k p ) = 0 . 93, n φ ( k p ) = 1 . 005, and ξ m ( k p ) = 0 . 1. We also fix n f = 0 . 001 so that mode coupling is weaker on superhorizon scales. As before, the upper left region shows where f obs NL glyph[greaterorsimilar] 10 in typical subvolumes. We see that adding the second source relaxes the constraint on f NL in the f NL 〈 σ 2 Gl 〉 1 / 2 glyph[lessmuch] 1 regime. This makes it possible to achieve a large shift ∆ n s for smaller values of 〈 σ 2 Gl 〉 and thus fewer superhorizon e-folds. ln k L H Figure 7 . Multifield parameter space for ξ m ( k p ) = 0 . 1, n σ = 0 . 93, n φ = 1 . 005, n f = 0 . 001. The black lines show ∆ n s glyph[similarequal] -0 . 03 for a +3 σ background fluctuation. The dotted-dashed line shows f NL ( k p ) 〈 σ 2 Gl 〉 1 / 2 = 10. The upper left region shows the Planck constraint on f CMB NL for a +3 σ background. Figure 7 . Multifield parameter space for ξ m ( k p ) = 0 . 1, n σ = 0 . 93, n φ = 1 . 005, n f = 0 . 001. The black lines show ∆ n s glyph[similarequal] -0 . 03 for a +3 σ background fluctuation. The dotted-dashed line shows f NL ( k p ) 〈 σ 2 Gl 〉 1 / 2 = 10. The upper left region shows the Planck constraint on f CMB NL for a +3 σ background. The condition ξ m ( k p ) = 0 . 1 makes the field φ G dominant on Planck scales, so from the perspective of the large volume, the power spectrum has a blue tilt n s ( k p ) glyph[similarequal] n φ = 1 . 005 on scale k p . However, for significant biasing (3 σ ) and a small (or zero) non-Gaussian running of the coupled field n f = 0 . 001, the black lines in Figure 7 denote where ∆ n s = -0 . 03, which would be consistent with Planck observations. Here the shift in ∆ n s is coming not from n f but from the difference in running of P σ,NG and P φ , n ( m ) f , as the red-tilted P σ,NG is amplified due to the strong background overdensity. It is also interesting to note that a cursory survey of background fluctuations reveals that biases less than | 3 σ | yield no ∆ n s corrections smaller than -0 . 03, which would seem to partly exclude these parameters for typical Hubble-sized subsamples. In the limit of very small ξ m ( k p ), φ G dominates the power and scale-dependence on observable scales, so unless the bias is extremely strong, any shift in the power and scale-dependence from the σ field will be too small to affect n obs s . Summary. In summary, a significant shift to the observed spectral index from correlations with long-wavelength background modes is possible under the following conditions: 1. A red tilt for the field with mode coupling , n σ ≤ 1 ( n ζ ≤ 1 in the single-source case), is necessary for the cumulative power 〈 σ 2 Gl 〉 on superhorizon scales to be large enough to significantly bias local statistics. 2. A blue bispectral index n f ≥ 0 for f NL ( k ) (assuming constant n f ) is needed to remove the power from the non-Gaussian term on large scales so that strong coupling of shortscale modes to background modes is consistent with weak global non-Gaussianity and ζ being perturbative, while having enough background modes to give a large bias. 3. In a two-source scenario, the ratio of power in the non-Gaussian field to total power should have a red spectrum ( n ( m ) f ( k p ) ≤ 0) so that the non-Gaussian field σ G grows relative to φ G on large scales, causing the background ζ l to be sufficiently correlated with local statistics. If φ G contributes on observable scales ( ξ m ( k p ) < 1), larger values of f NL ( k p ) are consistent with observational constraints on non-Gaussianity, so a smaller background σ Gl is needed to give the same shift to n obs s . Introducing scale-dependence into the spectral indices would relax the conditions for large | ∆ n s | . Although the scenario becomes more complicated in this case, the qualitative features remain valid: scale-dependence of power spectra and non-Gaussian parameters must allow for sufficient cumulative superhorizon power that a large background σ Gl from the source with mode coupling is typical. We note that for given large-volume statistics, the observed red tilt may not be equally consistent with a local overdensity or underdensity in σ G . In the single-source case with n f > 0, for example, an overdensity (underdensity) corresponds to an increase (decrease) of power on small scales. Thus, for a scale-invariant power spectrum in the large volume, the observed red tilt n obs s glyph[similarequal] 0 . 96 could be accounted for in terms of a blue-tilted global bispectrum and local underdensity. However, without information about the global power spectrum, it would be difficult to infer whether we sit on a local underdensity or overdensity. 3.3 The shift to the scale dependence of the bispectrum The bispectrum may also be shifted by mode coupling coming from the soft limits of the large-volume trispectrum and from any non-Gaussian shifts to power spectrum. We can define a spectral index for the squeezed limit of the bispectrum within any particular volume as n sq . ≡ d ln B ζ ( k L , k S , k S ) d ln k L -( n s -1) (3.8) where k L and k S are long wavelength and short wavelength modes, respectively. The small volume quantity, n obs sq . , should be calculated using the observed bispectrum and the observed spectral index. For a single source, scale-invariant local ansatz, n sq . = -3. For the single source, weakly non-Gaussian, scale-dependent scenario with g NL absent, the shift in this bispectral index between the large volume and what is observed in the small volume is Single Source : ∆ n sq . ( k ) ≡ n obs sq . ( k ) -n LargeVol . sq . ( k ) (3.9) ≈ -6 5 f NL ( k L ) σ Gl n f 1 + 6 5 f NL ( k L ) σ Gl . If 6 5 f NL ( k L ) σ Gl = 6 5 f NL ( k L ) 〈 ζ 2 G 〉 1 / 2 B glyph[lessmuch] 1, then ∆ n sq . ( k ) ≈ -6 5 f NL ( k L ) 〈 ζ 2 G 〉 1 / 2 Bn f . This shift is less than one in magnitude, but still relevant for interpreting bispectral indices of order slow-roll parameters. In the two source case, there can be additional scale dependence coming from the ratio of power of the two fields. Considering only the weak coupling case, 6 5 f NL ( k ) σ Gl glyph[lessmuch] 1 (and again setting g NL = 0 for simplicity), Two Source : ∆ n sq . ( k ) = 12 5 f NL ( k ) σ Gl 1 + 12 5 f NL ( k ) σ Gl n f -6 5 f NL ( k ) σ Gl 1 + 6 5 f NL ( k ) σ Gl n f (3.10) -12 5 ξ m ( k ) f NL ( k ) σ Gl 1 + 12 5 ξ m ( k ) f NL ( k ) σ Gl ( n f + n ( m ) f ) ≈ 6 5 f NL ( k ) σ Gl n f -12 5 ξ m ( k ) f NL ( k ) σ Gl ( n f + n ( m ) f ) . Reintroducing g NL and higher terms would lead to additional terms, introducing scaledependence even if f NL in the large volume is a constant. 3.4 Generalized local ansatz and single source vs. multi source effects The two source, weakly scale dependent local ansatz in Eq. (2.1) is representative of the properties of inflation models that generate local type non-Gaussianity. For example, the scale-dependence f NL ( k ) can come from curvaton models with self-interactions [24, 25]. The function ξ m ( k ) comes from the difference in power spectrum of two fields (eg, the inflaton and the curvaton) contributing to the curvature fluctuations. In typical multi-field models, the bispectral indices n f , n ( m ) f are of order slow-roll parameters (like the scale dependence of the power spectrum), and are often not constant. Generic expressions for the squeezed limit behavior of a multi-field bispectrum are given in [26]. The scale-dependent functions f NL ( k ) and ξ m ( k ) are observationally relevant for tests for primordial non-Gaussianity using the bias of dark matter halos and their luminous tracers (eg. quasars or luminous red galaxies). The power law dependence of the squeezed limit on the long wavelength, small momentum mode ( n sq . from Eq. (3.8)) generates the scale-dependence of the non-Gaussian term in the bias. The dependence on the short wavelength modes generates a dependence of the non-Gaussian bias on the mass of the tracer (which is absent in the usual local ansatz). In principle, if local non-Gaussianity is ever detected, it may be within the power of future large scale structure surveys to detect some amplitude of running [27]. However, as demonstrated above, the same shape of bispectrum can be generated locally by a single source for the curvature perturbations, so the presence of the non-trivial function ξ m in the observed bispectrum does not necessarily indicate that two fundamental fields contributed to the primordial curvature perturbations. On the other hand, the presence of one Gaussian source and one non-Gaussian source for the local curvature perturbations is in principle detectable by comparing power spectra that are sensitive in different ways to the total curvature field and to just the non-Gaussian part [28]. Eq. (2.12) shows that in a single source scenario the local background σ Gl can act as a second field to generate the full, multi-source shaped bispectrum, but σ Gl is constant within a single volume. This 'second field' does not have fluctuations on all scales, but its variations are relevant for considering a collection of subvolumes of a particular size. 4 Mode coupling effects from a non-local factorizable bispectrum We have considered the effect of superhorizon modes only for the case of nearly local nonGaussianity, but inflationary theory has generated an expanding space of models exhibiting different types of mode coupling. Intuitively, any scenario that does not couple modes of sufficiently different wavelengths should not lead to correlation functions whose amplitudes or shapes change under subsampling. As a first step towards considering the observational consequences of subsampling general non-Gaussian scenarios, it is straighforward to find corrections from the background to small-volume quantities in the case of a factorizable quadratic kernel in Fourier space with power-law dependence. Consider a curvature perturbation in the large volume given by ζ k = φ G, k + σ G, k + ∫ L -1 d 3 p 1 (2 π ) 3 d 3 p 2 (2 π ) 3 (2 π ) 3 δ 3 ( p 1 + p 2 -k ) F ( p 1 , p 2 , k ) σ G, p 1 σ G, p 2 + ..., (4.1) where F ( k 1 , k 2 , k 3 ) = ∑ j a NL ,j ( k p ) ( k 1 k p ) m 1 ,j ( k 2 k p ) m 2 ,j ( k 3 k p ) m 3 ,j (4.2) is a sum of factorizable terms with power law dependence on the momenta. On the right hand side the a j are amplitudes defined at a pivot scale k p . When ∑ i m i,j glyph[similarequal] 0 for every term j , the bispectrum is approximately scale-invariant. The kernel F ( k 1 , k 2 , k 3 ) can be chosen to generate a desired bispectrum with well behaved one-loop corrections to the power spectrum [29]. Splitting the modes into long and short, the locally defined short wavelength modes with shifts induced from coupling to long wavelength modes from one term in the series above are ζ k s = φ G, k s + σ G, k s + σ G, k s a NL ( k p ) [ ( k k p ) m 1 + m 3 σ ( m 2 ) Gl + ( k k p ) m 2 + m 3 σ ( m 1 ) Gl ] (4.3) + a NL ( k p ) ∫ M -1 d 3 p (2 π ) 3 σ G ( p ) σ G ( | k -p | ) ( k k p ) m 3 ( | k -p | k p ) m 2 ( p k p ) m 1 where σ ( m L ) Gl ≡ ∫ M -1 L -1 d 3 p (2 π ) 3 σ p ( p k p ) m L . (4.4) When the local field is weakly non-Gaussian, the second line is small and we can rewrite the first line as ζ k s ≈ φ G, k s + σ G, k s [1 + ∆ σ ( k )] (4.5) ∆ σ ( k ) = a NL ( k p ) [ ( k k p ) m 1 + m 3 σ ( m 2 ) Gl + ( k k p ) m 2 + m 3 σ ( m 1 ) Gl ] . The leading shift to the power spectrum P obs ζ in a subvolume from unobservable infrared modes in one term of the series above (and assuming weak non-Gaussianity) is: P obs ζ ( k ) = P ζ ( k ) { 1 + ξ m ( k ) [ 2∆ σ ( k ) + ∆ σ ( k ) 2 ]} ; (4.6) where ξ m ( k ) is still the ratio of power in the non-Gaussian source to the total power, defined in Eq.(2.7). In the two-field case with weak non-Gaussianity on all scales, the observed ratio of power in the two fields is related to the same ratio in the large volume by ξ obs m ( k ) = ξ m ( k ) [1 + ∆ σ ( k )] 2 1 + ξ m ( k )[2∆ σ ( k ) + ∆ σ ( k ) 2 ] . (4.7) The induced shift to the spectral index has two terms, but assuming that, say, the first term in the square brackets in ∆ σ is dominant and defining m S = m 1 + m 3 , m 2 = m L , and a NL ( k ) = a NL ( k p )( k/k p ) m S it is ∆ n s ( k ) ≈ 2 aξ m ( k ) a NL ( k ) σ ( m L ) Gl ( n ( m ) f + m S ) . (4.8) The bispectrum in the large volume is B ζ ( k 1 , k 2 , k 3 ) = a NL ( k p ) ( k 3 k p ) m 3 P ζ ( k 1 ) ξ m ( k 1 ) P ζ ( k 2 ) ξ m ( k 2 ) (4.9) × [( k 1 k p ) m 1 ( k 2 k p ) m 2 + ( k 1 k p ) m 2 ( k 2 k p ) m 1 ] +2perm . while the observed bispectrum is B obs ζ ( k 1 , k 2 , k 3 ) = a NL ( k p ) ( k 3 k p ) m 3 [ P obs ζ ( k 1 ) ξ obs m ( k 1 ) 1 + ∆ σ ( k 1 ) ][ P obs ζ ( k 2 ) ξ obs m ( k 2 ) 1 + ∆ σ ( k 2 ) ] (4.10) × [( k 1 k p ) m 1 ( k 2 k p ) m 2 + ( k 1 k p ) m 2 ( k 2 k p ) m 1 ] +2perm . Consider k 1 = k L glyph[lessmuch] k 2 ≈ k 3 . If m 2 < m 1 (so the second term in the second line of the equation above dominates), and m S ≡ m 1 + m 3 , then in the squeezed limit the large volume bispectrum has n sq . ( k ) = -3 + n ( m ) f + m 2 . (4.11) The shift to the observed running of the squeezed-limit bispectrum is ∆ n sq . ( k ) = ∆ n ( m ) f ( k ) -∆ σ ( k ) 1 + ∆ σ ( k ) m S ≈ ∆ σ m S -2 ξ m ∆ σ ( n ( m ) f + m S ) . (4.12) In the case of the generalized, two source local ansatz considered in Sections 2 and 3.3, a NL ( k ) = 3 5 f NL ( k ), m 3 = n f , and m 1 = m 2 = 0 so m S = n f , and both terms in the square brackets of ∆ σ , Eq.(4.5) contribute equally, so we recover the weakly non-Gaussian limits of Eqs. (3.1), (3.3), and Eq. (3.10). As a second example, consider single field inflation (with a Bunch-Davies vacuum and inflation proceeding along the attractor solution). In this case, the squeezed limit of the bispectrum diverges with the long wavelength mode no more strongly than [12-14, 30], B ζ ( k L , k S , k S ) ∝ O ( k L k S ) 2 P ζ ( k L ) P ζ ( k S ) . (4.13) A bispectrum with this squeezed limit can be obtained by using the equilateral template [31] to generate a kernel F ( p 1 , p 2 , k ) ∝ -3 -2 p 1 p 2 /k 2 +2( p 1 + p 2 ) /k +( p 2 1 + p 2 2 ) /k 2 [29]. This yields a squeezed-limit bispectrum with n sq . = -1 and m L = 2 in Eq.(4.4). That is, this bispectrum generates a bias B ∝ ∇ 2 ζ Gl , so there is no sensitivity of locally measured quantities to long wavelength, nearly constant modes. In single field inflation, there is a direct map between local observables and the parameters of the inflationary Lagrangian. Finally, suppose modes are coupled through a bispectrum with a very strong squeezedlimit (eg, n sq . = -4 and m L = -1). Then the biasing of local statistics may come predominantly from background modes farthest in the infrared, which are shared by many neighboring subvolumes. In other words, the dependence of the global bispectrum on the long wavelength mode is related to the average spatial gradient of the bias in the large volume. 5 Tensor mode running as a test of inflation? If the scale dependence of the tensor power spectrum, n t ≡ d ln P t d ln k , can someday be measured, a red tilt would be (nearly) definitive evidence for inflation and against a contracting or ekpyrotic scenario (an interesting special case is 'solid inflation' [32]). Would it be possible to induce a blue tilt n t > 0 in a subvolume the size of the observable universe when the larger volume exhibits a more typical red tilt? If so, a measurement of n t > 0 would not necessarily rule out standard scalar field models of inflation. Conversely, if a red tilt n t < 0 can be induced in a large fraction of subvolumes from non-Gaussianity in a contracting universe scenario, a measurement of n t < 0 may not be a smoking gun for inflation . Consider a three-point interaction 〈 χ k 1 γ s 1 k 2 γ s 2 k 3 〉 ≡ (2 π ) 3 δ 3 ( ∑ k i ) B ( k 1 , k 2 , k 3 ) δ s 1 s 2 (5.1) between two tensor modes γ k i with polarizations s i and one mode from a field χ (here, a scalar field for example). In the squeezed limit, this three-point function will induce a dependence of the local tensor power spectrum on superhorizon χ modes. Any choice of the Fourier space kernel that gives the correct squeezed limit of the bispectrum should show the correct shift to the local power spectrum. So, with a simple choice we find that the tensor power spectrum is shifted by the correlation with long wavelength modes p as γ s i k = γ s i G, k + ∫ L -1 d 3 p 1 (2 π ) 3 d 3 p 2 (2 π ) 3 (2 π ) 3 δ 3 ( p 1 + p 2 -k ) F ( p 1 , p 2 , k )( γ s i G, p 1 χ G, p 2 + γ s i G, p 2 χ G, p 1 ) + ..., (5.2) where we take F ( k 1 , k 2 , k 3 ) = f eff γγχ ( k p ) ∑ j a j ( k 1 k p ) m 1 ,j ( k 2 k p ) m 2 ,j ( k 3 k p ) m 3 ,j . (5.3) For long wavelength modes of the χ field, the tensor power spectrum is shifted by P obs γ = P γ [1 + ∆ χ ( k )] 2 (5.4) ∆ χ ( k ) = af eff γγχ ( k p ) [ ( k k p ) m 1 + m 3 χ ( m 2 ) Gl + ( k k p ) m 2 + m 3 χ ( m 1 ) Gl ] χ ( m L ) Gl ≡ ∫ M -1 L -1 d 3 p (2 π ) 3 χ p ( p k p ) m L . With this parameterization, long wavelength modes of the χ field can shift the locally observed tilt of the tensor power spectrum. In the case that the first term in ∆ χ dominates, we can again define m S = m 1 + m 3 , m L = m 2 and then the shift is approximately ∆ n t ( k ) ≈ 2∆ χ ( k ) m S . (5.5) The quantity m S is zero for an exactly scale-invariant, local type model and more generally cannot be too large if we want to require weak non-Gaussianity for all fields. Depending on the coupling of χ to the scalar curvature, this physics may also introduce a shift in the locally observed scalar power spectrum, the tensor-to-scalar ratio, and a 'fossil' signature in the off-diagonal part of the scalar power spectrum [33], which would be an interesting complementary observable. From these expressions, it looks possible to find scenarios where the locally observed tensor power spectrum would be shifted from red to blue and vice-versa, but a full analysis along the lines of Section 3 should be performed to check consistency with all observables. 6 Discussion and conclusions Non-Gaussianity that couples the statistics of fluctuations on observable scales to wavemodes spanning super-Hubble scales can bias cosmological statistics measured by an observer in a glyph[negationslash] local Hubble volume. Previous work showed that the relative amplitudes of the power spectrum and non-Gaussianity ( f local NL ) can vary in observable subvolumes. In this work we have shown that the spectral index can also vary by enough to be interesting, | ∆ n s | glyph[similarequal] 0 . 04. The scaling of the squeezed limit of the bispectrum can also be shifted, which is relevant for constraints on non-Gaussianity from galaxy bias. These results show that in spite of the excellent precision of the measurements from the Planck satellite (especially n obs s = 0 . 9603 ± 0 . 0073 [2] and constraints on non-Gaussianity), the door is open for a significant cosmic variance uncertainty in comparing our observed patch of the universe to any particular inflation theory even leaving aside issues with eternal inflation. Moreover, rather than just presenting a new source of uncertainty from the super-Hubble background, the correlation between bispectral running in a super-Hubble volume and subvolume power spectrum measurements reopens the door for inflationary models with flat or bluer super-Hubble spectral indices, n s = 0 . 96, provided they also have scale-dependent local non-Gaussianity. This may be particularly useful for hybrid inflation. The numbers measured by the Planck satellite are consistent with a range of levels of non-Gaussianity in a post-inflationary volume, given a model for the statistics in that volume. For example, we recover the observed power spectrum and spectral index, and satisfy current constraints on f CMB NL for a post-inflationary volume with · No local type non-Gaussianity, an arbitrary number of extra e-folds, and any behavior of the power spectrum on superhorizon scales. · Constant f local NL = 5. We observe f local NL = 8 if, for example, the spectral index is a constant n s = 0 . 96 over about 200 extra e-folds of inflation and our Hubble patch sits on top of a 2-sigma under density. · Local non-Gaussianity with constant f local NL = 15. We observe f local NL = 11 if, for example, the spectral index is a constant n s = 0 . 96 over about 150 extra e-folds of inflation and our Hubble patch sits on top of a 2-sigma over-density. · Scale-dependent non-Gaussianity with f NL ( k p ) = -2, n f = 0 . 04, n ζ = 0 . 93, and n s = 0 . 935. We would observe f NL ( k p ) = -1 and n obs s = 0 . 956 if our Hubble patch sits on top of a 2-sigma under density in a volume with about 190 extra e-folds. · Scale-dependent non-Gaussianity with f NL ( k p ) = 20, n f = 0 . 03, n ζ = 0 . 95, and n s = 1 . 005. We would observe f NL ( k p ) = 2 . 5 and n obs s = 0 . 975 if our Hubble patch sits on top of a 0.2-sigma over density in a volume with about 280 extra e-folds. In contrast, we could design an inflation model to have parameters roughly consistent with Planck data, say f NL ( k p ) = 5, n f = 0 . 1, n ζ = 0 . 98, and n s = 0 . 982. However, if the model allows about 400 extra e-folds of inflation, and our Hubble patch were to sit on a 2-sigma over density, we would observe f NL ( k p ) = 4 and n obs s = 1 . 013. These results demonstrate that predictions for our observations in any scenarios with local type non-Gaussianity must be given statistically. To turn the picture around, they also suggest a new route to understanding whether observations can give us any hints about the size of the universe beyond what is directly observable. Previous ideas focused on topologically finite universes (also significantly constrained by Planck [34]) or on evidence for or against a nonperturbatively connected multiverse from bubble collisions [35-37] or curvature [38, 39]. While observations will probably never tell us how long inflation lasted, our work suggests they may at least tell us if that uncertainty is relevant to our interpretation of the data we do have. From a cosmic variance point of view, we are fortunate that there is so far no detection of local type non-Gaussianity. We have shown that future observations could push the mode-coupling uncertainties we have considered here into irrelevance 5 if primordial local non-Gaussianity can be constrained to be | f NL | < 1. Even if | f NL | > 1 is observed, tests for the running of the spectral index, any scale-dependence of | f NL | , and any evidence for extra fields through isocurvature modes or 'fossil' relics hiding in the off-diagonal power spectrum could still limit the size of any subsampling uncertainty. For example, if a blue tilt to f NL is ruled out, biasing of the spectral index is unlikely for single-source models with n ζ and n f constant on all scales. Of course, making these observations statistically well-defined depends on comparing particular competing models. It would be particularly interesting if those models had other cosmological implications related to the size of the universe 6 [40]. It would also be worthwhile to investigate the generic behavior of the local ansatz beyond f NL alone with scale-dependent coefficients, along the lines of the analysis in [6]. It may be that there are statistically natural values for the spectral index in typical small subvolumes. Then, stronger conclusions about generic cosmic variance of the spectral index might be possible. However, it is already clear that if improved limits on the amplitude and scale-dependence of non-Gaussianity can be reached, we could close the window of observational access to a perturbatively connected larger universe. Acknowledgements We thank Chris Byrnes, Bhaskar Dutta, Louis Leblond and Marilena LoVerde for useful suggestions and discussions about this work. The work of J. B. is supported in part by Department of Energy grant DE-FG02-04ER41291. The work of J. K. is supported in part by Department of Energy grants DE-FG02-04ER41291 and DE-FG02-13ER41913. The work of S. S. is supported in part by the National Aeronautics and Space Administration under Grant No. NNX12AC99G issued through the Astrophysics Theory Program. In addition, S. S. thanks the organizers of the Primordial Cosmology Program at KITP for hospitality while this work was being completed and for support by the National Science Foundation under Grant No. NSF PHY11-25915. J. K. thanks the Center for Theoretical Underground Physics and Related Areas (CETUP* 2013) in South Dakota for its support and hospitality while this work was being completed. E. N. is supported by the Eberly Research Funds of The Pennsylvania State University. The Institute for Gravitation and the Cosmos is supported by the Eberly College of Science and the Office of the Senior Vice President for Research at the Pennsylvania State University. References [1] Planck Collaboration Collaboration, P. Ade et. al. , Planck 2013 results. I. Overview of products and scientific results , 1303.5062 . [2] Planck Collaboration Collaboration, P. Ade et. al. , Planck 2013 results. XXII. Constraints on inflation , 1303.5082 . [3] Planck Collaboration, P. Ade et. al. , Planck 2013 Results. XXIV. Constraints on primordial non-Gaussianity , 1303.5084 . [4] A. D. Linde and V. Mukhanov, The curvaton web , JCAP 0604 (2006) 009, [ astro-ph/0511736 ]. [5] V. Demozzi, A. Linde, and V. Mukhanov, Supercurvaton , JCAP 1104 (2011) 013, [ 1012.0549 ]. [6] E. Nelson and S. Shandera, Statistical Naturalness and non-Gaussianity in a Finite Universe , accepted for publication in PRL (2012) [ 1212.4550 ]. [7] S. Nurmi, C. T. Byrnes, and G. Tasinato, A non-Gaussian landscape , 1301.3128 . [8] M. LoVerde, E. Nelson, and S. Shandera, Non-Gaussian Mode Coupling and the Statistical Cosmological Principle , 1303.3549 . [9] C. T. Byrnes, S. Nurmi, G. Tasinato, and D. Wands, Implications of the Planck bispectrum constraints for the primordial trispectrum , 1306.2370 . [10] D. S. Salopek and J. R. Bond, Nonlinear evolution of long wavelength metric fluctuations in inflationary models , Phys. Rev. D42 (1990) 3936-3962. [11] E. Komatsu and D. N. Spergel, Acoustic signatures in the primary microwave background bispectrum , Phys.Rev. D63 (2001) 063002, [ astro-ph/0005036 ]. [12] J. M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models , JHEP 0305 (2003) 013, [ astro-ph/0210603 ]. [13] P. Creminelli and M. Zaldarriaga, Single field consistency relation for the 3-point function , JCAP 0410 (2004) 006, [ astro-ph/0407059 ]. [14] E. Pajer, F. Schmidt, and M. Zaldarriaga, The Observed Squeezed Limit of Cosmological Three-Point Functions , 1305.0824 . [15] P. McDonald, Primordial non-Gaussianity: large-scale structure signature in the perturbative bias model , Phys.Rev. D78 (2008) 123519, [ 0806.1061 ]. [16] M. LoVerde and K. M. Smith, The Non-Gaussian Halo Mass Function with fNL, gNL and tauNL , JCAP 1108 (2011) 003, [ 1102.1439 ]. [17] M. Gerstenlauer, A. Hebecker, and G. Tasinato, Inflationary Correlation Functions without Infrared Divergences , JCAP 1106 (2011) 021, [ 1102.0560 ]. [18] S. B. Giddings and M. S. Sloth, Cosmological observables, IR growth of fluctuations, and scale-dependent anisotropies , Phys.Rev. D84 (2011) 063528, [ 1104.0002 ]. [19] D. H. Lyth, The curvature perturbation in a box , JCAP 0712 (2007) 016, [ 0707.0361 ]. [20] L. Boubekeur and D. Lyth, Detecting a small perturbation through its non-Gaussianity , Phys.Rev. D73 (2006) 021301, [ astro-ph/0504046 ]. [21] J. Kumar, L. Leblond, and A. Rajaraman, Scale Dependent Local Non-Gaussianity from Loops , JCAP 1004 (2010) 024, [ 0909.2040 ]. [22] J. Bramante and J. Kumar, Local Scale-Dependent Non-Gaussian Curvature Perturbations at Cubic Order , JCAP 1109 (2011) 036, [ 1107.5362 ]. [23] N. S. Sugiyama, Consistency Relation for multifield inflation scenario with all loop contributions , JCAP 1205 (2012) 032, [ 1201.4048 ]. [24] C. T. Byrnes, K. Enqvist, and T. Takahashi, Scale-dependence of Non-Gaussianity in the Curvaton Model , JCAP 1009 (2010) 026, [ 1007.5148 ]. [25] Q.-G. Huang, Negative spectral index of f NL in the axion-type curvaton model , JCAP 1011 (2010) 026, [ 1008.2641 ]. [26] M. Dias, R. H. Ribeiro, and D. Seery, Scale-dependent bias from multiple-field inflation , 1303.6000 . [27] S. Shandera, N. Dalal, and D. Huterer, A generalized local ansatz and its effect on halo bias , JCAP 1103 (2011) 017, [ 1010.3722 ]. [28] D. Tseliakhovich, C. Hirata, and A. Slosar, Non-Gaussianity and large-scale structure in a two-field inflationary model , Phys.Rev. D82 (2010) 043531, [ 1004.3302 ]. [29] R. Scoccimarro, L. Hui, M. Manera, and K. C. Chan, Large-scale Bias and Efficient Generation of Initial Conditions for Non-Local Primordial Non-Gaussianity , Phys.Rev. D85 (2012) 083002, [ 1108.5512 ]. [30] T. Tanaka and Y. Urakawa, Dominance of gauge artifact in the consistency relation for the primordial bispectrum , JCAP 1105 (2011) 014, [ 1103.1251 ]. [31] P. Creminelli, A. Nicolis, L. Senatore, M. Tegmark, and M. Zaldarriaga, Limits on non-gaussianities from wmap data , JCAP 0605 (2006) 004, [ astro-ph/0509029 ]. [32] S. Endlich, A. Nicolis, and J. Wang, Solid Inflation , 1210.0569 . [33] D. Jeong and M. Kamionkowski, Clustering Fossils from the Early Universe , Phys.Rev.Lett. 108 (2012) 251301, [ 1203.0302 ]. [34] Planck Collaboration Collaboration, P. Ade et. al. , Planck 2013 results. XXVI. Background geometry and topology of the Universe , 1303.5086 . [35] S. Chang, M. Kleban, and T. S. Levi, Watching Worlds Collide: Effects on the CMB from Cosmological Bubble Collisions , JCAP 0904 (2009) 025, [ 0810.5128 ]. [36] S. M. Feeney, M. C. Johnson, J. D. McEwen, D. J. Mortlock, and H. V. Peiris, Hierarchical Bayesian Detection Algorithm for Early-Universe Relics in the Cosmic Microwave Background , 1210.2725 . [37] S. Osborne, L. Senatore, and K. Smith, Collisions with other Universes: the Optimal Analysis of the WMAP data , 1305.1964 . [38] M. Kleban and M. Schillo, Spatial Curvature Falsifies Eternal Inflation , JCAP 1206 (2012) 029, [ 1202.5037 ]. [39] A. H. Guth and Y. Nomura, What can the observation of nonzero curvature tell us? , Phys.Rev. D86 (2012) 023534, [ 1203.6876 ]. [40] D. B. Kaplan and A. E. Nelson, Inflationary Axion Cosmology Beyond Our Horizon , 0809.1206 . Gl Gl - 0 10 0 10 to a smaller (larger) number of e-folds in the case of a red tilt (blue tilt), as shown in Figure 2, so it is easier to realize a large shift to a global red tilt than to a global blue tilt. In fact, as previously noted in the discussion of Figure 2, imposing the requirement that P obs ζ be typical of subvolumes for scenarios with a blue tilt n ζ -1 causes 〈 ζ 2 Gl 〉 to converge to a particular value as N is increased. For these reasons single-source scenarios with a red power tilt in the large volume have the most significant range of cosmic variance due to subsampling. The solid black lines in Figure 3 show ∆ n s ( k p ) = -0 . 04 in subvolumes with a +0 . 5 σ background fluctuation ( ζ Gl = +0 . 5 〈 ζ 2 Gl 〉 1 / 2 ), and thus show part of the parameter space where | ∆ n s ( k p ) | can be observationally significant. Here we have neglected the subhorizon one-loop correction 〈 ζ 2 Gs ( k p ) 〉 glyph[lessmuch] ζ 2 Gl ; this breaks down for small N but is valid outside of the region of parameter space excluded by the requirement f CMB NL < 10. Rewriting Eq. (3.3) for a single source scenario ( ξ m = 1), Assuming n f = 0 . 1 and ζ Gl = 0 . 5 〈 ζ 2 Gl 〉 1 / 2 , we can solve this equation to show that ∆ n single source s = -0 . 04 when f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 = 0 . 94 or 5.9, which are the equations of the two black lines plotted in Figure 3. These lines assume positive f NL in the large volume, f NL > 0, but they remain the same for f NL < 0 and a -0 . 5 σ background fluctuation. For values of | n f | larger or smaller than 0 . 1, the distance between these lines grows or shrinks in parameter space. Of course, for the full expression of ∆ n s and a different set of parameter choices, there can be more than two solutions of | ∆ n s | = 0 . 04. For positive f NL ( k p ) ζ Gl (see below), the typical size of ∆ n s is largest in the region between these lines ( 6 5 f NL ( k p ) ζ Gl ∼ 1) and falls towards zero on either side. The upper dotted-dashed lines mark where 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 is large ( O (10)). When that quantity is large, ∆ n s glyph[similarequal] -n f 3 5 f NL 〈 ζ 2 Gl 〉 1 / 2 and thus approaches zero as indicated in Figure 3. Note that in this region the observed spectral index is n obs s glyph[similarequal] n s glyph[similarequal] n ζ +2 n f , so for the parameter choices in Figure 3 the Planck satellite excludes the region above the dotted-dashed lines. All lines and contours in Figure 3 assume that 6 5 f NL ( k p ) ζ Gl > 0 (eg, overdense fluctuations with positive f NL ). If this figure assumed 6 5 f NL ( k p ) ζ Gl < 0 (eg, overdense fluctuations with negative f NL ), the area in parameter space near the line 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 = 1 would be excluded. For further discussion of parameter space with 6 5 f NL ζ Gl < 0, see the discussion after Eq. (3.3). Figure 3 shows that, under the conditions we have imposed and the spectral indices considered, only scenarios where the bispectral tilt is not very red have typical subvolumes where the observed spectral index varies by an amount that is cosmologically interesting for us, | ∆ n s | glyph[greaterorsimilar] 0 . 01. A blue bispectral index may avoid the current observational constraints, which do not probe particularly small scales, and easily remain globally perturbative and weakly non-Gaussian (see paragraph below). In contrast, the bottom panels of Figure 3 illustrate that for either spectral index, a scenario with n f < 0 will be nonperturbative in the interesting part of parameter space where | ∆ n s | ∼ 0 . 04. (In addition, there is only a small window with strongly non-Gaussian but perturbative global statistics.) If both the power spectrum and non-Gaussianity increase in the IR, as in the lower right panel of Figure 3, the statistics will be strongly non-Gaussian across parameter space for a small number of superhorizon e-folds. The upper panels of Figure 3 illustrate a feature discussed in Section 2.2: 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 glyph[greaterorsimilar] 1 does not necessarily imply a large cumulative skewness, M 3 glyph[greaterorsimilar] 1. The dashed curves fix M 3 = 1 as a function of superhorizon e-folds, which are determined at each point in parameter space by the observed level of the power spectrum along with n f , f NL and 〈 ζ 2 Gl 〉 . In regions where M 3 < 1 but f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 glyph[greaterorsimilar] 1, there are a sufficient number of superhorizon modes with weaker coupling ( n f > 0) damp the total non-Gaussianity. To elaborate, in the limit n f ( N + N sub ) glyph[greatermuch] 1, Eq. (2.18) gives M 3 ∝ [ 〈 ζ 2 Gl 〉 /N ( N + N sub )] 1 / 2 . For f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 glyph[lessmuch] 1, N = 〈 ζ 2 Gl 〉 / P obs ζ and so M 3 becomes independent of 〈 ζ 2 Gl 〉 in the limit N glyph[lessmuch] N sub . For f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 glyph[greatermuch] 1, on the other hand, M 3 ∝ 1 /f NL ( k p ) 〈 ζ 2 Gl 〉 3 / 2 , so large f NL ( k p ) and sufficiently large 〈 ζ 2 Gl 〉 are needed to keep the total non-Gaussianity small, and P obs ζ ∼ 2 × 10 -9 typical in subvolumes, as seen in the upper left panel of Figure 3. Note that throughout this analysis, we have assumed n f is constant for all N sub = 60 subhorizon e-folds, so that for blue n f non-Gaussianity continues to grow on subhorizon scales where nonlinear evolution has taken over. If this condition is relaxed, the conditions from weak non-Gaussianity are less restrictive. Figure 4 shows the probability distribution for the shift ∆ n s for the parameters in part of the range of interest for the blue bispectral index shown in the top panels of Figure 3. Both panels show examples that (for appropriate choices of large volume parameters) give local power spectra amplitude and f CMB NL consistent with our observations. Notice that the distribution on the right is substantially less Gaussian than the distribution on the left. This trend continues if one considers larger 〈 ζ 2 Gl 〉 while keeping all other parameters fixed. In Figure 5 we show regions of parameter space in the ( 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 , n f ) plane that are consistent with the Planck measurement n obs s = 0 . 9603 ± 0 . 0073. Assuming that the scalar power spectrum in the full volume of the mode-coupled universe is completely flat, n ζ = 1, we see that 6 5 f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 must be at least O (10 -1 ) and for weakly non-Gaussian statistics, more than a hundred superhorizon e-folds are required. It is interesting to note that in the case of a blue-tilted f NL , a larger running non-Gaussianity n f loosens parameter constraints coming from requiring perturbative statistics 〈 [ f NL ( k ) ζ 2 Gl ] 2 〉 glyph[lessorsimilar] 0 . 1. Although the dotted lines in Figure 5 will shift to the left with more superhorizon e-folds, these curves exclude less parameter space as n f becomes larger. This is because we have assumed n f is blue and constant so f NL is driven to smaller values in the IR and 〈 [ f NL ( k ) ζ 2 Gl ] 2 〉 becomes smaller for larger n f . Notice the shift in the non-perturbative line in the right panel that occurs at n f > | n ζ -1 | : if the running of the power spectrum is larger than the running of f NL ( k ), then the running of the power spectrum will dominate the variance of the local quadratic term over superhorizon modes, because f 2 NL ( k ) P G ( k ) 2 ∝ k 2( n f + n ζ -1) . Lastly, the right panel of Figure 5 shows once again that for a blue tilted f NL , the weakly non-Gaussian parameter space enlarges with the number of superhorizon e-folds, because f NL is driven to very small values over more superhorizon e-folds, decreasing the value of M 3 . To conclude this section, Figure 6 illustrates a single-source scenario in which a power spectrum which appears blue-tilted in the large volume on short scales can appear red on the same scales in a subvolume. On scales where P ζ ( k ) glyph[similarequal] P G ( k ), n s ( k ) glyph[similarequal] n ζ , whereas on scales where the 1-loop contribution dominates P 1-loop ζ ( k ) glyph[similarequal] 36 25 f 2 NL ( k ) 〈 ζ 2 Gl 〉P G ( k ) and the spectral index will be n s ( k ) glyph[similarequal] n ζ + 2 n f . If the transition of power takes place on a scale near the observable range of scales ( f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 = O (1)), the observed spectral index can be shifted. For example, if ζ 2 Gl < 〈 ζ 2 Gl 〉 , the blue-tilted f 2 NL 〈 ζ 2 Gl 〉 contribution loses power in the subvolume, and if f NL ( k p ) ζ Gl > 0, the red-tilted piece gains power (compare Eqs. (2.5), (2.9)). This scenario is shown in Figure 6. Note that as long as f NL ( k p ) is not extremely large (which would violate the constraint on f CMB NL for the value of f NL ( k p ) 〈 ζ 2 Gl 〉 1 / 2 chosen here), ζ Gl glyph[greatermuch] 〈 ζ 2 Gs ( k ) 〉 1 / 2 and the 1-loop contribution to P obs ζ is very small, suppressed by a factor of 〈 ζ 2 Gs ( k ) 〉 /ζ 2 Gl .", "pages": [ 17, 18, 19, 20, 21 ] }, { "title": "Example III: Multiple sources with running ξ m ( k ) .", "content": "In the single-source case, a large shift to the observed spectral index could only occur if the 1-loop contribution to the power spectrum dominated on small scales. With two sources, a significant shift to n s can be consistent with weak non-Gaussianity ξ m ( k ) f NL ( k ) 〈 σ Gl 〉 1 / 2 < 1 on all scales. If the running of the 1-loop contribution lies between the runnings n σ ≡ d ln P σ ( k ) d ln( k ) and n φ ≡ d ln P φ ( k ) d ln( k ) of the Gaussian contributions to the total power, then it will be subdominant on large and small scales. The transition of power between σ G and φ G takes place over a finite range of scales, over H GLYPH<144> L H L glyph[negationslash] H GLYPH<144> L H GLYPH<144> L H L H GLYPH<144> L which n s changes from n σ to n φ . If the power spectrum of φ G is blue and dominates on small scales ( ξ m ( k glyph[greaterorsimilar] H 0 ) glyph[lessmuch] 1), and the Gaussian contribution from σ is red and dominates on large scales ( ξ m ( k << H 0 ) glyph[similarequal] 1), then the background ζ l glyph[similarequal] σ l for any subvolume couples to and biases the local statistics. For example, a globally flat or blue spectral index n s ( k > H 0 ) > 1 can again appear red, n obs s < 1, in a subvolume. The shift to n s can come only from the modulation of power in σ relative to φ G , and need not rely on running non-Gaussianity n f = 0. That is, a large running of the difference in power of the fields can be achieved without a large level of running non-Gaussianity. This becomes apparent upon inspecting the running of ξ m , If φ G is more red-tilted than σ G , the background is uncorrelated with short-wavelength modes because φ G dominates on large scales, ζ l glyph[similarequal] φ Gl , so local statistics are not biased. Thus, both n σ ≤ 1 and n φ > n σ are needed for a significant bias. In Figure 7 we show the parameter ln L H space for the two-source scenario described above, with n σ ( k p ) = 0 . 93, n φ ( k p ) = 1 . 005, and ξ m ( k p ) = 0 . 1. We also fix n f = 0 . 001 so that mode coupling is weaker on superhorizon scales. As before, the upper left region shows where f obs NL glyph[greaterorsimilar] 10 in typical subvolumes. We see that adding the second source relaxes the constraint on f NL in the f NL 〈 σ 2 Gl 〉 1 / 2 glyph[lessmuch] 1 regime. This makes it possible to achieve a large shift ∆ n s for smaller values of 〈 σ 2 Gl 〉 and thus fewer superhorizon e-folds. ln k L H The condition ξ m ( k p ) = 0 . 1 makes the field φ G dominant on Planck scales, so from the perspective of the large volume, the power spectrum has a blue tilt n s ( k p ) glyph[similarequal] n φ = 1 . 005 on scale k p . However, for significant biasing (3 σ ) and a small (or zero) non-Gaussian running of the coupled field n f = 0 . 001, the black lines in Figure 7 denote where ∆ n s = -0 . 03, which would be consistent with Planck observations. Here the shift in ∆ n s is coming not from n f but from the difference in running of P σ,NG and P φ , n ( m ) f , as the red-tilted P σ,NG is amplified due to the strong background overdensity. It is also interesting to note that a cursory survey of background fluctuations reveals that biases less than | 3 σ | yield no ∆ n s corrections smaller than -0 . 03, which would seem to partly exclude these parameters for typical Hubble-sized subsamples. In the limit of very small ξ m ( k p ), φ G dominates the power and scale-dependence on observable scales, so unless the bias is extremely strong, any shift in the power and scale-dependence from the σ field will be too small to affect n obs s .", "pages": [ 21, 22, 23, 24 ] }, { "title": "Summary.", "content": "In summary, a significant shift to the observed spectral index from correlations with long-wavelength background modes is possible under the following conditions: relative to φ G on large scales, causing the background ζ l to be sufficiently correlated with local statistics. If φ G contributes on observable scales ( ξ m ( k p ) < 1), larger values of f NL ( k p ) are consistent with observational constraints on non-Gaussianity, so a smaller background σ Gl is needed to give the same shift to n obs s . Introducing scale-dependence into the spectral indices would relax the conditions for large | ∆ n s | . Although the scenario becomes more complicated in this case, the qualitative features remain valid: scale-dependence of power spectra and non-Gaussian parameters must allow for sufficient cumulative superhorizon power that a large background σ Gl from the source with mode coupling is typical. We note that for given large-volume statistics, the observed red tilt may not be equally consistent with a local overdensity or underdensity in σ G . In the single-source case with n f > 0, for example, an overdensity (underdensity) corresponds to an increase (decrease) of power on small scales. Thus, for a scale-invariant power spectrum in the large volume, the observed red tilt n obs s glyph[similarequal] 0 . 96 could be accounted for in terms of a blue-tilted global bispectrum and local underdensity. However, without information about the global power spectrum, it would be difficult to infer whether we sit on a local underdensity or overdensity.", "pages": [ 24, 25 ] }, { "title": "3.3 The shift to the scale dependence of the bispectrum", "content": "The bispectrum may also be shifted by mode coupling coming from the soft limits of the large-volume trispectrum and from any non-Gaussian shifts to power spectrum. We can define a spectral index for the squeezed limit of the bispectrum within any particular volume as where k L and k S are long wavelength and short wavelength modes, respectively. The small volume quantity, n obs sq . , should be calculated using the observed bispectrum and the observed spectral index. For a single source, scale-invariant local ansatz, n sq . = -3. For the single source, weakly non-Gaussian, scale-dependent scenario with g NL absent, the shift in this bispectral index between the large volume and what is observed in the small volume is If 6 5 f NL ( k L ) σ Gl = 6 5 f NL ( k L ) 〈 ζ 2 G 〉 1 / 2 B glyph[lessmuch] 1, then ∆ n sq . ( k ) ≈ -6 5 f NL ( k L ) 〈 ζ 2 G 〉 1 / 2 Bn f . This shift is less than one in magnitude, but still relevant for interpreting bispectral indices of order slow-roll parameters. In the two source case, there can be additional scale dependence coming from the ratio of power of the two fields. Considering only the weak coupling case, 6 5 f NL ( k ) σ Gl glyph[lessmuch] 1 (and again setting g NL = 0 for simplicity), Reintroducing g NL and higher terms would lead to additional terms, introducing scaledependence even if f NL in the large volume is a constant.", "pages": [ 25, 26 ] }, { "title": "3.4 Generalized local ansatz and single source vs. multi source effects", "content": "The two source, weakly scale dependent local ansatz in Eq. (2.1) is representative of the properties of inflation models that generate local type non-Gaussianity. For example, the scale-dependence f NL ( k ) can come from curvaton models with self-interactions [24, 25]. The function ξ m ( k ) comes from the difference in power spectrum of two fields (eg, the inflaton and the curvaton) contributing to the curvature fluctuations. In typical multi-field models, the bispectral indices n f , n ( m ) f are of order slow-roll parameters (like the scale dependence of the power spectrum), and are often not constant. Generic expressions for the squeezed limit behavior of a multi-field bispectrum are given in [26]. The scale-dependent functions f NL ( k ) and ξ m ( k ) are observationally relevant for tests for primordial non-Gaussianity using the bias of dark matter halos and their luminous tracers (eg. quasars or luminous red galaxies). The power law dependence of the squeezed limit on the long wavelength, small momentum mode ( n sq . from Eq. (3.8)) generates the scale-dependence of the non-Gaussian term in the bias. The dependence on the short wavelength modes generates a dependence of the non-Gaussian bias on the mass of the tracer (which is absent in the usual local ansatz). In principle, if local non-Gaussianity is ever detected, it may be within the power of future large scale structure surveys to detect some amplitude of running [27]. However, as demonstrated above, the same shape of bispectrum can be generated locally by a single source for the curvature perturbations, so the presence of the non-trivial function ξ m in the observed bispectrum does not necessarily indicate that two fundamental fields contributed to the primordial curvature perturbations. On the other hand, the presence of one Gaussian source and one non-Gaussian source for the local curvature perturbations is in principle detectable by comparing power spectra that are sensitive in different ways to the total curvature field and to just the non-Gaussian part [28]. Eq. (2.12) shows that in a single source scenario the local background σ Gl can act as a second field to generate the full, multi-source shaped bispectrum, but σ Gl is constant within a single volume. This 'second field' does not have fluctuations on all scales, but its variations are relevant for considering a collection of subvolumes of a particular size.", "pages": [ 26 ] }, { "title": "4 Mode coupling effects from a non-local factorizable bispectrum", "content": "We have considered the effect of superhorizon modes only for the case of nearly local nonGaussianity, but inflationary theory has generated an expanding space of models exhibiting different types of mode coupling. Intuitively, any scenario that does not couple modes of sufficiently different wavelengths should not lead to correlation functions whose amplitudes or shapes change under subsampling. As a first step towards considering the observational consequences of subsampling general non-Gaussian scenarios, it is straighforward to find corrections from the background to small-volume quantities in the case of a factorizable quadratic kernel in Fourier space with power-law dependence. Consider a curvature perturbation in the large volume given by where is a sum of factorizable terms with power law dependence on the momenta. On the right hand side the a j are amplitudes defined at a pivot scale k p . When ∑ i m i,j glyph[similarequal] 0 for every term j , the bispectrum is approximately scale-invariant. The kernel F ( k 1 , k 2 , k 3 ) can be chosen to generate a desired bispectrum with well behaved one-loop corrections to the power spectrum [29]. Splitting the modes into long and short, the locally defined short wavelength modes with shifts induced from coupling to long wavelength modes from one term in the series above are where When the local field is weakly non-Gaussian, the second line is small and we can rewrite the first line as The leading shift to the power spectrum P obs ζ in a subvolume from unobservable infrared modes in one term of the series above (and assuming weak non-Gaussianity) is: where ξ m ( k ) is still the ratio of power in the non-Gaussian source to the total power, defined in Eq.(2.7). In the two-field case with weak non-Gaussianity on all scales, the observed ratio of power in the two fields is related to the same ratio in the large volume by The induced shift to the spectral index has two terms, but assuming that, say, the first term in the square brackets in ∆ σ is dominant and defining m S = m 1 + m 3 , m 2 = m L , and a NL ( k ) = a NL ( k p )( k/k p ) m S it is The bispectrum in the large volume is while the observed bispectrum is Consider k 1 = k L glyph[lessmuch] k 2 ≈ k 3 . If m 2 < m 1 (so the second term in the second line of the equation above dominates), and m S ≡ m 1 + m 3 , then in the squeezed limit the large volume bispectrum has The shift to the observed running of the squeezed-limit bispectrum is In the case of the generalized, two source local ansatz considered in Sections 2 and 3.3, a NL ( k ) = 3 5 f NL ( k ), m 3 = n f , and m 1 = m 2 = 0 so m S = n f , and both terms in the square brackets of ∆ σ , Eq.(4.5) contribute equally, so we recover the weakly non-Gaussian limits of Eqs. (3.1), (3.3), and Eq. (3.10). As a second example, consider single field inflation (with a Bunch-Davies vacuum and inflation proceeding along the attractor solution). In this case, the squeezed limit of the bispectrum diverges with the long wavelength mode no more strongly than [12-14, 30], A bispectrum with this squeezed limit can be obtained by using the equilateral template [31] to generate a kernel F ( p 1 , p 2 , k ) ∝ -3 -2 p 1 p 2 /k 2 +2( p 1 + p 2 ) /k +( p 2 1 + p 2 2 ) /k 2 [29]. This yields a squeezed-limit bispectrum with n sq . = -1 and m L = 2 in Eq.(4.4). That is, this bispectrum generates a bias B ∝ ∇ 2 ζ Gl , so there is no sensitivity of locally measured quantities to long wavelength, nearly constant modes. In single field inflation, there is a direct map between local observables and the parameters of the inflationary Lagrangian. Finally, suppose modes are coupled through a bispectrum with a very strong squeezedlimit (eg, n sq . = -4 and m L = -1). Then the biasing of local statistics may come predominantly from background modes farthest in the infrared, which are shared by many neighboring subvolumes. In other words, the dependence of the global bispectrum on the long wavelength mode is related to the average spatial gradient of the bias in the large volume.", "pages": [ 26, 27, 28 ] }, { "title": "5 Tensor mode running as a test of inflation?", "content": "If the scale dependence of the tensor power spectrum, n t ≡ d ln P t d ln k , can someday be measured, a red tilt would be (nearly) definitive evidence for inflation and against a contracting or ekpyrotic scenario (an interesting special case is 'solid inflation' [32]). Would it be possible to induce a blue tilt n t > 0 in a subvolume the size of the observable universe when the larger volume exhibits a more typical red tilt? If so, a measurement of n t > 0 would not necessarily rule out standard scalar field models of inflation. Conversely, if a red tilt n t < 0 can be induced in a large fraction of subvolumes from non-Gaussianity in a contracting universe scenario, a measurement of n t < 0 may not be a smoking gun for inflation . Consider a three-point interaction between two tensor modes γ k i with polarizations s i and one mode from a field χ (here, a scalar field for example). In the squeezed limit, this three-point function will induce a dependence of the local tensor power spectrum on superhorizon χ modes. Any choice of the Fourier space kernel that gives the correct squeezed limit of the bispectrum should show the correct shift to the local power spectrum. So, with a simple choice we find that the tensor power spectrum is shifted by the correlation with long wavelength modes p as where we take For long wavelength modes of the χ field, the tensor power spectrum is shifted by With this parameterization, long wavelength modes of the χ field can shift the locally observed tilt of the tensor power spectrum. In the case that the first term in ∆ χ dominates, we can again define m S = m 1 + m 3 , m L = m 2 and then the shift is approximately The quantity m S is zero for an exactly scale-invariant, local type model and more generally cannot be too large if we want to require weak non-Gaussianity for all fields. Depending on the coupling of χ to the scalar curvature, this physics may also introduce a shift in the locally observed scalar power spectrum, the tensor-to-scalar ratio, and a 'fossil' signature in the off-diagonal part of the scalar power spectrum [33], which would be an interesting complementary observable. From these expressions, it looks possible to find scenarios where the locally observed tensor power spectrum would be shifted from red to blue and vice-versa, but a full analysis along the lines of Section 3 should be performed to check consistency with all observables.", "pages": [ 28, 29 ] }, { "title": "6 Discussion and conclusions", "content": "Non-Gaussianity that couples the statistics of fluctuations on observable scales to wavemodes spanning super-Hubble scales can bias cosmological statistics measured by an observer in a glyph[negationslash] local Hubble volume. Previous work showed that the relative amplitudes of the power spectrum and non-Gaussianity ( f local NL ) can vary in observable subvolumes. In this work we have shown that the spectral index can also vary by enough to be interesting, | ∆ n s | glyph[similarequal] 0 . 04. The scaling of the squeezed limit of the bispectrum can also be shifted, which is relevant for constraints on non-Gaussianity from galaxy bias. These results show that in spite of the excellent precision of the measurements from the Planck satellite (especially n obs s = 0 . 9603 ± 0 . 0073 [2] and constraints on non-Gaussianity), the door is open for a significant cosmic variance uncertainty in comparing our observed patch of the universe to any particular inflation theory even leaving aside issues with eternal inflation. Moreover, rather than just presenting a new source of uncertainty from the super-Hubble background, the correlation between bispectral running in a super-Hubble volume and subvolume power spectrum measurements reopens the door for inflationary models with flat or bluer super-Hubble spectral indices, n s = 0 . 96, provided they also have scale-dependent local non-Gaussianity. This may be particularly useful for hybrid inflation. The numbers measured by the Planck satellite are consistent with a range of levels of non-Gaussianity in a post-inflationary volume, given a model for the statistics in that volume. For example, we recover the observed power spectrum and spectral index, and satisfy current constraints on f CMB NL for a post-inflationary volume with In contrast, we could design an inflation model to have parameters roughly consistent with Planck data, say f NL ( k p ) = 5, n f = 0 . 1, n ζ = 0 . 98, and n s = 0 . 982. However, if the model allows about 400 extra e-folds of inflation, and our Hubble patch were to sit on a 2-sigma over density, we would observe f NL ( k p ) = 4 and n obs s = 1 . 013. These results demonstrate that predictions for our observations in any scenarios with local type non-Gaussianity must be given statistically. To turn the picture around, they also suggest a new route to understanding whether observations can give us any hints about the size of the universe beyond what is directly observable. Previous ideas focused on topologically finite universes (also significantly constrained by Planck [34]) or on evidence for or against a nonperturbatively connected multiverse from bubble collisions [35-37] or curvature [38, 39]. While observations will probably never tell us how long inflation lasted, our work suggests they may at least tell us if that uncertainty is relevant to our interpretation of the data we do have. From a cosmic variance point of view, we are fortunate that there is so far no detection of local type non-Gaussianity. We have shown that future observations could push the mode-coupling uncertainties we have considered here into irrelevance 5 if primordial local non-Gaussianity can be constrained to be | f NL | < 1. Even if | f NL | > 1 is observed, tests for the running of the spectral index, any scale-dependence of | f NL | , and any evidence for extra fields through isocurvature modes or 'fossil' relics hiding in the off-diagonal power spectrum could still limit the size of any subsampling uncertainty. For example, if a blue tilt to f NL is ruled out, biasing of the spectral index is unlikely for single-source models with n ζ and n f constant on all scales. Of course, making these observations statistically well-defined depends on comparing particular competing models. It would be particularly interesting if those models had other cosmological implications related to the size of the universe 6 [40]. It would also be worthwhile to investigate the generic behavior of the local ansatz beyond f NL alone with scale-dependent coefficients, along the lines of the analysis in [6]. It may be that there are statistically natural values for the spectral index in typical small subvolumes. Then, stronger conclusions about generic cosmic variance of the spectral index might be possible. However, it is already clear that if improved limits on the amplitude and scale-dependence of non-Gaussianity can be reached, we could close the window of observational access to a perturbatively connected larger universe.", "pages": [ 29, 30, 31 ] }, { "title": "Acknowledgements", "content": "We thank Chris Byrnes, Bhaskar Dutta, Louis Leblond and Marilena LoVerde for useful suggestions and discussions about this work. The work of J. B. is supported in part by Department of Energy grant DE-FG02-04ER41291. The work of J. K. is supported in part by Department of Energy grants DE-FG02-04ER41291 and DE-FG02-13ER41913. The work of S. S. is supported in part by the National Aeronautics and Space Administration under Grant No. NNX12AC99G issued through the Astrophysics Theory Program. In addition, S. S. thanks the organizers of the Primordial Cosmology Program at KITP for hospitality while this work was being completed and for support by the National Science Foundation under Grant No. NSF PHY11-25915. J. K. thanks the Center for Theoretical Underground Physics and Related Areas (CETUP* 2013) in South Dakota for its support and hospitality while this work was being completed. E. N. is supported by the Eberly Research Funds of The Pennsylvania State University. The Institute for Gravitation and the Cosmos is supported by the Eberly College of Science and the Office of the Senior Vice President for Research at the Pennsylvania State University.", "pages": [ 31 ] } ]

[ { "title": "Effects of the imaginary inflaton component in supergravity new inflation", "content": "David Nolde 1 Department of Physics, University of Basel, Klingelbergstr. 82, CH-4056 Basel, Switzerland", "pages": [ 1 ] }, { "title": "Abstract", "content": "When models of new inflation are implemented in supergravity, the inflaton is a complex and not a real scalar field. As a complex scalar field has two independent components, supergravity models of new inflation are naturally two-field models. In this paper, we use the δN formalism to analyse how the two-field behaviour modifies the usual single-field predictions. We find that the model reduces to the single-field limit if the inflaton mass term is sufficiently small. Otherwise, the imaginary inflaton component reduces the amplitude A s and the spectral index n s of the scalar curvature perturbations. However, the perturbations remain nearly Gaussian, and the reduced bispectrum f NL is too small to be observed.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The recent measurement of the cosmic microwave background (CMB) by the Planck collaboration [1] has found excellent agreement with the paradigm of slow-roll, small-field inflation. In particular, the fluctuations seem to be adiabatic and Gaussian, with a slightly tilted power spectrum, confirming the generic predictions of slow-roll single-field inflation. Also, there are no signs of tensor perturbations, which is expected for small-field models, but puts tension on popular large-field models of inflation. New inflation [2], in which inflation is driven by a scalar field that slowly rolls from a hilltop towards its global minimum, is an attractive class of models which can generate CMB perturbations inside the Planck experiment's 1 σ bounds. It also allows interesting connections between inflation and particle physics. As the inflaton takes on a vacuum expectation value after inflation, it can be identified with a symmetry-breaking Higgs field which breaks e.g. an extended gauge symmetry [3] or some family symmetry [4]. If the symmetry breaking produces any dangerous topological defects, those are produced at the beginning of inflation and are automatically diluted away by the subsequent exponential expansion. Also, the field values during new inflation stay well below the Planck scale, which makes it possible to derive predictions within an effective field theory without making strong assumptions about Planck scale physics. However, when models of new inflation are implemented in supergravity [3, 4, 5, 6, 7, 8], the inflaton is a complex and not a real scalar field. As a complex scalar field has two independent components, supergravity models of new inflation are naturally two-field models. In this paper, we want to analyse how this two-field behaviour modifies the usual single-field predictions. We discuss for which model parameters the model reduces to the well-known single-field model, and how the predictions are modified otherwise. The paper is structured as follows. First we introduce the inflaton potential, including the superpotential and Kahler potential from which the inflaton potential is derived. Afterwards, we analytically study the initial conditions and field trajectories to understand under which conditions the model reduces to a single-field model, and under which conditions we expect that the multi-field dynamics should influence the inflationary predictions. Finally, we use the δN formalism to numerically calculate the predictions of the two-field model. We finish with a brief summary of our results.", "pages": [ 2 ] }, { "title": "2 Scalar potential", "content": "In this paper, we will study new inflation with the inflaton potential for the real scalar fields χ and ψ , which can be thought of as components of a complex scalar field Φ = 1 √ 2 ( χ + iψ ). Note that the inflaton potential (1) is invariant under transformations Φ → Φ ∗ and Φ → e 2 iπ/glyph[lscript] Φ. These symmetries are also obvious in fig. 1 where the potential is plotted for glyph[lscript] = 4. We can therefore assume that 0 ≤ ψ ≤ χ tan( π/glyph[lscript] ) without loss of generality, and we will do so throughout this paper. 2 We only consider the case β ≥ 0 in which the inflaton slow-rolls down the potential; for β < 0 the inflaton would be stuck at the local minimum at Φ = 0 and would have to tunnel to reach the global minimum.", "pages": [ 2, 3 ] }, { "title": "2.1 Construction from superpotential and Kahler potential", "content": "Inflaton potentials of the form of eq. (1) can be constructed from different superpotentials and Kahler potentials. One possibility, used e.g. in [4, 7, 9], is the superpotential with the superfields S and Φ and model parameters V 0 , M and glyph[lscript] . V 0 and M are chosen to be real, as their phases can be absorbed in the fields S and Φ. We use natural units with 8 πG = M -2 Pl = 1 to keep the notation simple. We also include the leading terms in an expansion of the Kahler potential The scalar F-term potential for chiral superfields can be calculated from W and K with the formula where K ij is the matrix inverse of the Kahler metric K ij , and For W and K given by eqs. (2) and (3), the scalar potential is 3 We assume that κ S > 1 12 so that the scalar component of S has a mass above the Hubble scale H and is stabilized at 0 during inflation. We can then neglect this field from now on. For the scalar component of Φ, we choose the decomposition into real and imaginary part: Inserting this decomposition in eq. (6), we arrive at the scalar potential given in eq. (1). Note that we only use the inflaton potential (1) for the calculations, so our results are also valid for other superpotentials and Kahler potentials which lead to an effective inflaton potential of the form (1), e.g. for [5, 6], and for [8] (in the limit c → 0 in the last cited paper).", "pages": [ 3, 4 ] }, { "title": "3 Initial conditions", "content": "In a multi-field model of inflation, the predictions can depend on the initial conditions. In this section, we want to discuss the initial conditions that we use for our analysis.", "pages": [ 4 ] }, { "title": "3.1 Φ i = 0 from preinflation", "content": "We start from the assumption that at some point in time, the field is very close to Φ = 0. Such a state can be generated dynamically e.g. by a period of preinflation [6, 7]. Preinflation is a preceding phase of inflation, driven by some other inflaton field Ξ, during which Φ has a large mass from the vacuum expectation value of the inflaton field Ξ. This large mass drives Φ → 0 and keeps it stabilized there. After preinflation has ended, Ξ → 0, and the Ξ-induced mass for Φ disappears. The inflaton Φ now sits at its 'initial value' Φ = 0 and can start slow-rolling down its potential.", "pages": [ 4 ] }, { "title": "3.2 Quantum diffusion boundary", "content": "When Φ = 0, the Friedmann equations imply that the field does not move away from Φ = 0 at all. However, the field Φ has quantum fluctuations which can be thought of as moving Φ randomly over time. Near Φ = 0, these quantum fluctuations dominate over the classical evolution; we call this area the quantum diffusion region. After some time, however, Φ has randomly walked away into some region where the potential is steep enough for the classical evolution to take over and the inflaton rolls away from Φ = 0, down the potential gradient towards the nearest minimum. We therefore choose initial conditions on the boundary where the classical evolution starts to dominate over the quantum diffusion, and calculate the evolution with the Friedmann equations from there. The boundary between the diffusion region and the region of classical field evolution can be estimated by comparing the classical evolution ∆Φ cl per Hubble time t H = H -1 to the growth of quantum fluctuations ∆Φ qu per Hubble time. where we used the Friedmann equations and in the slow-roll approximation ¨ χ glyph[similarequal] ¨ ψ glyph[similarequal] 0, 3 H 2 glyph[similarequal] V . To solve eq. (8), it is more useful to work in polar coordinates for Φ, with a radius φ and an angle θ : With these conventions, the scalar potential (1) is We must also be careful to note that where the factor 1 /φ appears in front of the derivative with respect to θ because θ is not a canonically normalized field. In these polar coordinates, and using the fact that V glyph[similarequal] V 0 during new inflation, eq. (8) can be written as We want to solve this equation for the initial field values χ i and ψ i separately for the cases β = 0 and β = 0. glyph[negationslash]", "pages": [ 4, 5 ] }, { "title": "3.2.1 Diffusion boundary for β = 0", "content": "For β = 0, the relevant derivatives are 4 With these ingredients, eq. (14) becomes very simple so the diffusion boundary is a circle in field space with the squared radius", "pages": [ 6 ] }, { "title": "3.2.2 Diffusion boundary for β > 0", "content": "For sufficiently small Φ, the mass term from β dominates over the other inflaton potential terms which are of higher order in the fields. Therefore, we can neglect the other interactions at the diffusion boundary 5 , and eq. (14) simplifies to √ We find that the diffusion boundary is a circle around the origin with radius H / ( 2 πβ ). To find out how large β must be so that we can neglect the other inflaton interactions, we can compare the size of the two interactions at the diffusion boundary given by eq. (18): For an order-of-magnitude estimate, we can insert the vacuum energy for single-field new inflation with β glyph[similarequal] 0, which is Then the condition (19) becomes and we find that eq. (18) is valid for any β glyph[greatermuch] 10 -5 if glyph[lscript] = 4. For larger glyph[lscript] , the threshold for β is even lower.", "pages": [ 6, 7 ] }, { "title": "4 Analytic estimate of field trajectory", "content": "In this section, we want to discuss the dynamics of the χ and ψ fields during inflation to estimate for which parameters the model reduces to the well-known single-field model of new inflation. We can rewrite the fields χ , ψ as φ , θ according to eq. (11). Using the partial derivatives and ignoring the negligible term proportional to φ 2 glyph[lscript] in the potential, we can calculate the partial derivatives of the scalar potential (12) during inflation: Using the Friedmann eqs. (9a)-(9b) and the slow-roll approximation ¨ χ glyph[similarequal] ¨ ψ glyph[similarequal] 0, we can calculate the inflaton trajectory in field space: We will discuss the behaviour of the field trajectory for two distinct cases: for a vanishing mass term ( β = 0), for which we will recover the single-field new inflation limit, and for a large mass term ( β glyph[greaterorsimilar] 10 -2 ), for which we show that the imaginary inflaton component cannot generally be neglected.", "pages": [ 7, 8 ] }, { "title": "4.1 Supergravity mass term vanishes ( β = 0 ): new inflation limit", "content": "If β = 0, the tachyonic mass term for Φ exactly vanishes. 6 In this case, we can show that the imaginary component ψ decays before the observable primordial fluctuations leave the horizon, and inflation reduces to single-field new inflation. With β = 0, eq. (24) simplifies to As explained in section 2, we can assume that 0 ≤ θ ≤ π/glyph[lscript] . Moreover, for large θ > θ thr , we have ∂ χ V > 0, so for such large θ , χ is rolling back towards 0 until the inflaton is so close to χ = ψ = 0 that quantum diffusion dominates. Eventually, the inflaton will randomly diffuse out of this diffusion region. Every time it leaves the diffusion region with θ > θ thr , it is pushed back, until it eventually leaves the diffusion region with θ < θ thr . Assuming 0 < θ < θ thr , we find that This implies that ψ drops off faster than 1 /χ glyph[lscript] -1 : for any initial values ψ 0 and χ 0 . If the initial displacement from ψ = χ = 0 is due to quantum fluctuations as in eq. (17), then eq. (27) implies that ψ is always negligible at horizon crossing. One can estimate this from the single-field results for new inflation, where the normalization of the CMB amplitude ensures that between the diffusion boundary and horizon crossing, χ grows by a factor of ( χ ∗ /χ i ) glyph[lscript] -1 = A -1 s . Then eq. (27) implies that 1+ 1 is already negligible ψ ∗ ψ i < A s , and so ψ ∗ χ ∗ < A glyph[lscript] -1 s ∼ 10 -5 , so at horizon crossing ψ compared to χ . We conclude that for β = 0, the model reduces to single-field new inflation in χ . y", "pages": [ 8, 9 ] }, { "title": "4.2 Supergravity mass term relevant ( β glyph[greaterorsimilar] 10 -3 )", "content": "If β > 0, ψ grows during the early stages of inflation. For sufficiently large β , ψ can be significantly large when the primordial fluctuations leave the horizon, and therefore the dynamics of ψ is expected to have an effect on the CMB spectrum. For small field values φ glyph[lscript] -2 glyph[lessmuch] βM 2 / (2 glyph[lscript] ), ψ grows proportionally with χ , as we can see from eq. (24): This linear growth of ψ with χ continues until the fields grow large enough to make the higher-order interactions dominate over the tachyonic mass term from β . From that point onwards, the dynamics are approximately given by eq. (27): ψ starts decaying faster than χ -( glyph[lscript] -1) while χ rolls towards the minimum at χ = M 2 /glyph[lscript] . Both the growing and the decaying phase for ψ can be clearly seen in fig. 2. If ψ decays before cosmological scales leave the horizon, then we recover the single-field inflation limit with χ as the inflaton. We can use this to estimate for which values of β we can expect significant deviations from the single-field limit. The transition from the growing to the decaying phase occurs at approximately χ glyph[lscript] -2 trans ∼ φ glyph[lscript] -2 trans ∼ βM 2 / (2 glyph[lscript] ), as can be seen from eq. (24). Calculating the field value χ ∗ at which cosmological scales leave the horizon is more involved. For a rough estimate, we assume that it is given by its single-field new inflation value: where N e is the number of e-folds before the end of inflation at which cosmological scales leave the horizon. Setting χ ∗ glyph[lessorsimilar] χ trans , we find that multi-field effects are expected to become important for β above the threshold value We therefore recover the single-field limit for β glyph[lessmuch] 10 -2 / ( glyph[lscript] -2), whereas for β glyph[greaterorsimilar] 10 -2 / ( glyph[lscript] -2) the predictions should be calculated in a multi-field formalism. This is done numerically in section 5.", "pages": [ 9, 10 ] }, { "title": "4.3 Generalization to initial field values outside the diffusion region", "content": "So far, we have assumed that inflation starts at Φ i glyph[similarequal] 0 inside the diffusion region given by eq. (17) or (18). Due to the field dynamics, our results can be immediately generalized to initial field values Φ i outside the diffusion region, as long as Φ i remains sufficiently small. For β > 0, we have ψ ∝ χ and therefore θ = const for small field values. This implies that our results, which depend on θ i , are valid even when the initial value Φ i is outside of the diffusion region, as long as Φ i starts in the growing phase in which ψ ∝ χ . A sufficient condition for the validity of our results is φ glyph[lscript] -2 i glyph[lessmuch] βM 2 2 glyph[lscript] . For β = 0, no growing phase exists. However, because ψ quickly decays according to eq. (27), we recover the single-field limit even for larger initial field values. We expect significant deviations from the single-field results only for initial values close to the horizon crossing scale, such that ψ has very little time to decay between the start of inflation and horizon crossing.", "pages": [ 10 ] }, { "title": "5 Numerical treatment", "content": "In the last section, we have discussed that the inflationary predictions can be changed by the imaginary inflaton component for β glyph[greaterorsimilar] 10 -2 / ( glyph[lscript] -2). In this section, we present a numerical calculation of the primordial perturbations using the δN formalism.", "pages": [ 10 ] }, { "title": "5.1 δN formalism", "content": "A powerful tool for calculating the primordial perturbations in multi-field models of inflation is the δN formalism [10, 11, 12, 13]. It is based on the fact that the curvature perturbation ζ ( x, t ) on a spatial uniform-density hypersurface is given by the difference δN in the number of e-folds between a flat initial hypersurface and the uniform-density final hypersurface: We choose the initial flat hypersurface at the time t ∗ at which cosmological scales leave the horizon. In this paper we want to calculate the perturbations at the end of inflation, so we choose the final hypersurface of uniform density at the end of slow-roll inflation. As the field perturbations are very small, one can expand δN in powers of the field perturbations δφ i on the initial flat hypersurface (the index i denotes the i -th inflaton field): where we introduced the notation Inserting the de Sitter space field perturbations in eq. (32), one can derive the leading-order expression for the amplitude A s of the primordial curvature perturbation: where a subscript star indicates that a quantity should be evaluated at the time of horizon crossing. With some extra work, one can also calculate the amplitude f NL of the reduced bispectrum [11] and the spectral index n s of the curvature perturbation [12] with the first slow-roll parameter For calculating the predictions for any set of parameters and initial conditions ( χ i , ψ i ), we took the following steps:", "pages": [ 10, 11 ] }, { "title": "5.2 Numerical results", "content": "Using the δN formalism, we have calculated the spectral index n s , the amplitude of the reduced bispectrum f NL and the vacuum energy V 0 during inflation. We assumed that inflation starts on the circle given by eq. (18) where the classical evolution starts to dominate over the quantum diffusion of the inflaton fields. 7 We calculated the predictions for various points on this initial surface, parametrized by the angle where the maximum angle is because as we explained in section 2, larger angles are related to angles in the range 0 ≤ θ ≤ θ max by symmetry transformations, so we can restrict ourselves to angles between 0 and θ max . The results for n s and V 0 are shown in fig. 3 for N e = 55 and M = 10 -5 . The numerical results confirm that we recover the single-field limit for β → 0, while for larger β the imaginary inflaton component reduces both n s and V 0 . Note that the green dots, which are quite close to the black single-field result, correspond to θ = 2 3 θ max . Therefore, even though the deviations become large for maximal θ , most initial conditions give results similar to the single-field limit. f NL is shown in fig. 4. It is generally in the range -1 < f NL < 0 which is too small to be observed, even for close-to-maximal θ . n s n s n l = 4 l = 6 0.96 0.97 0.96 0.95 0.94 0.93 0.92 0.96 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0 0.01 0.02 0.03 0.04 0 0.01 0.02 0.03 0.04 b 0 0.01 0.02 0.03 0.04 b b l = 8 4 Pl M 0.96 10 GLYPH<144> 0 V 4 Pl M 10 - 25 - 17 0.96 10 10 GLYPH<144> 0 V 4 Pl 10 - 19 0.96 10 10 M GLYPH<144> 0 V 10 - 17 - 15 - 16 - 18 - 24 0 0.01 0.02 0.03 0.04 b 0 0.01 0.02 0.03 0.04 b l = 4 l = 6 l = 8 f f We have checked that the β -dependence of our results is insensitive to changes in M . As in the single-field case, different choices of M only give a constant factor for V 0 , while n s does not depend on M at all. Note that while the results depend sensitively on θ i , they are valid for any initial φ i = √ χ 2 i + ψ 2 i as long as it is sufficiently close to zero (see section 4.3).", "pages": [ 12, 13, 14 ] }, { "title": "6 Summary and conclusions", "content": "In this paper, we have studied the effects of the multi-field dynamics of the complex scalar inflaton field in supergravity new inflation, providing a brief analytical discussion and a numerical calculation. We have found that for most of the parameter space, the model is well described by the usual single-field approximation, where only the real component of the inflaton is considered and its imaginary component is set to zero. In particular, this is the case if the mass term from the Kahler potential is very small or absent, in which case the imaginary component is quickly driven to zero before cosmological scales leave the horizon. For a sufficiently large mass term, the results become sensitive to the initial conditions. For these cases, we have numerically calculated the predictions using the δN formalism. For most initial conditions, we find that the results are still similar to the single-field results, but the deviations become significant for large initial values of the imaginary inflaton component (see fig. 3). Those deviations generally reduce the spectral index n s and the inflationary vacuum energy V 0 compared to the single-field case. The reduced bispectrum is within the range -1 < f NL < 0, which is in good agreement with the current data from the Planck experiment, but probably too small to ever be observed. l = 4 l = 8 Our conclusions are twofold. First, we want to emphasize that new inflation in supergravity is well-approximated by a single-field model if either Second, if both of these conditions are violated (if both the inflaton mass term and the initial value of the imaginary inflaton component are sufficiently large), the spectral index n s and the vacuum energy V 0 depend sensitively on the initial conditions, and the singlefield results should be interpreted as upper limits on n s and V 0 only.", "pages": [ 14, 15 ] }, { "title": "Acknowledgements", "content": "I want to thank Stefan Antusch and Stefano Orani for many valuable discussions.", "pages": [ 15 ] } ]

[ { "title": "Observational signatures of anisotropic inflationary models", "content": "Junko Ohashi, 1 Jiro Soda, 2 and Shinji Tsujikawa 1 1 Department of Physics, Faculty of Science, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku, Tokyo 162-8601, Japan 2 Department of Physics, Kyoto University, Kyoto 606-8502, Japan (Dated: August 27, 2018) We study observational signatures of two classes of anisotropic inflationary models in which an inflaton field couples to (i) a vector kinetic term F µν F µν and (ii) a two-form kinetic term H µνλ H µνλ . We compute the corrections from the anisotropic sources to the power spectrum of gravitational waves as well as the two-point cross correlation between scalar and tensor perturbations. The signs of the anisotropic parameter g ∗ are different depending on the vector and the two-form models, but the statistical anisotropies generally lead to a suppressed tensor-to-scalar ratio r and a smaller scalar spectral index n s in both models. In the light of the recent Planck bounds of n s and r , we place observational constraints on several different inflaton potentials such as those in chaotic and natural inflation in the presence of anisotropic interactions. In the two-form model we also find that there is no cross correlation between scalar and tensor perturbations, while in the vector model the cross correlation does not vanish. The non-linear estimator f NL of scalar non-Gaussianities in the two-form model is generally smaller than that in the vector model for the same orders of | g ∗ | , so that the former is easier to be compatible with observational bounds of non-Gaussianities than the latter.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The measurements of Cosmic Microwave Background (CMB) temperature anisotropies and large-scale structures have significantly improved in accuracy over the last decade [1-6]. In particular, the recently released Planck data [7-9] showed that the primordial power spectrum of curvature perturbations is slightly red-tilted from the exact scale-invariance. This is consistent with the theoretical prediction of standard slow-roll inflation driven by a nearly flat potential of a scalar field φ (called 'inflaton') [10]. While the WMAP and Planck data support the inflationary scenario overall, there are some anomalies in the data [2, 7] which are difficult to be addressed in the context of single-field slow-roll inflation. One of them is broken rotational invariance of the CMB perturbations [11-13]. The power spectrum of curvature perturbations ζ with broken statistical isotropy can be expressed in the form [14] where k is the comoving wave number, P (0) ζ ( k ) is the isotropic power spectrum, g ∗ characterizes the deviation from the isotropy, V is a privileged direction close to the ecliptic poles, and θ k , V is the angle between k and V . From the WMAP data, Groeneboom et al. [13] obtained the bound g ∗ = 0 . 29 ± 0 . 031 with the exclusion of g ∗ = 0 at 9 σ by including the CMB multipoles up to /lscript = 400. There is still a possibility that some systematic effect such as the asymmetry of the instrument beam accounts for broken rotational invariance [15] 1 . If broken statistical isotropy is really present for primordial perturbations, we need to go beyond the slow-roll single-field inflationary scenario to explain the origin of statistical anisotropies [17, 18]. For the models in which the inflaton φ has a coupling with a vector kinetic term F µν F µν , there exists an attractor-like solution along which an anisotropic vector hair survives even during inflation [19] (see Refs. [20] for early related works and Refs. [21, 22] for rich phenomenologies of anisotropic inflation). Even if the background energy density of the vector field is suppressed relative to that of the inflaton, the isotropic power spectrum P (0) ζ ( k ) is modified to have the form (1) with a negative anisotropic parameter g ∗ [23-25]. Moreover the non-linear estimator f NL of scalar non-Gaussianities can be as large as the order of 10 for the squeezed shape averaged over all directions of a wave number k 3 with k 1 /similarequal -k 2 [25, 26] (see also Refs. [27] for the large non-Gaussianities generated by vector fields). Recently, the present authors showed that anisotropic inflation can be also realized for the model in which the inflaton couples to a two-form field with the kinetic term H µνλ H µνλ [28] (see Ref. [24] for an early proposal). In this case, the anisotropic power spectrum is given by Eq. (1) with g ∗ > 0. Since the sign of g ∗ is opposite to that in the vector model, we can observationally distinguish between the two anisotropic inflationary scenarios. In Ref. [28] the bispectrum and trispectrum of curvature perturbations have been also evaluated in the two-form model by using the interacting Hamiltonian picture [25, 29]. It was shown that, in the strict squeezed limit, the non-linear estimator f NL vanishes and that, in the equilateral and enfolded limits, f NL can be larger than the order of 10. In this paper we place observational constraints on two anisotropic inflation models based on the vector and the twoform fields in the light of the recent Planck data [7-9]. We first derive the anisotropic power spectra of gravitational waves to evaluate the tensor-to-scalar ratio r correctly. Using the observational bounds of r as well as the scalar spectral index n s constrained by the joint data analysis of Planck and other measurements [8, 30], we test for several representative models such as chaotic and natural inflation in the presence of anisotropic corrections to the scalar and tensor power spectra. We also compute the cross correlations between curvature perturbations and gravitational waves. While the cross correlation survives in the vector model, it vanishes in the two-form model. This property is useful to distinguish between the two anisotropic inflationary scenarios from the correlation between the observed temperature perturbation and the B-mode polarization (TB correlation) [31]. We also revisit the estimation of the anisotropic scalar non-Gaussianities for several different shapes of momentum dependence (local, equilateral, and enfolded shapes) in both the vector and the two-form models. In fact, we show that the local non-linear estimators in the strict squeezed limit ( k 3 → 0 with θ k 1 , k 3 → π/ 2, θ k 2 , k 3 → π/ 2) vanish in both models. However, if we average over all directions of a wave number k 3 with k 1 /similarequal -k 2 for nearly squeezed shapes [25], we have non-zero values of f NL for | g ∗ | > 0. Taking this prescription, we show that the local non-linear estimator f local NL in the two-form model is smaller than that in the vector model by one order of magnitude for the same order of | g ∗ | . This paper is organized as follows. In Sec. II we review the background dynamics of anisotropic inflation for both the vector and the two-form models. In Sec. III we derive anisotropic corrections to the scalar/tensor power spectra and also evaluate the cross correlation between curvature perturbations and gravitational waves. In Sec. IV we put constraints on several different inflaton potentials by using the 68 % CL and 95 % CL observational contours in the ( n s , r ) plane. In Sec. V we compare two anisotropic inflationary models from the non-linear estimator f NL of scalar non-Gaussianities. Sec. VI is devoted to conclusions.", "pages": [ 1, 2 ] }, { "title": "II. INFLATION WITH ANISOTROPY", "content": "In this section we briefly review the background dynamics of anisotropic inflation for two classes of models in which the inflaton field φ couples to (i) a vector field A µ [19] and (ii) a two-form field B µν [28]. For suitable choices of couplings, the energy densities of the fields A µ and B µν can survive even during inflation. For details, we refer the reader to the review articles [17, 18]. Let us first discuss the vector model given by the action where g is the determinant of the metric g µν , M pl is the reduced Planck mass, and R is the scalar curvature. V ( φ ) and f ( φ ) are a potential and a kinetic function of the inflaton φ , respectively. The field strength of the vector field is characterized by Now, we consider cosmological solutions in this system. Without loosing the generality, one can take x -axis for the direction of the vector field. Using the gauge invariance, we can express the vector field as where v ( t ) is a function of the cosmic time t . Even if initial inhomogeneities and isotropies in v are present, it was shown that only the background field v ( t ) survives during anisotropic inflation in the Bianchi type I Universe [24]. The /similarequal - same conclusion also applies to more general anisotropic backgrounds [32]. There remains the rotational symmetry in the ( y, z ) plane. Hence, we can take the metric ansatz where e α ≡ a and σ are the isotropic scale factor and the spatial shear, respectively. It is easy to solve the equation of motion for v as where p A is a constant of integration and an overdot denotes a derivative with respect to the cosmic time t . The Friedmann equation and the inflaton equation of motion are given, respectively, by where H ≡ ˙ α is the Hubble expansion rate, and V ,φ ≡ dV/dφ , f ,φ ≡ df/dφ . We define the energy density of the vector field as In order to sustain inflation, the potential energy V ( φ ) of the inflaton needs to dominate over ˙ φ 2 / 2 and ρ A . Since the shear term Σ ≡ ˙ σ should be suppressed relative to H , the Friedmann equation (7) reads If f is a rapidly decreasing function in time, it happens that ρ A does not decay. In particular, for the coupling the energy density (9) stays nearly constant (under the approximation | σ | /lessmuch α ). In this case, neglecting the contribution of the vector field on the r.h.s. of Eq. (8), the field φ satisfies the slow-roll equation of motion 3 ˙ α ˙ φ /similarequal -V ,φ . Combining this equation with Eq. (10), it follows that dα/dφ /similarequal -V/ ( M 2 pl V ,φ ). Then, the critical coupling (11) can be expressed as The shear term obeys the equation of motion If an anisotropy converges to a nearly constant value Σ, then the quantity Σ /H reduces to where we used Eq. (10). In the following, let us estimate the ratio Σ /H during anisotropic inflation. Ignoring the ¨ φ term in Eq. (8) and combining it with Eq. (10), it follows that dφ dα 2 pl V V 2 p V 2 A ,φ where we used Eq. (12). Neglecting the variation of φ/M pl relative to that of α , we can integrate Eq. (15) to give M ,φ - 4 α - 4 σ - 4 ∫ V 2 pl V dφ + e M ,φ , (15) where α 0 > 0 is an integration constant. Substituting Eq. (16) into Eq. (9), we have where /epsilon1 ≡ -˙ H/H 2 , and we used the fact that /epsilon1 /similarequal ( M 2 pl / 2)( V ,φ /V ) 2 under the slow-roll approximation. Provided that α /lessmuch α 0 , the ratio r A is nearly constant. As we will see in Sec. III, the quantity r A is related to the anisotropic parameter g ∗ appearing in the power spectrum of curvature perturbations. From Eqs. (14) and (17) we obtain which means that the anisotropy survives during inflation for α /lessmuch α 0 . The above discussion can be generalized to the coupling of the form where c is a constant parameter. If the condition is satisfied, the energy density of the vector field grows as ρ A ∝ e 4( c -1) α during the slow-roll phase of the inflaton. Eventually, the vector field becomes relevant to the inflaton dynamics governed by Eq. (8). However, when the third term on the r.h.s. of Eq. (8) dominates over the second term, the inflaton does not roll down, which makes ρ A decrease. Hence the condition ρ A /lessmuch V ( φ ) is always satisfied. In this way, there appears an attractor where inflation continues even when the vector field affects the inflaton dynamics. For the coupling (19) the shear to the Hubble expansion rate approaches the value [19] Thus, inflation is slightly anisotropic and the energy density of the vector field never decays for the coupling (19) with c > 1. From Eqs. (14) and (21) it follows that the ratio r A defined in Eq. (17) is nearly constant, i.e., r A /similarequal ( c -1) / (2 c ). In the attractor regime where the vector field contributes to the dynamics of the system there is the relation dα/dφ /similarequal -cV/ ( M 2 pl V ,φ ) [18], in which case the coupling (19) evolves as f ( φ ) ∝ a -2 , i.e., the same as Eq. (11). In Sec. III we shall use this property for the evaluation of two-point correlation functions of primordial perturbations.", "pages": [ 2, 3, 4 ] }, { "title": "B. f ( φ ) 2 H µνλ H µνλ model", "content": "In the two-form model the scalar spectral index and the tensor-to-scalar ratio, which are averaged over all the angles, are given by Eqs. (107) and (109), respectively. For chaotic inflation characterized by the potential (115), these observables reduce to In the left panel of Fig. 2 the theoretical predictions of chaotic inflation are shown for g ∗ ranging in the region 0 ≤ g ∗ ≤ 0 . 5. For larger g ∗ both n s and r decrease, but the quadratic potential is outside the 68 % CL region. In the presence of anisotropic interactions the potentials with n = 1 and n = 2 / 3 enter the 95 % CL boundaries, but still these models are outside the 68 % CL region constrained by the Planck+WP+BAO+high/lscript data. This shows that, for the same value of | g ∗ | , the power-law potentials with n ≤ 1 in the two-form model are more difficult to be compatible with the data relative to the same potentials in the vector model. For natural inflation given by the potential (117), the observables (107) and (109) read which decrease for larger g ∗ and smaller F . The difference from the vector model is that, for the same value of | g ∗ | , the allowed region of the two-form model inside the 68 % CL observational contours is wider. When g ∗ = 0 . 5, for example, we find that the mass scale F is constrained to be 5 . 9 M pl < F < 10 . 1 M pl (68 % CL) in the two-form model. As long as g ∗ is smaller than 1 . 0, there are some allowed values of F compatible with the observational data at 68 % CL. For the potential (120) we have As we see in the right panel of Fig. 2, this potential is well within the 68 % CL region. We also find that hybrid inflation with the potential V ( φ ) = Λ 4 + m 2 φ 2 / 2 is outside the 95 % CL boundaries for g ∗ < 0 . 5. In summary, the observables n s and r in the two-form model exhibit similar properties as those discussed in the vector model, but there are some potentials which can be inside and outside the 68 % CL region depending on the values of g ∗ . The precise observational bounds of g ∗ (in particular the signs of g ∗ ) will be able to distinguish the two anisotropic inflationary models further.", "pages": [ 16, 17 ] }, { "title": "III. SCALAR AND TENSOR POWER SPECTRA AND THEIR CORRELATIONS", "content": "In order to study observational signatures of anisotropic inflation, we need to know the two-point correlation functions of curvature perturbations and gravitational waves as well as their cross correlations. For the vector model the power spectrum of curvature perturbations was derived in Refs. [23-25], whereas the anisotropic contribution to gravitational waves in the same model was discussed in Refs. [23, 24]. For the two-form field model the present authors obtained the anisotropic scalar power spectrum [28], but the tensor power spectrum has not been derived yet. We also note that in the vector model the correlation between the temperature perturbation and the B-mode polarization was studied in Ref. [31], but the cross-correlation between scalar and tensor perturbations in the two-form model has not been studied. Here we provide all the formulas of these observables convenient to confront with observations. Since the anisotropy of the expansion rate needs to be sufficiently small for the compatibility with observations, it is a good approximation to neglect the effect of the anisotropic expansion for the derivation of the perturbation equations [24]. The effect of the anisotropy appears in the interacting Hamiltonians between vector/two-form fields and scalar/tensor perturbations, by which the scalar/tensor power spectra are modified. Then, we consider a general perturbed metric with four scalar functions A,B,ψ,E and the tensor perturbation h ij about the flat FriedmannLemaˆıtle-Robertson-Walker (FLRW) background [35]: where τ = ∫ a -1 dt is the conformal time. After the end of inflation, the coupling f ( φ ) approaches a constant because the inflaton stabilizes at the potential minimum. In this case vector perturbations decay after inflation as in the standard scenario, so we neglect its contribution to the CMB observables relative to those of scalar and metric perturbations. We introduce the gauge-invariant comoving curvature perturbation [36] (see also Refs. [37]): where δφ is the perturbation of the inflaton φ . In the following we choose the spatially flat gauge ( ψ = 0), in which case ζ = -( H/ ˙ φ ) δφ . The curvature perturbation can be expressed in terms of the Fourier components with the comoving wave number k , as where the annihilation and creation operators a ( k ) and a † ( k ' ) satisfy the commutation relation [ a ( k ) , a † ( k ' )] = δ (3) ( k -k ' ). We define the scalar power spectrum P ζ in terms of the two-point correlation function of ζ , as We decompose ζ into the isotropic field ζ (0) and the contribution δζ coming from the anisotropic fields, as In what follows we shall focus on the couplings (12) and (30), i.e., c = 1. The situation is similar for the general couplings (19) and (35) with c close to 1. Then we can employ the usual slow-roll relations ˙ φ/H /similarequal -M 2 pl V ,φ /V and /epsilon1 = -˙ H/H 2 /similarequal ( M 2 pl / 2)( V ,φ /V ) 2 , so that ζ (0) /similarequal δφ/ ( M pl √ 2 /epsilon1 ). The solution to the Fourier mode ζ (0) ( k, τ ), which recovers the Bunch-Davies vacuum state for the field perturbation δφ in the asymptotic past ( kτ →-∞ ), is given by [29] The power spectrum can be written as the sum of the two contributions from ζ (0) and δζ , as P ζ = P (0) ζ + δ P ζ . Using the solution (43) long time after the Hubble radius crossing ( τ → 0), the isotropic power spectrum of ζ is given by In Secs. III A and III B we shall evaluate the anisotropic corrections to P (0) ζ in both the vector and the two-form field models. For the tensor perturbation h ij we impose the traceless and transverse conditions h ii = h ij,j = 0, as usual. The second-order action for h ij reads where the prime denotes the differentiation with respect to τ . We have two physical degrees of freedom for h ij which can be characterized by the symmetric polarization tensors e (+ , × ) ij ( k ) satisfying where s = + , × represent the polarizations. It is convenient to adopt the normalization where ∗ represents a complex conjugate. Remark that the following relation holds: Now, it is straightforward to quantize tensor perturbations. The mode expansion can be written as [39] where the creation and annihilation operators are normalized as [ a s ( k ) , a † s ( k ' ) ] = δ ss ' δ (3) ( k -k ' ). We define the tensor power spectrum P h , as When we study the polarization of tensor perturbations, we can take both the vectors k 1 and k 2 lying on the ( x, y )-plane without lose of generality (because of the momentum conservation k 1 + k 2 = 0). In this case we can take where θ represents the angle between k 1 and x -axis. For k 1 = ( k 1 , 0 , 0), i.e., θ = 0, the polarization tensors e ( s ) ij ( k 1 ) satisfying the relations (46)-(48) are To obtain the polarization for k 1 = k 1 (cos θ, sin θ, 0), we need to rotate the above one by θ as We write the Fourier mode ˆ h ij ( k , τ ) in Eq. (49), as Using Eq. (53), it follows that which will be used for the evaluation of the interacting Hamiltonians between gravitational waves and vector/two-form fields. We decompose the tensor perturbation ˆ h ij into the isotropic field ˆ h (0) ij and the anisotropic contribution δ ˆ h ij . The isotropic mode function u (0) k ≡ M pl √ k/ 2 h (0) s ( k ) obeys the following evolution equation where the canonical commutation relation leads to the normalization condition Once a set of mode functions satisfying this normalization is specified, the corresponding Fock vacuum is determined by a s ( k ) | 0 〉 = 0. The mode function in a de Sitter background is given by u (0) k ( τ ) = ( H/k ) (1 + ikτ ) e -ikτ , that is Using this solution and (54), (55) long after the Hubble radius crossing, the isotropic power spectrum defined by (50) reads In the following we evaluate the anisotropic corrections to P (0) h in both the vector and the two-form models. In doing so, it is convenient to notice the following commutation relations which can be derived by employing the solutions (43) and (58) in the super-Hubble regime ( | kτ | /lessmuch 1). For the model described by the action (2) we decompose the vector field A µ into the Fourier components by choosing the Coulomb gauge: where A (0) i ( τ ) = ( A (0) x , 0 , 0) is the background component, and /epsilon1 ( λ ) i ( k ) ( λ = 1 , 2) are polarization vectors satisfying the relations With the previous parametrization k i = k (cos θ, sin θ, 0), an explicit representation is given by The rescaled field V λ = fA λ obeys the equation of motion For the coupling f given by Eq. (11), we have f ∝ τ 2 on the de Sitter background ( a = -( τH ) -1 ) and hence f '' /f = 2 /τ 2 . In this case the resulting vector field perturbation is scale-invariant. The solution to Eq. (65), which recovers the Bunch-Davies vacuum in the asymptotic past, is It is convenient to define the electric components where E x and δE i correspond to the background and the perturbed values. The next step is to derive anisotropic contributions δ P ζ and δ P h to the isotropic scalar and tensor power spectra (44) and (59). The tree-level interacting Lagrangian is where in the first line 〈 〉 represents the background value and after the second line we picked up the second-order perturbation terms and dropped the symbol 〈 〉 . We also used ( ∂ 〈 f 2 〉 /∂φ ) δφ = 4 f 2 ζ , which follows from the relation ζ = -( H/ ˙ φ ) δφ and the slow-roll conditions. We decompose δE i ( x , τ ) into the Fourier components, as Using the solution (66) in the super-Hubble regime | kτ | /lessmuch 1, the mode function δ E i ( k , τ ) is given by The contributions to the interacting Hamiltonian H int = -∫ d 3 xL int , which come from the four interacting Lagrangians in Eq. (68), are given, respectively, by where θ k , x is the angle between the wave number k and the x -axis. In deriving the above Hamiltonians, we used Eqs. (40), (49), (55), (69), and replaced ˆ ζ ( -k , τ ) and ˆ h s ( -k , τ ) for the isotropic perturbations ˆ ζ (0) ( -k , τ ) and ˆ h (0) s ( -k , τ ), respectively. The two-point correction of scalar perturbations following from the interacting Hamiltonian (71) to the isotropic power spectrum P (0) ζ reads where we used the relation (60). In the first line of Eq. (75) the two integrals have been evaluated in the super-horizon regime characterized by -k i τ < 1, that is, τ min ,i = -1 /k i with i = 1 , 2. The choice of this contour is based upon the standard vacuum in the interacting field theory, that is, the change τ → τ -iε | τ | for large | τ | in the exponent e -ikτ in the mode function (66) [29]. This means that the oscillating term in the sub-horizon regime is exponentially suppressed, so that the main contribution to the integral (75) comes from the super-horizon mode ( -k i τ < 1) [25]. In fact, the direct computation of the oscillating contributions to the integrals appearing in the correlation functions shows that the prescription mentioned above leads to the similar results to those derived by the regularization (time averaging) of the oscillating terms (see Appendix B of Ref. [38]). In the second line of Eq. (75) the upper bound τ 1 of the second integral has been replaced by τ by dividing the factor 2! because of the symmetry of the integrand. We also used the property ∫ τ -1 /k i dτ i ( τ 3 -τ 3 i ) /τ 4 i /similarequal ln( aH/k i ) /similarequal N k i in the regime -k i τ /lessmuch 1, where N k i is the number of e-foldings before the end of inflation at which the modes with the wave number k i left the Hubble radius. Since k 1 = -k 2 , it follows that N k 1 = N k 2 ≡ N k . Thus, the total scalar power spectrum in the vector model is given by [25] where, in the second equality, we used the relation ρ A = E 2 x / 2 and the definition r A given in Eq. (17). For the parametrization (1), this result corresponds to a negative anisotropic parameter In order to avoid that the anisotropic contribution does not exceed the isotropic spectrum, we demand the condition | g ∗ | /lessorsimilar 1. From the WMAP data there is the bound g ∗ = 0 . 29 ± 0 . 031 [13]. Under the condition | g ∗ | /lessorsimilar 1 it follows that r A /lessorsimilar 10 -5 for N k ∼ 60. Since α in Eq. (17) corresponds to the number of e-foldings from the onset of inflation, we have r A /similarequal 1 / (8 α 0 ) = constant for α /lessmuch 10 4 . We recall that, for the coupling (19), the quantity r A is also constant. Thus, the scalar spectral index reads where η V ≡ M 2 pl V ,φφ /V . The momentum vector k 1 does not necessarily need to lie on the ( x, y )-plane, but it is generally given by k 1 = k 1 (sin θ 1 cos ϕ 1 , sin θ 1 sin ϕ 1 , cos θ 1 ), where 0 ≤ θ 1 ≤ π and 0 ≤ ϕ 1 ≤ 2 π . It then follows that cos θ k 1 , x = sin θ 1 cos ϕ 1 . The average of sin 2 θ k 1 , x integrated over all the angles of θ 1 and ϕ 1 is Using this property, the scalar spectral index (78) reads The anisotropic corrections to the two-point isotropic correlation of tensor perturbations ˆ h s ( s = + , × ) are where all possible combinations of interacting Hamiltonians H h 1 and H h 2 are taken, and Hence we obtain the total correction Therefore, the total tensor power spectrum is given by The tensor-to-scalar ratio can be evaluated by using the anisotropic parameter (77) as Taking the same average over angles as (79), it follows that From Eq. (84) the tensor spectral index reads The cross correlation between curvature perturbations and the plus mode of gravitational waves is given by ∣ where we neglected the anisotropic contributions because they are second order in slow-roll parameters. where 'perm.' represents the terms obtained by the permutations of H ζ and H A . Similarly we have We define the cross power spectrum P ζh ( k 1 ) by 〈 0 | ˆ ζ ( k 1 ) ˆ h + ( k 2 ) | 0 〉 = (2 π 2 /k 3 1 ) δ (3) ( k 1 + k 2 ) P ζh ( k 1 ). While there is no cross correlation without anisotropic interactions, it remains for the model (2) as This gives rise to the non-vanishing TB cross power spectrum of CMB anisotropies [31].", "pages": [ 6, 7, 8, 9, 10, 11 ] }, { "title": "IV. JOINT OBSERVATIONAL CONSTRAINTS ON ANISOTROPIC INFLATIONARY MODELS", "content": "In this section, we place observational constraints on each anisotropic inflationary model with concrete inflaton potentials. Using the Cosmological Monte Carlo (CosmoMC) code [40, 41], we carry out the likelihood analysis with the latest Planck data [7] combined with the WMAP large-angle polarization (WP) [2], Baryon Acoustic Oscillations (BAO) [42-44], and ACT/SPT temperature data of high multipoles (high/lscript ) [3, 45]. The flat ΛCDM model is assumed with N eff = 3 . 046 relativistic degrees of freedom and with the instant reionization. We also set the runnings of the scalar and tensor spectral indices to be 0. The pivot wave number is chosen to be k 0 = 0 . 05 Mpc -1 . We confirmed that the different choices of k 0 such as 0 . 002 Mpc -1 give practically identical likelihood results. From Eqs. (86), (87), and (109), (110), the consistency relations in the two anisotropic models are given by The presence of anisotropic interactions modifies the standard consistency relation r = -8 n t . If g ∗ = -0 . 5 and g ∗ = 0 . 5, then we have r /similarequal -6 . 0 n t for the vector model and r /similarequal -6 . 9 n t for the two-form model, respectively. We have run the CosmoMC code by using these consistency relations and found that the likelihood contours are very similar to those derived with the relation r = -8 n t . Therefore, we plot observational contours obtained by varying the three inflationary observables P ζ ( k 0 ), n s ( k 0 ), and r ( k 0 ) with the consistency relation r ( k 0 ) = -8 n t ( k 0 ). In the vector model, anisotropic interactions lead to the enhancement of the scalar power spectrum P ζ on larger scales because the amplitude (76) increases for larger N k . As a result, the spectral index n s gets smaller for any inflaton potentials. The power spectrum of gravitational waves is also enhanced in the presence of anisotropic sources, but its effect is small compared to that on P ζ . Hence the tensor-to-scalar ratio gets smaller irrespective of the inflaton potentials. We recall that the decreases of n s and r are controlled by the negative anisotropic parameter g ∗ given in Eq. (77). In the two-form model, anisotropic interactions also lead to the decrease of n s and r with the positive parameter g ∗ given in Eq. (106). Since the level of the enhancement of P ζ is different from that of the vector model, the observables n s and r exhibit some difference between the two anisotropic inflation models. In the following we study the vector and two-form models separately for several different inflaton potentials. Since we are considering the case c = 1, we can employ the standard slow-roll equations (10) and 3 H ˙ φ /similarequal -V ,φ at the background level. Under this approximation, the number of e-foldings is given by where φ f is the value of φ at the end of inflation determined by the condition /epsilon1 ( φ f ) = 1. For the comparison of the inflationary observables with the CMB data, we fix N k = 60. Let us study observational constraints on the vector model. First, we consider chaotic inflation characterized by the power-law potential [46] where n and λ are positive constants. In this case, we have that /epsilon1 = n 2 M 2 pl / (2 φ 2 ) and η V = n ( n -1) M 2 pl /φ 2 . The field value at the end of inflation can be estimated as φ f = nM pl / √ 2. From Eq. (114) the number of e-foldings N k is related to the field φ , as φ 2 /similarequal 2 n ( N k + n/ 4) M 2 pl . Then the observables (80) and (86), which are averaged over all the angles, reduce to In the left panel of Fig. 1 we plot the theoretical values of n s and r for the anisotropic parameter g ∗ ranging in the region -0 . 5 ≤ g ∗ ≤ 0 with N k = 60. The self-coupling potential V ( φ ) = λφ 4 / 4 is outside the 95 % confidence level (CL) observational boundaries even in the presence of anisotropic interactions. The quadratic potential V ( φ ) = λφ 2 / 2 is inside the 95 % CL boundaries, but it is still outside the 68 % CL contours. When g ∗ = 0, the linear potential V ( φ ) = λφ , which appears in the axion monodromy scenario [47], is outside the 95 % CL boundary constrained by the Planck+WP+BAO+high/lscript data, but the vector anisotropy with g ∗ < -0 . 4 allows the model to be inside the 68 %CL contour. A similar property also holds for another axion monodromy potential V ( φ ) = (3 λ/ 2) φ 2 / 3 [48], but larger values of | g ∗ | are required for the compatibility with the data. We also study natural inflation characterized by the potential [49] where Λ and F are constants having a dimension of mass. The relation between N k and φ is given by N k = (2 F 2 /M 2 pl ) ln [sin( φ f / (2 F )) / sin( φ/ (2 F ))], where φ f is known by solving the equation tan 2 [ φ f / (2 F )] = 2( F/M pl ) 2 . The observables (80) and (86) read For a given value of F we can numerically identify the field value φ corresponding to N k = 60. Then, we evaluate n s and r according to the formulas (118) and (119). In the limit that F → ∞ , these observables approach the values (116) of chaotic inflation with n = 2. In the right panel of Fig. 1, we show the theoretical values of n s and r for different values of F and g ∗ . For smaller F and larger | g ∗ | , both n s and r get smaller. When g ∗ = 0, the mass scale F is constrained to be 5 . 1 M pl < F < 7 . 9 M pl (68 % CL) from the Planck+WP+BAO+high/lscript data [30]. For larger | g ∗ | , the allowed parameter space inside the 68 % CL contours tends to be narrower. In particular, if | g ∗ | > 0 . 5, then the model is outside the 68 % CL boundary constrained by the Planck+WP+BAO+high/lscript data. Thus, in natural inflation, the presence of anisotropic interactions leads to the deviation from the observationally favored region. Let us also discuss the inflaton potential of the form where M is a constant having a dimension of mass. This potential arises in the Starobinsky's model f ( R ) = R + R 2 / (6 M 2 ) [50] after a conformal transformation to the Einstein frame with the field definition φ/M pl = √ 3 / 2ln[ ∂f ( R ) /∂R ] [51]. Recently there have been numerous attempts to construct the potential (120) in the context of supergravity and quantum gravity [52]. In the regime φ/M pl /greatermuch 1, the number of e-foldings is related to the inflaton, as e -√ 2 / 3 φ/M pl /similarequal 3 / (4 N k ) [53]. The slow-roll parameters are approximately given by /epsilon1 /similarequal 3 / (4 N 2 k ) and η V /similarequal -1 /N k , which means that /epsilon1 is much smaller than | η V | . Therefore, the observables (80) and (86) reduce to When g ∗ = 0 we have n s = 1 -2 /N k and r = 12 /N 2 k , which correspond to the values in the Starobinsky's model [54]. The anisotropic interactions lead to the decrease of n s , but still the model is well inside the 68 % CL contour, see the right panel of Fig. 1. We also study hybrid inflation characterized by the potential V ( φ ) = Λ 4 + m 2 φ 2 / 2 [55], where Λ and m are constants. When g ∗ = 0, this model gives rise to a blue-tilted spectrum ( n s > 1). In the presence of anisotropic interactions it is possible to have a red-tilted spectrum, but we find that n s is larger than 0.99 for | g ∗ | < 0 . 5. Hence the model is still outside the 95 % CL region.", "pages": [ 14, 15, 16 ] }, { "title": "V. STATISTICALLY ANISOTROPIC NON-GAUSSIANITIES", "content": "In this section, we estimate anisotropic contributions to the statistical non-Gaussianities of curvature perturbations. The three-point correlation functions of ζ have been already derived both in the vector model [25] and in the two-form model [28]. In the vector model, Bartolo et al. [25] considered the squeezed shape in which the angles θ k 1 , k 3 and θ k 2 , k 3 between the momentum vectors satisfying the relation k 1 + k 2 + k 3 = 0 are not necessarily close to π/ 2 and they took the average over angles to evaluate the local non-linear estimator f local NL . In the two-form model, the present authors [28] took the strict squeezed limit, k 3 → 0, θ k 1 , k 3 → π/ 2, θ k 2 , k 3 → π/ 2, and showed that f local NL vanishes. We clarify the difference between these two approaches and estimate the local non-linear estimators in both analyses. In the two-form model, the non-linear estimators have been also computed for other shapes of non-Gaussianities such as the equilateral and enfolded ones [28]. Since the similar analysis has not been yet done in the vector model, we evaluate f NL in the equilateral and enfolded limits. In order to calculate the three-point correlation function of curvature perturbations in the vector model described by the action (2), we need to derive the Hamiltonian H ζ 2 following from the third-order interacting Lagrangian L (3) int /similarequal 2 a 4 δE i δE j ζ , in addition to H ζ given in Eq. (71). On using Eqs. (40) and (69), this interacting Hamiltonian is given by Then we can evaluate the three-point correlation of ζ , as [25] The total anisotropic bispectrum B ζ is defined by δ 〈 0 | ˆ ζ ( k 1 ) ˆ ζ ( k 2 ) ˆ ζ ( k 3 ) | 0 〉 = B ζ δ (3) ( k 1 + k 2 + k 3 ). We introduce the non-linear estimator f NL in the form by which f NL for the vector model can be derived as [25] where, in the first line, we used ρ A = E 2 x / 2 and the quantity r A defined in Eq. (17). In the second line we employed the approximations ( P ζ ) 2 /similarequal ( P (0) ζ ) 2 , N k 1 /similarequal N k 2 /similarequal N k 3 ≡ N k , and the definition of g ∗ given in Eq. (77). Let us first take the same approach as that used in Ref. [28] for the two-form model. It is convenient to introduce the following parameters If we fix r 2 = 1 and define the angles β = π -θ k 1 , k 2 and γ = θ k 1 , x , it follows that The angle β is in the range 0 < β < π (i.e., 0 < r 3 < 2). In the strict squeezed limit ( r 3 → 0 and β → 0), the local estimator f local NL vanishes for any values of γ . This limit corresponds to the case in which the angles θ k 2 , k 3 and θ k 3 , k 1 approach π/ 2. Bartolo et al. [25] estimated f local NL in the following way. For the incomplete squeezed shape the angles θ k 2 , k 3 and θ k 3 , k 1 are not necessarily close to π/ 2. In other words, unless we take the strict squeezed limit k 1 → -k 2 , we can consider any angle between k 3 and k 1 , k 2 . Averaging over f NL in all the directions along the same line as Eq. (79), the local estimator for the squeezed shape ( k 3 /lessmuch k 1 /similarequal k 2 , θ k 1 , k 3 → π -θ k 2 , k 3 , and θ k 2 , x → π -θ k 1 , x ) reads where we used the fact that the average value of the function in the last square bracket integrated over all the angles is 4 / 9. In this analysis the local non-linear estimator does not vanish and it can be as large as the order of 10 for g ∗ ∼ -0 . 1. The Planck group placed the bound on the local non-linear estimator to be f local NL = 2 . 7 ± 5 . 8 (68 % CL) [9], but we cannot literally use this bound to constrain the anisotropic inflationary models. As we have seen above, the local non-Gaussianities depend on what kind of squeezed shapes to be taken in the data analysis. It is interesting that the strict or incomplete squeezed shapes do matter to estimate the level of non-Gaussianities correctly. The detailed data analysis based on different squeezed shapes is beyond the scope of our paper. Using the result (131), we can calculate the non-linear estimator for the shapes other than the squeezed one. In the equilateral ( β → π/ 3 , r 3 → 1) and the enfolded ( β → π , r 3 → 2) limits, we have Unlike the squeezed case, these estimators do not depend on how we take the limits of equilateral and enfolded shapes. The equilateral non-linear estimator is independent of the angle γ and it is typically of the order of 1 for | g ∗ | /lessorsimilar 0 . 1. Meanwhile, f enfolded NL depends on γ and it has a maximum at γ = π/ 2. The maximum value of f enfolded NL can be as large as 60 for | g ∗ | ∼ 0 . 1, which is an interesting signature of the vector model. In the two-form model the Hamiltonian following from the third-order interacting Lagrangian L (3) int /similarequal a 4 δE ij δE ij ζ/ 2 is given by The three-point correlation function of ζ can be computed by using the interacting Hamiltonians (100) and (135) in the way similar to the derivation of Eq. (127). In Ref. [28] this was already derived as From the definition (128) of f NL , it follows that where we used the anisotropic parameter g ∗ given in Eq. (106) with r B = E 2 yz / (2 /epsilon1V ). For the squeezed shape, if we take the average over angles as the derivation of Eq. (132), the local non-linear parameter reads where the value 1 / 9 is the total spatial average of the function in the last square bracket. Therefore, f local , average NL dose not vanish in this analysis. We note that f local , average NL in this case is smaller than that in the vector model by one order of magnitude, see Eq. (132). Thus the local non-Gaussianities averaged over all the directions can allow us to discriminate between the vector and the two-form models. On the other hand, in the strict squeezed limit characterized by k 3 → 0, θ k 2 , k 3 → π/ 2, and θ k 3 , k 1 → π/ 2, the non-linear estimator (137) vanishes, f local NL = 0 [28]. The non-linear parameters in the equilateral and the enfolded limits were already evaluated in Ref. [28] by considering a configuration with k 1 = k 2 and β = π -θ k 1 , k 2 , γ = θ k 1 , x . They are given, respectively, by The equilateral non-linear estimator is independent of γ , but the enfolded one depends on γ . Unlike the vector model, f enfolded NL has a maximum at γ = 0 or π . Both f equil NL and f enfolded NL are about half times smaller than those in the vector model. In general, for the same values of | g ∗ | , the two-form model is easier to satisfy observational bounds of non-Gaussianities relative to the vector model.", "pages": [ 17, 18, 19, 20 ] }, { "title": "VI. CONCLUSIONS", "content": "We have studied differences between two anisotropic inflationary scenarios paying particular attention to their observational signatures. In the presence of a vector or a two-form field coupled to the inflaton, there exist background solutions along which the anisotropic hairs survive during inflation. In the vector model the anisotropic shear Σ divided by the Hubble rate H is given by Eq. (18) for the coupling (12), while in the two-form model it is given by Eq. (33) for the coupling (30). The opposite signs of Σ /H between the two models reflect the fact that the types of anisotropies are different (either oblate or prolate). In the presence of anisotropic interactions, we have computed the power spectra of scalar/tensor perturbations and the resulting spectral indices convenient to confront with the CMB observations. The different types of anisotropies affect the sign of g ∗ appearing in the scalar power spectrum defined by Eq. (1), i.e., g ∗ < 0 in the vector model and g ∗ > 0 in the two-form model. The anisotropic contributions to the isotropic tensor power spectrum are suppressed relative to those to the isotropic scalar one. In particular, in the two-form model, we showed that anisotropic interactions do not give rise to any corrections to the isotropic tensor power spectrum. We have computed the two-point cross correlations between curvature perturbations and gravitational waves. While the cross correlation remains in the vector model, we find that it vanishes in the two-form model. The latter property may reflect the fact that, unlike the vector field, the two-form field can be mapped to the form of a scalar field. The different signatures of the two anisotropic models will be useful to discriminate between those models in future observations of the TB power spectrum. In the light of the recent Planck data, we have placed observational constraints on several different inflaton potentials from the information of the scalar spectral index n s and the tensor-to-scalar ratio r . In both the vector and two-form models anisotropic interactions lead to the enhancement of the scalar power spectrum on larger scales, by which both n s and r decrease for any inflaton potentials. In the vector model, we found that the potentials V ( φ ) = λφ and V ( φ ) = (3 λ/ 2) φ 2 / 3 show the better compatibility with the data for larger | g ∗ | and that the potential of natural inflation is outside the 68 % CL region constrained by the Planck+WP+BAO+high/lscript data for | g ∗ | /greaterorsimilar 0 . 5. In the two-form model, the level of the decreases of n s and r is less significant relative to the vector model for the same values of | g ∗ | . The potential (120), which originates from the Starobinsky's f ( R ) model, is well inside the 68 % CL region even in the presence of anisotropic interactions with | g ∗ | < 0 . 5. We also compared the three-point correlation functions of curvature perturbations between the two anisotropic inflationary scenarios. If the strict squeezed limit characterized by k 3 → 0, θ k 1 , k 3 → π/ 2, and θ k 2 , k 3 → π/ 2 is taken, the local non-linear estimators f local NL vanish in both the vector and the two-form models. However, for the squeezed shape where the angles θ k 1 , k 3 and θ k 2 , k 3 are not necessarily close to π/ 2, the non-linear estimator averaged over all the directions is given by Eq. (132) in the vector model and by Eq. (138) in the two-form model. The former is larger than the latter by one order of magnitude for the same order of the anisotropic parameter g ∗ . We also evaluated the non-linear estimators in the equilateral and enfolded limits and found that f NL in the vector model is about twice larger than that in the two-form model for the same values of | g ∗ | . In general, the two-form model is easier to satisfy observational bounds of non-Gaussianities relative to the vector model. We have thus shown that the two anisotropic inflationary scenarios can be distinguished from each other by evaluating several CMB observables. In particular, the precise measurements of g ∗ as well as the TB correlation will clarify which anisotropic model is favored over the other.", "pages": [ 20 ] }, { "title": "Acknowledgments", "content": "This work is supported by the Grant-in-Aid for Scientific Research Fund of the Ministry of Education, Science and Culture of Japan (Nos. 23 · 6781, 25400251, and 24540286), the Grant-in-Aid for Scientific Research on Innovative Area (No. 21111006).", "pages": [ 21 ] } ]

[ { "title": "ABSTRACT", "content": "Aims. We search for the long-term variability of the γ -ray sky in the energy range E > 1 GeV with 168 weeks of Fermi-LAT data. Methods. Weperform a full sky blind search for regions with variable flux looking for deviations from uniformity. We bin the sky into 12288 bins using Healpix package and use Kolmogorov-Smirnov test to compare weekly photon counts in each bin with a constant flux hypothesis. The weekly exposure of Fermi-LAT for each bin is calculated with the Fermi-LAT tools. We consider flux variations in the bin significant if statistical probability of uniformity is less than 4 × 10 -6 , which corresponds to 0.05 false detections in the whole set. Results. We identified 117 variable sources, variability of 27 of which has not been reported before. Among the sources with previously unidentified variability there are 25 AGNs belonging to blazar class (11 BL Lacs and 14 FSRQs), one AGN of uncertain type and one pulsar PSR J0633 + 1746 (Geminga). The observed long term flux variability of Geminga has a statistical significance of 5.1 σ . Key words. Methods: statistical-BL Lacertae objects: general-quasars: general-Gamma rays: galaxies", "pages": [ 1 ] }, { "title": "Variable gamma-ray sky at 1 GeV", "content": "M.S. Pshirkov /star 1 , 3 and G.I. Rubtsov /star/star 2", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Time domain astronomy at di ff erent wavelengths from radio to very high-energy gamma-rays is developing very rapidly nowadays. In the high energy (HE) range ( ≥ 100 MeV) great progress was achieved with advent of the gamma-ray telescope FermiLAT (Atwood et al. 2009). Its high sensitivity and almost uniform sky coverage allow one to study variability of large number of sources at these energies on time scales from seconds to years. Sources that demonstrate the most variable behaviour are active galactic nuclei (AGN), primarily of the blazar type (Abdo et al. 2010a). It is well known that these sources exhibit variability on di ff erent time scales and at di ff erent wavelengths (Carini et al. 1991; Ulrich et al. 1997; Welsh et al. 2011; Rani et al. 2009; Bonning et al. 2009; Soldi et al. 2008; Ghisellini & Tavecchio 2008; Raiteri et al. 2005; Ciprini et al. 2003; Urry et al. 1993). Studies of time variability of these sources are very important for better understanding of AGN engines; they are also essential for assessing the quality of spectral energy distributions obtained from multiwavelength observations made at di ff erent epochs. In this paper we perform a full sky blind search for variable sources. We bin the sky into equal area pixels and search for deviations of photon number counts from the uniformity in time.", "pages": [ 1 ] }, { "title": "2. Data and method", "content": "The LAT Pass 7 weekly all-sky data publicly available at Fermi mission website 1 were used in this work. The analysis covers the time period of 168 weeks from August 04, 2008 to October 18, 2011, corresponding to mission elapsed time (MET) from 239557417 s to 340622181 s. We use the 'Pass 7 Source' event class photons with E > 1 GeV and impose an Earth relative zenith angle cut of 100 · and rocking angle cut of 52 · . We bin the data week by week using HEALPIX package (G'orski et al. 2005) into a map of resolution N side = 32 in galactic coordinates with 'RING' pixel ordering. Total number of pixels is equal to 12288 and the area of each pixel is 3.6 sq. deg, chosen according to the size of Fermi-LAT point-spread function (PSF) above 1 GeV which is approximately 1 · . We estimate integral weekly exposure for each pixel using the standard FermiLAT tools gtltcube and gtexpcube (ScienceTools-v9r23p1-fssc20110726). For each pixel we count the number of photons in each of 168 weeks and consider corresponding values of weekly exposure. Typically there are 2-10 photons in a pixel per week except bins with the brightest sources. Cumulative distribution functions (CDFs) P ( t ) , E ( t ) for both photon counts and exposure are constructed. In the absence of variability, the photon counts would represent a random process with CDF proportional to E ( t ) and thus P ( t ) would follow E ( t ) with deviations caused by a finite number of observed photons. Otherwise P ( t ) would not be statistically compatible with E ( t ). A Kolmogorov-Smirnov (KS) test is a natural and straightforward way to examine statistical compatibility of the observed photon counts with the distribution given by CDF E ( t ). The probability that both sets represent the same distribution could be estimated from the maximal value of distance between the functions P ( t ) and E ( t ). An example CDFs for one of the pixels is shown in Figure 1 and the corresponding flux is shown in Figure 2. Implemented KS test is most sensitive to the variability at long scales (longer than a week), while transient bursts and flares at shorter time scales may be missed if they are not overwhelmingly strong. On the other hand, our method is sensitive to gradual moderate changes in photon fluxes without any prominent bursts, which could be missed by burst searching techniques. KSprobability is calculated for each pixel - we are interested in pixels with probabilities smaller than the threshold value P 0 = 4 × 10 -6 . This threshold value is set to allow for penalty coming from the large number ( N pix = 12288) of trials: the detection criterion is chosen in such a way that the entire search would give a single false detection with the probability of P 0 × N pix ∼ 0 . 05. A map of the probabilities is presented in Figure 4.", "pages": [ 1, 2 ] }, { "title": "3. Results", "content": "The total number of bins with probabilities smaller than the threshold value P 0 is 151. The source identification for each of these bins is performed as follows. We consider the photons that arrived during several weeks at the epoch of the maximal flux. The center of mass of the spatial distribution of these photons is used as an initial estimate of the source location. We search for sources from 2FGL (The Fermi-LAT Collaboration 2011) catalog in the circle with a radius equal to the error of center of mass estimation which is usually about 10'-15' for ∼ 20 photons). For 34 pixels no source is found with the initial estimate: in 31 of them the variability time pattern and photon spatial distribution leads us to identification of the source located in the neighbouring bin and already identified there. In three cases (pixels 2877 and 2749, PMN J1532-1315, and pixel 2299, SBS 0846 + 513) we have found no counterpart in 2FGL catalog and use Simbad astronomical database. Gamma-ray flares from the latter two sources (PMN J1532-1315 and SBS 0846 + 513) have been recently reported by Fermi-LAT (Gasparrini & Cutini 2011; Donato & Perkins 2011), but the activity period is outside of the time region of 2FGL catalog. In order to avoid false identifications we performed an additional check in four cases when two di ff erent sources are residing in adjacent pixels: significant di ff erence in the observed luminosity curves confirms that these detections are real. The results are presented in Table 1. The total number of identified sources is 117; variability of 55 of them was reported in (Abdo et al. 2010a; Tanaka et al. 2011; Schinzel et al. 2011), 35 additional detections were made in numerous Astronomers's telegrams (ATels, see Appendix) and on the Fermi-LAT blog 2 . That leaves us with 27 sources for which the variability has not been previously detected (see Table 2 and Figures 5-9). We have explicitly checked that flux from these sources is not contaminated with the contribution from the Sun. For convenience we assign a variability type to the source: a gradual change in the photon flux is referred to as 'rate', if the whole variability is dominated by one or several flares we call it 'flare' and if these flares are observed by more or less prolonged time span (typically, more than 20 weeks) we designate it as 'activity'. This morphological distinction is not totally unambiguous: several bright consequent flares could be defined as 'activity'. On the other hand, gradual changes could take place on a time scales of several tens of weeks, thus fitting the 'activity' type as well. The BL Lacs and the flat-spectrum radio quasars (FSRQs) are represented almost equally in the set of previously unidentified sources: there are 11 BL Lacs, 14 FSRQs, and one AGN of uncertain type (PKS 0644-671). Also there is one pulsar (Geminga) in the list (see Table 2 and Figures 5-9). Both BL Lacs and FSRQs demonstrate two types of variability: gradual change of photon flux (10 out of 25) and flares or increased activity (15 out of 25). Variability of several sources was observed before in other energy ranges: a flare in the near infrared region was observed for B2 1732 + 38A (ATel #3504 (Carrasco et al. 2011), 1 July 2011), VHE flares of 1ES0806 + 524 were observed by MAGIC (ATel #3192 (Mariotti 2011), 24 February 2011), and EGRET observed a very bright flare of PKS 2255282 in December 1997 (Macomb et al. 1999). Pulsed γ -rays from the Geminga pulsar were observed with 1 year of FermiLAT data (Abdo et al. 2010c), while the source were considered non-variable. In this paper the long term flux change of Geminga pulsar is detected with KS probability of 2 . 3 × 10 -7 . In present study Geminga is the only pulsar demonstrating variability above our threshold. For comparison the KS probabilities for bins containing Vela, PSR J1709-4429, and Crab pulsars are 25%, 78% and 1.2% respectively. In Crab case it was shown that the observed variability is caused by processes in the Crab nebula rather than the pulsar itself (Buehler et al. 2011). Fermi-LAT collaboration also presented results of observations of three other gamma-ray pulsars (J1836 + 5925, PSR J2021 + 4026, and PSR J0007 + 7303) where no flux variability was observed (Abdo et al. 2010b; Saz Parkinson et al. 2010; Abdo et al. 2012). That makes the case of Geminga even more intriguing. We note that while 2FGL catalog contains 577 unidentified sources (out of 1873), 153 of which have flux higher than 2 × 10 -9 photons cm -2 s -1 , none of them show variability above our threshold. It is also worth noting that sources not included in the Table 2 because of being reported either in ATels or on the Fermi blog could have the variability patterns that di ff er considerably from the reported one. As an example, the flare from the source MG2 J130304 + 2434 that took place on 3 July 2009 (week no. 56) was reported in ATel #2110 (Hays & Marelli 2009). On the other hand, Figure 3 shows that the flare occured during the high state of the source, with its flux slowly increasing from the start of the Fermi observations till approximately the 80-th week when it began to decrease.", "pages": [ 2, 3 ] }, { "title": "4. Conclusions", "content": "A method for variable sources detection is proposed that uses the KS statistical test. The method is implemented for a full sky blind search for regions with variable flux at energies above 1 GeVusing Fermi-LAT 168 weeks data. The search leads to identification of 117 variable sources, the variability of 27 of which has not been reported before. Among the sources with previously unidentified variability there are 25 AGNs belonging to blazar class (11 BL Lacs and 14 FSRQs), one AGN of uncertain type (PKS 0644-671), and one pulsar PSR J0633 + 1746 (Geminga). The observed long term flux variability of Geminga pulsar has a statistical significance of 5.1 σ .", "pages": [ 3 ] }, { "title": "Appendix", "content": "The following ATels are cited in the text: ATel #1933 (Corbel & Reyes 2009), ATel #2048 (Ciprini 2009c), ATel #2049 (Ciprini 2009a), ATel #2104 (Longo et al. 2009), ATel #2110 (Hays & Marelli 2009), ATel #2136 (Ciprini 2009b), ATel #2243 (Tanaka et al. 2009), ATel #2316 (Hays & Escande 2009), ATel #2402 (Sokolovsky et al. 2010), ATel #2413 (Hill & Vandenbroucke 2010), ATel #2539 (Wallace 2010), ATel #2583 (Donato 2010), ATel #2669 (Cutini 2010), ATel #2783 (D'Ammando 2010b), ATel #2829 (Schinzel 2010), ATel #2860 (D'Ammando 2010a), ATel #2907 (Cannon & D'Ammando 2010), ATel #2943 (Ciprini 2010), ATel #3002 (D'Ammando & Vandenbroucke2010), ATel #3026 (Allafort & D'Ammando 2010), ATel #3171 (Buson & Bastieri 2011), ATel #3192 (Mariotti 2011), ATel #3207 (Allafort 2011), ATel #3452 (Donato & Perkins 2011), ATel #3445 (Gasparrini 2011), ATel #3504 (Carrasco et al. 2011), ATel #3579 (Gasparrini & Cutini 2011), ATel #3670 (Schinzel & Ciprini 2011), ATel #3793 (Ojha et al. 2011). Acknowledgements. We are indebted to P. Tinyakov for numerous helpful discussions at all stages of this work. We thank M. Gustafsson, B. Stern, I. Tkachev and S. Troitsky for useful comments and suggestions. The work was supported in part by the RFBR grants 10-02-01406a, 11-02-01528a, 12-02-91323-SIGa (GR), by the grants of the President of the Russian Federation NS-5525.2010.2 (GR), MK-1632.2011.2 (GR), MK-1582.2010.2 (MP). The work of M.P is supported in part by the IISN project No. 4.4509.10 and the Belgian Science Policy (IAP VI-11). GR is grateful for the hospitality of ULB Service de Physique Theorique where this study was initiated. The analysis is based on data and software provided by the Fermi Science Support Center (FSSC). The numerical part of the work is performed at the cluster of the Theoretical Division of INR RAS. This research has made use of NASA's Astrophysics Data System and SIMBAD database, operated at CDS, Strasbourg, France.", "pages": [ 3 ] }, { "title": "References", "content": "Abdo, A. A., Ackermann, M., Ajello, M., et al. 2010a, ApJ, 722, 520 Abdo, A. A., Ackermann, M., Ajello, M., et al. 2010b, ApJ, 712, 1209 Abdo, A. A., Ackermann, M., Ajello, M., et al. 2010c, ApJ, 720, 272 Abdo, A. A., Wood, K. S., DeCesar, M. E., et al. 2012, ApJ, 744, 146 Allafort, A. 2011, The Astronomer's Telegram, 3207, 1 Allafort, A. & D'Ammando, F. 2010, The Astronomer's Telegram, 3026, 1 Atwood, W. B., Abdo, A. A., Ackermann, M., et al. 2009, ApJ, 697, 1071 Bonning, E. W., Bailyn, C., Urry, C. M., et al. 2009, ApJ, 697, L81 Buehler, R., Scargle, J. D., Blandford, R. D., et al. 2011, ArXiv e-prints Buson, S. & Bastieri, D. 2011, The Astronomer's Telegram, 3171, 1 Cannon, A. & D'Ammando, F. 2010, The Astronomer's Telegram, 2907, 1 Carini, M. T., Miller, H. R., Noble, J. C., & Sadun, A. C. 1991, AJ, 101, 1196 Carrasco, L., Carraminana, A., Escobedo, G., et al. 2011, The Astronomer's Telegram, 3504, 1 Ciprini, S. 2009a, The Astronomer's Telegram, 2049, 1 Ciprini, S. 2009b, The Astronomer's Telegram, 2136, 1 Ciprini, S. 2009c, The Astronomer's Telegram, 2048, 1 Cutini, S. 2010, The Astronomer's Telegram, 2669, 1 D'Ammando, F. 2010a, The Astronomer's Telegram, 2860, 1 Donato, D. 2010, The Astronomer's Telegram, 2583, 1 Donato, D. & Perkins, J. S. 2011, The Astronomer's Telegram, 3452, 1 Gasparrini, D. 2011, The Astronomer's Telegram, 3445, 1 Gasparrini, D. & Cutini, S. 2011, The Astronomer's Telegram, 3579, 1 Ghisellini, G. & Tavecchio, F. 2008, MNRAS, 386, L28 G'orski, K. M., Hivon, E., Banday, A. J., et al. 2005, ApJ, 622, 759 Hays, E. & Escande, L. 2009, The Astronomer's Telegram, 2316, 1 Hays, E. & Marelli, M. 2009, The Astronomer's Telegram, 2110, 1 Hill, A. B. & Vandenbroucke, J. 2010, The Astronomer's Telegram, 2413, 1 Longo, F., Iafrate, G., Hays, E., & Marelli, M. 2009, The Astronomer's Telegram, 2104, 1 Macomb, D. J., Gehrels, N., & Shrader, C. R. 1999, ApJ, 513, 652 Mariotti, M. 2011, The Astronomer's Telegram, 3192, 1 Ojha, R., Dutka, M., & Torresi, E. 2011, The Astronomer's Telegram, 3793, 1 Raiteri, C. M., Villata, M., Ibrahimov, M. A., et al. 2005, A&A, 438, 39 Rani, B., Wiita, P. J., & Gupta, A. C. 2009, ApJ, 696, 2170 Soldi, S., Turler, M., Paltani, S., et al. 2008, A&A, 486, 411 Tanaka, Y. T., Stawarz, Ł., Thompson, D. J., et al. 2011, ApJ, 733, 19 The Fermi-LAT Collaboration. 2011, ArXiv e-prints Ulrich, M.-H., Maraschi, L., & Urry, C. M. 1997, ARA&A, 35, 445 Urry, C. M., Maraschi, L., Edelson, R., et al. 1993, ApJ, 411, 614 Wallace, E. 2010, The Astronomer's Telegram, 2539, 1 Welsh, B. Y., Wheatley, J. M., & Neil, J. D. 2011, A&A, 527, A15 Table 1. List of pixels demonstrating variability exceeding threshold value ( P < 4 × 10 -6 ). l , b are the galactic coordinates of the center of the pixel, N phot is the total number of photons observed in the pixel, Φ -8 is the average flux from the pixel Φ -8 ≡ Φ / 10 -8 photons cm -2 s -1 : the total number of photons divided by the total exposure, P is the KS probability, the designations of the identified source in the literature and in the 2FGL catalog are in the 8th and 9th columns. Previous references to the variability of the source are presented in the last column: P stands for paper (Abdo et al. 2010a), P2 for (Tanaka et al. 2011), P3 for (Schinzel et al. 2011), ATNNNN for ATel #NNNN, and BNNN indicates that the outburst from the source was mentioned on the Fermi blog in the NNNth weekly report. ATNNNN with prefix VHE or IR indicate that flare was observed (and reported in corresponding ATel) in some other energy range: very high energy (larger than 100 GeV) or in the infrared. All the references for the ATels are listed in the Appendix.", "pages": [ 3, 6 ] } ]

[ { "title": "RN/CFT Correspondence From Thermodynamics", "content": "Bin Chen 1 , 2 ∗ and Jia-ju Zhang 1 † 1 Department of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, P.R. China 2 Center for High Energy Physics, Peking University, Beijing 100871, P.R. China", "pages": [ 1 ] }, { "title": "Abstract", "content": "Recent studies suggest that in the Kerr/CFT correspondence, much universal information of the dual CFT, including the central charges and the temperatures, is fully encoded in the thermodynamics of the outer and inner horizons of the Kerr(-Newman) black holes. In this paper, we study holographic descriptions of Reissner-Nordstrom (RN) black holes in arbitrary dimensions by using the thermodynamics method.We refine the thermodynamics method proposed in [19] by imposing the 'quantization' condition so that we can fix the ambiguity in determining the central charges of the dual CFT of RN black holes. Using the refined thermodynamics method, we find the holographic CFT duals for the RN black holes, and confirm these pictures by using conventional analysis of asymptotic symmetry group and the hidden conformal symmetry in the low-frequency scattering. In particular, we revisit the four-dimensional dyonic RN black hole and find a novel magnetic picture, besides the known electric CFT dual picture. We show how to generate a class of dual dyonic pictures by SL (2 , Z ) transformations.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The Kerr/CFT correspondence [1] asserts that there is a two-dimensional (2D) conformal field theory (CFT) to describe the Kerr black hole holographically. In setting up the Kerr/CFT correspondence, the conventional way is to obtain the central charges of dual CFT from the asymptotic symmetry group (ASG) of near-horizon geometry of extreme black hole in either Barnich-Brandt-Compere (BBC) formalism [1,2] or equivalently the stretched horizon formalism [3-5], and read the dual temperatures from the Frolov-Thorne vacuum [1] or the hidden conformal symmetry in the low-frequency scattering [6]. Kerr/CFT has many extensions and generalizations, and the reader can find details and more complete references in the nice reviews [7,8]. One remarkable feature in the holographic description of Kerr and multi-charged black holes is that the central charges of the dual CFT are written in terms of 'quantized' charges, angular momenta and U (1) charges, independent of the mass of the black holes. This feature could be related to the fact that the area product of the horizons S + S -of these black holes are also mass-independent. Actually, it was shown [9-12] that for general five-dimensional (5D) and four-dimensional (4D) multi-charged rotating black holes, the outer and inner horizon entropies could be written respectively as where N L , N R could be interpreted as the levels of the left- and right-moving sectors in a two-dimensional CFT. Therefore the entropy product should be quantized, as ( N L -N R ) must be integer due to the level matching condition in CFT. As a result, the entropy product S + S -must be mass-independent [13,14]. For other recent relevant studies on this issue, see [15-18]. Strictly speaking, the mass-independence condition breaks down in some cases, including various warped black holes in three-dimensional (3D) topologically massive gravity, but the relation (1.1) is always sound for the black holes with holographic descriptions [17]. From (1.1), one may find microscopical entropy of dual CFT. This suggests that the physics of the inner horizon of the black hole should be taken seriously. Very recently, the Kerr/CFT correspondence was investigated from the point of view of thermodynamics of both outer and inner horizons [19]. Firstly, it was proved that the first law of thermodynamics of the outer horizon always indicates that of the inner horizon, under reasonable assumption. Secondly, the mass-independence of the entropy product S + S -is equivalent to the condition T + S + = T -S -, which is much easier to check. More interestingly, it was found that the thermodynamics in the left- and right-moving sectors of the dual CFT could be obtained from the linear composition of the thermodynamics of the outer and inner horizons [13, 19]. This thermodynamics method provides us a simple way to read the information of the dual CFT. It has been checked in many well-established black hole/CFT correspondences, including 3D BTZ, 4D Kerr-Newman and 5D Myers-Perry black holes [5,20-27], and applied to the study of holographic descriptions of black rings [28]. It turns out to be quite effective, allowing us to read the central charges and the temperatures in all possible pictures. One of interesting generalizations of Kerr/CFT is the so-called RN/CFT correspondence [22,29,30], which states that there is a holographic 2D CFT description for the four-dimensional Reissner-Nordstrom (RN) black hole. The central charges of dual CFT have been computed either from a reduced two-dimensional effective gravity action or from a uplifted 5D metric point of view [30]. It is puzzling to see that the central charge could only be determined up to a scale factor c = 6 Q 3 /l , with l being an undetermined factor. Correspondingly, there seems to be an one-parameter class of CFTs dual to 4D RN black hole. Such an ambiguity looks strange if one apply the same techniques to the well-known multi-charged black holes in string theory, whose CFT duals have quantized central charges proportional to the product of the numbers of different branes. We try to solve this puzzle in this paper. The key point in our treatment is to impose the 'quantization' condition on the thermodynamics method, which allows us to get rid of the ambiguity. We find that this condition is actually in accord with the quantization condition on the angular momentum of the higher dimensional uplifted configuration. Another interesting issue in the RN/CFT correspondence is the holographic duals for dyonic RN black holes. It has been studied using the hidden conformal symmetry in [31]. In [31], the dual picture was obtained by using an electric-charged scalar to probe the geometry. This picture will be called as the electric ( E ) picture. As the dyonic RN black hole carries both electric and magnetic charges, it is interesting to inquire what one can get if using a magnetic charged probe. We will show that such an investigation gives us a magnetic dual picture of the dyonic RN black hole. Actually, the magnetic ( M ) picture could be easier to figure out from the thermodynamics method, as we will show in section 4. As shown in the case of KerrNewman black hole, when the black hole has two U (1) symmetries, there will be a CFT dual picture for every U (1) charge [22-24, 26, 31], and a whole class of novel CFT pictures could be generated by SL (2 , Z ) transformations acting on two elementary dual pictures [5,27]. We find that the similar phenomenon happens for 4D dyonic RN black holes. In this case, there is an electromagnetic duality group SL (2 , Z ) acting on the elementary electric and magnetic pictures. In this paper we investigate RN/CFT correspondence mainly using the thermodynamics method, and verify our results using conventional methods if possible. In Section 2 we consider RN black holes in all dimensions d ≥ 4, and find that T + S + = T -S -are always satisfied. We find their CFT duals using the thermodynamics method, and verify the pictures by re-deriving the results via ASG analysis and the hidden conformal symmetry. In Section 3, we consider RN-AdS black holes in all dimensions, and find that T + S + = T -S -breaks down, which suggests that there are no CFT duals for such black holes. In Section 4, we consider the four-dimensional dyonic RN black hole, and find a novel magnetic CFT dual. The picture is confirmed by the study of hidden conformal symmetry in low frequency scattering of various kinds of probe scalar and also ASG analysis of a 6D uplifted spacetime. In Section 5, we end with conclusion and discussion.", "pages": [ 2, 3, 4 ] }, { "title": "2 RN/CFT in arbitrary dimensions", "content": "In this section we consider the RN/CFT correspondence in spacetime of dimension d ≥ 4. We set up the general RN/CFT in three different ways, i.e. the thermodynamics method, ASG analysis, and the hidden conformal symmetry.", "pages": [ 4 ] }, { "title": "2.1 Black hole solutions", "content": "The charged spherically symmetric black hole solutions in d dimensions were found in [32]. We have c = /planckover2pi1 = 1 for convenience, but we set G d = /lscript d -2 p with /lscript p being the Planck length in d -dimensional spacetime. We use the convention here because the dimensional analysis plays a subtle role in our calculation. We consider the Einstein-Maxwell theory with the action where we have normalized the electromagnetic field so that Ω d -2 is the volume of a unit d -2 sphere S d -2 Note that in four dimensions we are using the Gauss convention with this action. The Einstein equation is now The d -dimensional RN black hole has the metric and the electromagnetic potential with The black hole has outer and inner horizons locating at r ± , with The mass of the black hole, the Hawking temperatures and the entropies of the outer and inner horizons are respectively The electric charge of the black hole is Q , which is just and the electric potentials at the outer and inner horizons are One can see that in d dimensions the electric charge has the dimension of length power d -4 2 , i.e. [ Q ] = L d -4 2 . It is dimensionless in four dimensions but is not so in higher dimensions. One can verify the first laws of thermodynamics of the outer and inner horizons which are equivalent to Smarr formulas of the two horizons", "pages": [ 4, 5 ] }, { "title": "2.2 Refined thermodynamics method", "content": "Let us apply the thermodynamics method proposed in [19]. First of all, we check that T + S + = T -S -, which means that the entropy product S + S -is independent of the mass M and there is CFT dual for the black hole with equal right- and left-moving central charges. We define the new quantities [11-13] such that the first laws and the Smarr formulas can be separated into the right- and left-moving sectors Explicitly, these quantities are We have to mention that, the Smarr formulas play no fundamental rule in our calculations and they are just convenient ways to verify the first laws. As what have been done in [19,28] for the BTZ black hole, 4D Kerr-Newman black hole, 5D Meyers-Perry black hole, doubly rotating and dipole black rings, one could rewrite the first laws of black hole as then one identify T J R,L as the J picture CFT temperatures. Since the angular momentum J and the entropies S R,L are dimensionless, the CFT temperatures T J R,L are dimensionless as required. This treatment works remarkably well to get the J pictures of various black holes with rotations. For the RN black holes, we may use the same strategy. From (2.13) we can get But now the left hand side of the equation is not dimensionless, and so a factor must be multiplied to both sides of the equation. In [19], we multiplied an arbitrary factor to read the Q picture of the Kerr-Newman black hole and found complete agreement with the results in the literature [30]. However, such an ambiguity makes us uncomfortable, especially considering the fact that for multi-charged black holes in string theory this treatment may give us bizarre results. Actually without dimension uplifting or reducing, the only natural scale is the Planck length 1 //lscript d -4 2 p , but still there is an ambiguity in introducing a numerical dimensionless factor. Note that in setting up the J picture, we should work with Eq. (2.16). The underlying reason is simple, as the angular momentum is quantized. In fact, Eq. (2.16) tells us how the black hole responds to the perturbation. As the angular momentum is quantized, the minimal variation due to the perturbation gives exactly Eq. (2.16). Similarly, for the equation (2.17), the left hand side should be quantized. It is more suggestive to recast the first laws into the form with N being an integer-valued quantized charge, then the temperatures of the N picture CFT dual are T N R,L without any ambiguity. For the RN black holes, we scale (2.17) as where λ is a numerical factor. The factor λ makes with e being the unit charge which is determined by the Maxwell theory. Then we can identify the size of the space where the two-dimensional CFT reside as [11-13] and the temperatures of the CFT as We suppose that the right- and left-moving entropies could be expressed in the form of the Cardy formula if there really exists a CFT dual then we obtain the right- and left-moving central charges For example, in four-dimensions we at last get with the numerical factor λ = 1 /e . Therefore, we resolve the puzzle on the undetermined scale factor in RN/CFT [30]. If the Maxwell theory is that in quantum electrodynamics (QED), then the unit charge e is related to the fine structure constant as Since the black hole charge is also quantized, we have the black hole Q = Ne with N being a possibly very large integer. Then the central charges become The appearance of the fine structure constant in the central charge is actually in accord with the discussion in [16]. Furthermore the first laws (2.13) could be written in a more suggestive way Under some perturbations dM = ω , dQ = k e e , with e being the unit charge (2.49) and so k e being an integer, we identify with ω Q R,L , q Q R,L , µ Q R,L being the frequencies, the charges, and the chemical potentials of the perturbation around the thermodynamical equilibrium of finite-temperature CFT. Explicit calculations show that These results will be compared with those obtained from the hidden conformal symmetry.", "pages": [ 6, 7, 8, 9 ] }, { "title": "2.3 ASG analysis", "content": "To do ASG analysis we uplift the d -dimensional RN black hole to ( d +1)-dimensional Einstein gravity The metric becomes with ds 2 d and A being defined as (2.4), and χ ∼ χ +2 π and /lscript d +1 being the scale of the extra dimension. Again the natural scale of /lscript d +1 is the Planck length /lscript p up to some numerical constant /lscript d +1 = λ/lscript p . From Kaluza-Klein reduction, we have and thus we have the identification of two theories I d +1 = I d with For uplifted RN black hole, the areas of the horizons satisfy so that we always have the relationships where we have G d = /lscript d -2 p and denote A ± = ( A ± ) d . The uplifting does not change the entropies of the black holes. Note also that, the electric charge Q has been transformed to the angular momentum along the angle χ . The angular momentum and angular velocities of the outer and inner horizons could be identified as It is remarkable that the quantization of the angular momentum J χ indicates the relationship with e being the unit charge of the theory. In other words, the quantization condition imposed on the thermodynamics method in the last subsection is equivalent to the quantization of the angular momentum in the uplifted spacetime. The requirement of a quantized angular momentum which is crucial to pin down the extra factor in the central charge has been ignored in the literature. To do ASG analysis, we take the extremal limit of the metric (2 . 32). We expand the following quantities at the horizon r + = r -, Explicitly, we have and we may also define Remember that in the extremal case, we have the areas of the horizons (in d -dimensional spacetime) and the electric charge It was demonstrated in [4, 5] that for an extremal black hole, there is always a CFT dual, whose information could be read from ASG analysis, no matter in the BBC formalism [2], or in the stretched horizon formalism [3,4]. The left-moving central charge of the CFT is And from the Frolov-Thorne vacuum the left-moving temperature of the CFT is read out Comparing the results with (2.24), (2.22), we see that in the extremal limit c Q L = c χ L and T Q L = T χ L , the results are in perfect match. Especially, the results justify the factor multiplied in (2.19) and thus the prescription (2.18). This shows that the thermodynamics method is an effective way of getting the CFT dual of the black hole.", "pages": [ 9, 10, 11 ] }, { "title": "2.4 Hidden conformal symmetry", "content": "We investigate the scattering of a complex scalar off the RN black hole. We can consider either a charged scalar in the d -dimensional RN black hole background (2.4), or equivalently a neutral scalar in the uplifted ( d +1)-dimensional black hole background (2.32). In the former case, we consider a scalar of mass µ d and charge k e e , with e being the unit charge (2.49) and k e being an integer. The equation of motion for such scalar Φ is We define ρ = r d -3 , and expand Φ = e -iωt R ( ρ )Θ Λ , with Θ Λ being the eigenfunction of the Laplace operator of the unit d -2 sphere S d -2 , i.e. ( D i D i +Λ)Θ Λ = 0. Then we could get the radial euqation In the later case we consider a scalar of mass µ d +1 , with its equation of motion being We expand Φ = e -iωt + ik χ χ R ( ρ )Θ Λ , and then it can be shown that the equations (2.45) and (2.47) are identical as long as we have (2.56) and It is crucial that the integer k e is identified with the integer k χ , whose quantization is due to the periodic nature of χ . The importance of this consistent identification has been ignored in the literature. Under some suitable approximations in the low-frequency limit, from the radial equation (2.46), one can arrive at with K being some constant. In the study of hidden conformal symmetry for the non-extreme black hole, the conformal coordinates could be defined as 1 [6] Here t is the time, ρ is the radial coordinate which is not necessarily r but can be a monotonically increasing function of r , and ψ ∼ ψ +2 π may be an angle of the spacetime, or an internal angle, or a supposition of some angles. Also we use the letter C to denote CFT , T C R,L are the right- and left-moving central charges of the CFT, and n C R,L have no immediate physical meaning now. With the conformal coordinates the vector fields could be locally defined as and These vector fields obey the SL (2 , R ) Lie algebra and similarly for ( ˜ H 0 , ˜ H ± 1 ). The quadratic Casimir is In terms of ( t, ρ, ψ ) coordinates, the Casimir becomes With the scalar field being expanded as Φ = e -iωt + ikψ R ( ρ ), the equation H 2 Φ = K Φ gives us the radial equation of motion where K is a constant. Note that k is the quantum number along the angle ψ and must be integer-valued. Identifying the above two radial equations (2.49), (2.56), we find The temperatures obtained here are in perfect accord with the results got from the thermodynamics method (2.22). Here, the remarkable point is that the integer-valued quantum number k is consistently identified with the integer-valued charge k e . This point has not been taken seriously in the former study of RN/CFT. It helps us to pin down the ambiguous factor in the central charge of RN/CFT. Again, the result justifies the prescription (2.18). The above radial equation can be solved in terms of hypergeometric functions and gives the retarded Green's function and the absorption cross section that agree with the predictions of the CFT side [24,33], ∣ ∣ ∣ ∣ The conformal weights, the frequencies, the charges, and the chemical potentials of the perturbations in the CFT side could be identified as These quantities got from the hidden conformal symmetry are in perfect match with the ones got in the thermodynamics methods (2.30). This agreement is remarkable. On one hand, the thermodynamics of the black hole tells us how it respond to the perturbation. On the other hand, the scatting amplitude of the probe scalar gives us the information of the black hole. It is amazing to see that the thermodynamics method gives us almost the same information on the dual CFT as the probe scalar: the same frequencies, the charges and the chemical potentials.", "pages": [ 11, 12, 13, 14 ] }, { "title": "3 RN-AdS black holes in arbitrary dimensions", "content": "The general RN-AdS black holes in arbitrary dimensions were found in [34], and the solutions are similar to their asymptotically flat cousins. The theory has the action with the Newton constant G d = /lscript d -2 d the cosmological constant The equation of motion becomes The d -dimensional RN-AdS black hole has the metric and the electromagnetic potential with From N 2 ( r ± ) = 0, we can get the location of the outer and inner horizons r ± , and represent the parameters of the black hole m,q in terms of r ± . The mass of the black hole, the Hawking temperatures and the entropies of the outer and inner horizons are respectively The electric charge of the black hole is Q , and the electric potentials at the outer and inner horizons are One can verify the first laws of the outer and inner horizons Note that there are no trivial Smarr formulas here. For example, in four dimensions, we have The first laws can be verified easily, and we can see the symmetry of the quantities under the exchange of r ± proposed in [19]. But now we have which is not vanishing. This suggests that the entropy product S + S -is mass-dependent and there should be no CFT dual for the four-dimensional RN-AdS black hole 2 . For the RN-AdS black holes in higher dimensions, we have similar conclusion. In five dimensions, we get We check that for d = 6 ∼ 30, we always have T + S + -T -S -nonvanishing, and we believe that the result holds in all higher dimensions. As a result, we conclude that there seems to be no CFT dual for the RN-AdS black holes in all dimensions. Actually, one may naively using the thermodynamics method proposed in Section 2 to the RN-AdS case, as now the first laws at both horizons are still well-defined. But one would find that the central charges of left-moving and right-moving sectors are different. The result contradicts with our expectation since in Einstein gravity without diffeomorphism anomaly the central charges in both sectors of candidate CFT should be the same, leading to T + S + = T -S -[19].", "pages": [ 14, 15, 16 ] }, { "title": "4 Four-dimensional Dyonic RN black holes", "content": "Four-dimensional RN black hole is special compared to its higher dimensional cousins in the sense that it not only can carry electric charge but can also carry magnetic charge, namely in four dimensions there are electromagnetically charged, i.e. dyonic, RN black hole. We investigate the holographic descriptions of the dyonic black hole using the thermodynamics method and the hidden conformal symmetry. We find that there are two elementary CFT duals, namely the known electric ( E ) picture [31] and a novel magnetic ( M ) picture, from which the other dyonic pictures could be generated by SL (2 , Z ) transformations. Since the embedding of the dyonic RN black hole in higher dimensions is nontrivial, we cannot use the ASG formalism in a straightforward way. However, as we show, the dyonic black hole geometry could be understood as the solution of a theory with two U (1) fields, which allows us to uplift the solutions to six dimensions and analyze its ASG.", "pages": [ 17 ] }, { "title": "4.1 Dyonic RN black hole", "content": "The dyonic RN black hole is a solution of the action with ∗ being Hodge duality. Here to discuss the full SL (2 , Z ) symmetry, we have introduced the θ -term for the gauge field. The two real constants e, θ could be combined into a complex coupling parameter Again, we use the convention c = /planckover2pi1 = 1 and G 4 = /lscript 2 p , and for the electromagnetic part we have used the Gauss convention. The metric of the dyonic RN black hole is of the form Here M is the mass of the black hole, and with Q e,m being the electric and magnetic charges of the black hole respectively. The gauge field of the theory can be written as The upper minus sign applies to the sphere with the south pole deleted, say 0 ≤ θ < π , and the lower plus sign applies to the sphere with the north pole deleted, say 0 < θ ≤ π . We denote F = dA and ∗ F = d ˜ A , then we have the dual electromagnetic potential Due to the Witten effect [35], the electric and magnetic charges are respectively with N e,m being integers. Note that for two dyons with charges Q e,m and Q e ' ,m ' , there should be the Dirac-Zwanziger-Schwinger quantization condition The horizons locate at r ± = G 4 M ± √ G 2 4 M 2 -G 4 Q 2 , and the temperatures, the entropies, the electric and magnetic potentials of the outer and inner horizons are respectively There are the first laws at the outer and inner horizons which are equivalent to the Smarr formulas", "pages": [ 17, 18 ] }, { "title": "4.2 Thermodynamics method", "content": "According to the discussion in Sect 2, Q e,m are not good quantum numbers and we should use the integers N e,m that appear in (4.7) to apply the thermodynamics method. We rewrite the first laws (4.10) as with N e,m being the integers appear in (4.7) and With the quantities defined as those in (2.12), the first laws could be recast into with Setting dN m = 0 in (4.15), we could get then we may identify the scale factor and the temperatures of underlying CFT in the N e picture, or electronic ( E ) picture as From the Cardy formula we read the central charges We see that the central charges and the temperatures of CFT in the E picture agree with the ones found in [31] up to the overall factor we fix here. Moreover from the first laws (4.15), we may set dM = ω, dN e = k e , dN m = 0 and get with the frequencies, the charges, and the chemical potentials of the perturbations in the electric picture CFT being identified as Similarly we can set dN e = 0 in (4.15) and get the CFT dual in the N m picture, or magnetic ( M ) picture. The scalar factor and the temperatures of the M picture CFT could be identified as and the central charges are Furthermore, the frequencies, the charges, and the chemical potentials of the perturbation in the magnetic picture CFT, which is dual to the perturbation dM = ω, dN e = 0 , dN m = k m in the gravity side, could be identified as Note that we call the electronic and magnetic pictures as N e,m pictures or E,M pictures, and try to avoid the name Q e,m pictures, because Q e.m are not good quantum numbers and they are not integers, but N e,m are integers. Now we have two CFT pictures for 4D dyonic RN black hole, namely the E and M pictures. From our experience in 4D Kerr-Newman and 5D Myers-Perry black holes, once there are two dual pictures, there could be a class of dual pictures related by SL (2 , Z ) transformations with each other [5,27]. In these cases, the SL (2 , Z ) group could be understood as the T-duality group. In the case of dyonic RN black hole, we may try to generate more dual pictures from SL (2 , Z ) transformations as well. Using the description in [19], we redefine the charges N e,m and their intensive quantities Ω e,m ± as with and so We stress that the justification of the redefinitions is that the charges N e,m are integers. Then the first laws (4.12) become From the first laws, a pair of generic dyonic pictures with temperatures T e ' ,m ' R,L and central charges c e ' ,m ' R,L could be obtained as We show in [36] that this SL (2 , Z ) symmetry originates from the electromagnetic duality in the four-dimensional Einstein-Maxwell theory. There is a simple way to understand various pictures in the thermodynamics method. The thermodynamics of the black hole tells us how the black hole responds with respect to the perturbations of the infalling particle carrying the mass and the charges. The electric and magnetic charges of the perturbation q e,m could be expressed in terms of two integers k e,m as suggested in (4.7) If the perturbation carries only an electric charge, or more accurately k m = 0, then the thermodynamics laws tell us how the right- and left-moving sectors changes with the charges. This gives us the electric picture of the black hole (4.19). While if the perturbation carries only a magnetic charge, or more accurately k e = 0, the thermodynamics laws tell us the magnetic picture (4.23). If the perturbation carries both the electric and magnetic charges, the thermodynamics laws give us the dyonic pictures (4.29). On the other hand, if we consider a probe scattering off the RN black hole, its scattering amplitude encodes the information of the black hole as well. As we will show in next subsection, if the probe is electrically (magnetically) charged, then it tells us the electric (magnetic) picture. While if the probe is dyonic, then it gives us a dyonic picture.", "pages": [ 18, 19, 20, 21, 22 ] }, { "title": "4.3 Hidden conformal symmetry", "content": "The hidden conformal symmetry of an electrically charged scalar scattering off the dyonic RN black hole was considered in [31], from which the electric CFT dual was found. Here we consider the scattering of a more general dyonic charged scalar, and try to find other CFT duals. Suppose that there is a complex scalar with the electric and magnetic charges (4.30) and mass µ . Its equation of motion is just with A, ˜ A defined as in (4.5) and (4.6). Note that the coupling of the magnetic charge with the background could be determined from the electromagnetic duality. Now the scalar has to be expanded as Φ = e -iωt + i [ k φ ∓ ( Q m q e -Q e q m )] φ R ( r )Θ( θ ) with upper sign applying to the north pole and the lower sign applying to the south pole [37]. Note that the scalar picks a factor e i 2( Q m q e -Q e q m ) φ when passing from north to south, and Dirac-Zwanziger-Schwinger quantization condition (4.8) could make the factor single-valued. The equation of motion could be decomposed into the angular part and the radial part with Λ as the separation constant. Since the black hole is static, the quantum number k φ does not appear in the radial equation. Under the conditions of low frequency, small mass, small electric and magnetic charges, and near region approximations [31], the radial equation could be written as with K being some constant. Note that from (4.7) and (4.30), in the above radial equation there is To get the electric picture CFT, we set the magnetic charge of the probe scalar vanishing in (4.34), which means k m = 0 not simply q m = 0. Comparing the radial equations (4.34) with (2.56), we find The temperatures are exactly the same as the ones found in the thermodynamics method. Just like in Section 2, we could get the retarded Green's function and the absorption cross section that agree with the those of the CFT. The conformal weights, the frequencies, the charges, and the chemical potentials of the perturbation in the CFT could be identified as The results here got from the hidden conformal symmetry are in perfect agreement with the ones got in the thermodynamics methods (4.21). Therefore, we see that the scattering of the probe scalar with the electric charge gives exactly the same electric picture as in the thermodynamics method. On the other hand, we may consider the probe scalar carrying only a magnetic charge. In this case, setting the electric charge of the scalar vanishing k e = 0, not q e = 0, in (4.34), the radial equation could still be compared with the Casimir (2.56). In this way, we find the magnetic picture with The temperatures are just (4.22). The conformal weights, the frequencies, the charges, and the chemical potentials of the perturbation in the CFT could be got as well. They are in perfect agreement with the ones got in the thermodynamics methods in the previous subsection (4.24). Furthermore, we can consider the probe dyonic scalar with both the electric and magnetic charges, and find the dyonic picture suggested before. The procedure is that we make the redefinition in the radial equation (4.34) Then from the redefined radial equation we could set k m ' = 0 or k e ' = 0 and get the generic CFT dual pictures (4.29). In summary, we see that the CFT duals got from the hidden conformal symmetry are in perfect match with the ones got in the thermodynamics method. This further verifies the robustness of the thermodynamics method of setting up the CFT duals of black holes.", "pages": [ 22, 23, 24 ] }, { "title": "4.4 ASG analysis", "content": "The usual way to obtain the central charges of the CFT dual for the RN black hole is to uplift the theory to a higher dimensional gravity theory [22] or reduce to a 2D effective action [30]. However, for the dyonic RN black holes, there is short of direct derivation. In [30], it was argued that the central charge in 2D effective theory should be proportional to Q 2 , up to an undetermined factor. This is in accord with what we found in the thermodynamics method. It is true that we can uplift the dyonic RN black hole into five dimension and may obtain the central charge of the electric picture, but we cannot read the central charge of the magnetic picture in a clear way. In this subsection, we provide another way to understand this problem. The essential point is that the spacetime (4.3) of dyonic RN black hole is also the solution of a gravity theory with two U (1) gauge fields, with We write the action in the analog of that of type IIB supergravity. As can be checked easily, the action is invariant classically under an SL (2 , R ) transformation with a, b, c, d ∈ R , ad -bc = 1. The pair of two forms ( F, H ) transforms under SL(2,R) the same way as the pair of one forms ( A,B ). Upon quantization N e,m are integers and the SL (2 , R ) becomes SL (2 , Z ). Now Q e,m in (4.3) are just the parameters in the black hole metric and are related to the integer-valued charges N e,m through Thus the U (1) 2 black hole could fully pertain the properties of the dyonic black hole. The thermodynamics quantities of the outer and inner horizons, and thus the first laws, are formally identical to the ones of the dyonic black hole. So the CFT duals, including the E and M pictures as well as the pictures generated by SL (2 , Z ), of the U (1) 2 black hole from thermodynamics are identical with the CFT duals of the dyonic black hole. This suggests that we may understand the CFT dual duals of dyonic RN black hole from the investigation of the identified CFT duals for the U (1) 2 black hole. The advantage of working with the U (1) 2 black hole is that it could be uplifted to six dimension in a simple way and from the uplifted metric we can do ASG analysis to read the information of dual CFTs. The uplifted six-dimensional metric is of the form with The above uplifted metric is the solution of the six-dimensional Einstein gravity with G 6 = 16 π 2 /lscript 4 p . The two extra dimensions form a torus with the modular parameter τ . The torus is invariant under the modular group SL (2 , Z ), with a, b, c, d ∈ Z and ad -bc = 1. Note that the uplift (4.44) cannot be done arbitrarily, and the uplifted metric has to be in accord with the quantization condition (4.43). From the geometry one could get the conserved angular momentum which must be integers. Thus the uplifted metric (4.44) is the only possible uplifting up to a possible SL (2 , Z ) redefinition of the modular parameter τ and the angles χ e,m (4.47). As we did in Section 2, we consider the extremal black hole and compute and thus As the horizon area of the four-dimensional extremal black hole is A + = 4 π/lscript 2 p Q 2 , we have the extremal version of the electric and magnetic CFT dual pictures They are in accord with the ones obtained in the thermodynamics method. After the redefinition of the modular parameter and the angles (4.47), one could get the SL (2 , Z ) generated pictures in accord with the results before. Now the SL (2 , Z ) duality could be understood as the geometric modular symmetry of the extra torus.", "pages": [ 24, 25, 26 ] }, { "title": "5 Conclusion and discussion", "content": "In this paper we further refined the thermodynamics method of setting up nonextremal black hole/CFT correspondence. The essential part of our improvement is to impose the quantization condition on the first laws. Physically, the quantization condition comes from the fact that the perturbation always carries integer units of angular momentum and/or charges. As a result, the first laws of the black hole, encoding the response of the black hole with respect to the perturbation, should obey the quantization rule as well. From the first laws of the outer and inner horizons we can in general have with N being an integer quantized charge and all other charges being kept invariant. Then the temperatures of the N picture CFT dual is T N R,L , and the central charges could be derived using the Cardy formula c N R,L = 3 π 2 S R,L T N R,L . On the other hand, taking (5.1) as the first laws for the underlying CFT requires reasonably the quantization condition. Certainly how to understand (5.1) in the underlying CFT is an important issue. We investigated the holographic descriptions of various RN black holes via the thermodynamics method. As showed in [19], the relation T + S + = T -S -could be taken as the criterion to see if a black hole may have a CFT dual in the Einstein gravity. We found that T + S + = T -S -holds for RN black holes in all dimensions, which means that RN/CFT correspondence could be generalized to all dimensions, and T + S + = T -S -breaks down for RN-AdS black holes, which forbids us finding their CFT duals. Moreover, we tried to set up CFT duals explicitly for RN black holes in various dimensions by using the first laws at the outer and inner horizons. It turned out that all the pictures we found are in agreement with the ones read from conventional ASG analysis and the hidden conformal symmetry. It is remarkable that the refined thermodynamics method resolve the puzzle on the ambiguity in determing the central charges of CFTs dual to the RN black holes. Starting from the first laws (5.1), there is no ambiguity in deciding the central charges. For example, for the four-dimensional RN black hole, the CFT dual has central charges c = 6 eQ 3 , where e is the unit electron charge. The quantization condition is actually reflected in the facts that the angular momentum along the extra circle in the uplifted spacetime must be quantized, and in the discussion of hidden conformal symmetry the identified angular quantum number should be integer-valued as well. These key points have been ignored in the literature. Besides the RN black holes in various dimensions, we also discussed 4D dyonic RN black holes and found the novel magnetic picture. This picture have not been discussed in the literature, partially due to the difficulty in deciding its central charge. The different kinds of pictures could be most easily read from the refined thermodynamics, while can be also seen from other points of view. For example, to read the magnetic picture from the hidden conformal symmetry, we had to consider the probe scalar field with magnetic charge scattering off the black hole. In the minimal coupling, while the electric charge couples to the gauge potential, the magnetic charge couples to the dual gauge potential by electromagnetic duality. Such a coupling indeed gives us a consistent magnetic picture. For convenience we call the electronic and magnetic pictures (1,0) and (0,1) picture respectively. As shown in (4.29) there are other pictures generated by SL (2 , Z ) group. In general we could obtain a ( a, b ) picture for every coprime integers a, b . This is a kind of duality among different CFT theories. How to understand this duality is an interesting issue. We tried to understand the dyonic black hole by relating it to a U (1) 2 two-charged black hole, from which we may read the central charges from ASG analysis of an uplifted six-dimensional metric and more remarkably we may interpret the underlying SL (2 , Z ) symmetry as the modular group of the extra torus. However, it would be much better to investigate the symmetry from the dyonic black hole itself. In [36], it has been shown that the SL (2 , Z ) duality originates from the electromagnetic duality of the theory. The basic point is that the dyonic black hole spacetime is invariant under the electromagnetic duality, even though the charges of the black hole and the gauge potential have to be transformed. Therefore it is feasible to describe the same black hole in different SL (2 , Z )-related theory. Note that the notion of electromagnetic duality is new in the context of 4D Einstein-Maxwell theory, but is well-known in string theory [38,39]. The effectiveness and rebustness of the thermodynamics method is remarkable. It not only allows us to read the temperatures and the central charges of dual pictures, it also helps to determine the the frequencies, the charges and the chemical potentials of dual operators if one consider a perturbation, in exact agreement with the ones got from the low frequency scattering. The underlying pictures of the thermodynamics method and the hidden conformal symmetry are different. Consider the situation that a particle falling into the black hole. The particle carries energy, possibly also an angular momentum and various charges. The response of the black hole with respect to the infalling particle is encoded in the thermodynamics laws, both at the outer and inner horizons. There is a dual operator in CFT corresponding to such a perturbation. The thermodynamics method shows how to read various information on dual CFT. The different pictures can be obtained by considering the response of the black hole with respect to different kinds of perturbations. For example, in the 4D dyonic Kerr-Newman case, if the perturbation carries only an angular momentum, the response gives us the J picture, while if the perturbations carries only an electric (magnetic) charge, then the response gives us the electric (magnetic) picture. In general, if the perturbation carries all the quantum charges, it may lead to a new picture, whose information could be obtained by SL (3 , Z ) transformation acting on the elementary pictures. On the other hand, a probe scattering off the black hole, especially at the low frequency limit in the near region, can tell us the information of the black hole as well. In a quite similar way, different probe could read out different dual pictures as we showed. In all the cases we studied the thermodynamics method is more powerful than the hid- den conformal symmetry in the sense that even for the cases in which the hidden conformal symmetry is not easy to find, say the black ring case [28], the thermodynamics method can still give us consistent pictures. Nevertheless, it would be interesting to understand better the relation of these two different methods [40]. The accumulated evidence supports that the thermodynamics method is universal to decide the dual holographic picture of black hole. The thermodynamics of outer and inner horizon seems encode all the information of the dual CFT. It would be important to have a better understanding of the physics underlying this method.", "pages": [ 26, 27, 28, 29 ] }, { "title": "Acknowledgments", "content": "The work was in part supported by NSFC Grant No. 10975005, 11275010. JJZ was also in part supported by Scholarship Award for Excellent Doctoral Student granted by Ministry of Education of China.", "pages": [ 29 ] } ]

[ { "title": "ABSTRACT", "content": "CQUeST-2012-0559", "pages": [ 1 ] }, { "title": "Fake Supersymmetry and Extremal Black Holes", "content": "Seungjoon Hyun 1 a , Jaehoon Jeong 2 b , Sang-Heon Yi 2 c Department of Physics, College of Science, Yonsei University, Seoul 120-749, Korea 2", "pages": [ 1 ] }, { "title": "Abstract", "content": "We derive the BPS type of first order differential equations for the rotating black hole solutions in the three-dimensional Einstein gravity coupled minimally with a self-interacting scalar field, using fake supersymmetry formalism. It turns out that the formalism is not complete and should be augmented by an additional equation to imply the full equations of motion. We identify this additional equation as a constraint by using an effective action method. By computing the renormalized boundary stress tensor, we obtain the mass and angular momentum of the black hole solutions of these first order equations and confirm that they saturate the BPS bound.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Supersymmetry, if confirmed experimentally, has a profound significance in our nature. It would give us various predictions and new perspectives for particle phenomenology and cosmology. Apart from these implications, it explains systematically many interesting analytic results which may be ad-hoc or difficult to understand otherwise. This analytic nature is more or less related to the so-called Bogomol'nyi-Prasad-Sommerfield (BPS) states in extended supersymmetric field theories, which preserve supersymmetry partially. One interesting and nice aspect of these BPS states is that they admit the Killing spinors which satisfy the Killing spinor equations (KSE). These KSE are usually lower order differential equations than the original equations of motion and therefore are easier to solve. Explicitly, in the usual two derivative theory, the bosonic equations of motion are given by second order differential equations, while BPS states can be described by first order equations. Typically these BPS states exist even in the model which contains only bosonic sector of the supersymmetric theory. In this reduced model they usually correspond to the states which minimize the energy, from which, once again, the lower order equations can be obtained. Inspired by this, fake supersymmetry method has been developed to obtain these BPS states which satisfy lower order equations of motion for non-supersymmetric, i.e. purely bosonic, model [1][2][3]. The basic idea is simple: One may consider a 'fake' supersymmetric extension of the bosonic model and introduce a spinor which satisfies fake KSE in the corresponding supersymmetric model. Since the EOM of original bosonic model are the same as the bosonic EOM of the supersymmetric model for vanishing fermions, these reduced order 'fake' KSE would almost imply the full EOM as in genuine supersymmetric theory. Along this line, various interesting BPS solutions of gravity with a minimally coupled scalar field have been found. They include domain wall solutions [1] and black hole solutions with a scalar hair [4]. They were found by considering some reduced EOM which are consistent with the full EOM. In the case of domain wall solutions and some static black hole solutions, those reduced EOM have been obtained by using fake supersymmetry formalism [3]. In this paper we would like to establish a systematic method to obtain these reduced order EOM by using fake supersymmetry formalism. Specifically, we consider the three-dimensional Einstein gravity with a minimally coupled and self-interacting scalar field. It would be considered as a bosonic sector of some fake supergravity. It was found that the model admits asymptotically anti-de Sitter black hole solutions with a scalar hair [5][6][7][8] as well as Banados-TeitelboimZanelli (BTZ) black holes [9]. We use the KSE of the fake supergravity to find the lower order EOM. It turns out that the KSE are not enough to uniquely determine the solutions. We find that it is due to the fact that the Killing vector associated with the fake Killing spinor is nulllike. We identify the missing equation and argue that this corresponds to the constraint equation rather than the dynamical EOM. In order to support the claim, we consider the effective action formalism. The resultant solutions are shown to correspond to quarter BPS solutions in the supersymmetric counter part. Since the solutions are asymptotically anti-de Sitter, these can be studied in the context of AdS/CFT correspondence. We determine the counter terms for the scalar and the metric fields and compute renormalized boundary stress tensor. From this we obtain the mass and angular momentum of the solutions and confirm that they really saturate the BPS bound.", "pages": [ 2, 3 ] }, { "title": "2 Einstein gravity with an interacting scalar field", "content": "The action of three-dimensional Einstein gravity with a minimally coupled scalar field is given by where we have taken the convention of the metric as mostly plus signs and the curvature tensors as [ ∇ µ ∇ ν ] V ρ = R µνρσ V σ and R µν = g αβ R αµβν . The EOM are composed of scalar field equation and the metric field equations as follows ; where As usual, the trace part of E µν can be used to rewrite EOM as which is the relevant form for our study in next sections. We are interested in the asymptotically AdS black holes with a scalar hair, which would be deformations of BTZ black holes. Our metric ansatz for rotating AdS black holes with axial symmetry in AdS-Schwarzschild-like coordinates is taken as where L denotes the radius of asymptotic AdS space. Accordingly, the scalar field φ is taken as a function only of the radial coordinate r . The asymptotic conditions on the metric functions A ( r ) , B ( r ) , C ( r ) are taken as The boundary condition for the scalar field consistent with this metric ansatz is given by We have taken our fall-off boundary conditions for the metric as the standard Brown-Henneaux type which allow us to obtain the central charge by Brown-Henneaux method [10]. One may note that the above metric admits a time-like Killing vector ∂ ∂t and a rotational Killing vector ∂ ∂θ , which would generate the full isometry group in the generic case as was shown in the rotating BTZ case [11]. Explicitly, EOM for the above ansatz are given by where ' denotes the differentiation with respect to the radial coordinate r . These equations are called the full EOM in the following. In Ref. [7] extremally rotating black hole solutions with a scalar hair were found as solutions of the above EOM. It has been known that extremal BTZ black hole solutions preserve partial supersymmetry in the context of supergravity. Since the extremal black hole solutions with scalar hair can be considered as a deformation of extremal BTZ, it is natural to expect that the supersymmetry-like argument might play some roles to the solutions.", "pages": [ 3, 4 ] }, { "title": "3 Fake Supersymmetry and Effective Action", "content": "In this section, by using the, so-called, fake supersymmetry technique, we obtain Bogomol'nyi type of first order differential equations which solve the full EOM. This can be considered as the generalization of the domain wall case to the extremally rotating AdS 3 black holes. This turns out to be the systematic derivation of the first order equations for extremal black holes [7]. It turns out that fake Killing spinor equations are not sufficient to obtain all of the first order equations. As in the case of genuine supersymmetric theory with null Killing spinors, the fake Killing spinors turn out to be null-like and should be augmented by a certain component of EOMs. In our case, by using effective action method, we show that this component of EOMs becomes effectively a first order equation and, in fact, it corresponds to a certain constraint not the dynamical equation.", "pages": [ 4, 5 ] }, { "title": "3.1 Fake supersymmetry", "content": "Our convention for Γ-matrices is taken such as { Γ ˆ a Γ ˆ b } = 2 η ˆ a ˆ b . Explicitly, 1 + 2 dimensional (lower indices) Γ-matrices may be taken as real and symmetric ones: where σ a 's are Pauli matrices. Note that /epsilon1 αβ Γ ˆ a βα = 0. Spinor indices are raised or lowered by rank two /epsilon1 -tensor as Then, Clifford algebra is realized as We also take /epsilon1 ˆ t ˆ r ˆ θ = 1 such that Though there is another inequivalent irreducible representation of Γ-matrices in three dimensions, one may deal with the inequivalent ones simply by taking ˜ Γ ˆ a ≡ -Γ ˆ a . In our case, the fake Killing spinors under 'fake' supersymmetry are determined by two equations, one of which corresponds to the (fake) dilatino variation and the other to the (fake) gravitino variation as where W = W ( φ ), the so-called superpotential, denotes a certain function of the scalar field φ and the curved index Γ-matrices are defined as Γ µ ≡ e µ ˆ a Γ ˆ a . The covariant derivatives in the above fake Killing spinor equations(KSE) are defined by where ω ˆ a ˆ b µ denotes the spin connection. The integrability conditions of the above fake KSE, after the contraction with a Γ-matrix, lead to the following conditions where ∂ φ denotes the differentiation with respect to the scalar field, φ , and the scalar potential V ( φ ) should be taken in the form of The above contracted integrability conditions show us that EOMs for metric and scalar fields are almost satisfied. However, as in the case of genuine Killing spinors, the fake KSE or their integrability conditions may not imply the full EOM. According to the nature of fake Killing spinors, one may need an additional condition to imply the full EOM as will be shown in the following. Now, let us solve the fake KSE explicitly. For our metric ansatz, dreibeins can be taken as The spin connection one forms, ω ˆ a ˆ b = ω ˆ a ˆ b µ dx µ , for these dreibeins are given by Firstly, let us solve the fake dilatino equation. Since the scalar field depends only on the radial coordinate r in our case, one can see that which leads to For definiteness, let us take Γ ˆ r /epsilon1 = /epsilon1 case, which may be regarded as a projection. By solving directly the KSE corresponding to the fake gravitino variation, it turns out that the fake Killing spinor is a function only of the radial coordinate r and given in terms of the metric function A ( r ) as Furthermore, it turns out that metric functions and the scalar field φ should be related through first order differential equations as It has been known that the KSE imply the full bosonic EOM if the Killing vector formed by genuine Killing spinors is time-like, while it doesn't if the corresponding Killing vector is nulllike. It is natural to expect the same behavior for the fake KSE. We show that in our case the Killing vector constructed from the fake Killing spinors is null-like and therefore the KSE are insufficient to satisfy the full EOM. Through the standard procedure, one can construct the one-form dual to Killing vector by the bilinear of the fake Killing spinors as 1 It is straightforward to check that ξ µ satisfies ∇ ( µ ξ ν ) = 0 by using fake KSE, which tells us that ξ µ is a Killing vector. Using the Fierz identity of three-dimensional Γ-matrices, it is also straightforward to see that which shows us that the Killing vector is null-like and the fake KSE is insufficient to imply full EOM. This manifests from three equations in Eq.(13) and (15) from KSE for four unknown variables. Following the standard way in the genuine KSE, let us identify the missing equation for KSE to imply the full EOM in our case. To achieve this, it is convenient to introduce null coordinates adapted to the above Killing vector as where f is a certain normalization function. By direct computation from the fake Killing spinor expression given in Eq.(14), one can take e ˆ + (with f ∼ e A ) as Then, our metric can be written as where It is interesting to note that the projection condition, Γ ˆ r /epsilon1 = /epsilon1 , for fake Killing spinor implies Γ ˆ + /epsilon1 = 0. Now, let us identify the missing equation. The following procedure is a direct adaptation of the genuine Killing spinor case [12][13] to the fake one. By the spinor contraction of ¯ /epsilon1 with the contracted integrability condition, 0 = E µν Γ ν /epsilon1 , one obtains Since ξ ˆ -is the only non-vanishing component of a Killing vector ξ = ξ ˆ + e ˆ + , the above condition implies that all the components E ˆ -µ should vanish. By multiplying E ρσ Γ σ to the contracted integrability condition, 0 = E µν Γ ν /epsilon1 , and symmetrizing the free indices, one also obtains Using this condition (or its flat space index form), one can see that all the components of EOM are implied by fake KSE except 0 = E ˆ + ˆ + . Therefore, to imply full EOM, fake KSE should be augmented by the equation 0 = E ˆ + ˆ + , which can be written in our case as Using the conditions from KSE or the automatically vanishing components of bosonic equations, the necessary condition to imply the full EOM is given by In the following we will show that this missing equation can be identified as a certain constraint not a dynamical equation in the effective action formulation. Collecting the previous results for fake Killing spinors given in Eq. (13) and Eq. (15) with the condition Eq. (23), one obtains the following first order differential equations, which satisfy the full EOM, These differential equations, called reduced EOM, were obtained by some educated guess in Ref. [7]. Some comments are in order. If we choose the other inequivalent representation for Γmatrices, ˜ Γ, and take the projection choice of fake Killing spinor as ˜ Γ ˆ r /epsilon1 = /epsilon1 , we obtain the same equations in Eq. (24) except for the third one which changes into Since the above equations in Eq. (24) was derived by solving KSE for the fixed representation of Γ-matrices with definite projection Γ ˆ r /epsilon1 = /epsilon1 , one may say that solutions of these reduced EOM preserves 1 / 4 fake supersymmetries just like extremal rotating BTZ black holes 2 . Note that the third equation in Eq. (24) can be integrated as where the integration constants C + can take any value consistently with the asymptotic boundary conditions. One of the convenient choices may be to take the integration constant as C + = 0, so that the metric function C ( r ) is simply given by Note that the standard choice, C + = -1, for instance for BTZ black holes in AdS-Schwarzschild coordinates, can be recovered by a simple coordinate transformation, θ → θ + C + t . One advantage of this choice is the fact that one of the null coordinates can be identified with θ coordinate. One can see that the Killing one-form, ξ , from fake Killing spinor becomes identified with rdθ as can be shown from Eq.(19). This explains partially the result that the equation, E θθ = 0, can be taken instead of the missing equation, E ˆ + ˆ + = 0 .", "pages": [ 5, 6, 7, 8, 9 ] }, { "title": "3.2 Effective Action", "content": "In order to clarify the nature of the missing equation in the fake Killing spinor formalism, we consider the effective action. By inserting the metric ansatz into the action (1), one obtains the effective action as whose EOM can be obtained, after rearranging results from the variation of the action with respect to A, B, C, φ , as One can verify that these equations are equivalent to the full EOM in Eq. (7). First of all, one may notice that there are just four equations rather than five compared with the original full EOM. However, one can see that one of the five equations in the full EOM is redundant as follows. Basically, the redundant equation corresponds to the one containing A '' term, for instance 0 = E rr in Eq. (7). Let us derive this equation from the above four equations. By differentiating the first equation with respect to the radial coordinate r , one can obtain an equation containing A '' term. Though this equation also has V ' term, this term can be eliminated through the equation obtained by multiplying the last equation by φ ' . By combining the resultant equation with the second and third equations in the above, one can derive a differential equation containing A '' term which can be shown to be equivalent to 0 = E rr . Up to total derivative, the effective action can be rewritten as in which it is clear that e A -B becomes a Lagrange multiplier and thus a variation with respect to this gives us a constraint equation, A ' + B ' -rφ ' 2 / 2 = 0. This equation is nothing but the missing equation obtained in Eq. (23). In appendix A we present preliminary study on the canonical formulation of our model to investigate the origin of this constraint. Let us try to extremize the above effective action by a complete square to obtain BPS like first order equations. By squaring the Lagrangian successively, one obtains One can see that the following conditions extremize the effective action partially which should be augmented by the constraint from Lagrange multiplier e A -B . After inserting the above first order equations (32) in the effective action with the constraint 3 , the effective action can be further reduced as This reduced effective action can be extremized by the following first order equation These seem to suggest that the effective action formalism may reproduce the first order equations obtained from fake SUSY formalism.", "pages": [ 9, 10 ] }, { "title": "4 Boundary Stress Tensor and Conserved Charges", "content": "It was shown that first order differential equations derived in the previous section describe extremally rotating AdS black holes by near horizon analysis and, moreover, some of analytic solutions for these first order equations, called reduced EOM, were also presented in Ref. [7]. In this section we obtain renormalized boundary stress tensor on the AdS black hole solutions for these reduced EOM, which is interpreted as the stress tensor of dual CFT on the asymptotic boundary by the standard AdS/CFT dictionary [16]. We also confirm the extremality of these black hole solutions by obtaining mass and angular momentum through renormalized boundary stress tensor. It is interesting to note that mass and angular momentum from the renormalized boundary stress tensor have contribution from both metric and scalar fields, while two contributions are obtained in one stroke through metric in the so-called ADT formalism [17][18][19]. The (holographically) renormalized boundary stress tensor is given by the subtraction of an appropriate counter term from quasi-local stress tensor introduced by Brown and York [20] [16]. This boundary stress tensor becomes finite after the subtraction and can be identified with the (renormalized) stress tensor in the dual field theory according to the AdS/CFT correspondence. Using these renormalized boundary stress tensor, one can compute conserved charges in dual field theory which can also be identified with those in the bulk gravity. In the following, we obtain the renormalized boundary stress tensor for our model and also verify the previous expressions of conserved charges. The aim of this section is two-fold. On the one hand we would like to obtain the contribution of a scalar hair to the boundary stress tensor and on the other we verify the conserved charge expression of our concerned black hole solutions in another way and confirm the extremality of those black holes. Solving reduced EOM in Eq. (24) perturbatively at the asymptotic infinity, one can see that the asymptotic fall-off behaviors of AdS black hole solutions are given by where constants a 1 , b 1 and φ 1 are related as a 1 + b 1 = -φ 2 1 / 4. Note that the integration constant are taken as C + = -1, which is more appropriate to obtain conserved charges correctly. For the superpotential W ( φ ) which is an even function of ( φ -φ ∞ ), one can show that the asymptotic form of the scalar field φ is given by In fact, by using reduced EOM one can show that the coefficients in the next leading term is given by [7] where ∆ 0 is a constant related to the horizon value of the superpotential as ∆ 0 = r 2 H W ( φ H ). As was mentioned in the previous section, these asymptotic boundary conditions for metric functions satisfy the so-called Brown-Henneaux boundary conditions [10]. Together with this metric fall-off boundary condition, the scalar field should satisfy the similar fall-off boundary condition to be consistent with the EOM. As an explicit example, by turning off the scalar field, that is to say, setting φ = φ ∞ , one obtains the extremal BTZ black holes, of which solutions are given in the above coordinates as For the boundary stress tensor computation it is very convenient to consider the metric foliated in the radial direction with the further decomposition of the boundary metric in the ADM form. Note that our metric ansatz is already in such a form. Explicitly, our metric ansatz can be written as where As is clear from the definition of the boundary stress tensor or its unregularized BrownYork tensor form, there are two contributions to the boundary stress tensor. One contribution comes from metric fields and the other from the scalar field. The metric contribution to the renormalized boundary stress tensor is well-known [16] and given in our case by the following form where K ij denotes the extrinsic curvature and K is its trace, K ≡ γ ij K ij . Our convention for the extrinsic curvature K ij is where ∇ i denotes the covariant derivative with respect to the metric γ ij . Therefore, we focus only on the scalar part in the action, in the following. Fortunately for our purpose, the scalar field contribution to the boundary stress tensor was already determined for a specific scalar potential in Ref. [21]. However, the fall-off boundary conditions and the scalar potential are different in our case from that. Therefore, we need to rederive the scalar contribution which is appropriate in our case. According to the standard construction of counter terms, they are chosen to cancel the unwanted divergent part of the on-shell action. To apply this procedure, let us consider the variation of the scalar part in our action. After inserting the bulk EOM in the variation of the action, one obtains where n µ denotes the unit outward normal to the hypersurface or the boundary surface 4 . To cancel this term, one needs to introduce the variation of counter term for the scalar field as In the second equality, we have expanded the integrand in powers of 1 /r according to the fall-off boundary conditions. Now, let us take the integrated version of the above variational form of counter term as where α and β are a certain constant and will be determined in the following. The variation of the above counter term leads to which should be matched to the above variational form of the counter term. This condition determines only the combination of α and β as which means that the counter term may not be unique. This is not so strange since this ambiguity does not affect the conserved charges, as will be shown in below. It is also useful to recall that counter terms in higher curvature gravity, which has additional degrees of freedom through higher curvature terms, have such ambiguity [22][23]. One can verify that conserved charges are independent of the ambiguity explicitly as follows. First, one may note that the renormalized boundary stress tensor is given by the sum of metric and scalar contributions as follows: with the condition α + β = 1 / 2. Note that the scalar contribution solely comes from the the counter term in (41). Then, the conserved charges can be computed by where u i and ξ j , defined on the boundary, denote the time-like unit vector normal to the hypersurface and a Killing vector for the conserved charge, respectively. To obtain the mass of our black hole solutions, one can take the time-like Killing vector as ξ = e A u with unit one form u = Le A dt . Then, one can see that the metric and scalar contributions are given, respectively, by in which α and β appear only through a combination, α + β = 1 / 2. The total mass of the black hole solutions is given by By taking the space-like Killing vector for angular momentum as ξ = Lrv with unit one form v = Lr ( dθ + e C dt ), one obtains metric and scalar contribution to angular momentum as which leads to the total angular momentum as The above results on mass and angular momentum show us that the ambiguity in the counter term is harmless. Furthermore, the expressions of conserved charges confirm the extremality of the considered black holes ML = J , which were shown independently by the so-called ADT formalism [7] [24]. As alluded in the above, it is crucial for the correct conserved charge that one should keep the appropriate coordinates or the appropriate integration constant C + = -1, which was also the case in the ADT formalism. Though the ambiguity in counter term is not physical, one may determine the counter term completely by considering more generic fall-off boundary condition for the scalar field as where ζ is a constant. Under this generalized fall-off condition for the scalar field, the required counter term variation becomes The variation of integration ansatz is given by where we have retained ζ as a constant during the variation. Comparing the above two expressions for the variation of the counter terms, one obtains Then the counter term in our case can be chosen uniquely as the limit of such a counter term by taking ζ → 0. This phenomenon such as less ambiguity for additional fall-off tail has also analogy in higher curvature gravity, where the more general fall-off solutions determine the counter terms with less ambiguity [22].", "pages": [ 10, 11, 12, 13, 14 ] }, { "title": "5 Conclusion", "content": "In this paper we have considered fake supersymmetry to derive first order differential equations for the rotating black hole solutions in the three-dimensional Einstein gravity with a minimally coupled self-interacting scalar field. It turns out that the fake Killing spinor is null in the sense that it leads to the null Killing vector, so that the fake KSE should be augmented by one of EOM, 0 = E ˆ + ˆ + in our convention, to imply the full EOM. We have also shown that this additional equation can be regarded as a certain constraint by using the effective action method. We also computed the renormalized boundary stress tensor from which we determined the mass and the angular momentum of our black hole solutions with a scalar hair. They saturate the mass bound for the angular momentum just like the usual extremally rotating BTZ black holes. It is somewhat unclear how to obtain all the first order equations in the effective action formalism while the fake supersymmetry formalism may not be complete in the case of null Killing spinor. It would be very interesting to investigate further the nature of the missing equation in the generic context of the fake supersymmetry formalism. Our investigation suggests that it may correspond to a constraint equation. In this context it would be nice if one can identify the missing equation through the canonical approach with light-cone foliation. The fake supersymmetry formalism has been a powerful tool to study the BPS states in gravity models. Since the theory itself is not supersymmetric, the solutions of fake KSE are not guaranteed to be stable. It would be an separate issue to determine the stability of those solutions. It would also be very interesting to extend the fake supersymmetry formalism to the higher derivative gravity with scalar fields.", "pages": [ 15 ] }, { "title": "Acknowledgments", "content": "We would like to thank Yongjoon Kwon, Soonkeon Nam and Jong-Dae Park for useful discussions. This work is supported by the National Research Foundation(NRF) of Korea grant funded by the Korea government(MEST) through the Center for Quantum Spacetime(CQUeST) of Sogang University with grant number 2005-0049409. SH is supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) with the grant number 2012046278. S.H.Y is supported by Basic Science Research Program through the NRF of Korea funded by the MEST(2012R1A1A2004410).", "pages": [ 15 ] }, { "title": "A Canonical Formalism", "content": "In this appendix, we describe the canonical formalism of our model. Since the canonical formalism for the scalar field is trivial, we focus on the formalism for the metric. The aim of this section is to indicate that the missing equation in the fake supersymmetry formalism may be connected with the Hamlitonian and momentum constraints. Here, we adopt the standard notation in the canonical formulation with time-like foliation, which will be used only in this appendix. Through the ADM decomposition of the metric as one can apply the canonical formalism to gravity. In this formulation, γ ij 's are taken as canonical variables and their conjugate momentums are given in terms of the extrinsic curvature K ij by In our convention the extrinsic curvature K ij is defined by where ∇ i denotes the covariant derivative with respect to the metric γ ij . By diffeomorphism invariance, one obtains constraints which are called as Hamiltonian and momentum constraints. These constraints can be written in our case respectively as where (2) R denotes the curvature scalar in two dimenisons for ( r, θ ) and denotes the conjugate momentum for the scalar field φ . Using our antatz for the black hole metric, one can see that these constraints lead to The canonical Hamiltonian is given by and dynamical equations in the canonical formalism are given by where the first equation is nothing but the condition determining the extrinsic curvature by ∂ t γ ij . In terms of the extrinsic curvature K ij , the second dynamical equation can be written as As is clear from this expression, this equation leads to two equations among EOM for the metric field as follows These equations correspond to 0 = E rr and 0 = E θθ in the full EOM. If one use the first order equations obtained from fake supersymmetry formalism, the combination of those constraints and the equation 0 = E θθ becomes the missing equation. This indicates that the constraint by the Lagrange multiplier e A -B in the effective action may appear in the light-cone foliation.", "pages": [ 16, 17 ] } ]

[ { "title": "Valentin Bonzom 1, ∗", "content": "1 Perimeter Institute for Theoretical Physics, 31 Caroline St. N, ON N2L 2Y5, Waterloo, Canada (Dated: December 26, 2017) The Schwinger-Dyson Equations (SDEs) of matrix models are known to form (half) a Virasoro algebra and have become a standard tool to solve matrix models. The algebra generated by SDEs in tensor models (for random tensors in a suitable ensemble) is a specific generalization of the Virasoro algebra and it is important to show that these new symmetries determine the physical solutions. We prove this result for random tensors at large N. Compared to matrix models, tensor models have more than a single invariant at each order in the tensor entries and the SDEs make them proliferate. However, the specific combinatorics of the dominant observables allows to restrict to linear SDEs and we show that they determine a unique physical perturbative solution. This gives a new proof that tensor models are Gaussian at large N, with the covariance being the full 2-point function. Keywords: Random tensor models, Schwinger-Dyson equations, generalization of the Virasoro algebra", "pages": [ 1 ] }, { "title": "INTRODUCTION", "content": "Matrix models have provided a good description of two-dimensional quantum gravity coupled to matter (equivalently, non-critical strings) [1]. In the scaling limit, they provide access to Liouville gravity coupled to critical (unitary or not) matter with central charge c < 1. While there exist various ways to solve matrix models, the Schwinger-Dyson Equations (SDEs, also known as loop equations) have become a standard powerful tool. They enable to probe the correlators at all orders in the 1 /N expansion and the topological expansion [2] has been developed as an intrinsic method to solve them. In the double-scaling limit, the SDEs are equivalent to the string equation, which in turn corresponds to some integrable hierarchies (depending on the model) [3-7]. However the SDEs should not be only considered as a tool, as they probe the key features of matrix models very deeply. In particular, they can be recast in terms of differential operators which generate (half) a Virasoro algebra. This naturally leads to the fermion gas formalism which makes the initially hidden, conformal symmetry manifest, and consequently clarify the relationship of matrix models to 2d conformal field theories [8-10]. It is natural to extend those results to higher dimensions and the proposal was made in the early 90s to use tensors instead of matrices (objects with more indices) [11-13]. Rank d tensor models indeed generate discrete spacetimes in dimension d and are thus of high interest in different approaches to quantum gravity [14-17], e.g. (causal) dynamical triangulations [18-22] and loop quantum gravity [23] (and more generally sit as a natural generalization of matrix models wherever the latter are relevant - see [24] for an application to disordered systems). A large N limit has been found only quite recently for tensor models [25-27] , but the subsequent developments have rapidly expanded. The large N contributions are specific discretizations of the d -sphere, known as melonic [28], which provide an analytic description of the universality class of the Branched Polymer (BP) phase of Euclidean dynamical triangulations [30, 31] (previously known from numerical simulations). Moreover, the multi-critical behaviors [29] can be interpreted as critical, non-unitary matter [32, 33], just like in 1-matrix models [34]. Tensor models not only reproduce the statistical properties of the BP phase, but provide a new way to understand dynamical geometries through the SDEs. The algebra they generate was found at large N in [36], and extended at all orders in [37]. Geometrically, while the loop equations of matrix models describe the disc amplitude, the tensor SDEs describe the ball amplitude. The generators are labeled by boundary triangulations. As a first step towards a better understanding of this new symmetry algebra, we prove that the SDEs admit a unique physical solution in perturbations. This means that the symmetries completely determine the solution of the model, similarly to the conformal symmetry in two dimensions. This unique physical perturbative solution is obviously the same as found in [29, 38], by means of scaling arguments and a precise investigation of the combinatorics of the leading order contributions in the Feynman expansion. Thus, our method has the advantage that it bypasses the Feynman expansion, just like solving the loop equations at large N does not require the knowledge of the planar sector in matrix models. The main difficulty compared to matrix models is the proliferation of observables. Indeed, at each order in the tensor entries, tensors allow more than a single U ( N ) invariant (i.e. several boundary triangulations with the same number of simplices) and the SDEs are precisely equations on the expectation values of these observables. However only a specific subset of observables is relevant at large N and our analysis will consider this family as an input. The structure of the relevant observables is such that we can focus on linear equations, in contrast with matrix models. This linearity provides an alternative way to [38] to explain why tensor models are Gaussian at large N , the covariance being the full 2-point function 1 . Though the sub-leading corrections to expectation values in the 1 /N expansion are not known, it is clear that non-linearities eventually come into the game. We will restrict here to the large N limit. The organization is as follows. In the Sec. I we briefly review tensor models and the universality theorem. The SDEs are derived in the Sec. II, including our fundamental subset of linear equations. Our main result (the unique physical perturbative solution) is given in the Sec. III and we offer a precise comparison with the loop equations of matrix models. The Sec. IV is an attempt to give a global view on the solutions of the SDEs by comparing the number of observables to the number of independent equations. This is done in a simple example where the relevant observables are shown to be 1-to-1 mapped to non-crossing partitions and we conjecture that an infinity of 'initial conditions' is required.", "pages": [ 1, 2 ] }, { "title": "I. A BRIEF REVIEW OF TENSOR MODELS", "content": "Building a tensor model requires first a suitable choice of tensor ensemble, defined by its invariance properties (analogously to the Gaussian Unitary Ensemble, or the Gaussian Orthogonal Ensemble in random matrices). A natural choice (the only one for which the large N limit is known to exist) is an independent unitary invariance on each tensor index . If T a 1 ··· a d are the components of a rank d tensor, a i = 1 , . . . , N for i = 1 , . . . , d , define the following transformation, where the matrices U ( i ) are independent unitary matrices of size N × N . The complex conjugated tensor ¯ T transforms with the complex conjugated matrices. We are interested in functions f over such complex tensors that are invariant under those unitary transformations, f ( T, ¯ T ) = f ( T ' , ¯ T ' ). They are generated by invariant monomials built in the following way [38]: take p copies of T and p copies of ¯ T and contract all indices in such a way that an index in the i -th position of a T is contracted with an index in the i -th position on a ¯ T . Those invariant monomials are conveniently mapped, in a one-to-one fashion, to d -colored bipartite graphs, usually referred to as bubbles . Each T is represented by a white vertex and each ¯ T by a black vertex, giving, say, p black and p white vertices. The indices of each tensor are represented by half-lines labeled by their position, which we call color , from 1 to d . When two indices are contracted between a T and a ¯ T , they must have the same position hence the corresponding half-lines have the same color and one simply joins them together to form a line labeled with that color. The invariant monomial obtained from a bubble B can be thought as the 'trace over the bubble' 2 , and we denote it B ( T, ¯ T ). Bubbles are naturally dual to colored triangulations of d -1 pseudo-manifolds. The idea is to associate a ( d -1)simplex to each vertex whose half-lines represent the d boundary ( d -2)-simplices of the ( d -1)-simplex. Note that the ( d -2) simplices inherit the color of the half-lines. Colors further allow to identify all lower-dimensional sub-simplices by considering the sub-bubbles with exactly k < d colors. A bubble line between two vertices describe the gluing of two simplices along a boundary simplex identified by its color. We refer to [29] for details. Notice that for d = 2, T is a complex matrix and there is only one bubble with 2 p vertices: the loop with alternating colors 1 and 2, associated to the trace invariant tr( TT † ) p . Geometrically, it is dual to a loop with 2 p lines. Let I be a finite set, { B i } i ∈ I a set of bubbles and { t i } i ∈ I a set of couplings. A generic action for tensor models is where T · ¯ T = ∑ a i T a 1 ··· a d ¯ T a 1 ··· a d is the quadratic part (associated with the bubble formed by two vertices connected together by d lines). The partition function Z and the free energy F are given by Such integrals are usually 3 understood as power series in the couplings (perturbed Gaussians). It can be shown that F has a 1 /N expansion which starts at order O (1), [29]. The natural observables are the bubbles and their expectation values read A distinguished set of bubbles which is of particular importance is the set of melonic bubbles. Definition 1. The elementary melon with external color c is defined as the 2-point graph made of two vertices V, ¯ V connected together by d -1 lines (having all colors but c ), with one external half-line of color c attached to the white vertex ¯ V and the other, of the same color, attached to the black vertex V (see Fig. 1). A melon is obtained by inserting recursively elementary melons on any (internal) line between V and ¯ V , starting from the elementary melon itself, as in Fig. 1. Melons with the same external color can be joined together so as to get closed connected graphs called melonic bubbles . Notice that melons in a melonic bubble are precisely the connected, 1-particle-irreducible, 2-point sub-graphs. A melon can also be identified by the two vertices V, ¯ V on which the two external lines are attached. This gives a canonical way to associate to a black vertex V a white vertex, denoted ¯ V . The free energy and the Bubble Expectation Values (BEVs) have Feynman expansions onto connected ( d + 1)colored graphs (bubbles connected together by propagators which are given a fictitious color). The way tensor model have been solved so far heavily relies on those expansions. The main results have been synthesized in [29]. We recall them briefly. K B is the large N amplitude, given by the theorem below. For a non-melonic bubble, lim N →∞ 〈 B ( T, ¯ T ) 〉 /N = 0. where | B | is the number of vertices of B and G = 〈 T. ¯ T 〉 /N . All dependence on the coupling constants t i are carried in G . The latter satisfies an algebraic equation (which comes from combining (6) with a Schwinger-Dyson equation) where 2 p i is the number of vertices of B i , and with the condition G = 1 when all the couplings go to zero. 1 In this paper, we will take the item 1 as granted, as it comes from scaling arguments and amounts to say that we have identified the dominant observables. For d = 2 it reduces to a quite trivial statement, as it is equivalent to say that the observables are 〈 tr( MM † ) p 〉 and that their expectation values start like O ( N ). In tensor models it however becomes a less trivial assertion. We have not found a way to bypass that argument (i.e. derive the dominance of the melonic bubbles independently) and we actually find it reasonable to start our study with a given set of relevant observables. Then the purpose of the present paper is to find the key equations (6) and (7) by relying only on Schwinger-Dyson equations and without any use of the Feynman expansion of the BEVS onto ( d +1)-colored graphs. In addition to the item 1, we will taken as granted the large N factorization This is the same assumption that is used in matrix models. It only relies on scaling arguments 5 .", "pages": [ 2, 3, 4 ] }, { "title": "II. THE SCHWINGER-DYSON EQUATIONS", "content": "The SDEs and their algebra have been presented in [37]. Since this is not quite standard yet, we re-derive them in this section.", "pages": [ 4 ] }, { "title": "A. Bubble insertions", "content": "The simplest SDE is derived from the identity By taking the derivative explicitly and simplifying by N d , one gets It is a (linear) equation which relates the BEVs together at all orders of the 1 /N expansion. It is however not closed and one way to close it at large N is to use of the Gaussian universality property (6), which turns (10) into an equation on G , namely (7). But we have decided not to use the Gaussian universality (and instead to derive it from the SD equations), which means we have to write other SD equations. Definition 2. Let V be a white vertex in B . The open bubble B /integerdivide V , with tensor components ( B /integerdivide V ) a 1 ··· a d , is obtained by removing the vertex V from the bubble (and the corresponding T a 1 ··· a d in the invariant monomial), so that there are d open half-lines carrying the tensor indices a 1 , . . . , a d . These half-lines hang out from black vertices, hence B /integerdivide V transforms like a ¯ T . An example is given in the Fig. 2. One defines similarly the bubbles B /integerdivide ¯ V open on a black vertex ¯ V , which transform like a T . If V ∈ B and ¯ V ' ∈ B ' , the open bubbles B /integerdivide V , B ' /integerdivide ¯ V ' can be contracted on their free indices to get an invariant under (1). We denote it This operation has been called bubble gluing in [37]. Open bubbles appear naturally in the derivatives of a bubble invariant, What happens when an open bubble is opened a second time? That gives an object with more indices (and possibly disconnected as a graph, i.e. which factorizes as the product of two open bubbles). This can be repeated several times. Conversely, one can take a bubble opened several times and sum over some indices as long as one contracts an index of a T with one of a ¯ T (to satisfy (1)). This operation has been called bubble contraction in [37]. Open bubbles are covariant objects that can be used as insertions to generalize (9). SDEs are thus labeled by open bubbles and come from the identity We have to distinguish three types of contributions. Here the sum over V i runs over the white vertices of B i . When B and B i are melonic, all the resulting bubbles ( B /integerdivide ¯ V ) · ( B i /integerdivide V i ) are melonic, with | B | +2 p i -2 vertices, and the scaling is the same as in (14). Each B /integerdivide ¯ V /integerdivide V ' is a graph with two vertices less than B , and is typically a disconnected set of bubbles (see below). /0/0/0 /1/1/1", "pages": [ 4, 5, 6 ] }, { "title": "B. The fundamental large N equations", "content": "One can write an exact equation, which holds at all orders in the 1 /N expansion, by summing (16), (14) and (15) together. But there is a simplification at large N , as we have to keep only the melonic contributions in (16) which scale like (14) and (15), i.e. O ( N d ). It turns out there is only one such contribution, and this can be proved as follows. For a melonic B , the vertex ¯ V is part of a canonical pair ( V, ¯ V ) and one can always draw B like on the left of the Fig. 3. The lines of color 1 , . . . , d carry 2-point insertions M 1 , . . . , M d that we first assume to be non-trivial. When ¯ V is removed, one gets the open bubble on the right of the Fig. 3, where M 1 , . . . , M d have been expanded to describe all typical white vertices. To get B /integerdivide ¯ V /integerdivide V ' , a white vertex V ' has to be removed. Each disconnected piece in B /integerdivide ¯ V /integerdivide V ' scales like N , so that to reach the scaling O ( N d ), B /integerdivide ¯ V /integerdivide V ' must contain d disconnected pieces. It can be checked explicitly using the figure that there is only one way to get d disconnected pieces, which happens for V ' = V , the vertex canonically associated with ¯ V . The bubbles which result from the contraction are simply the 2-point sub-graphs M 1 , . . . , M d which are closed by joining their two external lines (abusing the notation, we still denote the corresponding bubbles M c , for c = 1 , . . . , d ). Using the large N factorization (8), we are finally led to If some of the M c s are trivial, i.e. they have no vertices and just consist of lines of color c , closing them produces loops with no vertices and free sums on the indices a c , each of which simply producing a factor N . The main difficulties with this set of equations are: The melonic family of bubbles is constructed by recursive insertions of the elementary melon on any line. It turns out that this is the reason why it is going to be sufficient to focus on the SDEs for B /integerdivide ¯ V where ¯ V is the black vertex of an elementary melon. The equations are given in the following lemma. Lemma 1. Linear equations for elementary melons. A melonic bubble B always has at least one elementary melon M , with external vertices V, ¯ V , and without loss of generality, external color d . The SDE for B /integerdivide ¯ V involves the 2-point subgraphs M c , c = 1 , . . . , d -1, which are just closed lines with no vertices, hence contributing as N d -1 . Define the bubble B /integerdivide M d as B with the elementary melon M d replaced by a line of color d . Then (17) becomes These are linear equations which describe the creation/annihilation of an elementary melon M d . The fact that this is a complete set at leading order and that they are linear will quite directly lead to the universal property (6). An interesting, direct application of the above Lemma is given below (but it will not be used in the proof of the universality result). Application 1. Equalities between sums of BEVs. Let G be a 2-point (melonic) graph, say, with external color 1. Then, The proof goes as follows. Consider the bubble B obtained by closing the graph G with two additional elementary melons next to each other on the color 1. We call V and V ' the two white vertices of these melons, and compare the SDEs for B /integerdivide V and B /integerdivide V ' . Both contain the expectation value of B . Since in the two cases B is opened on elementary melons, the lemma 1 tells us that the SDEs generate a single bubble with two less vertices. This is the same bubble in both cases: B with one elementary melon replaced by a line of color 1. Therefore, the only difference comes from the gluings of B with the bubbles B i of the action. The proof ends by substracting one equation to the other. This application becomes more interesting in special cases. For instance if the bubbles in the action have the same number of vertices, then the equation compares the BEVs of bubbles which all have the same number of vertices. It is even better for instance if there are some symmetries on the bubbles which reduce the number of terms of the equation.", "pages": [ 6, 7 ] }, { "title": "A. The Gaussian model", "content": "Before analyzing the generic model, it is useful to understand the behavior of the Gaussian one. For a covariance g , the action is The SDEs of the Lemma 1 are particularly simple, for any elementary melon M in B . As any melonic bubble comes from inserting an elementary melon on a smaller bubble, we get with the initial condition 〈 1 〉 = 1 that where p is the half number of vertices. This result has a combinatorial interpretation. Indeed a Gaussian measure is standardly defined by the fact that the expectation values are sums over Wick pairings weighted by the covariance. However, in matrix and tensor models not all Wick contractions have the same scaling with N , which means that some are suppressed at large N . As well-known, in the Gaussian matrix model, only planar contractions survive at large N (and are counted by the Catalan numbers). For the Gaussian tensor model, the BEV (22) shows that there is a single Wick pairing which survives the large N limit. It has been shown to be the pairing which connects the canonical pairs ( T, ¯ T ) of the melons in B 6 .", "pages": [ 7 ] }, { "title": "B. Solving the generic 1-tensor model", "content": "In this section I is a finite set, { B i } i ∈ I a set of bubbles with associated couplings { t i } i ∈ I . We denote the half-number of vertices of each bubble by p i = | B i | / 2. We will need the following Proposition. and At this stage, we would like to apply the lemma for the set I as our induction hypothesis. However the right hand side of (25) is a priori not a sequence since it seems that it depends on gluings of bubbles. However, we now show by induction on the order of the perturbation that it only depends on the number of vertices of B and that the lemma for the set I can be applied to (25). Therefore, the full expansion (24) of E B only probes bubbles through their number of vertices, which proves the desired property for the set { B i } i ∈ I supplemented with B 0 . We now apply the above proposition to the tensor model with the generic action One sets E B = 〈 B ( T, ¯ T ) 〉 , F = 0 and the large N BEVs satify the equation (23) (the Lemma 1). Corollary 1. The bubble dependence of the BEVs is just through their number of vertices. For any bubble with 2 p vertices, denote 〈 B ( T, ¯ T ) 〉 = N G p . Then the linear SD equations reduce to Proposition 1. Let ( F n ( { t i } )) n ∈ N be a sequence of functions such that each F n has a power series expansion in each t i . Let E be a map which associates to each bubble B a series E B ( { t i } ) in the couplings { t i } such that the empty bubble is mapped to the constant 1. Assume that for any vertex V of an elementary melon M in a bubble B with 2 p vertices, E satisfies the equation Then the evaluation E on two bubbles with the same number of vertices gives the same function, i.e. E only depends on the number of vertices of the bubbles B and not their specific structure. Proof. We proceed by induction on the number of elements in I . The function F p has a similar expansion as F p = ∑ n t n 0 F ( n ) p . The equation satisfied by E becomes at order n ≥ 1 Let us re-organize the set of bubbles { B i } i ∈ I according to the number of vertices of the bubbles. Set I = ∪ p ∈ P I p , where P is a finite set of integers greater or equal to 2, such that { B i } i ∈ I p is the subset which contains the bubbles with 2 p vertices. We define the coupling at 2 p vertices as t p = ∑ i ∈ I p t i . Then the recursion reads This linear recursion can be solved in two (equivalent) ways. Here p ∗ = Sup P is the maximal number of vertices among the bubbles in the action. When the couplings { t p } go to zero, there is a single root X = 1. We have denoted G the root which goes to 1 when the couplings go to 0 and we have isolated it on purpose as it is the physical root. Assuming that the P roots G, ( G ( α ) ) of this polynomial are distinct, it comes for any B with 2 p vertices for some constants c and c α . The physical interpretation is clear from the analysis of the Gaussian model in the Sec. III A: each geometric contribution G p ( α ) , G p , is a large N , Gaussian channel with covariance G ( α ) , G . But only one of them goes to 1 when the couplings go to 0 and it is the physical Gaussian channel with covariance G . Therefore 〈 B 〉 = N G n , and G is the full 2-point function. Note that this argument avoids the discussion of the initial conditions which are in principle necessary to solve the recursion. In matrix models it turns the SDEs into an algebraic equation on ω ( z ). We can proceed similarly here, to get where G k for k = 0 , . . . , p ∗ -1 are the required p ∗ initial conditions. To avoid setting the initial conditions by hand, we need to identify the physical Gaussian of covariance G (the one which goes to 1 when the couplings go to zero) in the language of the resolvent. This is done by analyzing the structure of its singularities. We observe that the denominator is the characteristic polynomial (29), whose roots are G, ( G ( α ) ), hence Since the numerator is a polynomial of degree p ∗ -1, the resolvent has at most p ∗ poles, z = G,z = G ( α ) , and at least one pole. As only G is physically relevant, we make a 1-pole hypothesis : the polynomial in the numerator must remove the p ∗ -1 non-physical singularities located on ( G ( α ) ). This fixes the initial conditions G k , k = 0 , . . . , p ∗ -1, up to a global scale. The latter is fixed by the trivial condition G 0 = 〈 1 〉 = 1. This reasoning leads to the final form of the resolvent The associated 'eigenvalue distribution' ρ ( λ ) is obtained as usual by taking the discontinuity of the resolvent across the real line, with The two ln λ terms are due to the fact we work with a rectangular matrix (they cancel each other for a square matrix). One recognizes 1 N ∑ i 2. Consequently the saddle point equation decouples the eigenvalues and reads 2. Consequently the saddle point equation decouples the eigenvalues and reads One recognizes the characteristic polynomial (29) of the BEV recursion. Thus, the large N Gaussian channels with covariance G, ( G ( α ) ) identified in the tensor model analysis naturally appear here as the solutions of the saddle point approximation. Let us now show how the loop equations describe the above phenomenon. We start with the identity", "pages": [ 7, 8, 9, 10 ] }, { "title": "C. Comparison with matrix models", "content": "The above resolvent and 'eigenvalue distribution' describe a system of non-interacting particles, all falling in the same potential well at z = G . This is in contrast with matrix models, but it is actually what would happen in a matrix model if the famous Vandermonde contribution, which acts like a Coulomb gas repulsion between the eigenvalues, could be removed. Such a matrix model has actually been constructed in [35]. Further, the universality property (6) ensures that all physical quantities of the generic 1-tensor model can be evaluated with this model. It is a matrix model for 'very rectangular' matrices, of size N × N d -1 , that is built as a tensor model with very specific bubbles. The bubbles entering the action are chosen such that the indices a 2 , . . . , a d of T a 1 a 2 ··· a d are always contracted all at the same time with the corresponding indices of a single ¯ T , as in the Fig. 4. The associated bubbles look like loops with some lines of color 1 and the others being 'fat lines' with colors 2 , . . . , d . These invariants can be written in a matrix form. Introduce a 'fat index' of size N d -1 , denoted by a capital letter, A = ( a 2 , . . . , a d ) and write the tensor like a matrix ( T aA ) of size N × N d -1 and its complex conjugate ¯ T aA = ( T † ) Aa . The corresponding action is This model was analyzed in details in [35]. Let us reproduce briefly the points that are most relevant to us. Upon introducing the eigenvalues ( λ i ) i =1 ,...,N of TT † , the partition function reads Next we evaluate explicitly the derivatives. Using the large N factorization 〈 tr( TT † ) k tr( TT † ) p 〉 = 〈 tr( TT † ) k 〉〈 tr( TT † ) p 〉 , one gets All quantities into brackets 〈 〉 are of order O (1), hence for d > 2 the non-linear terms are suppressed. In other words, the presence of the Vandermonde contribution to the saddle point equation translates into non-linearities in the SD equations. For d > 2 the natural scaling of tensor models removes those non-linearities and the linear equations of the Lemma 1 are recovered.", "pages": [ 10, 11 ] }, { "title": "IV. ABOUT OBSERVABLES AND OTHER SOLUTIONS OF THE SCHWINGER-DYSON EQUATIONS", "content": "We proved in the previous section that there is a unique physical perturbative solution to the SDEs at large N . However, nothing has been said about the full set of solutions to the equations. We propose some preliminary analysis in this Section, based on a specific, simple example, and conjecture that the set of solutions is determined by an infinity of 'initial' conditions. We consider two bubbles B 1 , B 2 , both with four vertices but different color labels, and two different coupling constants t 1 , t 2 , Note that the bubbles above have been drawn for d = 3. To get them for generic d > 3, it is sufficient to add lines with colors 4 , . . . , d whenever there is already a line of color 3, i.e. between all canonical pairs ( V, ¯ V ). The model gives a special role to the colors 1 and 2. As a consequence, it is natural to define a specific sub-set of melonic bubbles. Definition 3. A bubble is said to have melons on the colors 1 and 2 only if it can be built from the 2-vertex bubble by recursive insertions of elementary melons on the colors 1 and 2 only. Let B 12 ( p ) be the set of such bubbles with exactly 2 p vertices, |B 12 ( p ) | the corresponding number of bubbles, and B 12 = ∪ p ∈ N B 12 ( p ). To write the SDEs, we need to know the open bubbles obtained from B 1 and B 2 . The two black vertices of B 1 (and B 2 ) give the same open bubble B 1 /integerdivide ¯ V 1 (and B 1 /integerdivide ¯ V 1 ), depicted in Fig. 2. Therefore, for any choice of B , the SDE generates two larger graphs, one with an additional elementary melon on a line of color 1 on B and the other with an additional elementary melon on a line of color 2. This implies that if one writes a SDE for an open bubble B /integerdivide V which has melons of external colors 1 and 2 only, the equation only generates other bubbles in B 12 . We can write the SDEs graphically, Quite clearly, these equations actually generate the whole set B 12 . We will first count the number of observables generated by B 1 and B 2 from the trivial bubble, and then compare with the number of independent SDEs, for graphs with up to fourteen vertices.", "pages": [ 11 ] }, { "title": "A. Melonic bubbles and non-crossing partitions", "content": "As mentioned above, the sets B 12 ( p ) are actually independent of d ≥ 3. If d > 3, then for any line of color 3 between two canonically associated vertices V, ¯ V , there are also lines of colors 4 , . . . , d between V and ¯ V , and this situation accounts for all lines of colors 3 , . . . , d . By removing the lines of colors 4 , . . . , d , one obtains an element of B 12 for d = 3. Reciprocally, given an element of B 12 for d = 3, one can add lines of colors 4 , . . . , d between all pairs ( V, ¯ V ) to go back to the generic case. Therefore we focus on d = 3 in the following. 4a 4b 4 One can represent an element of B 12 ( p ) as a bipartite, planar contraction among 2 p elements. Indeed, one first places the 2 p vertices of the bubble on a circle, alternating black and white vertices. Using the clockwise convention, we put the color 1 on the p arcs which go from a black vertex to a white vertex, and the color 2 on the p other arcs. Each black vertex must then be connected to a white vertex by a line of color 3, the pairings being allowed only if they result in a planar graph. The vertices of the bubbles in B 12 are not labeled. We introduce B lab 12 as the set of bubbles whose melons have external colors 1 and 2 only, but now with labeled vertices. We label them (1 a , 1 b , . . . , p a , p b ) going clockwise around the circle, where the vertices ( n a ) are the white ones and ( n b ) the black ones, as shown on the right of the Fig. 5. Then B 12 can be described as the equivalence classes of these labeled planar contractions under the action of the rotations ( n a ↦→ ( n +1) a , n b ↦→ ( n +1) b ) (with periodic boundary conditions), and reflections. Let us now introduce the Non-Crossing Partitions (NCPs) of [ p ] = { 1 , . . . , p } in a graphical way. A partition π can be pictured by putting the elements of [ p ] on a circle, going from 1 to p clockwise, and adding links between the elements of each sub-set as follows. If n, m form a sub-set { n, m } with two elements, a line is drawn to connect them. When a sub-set has at least three elements { k, l, m, . . . } , we draw the unique convex polygon whose vertices are k, l, m, . . . . Finally singlets { n } are identified as isolated vertices on the circle. The set NCP( p ) of non-crossing partitions is the set of partitions π of [ p ] with no crossing inside the circle. An example is displayed on the left of the Fig. 5. We will describe a bijection between B lab 12 ( p ) and NCP( p ) and then mod out rotations and reflections. First we present the map from NCP( p ) to B lab 12 ( p ). We start by splitting each vertex n = 1 , . . . , p into two, denoted n a , n b , clockwise ordered, and label the arcs between n b and ( n +1) a with the color 1 and the arcs between n a and n b with the color 2. We can distinguish three types of elements in π ∈ NCP( p ): To get the inverse map, one simply reverses the above three rules, which account for all possible patterns inside the bubbles. The correspondance is detailed graphically in the Fig. 6. Then, it is easy to see using this map that the rotations ( n a ↦→ ( n + 1) a , n b ↦→ ( n + 1) b ) are mapped to usual rotations n ↦→ n + 1 on NCP( p ), and reflections to reflections. As a consequence, the number of bubbles |B 12 ( p ) | is the number of non-crossing partitions of [ p ] up to rotations and reflections (i.e. dihedral classes). They have been studied in [39] and are known as A111275( p ). The first values, from p = 1 up to p = 7, are 1 , 2 , 3 , 6 , 10 , 24 , 49.", "pages": [ 11, 12 ] }, { "title": "B. Independent SD equations", "content": "Now that we have a good control on the family B 12 , we come back to the SDEs. The key question is: do the SDEs determine the BEVs up to a finite number of initial conditions? While we have not been able to give a rigorous answer, we conjecture that the answer is no. It is based on the explicit analysis of the equations on B 12 ( p ) up to p = 7. Each SDE relates the BEVs of two bubbles with 2( p +2) vertices to the BEVs of bubbles which have less vertices. From the point of view of the two bubbles in B 12 ( p +2), the equations form a linear system. We will be interested in its rank, to compare it in particular with the number of bubbles |B 12 ( p +2) | . Our conjecture is that the rank of the system is precisely |B 12 ( p +2) | -1, for all p ≥ 0. To support it, we first write down the equations for a few values of p . For p = 0, There are two bubbles in B 12 (2), but only one equation. For p = 1, there are two independent equations on three bubbles, For p = 2, there is again one equation less than the number of bubbles |B 12 (4) | = 6 (the notation 1 ↔ 2 means that another equation is obtained by exchanging the colors 1 and 2, and t 1 with t 2 ), However, when p = 3 is reached, we get 12 equations, which is more than the number of bubbles on ten vertices |B 12 (5) | = 10, (47) But the system is of rank 9 = |B 12 (5) | -1 only. A consequence is that it is possible to extract 3 equations which do not involve the bubbles of B 12 (5) at all and appear as constraints on the BEVs of smaller bubbles. Then they might supplement the equations (44), (45) and (46) and determine some of the previously left undetermined BEVs. However, this is not the case. We have checked that these constraints are trivially satisfied if (44), (45) and (46) hold. We will here refrain ourselves from writing the 35 equations for p = 4 and the 102 equations for p = 5 (though quite tedious, it is actually really straightforward to write them in this model since B 1 , B 2 only add elementary melons). In those two cases, the rank is again |B 12 ( p +2) | -1. This inspection thus suggests that one BEV must be specified at each order in the number of vertices to determine all the others, and that the set of solutions to the SDEs is parametrized by an infinity of 'initial conditions'. To see how that could work in practice, we propose to re-organize (some) SDEs according the number of melons on the color 2. When t 2 = 0, the model can reformulated as a matrix model like in the Section III C with a simple 1-bubble potential. Then the loop observables with melons on the color 1 obey a special case of the recursion (28). Therefore, it can be instructive to understand how a non-zero coupling t 2 affects this recursion. Denote G n the expectation value of the 'loop' bubble with exactly n elementary melons on the color 1. At t 2 = 0, for any n ≥ 0 /negationslash We denote G n, 1 the expectation value of the bubble with n melons on the color 1 plus one elementary melon on the color 2 (hence n -1 elementary melons on the color 1). This is the term generated for t 2 = 0, Let us now focus on the family ( G n, 1 ). The bubble corresponding to G n, 1 has precisely n -1 elementary melons on the color 1, and another melon on the color 1 whose internal line of color 2 carries an elementary melon. By opening the graph on an elementary melon of external color 1, we see that at t 2 = 0 it satisfies the recursion (48) for n ≥ 1, the reason being that the graphs simply differ by the number of elementary melons on the color 1, /negationslash When t 2 = 0, the recursion picks up an additional term, involving graphs which have two elementary melons of external color 2. The BEVs of these graphs (for arbitrary t 2 ) only depend on the number of vertices. Indeed, this is a consequence of the Application 1 in the present model. The contribution of the bubble B 1 being the same on both sides of (19), it gives the following equality 7 We denote the corresponding BEV G n, 2 , for n ≥ 2 (they have exactly n melons on the color 1 and two on the color 2), so that These results are similarly extended to the family of graphs ( G n,m ) n ≥ m obtained by choosing m elementary melons on the graph G n and adding an elementary melon on the color 2 on each of them. The equality (51) extends to this family, stating that their BEVs are independent of the location of the m elementary melons on the color 2 (as long as each of them is added on a different elementary melon with external color 1). This family satisfies a double recursion, This double recursion does not take into account all SDEs on B 12 . However, it is consistent with our conjecture. It is clear it needs an infinite number of initial conditions, namely the family ( G n ), which can be freely chosen. It then determines the family ( G n, 1 ) via (49), which determines in turn the family G n, 2 via (52), and so on. Again, the conjecture is then that all other SDEs either determine some BEVs outside the family ( G n,m ), or are redundant.", "pages": [ 12, 13, 14, 15 ] }, { "title": "V. CONCLUSION", "content": "Our main result is that the large N SDEs of tensor models admit a unique physical, perturbative solution. This provides an alternative proof of the universality theorem (6) with less combinatorics than in the original proof [38]. If we think of the SDEs as differential operators acting on the partition function and generating a specific generalization of the Virasoro algebra [36, 37], our result actually means that these new symmetries completely determine the physical solution. While that was a necessary step, the full representation theory of this new algebra is still to be understood. Interestingly, the representation carried by the partition function of tensor models is such that it reduces the algebra to a Virasoro algebra, as already noticed in [29]. Beyond the large N limit, we expect the SDEs to give access to sub-leading orders, and possibly to a double-scaling limit (where N goes to infinity and the couplings go to their critical values simultaneously). We already know that it is possible for the tensor model used in the Sec. III C since it is actually a 'very rectangular' matrix model which has been shown to possess a double-scaling limit [35] (and the resolvent goes like ω ( z ) = 1 / ( z -G ) + a/N d -2 ( z -G ) 2 + b/N d -2 ( z -G ) 3 + · · · ). This project is quite interesting since at sub-leading orders the SDEs start to distinguish bubbles that have the same number of vertices. It would be useful to find a suitable generating function for bubbles, such as the one proposed in [40] which takes colors into account (but it is not clear whether that one is directly relevant). Another difficult part is to identify the non-melonic bubbles whose first non-zero contribution starts at a given sub-leading order. Using those ideas, we hope that will be possible in the future to recast tensor models in a way which puts the emphasis on the new symmetries encoded by the SDEs, similarly to the fermion gas formalism of matrix models which makes the conformal field theory content clearer. /negationslash", "pages": [ 15 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.", "pages": [ 16 ] } ]

[ { "title": "Alexander Westphal", "content": "Deutsches Elektronen-Synchrotron DESY, Theory Group, D-22603 Hamburg, Germany We attempt an estimate for the distribution of the tensor mode fraction r over the landscape of vacua in string theory. The dynamics of eternal inflation and quantum tunneling lead to a kind of democracy on the landscape, providing no bias towards large-field or small-field inflation regardless of the class of measure. The tensor mode fraction then follows the number frequency distributions of inflationary mechanisms of string theory over the landscape. We show that an estimate of the relative number frequencies for small-field vs large-field inflation, while unattainable on the whole landscape, may be within reach as a regional answer for warped Calabi-Yau flux compactifications of type IIB string theory.", "pages": [ 1 ] }, { "title": "1 Introduction and Motivation", "content": "String theory is a candidate for a fundamental theory of nature, providing at the same time a UV-finite quantum theory of gravity and unification of all forces and fermionic matter. Mathematical consistency requires string theory to live in a ten dimensional space-time, and a description of our large four-dimensional physics thus necessitates compactification of the additional six dimensions of space. The need for compactification confronts us with two formidable consequences: Firstly, even given the known internal consistency constraints of string theory, there are unimaginably large numbers of 6d manifolds available for compactification. Secondly, many compact manifolds allow for continuous deformations of their size and shape while preserving their defining properties (such as topology, vanishing curvature, etc) - these are the moduli, massless scalar fields in 4d. This moduli problem is exacerbated if we wish to arrange for low-energy supersymmetry in string theory, as compactifications particularly suitable for this job - Calabi-Yau manifolds - tend to come with hundreds of complex structure and Kahler moduli. Therefore, a very basic requirement for string theory to make contact with low-energy physics is moduli stabilization - the process of rendering the moduli fields very massive. Moreover, as supersymmetry is very obviously broken - and so far has not been detected ideally, moduli stabilization should tolerate or even generate supersymmetry breaking. And finally, the process should produce a so-called meta-stable de Sitter (dS) vacuum with tiny positive cosmological constant, so as to accommodate the observational evidence for the accelerated expansion of our universe by dark energy [1, 2, 3]. The task of moduli stabilization and supersymmetry breaking has recently met with considerable progress, which is connected to the discovery of an enormous number [4, 5, 6, 7, 8] of stable and meta-stable 4d vacua in string theory. The advent of this 'landscape' [7] of isolated, moduli stabilizing minima marks considerable progress in the formidable task of constructing realistic 4d string vacua. A large aspect of these recent advances relies on the use of quantized closed string background fluxes in a given string compactification. These flux compactifications can stabilize the dilaton and the complex structure moduli of type IIB string theory compactified on a Calabi-Yau orientifold supersymmetrically [5]. The remaining volume moduli are then fixed supersymmetrically by non-perturbative effects, e.g. gaugino condensation on stacks of D7branes [6]. The full effective action of such fluxed type IIB compactifications on Calabi-Yau orientifolds was derived in [9]. In type IIA string theory on a Calabi-Yau manifold all geometric moduli can be stabilized supersymmetrically by perturbative means using the larger set of fluxes available [10]. Moduli stabilization itself is a necessary prerequisite for a successful description of cosmological inflation in string theory. For slow-roll inflation requires a separation of scales between the inflationary scalar degrees of freedom with masses lighter than the inflationary Hubble scale and every other scalar field which needs to be heavy. Beyond that, the slow-roll flatness of the scalar potential required during the last observable 60 e-folds of inflation requires a substantial amount of control over higher-dimension operators in the effective field theory. This further motivates embedding inflation into string theory as a UV-complete candidate for quantum gravity. Counting numbers of models, most successful inflationary model building in string theory has focused on the corner of the landscape described by type IIB flux compactifications on orientifolds of warped Calabi-Yau threefolds, and has produced small-field models of slowroll inflation. A small-field model is characterized by the fact that the field range ∆ φ 60 the inflaton traverses during the observable last 60 e-folds of inflation is less than M P . Planckian field traversal during inflation marks a critical boundary, as here one transitions from the need to control chiefly dimension-six operators in small-field inflation to the need to control correction to all orders in large-field inflation. The inflaton candidate in these constructions is often chosen to be the position of a mobile D-brane [11, 12], a combination of the geometric volume moduli of the Calabi-Yau [13, 14, 15, 16, 17], or an axion originating in the higher p -form NSNS and RR gauge fields of string theory [18, 19, 20, 21]. There are several alternatives this plethora of slow-roll small-field models. One comes in the form of DBI inflation, where the specific form of the interactions of the inflaton dictated by the DBI action on a D-brane serve to slow down the field on a steep potential [22]. Another one consists of the idea of using a 'coherent' assistance effect of typically hundreds of string theory axions with sub-Planckian field range to yield an effective large-field model, called 'N-flation' [23]. Finally, there are recent constructions harnessing monodromy of the potential energy sourced by branes or fluxes with respect to D-brane position or p -form axions. This monodromy inflation mechanism allows for parametrically large-field inflation in string theory driven by a single field [24]. In the case of axion monodromy, the powerful shift symmetries of some of the p -form axions allow for a well-controlled and potentially large class of large-field models on warped type IIB Calabi-Yau compactifications [25, 26, 27, 28]. For some recent reviews on flux compactifications and the associated questions of the landscape of meta-stable dS vacua and inflation in string theory, with a much more complete list of references, please see [29, 30, 31]. Observationally, the ∆ φ 60 ∼ M P boundary between small-field and large-field models has a second tantalizing aspect. This is the case because of the Lyth bound [32, 33] shows that for any single-field model of inflation, observable tensor modes in the CMB require a super-Planckian field range. Upcoming CMB observations, both ground and space based, are projected to detect or constrain the tensor to scalar ratio r at the level r glyph[greaterorsimilar] 0 . 01 (see for example [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48]), while existing and upcoming satellite experiments also significantly constrain the tilt of the spectrum [49, 3, 50]. The tensor-to-scalar ratio r is a single dimensionless number with a lot of discriminative power, separating single-field slow-roll inflation into two classes clearly distinguished by their dramatically different sensitivity to UV physics. Each of the two classes encompasses a large set of individually different models of inflation. Therefore, one may hope by summing over sizable samples of each class - each sample averaging widely over many different model construction corners of the landscape - and accounting for the influence of their respective population likelihood by cosmological dynamics, like tunneling, one can provide a statistical expectation as to whether r = 0 or r > 0 (in the observationally accessible sense of r glyph[greaterorsimilar] 0 . 01). If successful, this would provide a second example where statistical reasoning on the landscape leads to a modest prediction akin to the weakly-anthropic explanation of the present-day small vacuum energy by scanning over N vac. glyph[greaterorsimilar] 10 500 flux vacua [4]. The present work is an attempt to do so. We will show that an answer to this question can be reduced to the question of the number frequency distributions of small-field and largefield driving regions of the landscape, and of their vacuum energies - that is, counting. Then we will fail, as we do not know how to count across the whole landscape, for its largest part is terra incognita still. A much more modest version of the counting task can be formulated for the region of the landscape described by flux-stabilized warped CY 3-fold compactifications of type IIB string theory in the description of F-theory on elliptically fibered 4-folds. A large sample (several millions) of such potentially low-energy supersymmetric compactifications are fully computationally accessible in terms of hypersurfaces in toric ambient space described completely the discrete data of the associated gauged linear-sigma models (GLSMs). We then outline for this sector of the landscape how to formulate the counting problem, and discuss the implications of a possible answer. For this purpose, we start in Section 2 with an analysis of those premises of possible arguments which follow from what is currently known about scalar fields in compactified string theory. Section 3 proceeds from these premises with an argument which essentially states that populating large-field models of inflation in string theory would cost exponentially dearly compared to seeding small-field models if populated from the lowest-lying de Sitter (dS) vacuum of the whole landscape. Section 4 reviews the interplay of the dynamics of eternal inflation and tunneling as well as the anthropic requirements for a statistical explanation of the observed small cosmological constant (c.c.) to work. Its central outcome is the observation that the progenitor vacua of cosmologically and anthropically viable descendant regions of the landscape are of high-scale vacuum energy, and ultimately lead to democracy on the landscape. Hence the reduction to counting. Section 5 then closes with a discussion of these arguments.", "pages": [ 3, 4, 5, 6 ] }, { "title": "2 Assumptions", "content": "Let us being by collecting some of the known results on scalar fields obtained from compactification of string theory to four dimensions. These results and properties of the types of scalar fields, the moduli, coming from a given compactification will form the premises of our later discussion of the prevalence of small-field versus large-field models of inflation in string theory.", "pages": [ 6 ] }, { "title": "2.1 Need for symmetry", "content": "We will start our walk through the premises by looking at the the need for a symmetry if large-field inflation driven by a single scalar field is to be embedded into string theory. Large-field slow-roll inflation driven by a single scalar field is stable under radiative corrections only in presence of an effective symmetry suppressing higher-dimension operators to all orders, which must be dominantly broken by just the inflaton potential itself. This is immediately clear from the known argument to the effect of the so-called 'eta problem'. The classical background dynamics of single-field large-field inflation requires an inflaton potential dominated by a monomial Such a potential requires a minimum field range needed to generate the last observationally required N e glyph[similarequal] 60 e-folds of inflation of The minimal distance in field space traversed during the observationally accessible period of inflation is super-Planckian, that is 'large-field', in these models. In absence of any further information constraining the effective field theory below M P dimension-6 operators will be generated by quantum corrections. Among them we will generically operators of the type Such a correction will shift, in particular, the 2nd slow-roll parameter where () ' ≡ ∂/∂φ by a piece which destroys slow-roll inflation as soon as the inflaton moves about a Planck unit. The only known way to forbid these dangerous higher-dimension operators to all orders in perturbation theory for an elementary scalar field (besides supersymmetry or conformal symmetry which, however, are incompatible with positive vacuum energy) is a shift symmetry. An unbroken shift symmetry requires a constant scalar potential. Assume now, that the dominant source of soft breaking of the shift symmetry is the field-dependence of the scalar potential itself, and its potential energy density is sufficiently below the cutoff of the effective field theory. Then radiative corrections induced by this inflationary soft shift symmetry breaking are proportional to powers of the order parameter of the symmetry breaking, namely So they become large only at extremely large super-Planckian field values if V 0 ( φ 60 ) glyph[lessmuch] M 4 P . This is the original idea of large-field chaotic inflation [51]. Parametrically large-field inflation with a single scalar field in string theory will thus constrain us to finding candidate scalar field protected by a very good shift symmetry. This essentially limits us to the axions of string theory, as we will see.", "pages": [ 6, 7, 8 ] }, { "title": "2.2 Properties of scalar fields in compactified string theory", "content": "Let us now look at the general properties of scalar fields arising from compactification of string theory to four dimensions. 1 These consist of the geometrical closed string moduli related to massless deformations of the internal compact manifold, open string moduli such as transverse positions of D-branes mutually supersymmetric and BPS with respect to the background manifold, and pseudoscalar 'axions' from the NSNS and RR p -form gauge fields. The geometrical closed string moduli, such as volumes or shapes of the internal manifold, are not protected by fundamental shift symmetries. 'Fundamental' is used here in the sense, that it describes a shift symmetry of a pseudoscalar field which is inherited from a gauge symmetry already present on the worldsheet. Geometric moduli therefore are generically not useful for parametrically large-field behaviour, although they allow very well for a plethora of small-field string inflation models. 2 Open string moduli, i.e. the position of the D-branes, have been shown in general to possess a sub-Planckian field-range for the canonically normalized scalar field corresponding to a given D-brane position modulus. Canonical normalization of the open string moduli involves inverse powers of the size of internal manifold which for control reasons must be always larger than the string length √ α ' [52]. Therefore which resides at the borderline between small-field and large-field models (e.g. fibre inflation in type IIB [16]). cannot be parametrically super-Planckian. Besides that, open string moduli generally do not inherit good shift symmetries. A similar parametric argument on the field range holds for the geometrical closed string moduli as well. The remaining class of scalar fields from compactification are axionic fields which arises from integration of the NSNS 2-form gauge field B 2 or RR p -form gauge fields C p over cycles of the internal manifold. This gives rise to axion fields where Σ i 2 denotes i th 2-cycle, and Σ α p denotes the α th p -cycle. The gauge symmetries of the p -form gauge fields from the worldsheet translate into shift symmetries of the dual pseudoscalar 4d axion fields. These continuous shift symmetries are broken to a discrete subgroup by instanton effects, and sometimes also by the effects of orientifold projections introduced to further break supersymmetry. An example of the latter is type IIB compactified on an O7 orientifolded Calabi-Yau manifold with fluxes and D3-branes, where the orientifolding breaks the shift symmetry of axions coming from B 2 [24, 25, 26, 27, 28]. The p -form induced axions in string theory, denoted collectively with a I , are therefore periodic on fundamental domain with limited field range [53, 54, 20] Converting this into canonically normalized scalar fields involves again inverse powers of the size of the internal manifold. The result behaves similar to the case of the open string moduli [52] These results have lead to the notion of no-go statement: there are no scalar fields with an intrinsically super-Planckian field range coming out of 4d string compactifications [53, 54, 52, 20].", "pages": [ 8, 9 ] }, { "title": "2.3 Populating vacua - tunneling and quantum diffusion", "content": "The final premise we need to discuss concerns the mechanism to populate meta-stable a given dS vacuum, and by extension a string theoretic inflationary region, in the landscape. The notion of vacua as the (meta)stable ground states of a local QFT as an effective field theory derived from string theory exists only the regime of controlled 4d low energy approximation to string theory. This is realized only in the large-volume and weak string coupling regime, when the supergravity approximation supplemented by the leading string-loop, α ' - and nonperturbative corrections is valid. Within this region of controlled approximations the only known mechanism of vacuum transitions at zero temperature proceeds via field theoretic tunneling. There are basically two known Euclidean instantons describing tunneling in QFT, and one process based on the quantum fluctuations of a light scalar field in dS space. The string landscape consists of a moduli scalar potential for an O (100 . . . 1000)-dimensional scalar field space which has upwards of O (10 500 ) isolated local minima. The local situation of tunneling between two adjacent vacua in moduli space is the often described by a scalar potential V ( χ i ) of the canonically normalized moduli χ i which has two local minima χ i, ± separated by a finite potential barrier. Let us call χ i, + and χ i, -the false, and true vacuum, respectively. These minima are separated by generically sub-Planckian distances in field space If in this general situation the barrier height is non-negligible compared the vacuum energy difference ∆ V = V ( χ i, + ) -V ( χ i, -), then the dominant Euclidean instanton contributing to tunneling is the Coleman-DeLuccia (CDL) instanton [55, 56]. In flat space this instanton is described by the so-called SO (4) symmetric 'bounce solution' to the Euclidean field equations in the 'inverted' scalar potential -V ( χ i ) Here ρ = √ τ 2 + | glyph[vector]r | 2 denotes the SO (4) symmetric radial variable with τ = it Euclidean time. The boundary conditions on the bounce solution require In terms of ρ , χ i ( ρ ) they read as One then computes the Euclidean action on the bounce solution, called here χ ( b ) i ( ρ ), and the tunneling rate is given by The presence of gravity yields corrections to this result which become important in two situations. For one, this happens if the barrier thickness becomes super-Planckian, which is generically not the case for next-neighbour local minima in the landscape moduli potential. The other situation occurs when the false vacuum approaches zero vacuum energy with the true vacuum being AdS. Then gravitational suppression of tunneling can happen. This case is irrelevant for our situation of having to populate a possible inflationary region of the landscape from a prior, higher-lying nearby dS vacuum. Therefore, CDL tunneling in our dS-to-dS situation with sub-Planckian barrier thickness can be described by flat space CDL tunneling neglecting gravity. This generic picture of a landscape populated CDL tunneling has been studied extensively in the literature. In certain controlled constructions such as warped Calabi-Yau compactifications of type IIB string theory with imaginary self-dual 3-form fluxes [5] and anti-D3-branes provide a string theoretic realization of the effective CDL tunneling description in terms of the derived moduli potential [57, 58]. The exception to this situation is the case where the barrier becomes very shallow and flat. In that case the gravitational corrections to tunneling become very important, and the Euclidean solution with the smallest action B mediating tunneling switches to the HawkingMoss instanton [59]. This describes tunneling from the false vacuum at χ i, + to the barrier top at χ i,T with a rate Finally, one can show that this process is essentially equivalent to a description where the light scalar fields χ i close to the false dS vacuum at χ i, + are driven by the dS quantum fluctuations up the barrier onto its top. The magnitude of the dS quantum fluctuations which form a Gaussian random field are given by (see e.g. [60]) where H + = √ V ( χ i, + ) denotes the Hubble constant of the false dS vacuum. The diffusion probability to reach the barrier top behaves like the one derived from the Hawking-Moss instanton. For all processes it is visible that a possible minimal field displacement will be expensive in terms of tunneling rate suppression if such a displacement due to tunneling were required by the initial conditions of a given inflation model.", "pages": [ 9, 10, 11, 12 ] }, { "title": "3 An almost argument - field displacement is expensive", "content": "We will now explore the consequences of the premises outlined in the last section. We know from the discussion in subsection 2.1 that large-field inflation driven by a fundamental scalar field needs an effective shift symmetry to be radiatively stable. The only other known mechanism for curing the radiative instability of generic scalar field theories, supersymmetry, forbids positive vacuum energy which renders it useless for inflationary purposes. From subsection 2.2 we have that essentially all scalar fields from compactifying string theory to four dimensions will have a sub-Planckian, or in some case just-so Planckian, intrinsic field range. Now consider that, in particular, the fundamental domain of all scalar fields with good shift symmetries from string theory, the p-form axions, is limited. Then by the very definition of a fundamental domain the only way beyond this point consists of finding an effect which unwraps the fundamental domain onto its covering space. This is called monodromy. For inflationary purposes this indicates two things. Firstly, the required effect has be something which gives potential energy to the candidate inflaton field in a given string compactification. Secondly, it must possess monodromy in the inflaton in order to see its covering space instead of its limited fundamental domain. This would result in parametrical large-field inflation (kinematically at least, up to back reaction constraints). Some form of non-trivial monodromy of the potential energy of the candidate inflaton with respect that very same inflaton is thus necessary for large-field inflation. We see, that this follows from the very definition of the limitation of the field range for those fields, the axions, which possess potentially good shift symmetries. As parametrical large-field inflation in string theory can only proceed given a good shift symmetry, the only viable candidates are the p -form axions and we are left with trying to generate axion monodromy in some form of potential energy for the axion. A potential energy for the p -form axions - which spontaneously breaks their shift symmetry - is generated by instanton effects, branes or fluxes. Monodromy in the potential energy with respect to the p -form axions exists for ( p + 3)-branes wrapped on p -cycles [25], or non-topological fluxes involving the p -form gauge field on a p -cycle [28]. The crucial aspect here is the fact that the monodromy-carrying objects, the branes and non-topological fluxes, spontaneously break the axionic shift symmetry the same way they also spontaneously break supersymmetry in a regime controlled typically by warping. Therefore, parametrical large-field inflation in string theory will come from some form of axion monodromy. For the case of axions from c = ∫ Σ 2 C 2 in type IIB on a homologous pair of 2-cycles in a pair of warped throats wrapped by an NS 5 NS 5-brane pair [25] this leads to a large-field potential The back reaction of the inflationary vacuum energy stored in the wound-up axion on the moduli potential as well as differing types of potential energy with axion monodromy such as other branes or fluxes will generically lead to a flattening-out of the axion inflaton potential [28]. In general we expect large-field potentials from axion monodromy to behave like There is a crucial observation to be made here. The scalar potential during axion monodromy inflation is driving both the leading shift symmetry breaking and the back reaction. Therefore, the point of vanishing axion vev being also the point of vanishing axion-induced D 3-brane charge and potential energy is a minimum of the inflaton potential. This holds to high accuracy independently of the shape or structure of the moduli potential, or the back reaction of the compactification geometry. If it were otherwise, the non-universality of the back reaction from the moduli potential or the geometry would destroy the shift symmetry in a non-universal way to begin with. This, by construction, cannot happen as the sole source of back reaction is controlled by the same parametrically weak effect which spontaneously breaks the shift symmetry to leading order in the first place. Let us denote the moduli sector other than the inflaton axion collectively with field(s) χ . Then we can write the full scalar including both moduli and large-field inflation from string theory following from the premises of section 2 and the discussion above as The final aspect of our discussion below will involve the dynamics of populating a potential large-field inflation model in string theory. The only way known to exist proceeds via quantum tunneling from a prior meta-stable dS vacuum (see subsection 2.3). Thus, we will only need to discuss the immediate neighbourhood of the moduli potential with respect to our current vacuum. Tunneling will proceed dominantly from the closest-by higher-lying dS minimum of the moduli potential with the smallest barrier height at the point where the Euclidean bounce solution crosses the barrier. Again, this relies on the decoupling between the position of the minimum of the inflaton potential and the moduli potential due the strong axionic shift symmetry. Neglecting the generally curved trajectory of the multi-field bounce solution, we will take the moduli potential as approximated by a '1d section along the bounce'. Then the existence of two close-by non-degenerate dS minima can be modeled by a quartic polynomial Such a local neighbourhood structure of the moduli potential relies on two properties. Firstly, the string landscape admits an extremely large number of isolated minima of the moduli potential. Second, the intrinsic field range limitation of all moduli fields implies that most of these minima must have distance glyph[lessmuch] M P . This justifies Taylor expanding the potential around a given minimum towards the closest neighbour. At last, there are subleading sources of shift symmetry breaking, typically non-perturbative effects. The presence of these instanton effects generates a periodic potential for the p -form axions. Its period is given by the extent of their fundamental domain, 2 πf in terms of the axion decay constant f . The magnitude of the instanton-induced axion potential is exponentially suppressed at large volumes [61, 25]. This gives us where vol Σ denotes the volume of the cycle threaded by the axion. We can thus easily dial them negligibly small. Yet their presence will be crucial for the relative count of small-field inflation models in string theory compared to the axion monodromy based large-field models. The full scalar potential for large-field inflation in string theory in presence of the local moduli potential, under the premises of section 2, thus reads We will consider for the sake of explicitness V 0 ( φ ) of C 2 axion monodromy from NS 5-branes, but the conclusions drawn from here will be general and do not depend on the precise choice V 0 ( φ ) This form introduces the slope parameter True large-field behaviour requires b glyph[lessmuch] 1, or equivalently vol Σ 2 glyph[greatermuch] 1. For this case, the example potential is shown in Fig. 1. Let us label again the two adjacent (meta)stable minima of the moduli potential by χ ± with χ + < χ -and place χ -= 0. Conversely, fine-tuning b around b glyph[similarequal] 1 will have near-flat inflection points appearing in the potential [26]. As f < M P , these inflection will be spaced with sub-Planckian distances. Therefore, moderately fine-tuning one of them to slow-roll flatness around the inflection point by using b will result in small-field inflation. This leads to a crucial result: Under the premises of section 2, there will be at least one model of small-field inflation contained in every working model of large-field inflation in string theory. The small-field and large-field parameter regions of axion monodromy occupy different volumes of microscopic parameter space, as the occurrence of a slow-roll flat inflection point needs a significant tuning in b glyph[similarequal] 1 in terms of the microscopic parameters, such as fluxes. Such fine-tuning can range from moderate O (10 -2 ) [18, 12] to more severe values of O (10 -8 ) [15]. However, the number frequency hierarchy deriving from the tune is finite, and can be easily dominated by exponential ratios from the dynamics of populating all these different models, which is tunneling. We will now take a look at the process of tunneling into the inflationary valley of χ -= 0 from the close-by valley at χ + < 0. Behind this is the premise of subsection 2.3 that tunneling is the only process for cold vacuum transitions in the landscape which is known. For this purpose, one more property of the full large-field scalar potential eq. (3.25) is crucial. Generically, it is the instanton-induced inflection point closest to post-inflationary minimum at χ -= φ -= 0 which is the only one suitable for small-field inflation. The reason is the upper bound on f . For typical values of f several of the inflection points will sit within the quadratic region φ glyph[lessorsimilar] M P close to the origin of V 0 ( φ ). Unless the lowest-lying, and thus closest-to-origin inflection point is tuned inflationary flat using b glyph[similarequal] 1, then there will be local minima at lower-lying inflection points which would trap the inflaton in false vacua. Conversely, the higher-lying inflection points above the fine-tuned one are too steep to support small-field inflation. An example of such an instanton-induced small-field contained in every stringy large-field model is shown in Fig. 2. The inflationary inflection points are the two ones closest to φ = 0. We will first look at tunneling mediated by the CDL instanton in the flat space approx- imation [55, 56]. As discussed in subsection 2.3, the gravitational correction factor from including gravity is typically not important for tunneling from one dS to another lower dS vacuum. glyph[negationslash] Let us first discuss which locale in the false-vacuum valley close to φ + = 0 we expect to be the most likely starting position for any tunneling process. We may a priori expect the false valley to get populated by an even earlier tunneling event. That event may exit in particular at some φ = 0 up the valley from where a further tunneling could start. However, the false valley is slow-roll in φ even more than the true valley close to χ -= 0, and it supports false-vacuum eternal inflation at its false minimum χ + < 0 , φ + = 0. Therefore, the ambient space-time residing initially in the false valley will have a fraction exponentially close to unity which actually sits at the false vacuum. The false vacuum inside the false valley thus exponentially dominates the initial state for any subsequent tunneling towards the true valley. Next, getting slow-roll inflation in the true valley after tunneling there places a constraint on the exit state directly after tunneling. We must have the position of φ after tunneling, called φ 0 , supporting at least 60 e-folds of slow-roll inflation in the true valley For the large-field case b glyph[lessmuch] 1 this implies while in the small-field situation with b glyph[similarequal] 1 we have Note, that the fields may exit at χ 0 < 0 and | φ 0 | glyph[greatermuch] φ 60 and possess finite speeds φ ' ( ρ ) , χ ' ( ρ ) as well, even in the small-field case. This is due the the negative spatial curvature inside a freshly formed CDL bubble. Negative curvature will serve to slow scalar fields regardless of their initial conditions enough to track them into slow-roll even on a small-field inflection point [62, 63] 3 . A viable bounce thus requires boundary conditions according to eq. (2.13) and for reasons of regularity also There is also an energetics constraint which reads to get down tunneling. If we look now at the inverted potential -V ( φ, χ ) driving the Euclidean dynamics, eq. (2.11), we see that such a bounce which originates downhill from the ridge and ends uphill on the ridge at χ + < 0 , φ + = 0 does not exist. The gradient of -V ( φ, χ ) always points away from φ = 0. Even allowing finite initial speed φ ' (0) , χ ' (0), neglecting the regularity boundary conditions eq. (3.29), does not avoid this fate, as the friction term in the bounce e.o.m. eq. (2.11) immediately destroys any initial speed [63]. This forces us to consider processes which move the field φ uphill in the false valley to values | φ | > φ 0 > φ 60 . A CDL bounce ending here and thus curving downhill (in the inverted potential -V ( φ, χ )) from its starting point φ 0 , χ 0 is then possible after φ has moved uphill (in V ( φ, χ )) in the false valley. How do we move up the false valley? The false vacuum at χ + < 0 , φ + = 0 drives falsevacuum eternal inflation. As m φ < H , φ undergoes scale-invariant dS quantum fluctuations. These can be thought of as φ performing a Gaussian random walk with variance eq. (2.17) This quantum diffusion has small probability for large jumps ∆ φ given in eq. (2.18) As m φ glyph[lessorsimilar] 10 -5 M P from the inflationary constraints in the true valley, we see that as long as V ( φ + , χ + ) 1 / 4 glyph[lessorsimilar] 0 . 1 M P . Therefore we arrive at a combination of two results. For every large-field inflation model in string theory there exists at least one small-field model contained within via moderate tuning. However, the relative probability of realizing them dynamically via first quantum diffusing uphill in the false valley and then CDL tunneling out, is given by for large-field models V 0 ( φ ) ∼ φ p , p > 1 / (2 N e ). Here we have used the large-field model N e e-folds interval and typically N e glyph[similarequal] 60 observationally. We can check this result by replacing the quasi-rectangular path just considered by looking at quantum diffusion directly from the false vacuum at ( φ + , χ + ) to a point ( | φ | > φ 60 , χ T on the top of the potential barrier ridge at χ T (which is then followed by classical rolling into the true valley). This process, as discussed in subsection 2.3, is equivalent to a Hawking-Moss instanton [59] which tunnels to said point on the barrier ridge. Both descriptions yield parametrically the same result as found above in eq. (3.34). This exponential advantage of the instanton-induced small-field regime crucially depends on the the necessity of uphill tunneling in the false-vacuum valley prior to the barriertraversing tunneling. Uphill tunneling in the false valley to provide a starting point for the barrier-crossing tunneling in moduli space is necessary only if the magnitude of the instanton correction is constant over moduli space. However, this assumption in eq. (3.25) does not follow from the premises of section 2. Indeed, the very arguments used above which imply the shift symmetry of the inflaton axion forcing a decoupling between the large-field potential V 0 ( φ ) and the moduli potential U mod. ( χ ), imply also that b can vary over moduli space. The instanton corrections break the continuous shift symmetry to a discrete subgroup. This fixes the argument of their sinusoidal dependence completely. In particular, the decoupling argument from the shift symmetry dictates that any phase ϕ in the sinusoidal dependence of the instanton correction is constant over moduli space to a high degree glyph[negationslash] However, its magnitude can and will generically vary widely over moduli space. This happens because changing b will preserve the positions of the critical points of the instanton correction which is what the discrete remainder of the shift symmetry demands glyph[negationslash] An example of a potential from the axion monodromy large-field mechanism of string theory, which represents this generic situation on the landscape is depicted in Fig. 3. This changes the picture in one crucial aspect. Tunneling in moduli space can now easily provide for access to the true-vacuum valley of slow-roll inflation, where b glyph[lessorsimilar] 1, from a falsevacuum valley with b > 1, as b will generically change upon barrier traversal. Therefore, in the false-vacuum valley there will be generically often a wash-board potential for φ from the repeating multiple minima at b > 1. Because of f < M P this will typically provide for false vacua close enough to any φ 60 needed, regardless whether φ 60 glyph[lessorsimilar] M P or φ 60 glyph[greatermuch] M P . 4 These false vacua (naively) are on equal footing 5 as initial states for subsequent tunneling through the moduli potential barrier, as they do not require uphill tunneling for their respective population. We end up in the following situation. In all regions of the landscape where b varies freely over moduli space between b < 1 and b > 1, the small-field and large-field regime contained as one-to-one in all large-field models of string theory are populated equally by tunneling. Therefore, the large-field mechanism by itself does not predict any bias towards small-field or large-field inflation, respectively. The exception to this statement are possibly special regions in the landscape where microscopic constraints may impose b < 1 for, say, certain classes of compactifications. There is no small-field inflation at all in the large-field models in such a landscape region. However, the population of all possible large-field models in this region must proceed via uphill tunneling from φ + = 0. We can then apply the same type of calculation as before to axion monodromy large-field models with varying monomial power, and suppressed instanton corrections. In these special landscape regions the above result implies a hierarchy among the large-field models in string theory. This hierarchy exponentially favors the models with the smallest monomial power p The question whether small-field or large-field inflation is dominant in the landscape therefore has no answer within the large-field mechanism of string theory itself. The situation is better only for those presumably small regions of the landscape, which are characterized by bounded instanton corrections such that among them the large-field mechanism does not contain a small-field regime. We are thus forced to look to the far wider class of small-field models outside the range of the large-field mechanism, and we need to count - and eventually weight that count by the combined dynamics of tunneling and eternal inflation. We may now ask about conceivable loopholes in the line of thought of this section. One known alternative to extending the field range seen by the potential energy of an otherwise periodic single axion, by definition, via monodromy, is an assistance effect of many subPlanckian axions, known as 'N-flation' [23]. For this proposal to succeed, however, all of the axions used need a shift symmetry of similar quality as in single-field monodromy. The minimum of the combined effective multi-axion potential thus is similarly decoupled from vacuum-hopping in the moduli potential as in the single-field case - again requiring superPlanckian uphill quantum diffusion in the axion valleys to enable entering the inflationary multi-axion valley via tunneling. This results in the same exponential bias against N-flation large-field models, as for single field axion monodromy setups. Next, it is clear that the minimum of the inflaton potential in large-field models is almost decoupled from the moduli potential by virtue of the shift symmetry, but not completely so. Assume therefore that the minimum of the inflaton potential shifts by δφ glyph[lessmuch] M P each time a next-neighbour vacuum transition executes in the moduli potential. Then there will possibly be multi-tunneling paths through the landscape which transport us into a progenitor falsevacuum axion valley with a minimum of the axion potential φ min. glyph[similarequal] φ 60 compared to the minimum of the final 'our-world' axion valley with φ min. = 0. A tunneling jump from such a progenitor valley will not require φ quantum diffusing up the hill by a super-Planckian field range, and would thus cause no relative suppression compared to the instanton-tuned small-field sub-setup. However, as the shift of the axion minimum δφ glyph[lessmuch] M P is very small, we will need many such tunneling jumps to get us into the right progenitor false-vacuum axion valley. As the maximum total potential difference ∆ V tot crossed by N such jumps prior to arriving in the right progenitor valley in the controlled region of the landscape is ∆ V tot. < M 4 P we have an average potential energy difference per jump of In the limit of large N we get ∆ V N → 0. Then the thin-wall limits holds for the N successive CDL bounces which implies As the full amplitude will be this will be exponentially suppressed for a super-Planckian ∆ φ 60 glyph[greatermuch] M P compared to a smallfield model requiring ∆ φ 60 glyph[lessorsimilar] M P , as then N large -field glyph[greatermuch] N small -field glyph[greatermuch] 1. This last hierarchy holds because the shift of the axion minimum due to a single-jump in the moduli potential is δφ glyph[lessmuch] M P . We have also assumed here that each of the N jumps traverses on average the same typically sub-Planckian field interval in moduli space, with the interval length not or only weakly depending on N . This seems to be reasonable as long as the number of jumps N is small compared to number N vac. glyph[greaterorsimilar] 10 500 the landscape must possess in order to allow for a weakly-anthropic explanation of the present-day small vacuum energy.", "pages": [ 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24 ] }, { "title": "4 Counting ...", "content": "We have seen in the last section that within the largest part of the landscape the mechanism for large-field inflation in string theory provides for large-field and small-field inflation regimes with comparable likelihoods because they get populated evenly by CDL tunneling. We will see that such a democracy between large-field and small-field inflation seems to hold generally across the landscape as far as the dynamics of eternal inflation and tunneling are concerned as long as we post-condition our analysis on the regions of parametrically small-c.c. dS vacua and inflationary regions which are cosmologically viable and permit an anthropic explanation of the observed extremely small c.c. along Weinberg's argument. We start with discussing the effects of tunneling and eternal inflation together with anthropic post-conditioning, and then proceed to discuss attempts at counting the different inflationary realizations on the landscape.", "pages": [ 25 ] }, { "title": "4.1 Eternity", "content": "There a few salient facts concerning the dynamics of tunneling and false vacuum eternal inflation which seem to hold across the whole landscape: These will determine the ensuing sketch of an argument for inflationary democracy on the landscape. Let us begin with the properties I and II. These together rule out population of cosmologically viable small-c.c. vacua by up-hill tunneling from an extremely small c.c. dS vacuum, as such up transitions are punished by an an exponentially suppressed transition rate This situation could only reverse itself if large-c.c. dS vacua are more than exponentially rare, or down-hill tunneling from large-c.c. dS vacua were forbidden. Property III prevents this from happening. Furthermore, the characteristics III and IV combined tell us that the immediate field space neighbourhood of any small-c.c. dS vacuum will almost everywhere consist of large potential barriers and adjacent large-c.c. dS vacua. Therefore, population of the small-c.c. dS vacua will happen almost everywhere by direct and (compared to up transitions from extremely small-c.c. vacua) fast down-hill tunneling transitions from adjacent large-c.c. dS vacua. Combined with I and II we also see, that such fast down-hill tunneling is necessary in order to have a shot at efficiently populating an exponentially large number of small-c.c. dS vacua for Weinberg's argument to work. We can now compare this with the dynamics of eternal population. Appendix B concerns itself exclusively with the one relevant aspect here - the typical vacuum energy of the progenitor dS vacua. The discussion of this aspect follows in particular [67]. The upshot can be summarized as this: For global measures with full volume weighting the progenitor is the largest-c.c. metastable dS vacua. This is the result of the exponential 3-volume reward driving the progenitor Hubble parameter to be as large as possible. All other measures free of obvious paradoxa (global measures without exponential volume reward, such as the scale factor measure, or local measures, such as the causal diamond measure) will see the the landscape populated from the longest-lived progenitor dS vacuum. From the discussion above, and in Appendix A we know that parametrical longevity of dS vacua is achieved once it has no down-hill tunneling paths accessible, and an exit thus will proceed by an up transition. Therefore, the longest-lived progenitor of a given region in the landscape will be a dS vacuum of somewhat small c.c. which can exit only via up-hill tunneling to another dS vacuum of near-Planckian c.c. Note that typically the c.c. of the progenitor will be just somewhat small compared to the Planck (or string) scale. In most cases it will not have the extremely small values relevant for cosmologically viable vacua because the up transition rate behaves as Note, that the remaining extreme cases where the longest-lived progenitor has extremely small-scale c.c. and thus does not need a near-Planckian intermediate state to guarantee longevity of the up transition, are taken care of by the requirement of Weinberg's argument to work. Namely, those extremely small-c.c. progenitors which do not exit via up-hill tunneling into very high-scale dS vacua, will not efficiently populate an exponentially large number of very small-c.c. descendant dS vacua, and thus are ruled out anthropically. We therefore expect in all relevant classes of inflationary measures that the population of an exponentially large number of cosmologically viable descendant vacua in the landscape involves as their immediate predecessor a meta-stable dS of very high-scale c.c.", "pages": [ 25, 26, 27 ] }, { "title": "4.2 Democracy - tumbling down the rabbit hole", "content": "We know at this point that all cosmologically and anthropically viable small-c.c. regions of the landscape have a very high-scale c.c. dS vacuum as their immediate progenitor. If we can establish in addition that the down-hill tunneling rate from a high-scale c.c. vacuum does not depend on the vacuum energy of a small-c.c. target dS vacuum, then we can show that tumbling down from the anthropically selected high-scale c.c. immediate progenitor into the small-c.c. regions of the landscape provides a flat prior for those vacua. Tunneling into the small-c.c. part of the dS landscape from the anthropically selected high-scale progenitor proceeds democratically. The independence of the Euclidean bounce action B = S E ( φ ) -S E ( φ + ) for CdL tunneling from small changes of the vacuum energy V -of the target small-c.c. dS vacuum can be demonstrated for a general potential as well as for the thin-wall limit. The outcome of the preceding discussion dictates a hierarchy of vacuum energies V T > V + glyph[greatermuch] V -≥ 0. V T denotes the height of the potential barrier which separated the progenitor dS vacuum with large c.c. V + from one of the cosmologically viable descendant dS vacua with very small c.c. V -. According to [6] we can write We tunnel into different descendant small c.c. dS vacua whose vacuum energies V -, V ' -differ by a still parametrically small amount | ∆ V -| glyph[lessmuch] V + . Hence, we immediately get that the negative term in B gets only a negligible correction from the variation of the vacuum energy among the many small-c.c. descendant vacua. The Euclidean action for the bounce solution itself we will approximate from above, and find the same parametrical result. We have roughly where ∆ ξ is the Euclidean time elapsed while ρ ( ξ ) increases from zero to ρ max and decreases back to zero. From the CdL Einstein equation we get ρ max at dρ/dξ = 0 to be Moreover, we have ∆ ξ glyph[lessorsimilar] H -1 + = √ 3 /V + . Hence, we arrive at with the same parametrically suppressed correction. One can further show using the field e.o.m. that also the bounce solution φ ( ξ ) gets parametrically small corrections of O (∆ V -/V + ). In the thin-wall limit where we have not just V T > V + glyph[greatermuch] V -≥ 0 but V T glyph[greatermuch] V + glyph[greatermuch] V -≥ 0, we can calculate the influence of a shift | ∆ V -| glyph[lessmuch] V + in V -on B exactly and get in agreement with the general result above. Summing up, we know that the anthropically efficient population of a cosmologically viable small-c.c. region of the landscape proceeds via tunneling from a very high-scale c.c. dS vacuum, and the down-hill tunneling rate into such small-c.c. vacua is independent from the varying vacuum energy of the many small-c.c. vacua. This implies that the tunneling rates get their variation from the distribution of barrier shapes - with the two main parameters height V T and thickness ∆ φ - across the many down-hill tunneling paths from the high-scale progenitor to the exponentially many small-scale c.c. descendants. Since the rates are independent from the varying c.c. of the low-scale dS vacua, we can thus average over the barrier shapes. Effectively, for small-c.c. regions satisfying anthropically efficient population the neighbourhood of each small-c.c. dS vacuum will show the same statistical distribution of high-scale progenitors and barrier shapes which allows for averaging over them. We can thus write the down-hill tunneling rate for small-c.c. regions satisfying anthropically efficient population as where the barrier shape average 〈 Γ( V + ) 〉 barrier shapes will be a function of the immediate progenitor's vacuum energy V + only. Again, the immediate progenitor denotes either the highest dS minimum for fully volume-weighted global measures, or the intermediary exit vacuum of the longest-lived dS vacuum for all other measures. Eq. (4.52) describes a central result which we expect to hold across all of the landscape: Namely, to be efficient enough for Weinberg's argument to work, the population of cosmologically viable small-c.c. regions of the landscape proceeds via down-hill tunneling from very high-scale c.c. progenitors, and this process populates the small-c.c. vacua democratically, placing no prior due to tunneling. Let us compare this to the discussion in section 3. From there we know that the smallfield regime inside stringy large-field models has to use the lowest lying instanton-induced inflection point, because doing otherwise would get the inflaton trapped in local minima at lower-lying inflection points. However, recently the absence of the overshoot problem was shown for tunneling-born small-field models [63]. This implies that we can tunnel into the small-field regime of a large-field model as far out and high (in potential energy) above the inflection point as we wish to. We can therefore, using the notation of section 3, seed both a small-field regime within a large-field model of string theory, as well as its own large-field regime by tunneling from an instanton-induced local minimum close to φ + glyph[similarequal] ∆ φ large -field glyph[greatermuch] M P . Now take into account the anthropically required democracy in down-hill tunneling which feeds our slow-roll inflationary regions in a cosmologically viable region of the landscape. We are in the regime of a flat tunneling prior, because cosmological viability forces all slow-roll inflationary vacuum energies by COBE normalization to have V glyph[lessorsimilar] 10 -10 which is already in the small-c.c. regime compared to the progenitor vacua. Hence, down-hill tunneling from the high-scale progenitors will populate all false minima in the false vacuum valley, and also all starting points in the true vacuum valley φ + evenly. As this includes values φ + glyph[similarequal] ∆ φ large -field glyph[greatermuch] M P , and we can seed both the small-field and the large-field regime from the same φ + glyph[similarequal] ∆ φ large -field glyph[greatermuch] M P , this will populate both the small-field and the large-field regime of every string theory large-field model equally. At the same time, the same argument leads to equal population of all other small-field saddle points outside the large-field mechanism class. Hence, we conclude that the dynamics of eternal inflation and vacuum tunneling transitions realize both small-field and large-field inflation with a flat prior, when conditioned on cosmologically and anthropically viable descendant regions. Fig. 4 displays two examples which show schematically the similarity of the global and local measures, and the democracy in down-hill tunneling that ensues from requiring cosmologically and anthropically viable descendant vacua, and the vacuum structure the shift symmetry enforces on the progenitors in the axion direction. As presented, this argument still has a possible loophole. Everything we said holds strictly true if the lowest-lying 1st inflection point is the only one suitable for small-field inflation. However, the latter statement is only valid for convex large-field potentials V 0 ( φ ) ∼ φ p , p ≥ 1. For concave potentials V 0 ( φ ) ∼ φ p , 0 < p < 1, however, which are still largefield for p glyph[greaterorsimilar] 0 . 1, you could equally well have small-field on the n th inflection point at φ n th inflection point > M P . This follows because almost all (except for the first few ones at φ glyph[lessmuch] M P where V 0 ( φ ) ∼ φ 2 again) of the lower-lying inflection points will have b < 1 as V 0 ( φ ) is concave. The number of inflection points generated by the instanton correction within ∆ φ 60 ,large -field is given by We have already discussed that f glyph[lessorsimilar] M P in string theory, and in concrete models of 5-brane axion monodromy one gets e.g. [26] Here V is the warped volume of the internal manifold in units of α ' . Thus, N inflection point glyph[greatermuch] 1 typically, and for concave models each of them is available for slow-roll tuning. This might Right: The schematic case for a local measure of eternal inflation is shown. The left-most valley of vacua forms the longest-lived progenitor vacua. The instanton-induced ripples are depicted larger here, as the longest-lived progenitor typically have high-scale c.c. but are not necessarily almost Planckian. The progenitor vacua exit by passing via up-hill tunneling through the metastable mediator valley in the middle. We see that largely equivalent parallel paths connect the equidistantly space local minima which are produced by the instanton effects in the direction of the axion in both the progenitor and the mediator valley. Downhill tunneling again proceeds from the mediator valley with equal rates for all the local minima in the axion direction as the rate is independent of the tiny vacuum energy of the descendant valley at the right or its tiny vacuum energy variation. lead us to conclude towards a counting bias towards small-field models. However, the same fact N inflection point glyph[greatermuch] 1 also implies that the inflection points are spaced densely compared to the evolution of V 0 ( φ ). Therefore, if the n th , n glyph[greatermuch] 1 inflection point is tuned flat ( b = 1), than its neighbours will have b glyph[similarequal] 1 to very good degree, too. This leads to a wide field range over which the slow-roll parameters glyph[epsilon1], η have sizable oscillations. They, in turn, imprint themselves as large oscillations on the 2-point function power spectrum of the curvature perturbation generated, which are severely bounded by the observed the CMB [26]. In particular, these limits imply b glyph[lessmuch] 1 [26], and therefore small-field inflation starting from a high-lying b = 1-tuned inflection point would give a universe with the wrong CMB. This removes all such small-field candidates except the ones starting from the few lowest-lying inflection points from the comparison with the large-field regime. So, demanding consistency with the observed CMB leads us back to the conclusion already drawn above.", "pages": [ 27, 28, 29, 30, 31, 32 ] }, { "title": "4.3 Vacuum energy distribution", "content": "This democratic result is to be compared with the product of the number frequency distribution of the vacuum energy of inflationary regions in the landscape, and the number frequency distribution for different inflationary model classes on the landscape. We start here the vacuum energy distribution. This prior is relevant as the field range of a given inflationary region implies a posterior constraint on the admissible vacuum energy range from the COBE normalization of the CMB fluctuations. Otherwise one could average over all occurring vacuum energies because the mechanisms for realizing small-field or large-field inflation in string theory do not depend strongly on the potential energy scale realized. This poses a danger if the prior number frequency distribution of the vacuum energies were to scale like as this would offset the hierarchy introduced by CdL tunneling discussed before. We lack calculational access to large swaths of the landscape, so we can only look at estimates of number frequency distributions of vacuum energies in corners where we have access. In one such corner, flux compactifications of type IIB string theory on warped Calabi-Yau threefolds, space-time supersymmetry can be used to estimate the distribution of vacuum energies among supersymmetry breaking vacua where the fluxes stabilize the moduli. The relevant distribution computed there is the distribution of the supersymmetry breaking scale M 2 S [68, 69, 70, 71, 72] (for a review, see e.g. [29]). We have M 2 S ∼ F in terms of the supersymmetry breaking F-terms, and the upper limit V max of the positive vacuum energy of a given vacuum is related to the F-terms as V max ∼ F 2 . Hence, we can estimate the largescale distribution of vacuum energies as the one given for M 2 S provided that no strong tuning of the vacuum energy has been selected for (the situation relevant for inflation). According to the results of [69, 70, 72], this leads to In flux compactifications we expect F ∼ W to have a flat distribution. This implies a flat number density distribution for the vacuum energy.", "pages": [ 32, 33 ] }, { "title": "4.4 Multitude", "content": "We are thus left with estimating the number frequency distributions of small-field and largefield inflation mechanisms in the string landscape. Let us start with the generic small-field models on the landscape (i.e. those which do not arise from the instanton contributions to large-fields models with axion monodromy). Most of these occur 'accidentally', that is, in vacua where the microscopic parameters such as fluxes, result in a local inflationary slow-roll flat dS saddle point of the moduli potential which can drive inflation. Barring further constraints, we can as a very rough approximation model the landscape (outside the symmetry-protected large-field mechanism occurrences) as a random potential for an N -dimensional scalar field space. 7 We then need to determine how many slow-roll flat dS saddle points we statistically expect in such a description. This question has been dealt with in a work by Aazami & Easther [73], where the propose to model the landscape as a random potential given by glyph[negationslash] We then have two cases. To describe them, let us estimate the scale of the cross couplings as c ij ∼ M 4 /M 2 P . At first, we can now look at the case where M glyph[lessmuch] M P of very small cross couplings. If each of the functions f i has α i ≥ 1 extrema, then the total number of extrema due to the lack of cross-terms is given by where α denotes the geometric mean of the α i . With N easily being of O (10 3 ) in the moduli space of string theory, even an α as close to unity as, say, 1 . 1 would imply O (10 100 . . . 10 1000 ) critical points in the landscape. Saddles among these critical points are classified by a Hessian which does not have all positive eigenvalues. The theory of random matrices then tells us that for an N × N symmetric almost diagonal matrix, each choice and permutation of eigenvalue signs occurs statistically with a frequency approaching 1 / 2 N [73]. A local minimum, represents just a single choice among all possible choices and permutations of eigenvalue signs. Thus, the number of saddle points (including local maxima) in our model landscape is while we get only Almost all of the critical points are saddles. For inflationary purposes we may wish to restrict our attention to class of saddles with just one negative eigenvalue, as these guarantee singlesmall-field inflation. For small cross couplings their number is There is by now ample evidence that the landscape contains, even in the small calculable sectors, more than O (10 100 ) local minima. This tells us that we have to put α = O (4), and thus there are easily more than O (10 100 ) single-field saddle points available. Keeping this estimate for α , the number of local minima and single-field saddle points begins to decrease super-exponentially compared to the above results only in the extreme opposite case where M ∼ M P (i.e. when the cross couplings are of the same order as the f i themselves). In this case the Hessian of the extrema of V becomes a general symmetric matrix. If the coefficients in the f i and the c ij are drawn from a normal distribution, then the Hessian of the extrema of V is a symmetric matrix drawn from Gaussian Orthogonal Ensemble. Its eigenvalue distribution obeys the Wigner semi-circle law, i.e the eigenvalue density E ( λ ) is One can then show [73] (see also more recently [76, 77]) that the probability to have a local minimum (i.e. all positive eigenvalues of the Hessian) or a single-field saddle point is given by As shown in [73], already a separation of scales between the f i and the cross couplings as small as 2 orders of magnitude is enough to sit safely within the first case discussed above, giving exponentially many local minima and single-field saddles potentially suitable for inflation. In general, one expects these two cases to appear mixed together in that a few cross couplings may appear with M glyph[greaterorsimilar] 10 -2 M P while most of them will be at smaller scales. The corresponding Hessian will then be approximately band diagonal, but the count of single-field saddle points will remain exponentially large of O (10 N ), because band width is generically expected to be small compared to N . One may condition this analysis on subsectors of the landscape which potentially allow for low-energy space-time supersymmetry. In such a situation the random potential over moduli space should be replaced by a random supergravity, i.e. random choices for the Kahler and superpotential of the moduli. Moreover, in such a supersymmetric sector of the landscape we should envision for a large number N H < N of moduli being stabilized supersymmetrically at a large mass scale (flux stabilization of complex structure moduli and the axio-dilaton in type IIB on a warped Calabi-Yau provides a large class of examples), while supersymmetry breaking occurs together with the stabilization of the small number N L of remaining moduli at a parametrically smaller mass scale. Both effects have been studied in detail in [76] with the result that the probability to have a local minimum (i.e. all positive eigenvalues of the Hessian) is given by Here 1 < p < 2 and c L is an O (1) number which can be estimated with random matrix methods [76]. The total number of local minima (i.e. all positive eigenvalues of the Hessian) in a given sector with N > N H glyph[greatermuch] N L moduli then remains still exponentially large The upshot is that we will get in a landscape with N scalar degrees of freedom typically O (10 N ) meta-stable dS minima. The fraction β saddle of them which constitute single-field saddle points potentially suitable for inflation we do not know so far. From the existing random matrix studies [76] so far it is not clear whether there will be more or less single-field saddle points than meta-stable dS minima. What we do know about is the cost of flatness of such a saddle point. The fraction of them which are locally flat enough to support 60+ e-folds of slow-roll inflation will be determined by the fraction of volume in microscopic parameter space, such as fluxes, which yields sufficiently flat saddle points. According to work done by [18, 12, 15, 75] this imposes a fine-tuning cost of typically O (10 -8 . . . 10 -2 ) in the space of single-field saddle points, with [75] most recently finding this suppression for warped D3-brane inflation to be of O (10 -5 . . . 10 -3 ). This cost is negligible compared to the quasi double-exponential number O (10 N ) of dS minima, as N = O (100 . . . 1000). The next step consists of counting the realizations of the large-field mechanism in the landscape. The crucial differences to the small-field count above reside in the need for a shift symmetry and the functional 'fine-tune' characteristic for large-field inflation. The combination of both requires a realization of the large-field mechanism to have a distinctly projected-in axion field, with a distinctly chosen 'discrete' source of potential energy with non-trivial axion monodromy. This can work only for each given axion direction once-atime, and can not, by definition, yield multiple locally-flat regions in each axion field space direction, because the potential energy source has to have monodromy and thus is of a fixed large-field functional form. However, this leads to a crucial difference in counting. Projecting in a suitable RR-form axion field, and supplying it a source of potential energy with axion monodromy, constitute discrete choices selecting a whole manifold for compactification. On each such manifold there is still a potentially large discretuum of vacua generated by the available choice of fluxes used in moduli stabilization. If the number of moduli is large the available flux discretuum will be only insignificantly changed by imposing the condition of e.g. projecting in a suitable axion. Therefore, a large fraction of all available flux dS vacua on a given manifold of compactification will lead to a axion monodromy large-field inflation 8 , if the manifold itself was chosen correctly, while on the same manifold only a certain fraction of all available flux dS vacua will constitute a sufficiently tuned small-field inflationary saddle point. We do not yet know whether the latter are more abundant than dS minima or not. The estimation of the number frequency distributions of generic small-field saddle points and axion monodromy large-field regions requires us therefore to determine the abundance of small-field single-field saddle points relative to the one of the dS minima, sum over all compactification manifolds, and determine the fraction of them which allow for projecting in a suitable axion and supplying it potential energy with axion monodromy. In general, we do not have (yet) sufficient calculational access into the landscape to do so.", "pages": [ 33, 34, 35, 36, 37 ] }, { "title": "4.5 An accessible sector of landscape", "content": "Still, it is potentially possible to answer a more modest form of the same question for a known and calculable sector of the landscape. One such example is the landscape of flux vacua on warped Calabi-Yau (CY) orientifold compactifications of type IIB string theory. This sector of the landscape may be of additional interest, as any possible strong number frequency bias arising there would tie the result to a possible detection of low-energy supersymmetry by virture of being most naturally realized in Calabi-Yau compactifications. On this sector of warped fluxed CY compactifications of type IIB we can now specify the parameters entering the number frequency distributions of inflationary mechanisms a bit more precisely. In particular, we have N H ≥ h 2 , 1 +1 and N L ≤ h 1 , 1 + , where h 2 , 1 denotes the number of complex structure moduli on a given CY 3-fold stabilized supersymmetrically at a high mass scale by fluxes together with the axio-dilaton, while h 1 , 1 + counts the number of Kahler moduli. From the last section we have on each CY an estimate for number of all critical points of the moduli potential while the fraction of meta-stable minima is In terms of these we can now estimate the number of local minima N i, min. on a given CY 3-fold i and the fraction of those which are sufficiently fine-tuned inflationary single-field saddle points Here β i,flat saddle denotes the ratio of the number of inflationary flat single-field saddle points to the number of meta-stable dS minima. Note that we do not know a priori whether β i,flat saddle < 1 or β i,flat saddle > 1. A more detailed study of random matrix models along the lines of [76] may yield an answer to this question. The quantity of 1 -β i,V 1 4 > 10 16 GeV denotes the fraction of such inflationary single-field saddle points with an energy scale small enough to support observationally viable inflation on a sub-Planckian field range. Finally, we now have to sum this over all CY 3-folds. If we denote averages of the above quantities over the number of CY manifolds by dropping the label i , then we get To determine the fraction of manifolds with axion monodromy inflation we have to multiply each term eq. (4.69) with a factor δ which is either zero or unity depending on whether the given CY 3-fold has a projected-in RR C 2 -form axion (or equivalently, h 1 , 1 i, -= 1) and suitable source of potential energy with axion monodromy, everything placed inside a warped throat etc. For a conservative estimate we may ask to bound the number of CY's supporting axion monodromy by counting all those with h 1 , 1 i, -≥ 1 as the most basic requirement ( δ = δ h 1 , 1 i, -≥ 1 ∈ 0 , 1). We still do not know how to do this for all CY 3-folds. But, we may be able to do this for a large set (several million CY 3-folds) of examples given by their corresponding F-theory compactifications on an elliptically fibered CY 4-fold which are given as hypersurfaces in ambient toric spaces. This class of fluxed warped CY compactifications of type IIB is specified completely in terms of the discrete data of the GLSM description of the ambient toric spaces and hypersurfaces therein together with 4-form flux data. The discrete GLSM data then allows for determining for each choice whether h 1 , 1 i, -≥ 1, and thus to determine the fraction β h 1 , 1 -≥ 1 of all CY's within this sample which support the basic requirement of large-field inflation. Next, we denote with 〈 h 1 , 1 -〉 the average number of RR 2-form axions projected in on the elliptically fibered toric ensemble. Moreover, we have to restrict to the fraction β i,V 1 4 > 10 16 GeV of axion monodromy realizations with sufficiently large energy scale to drive to correct amount of curvature perturbations. Hence, we write N large -field ≤ N toric F -theory CY ' 4 s h 1 , 1 -≥ 1 and Plugging in we thus get what we may call the 'landscape Drake equation' [78] The sums in these expressions run over the set of CY's denoted by toric F -theory CY ' 4 s . This result assumes a flat number frequency distribution of vacuum energy. Arguments for this flat prior to arise in the context of type IIB flux compactification were reviewed above in section 4.3. 9 Note that we do not know a priori whether β flat saddle < 1 or β flat saddle > 1. An estimate of the axionic in-projection cost β h 1 , 1 -≥ 1 seems feasible for the large sample of CY 3-folds described in F-theory as elliptic 4-folds given in terms of their GLSM data. It is conceivable that a study of random matrix models along the lines of [76] may yield in fact β flat saddle > 1. Then in virtue of β h 1 , 1 -≥ 1 bounding the number frequency of axion monodromy inflation from above, this would tell us to expect a negligible tensor fraction r in the type IIB CY landscape to the extent that the occurrence of an observable tensor-to-scalar ratio r glyph[greaterorsimilar] 0 . 01 is tied to the inflationary scale and thus to the existence and realization of large-field models of inflation. 10 10 The link between r glyph[greaterorsimilar] 0 . 01 and large-field inflation is not watertight. On the one hand axion inflation can lead to additional highly non-Gaussian scalar perturbations sourced through the axion-photon coupling, which effectively suppresses r even for large-field models [80, 81]. Next, small-field inflation models can be (severely!) fine-tuned to produce r glyph[greaterorsimilar] 0 . 01 [82, 83]. And finally, a small-field inflaton can source additional scale-invariant B-mode power through couplings to degrees of freedom (particles or strings) which get light at points of enhanced symmetry [84], similar to the trapping mechanism [85, 86].", "pages": [ 37, 38, 39 ] }, { "title": "5 Discussion", "content": "Let us stop here to summarize again the crucial aspects of the story just told. Firstly, the premises laid out in subsections 2.1, and 2.2 together imply that the properties of the scalar field from string compactification require large-field inflation in string theory to take the form of axion monodromy. The axionic shift symmetry, rooted in the p -form gauge symmetry on the worldsheet, decouples the position of the minimum of the axion monodromy inflaton potential from the moduli potential. If this were otherwise, it would imply sizable nongravitational couplings between the inflaton axion and the moduli which would invalidate the shift symmetry in the first place. Therefore, the many local minima of the moduli potential landscape share (almost) the same minimum of the axion inflaton potential. Secondly, the population of the inflationary axion valley is only known to proceed within the semi-classical regime, and within parametrically controlled approximations, via quantum tunneling, the last premise of subsection 2.3. Entering the inflationary axion valley of largefield inflation while providing at least 60 e-folds of slow-roll inflation after tunneling thus requires tunneling from a local dS vacuum to super-Planckian inflaton vev post-tunneling. If the only local dS vacuum in the false is the one at zero inflaton VEV, then a direct Euclidean bounce with such boundary conditions is impossible, requiring the inflaton axion to first quantum diffuse uphill in the false vacuum axion valley. This leads to exponential suppression in the population of large-field inflation in string theory compared to the small-field setup contained in every large-field model via tuning generically present instanton corrections. However, the instanton correction may induce multiple local false dS vacua in the falsevacuum inflaton axion valley, while being absent in the true-vacuum valley. This is generic in the landscape, the instantons being allowed to vary in size over moduli space. Then the population of the large-field and small-field regimes can proceed from a local dS vacuum of the false-vacuum valley which is close to the 60 e-fold point of the large-field regime. Due to the absence of overshoot post-tunneling this populates the small-field and the large-field regime evenly. The next crucial fact is the indifference of the dynamics of eternal inflation and tunneling to the vacuum energy of regions of cosmologically viable slow-roll inflation (i.e. satisfying COBE normalization) and very small-c.c. descendant dS vacua. Both global volumeweighted and local measures combine with the high-dimensionality of the moduli space and the anthropic requirement of efficient population of an exponentially large number of such descendant vacua such, that the immediate progenitor vacua are of very high-scale to almost Planckian vacuum energy. Down-hill tunneling into the descendant vacua of parametrically small c.c. then proceeds democratically which allows us to reduce the question of the relative prevalence of large-field and small-field inflationary regions to one of mere counting. This counting is hard in general due to lack of calculational access. However, if we restrict the scope to first obtaining an answer for a region of the landscape with established control, counting may be feasible. As an example we gave a sketch of the discussion for the landscape of elliptically fibered 4-folds in F-theory. A large sample (several millions) of such potentially low-energy supersymmetric compactifications are fully computationally accessible in terms of hypersurfaces in toric ambient space described completely by the discrete data of the associated GLSMs. Hence, in this sector of the landscape we may be able to get an estimate of the fraction β h 1 , 1 -≥ 1 of CY manifolds in the sample which have the RR-form axions required for axion monodromy large-field inflation in the first place (which we leave for future work). As such, an estimate of β h 1 , 1 -≥ 1 < 1 would provide an upper bound on the fraction of CY's which carry axion monodromy inflation. It is conceivable that a study of random matrix models along the lines of [76] may yield in fact that small-field models are more abundant than dS minima themselves. Combined with β h 1 , 1 -≥ 1 < 1 this would imply the absence of detectable tensor modes if a detection of low-energy supersymmetry pointed towards CY's. Finally, if there is a way of shifting around the axion minimum as a function of the moduli without spoiling the shift symmetry, or if there is a mechanism to protect large-field models without relying on an effective shift symmetry, the argument as it is fails.", "pages": [ 40, 41 ] }, { "title": "Acknowledgments", "content": "I am deeply indebted for many crucial and insightful discussions with R. Bousso and E. Silverstein. I am grateful to M. Aganagic, A.R. Brown, A. Dahlen, S. Kachru, M. Larfors, A. Linde, D. Lust, L. McAllister, M. Rummel, S. Shenker, V. Vanchurin, and P.M. Vaudrevange for many elucidating comments. This work was supported by the Impuls und Vernetzungsfond of the Helmholtz Association of German Research Centres under grant HZ-NG-603, and German Science Foundation (DFG) within the Collaborative Research Center 676 'Particles, Strings and the Early Universe'.", "pages": [ 41 ] }, { "title": "A Suppression of uphill tunneling", "content": "Here we shall shortly discuss the process of tunneling uphill from a lower-lying dS vacuum into a higher-lying one. This process is highly exponentially suppressed compared to downhill tunneling, and as a function of the vacuum energy of the final higher-lying minimum. One can see this explicitly in three different regimes of CdL tunneling. As we have the hierarchy V small -field , V large -field glyph[greatermuch] V lowest dS glyph[similarequal] 0, we can approximate the tunneling bounce by putting V lowest dS = 0.", "pages": [ 42 ] }, { "title": "A.1 CdL tunneling in the thin-wall approximation", "content": "At first we look at the case of a high potential barrier V T glyph[greatermuch] V small -field , V large -field , V lowest dS which places us into the regime of the thin-wall approximation. For this situation, the Euclidean bounce action including the effects of gravity reads where V + = V ( χ + ) = V small -field or V large -field , respectively. denotes the tension of the CdL bubble wall, with χ -denoting the position of lowest-lying dS minimum V -= V ( χ -). The ratio T 2 /V + controls the importance of the gravitational correction inside the rectangular bracket. If we approximate the potential barrier separating χ ± as being of height V T glyph[greaterorsimilar] V + and thickness ∆ χ we can write T ∼ ∆ χ √ V T . Gravity is important for V + glyph[lessmuch] T 2 , or equivalently ∆ χ glyph[greatermuch] √ V + /V T , resulting in For sub-Planckian barrier thickness we expect that the leading terms in the scalar potential which are responsible for the two adjacent local minima at χ ± will also produce the barrier separating them. Therefore, if V + is not subject to specific tuning, we expect the barrier height to vary roughly together with V + as V + glyph[lessorsimilar] V T . This leads to with some c > 0. This is a regime where a hierarchy is valid. If the barrier thickness ∆ χ is sufficiently sub-Planckian, increasing V + while keeping V T fixed will eventually take us into opposite regime ∆ χ glyph[lessmuch] √ V + /V T of weak gravity where Thus, we see that increasing V + eventually leads to a regime where eq. (A.78) is again valid. In summary, CdL tunneling in the thin-wall approximation yields a hierarchy leading to an exponential suppression of uphill tunneling scaling as Γ ' / Γ ∼ exp( -c/V + ) for V + < V ' + . The exception is a situation where V + > 0 is tuned to be extremely small compared to the barrier height V T . However, this limit is irrelevant for the discussion here, as the discrimination between large-field and small-field inflation around ∆ φ 60 glyph[similarequal] M P corresponds to a change of the inflationary potential energy by about 2 orders of magnitude around the GUT scale.", "pages": [ 42, 43 ] }, { "title": "A.2 CdL tunneling away from the thin-wall approximation", "content": "We saw in the last section how lifting V + towards V T shuts down the gravitational correction in the thin-wall limit. However, eventually this limit will also leave the thin-wall approximation itself. There are no general explicit results for the bounce action away from the thin-wall approximation known for generic potentials. However, one may approximation any given smooth potential with two local minima by triangulating it with linear functions. Coleman tunneling in such an approximative piecewise linear potential can be solved exactly by analytical methods without using the thin-wall approximation [87]. In the case where ∆ V + < ∆ V -/ 4, the bounce action can be found to be [87] Here it is For constant barrier thickness parameters ∆ χ ± taking V + → V T implies ∆ V + → 0, and c glyph[greatermuch] 1 which yields From the discussion of the gravitational correction factor in the regime of V + glyph[lessorsimilar] V T of the last section we expect gravitational corrections to the flat space result just given to be small. In summary, we again find the hierarchy demanded by eq. (A.78) which leads to an exponential suppression of uphill tunneling scaling as Γ ' / Γ ∼ exp( -c/V + ) for V + < V ' + .", "pages": [ 43, 44 ] }, { "title": "A.3 Hawking-Moss tunneling - the 'no-wall' limit", "content": "Finally, we can discuss tunneling in the limit of a wide and flat barrier with V '' ( χ T ) /H ( V T ) 2 < 1. This process is mediated by the Hawking-Moss instanton, and can be understood as uphill quantum diffusion of the scalar field χ , see the end of section 2.3. In our context of uphill tunneling from V -towards V + > V -this gives a tunneling rate (see eq. (2.16)) The ratio of tunneling rates for tunneling uphill from V -into two different higher-lying vacua with vacuum energies V + glyph[lessorsimilar] V T glyph[lessmuch] V ' + glyph[lessorsimilar] V ' T thus comes out to be Again, we find the hierarchy of eq. (A.78), and therefore tunneling uphill is severely punished for increases in the potential energy V + glyph[lessorsimilar] V T of the tunneling destination.", "pages": [ 44 ] }, { "title": "B Progenitor dS vacua - global vs. local measures of eternal inflation", "content": "There are quite a number of measures of eternal inflation which have been proposed to this date (for a by no means complete list of recent works see e.g. [88, 89, 67, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108]), even if we only count those which do not immediately run into paradoxa such as the youngness, or the Boltzmann brain problems. However, for the purpose of our discussion their most important property is that they separate into two classes with respect to the one relevant aspect here - the typical vacuum energy of the progenitor dS vacua. The discussion of this aspect follows in particular [67]. eternal phase. Correspondingly, in volume-weighted global measures all the vacua and slow-roll inflationary regions of the landscape are seeded ultimately by the highest-lying meta-stable dS vacuum of the landscape. Let us illustrate this at the example of a simple toy landscape with 3 vacua V s < 0 < V 1 < V 2 which is depicted in Figure 5. This example features the general structure of the string landscape - neighbouring vacua tend to have large differences in vacuum energy, the width of the potential barriers are glyph[lessorsimilar] M P , and there are AdS vacua. Therefore, we choose the vacuum S to be an AdS vacuum of negative cosmological constant V s < 0 . Tunneling from vacua with positive vacuum energy, such as the vacua 1 and 2 into vacuum S will create AdS bubbles within which space-time ends in a big brunch. Therefore, the AdS vacuum acts as a sink, destroying probability current flowing from the eternal inflating vacua 1 and 2. The vacuum population dynamics of this system is governed by differential rate equations. They determine the rate of change of probability ˙ P i of realizing vacuum i by the probability currents J ij which feed or drain vacuum i [67, 90] The probability currents are given as J ij = P i Γ ij , and J i,vol = P i · 3 H i . Here, Γ ij denotes the decay rate for forming bubbles of vacuum j in a sea of vacuum i . Note, that in a global measure the volume growth ∼ e 3 H i t of each vacuum i is weighted for by adding J i,vol . The dS-dS vacuum decay rates are given from CdL tunneling as while we denote the decay of vacuum 1 into the AdS vacuum S by Γ 1 s = e -C 1 . From now on, we will set Γ 2 s = 0 for simplicity. Then the vacuum dynamics reads Wewill assume Γ 21 glyph[greatermuch] Γ 12 as usually V 2 > V 1 , i.e. up-hill tunneling is highly suppressed. Furthermore, in most cases we have overwhelmingly H i glyph[greatermuch] Γ ij . With these inputs, eq.s (B.87) has a solution [67] All vacua grow with the volume growth of the highest-lying meta-stable dS vacuum whose population dominates everything else. This does not depend on the decay rate Γ 1 s into the AdS sink, as long as H 2 > Γ 1 s . In this case, the vacuum dynamics is governed by Note that the volume growth rate contributions are absent by definition of the local nature of this class of measures. There is one variation to this class of measures, in that there is a local-global duality which links the local causal patch measures with the global 'scale-factor' measure. The scale-factor measure adds back volume growth terms J i,vol = 3 P i However, these lead to universal volume growth ∼ e 3 t of all dS vacua. This implies, that the overall volume growth can factored out unambiguously, so that ratio of vacuum population probabilities behave exactly as in local causal patch measures (this is a manifestation of the 'global-local duality' between causal patch measures and the scalefactor measure [101]). The asymptotic behaviour of the ratio P 2 /P 1 does now have two distinct regimes, depending on whether Γ 1 s glyph[lessmuch] Γ 21 (a 'narrow' sink) or the opposite Γ 1 s glyph[greatermuch] Γ 21 (a 'wide' sink) is realized [67, 90]. For a narrow sink we find The opposite case of a wide sink yields Both cases share a common property - for a narrow sink, the vacuum populations are dominated by vacuum 1, while for a wide sink vacuum 2 dominates - in each cases it is the longest-lived dS vacuum which dominates the landscape in the stationary limit, and in turn then feeds everyone else [67, 90].", "pages": [ 44, 45, 46, 47 ] } ]

[ { "title": "The Navier-Stokes equation and solution generating symmetries from holography", "content": "Joel Berkeley 1 and David S. Berman 2 , Queen Mary University of London, Centre for Research in String Theory, School of Physics, Mile End Road, London, E1 4NS, England", "pages": [ 1 ] }, { "title": "Abstract", "content": "The fluid-gravity correspondence provides us with explicit spacetime metrics that are holographically dual to (non-)relativistic nonlinear hydrodynamics. The vacuum Einstein equations, in the presence of a Killing vector, possess solution-generating symmetries known as spacetime Ehlers transformations. These form a subgroup of the larger generalized Ehlers group acting on spacetimes with arbitrary matter content. We apply this generalized Ehlers group, in the presence of Killing isometries, to vacuum metrics with hydrodynamic duals to develop a formalism for solution-generating transformations of Navier-Stokes fluids. Using this we provide examples of a linear energy scaling from RG flow under vanishing vorticity, and a set of Z 2 symmetries for fixed viscosity.", "pages": [ 1 ] }, { "title": "5 Discussion", "content": "We have demonstrated how solution-generating transformations of the Einstein equations in the presence of a Killing vector may be applied to spacetime holographically dual to hydrodynamics. Our focus has been on the incompressible Navier-Stokes fluid dual to vacuum Rindler spacetime, where we have uncovered a selection of fluid transformations: a linear energy scaling symmetry for solutions with vanishing vorticity (this constraint may be relaxed to (44)), deriving from RG flow of the fluid hypersurface through the bulk, and a Z 2 group of transformations for fixed viscosity (boundary) with explicit examples of reflection-like symmetry in translational and rotational fluid isometries. These transformations may not be remarkable from the perspective of hydrodynamics but it shows how part of the generalized Ehlers transformations can survive holography and give rise to transformations in the fluid dual. These fluid transformations, when applied to fluid metrics, will produce solutiongenerating transformations in the vacuum Einstein equations. However, the transformed metrics produced directly by our method are not necessarily vacuum. This apparent contradiction may be explained as follows. It is discussed in [42] how the electromagnetic field strength contribution to the Navier-Stokes equations (5) is determined by the projection of the electromagnetic field strength of the bulk spacetime along the unit normal to the hypersurface. If in the transformed spacetime this projected field strength vanishes, one will still recover the Navier-Stokes equations (32) (without forcing terms) in the dual parameters. In this way, it is not strictly necessary that the fluid metrics be vacuum to recover solution-generating transformations of the unforced incompressible Navier-Stokes equations. One can then be inspired to try the Harrison transformation which is a solutiongenerating transformation in Einstein-Maxwell theory to give new transformations in magnetohydrodynamics. A holographic relation of the sort described here has been constructed for magnetohydrodynamics in [41, 42, 43]. The Harrison transformation in the bulk may then lead to nontrivial transformations in magnetohydrodynamics transforming between fluid velocity and magnetic potentials; this is the subject of current work. One can also study the dimensional dependence of these solution-generating transformations. In recent work [44], the difference in scaling for turbulence between three and four dimensions was studied holographically with a large difference in qualitative behaviour. One could also examine backgrounds associated with nontrivial chemical potential in the dual; for example rotating or charged black hole backgrounds. Essentially, in this paper we wish to open up the use of gravitational solution generating symmetries in holography. It is encouraging that this did not immediately fail and one could preserve the fluid metric ansatz with some residual transformations surviving, yet it is intriguing that these transformations did not give anything particularly new. The results for magnetohydrodynamics may prove more significant.", "pages": [ 19, 20 ] }, { "title": "1 Introduction", "content": "In 1974, Damour [1], and later in 1986, Thorne et al. [2], considered an observer outside a black hole, interacting with (perturbing) the event horizon. Surprisingly, they found that the observer will experience the perturbations of the 'stretched' horizon as modes of a viscous fluid possessing electric charge and conductivity. This inspired a host of works over the years but the connection between gravitational physics and fluids became sharper with the advent of the AdS/CFT correspondence when Policastro et. al [5] related the shear viscosity of N = 4 super Yang-Mills theory to the absorption of energy by a black brane. This was the start of using the holographic principle, that is the correspondence between gravitational theories on ( d +1)-dimensional manifolds and d -dimensional quantum field theories, as a tool for calculating hydrodynamic properties. More recently, there has been a set of works that directly relate solutions of Einstein's equations of a particular type to solutions to the Navier-Stokes equations in one dimension less [6, 7, 8, 9, 10, 11, 12, 13, 14]. Later we will review the details of how this correspondence is derived but the essential flavour is as follows. One writes down a very particular ansatz for the metric in d +1 dimensions which has undetermined functions, v i ( x, t ) , P ( x, t ) and parameter, ν with the index i = 1 , .., d -1 i.e. over a ( d -1) subset of the d spacetime dimensions. Solving the Einstein equations then constrains the functions v i ( x, t ) , P ( x, t ) to give a set of second order nonlinear differential equations for v i ( x, t ) , P ( x, t ). This set of equations are the Navier-Stokes equations describing a fluid in d dimensions with pressure, P ( x, t ), fluid velocity field v i ( x, t ) and viscosity ν . Thus particular solutions to the Navier-Stokes equations provide particular solutions to the Einstein equations. Other recent work on the fluid/gravity correspondence may be found in [20, 21, 22, 23, 24, 25, 26, 27, 28]. It has been known since Buchdahl [17] that for manifolds with isometries Einstein's equations have solution generating symmetries. That is there are 'hidden' symmetries of the equations that map from one solution to the other. In fact there is a vast set of these as described by Ehler [18] and Geroch [19]. The question we wish to pose in this paper is whether the solution generating symmetries of Einstein's equations can lead to solution generating symmetries in the Navier-Stokes equations? The procedure to determine this will be as follows: The procedure could immediately fail if there were no generalized Ehler's transformations that preserve the metric ansatz required for the fluid/gravity correspondence. We will find that there are a finite set and we will be able to explore the transformations on the Navier-Stokes fields for different choices of Killing directions in spacetime. Along the way we will show that they are not part of the usual spacetime Ehler's transformations and yet they do produce solution generating transformations for the Navier-Stokes fields. The paper will try to be as self contained as possible and so we begin with a review of the necessary ideas in fluids; the Navier-Stokes equation; the fluid gravity correspondence; and solution generating symmetries in general relativity. We will then carry out the procedure described above for spatial, timelike and null Killing vectors to see what solution generating symmetries they correspond to in the Navier-Stokes equation. We end with some comments and ideas for future work. A reader familiar with the formalism of hydrodynamics and the NavierStokes equation may wish to skip directly to section 3 where we carry out the solution generating transformation in the gravity dual to see the resulting induced transformations on the solutions of the Navier-Stokes equation.", "pages": [ 2, 3, 4 ] }, { "title": "2 Hydrodynamics", "content": "The study of hydrodynamics is fundamental to vast areas of physics and engineering, owing to its origin as the long-wavelength limit of any interacting field theory at finite temperature. Such a limit needs a consistent definition. Consider a quantum field theory where quanta interact with a characteristic length scale glyph[lscript] corr , the correlation length. The long-wavelength limit simply requires that fluctuations of the thermodynamic quantities of the system vary with a length scale L much greater than glyph[lscript] corr , parameterized by the dimensionless Knudsen number For a fluid description to be useful in non-equilibrium states, we naturally require that L remain small compared to the size of the system. This is usually satisfied trivially by considering systems of infinite size. The long-wavelength limit allows the definition of a particle as an element of the macroscopic fluid, infinitesimal with respect to the size of the system, yet containing a sufficiently large number of microscopic quanta. One mole contains an Avogadro's number of molecules, for example. Each particle defines a local patch of the fluid in thermal equilibrium, that is, thermodynamic quantities do not vary within the particle. Away from global equilibrium quantities vary between particles as function of time τ and spatial coordinates glyph[vector]x , combined as x a = ( τ, glyph[vector]x ). The evolution of particles in the fluid is parameterized by a relativistic velocity u b ( x a ), which refers to the velocity of the fluid at x a . It is well known [29] that the thermodynamic quantities, such as the temperature T ( x a ) and the density ρ ( x a ), are determined by the value of any two of them, along with the equation of state. The evolution of the system is then specified by the equations of hydrodynamics in terms of a set of transport coefficients, whose values depend on the fluid in question. Fluid flow is in general relativistic in that the systems it describes are constrained by local Lorentz invariance, and velocities may take any physical values below the speed of light. Applications at relativistic velocities are multitudinous: the dust clouds in galaxy and star formation; the flow of plasmas and gases in stars supporting fusion; the superfluid cores of neutron stars; the horizons of black holes are all described by hydrodynamics. Modelling black holes (and black branes in M/string theory) with hydrodynamics has now developed into a fundamental correspondence of central importance to our present study, as discussed in § 1. Quarkgluon plasmas behave as nearly ideal fluids and are expected to have formed after the inflationary epoch of the big bang, are reproduced in collisions at the RHIC and LHC. Non-relativistic fluids are equally ubiquitous, somewhat more familiar, and constitute an endless list of phenomena from the atmosphere to the oceans.", "pages": [ 4, 5 ] }, { "title": "2.1 The fluid equations", "content": "We begin with a discussion, adapted from [30], of the relativistic fluid described by the stress energy tensor T ab and a set of conserved currents J a I where I indexes the corresponding conserved charge. The dynamical equations of the d -spacetime dimensional fluid are For an ideal fluid, with no dissipation, the energy-momentum tensor and currents may be expressed in a local rest frame in the form where p is the pressure, q I are the conserved charges and g ab is the metric of the space on which the fluid propagates. The velocity is normalised to u a u a = -1. The entropy current is given by (3b) with the charge q being given by the local entropy density. The conservation of the entropy current illustrates the non-dissipative nature intrinsic to zero entropy production. In a dissipative fluid, there are corrections to (3). We must first take into account the interrelation between mass and energy to define the velocity field more rigorously. This is achieved by using the Landau gauge, which requires that the velocity be an eigenvector of the stress-energy tensor with eigenvalue the local energy density of the fluid (this is satisfied by the velocity normalisation for the ideal fluid). If the stress energy tensor gains a dissipative term Π ab , and the current a term Υ a I , this reads Dissipative corrections to the stress tensor are constructed in a derivative expansion of the velocity field and thermodynamic variables, where derivatives implicitly scale with the infinitesimal Knudsen number (1). Recalling that the equations of motion for the ideal fluid are composed of relations between these gradients, we may express Π ab purely in terms of the derivative of the velocity (when charges are present this is only true to to first order). This can be iterated to all orders in the expansion. Now, the derivative of the velocity may be decomposed using the acceleration A a , divergence θ , a symmetric traceless shear σ ab , and the antisymmetric vorticity ω ab into the form where and P ab = g ab + u a u b is a projection operator onto spatial directions. In the Landau frame, only the divergence and shear can contribute to first-order stressenergy tensor. A similar analysis for the charge current retains the acceleration, and if one includes the parity-violating pseudo-vector contribution the leading order dissipative equations of motion for a relativistic fluid are (2) with where η and ζ are the shear 1 and bulk viscosities respectively, χ IJ is the matrix of charge diffusion coefficients, γ I indicates the contribution of the temperature gradients and Θ I the pseudo-vector transport coefficients. The transport coefficients have been calculated in the weakly coupled QFT in perturbation theory, whereas in the strongly coupled theory, a dual holographic description may be employed, see e.g. [3].", "pages": [ 5, 6 ] }, { "title": "2.1.1 The incompressible Navier-Stokes equations", "content": "In the non-relativistic limit defined by long distances, long times and low velocity and pressure amplitudes (see e.g [12]), the fluid equations (2) with (4) become the incompressible non-relativistic Navier-Stokes equations. In flat space and in the presence of an external electromagnetic field a i , these are where f ij = ∂ i a j -∂ j a i is the field strength of a i . Ideal fluids are described by Euler's equations, where the kinematical viscosity ν (related to the shear viscosity) vanishes. We will mostly be concerned with fluid flow in the absence of external forces, where a i is zero.", "pages": [ 6 ] }, { "title": "2.2 The Navier-Stokes fluid on a Rindler boundary", "content": "A metric dual to the non-relativistic incompressible Navier-Stokes equations was first developed in [8] on the Rindler wedge, up to third order in the non-relativistic, small amplitude expansion detailed later in this section. An algorithm for generalising this metric to all orders was subsequently developed in [9], though terms calculated beyond third order are not universal. They receive corrections from quadratic curvature in Gauss-Bonnet gravity [11]. We summarise the construction in [9] here. Consider the surface Σ c with induced metric where the parameter √ r c is an arbitrary constant. One metric embedding this surface is which describes flat space (fig. 1) in ingoing Rindler coordinates x µ = ( τ, x i , r ), defined in terms of the Cartesian chart ( t, x i , z ) by The hypersurface Σ c is defined by r = r c where r is the coordinate into the bulk. Allowing for a family of equilibrium configurations, consider diffeomorphisms satisfying the three conditions The allowed set is reduced to the following boost, shift and rescaling of x µ . First, a constant boost β i , where γ = (1 -β 2 ) -1 / 2 and β i ≡ r c -1 / 2 v i . Second, a shift in r and a rescaling of τ , These yield the flat space metric in rather complicated coordinates, The Brown-York stress tensor on Σ c (in units where 16 πG = 1) is given by where are the extrinsic curvature and its mean, and n µ is the spacelike unit normal to the hypersurface. By imposing that the Brown-York stress tensor on Σ c gives that of the stressenergy tensor of a fluid we can identify the parameters of the metric (11) with the density, ρ , pressure, P and four-velocity u a of a fluid, as follows: The Hamiltonian constraint on Σ c yields a constraint on the Brown-York stress tensor When this constraint is applied to the equilibrium configurations described above, one finds the equation of state is ρ = 0 (as above), or ρ = -2 d ( d -1) p which occurs for a fluid on the Taub geometry [14]. Promoting v i and p to slowly varying functions of the coordinates x a , and regarding v i ( τ, x j ) and p = r -1 / 2 c + r -3 / 2 c P ( τ, x i ) as small perturbations, which scale as about equilibrium yields the metric which satisfies the Einstein's equations to O ( glyph[epsilon1] 3 ) if v i satisfies incompressibility, ∂ i v i = O ( glyph[epsilon1] 3 ). Corrections appear in powers of glyph[epsilon1] 2 , so this is the complete metric to second order. The metric may now be built up order by order in the hydrodynamic scaling. Assume one has the metric at order glyph[epsilon1] n -1 , where the first non-vanishing component ˆ R ( n ) µν of the Ricci tensor appears at order n . By adding a correction term g ( n ) µν to the metric at order n , resulting in a shift in the Ricci tensor δR ( n ) µν , and requiring the vanishing of the Ricci tensor is guaranteed to order n . Recalling that, in the hydrodynamic scaling, derivatives scale thus, one sees that corrections δR ( n ) µν at order n will appear only as r derivatives of g ( n ) µν . It is shown in [9] that, using the Bianchi identity and the Gauss-Codacci relations, integrability of the set of differential equations (16) defining δR ( n ) µν in terms of g ( n ) µν is given by imposing the momentum constraint, equivalent to the conservation of the stress tensor on Σ c , which is precisely the fluid equations of motion, to order n . The perturbation scheme contains several degrees of freedom. The gauge freedom of the infinitesimal perturbations for some arbitrary vector ϕ µ ( n ) ( τ, glyph[vector]x, r ) at order glyph[epsilon1] n , which may be fixed by demanding that g rµ is that of the seed metric to all orders in glyph[epsilon1] . The x a -dependent functions of integration from equation (16) may be fixed by imposing the boundary form (6) of the metric on Σ c , and also requiring regularity of the metric at r = 0, which in this construction translates to the absence of logarithmic terms in r . Corrections to the bulk metric under these conditions then become where the F ( n ) a ( τ, glyph[vector]x ) comprise of the remaining integration functions, and the final degree of freedom; field redefinitions of δv ( n ) i and δP ( n ) at order glyph[epsilon1] n . F ( n ) i ( τ, glyph[vector]x ) is related to redefinitions of the fluid velocity and is fixed by the isotropic gauge condition P b a T bc u c = 0. F ( n ) τ ( τ, glyph[vector]x ) is related to redefinitions of the pressure and is fixed by defining the isotropic part of T ij to be to all orders. Applying the perturbation scheme to the seed metric yields to third order, which satisfies the vacuum Einstein equations if which are the Navier-Stokes equations with kinematical viscosity The corresponding corrections to the Navier-Stokes equations follow from conservation of the stress tensor on Σ c . Vector and scalar quantities are odd and even orders respectively in the scaling glyph[epsilon1] . Accordingly, corrections to the scalar incompressibility equation appear at even orders, and to the vector Navier-Stokes equations at odd orders.", "pages": [ 7, 8, 9, 10, 11 ] }, { "title": "3 Duality in the context of holography", "content": "The defining equations in general relativity are the Einstein field equations, and in the non-relativistic limit of hydrodynamics, the Navier-Stokes equations (5). Each is a set of non-linear partial differential equations whose solutions exhibit fantastically varied phenomenology. When approaching any complex physical system with a view to finding solutions, it is often advantageous to consider the symmetries, intensively studied in both of these systems since their conceptions. Beyond diffeomorphisms, the search in gravity has in general been somewhat limited [15, 36], however in the presence of a spacetime isometry, the symmetry group becomes remarkably large [35], particularly for vacuum spacetimes. For symmetries of the Navier-Stokes equations see [31], and with regards to the conformal group [12, 32]. In the light of the fluid/gravity correspondence, one may ask whether the symmetries of these systems are linked. In [37, 38, 39], they apply known symmetries of the Einstein equations to spacetimes with perfect fluid sources, constructing new spacetimes with the same equation of state, though not within a holographic framework. By drawing on the tools provided by these works and those in holography, we hope to develop a more general approach to the problem. The bulk provides an additional valuable degree of freedom, where the boundary sets the scene for the fluid evolution on the induced geometry. Moreover, we are now free to exploit the symmetries of the more extensive yet simpler vacuum geometries. It is these symmetries which we intend to holographically project to the fluid. In particular, we are interested in transformations between solutions to the Navier-Stokes equations arising from transformations between solutions to the vacuum Einstein equations: transformed metrics yield transformed fluid configurations. In this section, we discuss the work leading up to the spacetime Ehler's symmetry group of the vacuum Einstein equations with zero cosmological constant, itself contained within the generalized Ehlers group. We continue in § 4 to apply the latter, in the presence of a Killing isometry, to fluids on the boundary of the Rindler space, thus deriving solution generating transformations of the fluid velocity, pressure and viscosity (the latter defining the RG flow). We offer in § 4.1.1 a selection of example transformations including RG flow for zero vorticity fluids (where one may relax this constraint), and Z 2 transformations for fixed viscosity which we show in § 4.3 in fact lie outside the spacetime Ehler's group.", "pages": [ 11, 12 ] }, { "title": "3.1 Symmetry groups of the Einstein equations", "content": "Understanding the properties of the Einstein field equations has long been a subject of great theoretical interest, a sensible starting point being the inherent symmetries involved. To this end, Buchdahl [17] derived a form of duality in vacuum spacetime metrics, where an n -dimensional vacuum metric static with respect to a coordinate x s : generates a dual vacuum metric glyph[negationslash] It is this solution-generating property of spacetime isometries we wish to apply to solutions of the Einstein equations and holographically map to hydrodynamics. We have, however, a considerably larger symmetry group at our disposal. The authors of [16, 18, 19] develop the concept culminating in the generalized Ehlers symmetry group of the Einstein equations also for non-vacuum spacetimes. An extension exists [33, 34] to dualities between vacuum spacetimes and those with electromagnetic backgrounds described by the Einstein-Maxwell equations, of relevance for magnetohydrodynamics.", "pages": [ 12 ] }, { "title": "The generalized Ehlers group", "content": "Define a vector field ξ = ξ µ ∂ µ and one-form W = W µ d x µ on a manifold with metric g = ( g µν ). The generalized Ehlers group is defined in [16] by the transformation where Ω 2 ≡ ξ α W α + 1 ≥ 1, and the inequality holds over the whole geometry. This group does not send vacuum metrics to vacuum metrics in general, but such transformations may be found in the spacetime Ehlers subgroup.", "pages": [ 12 ] }, { "title": "The spacetime Ehlers group", "content": "Let us restrict g to be some (3 + 1)-dimensional Lorentzian metric satisfying the vacuum Einstein equations and exhibiting some Killing isometry. Let us restrict ξ to define this Killing isometry, which is equivalent to the condition that the Lie derivative of the metric along ξ vanishes: The twist potential and Killing vector norm give the Ernst one-form for some scalar ς (exactness is guaranteed by vanishing Ricci tensor, see [40] p.164). Define a self-dual two form where ∗ is the Hodge dual operator. The spacetime Ehlers group is defined for (3 + 1)-dimensional Lorentzian metrics by (24) for W satisfying where a bar denotes complex conjugation, and γ and δ are non-simultaneously vanishing real constants, which as a pair fix the gauge of W . The transformation defines an SL (2 , R ) group action on the Ernst scalar by the Mobius map", "pages": [ 12, 13 ] }, { "title": "4 Solution-generating transformations on the NavierStokes fluid", "content": "Consider those transformed metrics h ( ξ, W, g ) which preserve the functional form g (it clear that this is not in general the case). In the case of the Rindler metric dual to the incompressible Navier-Stokes fluid, we define the parameters of g by the fluid velocity v i , pressure P , and boundary position r c within the bulk. In the transformed metric h , we define the transformed parameters by ˜ v i , ˜ P and ˜ r c , denoted by ' ∼ '. On satisfying the vacuum Einstein equations on ˜ Σ c , now at r = ˜ r c in the transformed geometry, the transformed metric will yield the incompressible Navier-Stokes equations in the transformed parameters Vitally, if ( v i , P ) satisfy the Navier-Stokes equations with viscosity ν = r c , then the transformed velocity and pressure (˜ v i , ˜ P ) represent a new set of solutions for viscosity ν = ˜ r c . That is, we look for a subset of the generalized Ehlers transformation acting on the fluid metric (21), obeying some Killing isometry, which corresponds to solution-generating transformations of the velocity and pressure, and RG flow parametrised by r c , of an incompressible Navier-Stokes fluid. The Rindler metric is just one fluid metric supporting flat background geometries on the boundary. We therefore only wish instead to retain the common features of such metrics; the metric gauge g µr , and the flat boundary metric of the form (6). The equation we wish to solve is thus where", "pages": [ 13, 14 ] }, { "title": "4.1 Transforming the fluid", "content": "We are provided in (34) with sufficient information to derive the possible fluid transformations via the form of the one-form W . Preserving the vanishing of the ˜ g rr = g rr = 0 component of the metric we find, directly from (24), the two possibilities One may obtain an expression for W a by contraction of (24) with the boundary indices ( a, b, . . . ) of the Killing vector: (Note, here and in what follows ξ µ = g µν ξ ν , ie. it is lowered with the metric g µν and never with ˜ g µν ). This expression is uniquely defined only at the dual boundary ˜ Σ c , where we have defined the form of ˜ g ab and W a becomes independent of the dual fluid velocity and pressure. These expressions diverge for null Killing vectors, where λ = 0. We cover this case shortly. One can see that the parameters of the fluid is determined, to all orders in glyph[epsilon1] , by g ar = g ar | r c . Consequently, the transformation in the fluid will be given by the transformation of these components. Evaluation at ˜ r c is necessary in order to circumvent the ambiguity in the dual metric, and also provides explicit fluid transformations. We begin with Killing vectors null at the dual boundary, λ | ˜ r c = 0, where one finds from contraction of (24) with the Killing vector, which yields the following transformation, accompanied by the preservation of a null Killing vector, ξ µ ξ ν ˜ g µν = 0. For non-null Killing vectors we employ the relations derived by comparing (35) and (37), and found from contraction of (24) twice with the Killing vector. Inserting W r (35) and W a (36) into the Ehlers transformation (24) and employing (39) and (40), one finds", "pages": [ 14, 15 ] }, { "title": "4.1.1 Energy scaling invariance from a bulk isometry", "content": "We begin with an example of a (null) Killing vector into the bulk, ξ = ξ r ( x µ ) ∂ r . The Killing equation components ( L ξ g ) ai = 0 yield Integrability of these equations requires firstly where we have used the Navier-Stokes equations to express the constraint in this form. Additionally, integrability requires vanishing vorticity to first order, which with incompressibility implies (44). Transformation (38) yields ˜ g ar = g ar , or which is exact to all orders. It is trivial to show that the pair (˜ v i , ˜ P ) satisfy the incompressible Navier-Stokes equations (with viscosity ˜ r c ) if ( v i , P ) do so (with viscosity r c ) for velocities satisfying (44) alone - vanishing vorticity imposes unnecessary constraint and removes the dissipative term from the fluid equations. It is interesting to consider the problems of existence, uniqueness and regularity of the Navier-Stokes in this case. The divergence of (44) yields a vanishing mean square vorticity which ensures the class of solutions ( v i , P ) generated by (45) are regular. With respect to existence, the kinetic energy scales by a factor ˜ r c /r c and thus is bounded if there exists any solution satisfying (44) where the energy is finite.", "pages": [ 15, 16 ] }, { "title": "4.1.2 The timelike Killing vector", "content": "One might expect, in the presence of a timelike Killing vector ξ = ∂ τ (it is sufficient for this discussion to consider stationary solutions), a transformation of the form enacting time-reversal of the fluid but this is not the case. This is explained by noting that time-reversal is enacted by redefining the viscosity by ν = ± r c [8] rather than by changing r c itself. This is because sending r c → -r c brings the fluid outside the causal region of the spacetime.", "pages": [ 16 ] }, { "title": "4.2 Fixed viscosity transformations", "content": "We turn to fixed boundary (viscosity) transformations, where ˜ r c = r c . For Killing vectors null at the dual boundary then α = 0, and one recovers the identity. For non-null Killing vectors with α = 1, one finds which defines a Z 2 group.", "pages": [ 16 ] }, { "title": "4.2.1 Spacelike Killing vectors", "content": "Consider a generic space-like Killing vector ξ = ξ k ∂ k . Under (47), the pressure is preserved, while the velocity transforms as which is a reflection in the hyperplane normal to the Killing vector and containing the point at which the velocity is defined.", "pages": [ 17 ] }, { "title": "Translational isometry", "content": "Consider ξ = c k ∂ k where the constants c k are normalised to ∑ k c 2 k = 1, and the corresponding isometries are c k ∂ k v i = c k ∂ k P = 0. The dual fields are The incompressibility condition and Navier-Stokes equations are satisfied by the incompressible Navier-Stokes equations in the original fluid parameters along with the isometries. This is valid for fluids of arbitrary dimension.", "pages": [ 17 ] }, { "title": "Rotational isometry", "content": "Consider a Killing vector ξ = -x 2 ∂ 1 + x 1 ∂ 2 corresponding to a rotational isometry in a d -dimensional fluid. In polar coordinates where primed ( ' ) indices run from 3 to ( d -1), and the Killing vector becomes ξ = ∂ θ , the isometries are Solutions satisfy with dual velocities that is, the transformation sends η →-η (equivalently θ →-θ ). The incompressible Navier-Stokes equations for the original fluid may be expressed as - r c (3 ρ ∂ ρ µ + ∂ 2 ρ µ + ∂ ∂ k ' µ ) It is clear from the parity of these equations in η that if there exists a fluid solution defined in terms of a pair ( µ, η ) by (54), then there also exists a solution parameterized by the pair ( µ, -η ). That is, the transformed fluid satisfies the incompressible Navier-Stokes equations. Again, this is valid for fluids of arbitrary dimension. We provide an example with the three-dimensional fluid solution which satisfies the isometries (53). Here, A , B and τ 0 are arbitrary non-vanishing constants and q ( τ ) is an arbitrary function of time. The duality is equivalent to sending τ 0 → τ 0 + iπ/ 2 A .", "pages": [ 17, 18 ] }, { "title": "4.3 Generalized versus spacetime Ehlers", "content": "In this section we discuss whether the fluid transformations of § 4.2.1, derived directly from the generalized Ehlers transformation (24) on the basis of definition (34) of the dual metric components, lie within the spacetime Ehlers group. The spacetime Ehlers map is defined only in (3+1) spacetime dimensions, so we assume this number of dimensions for all calculations in this section. Consider the fixed boundary transformations (47). For those examples calculated in § 4.2.1, the Ernst scalar transforms as - 1 k ' In the case of the vacuum-to-vacuum spacetime Ehlers group, the Ernst scalar transforms according to the Mobius map (31). If conjugation of the Ernst scalar is to belong to this map, we must have which is satisfied if and only if Ω 2 = 1. Explicit calculation for those examples in § 4.2.1 shows that Ω 2 = 1 requires ( γ, δ ) = (0 , ± 1). The Mobius map (31) with these parameter values reduces to That is, complex conjugation and the Mobius map are equivalent only where they describe a constant complex shift. It follows from the definition of complex conjugation that this demands a vanishing twist potential (26). For those fluid transformations of § 4.2.1, the twist potential is non-vanishing, so complex conjugation of the Ernst scalar does not here correspond to the Mobius map. Therefore these fluid transformations, generated by the generalized Ehlers group on the basis of the dual metric definition (34), do not correspond to transformations within the spacetime Ehlers group.", "pages": [ 18, 19 ] }, { "title": "Acknowledgements", "content": "We would like to thank Cynthia Keeler and Andrew Strominger for their illuminating and insightful thoughts and comments on the fluid gravity correspondence and Shiraz Minwalla whose lectures on hydrodynamics at the Isaac Newton Institute programme on M-theory were an inspiration. This work is partially supported by STFC consolidated grant ST/J000469/1.", "pages": [ 20 ] } ]

[ { "title": "Petrov type I Condition and Dual Fluid Dynamics", "content": "Rong-Gen Cai ∗ , Li Li † , Qing Yang ‡ , Yun-Long Zhang § State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, People's Republic of China. February 11, 2014", "pages": [ 1 ] }, { "title": "Abstract", "content": "Recently Lysov and Strominger [arXiv:1104.5502] showed that imposing Petrov type I condition on a ( p +1)-dimensional timelike hypersurface embedded in a ( p +2)dimensional vacuum Einstein gravity reduces the degrees of freedom in the extrinsic curvature of the hypersurface to that of a fluid on the hypersurface, and that the leading-order Einstein constraint equations in terms of the mean curvature of the embedding give the incompressible Navier-Stokes equations of the dual fluid. In this paper we show that the non-relativistic fluid dual to vacuum Einstein gravity does not satisfy the Petrov type I condition at next order, unless additional constraint such as the irrotational condition is added. In addition, we show that this procedure can be inversed to derive the non-relativistic hydrodynamics with higher order corrections through imposing the Petrov type I condition, and that some second order transport coefficients can be extracted, but the dual 'Petrov type I fluid' does not match the dual fluid constructed from the geometry of vacuum Einstein gravity in the nonrelativistic limit. We discuss the procedure both on the finite cutoff surface via the non-relativistic hydrodynamic expansion and on the highly accelerated surface via the near horizon expansion.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "In the non-relativistic hydrodynamic limit, a correspondence between the nonlinear solutions of the Einstein equations and incompressible Navier-Stokes equations is constructed in [1, 2, 3] where an intrinsically flat finite cutoff surface and regularity on the future horizon are imposed. Two equivalent presentations of the non-linear perturbed gravity solution and dual fluid expansion are given, one is for the dual fluid living on a finite cutoff surface via non-relativistic hydrodynamic expansion, the other is on the highly accelerated surface via near horizon expansion. This relation is further shown to be universal for the geometry with sphere horizon [4, 5] and with higher curvature corrections [6, 7, 8, 9, 10]. And the dual incompressible Navier-Stokes equations are found to be corrected at leading order when a non-trivial gravitational Chern-Simons term appears in the bulk [11]. More generally, the gravity is related with a fluid without gravity in one lower dimension, and related works can also be found in [12, 13, 14, 15, 16, 17, 18, 19, 20], which show their close relation with the fluid dynamics from membrane paradigm [21, 22, 23, 24, 25], as well as the fluid/gravity correspondence from holography [26, 27, 28, 29, 30]. It was noted in [2] that the nonlinear solution of vacuum Einstein gravity is of an algebraically special Petrov type [31, 32, 33], and the procedure was reversed via the near horizon expansion in [34] to derive the dual hydrodynamics. The Petrov type I condition is imposed to reduce the Einstein equations to the incompressible Navier-Stokes equations in one lower dimension. The universal fixed-point behavior of the near-horizon scaling in general relativity is shown to be the same as that of hydrodynamic scaling in fluid dynamics [34]. This condition is expected to be equivalent to the regularity on the future horizon, and the framework has also been generalized to the highly accelerated surface which is spatially curved, and to the case with cosmological constant and Maxwell field in the bulk [35, 36, 37]. Note that in those works only the nontrivial leading order has been considered, we are here going to generalize the procedure to higher order to see whether the equivalence still holds or not. In the frame which is associated with a hypersurface where the dual fluid lived on, we find that the non-relativistic fluid dual to the non-linear solution of vacuum Einstein gravity from boost transformation does not satisfy the Petrov type I condition at the next order, unless additional constraint is added such as the irrotational condition. We also inverse this procedure by imposing the Petrov type I condition on the fluid stress tensor, and then obtain the non-relativistic hydrodynamics with higher order corrections. But we see that the dual 'Petrov type I fluid' can not match the dual fluid of vacuum Einstein gravity constructed in the non-relativistic limit. We study the procedure in two equivalent expansions: one is the non-relativistic hydrodynamic expansion associated with a finite cutoff surface, the other is the near horizon expansion associated with a highly accelerated surface. This paper is organised as follows. In section 2, a simple review of the Petrov type I condition is given. In section 3, the higher order non-relativistic stress tensor dual to vacuum Einstein gravity is used to check the Petrov type I condition. Then the logic is turned around and the Petrov type I condition is imposed to reduce the gravity to the dual non-relativistic hydrodynamics. In section 4, an alternative presentation of this procedure in the near horizon expansion is discussed. The results and discussions are given in section 5.", "pages": [ 2, 3 ] }, { "title": "2 Petrov type I condition", "content": "Firstly, we give a simple review of the Petrov type I condition with respect to the ingoing and outgoing pair of null vectors whose tangents to a timelike hypersurface generate time translations [34]. Introducing the ( p +2) Newman-Penrose-like vector fields, the spacetime is Petrov type I [32, 33] if for some choice of frame, Consider a timelike ( p +1)-dimensional hypersurface Σ c with flat intrinsic metric and extrinsic curvature K ab . The hypersurface is embedded in a ( p +2)-dimensional vacuum Einstein spacetime that Choosing the frame that where n is the spacelike unit normal to the hypersurface, and ∂ i , ∂ 0 are the tangent vectors to Σ c [34], one has where the following projections to Σ c have been used with γ α a = δ α a -n a n α . The Petrov type I condition (2) imposes ( p -1)( p +2) / 2 constraints on the ( p +1)( p +2) / 2 components of K ab , or determines the trace-free part of K ij in terms of K, K 00 and K 0 i . This leaves ( p + 2) independent components, which are exactly the number of components of a fluid with a local energy density, pressure and velocity. The dual fluid is described by the Brown-York stress tensor on the hypersurface, The Hamiltonian constraint of vacuum Einstein equations can be viewed as the equation of state for the dual fluid relating the pressure and energy density. On the other hand, the ( p +1) momentum constraint equations give us the equations of motion for the dual fluid.", "pages": [ 3, 4 ] }, { "title": "3 On finite cutoff surface", "content": "In this section, with the non-relativistic stress tensor of fluid dual to vacuum Einstein gravity at finite cutoff surface given in [3], we will firstly check whether the Petrov type I condition is satisfied or not at higher orders. Then we impose the Petrov type I condition to reduce the gravity to the dual non-relativistic hydrodynamics. With the ingoing Rindler metric the induced metric at the finite cutoff surface r = r c is The Hamiltonian constraint becomes H = 0, where Defining P ij = 4 C ( /lscript ) i ( /lscript ) j and using equations (6) and (8), the Petrov type I condition turns out to be P ij = 0, where", "pages": [ 4, 5 ] }, { "title": "3.1 Non-relativistic fluid and Petrov type I condition", "content": "Take the non-relativistic expansion in [2, 3] the Brown-York stress tensor up to order /epsilon1 4 can be expressed as [3] where the fluid shear σ ij and vorticity ω ij are given by 1 Comparing this stress tensor with the non-relativistic fluid stress tensor given in Appendix B.1, one can read off some transport coefficients as The equations of motion of the dual fluid ∂ a T ab = 0 turn out to be the incompressible Navier-Stokes equations with higher order corrections given in (80), and the stress tensor satisfies the Hamiltonian constraint H = 0 consistently. Inserting the stress tensor (16)-(19) into P ij and expanding in powers of parameter /epsilon1 , one has Taking into account the equations of motion (80), one can see that P (0) ij and P (2) ij vanish identically, but This result can also be obtained through substituting the nonlinear solution of vacuum Einstein gravity given in Appendix A.1 into the Weyl tensor (2) directly. And it is independent of the gauge transformation that v i → v i + δv i or T ij → T ij + δPδ ij , where δv i ∼ /epsilon1 3 , δP ∼ /epsilon1 4 . Thus the perturbed stress tensor (16)-(19) on the finite cutoff surface does not satisfy the Petrov type I condition at order /epsilon1 4 , if we choose this frame (5) associated with the finite cutoff hypersurface. Or in other words, the non-linear solution of vacuum Einstein gravity constructed by boost transformation, up to order /epsilon1 4 , does not satisfy the Petrov type I condition. But we can additionally require the constraint P (4) ij = 0 holds. For example, if we take the irroational condition with ω ij ∼ O ( /epsilon1 4 ), then in view of θ ≡ ∂ i v i ∼ O ( /epsilon1 4 ), one has Thus P (4) ij vanishes at this order and T ij is reduced to In this case, comparing (26) with the non-relativistic fluid stress tensor in Appendix B.1, we can read off The incompressible Navier-Stokes equations with higher order corrections (80) are reduced to where the higher order corrections become f ( σ ) i = - - - 3 r c ∂ i ( σ kl σ kl ) + 4 r c σ kl ∂ k σ li 2 v k ∂ k ∂ i P 2( ∂ k v i ) ∂ k P Here according to (24), the term r c ∂ 2 v i ∼ O ( /epsilon1 5 ), therefore we move this term to the right hand side of the Navier-Stokes equations in (28).", "pages": [ 5, 6 ] }, { "title": "3.2 From Petrov type I condition to dual fluid", "content": "At the finite cutoff surface, if we impose the Petrov type I condition P ij = 0 firstly, and consider the non-relativistic hydrodynamic scaling laws in (15), then the Brown-York stress tensor can be expanded in powers of the non-relativistic hydrodynamic expansion parameter /epsilon1 as Here superscript in round brackets stands for the expansion order, such as T τ (1) i ∼ /epsilon1, T τ (3) i ∼ /epsilon1 3 , and so on. The Brown-York stress tensor at the cutoff surface r = r c of the metric (11) gives We now put the expansions (31) into the Hamiltonian constraint equation (13) and the Petrov equations (14), which both can be expanded in powers of the parameter /epsilon1 . The first non-trivial order appears at order /epsilon1 2 , where the Hamiltonian constraint H (2) = 0 and Petrov type I condition P (2) ij = 0 lead to respectively. Following [34], if we assume that we can recover the stress tensor (16)-(19) up to order /epsilon1 2 . The next non-trivial Hamiltonian constraint H (4) = 0 and Petrov type I condition P (4) ij = 0 give respectively. To give assumptions at higher orders, we choose the Landau frame which gives where u a = γ v (1 , v i ) and γ ab u a u b = -1 [3]. At order /epsilon1 3 , its spatial components lead to where the energy density e ≡ T ab u a u b . With the recovered stress tensor up to /epsilon1 2 , one can show e (2) = 0. Putting (34) and (35) into the above equation, we obtain Then T τ τ in (17) can be recovered up to order /epsilon1 4 with the Hamiltonian constraint which leads to (33) and (36). On the other hand, putting (34) (35) and (40) into (37), one finds that at order /epsilon1 4 , there is only one term T (4) δ ij proportional to δ ij . Thus, we can choose the isotropic gauge with T (4) = 0 as in [3], and finally T (4) ij is given by Compare (41) with the terms in (18) at order /epsilon1 4 , we obtain the additional terms Thus, the incompressible Navier-Stokes equations with higher order corrections from the equations of motion of the fluid ∂ a T ab = 0 become where θ and f i are given in (81) and (82), respectively, and Comparing (41) with the non-relativistic fluid dual to vacuum Einstein gravity constructed in Appendix B.1, one can extract the second order transport coefficients as which implies that the correction terms in (23) do not contribute to the terms associated with second order transport coefficients. Thus, such kind of higher order fluid reduced from the Petrov type I condition, which we name as 'Petrov type I fluid', does not satisfy the non-relativistic fluid that constructed in Appendix B.1. However, if additionally requiring that the terms in (23) vanish at this order, we can again recover the previous stress tensor (16)-(19), up to order /epsilon1 4 . In particular, taking the irrotational condition that ω ij ∼ O ( /epsilon1 4 ), we can recover equations (26)-(30).", "pages": [ 7, 8 ] }, { "title": "4 On highly accelerated surface", "content": "An alternative presentation of the procedure discussed in the previous section can also be realized with the near horizon expansion. Introducing the expansion parameter λ = r 1 / 2 c via the transformation τ → λ -2 ˆ τ, r → λ 2 ˆ r, x → ˆ x , the ingoing Rindler metric (11) becomes which gives the first three terms in (85). The induced metric (12) changes into In the hatted coordinates, the Hamiltonian constraint becomes ˆ H = 0, where The Petrov type I condition turns out to be ˆ P ij = 0, where", "pages": [ 8, 9 ] }, { "title": "4.1 Near horizon fluid and Petrov type I condition", "content": "In the near horizon expansion, with the transformations (83),(84) and (100), the stress tensor (16)-(19) becomes where the fluid shear ˆ σ ij ≡ ˆ ∂ ( i ˆ v j ) and vorticity ˆ ω ij ≡ ˆ ∂ [ i ˆ v j ] . Comparing the stress tensor with the one of dual fluid given in Appendix B.2, one has The equations of motion ˆ ∂ a ˆ T ab = 0 turn out to be (87), and the stress tensor satisfies the Hamiltonian constraint ˆ H = 0 consistently. Inserting equations (50)-(53) into ˆ P ij with expansion in powers of λ , we have We see that ˆ P ( -2) ij and ˆ P (0) ij This is independent of the gauge transformation with ˆ v i → ˆ v i + λ 2 δ ˆ v i or ˆ T ij → ˆ T ij + λ 3 δ ˆ Pδ ij . Thus the perturbed stress tensor (50)-(53) does not satisfy the Petrov type I condition at order λ 2 , if we choose this frame (5). Again, we can also additionally require ˆ P (2) ij = 0. For example, if we add the irroational condition that ˆ ω ij ∼ O ( λ 2 ), then ˆ P (2) ij vanishes at this order and ˆ T ij is reduced to Comparing this with the stress tensor of dual fluid given in Appendix B.2, we have In this case, the incompressible Navier-Stokes equations with higher order corrections (87) are reduced to where the higher order corrections are given by Since the term ˆ ∂ 2 ˆ v i ∼ O ( λ 2 ), it is therefore put on the right hand side of the equation (59).", "pages": [ 9, 10 ] }, { "title": "4.2 From Petrov type I condition to dual fluid", "content": "In this subsection we will inverse the procedure and expand the Brown-York stress tensor in powers of the parameter λ with the background metric (47), Note that here only the odd order terms are selected. The even order terms can also be added, because it can be shown that they give no further information of the higher order fluid, and thus are set to be vanished to satisfy the constraint equations as well as Petrov type I condition. We now put the expansions (62) into the Hamiltonian equation (48) and the Petrov equations (49), which both can be expanded in powers of the parameter λ . The first non-trivial order appears at λ 0 , where the Hamiltonian constraint ˆ H (0) = 0 and Petrov type I condition ˆ P (0) ij = 0 lead to respectively. Again, following [34], if assuming that we can recover the stress tensor (50)-(53) up to order λ . The next non-trivial Hamiltonian constraint ˆ H (2) = 0 and Petrov type I condition ˆ P (2) ij = 0 give respectively. To give assumptions at higher order, we choose the Landau frame which gives where ˆ u a = ˆ γ v (1 , ˆ v i ) and ˆ γ ab ˆ u a ˆ u b = -1. At order λ , the spatial components give us with where ˆe ≡ ˆ T ab ˆ u a ˆ u b . From the recovered stress tensor up to order λ we have e (1) = 0. Putting (64) and (65) into the above equation we get Then ˆ T ˆ τ τ in (51) can be recovered up to order λ 3 via the Hamiltonian constraint which leads to (63) and (66). On the other hand, putting (64)(65) and (70) into (67), one finds that at order λ 3 , there is only one term ˆ T (3) δ ij proportional to δ ij . Thus, we can choose the isotropic gauge so that ˆ T (3) = 0 and ˆ T (3) ij can be expressed as Comparing (71) with the terms in (52) at order λ 3 , one can find that the additional terms are Thus, the incompressible Navier-Stokes equations with higher order corrections from the equations of motion of the fluid ˆ ∂ a ˆ T ab = 0 become where ˆ θ and ˆ f i are given in (88) and (89), respectively, and Comparing (71) with the stress tensor of fluid given in Appendix B.2, one can obtain the second order transport coefficients as Thus, we have shown that the additional corrections do not make contribution to the second order transport coefficients. Such kind of higher order Petrov type I non-relativistic fluid does not match the fluid constructed in Appendix B.2. However, if we additionally require that the terms in (72) vanishes at this order, the stress tensor (50)-(53) can be recovered. In particular, taking the irrotational condition with ˆ ω ij ∼ O ( λ 2 ), we can still recover equations (57)-(61).", "pages": [ 10, 11, 12 ] }, { "title": "5 Conclusion and Discussion", "content": "In Einstein gravity, the Petrov type I condition relates the gravity theory to a dual fluid without gravity in one lower dimension. It reduces the the extrinsic curvature of a time-like hypersurface to p +2 components, which can be interpreted as the energy density, pressure and velocity field of a dual fluid living on the hypersurface, constrained by equation of state and p + 1 evolution equations (incompressible Navier-Stokes equations) that come from the Einstein constraint equations [34]. To the leading order there are two equivalent presentations, one is for the dual fluid living on a finite cutoff surface via non-relativistic hydrodynamic expansion, and the other on a highly accelerated surface via the near horizon expansion. Imposing the Petrov condition is mathematically much simpler than imposing regularity on the future horizon. Via appropriate gauge choice, we generalized this procedure to the next order and obtained the incompressible Navier-Stokes equations with higher order corrections and associated second order transport coefficients. More higher order hydrodynamics can also be obtained order by order with appropriate expansion parameters. We can recover the nonrelativistic fluid stress tensor dual to vacuum Einstein gravity from boost transformation up to order /epsilon1 4 , only if by imposing additional constraint such as the irrotational condition. In other words, the non-linear solution of vacuum Einstein equations constructed by boost transformation does not satisfy the Petrov type I condition up to order /epsilon1 4 , although it holds at the order /epsilon1 2 . As the dual fluid constructed in Appendix B is reduced from the relativistic hydrodynamics, while the Petrov type I condition singles out a preferred time coordinate and thus breaks Lorentz invariance of the hypersurface [34], it might be not surprised that the 'Petrov type I fluid' does not match the boosted fluid at higher orders. In this sense it would be interesting to construct the non-relativistic hydrodynamics of this special higher order fluid directly, with the corresponding non-linear gravitational solution. In addition, note that the Petrov type I condition (2) looks different depending on choice of frame. In this sense it should be instructive to consider a different frame instead of (5), such as m i = h i , √ 2 /lscript = u -n, √ 2 k = -u -n , where u is the fluid velocity and h i (with i = 1 , ..., p ) are the spatial basses orthogonal to both u and n . 2 In general, these basses can be written as where the fluid velocity u has be defined as ( γ, γβ i ) with γ = (1 -β 2 ) -1 / 2 , and β i ≡ r -1 / 2 c v i if we use the induced metric (12) on a finite cutoff surface. Taking the non-relativistic hydrodynamic limit in (15) one can show that the result in (23) becomes P (4) ij = ∂ 2 σ ij , which is the third order in derivative expansion of the velocity. Inversely, if the condition P ij = 0 is imposed, only one term f ( ω ) i = -r 2 c 2 ∂ 4 v i is left in the correction terms (44). Thus, the frame given in (76) has better properties than the one in (5), and it is expected that in the case of derivative expansion with the relativistic fluid solution [15, 16], the Petrov type I condition in this frame holds at least up to second order [38]. In the nonrelativistic hydrodynamic expansion discussed in this paper, the additional term P (4) ij = ∂ 2 σ ij might be reduced from the third order in derivative expansion of the relativistic fluid. In the near horizon expansion, the situation is similar. Thus it would be an interesting question whether one can find a frame such that the Petrov type I condition on finite cutoff surface holds up to order /epsilon1 4 . In addition, changing the boundary conditions of the hypersurface to see their effects at higher orders, and generalizing to other bulk geometries would also be valued for further works. Note added: During the preparation of this work, we were informed that the leading order calculation in section 3.2 might have some overlap with the work in [39]. After finishing this work, we were told that the authors in [3] also reached the conclusion that the Petrov type I condition does not hold at higher orders for the non-linear solution of vacuum Einstein gravity (unpublished, May 2011).", "pages": [ 12, 13 ] }, { "title": "Acknowledgements", "content": "This work was supported in part by the National Natural Science Foundation of China (No.10821504, No.10975168 and No.11035008), and in part by the Ministry of Science and Technology of China under Grant No. 2010CB833004. We thank Professors Yi Ling, Yu Tian and Xiao-Ning Wu for useful discussions. In particular, we thank Professor Kostas Skenderis for helpful correspondence and valuable comments on this manucript. R. -G. Cai thanks the long-term workshop YITP-T-12-03 on 'Gravity and Cosmology 2012', Y. -L. Zhang thanks the workshop on 'Gauge/Gravity Duality' (YIPQS (2012)). Both workshops were held at Yukawa Institute for Theoretical Physics at Kyoto University.", "pages": [ 14 ] }, { "title": "A Nonlinear metric solution", "content": "In this Appendix, we briefly give the nonlinear solution of vacuum Einstein equations G µν = 0, which is obtained via the non-relativistic hydrodynamic expansion and near horizon expansion, respectively [2, 3].", "pages": [ 14 ] }, { "title": "A.1 Non-relativistic hydrodynamic expansion", "content": "Associated with the non-relativistic hydrodynamic expansion the metric which solves Einstein equations (4) up to order /epsilon1 4 is given as [3], where The dual fluid satisfies the incompressible Navier-Stokes equations with higher order corrections where", "pages": [ 14, 15 ] }, { "title": "A.2 Near horizon expansion", "content": "An alternate presentation of the metric (78) was given in [2], through taking the coordinate transformations so that ˆ ∂ i ≡ ∂ ∂ ˆ x i ∼ /epsilon1 0 , ∂ ˆ τ ∼ /epsilon1 0 , and ∂ ˆ r ∼ /epsilon1 0 . In the new coordinates one defines and ˆ v 2 ≡ ˆ v i δ ij ˆ v j . Considering the rescaled metric dˆ s 2 p +2 = /epsilon1 2 r -2 c d s 2 p +2 and defining λ 2 = /epsilon1 2 r -1 c , one finds where The incompressible Navier-Stokes equations (80) change into where With these constraints the metric (85) solves the vacuum Einstein equations (4) up to order λ 0 consistently. To solve the next non-trivial order that at λ 2 , especially the ˆ τ ˆ τ and ˆ τi components, the terms ˆ g (4) ˆ τi and ˆ g (4) ˆ τ ˆ τ are needed. We do not intend to find their explicit expressions here, as it is found that at order λ 2 , they do not contribute to the Petrov type I equation in (2).", "pages": [ 15, 16 ] }, { "title": "B The dual Fluid", "content": "To discuss the fluid dual to vacuum Einstein gravity, the theory of relativistic hydrodynamics up to second order in fluid gradients was presented in [3, 7, 15]. Choosing the Landau frame of the relativistic fluid with velocity u a so that its stress tensor is written as where e and p represent the energy density and pressure of the fluid in the local rest frame. The induced metric h ab = γ ab + u a u b , with γ ab the intrinsically flat metric and γ ab u a u b = -1. The dissipative corrections can be written down through taking the isotropic gauge so that Π ⊥ ab does not contain terms proportional to h ab . Up to second order in gradients, where D ⊥ a ≡ h b a ∂ b , D ≡ u a ∂ a have been defined, η is the relativistic kinematic shear viscosity, and c 1 , ..., c 6 are the corresponding transport coefficients at the second order. The equations of motion ∂ b T ab at the lowest order have been considered in writing down the above form, and the relativistic shear and vorticity are defined as The energy density which vanishes for equilibrium configurations can also be expanded as where ζ ' is an alternative first order transport coefficient which is similar to the bulk viscosity that measures variations of the energy density, and d 1 , ..., d 5 are the corresponding second order transport coefficients. However, these six coefficients are not independent [15], if we consider the equation of state of this special fluid dual to vacuum Einstein gravity that T 2 -pT ab T ab = 0, which comes from the Hamiltonian constraint (9). Taking account of the expansions (90) and (91), one finds the energy density e can be expressed as Comparing (93) with (94), one can read off Thus, in this paper we only consider the independent transport coefficients in (91).", "pages": [ 16, 17 ] }, { "title": "B.1 Non-relativistic hydrodynamic expansion", "content": "With the pressure p = r -1 / 2 c + r -3 / 2 c P , the full fluid stress tensor (90) can be expanded up to order /epsilon1 4 through the non-relativistic hydrodynamical expansion (15) as where the equations of motion ∂ b T ab = 0 at lower orders have been employed.", "pages": [ 17 ] }, { "title": "B.2 Alternate presentation", "content": "With the coordinates in (83), considering the re-scaled stress tensor one finds the stress tensor (96)-(99) can be transformed into This is also used to compare with the Brown-York stress tensor dual to the metric (85), which is mathematically equivalent to the metric with the near horizon expansion [2].", "pages": [ 17, 18 ] } ]

[ { "title": "Kyle Allison", "content": "Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, United Kingdom E-mail: [email protected] Abstract: We consider an extension of the ν MSM in which sterile neutrino masses originate from the VEV of a Higgs singlet φ and dark matter is produced through the decays of φ rather than through active-sterile neutrino mixing. This model, which we refer to as the ν NMSM, can readily satisfy or escape the constraints on warm dark matter from the Lymanα forest and other small scale structure. However, it requires a particular hierarchy of Majorana masses and Yukawa couplings without an obvious origin. We show that the hierarchical parameters of the ν NMSM can arise from symmetries broken at or near the Planck scale for two specific examples of this model: one in which φ helps stabilize the electroweak vacuum through a scalar threshold effect and one in which φ is a light inflaton. Both examples require a complex φ and have several experimental signatures that are distinct from the ν MSM. These signatures include additional dark radiation that is relativistic at both primordial nucleosynthesis and CMB decoupling and, for the former, a large invisible branching ratio of the Higgs. ArXiv ePrint: 1210.6852", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The ν MSM [1] is an extension of the Standard Model (SM) that attempts to explain all observed phenomena beyond the SM using only three sterile neutrinos with Majorana masses below the electroweak scale. In the ν MSM, one sterile neutrino, N 1 , is responsible for dark matter [2] while two additional sterile neutrinos, N 2 and N 3 , are responsible for baryon asymmetry production [3]. Moreover, the Higgs boson with a non-minimal coupling to gravity is responsible for inflation [4]. Although a detailed study of the ν MSM (see [5] for a recent update) shows that this minimal model can explain most of the observed phenomena beyond the SM, there are several indications that an extension of the ν MSM, such as by a Higgs singlet, may be necessary: The fact that a Higgs singlet can both provide an origin for the Majorana masses of the sterile neutrinos and allow a simple dark matter production mechanism that, unlike the non-resonant production of dark matter in the ν MSM, is consistent with the Lymanα forest bound is a strong motivation for considering singlet extensions of the ν MSM (e.g. [16-20]). It is then natural to ask whether such extensions can also address the issues with Higgs inflation, as in [16-18], or help stabilize the electroweak vacuum, if necessary. These singlet extensions of the ν MSM, like the original model, require a particular hierarchy of Majorana masses and Yukawa couplings without an obvious origin. An important open question for these extensions is whether it is possible for such structure to come from an underlying symmetry. In the context of the ν MSM, models employing a U(1) flavour symmetry [31, 32], discrete flavour symmetries [33], the split seesaw mechanism [34, 35], and the Froggatt-Nielsen mechanism [36-38] have been suggested for producing a hierarchical pattern of Majorana masses and Yukawa couplings. Similar techniques should also be able to produce the necessary pattern of masses and couplings in singlet extensions, but this has not been demonstrated explicitly. In this paper, we consider extensions of the ν MSM by a Higgs singlet φ that address some of the model's possible phenomenological problems and demonstrate how underlying symmetries can give the necessary pattern of Majorana masses and Yukawa couplings in these extensions. In particular, our starting point is a generic model in which the decays of φ allow for primordial dark matter production that is consistent with the Lymanα forest bound and in which the VEV of φ provides an origin for the Majorana masses of the sterile neutrinos. We then construct symmetries broken at or near the Planck scale that can produce the hierarchy of parameters for two specific examples of this model: one in which φ helps stabilize the electroweak vacuum through a scalar threshold effect [26] and one in which φ is the inflaton [16-18]. Both examples require a complex φ to be realized with underlying symmetries and have several experimental signatures that are distinct from the ν MSM. The paper is organized as follows. In section 2, we review the constraints on the ν MSM and primordial dark matter production from a Higgs singlet. In section 3, we develop symmetries broken at or near the Planck scale that can produce the required pattern of Majorana masses and Yukawa couplings for two examples of this model. Section 4 gives the conclusions.", "pages": [ 2, 3, 4 ] }, { "title": "2 The ν MSM and dark matter production from a Higgs singlet", "content": "In this section, we first review the constraints on the ν MSM and motivate the scenario of dark matter production from a Higgs singlet. We then discuss the constraints on dark matter production from a Higgs singlet.", "pages": [ 4 ] }, { "title": "2.1 The ν MSM", "content": "The Lagrangian of the ν MSM is given by where L SM is the SM Lagrangian, N I ( I = 1 , 2 , 3) are the sterile neutrinos, L α ( α = e, µ, τ ) are the lepton doublets, H and φ are the Higgs doublet and singlet, respectively, F αI are the Yukawa couplings for neutrinos, and M IJ are the Majorana masses for the sterile neutrinos. One of the striking features of the ν MSMis the highly constrained and hierarchical pattern of parameters required for successful baryogenesis and dark matter production. These constraints are often best expressed not in the basis N I of (2.1) but in the basis of the physical mass eigenstates N m I with masses M I and Yukawa couplings ˜ F αI . The two bases are related by the unitary transformation given in [31]. First, consider the constraints on N m 2 and N m 3 . The oscillations between N m 2 and N m 3 above T EW produce a lepton asymmetry in the active neutrinos that is converted into a baryon asymmetry by sphalerons [3]. 2 N m 2 and N m 3 cannot enter thermal equilibrium at temperatures much above T EW or else the lepton asymmetry produced in their oscillations is wiped out, giving the constraint [31] where F 2 I ≡ ( F † F ) II and, by convention, F 2 is taken to be larger than F 3 with /epsilon1 ≡ F 3 /F 2 ≤ 1. Similarly, masses M 2 , M 3 /lessmuch T EW are required so that lepton number violating processes are negligible for T /greaterorsimilar T EW ; masses satisfying are generally considered acceptable [3, 39]. Meanwhile, effective baryon asymmetry production requires M 2 , M 3 /similarequal M to be highly degenerate with a mass difference ∆ M ≡ M 3 -M 2 /lessmuch M [3]. The baryon asymmetry produced can be expressed as a function of F 2 , /epsilon1, M, ∆ M , and the neutrino hierarchy. Since active neutrino masses are generated via the seesaw mechanism, one of F 2 , /epsilon1, and M (typically F 2 ) can be expressed in terms of the others with the relation [31] where ∆ m atm /similarequal 0 . 05 eV, v = 246 GeV, and κ = 1(2) for the inverted (normal) hierarchy. Analytic expressions for the baryon asymmetry are given in [3, 40] while a numerical study has been carried out in [39]. The allowed range of each parameter individually is [39] though the combination must produce the observed asymmetry n B /s /similarequal (8.4-8.9) × 10 -11 [41]. Note that the lower bound (2.5) comes from demanding that N m 2 and N m 3 decay before Big Bang nucleosynthesis (BBN) [42, 43] 3 and that a significant amount of parameter space for M /lessorsimilar 500 MeV is ruled out by the CERN PS191 experiment and other accelerator bounds [39, 43]. For the parameter space allowed by (2.5)-(2.7), a lower bound on F 2 is approximately Now consider the constraints on the dark matter candidate N m 1 . Mixing with active neutrinos below T EW allows the 1-loop decay N m 1 → ν m γ with width [1, 46] where θ 2 1 = v 2 ˜ F 2 1 / ( 2 M 2 1 ) and ˜ F 2 1 is evaluated with [31] The second and third terms on the right hand side of (2.10) are contributions to ˜ F α 1 induced by the mixing of N 1 with N 2 and N 3 to form the mass eigenstate N m 1 . Direct searches for the X/ γ -ray line corresponding to this decay provide the strongest limits on θ 1 (as a function of M 1 ) for the mass range relevant to the ν MSM. A summary of these limits is given in [9]. In general, must be satisfied for 0 . 5 keV /lessorsimilar M 1 /lessorsimilar 14 MeV, though the constraint is typically 100 times stronger than (2.11) for masses outside the 12-40 keV range [9]. For N m 1 produced entirely from active-sterile neutrino mixing, M 1 can be bounded above by combining the X-ray constraints with the requirement of sufficient dark matter production ( ∝ θ 2 1 ). The bound obtained depends on the lepton asymmetry at the time of N m 1 production: a negligible lepton asymmetry is called the non-resonant production (NRP) scenario while a large lepton asymmetry is called the resonant production (RP) scenario. The bounds for these two scenarios are [9, 10, 12] Meanwhile, M 1 can be bounded below by phase-space density arguments for dwarf spheroidal galaxies [47, 48], the Lymanα forest data [6, 49], studies of gravitationally lensed QSOs [7], and N-body simulations of the Milky Way [8]. The bounds from the Lymanα forest data and N-body simulations of the Milky Way are the strongest and give 4 Combining (2.12) and (2.13) rules out the simpler NRP scenario, even with a possibly large entropy dilution from the decays of N m 2 and N m 3 [12]. The RP scenario is still allowed for a range of M 1 ; it requires an even larger degeneracy than (2.6), on the order ∆ M /lessorsimilar 10 -7 eV, to produce the required lepton asymmetry for enhanced dark matter production [15]. This level of degeneracy is unstable in the presence of radiative corrections and must be achieved with either fine-tuning or an extension of the model by a Planck-scale symmetry and nonrenormalizable operators [15].", "pages": [ 4, 5, 6 ] }, { "title": "2.2 Dark matter production from a Higgs singlet", "content": "An alternative dark matter production scenario that is capable of satisfying the Lymanα forest bound for warm dark matter (or allows for heavier cold dark matter) uses a real Higgs singlet φ and its decays to N m 1 [16]. This scenario is arguably simpler than the RP scenario and has the advantage that Majorana masses originate from the VEV of φ rather than as bare mass terms. This extension of the ν MSM, which we will call the neutrino Next-to-Minimal Standard Model ( ν NMSM), is the basis of this paper. In the ν NMSM, the Majorana mass term in the Lagrangian (2.1) is modified to where M IJ = λ IJ 〈 φ 〉 once φ acquires a VEV. In the mass basis N m I , λ IJ → λ I where M I = λ I 〈 φ 〉 . The mixing angle θ 2 1 is assumed small enough that dark matter production from active-sterile neutrino mixing is negligible and the X/ γ -ray constraint (2.11) is satisfied. 5 Assuming no miraculous cancellations of terms in (2.10), this requires With (2.15), one can show that the induced contributions to M 1 from M 12 and M 13 are small [31] and hence λ 1 /similarequal λ 11 . Dark matter production then proceeds via the decays φ m → N m 1 N m 1 with the partial width [16] where m φ > 2 M 1 is the mass of the physical mass eigenstate φ m . 6 This production depends on the thermal history of φ m , specifically the ratio of its mass to its freeze-out temperature, r f ≡ m φ /T f [20]. For the case that φ m is in thermal equilibrium down to temperatures T /lessmuch m φ (i.e. r f /greatermuch 1), the dark matter abundance is given by [16] where f ( m φ ) /similarequal (10 . 75 /g ∗ ( m φ / 3)) 3 / 2 and 1 ≤ S /lessorsimilar 2 is a factor that accounts for entropy production from the decays of N m 2 and N m 3 after N m 1 is produced. 7 Using M 1 /similarequal λ 11 〈 φ 〉 in (2.17), the appropriate dark matter abundance Ω N m 1 /similarequal 0 . 23 is generated when For the case that φ m is a thermal relic decaying out of equilibrium (i.e. r f /lessmuch 1), the dark matter abundance is given by [20] where B ≡ Γ / Γ tot φ is the branching ratio of φ m → N m 1 N m 1 . 8 Analytic expressions relevant to the intermediate case r f ∼ 1 can be found in [20], and the result is a combination of (2.17) and (2.19). The Lymanα forest bound for this dark matter production mechanism can be estimated by rescaling the NRP bound, giving [12, 20] where T prod is the temperature at which N m 1 is produced. Further constraints come from the requirement that the interactions φ m ↔ N m 2 N m 2 and φ m ↔ N m 3 N m 3 (and any interactions SM ↔ N m 2 N m 2 and SM ↔ N m 3 N m 3 mediated by φ m ) do not bring N m 2 and N m 3 into thermal equilibrium at the characteristic temperature of leptogenesis [52] where M 0 /similarequal 7 × 10 17 GeV, and spoil baryogenesis. 9 Moreover, the addition of φ must not open an invisible branching ratio of the Higgs greater than 30% at 2 σ [53]. These constraints are discussed further in section 3 for specific models of the scalar sector. Although we have assumed that φ is real in the discussion above, it is also possible (with some restrictions) to have a complex φ . (We parametrize φ = ( ρ + iχ ) / √ 2 for a complex φ but continue to use m φ and φ m instead of m ρ and ρ m to maintain consistency with the notation for a real φ .) In previous studies of the ν NMSM, which do not attempt to explain the origin of its parameters, φ is typically assumed real to avoid a massless Goldstone boson χ and hence the unsuitably fast decay channel N m 1 → ν m χ for dark matter [16-20]. We have found it very difficult, however, to explain the parameters of the ν NMSM with an underlying symmetry if φ is real and hence uncharged. To construct such a symmetry, we must therefore consider a complex φ and address the problems and constraints associated with a Goldstone boson. There are several ways to avoid the decay N m 1 → ν m χ for a complex φ . First, if φ is charged under a discrete symmetry and terms of the form φ n + φ † n are allowed, these terms give χ a mass and can kinematically forbid the decay N m 1 → ν m χ . If the analogous decays N m 2 , N m 3 → ν m χ are still allowed, they can relax the constraint (2.5) to M /greaterorsimilar few MeV [31]. Alternatively, if χ remains light enough to allow N m 1 → ν m χ then the mixing of N 1 with other neutrino species can be suppressed or forbidden by a symmetry, thereby suppressing the decay. This case is particularly interesting since χ can contribute to the effective number of neutrino species and give a value of N eff above the SM or ν MSM prediction, as recent measurements prefer (see [54] and references therein). 10 The contribution of χ to N eff depends on the freeze-out temperature T f : it can be as large as ∆ N eff ∼ 1 for a thermal distribution of χ or much smaller if χ decouples early. The Planck experiment and other future cosmic microwave background (CMB) experiments will therefore be able to constrain these models with a complex φ [56].", "pages": [ 6, 7, 8 ] }, { "title": "3 Symmetries and the ν NMSM", "content": "The ν NMSM, like the ν MSM, requires parameters that are constrained to be hierarchically small. An important question is whether it is possible for such structure to come from an underlying symmetry. In the context of the ν MSM, flavour symmetries [31-33], the split seesaw mechanism [34, 35], and the Froggatt-Nielsen mechanism [36-38] have been explored for producing the required pattern of Majorana masses and Yukawa couplings. Following this approach, we would like to demonstrate explicitly how the parameters of the ν NMSM can arise from symmetries broken at or near the Planck scale. Since the values of some parameters (e.g. λ 11 in (2.18)) depend on an unspecified scalar sector, we first keep the discussion general and then consider two specific models of the scalar sector: one in which φ helps stabilize the electroweak vacuum [26] and one in which φ is the inflaton [16-18]. These models of the scalar sector, though motivated as minimal solutions to other possible problems with the ν MSM, are meant only to provide definite examples for the symmetries used in the flavour sector; other models may certainly be considered. We do not provide an explanation for the values of parameters in the scalar sector or the associated hierarchy problems since little is known about their origin.", "pages": [ 9 ] }, { "title": "3.1 Symmetries in the flavour sector", "content": "First consider how the structure of the ν NMSM Lagrangian, can arise from an underlying symmetry without regard to the size of the couplings F αI and λ IJ . There are several ways this structure can arise: or Green-Schwarz anomalous [64, 65]. An anomaly-free example in which matter fields have U(1) B -L charges is given in table 1. Although either a global U(1) or discrete Z N symmetry can give the desired Lagrangian structure (3.1), we use a global U(1) symmetry to avoid introducing the problems associated with domain walls. Now consider the hierarchy of Majorana masses and Yukawa couplings in the ν NMSM. To explain the small Yukawa couplings ˜ F α 1 /lessorsimilar 10 -13 and, for a complex φ , to prevent the fast dark matter decay channel N m 1 → ν m χ , we introduce a Z 2 symmetry under which only N 1 is charged (see table 2). 12 This symmetry allows only the couplings and hence forbids mixing of N 1 with the other neutrinos, making N m 1 completely stable ( θ 1 = 0) and one active neutrino exactly massless. The required pattern of Majorana masses and Yukawa couplings can then be produced if there are strong hierarchies in the remaining λ IJ and F αI , specifically if We consider two possibilities for generating these hierarchies from an underlying symmetry, in which case the small couplings in (3.3) are preserved under the renormalization group flow: Although either mechanism may be used to generate the hierarchical parameters (3.3) for the same charge assignment, the non-perturbative symmetry breaking mechanism does not require additional symmetry breaking or scalar particles below the Planck scale and therefore adheres closer to the 'minimal' philosophy of the ν MSM. To fix the absolute scale of the couplings λ IJ and hence construct an explicit model of symmetries in the flavour sector, the values of m φ and 〈 φ 〉 must be fixed (see (2.18) and (3.3)) by some model of the scalar sector. We now consider two models of the scalar sector that are motivated as solutions to other possible problems with the ν MSM.", "pages": [ 9, 10, 11 ] }, { "title": "3.2 Stabilization of the electroweak vacuum", "content": "For a Higgs mass m h /similarequal 125-126 GeV, the SM (and hence ν MSM) potential develops an instability below the Planck scale unless the top mass is about 2 σ below its central value [23]. While it is possible that more precise measurements of the top mass will lower its central value and relieve this tension, we first consider a model of the scalar sector in which the Higgs singlet can, for the central value of the top mass, stabilize the electroweak vacuum through a scalar threshold effect. This model, described in [26], considers a complex φ and scalar potential of the form which is the most general renormalizable potential that respects a global abelian symmetry under which only φ is charged. Values of λ h , λ φ > 0 and λ 2 hφ < λ h λ φ are assumed so that the minimum of this potential is given by where v = 246 GeV. The mass matrix for the real components of H and φ is then while the imaginary component of φ (i.e. χ ) remains massless. In contrast to other models that use a Higgs singlet to stabilize the electroweak vacuum (e.g. [24, 25]), this model assumes w /greatermuch v . The two eigenstates of (3.6) then have masses with a mixing angle θ hφ ∼ v/w . Integrating out the heavier state for scales below m φ gives the effective potential where the matching condition for the Higgs quartic coupling gives a tree-level shift δλ ≡ λ 2 hφ /λ φ from λ just below m φ to λ h just above m φ . Provided m φ is below the instability scale Λ /similarequal 10 9 -10 10 GeV [23], a value of δλ /similarequal 0 . 01 can push the instability beyond the Planck scale. Due to the massless Goldstone boson χ , the value of λ hφ is constrained by limits on the invisible branching ratio of the Higgs. For m h /similarequal 125 GeV, the total SM decay width of the Higgs is [77] while the invisible decay width for h m → χχ is [78] Allowing an invisible branching ratio of up to 30% [53] gives the constraint A value of δλ that stabilizes the electroweak vacuum and is consistent (3.12) can then be obtained for λ φ /lessorsimilar 0 . 01 (the running of λ hφ and λ φ is small for these values). We illustrate this by constructing a model with λ hφ , λ φ ∼ 0 . 01 and hence an invisible branching ratio of the Higgs of about 30%. 13 For these values, one can show that χ remains in thermal equilibrium down to temperatures just below m µ . The model therefore has a ∆ N eff /similarequal 4 / 7 contribution to the effective number of neutrino species from χ and hence a total value of N eff /similarequal 3 . 6. This value can be tested by the Planck experiment and other future CMB experiments [56]. Now consider the flavour sector of the ν NMSM for this model of the scalar sector. For λ hφ ∼ 0 . 01, the interactions H † H ↔ φ m φ m keep φ m (the real component of φ ) in thermal equilibrium down to temperatures T /lessmuch m φ for any mass m φ /lessorsimilar Λ. We are therefore in the dark matter production case r f /greatermuch 1. For λ φ ∼ 0 . 01, the ratio m φ / 〈 φ 〉 is fixed by (3.8) and (2.18) gives a value of to produce the correct dark matter abundance. 14 The Lymanα forest bound (2.20) is therefore satisfied for a choice 〈 φ 〉 /greaterorsimilar 500 GeV. Taking 〈 φ 〉 /similarequal 10 8 GeV, a pattern of masses M IJ and couplings F αI that gives the correct dark matter abundance and baryon asymmetry can be achieved with two fields ϑ 1 , ϑ 2 , the values η 1 /similarequal 10 -8 , η 2 /similarequal 10 -7 , and the charge assignments given in table 2. We stress that this is the simplest anomaly-free model we could find, though other charge assignments are possible. 15 For the sake of definiteness, suppose that the non-perturbative symmetry breaking mechanism is used for generating the hierarchies in F αI and λ IJ ; that is, ϑ i = e -T i /M Pl and η i = e -〈 t i 〉 /M Pl for i = { 1 , 2 } . From table 2, the Lagrangian for the flavour sector is then where f αI and h IJ are O (1) couplings. Meanwhile, the scalar potential is given by (3.4) plus the additional terms ϑ † i ϑ i H † H , ϑ † i ϑ i φ † φ , and ϑ 1 φ 3 + ϑ † 1 φ † 3 involving ϑ 1 and ϑ 2 . Note that we must assume these additional terms, which are allowed by the symmetries, have sufficiently small coefficients to preserve (3.4). 16 For the former two terms, this assumption is a further aspect of the hierarchy problem in the scalar sector. An explanation for these small coefficients may arise from the solution to the hierarchy problem, but providing such an explanation goes beyond the scope of this paper. It is interesting to see, however, that in order to produce hierarchical parameters in the flavour sector of the ν NMSM with symmetries the hierarchy problem in the scalar sector may be made worse. 17 For the latter terms ϑ 1 φ 3 + ϑ † 1 φ † 3 , we similarly accept a small parameter in the scalar sector without explanation, but note that these terms could also be forbidden by an additional U(1) symmetry under which φ and ϑ 1 have opposite charges. After the spontaneous symmetry breaking associated with ϑ 1 and ϑ 2 , (3.14) reduces to (3.1) with the textures The parameters of the ν NMSM are then up to O (1) constants. This example shows that, in contrast to the ν MSM, dark matter in the ν NMSM can be much heavier than the keV scale. Active neutrino mixing in this model is anarchical (up to charged lepton corrections) while the charged lepton and quark Yukawa couplings remain unsuppressed. Therefore additional flavour symmetries using the Green-Schwarz anomaly cancellation mechanism, such as in [36, 80], must be used to produce hierarchies in the charged lepton and quark sectors. As a consistency check on this model, we must verify that N m 2 and N m 3 are out of thermal equilibrium at the characteristic temperature of leptogenesis T L ∼ 3 × 10 3 GeV. Since m φ /similarequal 2 × 10 7 GeV /greatermuch T L , φ m has decayed away by leptogenesis 18 and only the scattering processes H † H ↔ N m 2 N m 2 and H † H ↔ N m 3 N m 3 mediated by φ m and χ need to be considered. For λ hφ ∼ 0 . 01, these processes are out of equilibrium at T L for λ 2 , λ 3 /similarequal λ 23 /lessorsimilar 10 -5 , which is satisfied by (3.15). This model demonstrates that it is possible to use symmetries broken at or near the Planck scale to obtain the hierarchical pattern of Majorana masses and Yukawa couplings required for successful baryogenesis and dark matter production in the ν NMSM. The model obeys all phenomenological constraints and allows for the possibility of Higgs inflation by ensuring that the Higgs potential does not develop a second minimum before the Planck scale. Of course, the symmetries used do not address the hierarchy problem associated with radiative corrections to the scalar sector. To do so would involve implementing a supersymmetric version of the theory, which departs from the underlying philosophy of the ν MSM, or implementing a conformal symmetry solution, which requires an understanding of how to include gravity in such a theory. This is something we cannot do at present.", "pages": [ 11, 12, 13, 14, 15 ] }, { "title": "3.3 φ Inflation", "content": "Although the Higgs inflation of the ν MSM has not been ruled out, it relies on the questionable assumption that new strong dynamics appearing at the scale of perturbative unitarity breakdown, M Pl /ξ , preserve the intact shape of the Higgs potential even above M Pl /ξ [81]. We now consider another model of the scalar sector for the ν NMSM, given in [16] and developed further in [17, 18], in which the Higgs singlet φ can be a light inflaton ( m φ < m h ) and thus provide an alternative to Higgs inflation. The scalar potential of this model is where it is assumed that m /lessmuch √ βM Pl so that chaotic inflation proceeds via the quartic term and, in contrast to [16-18], we require φ to be complex to explain the hierarchical parameters of the ν NMSM with an underlying symmetry. The potential (3.17) is then the most general renormalizable potential that respects a global U(1) symmetry under which only φ is charged, assuming the bare mass term for the Higgs is negligible. 19 Successful chaotic inflation requires β /similarequal 1 . 5 × 10 -13 to give the correct amplitude of adiabatic scalar perturbations and α /lessorsimilar 10 -7 , λ IJ /lessorsimilar 1 . 5 × 10 -3 to ensure that the flatness of the potential is not spoiled by radiative corrections from the loops of SM particles and sterile neutrinos [18]. 20 Achieving a sufficiently high reheating temperature T r > T L for baryogenesis requires α /greaterorsimilar 7 × 10 -10 [17]. Moreover, a value of λ /similarequal 0 . 13 is required for m h /similarequal 125 GeV. For these parameters, expanding the potential (3.17) about its minimum gives the relations where v = 246 GeV. The upper bound on α can be further strengthened by limits on axion searches in the CHARM experiment [18]. The mass range allowed by this experiment, 270 MeV /lessorsimilar m φ /lessorsimilar 1 . 8 GeV, corresponds to 2 × 10 -10 /lessorsimilar α /lessorsimilar 8 × 10 -9 for m h /similarequal 125 GeV. Note that we do not provide an explanation for the small values of α and β in the scalar potential; we simply take their values to be within the range allowed by successful inflation. Also note that, for α /lessorsimilar 8 × 10 -9 , the invisible branching ratio of the Higgs is negligible. Now consider the flavour sector of the ν NMSM for this model of the scalar sector. As in [16], we assume an inflaton mass m φ /greaterorsimilar 300 MeV so that the mixing angle θ hφ is large enough to keep φ m in thermal equilibrium down to temperatures T /lessmuch m φ via the interactions φ m ↔ e -e + , φ m ↔ µ -µ + , etc. We are therefore in the dark matter production case r f /greatermuch 1. The ratio m φ / 〈 φ 〉 = √ β is fixed by (3.18) and (2.18) gives a value of to produce correct dark matter abundance. 21 The absolute scale of 〈 φ 〉 , however, is not fixed. There is a relatively narrow window 7 × 10 5 GeV /lessorsimilar 〈 φ 〉 /lessorsimilar 2 × 10 6 GeV that is consistent with the constraints on α , the assumption m φ /greaterorsimilar 300 MeV, and the Lymanα forest bound. Taking 〈 φ 〉 /similarequal 10 6 GeV, a pattern of masses M IJ and couplings F αI that gives the correct dark matter abundance and baryon asymmetry can be achieved with two fields ϑ 1 , ϑ 2 , the values η 1 /similarequal 2 × 10 -3 , η 2 /similarequal 5 × 10 -5 , and the charge assignments given in table 3. Again, this is the simplest anomaly-free model we could find, though other charge assignments are possible. Suppose this time that the Froggatt-Nielsen mechanism is used for generating the hierarchies in F αI and λ IJ , and hence η 1 = 〈 ϑ 1 〉 /M Pl and η 2 = 〈 ϑ 2 〉 /M Pl . From table 3, the Lagrangian for the flavour sector is then where f αI and h IJ are O (1) couplings. Meanwhile, the scalar potential is given by (3.17) plus the additional terms ϑ † i ϑ i H † H , ϑ † i ϑ i φ † φ , and ϑ † 2 1 φ 3 + ϑ 2 1 φ † 3 involving ϑ 1 and ϑ 2 . Again, we must assume that these additional terms in the scalar sector, which are allowed by the symmetries, have sufficiently small coefficients to preserve (3.17). Once ϑ 1 and ϑ 2 acquire VEVs, (3.20) reduces to (3.1) with the textures The parameters of the ν NMSN are then up to O (1) constants. As before, active neutrino mixing is anarchical (up to charged lepton corrections) and additional flavour symmetries must be used to produce hierarchies in the charged lepton and quark sectors. We also have the parameters For these values, it can be shown that χ remains in thermal equilibrium roughly while φ m does (to temperatures below m µ ) via the interactions φ m ↔ χχ . The near massless χ therefore contributes ∆ N eff /similarequal 4 / 7 to the effective number of neutrino species. As a consistency check on this model, we must verify that N m 2 and N m 3 are out of thermal equilibrium at the characteristic temperature of leptogenesis T L ∼ 2 × 10 3 GeV. Since m φ < 2 M , the processes φ m → N m 2 N m 2 and φ m → N m 3 N m 3 are kinematically forbidden and the dominant processes are H † H ↔ N m 2 N m 2 and H † H ↔ N m 3 N m 3 . These are out of equilibrium at T L for λ 23 /lessorsimilar 0 . 01, which is satisfied by (3.21). One can also verify that the reheating temperature for α /similarequal 4 × 10 -9 can be as large as T r /similarequal 5 × 10 3 GeV [17], which is above the leptogenesis temperature. This model demonstrates that, for a scenario in which φ is a light inflaton, it is again possible to use symmetries broken at or near the Planck scale to obtain the pattern of Majorana masses and Yukawa couplings required for successful baryogenesis and dark matter production in the ν NMSM. This model obeys all phenomenological constraints and provides an alternative to the Higgs inflation of the ν MSM, but it requires small parameters in the scalar potential without explanation (a problem that plagues virtually all inflationary models) and does not improve the stability of the electroweak vacuum.", "pages": [ 15, 16, 17 ] }, { "title": "4 Conclusion", "content": "The ν MSM is an extension of the SM that attempts to explain neutrino oscillations, dark matter, the baryon asymmetry of the universe, and inflation using only three sterile neutrinos with masses below the electroweak scale. Despite the phenomenological successes of the ν MSM, a further extension may be necessary to accommodate the Lymanα forest bound, stabilize the electroweak vacuum, and allow for inflation. In this paper, we have studied extensions of the ν MSM by a Higgs singlet φ that can address these issues and have demonstrated how the required pattern of masses and couplings in such models can arise from an underlying symmetry. Our starting point has been an extension of the ν MSM in which the decays of φ give a primordial production of dark matter that is readily consistent with the Lymanα forest bound and in which the VEV of φ produces the Majorana masses of the sterile neutrinos. For this next-to-minimal model, or ν NMSM, we have considered two specific models of the scalar sector: one in which φ helps stabilize the electroweak vacuum through a scalar threshold effect and one in which φ is a light inflaton. For these definite examples, we have demonstrated that symmetries broken at or near the Planck scale can produce the required hierarchical pattern of Majorana masses and Yukawa couplings. The former model uses a U(1) × Z 3 × Z 2 symmetry while the latter uses a U(1) × Z 4 × Z 2 symmetry; both require a complex φ rather than, as typically assumed, a real φ . We have not, however, provided an explanation for the parameters of the scalar sector or addressed the hierarchy problem associated with radiative corrections to the scalar sector. The models presented in this paper satisfy all phenomenological constraints and make several experimental predictions that are distinct from the ν MSM. These predictions include completely stable N m 1 dark matter (hence no visible X/ γ -ray signals from its decays) as well as anarchical active neutrino mixing angles (up to charged lepton corrections) with one active neutrino exactly massless. Moreover, due to the complex φ , both models have N eff /similarequal 3 . 6 for the effective number of neutrino species while the former model has an invisible branching ratio of the Higgs of about 30%. It will therefore be possible to test these models with the Planck experiment and the LHC in the near future.", "pages": [ 17, 18 ] }, { "title": "Acknowledgments", "content": "I am grateful to Graham Ross for proposing this investigation and for much valuable input, as well as to Subir Sarkar for helpful discussions. This work was supported by the European Commission under the Marie Curie Initial Training Network UNILHC 237920 (Unification in the LHC era). Contents reflect only the author's views and not the views of the European Commission.", "pages": [ 18 ] } ]

[ { "title": "Hawking Radiation in a Pleba'nski-Demia'nski Black Hole", "content": "Jes'us A. C'azares ∗ Rudjer Boˇskovi'c Institute, P.O.Box 180, HR-10002 Zagreb, Croatia March 9, 2021", "pages": [ 1 ] }, { "title": "Abstract", "content": "In this paper, we show the flux of Hawking radiation in a Pleba'nski-Demia'nski black hole from the point of view of gauge and gravitational anomalies. We will use the consistent anomaly method to guarantee that our results are valid in the de Sitter space. This is because we are including the cosmological constant into our parameters and the covariant anomaly method gives a wrong value for the Hawking temperature. We also show that these calculations are a general result. In order to verify the consistence of our results, we can reproduce earlier known results as certain limiting cases.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Black hole radiation, also called Hawking radiation, was originally reported by Zeldovich and Starobinsky in 1971 [1] and it is one of the more interesting known effects. This effect is a consequence of the combination of two modern theories: quantum field theory and general relativity. The Hawking radiation originates by the quantization of matter in a background space-time with an event horizon, like a black hole. It has been found that the occupation number spectrum of quantum field modes in the vacuum state corresponds to the blackbody at a fixed temperature given by the surface gravity of the horizon. There are several explanations for Hawking radiation. The derivation by Hawking [2, 3] is direct and physical. He calculates the Bogoliubov coefficients between the in and out states of fields in the black hole background. The derivation based in quantum gravity [4, 5] is fast and elegant, but needs microscopic foundation. The derivation based on string theory [6, 7] has a logical consistency foundation but can be applied only to special solutions and does not explain the simplicity and generality of the results inferred from other methods. Christensen and Fulling [8] showed that the magnitude of the Hawking blackbody effect at infinity is directly proportional to the magnitude of the trace anomaly. With this starting point, Robinson and Wilczek [9] have recently shown a procedure for calculating Hawking radiation via gravitational anomaly cancelation in a Schwarzschild black hole metric. This original idea soon was extended to the Reissner-Nordstrom metric [10], to rotating black holes [11, 12, 13, 14, 15] and even to metrics with NUT parameter [16]. This leads to the important question: What are the fluxes like for Hawking radiation in the most general black hole? As it is known, the most general of Petrov D-type solutions of the Einstein-Maxwell equations is the socalled Pleba'nski-Demia'nski solution [17]. In the present article, we calculate the Hawking radiation for the Pleba'nski-Demia'nski black hole in the spirit of Robinson and Wilczek method and show the form of the fluxes and the Hawking temperature. It is necessary to comment that the method based on the gravitational and gauge anomalies reproduce succesfully the Hawking fluxes, but this anomalies contain only the information of fluxes of energy and charge. These fluxes correspond to the zero-th and first moments of the thermal distribution of radiation. To obtain the full information, it is necessary to calculate all the other higher moments. Iso, Morita and Umetsu [18, 19, 20, 21] attributed these higher fluxes to phenomenological higher spin currents, i.e. higher spin generalizations of the energy-momentum tensor. Bonora, Cvitan, Pallua and Smoli'c [22, 23, 24] have shown that such higher currents describe the higher spin fluxes of the Hawking radiation and that these higher spin currents cannot have trace anomalies neither have diffeomorphism anomalies. They also showed that the thermal spectrum of the Hawking radiation is induced by the underlying W 1+ ∞ algebra structure of the higher spin currents. That is, the Hawking radiation and in particular its thermal spectrum, points toward the existence of a symmetry much larger than the Virasoro algebra in the near horizon region. That is a W ∞ or a W 1+ ∞ algebra, an extension of the W ∞ algebra to include a U (1) current. The article is organized as follows: In the second section we introduce the Pleba'nski-Demia'nski metric and by a partial wave decomposition of the scalar field in terms of spherical harmonics, we get the effective action, corresponding to a (1+1)-dimensional metric in a dilaton background and a gauge field. In the present situation, we will get three different gauge fields: one related to the mass and the other two are related to the electric and magnetic charges, respectively. In sections 3 and 4, using the generalization made by Vagenas and Das [25, 26], we calculate the gauge and gravitational anomalies from which the Hawking radiation arises at the event horizon and cosmological horizon, respectively. It is necessary to comment at this point that we shall use the consistent anomaly method because we are including the cosmological constant into our parameters, since the covariant anomaly method gives a wrong value for the Hawking temperature [27]. Recently, a new method has been found by Zampeli, Singleton and Vagenas [28] that works perfectly for de Sitter and Rindler space-time using either the covariant or the consistent formalism. In order to verify that our results have mathematical and physical consistence, we reduce several parameters of our metric in section 5 to get two well-known metrics which in the limiting cases correspond to the Kerr-Newman-de Sitter black hole metric and the NUT-Kerr-Newman-de Sitter. When this parameters are also reduced in the obtained fluxes, we recover the same results obtained previously in the literature [14, 16]. From this two metrics, we can vanish all the parameters except the cosmological constant and get the de Sitter metric. When we do this, we get the right Hawking temperature [29, 27]. Finally, the conclusion is presented in section 6.", "pages": [ 1, 2, 3 ] }, { "title": "2 The Effective action of Pleba'nski-Demia'nski black hole", "content": "In the Petrov clasification, the most general D-type metric is the Pleba'nski-Demia'nski metric [30]. It is possible to get, by certain transformations and limiting procedures, many particular and well known D-type metrics, such as the Pleba'nski-Carter, the Kerr-Newman, the Kerr and Schwarzschild solutions among others. The Pleba'nski-Demia'nski metric is [17] with the 1-form potential the functions P and Q in the metric, are the related quartic functions and ˆ m, ˆ n, ˆ e, ˆ g, ˆ /epsilon1, ˆ k, Λ are arbitrary real parameters. It is habitually assumed that ˆ m and ˆ n are the mass and NUT parameters, respectively, but this is not generally the case. To describe the complete family of black hole-like space-time, recently Griffiths and Podolsky [30] have shown that it is possible to transform the Pleba'nski-Demia'nski metric in the following form where A µ is the electric potential, B µ is a magnetic-like potential, and the functions that appear in the metric are given by Ξ = 1+ Λ 3 ( a - l ) , (10) Now m, Q, G, a, l, α, Λ represent the mass, electric and magnetic charge, angular momentum, NUT parameter, acceleration and cosmological constant, respectively. That is, they are now physical parameters. It is assumed that | a 3 | and | a 4 | are sufficiently small to guarantee that ˆ P has no additional roots in θ ∈ [0 , π ]. The equations (15-17) define the parameters /epsilon1, n, k and give a strong restriction to ω . It can take a convenient value if a and l are not both zero. For simplicity in our calculations, it is possible to write the metric (5) in the following form where we have introduced the functions Θ = Θ( θ ) and R = R ( r ) given by The outer horizon ( r = r H ) is determined by ˆ Q ( r H ) = 0. The term 1 / Ω 2 is a conformal factor, so, this term does not contribute to the Hawking radiation, and we can omit it in our calculations. We will consider matter fields in the Pleba'nski-Demi'anski black hole background. As we have both electric and magnetic charge, we can take into consideration that the covariant derivative is [31, 32] where e and g are the electric and magnetic charges, respectively. With all this taken into in consideration, the 2 2 action is given by where the elements of the metric are Performing the partial wave decomposition of the scalar field Φ in terms of the spherical harmonics Φ = ∑ l,m φ lm ( r, t ) Y lm ( θ, ϕ ) this action can be reduced to a two dimensional effective theory. To do this it is necessary to transform it to the r ∗ tortoise coordinate defined by and considering the region near the horizon. After this process, the action (22) can be simplified to that is, each partial wave of the scalar field can be considered as a (1 + 1)-dimensional complex scalar field in the background of the dilaton Ψ, where the elements of this metric ˜ g µν and gauge fields ˜ A µ are given by where m, e, g are the charges of the gauge fields ˜ A t 1 , ˜ A t 2 , ˜ A t 3 ; respectively. In this (1+1)-dimensional reduction, the effective field theory is based on the observable physics and defined outside the horizon of the black hole. This means that the ingoing modes are omitted at the horizon making the theory chiral there. With this, each partial wave becomes anomalous with respect to gauge and general coordinate symmetries. So, in order to have gauge invariance and diffeomorphism covariance it is necessary that the fluxes of the U (1) gauge current and the energy-momentum tensor cancel the gauge and gravitational anomaly at the horizon, respectively. We will show this procedure in the next section.", "pages": [ 3, 4, 5, 6 ] }, { "title": "3 Anomalies and Hawking radiation", "content": "An anomaly in QFT is a conflict between a symmetry from the classical action and the quantization. There exist anomalies in global symmetries and gauge symmetries [33, 34]. The gauge current must satisfy the conservation equation ∇ µ J µ = 0. However, near the horizon, the U (1) gauge current satisfies an anomalous equation [33] where we have used α to denote the gauge charge of the U (1) gauge field A ν . So, in the region r ≥ r H + /epsilon1 , as there is no anomaly, the U (1) gauge current must satisfy the conservation equation ∂ r J r ( out ) = 0 but, near the horizon it must satisfy ∂ r J r ( H ) = α∂ r ˜ A t . Thus we can get 4 π where c o and c H are integration constants. c o is the value of the current at r = ∞ and c H is the value of the consistent current of the outgoing modes at the horizon. To get the values of c o and c H , we can use the consistent current where the scalar step function is  Since we have omitted the ingoing modes near the horizon, this current is only a part of the total current. If a classical action S [Φ , g µν ] is quantized, we get and, under gauge transformations, the variation of the quantum effective action is where λ is a gauge parameter. By integration by parts we have The total effective action must be gauge invariant. So, the last term would vanish by quantum effects of the classically irrelevant ingoing modes. The coefficient of the delta function would also cancel, and we get To ensure the regularity requirement at the horizon, the covariant current must also vanish there. Since the covariant current is ˜ J r = J r + α 4 π ˜ A t [1 -Θ( r -r H -/epsilon1 )], the condition at the horizon ˜ J r = 0 determines the flux of the U (1) gauge current as We have three gauge charges for a Pleba'nski-Demia'nski black hole; thus, the U (1) gauge charge flux, the electric current flux and the magnetic current flux are determined by * * * The anomalies in global symmetries are theoretical inconsistences. For this reason their cancellation gives important restrictions [33, 34]. A gravitational anomaly is a gauge anomaly in general covariance, making nonconservative the energy-momentum tensor. This anomaly can only happen in theories with chiral matter coupled to gravity in a (4 k +2)-dimentional space-time, with k being an integer. This chiral matter can be a fermion and can also be a 2 k -form with an (anti-) autodual field. An important situation is the (1 + 1)-dimensional scalar autodual field; this field obeys that is, it only has modes that are moving to the right, so it is chiral. The anomaly is [35, 36] that is, the energy-momentum tensor is not conserved in a curved space-time. As discussed previously, the effective field theory is defined outside the event horizon. In the region r ≥ r H + /epsilon1 there is an effective background gauge potential, but without anomaly, so the energy-momentum tensor satisfies the modified conservation equation ∂ r T r t ( out ) = c o ∂ r ˜ A t . Near the horizon r H ≤ r ≤ r H + /epsilon1 the energy-momentum tensor exhibits an anomaly and satisfies the Ward identity, that is ∂ r T r t ( H ) = J r ( H ) ∂ r ˜ A t + ˜ A t ∂ r J r ( H ) + ∂ r N r t , where N r t = ( f ' 2 + ff '' ) / 192 π and in our situation this gives where the functions C and M are given explicitly by We can use the energy-momentum tensor which combines contributions from these two regions. Under general coordinate transformations x µ -→ x µ -λ µ , the variation of the quantum effective action (38) becomes where the relation J r ( H ) = c o + α 4 π ˜ A t has been used. The effective action must vanish if we demand the covariance under the diffeomorphism transformation. The first term of the effective action is the classical effect of the background electric field. The second term is cancelled by the quantum effect of the classically irrelevant ingoing modes. The third one would be also cancelled, leading to the condition where are the values of the energy flow at infinity and at the horizon, respectively. To ensure the regularity requirement at the horizon, the covariant energy-momentum tensor must also vanish there. Since the energy-momentum tensor is ˜ T r t = T r t + 1 192 π ( ff '' -2 f ' 2 ), it will take the explicit form The condition at the horizon ˜ T r t = 0 gives us where the surface gravity of the black hole is Then, the flux of the energy-momentum tensor required to restore general coordinate covariance at quantum level in the effective field theory is where T h = κ 2 π is the Hawking temperature of the black hole. As it is known, the Planck distribution at an inverse temperature β with a chemical potential µ is for bosons and fermions respectively. In the zero temperature limit, if ω ∓ µ it is negative, the distribution become ∓ 1 for bosons or fermions. In the bosonic case, this result leads to the effect of superradiance. But, in the fermionic case, the occupation numbers become 1 when temperature goes to 0 for these low frequency modes. This leads to zero flux of radiation even at the extremal case [10, 11]. We will take in consideration the fermionic case. The Hawking distribution with chemical potential µ = m ˜ A t 1 + e ˜ A t 2 + g ˜ A t 3 of the Pleba'nski-Demia'nski black hole is given by From this distribution (eq. 66), the angular momentum flux (that is, the U (1) gauge current flux), the electric current flux and the magnetic current flux can be obtained as and the energy-momentum tensor current flux is Comparing equations (43-45, 64) with equations (68, 70, 72, 74), we can conclude that the fluxes of the U (1) gauge current, electric current, magnetic current and the energy-momentum tensor required to cancel gauge or gravitational anomalies at horizon are identical to that of Hawking radiation.", "pages": [ 6, 7, 8, 9, 10 ] }, { "title": "4 Hawking radiation at Cosmological horizon", "content": "Near the cosmological horizon (CH), we must take the ingoing modes into consideration. Inside the CH ( r c -/epsilon1 < r < r c ), the modes need to satisfy the equation ∂ r J r ( c ) = -α∂ r ˜ A t 4 π , but in r < r c -/epsilon1 there is no anomaly, so we have a conservation equation ∂ r J r ( c -out ) = 0. We can get from these equation, the consistent current Under gauge transformation, the variation of the quantum effective action (38) becomes Because the covariant current must vanish at the CH, we obtain the U (1) gauge charge flux, the electric current flux and the magnetic current flux at the CH, are determined by Similarly, in r c -/epsilon1 < r < r c , the energy-momentum tensor satisfies the Ward identity ∂ r T r t ( c ) = J r ( c ) ∂ r ˜ A t + ˜ A t ∂ r J r ( c ) -∂ r N r t , and, in r < r c -/epsilon1 , the energy-momentum satisfies ∂ r T r t ( c -out ) = ˜ a o ∂ r ˜ A t . If we use the energy-momentum tensor after general coordinate transformations, the variation of the effective action (38) become and, because the covariant energy-momentum tensor must vanish at the CH, the flux of the energy-momentum tensor required to restore general coordinate covariance at quantum level in the effective field theory is ∣ The Hawking radiation spectrum of the Pleba'nski-Demia'nski black hole at CH is given by the Fermi-Dirac distribution where T c = -1 4 π C R 2 ∣ ∣ r = r c is the temperature of black hole. From this distribution, the U (1) gauge current flux, the electric current flux and the magnetic current flux can be obtained as and the energy-momentum tensor current is Comparing equations (77-79, 82) with equations (85, 87, 89, 91), we can conclude that the fluxes of the U (1) gauge current, electric current, magnetic current and the energy-momentum tensor required to cancel gauge or gravitational anomalies at cosmological horizon are identical to that of Hawking radiation.", "pages": [ 10, 11, 12 ] }, { "title": "5 Limiting cases", "content": "In this section, we will reduce the parameters to obtain more simple and known metrics as certain limits of PD metric (18). We will first proceed with vanishing the NUT parameter l = 0 and then we will vanish the acceleration α and the magnetic charge g . This way we obtain the Kerr-Newman-de Sitter black hole. This black hole was studied by Jiang and Wu [14]. In the second case, we will first vanish the acceleration and the magnet charge, obtaining the NUT-Kerr-Newman-de Sitter black hole. This black hole was studied in [16]. Finally, we will test our results obtaining the de Sitter space from this two limit cases. We get the correct Hawking temperature [29, 27].", "pages": [ 12 ] }, { "title": "Black holes without NUT parameter", "content": "An interesting case is obtained if there is no NUT parameter ( l = 0). With this value, the equation (17) reduces to ω 2 a 2 k = 1. With this restriction, the parameter ω can take the value of the angular momentum a and the values for the parameters k , /epsilon1 and n (eqs. 15-17) are the coefficients a 3 and a 4 take the values a 3 = 2 αm, and the metric functions (8-12) of the PD metric become Ω = 1 - αr cos θ, This way we have obtained the accelerated Kerr-Newman-de Sitter with magnetic monopole black hole metric. If we consider α = 0 and G = 0, we will get and the metric becomes the Kerr-Newman-de Sitter one As we assumed there is not magnetic charge, then the value of g = 0. When we take these values in the equations (29-30), the gauge field will be as follows and the U (1) gauge charge flux, electric current flux, energy-momentum flux and Hawking temperature are respectively; these values correspond to the values obtained in [14].", "pages": [ 12, 13, 14 ] }, { "title": "Non accelerated black holes", "content": "Another interesting case is obtained if the black hole is not accelerated ( α = 0). The equation (17) takes the form ω 2 k a 2 -l 2 = 1 -l 2 Λ. In this case, for the parameters /epsilon1 and n we get while the vale α = 0 implies that the coefficients a 3 and a 4 become and the metric functions (8-12) of the PD metric (5) take the form and we obtain the NUT-Kerr-Newman-de Sitter with magnetic monopole black hole metric. If we have no magnetic monopole, we get and the metric becomes the NUT-Kerr-Newman-de Sitter one Again the value of g = 0 because we assumed there is not magnetic charge. When we take these values in the equations (29-30), the gauge field will be as follows and the U (1) gauge charge flux, electric current flux, energy-momentum flux and Hawking temperature are respectively; these values were reported in [16]. It is important to observe that expresions (105-111) reduce to (95-101) when l = 0.", "pages": [ 14, 15 ] }, { "title": "Black holes in de Sitter space", "content": "As we commented in the introduction, the strongest reason to use the consistent anomaly and not the covariant anomaly is because the covariant method can give wrong results [27]. In order to show that our calculations are general, we can find the Hawking temperature for a black hole in de Sitter space. To get the de Sitter metric, we can vanish all the parameters in equation (94) or in equation (104) except Λ. We will get in both process the final values Then, the PD metric will take the form of the de Sitter metric where the function f ( r ) in the metric is given by f ( r ) = ˆ Q R = 1 -Λ 3 r 2 . It is trivial to see that the cosmological horizon is r c = √ 3 / Λ. When we vanish all the parameters in the gauge fields, except Λ, we find that these gauge fields take the value of zero. So, the only flux different from zero is the total energy momentum tensor flux (eq. 91) and is a o = -π 12 T 2 h , where the Haking temperature is ∣ this value was reported by [29] and [27]. This means that the equations (43-45, 64) can be used to determine the fluxes of Hawking radiation at the EH of any Petrov D-type metric while the equations (77-79, 82) give completely the fluxes at CH.", "pages": [ 15, 16 ] }, { "title": "6 Conclusions", "content": "After the Robinson-Wilczek article [9], where it was shown how to get the Hawking radiation for the Schwarzschild black hole via anomalies, several groups analyzed the Hawking radiation for more general black holes. In this process it was necessary to make several modifications to the original method in order to get a correct expresion for the Hawking fluxes. In this paper, we used this analysis of Hawking radiation based on gauge and gravitational anomalies and applied it to the case of the most general D-type metric in Petrov clasification of the Einstein-Maxwell equations; that is, the so-called Pleba'nski-Demia'nski black-hole. It is important to remark that any particular combination of parameters included in the PD black hole (mass, electric and magnetic charge, angular momentum, NUT parameter, acceleration and cosmological constant) can be obtained from our calculations and we have shown this by obtaining special cases previously reported: the Kerr-Newman-de Sitter black hole and the NUT-Kerr-Newman-de Sitter black hole. It is necessary to stress out that both black holes were obtained by reducing in two different ways but, vanishing the NUT parameter the last one reduce to the first one. We finally showed that, starting from any of this two metrics it is possible to get the de Sitter space and our calculations give the correct temperature Hawking for it. With this generalization, we are showing that near the event horizon of the PD black hole there is quantum vacuum fluctuation effect and virtual particle pairs can appear. The negative energy particle enter the EH (or exit the CH) and become ingoing modes (or outgoing modes); but the positive energy particle exit the EH (or enter the CH) and become outgoing modes (or ingoing modes). With this process the theory becomes anomalous and the anomaly can be cancelled, thus, originating Hawking radiation as a consequence. That is, the cancelation anomalies method works even for the most general Petrov D-type metric. It is necessary, of course, to find the higher spin currents to determine the complete thermal distribution of Hawking radiation and verify the existence of an underlying W 1+ ∞ algebra structure that induce the thermal spectrum of the Hawking radiation.", "pages": [ 16 ] }, { "title": "Acknowledgements", "content": "I am very thankful to the Physics Department at Centro de Investigaci'on y de Estudios Avanzados del IPN (CINVESTAV-IPN) for all the support in the elavoration of this article. I am very thankful to I. Smoli'c, D. Singleton, E.C. Vagenas, S. F. Psihas and especially to T. Juri'c for their discussions, comments and reviews. I am especially thankful to Prof. S. Meljanac, for his comments and advises. This work was supported by the National Counceil for Science and Technology (CONACyT-M'exico) under the scholarship 130881 and under the project 000000000187155.", "pages": [ 17 ] } ]

[ { "title": "Predrag Dominis Prester", "content": "Department of Physics, University of Rijeka, Radmile Matejˇci'c 2, HR-51000 Rijeka, Croatia Email: [email protected]", "pages": [ 1 ] }, { "title": "Abstract.", "content": "The large D limit of AdS 2 × S D -2 solutions in the particular higher-derivative Lovelock-type theory is analyzed. The theory and the solutions were originally considered in an attempt to effectively describe near-horizon behavior of D -dimensional spherically symmetric 2-charge small extremal black holes which in superstring theory context are assumed to correspond to configurations in S 1 × T 9 -D compactification schemes in which fundamental string is wound around circle S 1 . Though in D → ∞ limit the action contains infinite number of higher-derivative terms, their contributions to equations of motion sum into simple exponential form which allows us to find explicit solutions. A simplicity of this example gives us the opportunity to study some connections between α ' and 1 /D expansions. In the leading order in 1 /D the relation between the string parameter α ' and the radius of the horizon r h (in the string frame) satisfies r h ∝ D √ α ' , i.e., we obtain an explicit realization of the relation inferred by Emparan et al. in the different context of large black holes in the ordinary Einstein gravity where α ' is not manifestly present.", "pages": [ 1 ] }, { "title": "1 Introduction and summary of the results", "content": "The idea of studying gravity in the limit of large number of spacetime dimensions D dates back at least to [1]. Original attempts were focused to quantization of gravity, motivated by the possible analogy with large N limit of SU ( N ) gauge theories [1-3]. Though this original application so far did not meet the expectations, it was shown that the idea may be useful in other contexts [4-15]. Recently a new fresh blood was injected to the idea. A detailed study of black hole solutions in classical General Relativity in the large D limit was initiated in [16]. It was shown that General Relativity in D → ∞ limit drastically simplifies, which makes a good starting point for a perturbative expansion in 1 /D . The hope is that this expansion may be useful for answering some realistic questions in D = 4. In the follow-up paper [17] it was shown that in D →∞ limit of GR a broad class of (large) neutral black holes has a universal near-horizon limit, with the limiting geometry being the two dimensional black hole of string theory with a two dimensional target space. To achieve a connection with string geometry, one needs to make identification where r 0 is the radius of the horizon and α ' is the string parameter. In [18] similar large D limit was applied to the special class of higher-derivative Lovelock gravity theories. In this letter we want to study large D limit of AdS 2 × S D -2 backgrounds which describe near-horizon region of spherically symmetric 'small' extremal black holes, i.e., zero-temperature black holes whose horizons are resolved by higher-curvature terms present in the action. In the string theory context, small black holes are assumed to describe 'stringy' objects with the size of the order of the string length l s = √ α ' . As in some regimes effective string coupling constant is small, such objects have potential to offer us a window into small distance behavior of gravity without worries about effects of quantum corrections on the tree-level result. However, there is a problem with this program - because spacetime curvature in the near-horizon region is of the order of 1 /α ' low-energy expansion is expected to break down and one cannot trust low-energy string effective actions for studying such backgrounds. Indeed, except for some 4-dimensional cases, low-energy effective actions are producing wrong results for the entropy of such small extremal black holes. For the development and status of the current understanding of small extremal black holes in the heterotic string theory see, e.g., [19-34] (for a more complete set of references one can consult reviews [35-37]). In the following we shall take standard assumption that effective space-time near-horizon description for heterotic string configurations of our interest is possible, and in particular that near-horizon geometry is AdS 2 × S D -2 . To study large D limit of small black holes in the classical regime of gravity theory, while also keeping the contact with the string theory, we propose to start with a simple and accessible example. Such example is provided by small spherically symmetric extremal 2-charge black holes in the particular 'Lovelock-type' higher-curvature action, whose near-horizon AdS 2 × S D -2 solution was constructed in [24]. 1 This action was invented in an attempt to provide effective near-horizon description of the supersymmetric black hole configurations in the heterotic string theory compactified on S 1 × T 9 -D , where 4 ≤ D ≤ 9, which correspond to fundamental string wound w times around S 1 which has momentum in the direction of S 1 characterized by the natural number n . Indeed, it was shown in [24] that this rather simple effective action provides near-horizon extremal black hole solutions with the Wald entropy matching statistical entropy (obtained by microstate counting inside the string theories) at the lowest order in the string coupling g s (but exactly in α ' ). This Lovelock-type action is not equal to the tree-level effective lowenergy action of the heterotic string theory in any D , so its meaning in the string theory context is not clear. In any case, it is at least an interesting toy model to study large D limit. Though this particular stringy interpretation is meaningful only for D ≤ 9, the theory and solutions can be extended to all D ≥ 4, which allows us to study formally the D →∞ limit. Also, in D →∞ limit this theory has infinite number of higher-derivative terms with unlimited order, which is a property of string low-energy effective actions (but already at finite D ). Though in the course of the paper we frequently refer to string theory connection (and even borrow a notation), we would like to emphasize that our analysis is not restricted to this context and does not rest on the validity of this connection. Lovelock-type actions are interesting by themselves as alternative theories of gravity - they are higher-derivative gravity theories which lead to normal second-order EulerLagrange equations and which do not induce new degrees of freedom (ghosts) when perturbed around large classes of backgrounds. Our AdS 2 × S D -2 configurations are exact classical solutions in one such theory, which makes them valuable topic of interest, even aside mentioned applications in string theory or in descriptions of small black holes. Our study has an important difference from [17]. The black holes we study are small, with the horizon of the size of the order of √ α ' (singular in the lowest, i.e., two-derivative, order), in which contributions to near-horizon properties of all higher-derivative terms are of the same order in α ' . 2 So, in our case α ' is already present from the start and we are interested in the 'interplay' between D and α ' (and possibly other parameters). Let us summarize our main results here: (i) We find that AdS 2 × S D -2 solutions of our Lovelocktype theory take remarkably simple form in D →∞ limit. Though higher-derivative terms of all orders contribute to equations, this infinite sum takes a simple form of the exponential function. This allowed us to find simple analytic solutions, which was not the case for finite D > 7. Our results support the expectations that large D limit may be a useful tool, at least for classical analyzes. (ii) We show that the relation between α ' and D in D →∞ limit is consistent with the relation (1) from [17]. More precisely, it takes the form where r h is the proper radius of the black hole horizon in the string frame metric (all our statements here refer to the string frame). (iii) All Euler densities E k (where k -order density has 2 k -derivatives) have the same leading order at large D , which is O (1) (they are finite in D → ∞ limit). A similar statement holds for α ' -expansion (expansion in derivatives) where all higher curvature terms multiplied by the corresponding power of α ' (i.e., α ' k R k , which is what one has in the effective action) evaluated on near-horizon extremal small black hole solution are O (1) in α ' . We see here explicit realization of the connection between α ' and 1 /D expansions hinted in [17]. However, in our example we can spot a notable difference - in S D -2 spacetime block only Ricci scalar factors contribute in the leading order (contractions containing Ricci tensor and/or Riemann tensor are subleading in 1 /D , though they are of the same order in α ' ). (iv) Conclusions (i)-(iii) are not generic, e.g., they are not satisfied if we truncate the action to fixed finite order in higher-derivative expansion while taking D → ∞ . We demonstrate this on an example of 4-derivative theory in which higher-derivative part of the Lagrangian contains only second Euler density (so called Gauss-Bonnet term). In this theory instead of (1) we have α ' ∼ 1 /D . The fact that theory contains infinite number of higher-derivative terms of unlimited order when D →∞ appears to crucially affect its large D properties. The outline of the paper is as follows. In Section 2 basic properties of Lovelock-type theory and its two charge AdS 2 × S D -2 solution (for finite D ) are reviewed, together with its connection with small extremal black holes and heterotic string theory. In Section 3 we perform large D limit on this solution and analyze its properties. In Section 4 we perform the same calculations inside the theory obtained by truncating the higher-derivative sector to 4-derivative part and show that the solution in large D limit has markedly different properties from the corresponding one from Section 3.", "pages": [ 2, 3 ] }, { "title": "2 Lovelock-type action and small black holes", "content": "In [24] we introduced the following higher-derivative action where E k are Euler densities (or extended Gauss-Bonnet densities) and A 0 denotes lowest-order (2-derivative) part given by Here Φ and T are scalar fields, F (1) and F (2) two U (1) gauge fields, and G D is the Newton constant. The coefficients λ k in (3) are given by where α ' is a constant which in stringy interpretation becomes (squared) string length parameter. The theory defined by (3)-(6) can be defined in any number of spacetime dimensions D . Though we left the upper bound in the sum in (3) unlimited, with intent to show that the action has the same form in all D , it is well-known that for k > [ D/ 2] Euler densities vanish. So the upper limit of the sum can be put to [ D/ 2] and we see that there are finite number of terms in the action (having up to 2[ D/ 2] derivatives). /negationslash For D satisfying 4 ≤ D ≤ 9 the action A 0 (5) is equal to the lowest-order (2-derivative) bosonic part of the effective action of the heterotic string theory compactified on S 1 × T 9 -D , consistently truncated to a sector in which the only nonvanishing fields beside metric are Kaluza-Klein fields coming from S 1 compactification: two gauge fields and one scalar modulus field T . In 2-charge black hole solutions of A 0 mentioned above, one charge is proportional to the winding number w of the fundamental string on the compactification circle S 1 , while the other charge is proportional to the momentum number n along the same S 1 . BPS condition imposes nw > 1. One can calculate the statistical entropy of such BPS states, which are characterized by fixed n and w with nw > 0, by direct counting of microstates, and the result in the leading order in nw is Let us first neglect higher-curvature part of the action. Then we are left with the action A 0 which is known to be the bosonic part of a particular N = 2 supergravity. This theory has supersymmetric (BPS) solutions with the geometries of asymptotically flat spherically symmetric extremal black holes [38]. However, these black holes have singular horizons which have vanishing area. When higher-curvature terms are switched on (by taking α ' = 0 in (3)) one expects that black holes get 'regularized' - the horizon becoming regular, with nonvanishing area and the radius of the order √ α ' . Such black holes are referred to as small. Assuming that these black holes are still extremal ( T bh = 0) one expects that the near-horizon geometries are AdS 2 × S D -2 . When one tries to use full heterotic low-energy effective action to find regulated small black hole solutions, one immediately hits the following problem - due to the fact that curvature around the horizon goes as 1 /α ' , all higher-derivative terms contribute in same order in α ' so the low-energy effective expansion breaks down. Even the techniques of summing up all α ' -corrections, which are working for large extremal black holes in D = 4 and D = 5, are not producing correct results for black hole entropy (7) in the small black hole limit in D = 5 [29,30,37,41]. A proper construction of the near-horizon description for the 2 charge extremal small black holes in heterotic string theory is an interesting issue which is still not completely settled. It was our goal in [24] to try to use the relatively simple Lovelock-type action (3), as a sort of a toy theory, 3 to obtain reasonable near-horizon solutions with the correct expression for the black hole entropy in all 4 ≥ D ≤ 9. 4 Near-horizon geometry of extremal black holes in D dimensions is expected to be isometric to AdS 2 × S D -2 , which in the case of the theory (3)-(6) is where d Ω k denotes standard metric on the unit k -dimensional sphere, and v 1 , 2 , u S,T , and e 1 , 2 are constants to be determined from equations of motion. By using Sen's entropy function formalism [35, 43] we found [24] ∣ ∣ Normalized electric charges n and w , which in the heterotic string theory interpretation correspond to momentum and winding number, are connected with charges q 1 and q 2 , corresponding to U (1) gauge fields F (1) and F (2) (and defined by standard use of Sen's entropy function formalism), through the relations Now, v 2 is the real positive root of an equation which is a polynomial equation in 1 /v 2 of ([ D/ 2] -1)-th order. The equation can be analytically solved for D ≤ 9. For D = 4 and D = 5 the solutions are given by 3 Now, one can still be puzzled how a simple action like (3) can 'substitute' full heterotic low-energy effective action (HLEEA) which is very different and much more complicated (see., e.g., [39, 40]). In particular: (A) HLEEA contains infinite number of higher-derivative terms, organized as expansion in string parameter α ' , some of which contain covariant derivatives and gauge fields, while higher-derivative terms in (3) are finite in number and contain just the metric and no covariant derivatives; (B) HLEEA does not contain α ' 2 (6-derivative) terms which are field-redefinition invariant, while (3) contain such terms inside Euler density term E 3 ; (C) HLEEA contain gravitational Chern-Simons terms, while in (3) there are no such terms. Part of the explanation for (A) lies in SO (1 , 1) × SO ( D -2)) symmetry of the solutions we consider here, which makes all terms which include covariant derivatives irrelevant. Moreover, in case of large extremal geometries it was shown that in a particular scheme, due to the special property of solutions ('parallelizable torsion'), only finite number of terms in the action are relevant [37, 41, 42]. So, it is not unreasonable to expect that something similar happens for small geometries. To answer (B), we note that analysis from [24] implies that only relevant higher-derivative terms in (3) are not invariant on field redefinitions. The objection (C) is the trickiest, especially because for large extremal geometries the relevant higher-derivative terms in HLEEA are exactly those originating from gravitational Chern-Simons term [37,41,42]. On this we just note that both gravitational Chern-Simons terms and Euler terms are closely connected with anomalies. It is important to remember that LEEA are by construction not appropriate for addressing small geometries, so use of (3) is an attempt of a different type of an effective description. Let us also add here that there are also interesting similarities between HLEEA and (3). For example, if we identify parameter α ' in (3) with the string tension, then the coefficient λ 2 = α ' / 8 is the same in the two actions, and in general the coefficients λ k have dependence on k which roughly corresponds to the behavior of coefficients multiplying Riemann k -type terms in the heterotic actions. 4 Note that we cannot claim that full (i.e., in the whole space) asymptotically flat extremal black hole solutions with AdS 2 × S D -2 near-horizon geometry indeed exist in the Lovelock-type theory. In [44] it was claimed, based on the numerical analysis, that in D = 4 it indeed does not exist. However, this analysis is not conclusive because some 4-derivative Lagrangian terms were neglected in the calculation, so this result is not obtained for (3). Beside, outside the near-horizon region there is no reason to believe that action (3) can be used as effective description of heterotic string theory, so this question is meaningless in string theory context while for D = 6 and D = 7 they are given by We have not found analytic form of the solution for general D . Sen's entropy function formalism allows one to find the black hole entropy as defined by Wald formula [45]. The result, valid in all D , is [24] S bh = 4 π √ | nw | (14) which matches microscopic result (7) for BPS configurations. In fact, requirement of this matching fixes the coefficients λ k to the form (6) uniquely. 5 Lovelock-type theory (3)-(6) gives the same result for black hole entropy (14) for configurations with nw < 0. This is not so in the heterotic string theory interpretation where 2-charge configurations with nw < 0, which are not supersymmetric (non-BPS), have statistical entropy given by The same expression for the microscopic entropy one gets for corresponding 2-charge configurations with the elementary string in type-II superstring theories compactified on S 1 × T 9 -D , which are 1/4-BPS, regardless of the sign of nw . We note that the entropy (15) can be reproduced by Lovelock-type action (3) by taking the coefficients λ k to be instead of (6). With this choice AdS 2 × S D -2 solutions can be obtained from those in (9) through In the following we shall focus on solutions (9). Using (17) one can easily extend all results to solutions of the theory with λ k given by (16). 6", "pages": [ 4, 5, 6 ] }, { "title": "3 D →∞ limit", "content": "The simplicity of the near-horizon solution from the previous section makes it interesting toy model for analyzing large D limit. In particular, we see from (9) that all dependence on D is contained in parameter v 2 , which is also the only parameter which is not trivial to find. On the other hand, in the D →∞ limit number of terms in the action becomes infinite (of the order D/ 2), which makes the limit a priori non-trivial (and also mimics behavior of effective string theory actions). We take D →∞ while keeping other parameters ( G D , α ' ) and charges ( n and w ) fixed.In the leading order in large D limit polynomial on the right-hand side of (11) can be summed which means that equation (11) for v 2 takes the simple form in D →∞ limit That infinite series can be summed to give simple exponential function is a remarkable result, which may be connected to the conjecture from [24] that the gravitational part of the action (3)-(6) can be written in an exponential form (see Eq. (5.1) from [24]). The solution of (19) is Note that v 2 is the square of the proper radius of the S D -2 sphere, so of the black hole horizon, by which we mean that a proper area of the horizon in the string frame metric is given by where Ω D -2 is the area of the standard ( D -2)-sphere with unit radius. By putting r h ≡ √ v 2 in (20) we obtain the relation (2). Interestingly, the behavior r h ∼ D √ α ' was previously obtained in [17] by analyzing large D limit of large black holes in the ordinary general relativity, which is a rather different context in which parameter α ' is not present but inferred through the comparison with string theory calculations. Let us mention here that the property that α ' v 1 ∼ 1 and α ' v 2 ∼ D 2 is such that it guarantees that curvature scalars are generically finite and nonvanishing in D → ∞ limit (i.e., they are O (1) in D ), which means that every Euler terms E k , when evaluated on our solution, gives finite contribution in the Lagrangian density in D →∞ limit both in AdS 2 and S D -2 block. This shows that in this theory there are some similarities between 1 /D and α ' expansions. We postpone more detailed discussion on this to the next section, where we shall also show that the things differ when one truncates higher-derivative part of the action to the fixed order. To complete large D limit of the solution (9) we have to calculate u S . For this we need large D behavior of Ω D -2 which is given by Using (20) and (22) in expression for u S in (9) we obtain Together with (9) and (20) this completes large D limit of our solution. In the fully quantized theory one expects that the field Φ is connected with the effective quantum coupling constant g eff through a relation of the form From (23) and (24) follows that if we want to have sensible theory in large D limit, with meaningful perturbative expansion in which g eff is finite and nonvanishing, we have to scale Newton constant G D and/or α ' such that where ζ and b are D -independent dimensionless parameters. 7 Then the effective coupling constant becomes Now we see that there is a critical value of ζ defined by which separates the two phases in the D → ∞ limit: ζ < ζ c for which g eff → 0, and ζ > ζ c for which g eff →∞ . In the critical point ζ = ζ c we obtain that g eff is finite in D →∞ limit and In the critical point the effective coupling constant is finite in D → ∞ limit so there is a hope that in this case one can define sensible quantum theory. For us here it is important that from (28) follows that g eff can be made arbitrarily small by taking | nw | /greatermuch 1 (which is the relevant regime in the string theory interpretation) which makes tree-level approximation credible. Let us say a few words more on the near-horizon geometry in D →∞ limit. Putting (22) and (20) in (21) one gets that in the 'string frame' horizon area is If we use the critical scaling (i.e., (25) with ζ = ζ c ) we obtain If g 0 is D -independent the dimensionless horizon area, when measured in the units of G D , is finite and nonvanishing in D →∞ limit. For completeness, let us briefly analyze D →∞ limit of the geometry in the Einstein frame, in which metric is The horizon area in the Einstein frame is then while for the square of the AdS 2 radius one gets If we use again the critical scaling we obtain If g 0 is D -independent one obtains v (E) 1 → α ' / 2 = v 1 . We see that in critical scaling, combined with an assumption that g 0 ∼ O (1) in D , leads to the same O (1) behavior of geometries, both in string frame and Einstein frame, if one properly defines units of geometrical objects.", "pages": [ 6, 7, 8 ] }, { "title": "4 Finite- vs. infinite- derivative action in D →∞ limit", "content": "The natural question to ask is are the results presented in Sec. 3 special or general, i.e., does the choice of particular higher-derivative corrections, given by (3) and (6) (or (16)), have some special consequences not shared by generic higher-derivative theories in D →∞ . In particular, how important is the property that in D →∞ limit the Lagrangian of our Lovelock-type action has an infinite number of higher-derivative terms? We would ideally also To throw some light one these issues, we take another simple example of a different kind - an action whose higher-derivative sector is terminated at 4-derivative (i.e., R 2 ) order and analyze its large D limit. To achieve easy comparison with the results from the previous section, we take the new action to be where E 2 is the second Euler density, known as the Gauss-Bonnet density, which by (4) is and A 0 is as in (5). This action is obtained from the action (3)-(6) by terminating the sum over higherderivative terms in (3) at the 4-derivative level (i.e., by keeping just the first element k = 2 for all D ). From now on we shall refer to this action as Gauss-Bonnet-type (GB-type) theory. Again, one can use Sen's entropy function formalism to find solutions of GB-type theory with AdS 2 × S D -2 geometry (8) and the entropy of small extremal black holes with such near-horizon configurations. This calculation was already made in [23] and the results are: where we again used the same (string theory motivated) normalization for electric charges (10). The black hole entropy is We see that for D = 4 and D = 5 (37)-(38) is equal to (9), (12) and (14), as it should. The simple 4derivative Gauss-Bonnet term is also capable of regularizing small black holes in all D , though in D > 5 dimensions it does not lead to black hole entropy which matches string theory statistical entropy (7). So, the GB-type theory, unlike Lovelock-type theory, does not have potential to offer effective description of 2-charge configurations in string theories when D > 5. We now perform the large D limit. In the leading order the solution (37) becomes ∣ while the black hole entropy is It is obvious that the D dependence of the solution (39) in GB-type theory is different from the one obtained in the Lovelock-type theory presented in Eqs. (9), (20) and (23). In particular, instead of the relation (2) one now obtains where again r h = √ v 2 is the proper radius of the horizon. The relation (41) has different scaling in D from (1). So, our first conclusion is that the scaling (1) is not a generic result (in the setup we consider), and that the scaling depends crucially on the properties of the higher-derivative sector. Let us now analyze the geometry in GB-type theory more closely. In the string frame the area of ( D -2)-sphere (horizon area) is now In the Einstein frame we obtain that the squared radius of AdS 2 factor and proper area of ( D -2)-sphere are Again, one obtains completely different D -dependence compared to those in Lovelock-type theory. In particular, we note that factors of the type D D/ 2 are completely absent in (39)-(43), if they are not introduced through parameters ( G D and α ' ) or charges ( n and w ). From the expression for u S in (39) we obtain that the analogue of the critical scaling condition (25) and (27) is now In the critical scaling the horizon area, both in string frame and Einstein frame, are finite in D → ∞ limit when measured in units of (2 πeα ' ). A mayor difference between solutions in Lovelock-type and GB-type actions is in the D → ∞ limit of curvature scalars. To show this, first not that all irreducible curvature scalars (those which are not products of two scalars) for AdS 2 × S D -2 geometry are sum of AdS 2 contribution (denoted with subscript A ) and S D -2 contribution (denoted with the subscript S ), e.g., In the Lovelock-type theory (3) generic curvature scalars are O (1) in D . More precisely, all AdS 2 scalars are D -independent (because v 1 = α ' / 2), while of S D -2 irreducible scalars only R S ∼ O (1) while all others are subleading in 1 /D , e.g., As a consequence every Euler term in the Lovelock-type action produces a finite contribution in D →∞ limit. We can be even more precise. Already for finite D we know that AdS 2 contribution of curvature terms to equations of motion is solely through R A (and linearly). When combined with previously said, we obtain that in the leading order of large D limit contributions from gravity sector to AdS 2 × S D -2 solutions are just factors of Ricci scalars R A and R S , which are linear in R A . In the GB-type theory (35) one obtains in the large D limit the following expressions We see that here the leading order in D comes solely from the 4-derivative terms (which are highestderivative terms) and that curvature scalars are of the order D 2 , which is completely different behavior then in the Lovelock-type theory. 8 From the analysis of this particular GB-type theory some generic conclusions can be drawn. One is that infinite number of terms (of the order D ) with unlimited number of derivatives (of the order D/ 2) present in the Lovelock-type action is responsible for markedly different large D behavior of solutions, compared with the theories which have fixed order in derivatives. However, our analysis does not reveal are there some properties of D →∞ limit which are special for our Lovelock-type action, when applied to AdS 2 × D -2 configurations. We plan to investigate this in the future.", "pages": [ 9, 10, 11 ] }, { "title": "Acknowledgements", "content": "We thank Zdravko Lenac for stimulating discussions. The research was supported by the Croatian Ministry of Science, Education and Sport under the contract no. 119-0982930-1016.", "pages": [ 11 ] } ]

[ { "title": "Guillaume Bossard a and Stefanos Katmadas a , b", "content": "guillaume.bossard [at] cpht.polytechnique.fr , stefanos.katmadas [at] cpht.polytechnique.fr Abstract: We present a manifestly duality covariant formulation of the composite nonBPS and almost-BPS systems of multi-centre black hole solutions in four dimensions. The method of nilpotent orbits is used to define the two systems in terms of first order flow equations that transform covariantly under the duality group. Subsequently, we rewrite both systems of equations in terms of real, manifestly duality covariant, linear systems of Poisson equations. Somewhat unexpectedly, we find that the two systems are naturally described by the same equations involving space dependent abelian isometries that are conjugate to T-dualities by similarity transformations. Keywords: Black Holes in String Theory, Supergravity Models.", "pages": [ 1 ] }, { "title": "1. Introduction and overview", "content": "The structure of black hole solutions in the supergravity effective description of string/Mtheory compactifications has long been a useful tool in understanding their microscopic realisation in string theory. In the supersymmetric (BPS) case, the supergravity black hole solutions [1, 2, 3] have been understood microscopically to be described by D-branes wrapping supersymmetric cycles [4, 5]. Nonetheless, all microscopic BPS configurations cannot correspond to single centre solutions in the effective N = 2 supergravity description, and it has been understood [6] that the latter then describe composite bound states of BPS black holes [7, 6, 8]. The detailed description of these solutions and their domain of stability in moduli space was very important in order to correctly reproduce the corresponding microscopic results [9, 10]. In the aim of generalising this understanding to more realistic non-supersymmetric black holes, the first non-trivial step is to consider non-supersymmetric extremal black holes. One can classify extremal black hole solutions into two main classes: the overrotating class, for which the angular momentum saturates the extremality bound, and the under-rotating class, for which the electro-magnetic charges saturate the extremality bound, and which includes the supersymmetric solutions. Given that a stationary spacetime defines a time fibration over a space-like three-dimensional base, one may distinguish the various classes by their corresponding base space. The characteristic of all known underrotating extremal solutions is that the three-dimensional base space is flat, i.e. R 3 with the Euclidean metric, whereas the over-rotating extremal solutions all admit the singular three-dimensional base of the extremal Kerr solution. In this paper we will only discuss under-rotating solutions admitting a flat three-dimensional base space. 1 Within the class of under-rotating solutions with an R 3 base, there are two known interacting systems of non-BPS black holes. The composite non-BPS system [11, 12, 13] describes the interactions of black holes that are non-BPS in isolation, while the almostBPS system [14, 15, 16], allows for configurations of centres that are be both BPS and non-BPS in isolation. Both these systems have been studied in some detail and, in several cases, the known explicit solutions are general enough to allow for the most general solution to be obtained by dualities ( i.e. symmetries of the three-dimensional theory). An important feature of the existing formulations of both these non-BPS systems is the existence of distinguished directions in charge space, such that some are associated to harmonic functions as in the BPS system, whereas others are associated to functions solving Poisson equations. The different directions being then intrinsically inequivalent, one cannot straightforwardly rotate one into another by duality, as for BPS solutions. It is therefore customary to solve these equations and construct solutions parametrised by integration constants that can only be related to charges and asymptotic moduli a posteriori. Of course, one may straightforwardly apply a duality rotation on the solutions to change the identification of pre-determined charges, thus covering all possible charge configurations. This process is however not only cumbersome, but more importantly it obscures the overall structure of these systems, which should be manifest in a fully covariant formulation. A further motivation for generalising the existing formulations is that non-supersymmetric under-rotating extremal black holes seem to admit a microscopic description similar to their BPS cousins [17, 18]. Understanding the domains of existence and stability of these solutions will eventually be important in order to understand in more detail their microscopic description. However, despite the existence of explicit solutions describing bound states of such non-BPS black holes, their domain of existence in moduli space has not been studied in detail. One of the main obstacles in carrying out this program originates from the property that these solutions are described in terms of parameters that are not the physical charges themselves, as explained above. In order to solve this problem, it is important to obtain the general solutions associated to fixed charge configurations. Being able to do this in a straightforward manner requires the definition of covariant equations that are not constrained to a fixed duality frame. In this paper, we provide a manifestly duality covariant formulation of both the composite non-BPS and the almost-BPS systems that unifies their description, by rewriting these systems of linear differential equations in terms of duality covariant quantities, similar to the BPS system [6]. Although these systems of equations turn out to be more complicated than the BPS one, the formalism permits to compute the most general solutions associated to fixed charge configurations. The construction of explicit solutions within these systems will be discussed in a forthcoming publication. Somewhat surprisingly, we find that both systems are naturally formulated by considering space dependent translations along abelian isometries of the scalar manifold. As all equations are written in terms of objects transforming linearly under electromagnetic dualities, it is natural to introduce local generators for the abelian subgroups corresponding to these isometries. The relevant equations in both the composite non-BPS and the almostBPS can then be expressed in terms of a covariant derivative that contains a nontrivial connection in the Lie algebra of the relevant abelian isometries. In a specific frame, these abelian isometries coincide with the isometries that generalise the action of T-dualities combined with large gauge transformations of the p -form gauge fields present in string theory, for continuous parameters. In general, they will define abelian subgroups that are homomorphic to the latter by similarity transformations. For simplicity, we will refer to them as T-dualities in this paper, despite the fact that they are not in general associated to any duality in string theory. The set of all abelian isometries in the scalar target space of symmetric models has been considered before in the context of (non-)BPS black holes in different guises, see e.g. [19, 20]. The paper is organised as follows. The remainder of this introductory section is devoted to an informal presentation of the method we use to obtain the duality covariant formulation of the composite non-BPS and almost-BPS systems. In doing so, we also explain in more detail way how the T-dualities arise in both systems and in fact allow us to define their action on the relevant quantities without introducing explicit matrix representations. In section 2 we present the general action of arbitrary T-dualities in terms of symplectic vectors and discuss the corresponding decomposition of the charge vector space. In section 3 we consider the three dimensional Euclidean coset non-linear sigma model describing stationary black hole solutions. We then explicitly solve the nilpotency conditions on the scalar momentum as a Lie algebra element for both the composite non-BPS and the almostBPS system, to obtain the first order flow equations describing each class. We then go on in sections 4 and 5 to rewrite these flow equations as a linear system of Poisson equations for a set of symplectic vectors parametrising the solutions in an arbitrary duality frame. The reader interested in applications can find summaries of the two systems in sections 4.5 and 5.5, which are self contained and only require the definition of T-dualities in section 2.2. We conclude in section 6, where we discuss some of the implications of the local T-dualities acting on both systems and comment on further generalisations.", "pages": [ 3, 4, 5 ] }, { "title": "1.1 Overview of results", "content": "In order to find systems describing stationary black hole solutions, we consider the reduction along time to a three-dimensional Euclidean theory. In this setting, one can straightforwardly dualise all vector fields to scalars, to obtain a non-linear sigma model coupled to Euclidean gravity. In this paper, we only consider N = 2 supergravity theories with a symmetric scalar space, so that the scalar fields of the resulting non-linear sigma model over a pseudo-Riemannian symmetric space. In the case of solutions with a flat base space, there is a powerful method for obtaining first order flow equations that solve the full equations of motion, starting from the observation that the Einstein equation can be solved by assuming the scalar momentum, P , to be a nilpotent element of the Lie algebra. It appears then that the scalar fields equations of motion reduce to a solvable system of differential equations. Based on standard group theoretical considerations, one shows that the nilpotency of P implies that there exists a second element of the algebra, h , such that an eigenvalue equation of the type holds, where h P ≡ [ h , P ] and n is a positive integer. It then follows that P must be a linear combination of eigenvectors of h with positive eigenvalues. In section 3 we consider the relevant eigenvalue equations for both the composite non-BPS and almost-BPS systems, giving the precise solutions for both the momentum P and the auxiliary fields h . For the composite non-BPS system, this analysis has been given in [13], but is included here for completeness. The corresponding flow equations for the almost-BPS system have been derived for the STU model in [21] by reduction of the second order equations of motion, while we present the general analysis for any symmetric model. The presence of the aforementioned auxiliary fields is a central feature of both systems, as will become clear from our treatment. In fact, we find it most convenient to keep them throughout, as variables characterising the solutions, despite the fact that one may in principle solve for them in terms of physical fields. In the systems we shall consider, h can be parametrised in terms of a distinguished direction in the charge vector space (plus a phase for the almost-BPS system). This distinguished direction is associated to a constant so-called very small vector, or in other words a one-charge vector, in the sense that it can always be brought by dualities to a canonical form where it has only one charge, as a pure D 6 charge for example. However, one must keep this very small vector arbitrary in order to be able to consider generic charge configurations. Treating the system in terms of electromagnetic charges and auxiliary vectors implies that one has to work with real quantities, rather than the complex quantities that appear naturally in the Euclidean 3-dimensional non-linear sigma model. It turns out that this is essential in solving the system in terms of local functions, since a linear structure only appears after writing all flow equations in the real basis. As mentioned above, this change of basis, that permits to solve non-linear first order equations in terms a solvable system of linear differential equations, is the main technical result of this paper. In order to appreciate the importance of the real formulation, the simple example of the analogous situation for the BPS system is instructive. In this case the auxiliary element h in (1.2) is much simpler, as it is parametrised by a single phase, e iα . Taking the static case for simplicity, (1.2) leads to the standard BPS equations Here, r is the distance from the horizon, e U is the single function parametrising the metric, t i are the vector multiplet scalars and Z (Γ), Z i (Γ) are the central charge of the charge vector Γ, and its Kahler derivative. In this form, it seems a rather non-trivial task to solve (1.3) explicitly. It is however possible to combine the scalar degrees of freedom in a single vector, e -U e -iα V , where V is the so called symplectic section, which parametrises the scalars t i . The two equations (1.3) can then be written as in [6], using standard special geometry identities. It is now trivial to solve the last equation as where H is a vector of harmonic functions, whose poles are identified with the charges Γ. One still has to solve a set of algebraic equations to obtain the scalars t i and e U in terms of harmonic functions [8], but it is important to stress that (1.5) is equivalent to a solution for the physical scalars. The analogous computation for the non-BPS case includes the auxiliary vector parametrising the element h and is therefore considerably more involved. For the restricted case of single centre under-rotating black holes, this was done in detail in [13] and the result was given in terms of the objects appearing in (1.5), together with the vectors in (1.2) above. This class is naturally part of both the multi-centre systems considered in this paper and its description is crucial for understanding the structure of the more general classes. Explicitly, for the single centre class we have [22, 13] where H is again a vector of harmonic functions describing the charges of the black hole, while three new objects appear, namely the function M and the two constant very small symplectic (pseudo-charge) vectors R and R ∗ , which are not mutually local, i.e. 〈 R,R ∗ 〉 /negationslash = 0. The presence of two different very small vectors in (1.6) can be somehow surprising, but it is important to note that they do not define independent parameters. Indeed, we find that these vectors are determined by the electromagnetic charges Γ, up to duality transformations leaving Γ invariant [13], so that they are determined in terms of Γ and the asymptotic scalar fields t i ∞ . Conversely, one can view the charges, Γ, and the harmonic vector H as being constrained to lie in a Lagrangian subspace determined by R and R ∗ . The terms proportional to the two very small vectors are worth discussing in more detail. First, note that the term proportional to R is simply a projection on that component of H with an additional factor of 2, implying that this particular component appears with a flipped sign. This is a very general feature of non-BPS solutions that has been observed in many examples in the literature [23, 24, 25]. On the other hand, the component along R ∗ in (1.6) contains the function, M , which represents the only genuinely new term in the expression for the scalars, and is constrained to be a dipole harmonic function characterising the angular momentum of the under-rotating non-BPS black hole [15]. In sections 4 and 5, we explicitly solve the composite non-BPS and almost-BPS nonlinear first order systems derived in section 3 in terms of real vectors of local functions. As we shall see, the generalisation to multi-centre systems can be performed by allowing one of R or R ∗ to vary in space, while keeping their symplectic product fixed 〈 R,R ∗ 〉 = 4. The space dependence of the non-constant vector can then be reabsorbed into a space dependent duality transformation that leaves the constant very small vector invariant. Because this duality transformation lies in an abelian subgroup that is conjugate to the group of Tdualities by similarity transformations, we shall simply refer to them as T-dualities. In particular, the composite non-BPS system is described by the T-dualities, T + , defined as leaving R invariant, while the almost-BPS system is described by the T-dualities, T -, leaving R ∗ invariant. One then shows that the symplectic section V still takes the form (1.6), with H constrained to lie in the same Lagrangian subspace determined by R and R ∗ . Since one of the very small vectors is not constant, H is not harmonic anymore, but satisfies the Poisson equation where the second equation imposes that the n v parameters of the T-dualities are given by arbitrary harmonic functions. The function M in (1.6) is also specified by a T-duality covariant equation, given by which also fixes the angular momentum one-form, ω . Although it is not manifest from these general equations, the graded structure of the vector space is such that the system is solvable. Because the two T-dualities in the right hand side of (1.7) have a nontrivial kernel, it follows that the source in (1.7) does not contain some of the components of exp[ -T ± ] H , which therefore includes both harmonic and non-harmonic components. It turns out that T ± act as raising or lowering operators, so that only the harmonic components turn out to source the non-harmonic ones, i.e. the non-harmonic components of exp[ -T ± ] H do not source themselves. These properties will be discussed in detail in sections 4 and 5. By definition, T-dualities are represented in terms of symplectic matrices in the 2( n v + 1)-dimensional vector space of charges. However, as will be shown in detail in the following sections, the n v parameters of a T-duality can be arranged into a symplectic vector obeying a number of constraints. One can therefore write the action of the corresponding T-duality in terms of this vector, together with R and R ∗ , using the symplectic product and the quartic symmetric invariant I 4 (which defines the entropy of extremal static black holes). In this way, we obtain explicit expressions for the sources in (1.7), in terms of exp[ -T ± ] H and the parameters of the T-dualities, that can be evaluated explicitly once a model and its associated quartic invariant are specified. These equations reduce to the systems introduced in [11, 14], for a particular choice for the two very small vectors above. This concludes our short presentation of the main results in this paper. For the convenience of the reader, we provide an account of the results appearing in the following sections, which can be read independently of each other with the exception of section 2, that is basic to most applications. In section 2 we give a detailed discussion of the general T-dualities. We show that given two very small vectors R and R ∗ that do not mutually commute, one can explicitly define a graded decomposition of the symplectic vector space, on which the T-dualities T ± can be respectively defined as raising and lowering operators. The structure of this decomposition is essential for all applications in this paper. We then go on in section 3 to define the two systems of non-BPS multi-centre black hole solutions in terms of the non-linear sigma model in three dimensions, obtained after dimensional reduction over the time direction. After an overview of the main properties of this three-dimensional Euclidean theory, we discuss in detail the eigenvalue equations (1.2) for the two systems at hand. This results to two sets of first order flow equations that completely describe the composite non-BPS and almost-BPS systems in four dimensions. Sections 4 and 5 are mirror copies of each other, wherein we present in detail the change of variables that transforms the flow equations of section 3 to two linear systems. The reader can find a concise summary of the two multi-centre systems in the real formulation in sections 4.5 and 5.5 respectively, where we also discuss some general properties of the solutions, deferring a more detailed presentation and explicit examples for a forthcoming publication [26].", "pages": [ 5, 6, 7, 8 ] }, { "title": "2. T-dualities", "content": "In this section, we provide a detailed discussion of the abelian isometries on the scalar target space of N = 2 supergravity, 2 that leave a given charge vector invariant. As will be shown in later sections of this paper, the precise action of these isometries, in their most general form, is of central importance in the construction of multi-centre non-BPS black holes. We start with an informal discussion of the simplest example of such abelian isometries. Subsequently, we review some properties of symmetric special Kahler spaces with a cubic prepotential in section 2.1. Section 2.2 is devoted to the definition of the abelian isometries we shall refer to as T-dualities and their explicit action in terms of the associated real vectors. In section 2.3 we discuss the realisation of the same isometries in the complex basis defined by the central charge and its Kahler derivatives. Throughout this paper, we study extremal multi-centre black hole solutions in N = 2 supergravity coupled to n v vector multiplets labeled by an index i = 1 , . . . , n v , whose scalar fields, t i , parametrise a symmetric special Kahler target space, M 4 . These spaces were classified some time ago [27] and include minimally coupled vector multiplets, which are not of interest in this work, and theories with a cubic prepotential, specified by a completely symmetric tensor c ijk [28] (cf. (A.5)). The target space geometry is governed by a Kahler potential (cf. (A.7)), which manifestly depends only on the imaginary part of the scalars, t i . It follows that the real parts of the scalars are coordinates along n v isometries of the scalar manifold, acting as where k i is a vector of n v constant real parameters. These isometries are generic in all cubic models and clearly form an abelian algebra. For theories originating from Calabi-Yau string compactifications, the operation (2.1) can be viewed as large gauge transformations on the higher dimensional tensor gauge fields along internal cycles, combined with T-dualities. While this description is useful in characterising the symmetries themselves, for reasons that will become clear below, in this work we are interested in the embedding of these isometries in the symplectic group, which acts on the electric and magnetic gauge fields in four dimensions. The convenient variable to use in order to make the action of the isometries in (2.1) transparent is the so called scalar symplectic section, V , which is a somewhat redundant way of repackaging the scalars, as where F I are the derivatives of the prepotential with respect to the X I . The index I = { 0 , i } , runs over one more entry than n v and enumerates all the gauge fields in the theory, i.e. the vector multiplet gauge fields and the graviphoton. Note that V changes under Kahler transformations by a phase and is subject to the constraint (A.6), so that it encompasses only 2 n v degrees of freedom, identified with the physical scalars t i s. The advantage of this variable is that, unlike the physical scalars, it transforms linearly under electric/magnetic duality transformations, in exactly the same way as the electromagnetic charges. For instance, the isometries in (2.1) are described by a linear transformation acting on the charges as where we defined the abelian generators T k for later convenience. One can now easily verify that the same operation (2.3) acting on the section in (2.2) leads to (2.1) for the physical scalars. In this formulation, the connection of the isometries (2.1) to higher dimensional gauge transformations is more transparent, since it has a natural action on the electromagnetic gauge fields. Moreover, this particular set of abelian transformations also arises in the form of spectral flows in conformal field theories describing black holes microscopically. The crucial feature of the symplectic embedding of the abelian isometries is, however, that one may generate an infinite number of inequivalent sets of abelian isometries by conjugating the matrix exp[T k ] in (2.3) by a general U -duality transformation, as in [20]. 3 These sets of isometries are more complicated than the one in (2.1) and do not commute with it. From a higher dimensional point of view, some of these more general isometries can also be viewed either as large gauge transformations conjugated with generic T - and/or U -dualities, or as (generalised) spectral flows in a dual conformal field theory. Here, we refer to them simply as T-dualities for brevity and we focus on the case of symmetric scalar manifolds, which allows for the most general transformations to be described explicitly. The representation of a generic T-duality in terms of matrices is of course a rather tedious task, which can be circumvented in a natural way, intrinsically tied to the systems of non-BPS black holes we consider. The crucial observation is that there is always a graded decomposition of the vector space in four components, generalising the clear distinction between the various components in (2.3), based on their transformation rule under Tdualities. Indeed, general T-dualities act consistently on each component of the charge space with a fixed homogeneity in the parameters k i , which can never exceed three. In particular, there is a distinguished direction that is invariant under the action of any given T-duality, as for example the electric charge q 0 is left invariant in (2.3). The q 0 charge is rather special, since it is an example of a so-called very small vector. We will recall the precise definition of such a vector in what follows, but loosely speaking a very small vector can be defined as a 'one charge vector', in the sense that it is U -dual to a pure q 0 charge. Clearly, the distinguished direction that is left invariant under a generic T-duality must then always be a very small vector, given that all such transformations are U -dual to the above example. The relevance of very small vectors for extremal non-BPS solutions arises already in the single centre class [22, 13, 29], which is naturally described in terms of an auxiliary pair of mutually nonlocal constant very small vectors, constrained by the physical charge vector. As we will show explicitly in sections 4 and 5 below, the corresponding multicentre systems are naturally described by promoting one of these very small vectors to be not constant. In order to see the connection to T-dualities, consider the very small vector ˆ R defined such that its only non-vanishing component is q 0 = 4. A general very small vector can be parametrised as where c and s i are allowed to take singular values as long as the components of S are well defined in the limit. 4 It follows that a general very small vector ˆ R ∗ satisfying 〈 ˆ R, ˆ R ∗ 〉 = 4 can be parametrised as where the only non-vanishing component of R ∗ 0 is p 0 = 1. In the case of a constant vector ˆ R ∗ , (2.5) is simply a convenient parametrisation, but in the more general case when ˆ R ∗ is not constant, one can assume the parameters s i to be functions of space, to obtain where we used the abelian property of (2.3). As we will show explicitly in later sections, the composite non-BPS system is naturally characterised by two very small vectors, one constant ˆ R and one non-constant ˆ R ∗ , which have a non-vanishing constant symplectic product 〈 ˆ R, ˆ R ∗ 〉 = 4. The almost-BPS system is similarly described by two very small vectors, only the role of ˆ R and ˆ R ∗ are interchanged. The constant very small vector is left invariant by the relevant T-dualities (as for example (2.3)), which are therefore different for each system. The non-constant very small vector can be expressed as in (2.5) for a constant vector R ∗ 0 in the composite non-BPS system (or respectively R 0 for the almostBPS system), and it follows that it satisfies (2.6). In the rest of the paper ˆ R and ˆ R ∗ will be generic very small vectors, and the associated T-dualities will define abelian subgroups conjugate to the one described in (2.3).", "pages": [ 8, 9, 10, 11 ] }, { "title": "2.1 Symmetric special Kahler spaces", "content": "In this paper we consider N = 2 supergravity theories defined in [28] for which the special Kahler target space, M 4 , is a symmetric space and that can be obtained as Kaluza-Klein reductions of corresponding five dimensional theories. 5 In this case, M 4 is a coset space of the four-dimensional duality group, G 4 , by its maximal compact subgroup U (1) × K 4 For the class of theories we consider, the scalar target space is a symmetric space even after dimensional reduction/oxidation to three/five dimensions, so that (2.7) is part of the sequence of embeddings where by G d , K d we denote the duality group and (part of) the isotropy group in d dimensions respectively. Note that the divisor group in three dimensions is non-compact because we consider the time-like reduction to three dimensions, 6 as that is relevant for the applications we consider later on. Note that K 4 is the compact real form of G 5 , by property of N = 2 supersymmetry. One can always define a set of vielbeine associated to the Kahler metric g i ¯  on M 4 such that the constant symmetric tensor where c ijk is the G 5 invariant tensor defining the prepotential (cf. (A.5)), is left invariant by K 4 . Then, the contravariant symmetric tensor c abc in the conjugate representation satisfies the Jordan identity [28] In a complex basis, the Lie algebra of G 4 , denoted g 4 (and respectively k 4 for K 4 ), naturally decomposes as It follows that the relevant parameters are given by those corresponding to the elements of k 4 , denoted by G a b , a real scalar γ and a complex vector Λ a . The corresponding algebra is realised in terms of anticommuting parameters with the nilpotent differential 7 Note that the statement of invariance of the tensor c abc under K 4 implies that N [ ¯ Z ] ≡ 1 6 c abc ¯ Z a ¯ Z b ¯ Z c is K 4 invariant for any vector ¯ Z a transforming in the relevant n v -dimensional complex representation of K 4 . One can check that the variation of G a b in (2.13) indeed leaves invariant the cubic norm N [ Z ] ≡ N [ ¯ Z ] for an anticommuting Λ a . The invariance of the cubic norm N [ Z ] can be used to define duality invariants and restricted charge vectors, a concept that is of central importance for the applications we consider later in this paper. First, we introduce the quartic invariant for a charge vector Γ, in terms of its central charges, Z ≡ Z (Γ), Z a ≡ Z a (Γ), as This expression can be verified to be invariant under the g 4 generators of (2.13), so that it is moduli independent. This is manifest by the corresponding real form of this invariant, which is given solely in terms of charges by where we also defined the completely symmetric tensor t MNPQ for later reference. Again, one can easily check that (2.15) is invariant under the example T-duality in (2.3), and it is more generally invariant under an arbitrary G 4 transformation. /negationslash We are now in a position to introduce the concept of charge vectors of restricted rank. A generic vector leads to a nonvanishing invariant (2.14)-(2.15) and is also referred to as a rank-four vector, due to the quartic nature of the invariant. Similarly, a rank-three vector, Γ 3 , is a vector for which the quartic invariant vanishes, but not its derivative. An obvious example is a vector with only p i = 0 and all other charges vanishing, so that the derivative I ' 4 (Γ 3 ) is nonzero and proportional to the cubic term N [ p ]. There are two more classes of restricted vectors, defined analogously as rank-two (small) and rank-one (very small) vectors. A rank-two vector, Γ 2 , is defined such that both I 4 (Γ 2 ) = I ' 4 (Γ 2 ) = 0, and a simple example is provided by a vector with all entries vanishing except the p i , with the additional constraint that N [ p ] = 0. Finally, a very small vector, Γ 1 , is defined such that for any vector Γ. In this case, we can also give a general definition in terms of the complex basis, which is in fact independent of the values of the scalar fields. In this paper we will often make use of a rank one vector, R , that we choose without loss of generality such that | Z ( R ) | = 1. One shows that such a very small vector satisfies where N [Ω] is a phase by construction. The remaining central charges Ω a are such that A general very small vector can be obtained by rescaling both N [Ω] and Ω a by a real function. Examples of very small vectors were already given above, as vectors where only the q 0 or p 0 component is nonzero, while the parametrisation given in (2.5) is generic up to a possibly singular rescaling.", "pages": [ 11, 12, 13 ] }, { "title": "2.2 Freudenthal ternary algebra realisation of G 4", "content": "We now proceed in describing the duality group G 4 , as defined above, in terms of real vector parameters. This is essential for discussing the T-dualities, which are contained in G 4 as subgroups and can therefore also be described in terms of real vector parameters in the general case, similar to the example (2.3) above. The central object for the definition of G 4 in the real basis is the quartic invariant in (2.15) and its derivatives. It is convenient to define a symplectic vector out the first derivative, I ' 4 (Γ), of the quartic invariant so that Using the definition (2.15) of the quartic invariant and the properties of the rank-three symmetric tensor c ijk , one shows the following quintic identity for a generic charge vector Γ. This identity is equivalent to the property that the Freudenthal ternary product satisfies the four axioms defined in [30], and inversely, any Freudenthal ternary product is necessarily of the form (2.21), for a completely symmetric rank four tensor satisfying (2.20). One can therefore define the g 4 Lie algebra as in [30]. We shall not use the Freudenthal ternary product, but rather the quintic identity (2.20). We refer to [30] for the more formal definition of the g 4 Lie algebra from the ternary product itself. It is straightforward to combine (2.20) with the symmetry properties of the sextic invariant 〈 I ' (Γ 1 ) , I ' (Γ 2 ) 〉 to show that for any two vectors J 1 and J 2 , the linear transformation preserves both the symplectic product and the quartic invariant. One concludes that (2.22) defines a generator of g 4 , and all g 4 generators can in fact be defined in this way. It follows that the Lie algebra takes the form as shown in [30]. A special case arises for two rank 1 vectors, denoted R and R ∗ , which are assumed to be mutually non-commuting. In this case, one can define the corresponding g 4 generator as in (2.22) which is central in the description of T-dualities. It is clear from (2.16) that for any rank 1 vector R (or respectively R ∗ ) one has This generator admits therefore R and R ∗ as eigenvectors, with eigenvalues +3 and -3 respectively, and the remaining eigenvectors of h T can be characterised as follows. Using (2.20) and (2.16), one can show that (2.26) from which follows the action of the square of h T , as This equation implies that R and R ∗ are the unique eigenvectors with eigenvalues +3 and -3 respectively, and also leads to the characteristic equation In view of the fact that h T is symplectic, it follows from (2.27), (2.28) that the 2 n v + 2 electromagnetic charge vector space decomposes into where the two distinguished vectors R and R ∗ are by definition the components of grade 3 and -3, respectively. This decomposition is clearly relevant to the T-dualities as described in (2.29), as it allows to identify four eigenspaces, based on two very small vectors. The n v eigenvectors of eigenvalue +1 and the n v eigenvectors of eigenvalue -1 can be obtained by defining the corresponding projectors to the four eigenspaces of h T , as h T Γ ( n ) = n Γ ( n ) for n = -3 , -1 , 1 , 3, i.e. Note that the Γ ( ± 1) can simply be identified as the solutions to These expressions will be very useful in evaluating the action of T-dualities on general symplectic vectors in the following sections. A further practical advantage of this decomposition is the fact that all inner products must respect the grading, leading to strong constraints on the possible nontrivial combinations. For instance, the grading implies that since there is no vector of weight ± 5 that these cubic terms could be equal to. Similar considerations apply to scalar products, which necessarily vanish unless the sum of grades of the vectors involved vanishes. In addition to the decomposition (2.29) of the vector space, the generator h T implies a corresponding decomposition of the duality group generators. Indeed, h T commutes with g 5 ⊂ g 4 and defines the following graded decomposition of g 4 where the gl 1 corresponds to h T itself. Clearly, the 2 n v generators of eigenvalue ± 2 with respect to h T can be used as raising and lowering operators on the eigenspaces in (2.29). As the reader might already understand, these grade 2 generators are related to the transformations (2.3) by similarity transformations in G 4 . In terms of the explicit expression (2.22) for the action of g 4 , one may consider any grade -1 vector of parameters k (-1) to define the grade 2 generators as This generator is manifestly of grade 2 because of the grading of R and k (-1) themselves and the algebra (2.23). It is convenient to write it in a way that makes the grading explicit where we used the projections in (2.30) and the fact that k (-1) is of grade ( -1). All these generators clearly commute between themselves for different k (-1) 's. Similarly, one defines the grade -2 generator in terms of a grade 1 vector k (1) The normalisations we have chosen are such that while one easily computes that In this form, one easily computes that these generators are nilpotent of order 4, as consistent with the grading (2.29), which only allows for four eigenspaces. Explicitly, we find the following expressions for the two sets of generators and moreover as in (2.33). Finally, one also computes using (2.23) and the grading that for any grade -1 vector e and grade 1 vector f One can straightforwardly check that so that the latter transformation lies in the g 5 ⊂ g 4 subalgebra, consistently with the graded decomposition (2.33). One can indeed check that these transformations preserve the cubic norm I 4 ( R, Γ (-1) , Γ (-1) , Γ (-1) ) for an arbitrary grade -1 vector Γ (-1) . It turns out that the identity (2.20) implies the associated Jordan identity which generalises (2.11). These equations are clearly valid upon replacing R with R ∗ and Γ (-1) by a grade (+1) vector Γ (1) throughout. One may now use the above formulae to identify T ± with T-dualities explicitly. Indeed, one can easily check that upon identifying R with the very small vector whose only nonvanishing component q 0 and R ∗ with its magnetic dual along p 0 , the exponentiated transformations are identical to the spectral flow shown in (2.3). The corresponding set of generators T -then generate the T-dualities one obtains by conjugating (2.3) by an electric/magnetic duality and leave R ∗ invariant. The grade ( -1) and (+1) components are then easily seen to be given by the magnetic, p i , and electric components, q i , respectively. In the general case, we can identify all possible sets of T-dualities as given by a choice of R or R ∗ , as above, as the generators T ± k are entirely determined by the rank 1 vector they leave invariant. Indeed, the characteristic feature of these abelian subgroups is that there is always a unique (up to rescaling) very small vector ( e.g. R ) that they leave invariant, whereas they act transitively (up to a rescaling) on the set of very small vectors ( e.g. R ∗ ) that are not mutually commuting with the former. In the specific example of (2.3), any very small vector that is not mutually local with R (along q 0 ) can be obtained by acting with a finite transformation exp(T + k ) on R ∗ (along p 0 ). In the following sections, we will consider the action of general T-dualities, as we find it convenient to describe multi-centre black hole solutions in terms of two auxiliary very small vectors R and R ∗ that arise naturally from the equations of motion, as mentioned below (2.6). Therefore, we will always consider a T-duality as given explicitly by an exponential as in (2.48), where the explicit action of each order is given by (2.35)-(2.43) above, rather than the equivalent matrix similar to the one in (2.3) that has to be defined explicitly. This concludes our discussion of T-dualities in the real basis. In the next section, we consider the same transformations in the complex basis, for later use. The reader interested in constructing solutions can however safely skip this technical discussion.", "pages": [ 13, 14, 15, 16, 17, 18 ] }, { "title": "2.3 T-dualities in the complex basis", "content": "In this section we discuss the realisation of T-dualities in the complex basis defined by the central charge Z = Z (Γ) and its Kahler derivative Z a = Z a (Γ). The discussion here is parallel to the one of the previous section, in the real basis, and is complementary to it. However, the construction of the T-duality generators in the complex basis will be necessary to solve the first order equations describing black hole composites in the following. For this purpose, we define the very small vector R as in (2.17)-(2.18), while R ∗ is defined from R using an arbitrary phase, e iα = N [Ω], as /negationslash where the choice of the phases is done for later convenience. It will also be useful to define the complex function Y that has unit real part and the specific parametrisation of the imaginary part will become meaningful in the following. The vector (2.49) is by construction mutually nonlocal with R because and defines a natural magnetic dual to R . We will first determine the T-dualities T + that leave R invariant. We note that Ω a is by construction (2.18) invariant with respect to a subgroup K 5 ⊂ K 4 . In order to describe the action of g 4 in (2.12) on this vector, we will parametrize the remaining n v -1 generators of k 4 , which describe the coset component k 4 /circleminus k 5 , in terms of a vector Q a . Requiring the matrix to be anti-Hermitian and to preserve N [ Z ] fixes the relative coefficients and implies the constraints Similarly, we parametrize u (1) in (2.12) by γ and C n v by a complex vector P a , such that the final result we find that the action of g 4 on a general vector reads In order to describe T-dualities, we must impose that these transformations leave R invariant, which can be shown to hold if 8 One can now verify that the resulting transformations leave R invariant and commute with each other. Furthermore, one shows that T + γ,Q is nilpotent of order four, as is clear from the example (2.3) where terms at most cubic in the parameters k i appear. Indeed, T + γ,Q can be identified with the corresponding generators in (2.35), which act as raising operators on the decomposition (2.29) and the vector R is the highest weight vector, to which we assign weight 3. At this point it is important to appreciate the fact that, while we used the complex scalar dependent basis to define T-dualities, the following relations hold where T + k denote the representation of these generators in the real basis, parametrised in terms of a grade -1 vector k , as in (2.34)-(2.35). It follows that the parameters γ and Q a depend on the n v constant parameters k and the scalar fields. As alluded to above, the second very small vector (2.49) plays a role dual to that of R , as one can check that R ∗ is never a zero mode of the T-duality operator defined in (2.56), and in particular where we also used (2.37) to given the explicit relation of the vector k to the parameters γ and Q a . In addition, one can verify that (T + γ,Q ) 3 R ∗ ∝ R , as in (2.41), so that the vector R ∗ can be identified with the lowest weight vector of the operators T + γ,Q , with assigned weight -3. As expected from the example in (2.3), the parameters k i must be a rank-three vector for a general T-duality. Indeed, the vector defined by (2.58) is of rank three, as can be shown by computing its quartic invariant. Furthermore, one can verify that this vector satisfies the reality constraint These generators, along with the K 5 subgroup of K 4 and the generators described by (2.55), account for the full G 5 /multicloseleft R n v subgroup of G 4 leaving invariant a given very small vector [31]. introduced in [13] in the study of single centre solutions. This defines a Lagrangian subspace that includes the small vector R , whereas one also verifies that which implies that 〈 T + R ∗ , R ∗ 〉 = 0. Making use of the above structure based on the original very small vector R , one can proceed to define similar structures for the dual very small vector R ∗ , in exactly the same way. This seems redundant at first sight, since one can always identify the small vector invariant under the T-dualities with R , as above. However, this is more natural in view of the discussion in the real basis in the previous section, as well as for the applications we are interested in, where both vectors appear simultaneously. It is therefore useful to have a dual description in terms of R ∗ throughout. The T-dualities leaving R ∗ invariant can then be shown to be defined as in (2.54) with parameters given by where the Q a are again constrained by (2.53), so that the resulting transformations read As expected, R is never invariant under (2.62), with transformation rule where we show the relation of the vector k in (2.37) to the parameters in the complex basis. Again, one can compute that (T -γ,Q ) 3 R ∝ R ∗ , consistent with (2.43). It follows that T -γ,Q are lowering operators with R ∗ and R as their lowest and highest weight vectors respectively. Furthermore, one can define a Lagrangian subspace that includes R ∗ and T -R , through the constraint dual to (2.59), as In addition, the vector (2.63) satisfies the relations which imply that 〈 T -R,R 〉 = 0. One can obtain the action of the relevant generator h T defined in (2.24) in the complex basis, as the commutator of a T-duality leaving R invariant and a T-duality leaving R ∗ invariant. In general, such a commutator will also give rise to an element of the grade zero component g 5 , as in (2.33). Choosing the parameters γ and Q a for the two transformations to be identical, one obtains where G ( γ, Q a ) is a generator of gl 1 ⊕ g 5 bilinear in γ and Q a . The gl 1 component, which is to be identified with the operator h T , corresponds to the transformation of parameter 1 2 γ 2 + 1 3 Q a ¯ Q a , whereas the g 5 transformation is parametrised by γQ a and Q a ¯ Q b -1 n v -1 ( δ a b -1 3 ¯ Ω a Ω b ) Q c ¯ Q c . To project to the gl 1 component, one can simply identify the terms quadratic in Q a as so as to cancel all the terms in g 5 . Equivalently, this identification can be understood as an average obtained by acting on the parameter Q a with the K 5 ⊂ K 4 subgroup leaving Ω a invariant, and integrating out the result over K 5 . By definition, none of the generators of g 5 are K 5 singlets, and the resulting expression is necessarily proportional to the gl 1 generator h T . Because of the reality constraint (2.53) on the Q a , this average furthermore implies and therefore where we used (2.11) to show that c acd c bcd = n v +3 3 δ a b . In practice, (2.69) is the only constraint one needs to use when computing the commutator in (2.66). After imposing the relations above, one finds One can now check that (2.44) is indeed satisfied in the complex basis. Similarly, the vector space of charges splits into 4 subspaces of eigenvalue {-3 , -1 , 1 , 3 } with respect to this linear operator. By construction, R and R ∗ are the unique vectors of eingenvalue 3 and -3 respectively, up to an overall rescaling. The remaining 2 n v directions can then be simply identified with the parameters of the two T-dualities T ± , as given above. It follows that the eigenspace of eigenvalue -1 is spanned by the n v vectors that satisfy the constraint (2.59) and are mutually local 9 with R ∗ , as 〈 R ∗ , Γ (-1) 〉 = 0, which is equivalent to Similarly, the eigenspace of eigenvalue 1 is spanned by the n v vectors that satisfy the dual constraint (2.64) and 〈 R, Γ (1) 〉 = 0, which lead to These equations can be identified from (2.30) in the real basis as (2.31).", "pages": [ 18, 19, 20, 21, 22 ] }, { "title": "3. The c ∗ -map, nilpotent orbits and first order systems", "content": "In this section we consider the first order systems describing multi-centre non-BPS black holes in N = 2 supergravity coupled to n v vector multiplets labelled by an index i = 1 , . . . , n v . We refer to the appendix for a short overview of our conventions on N = 2 supergravity, which coincide with the ones in [13], to which we refer for further details. The systems of black holes we are interested in can be constructed systematically in the special case when the special Kahler manifold, M 4 , parametrised by the vector multiplet scalars, t i , is symmetric. Moreover, we exclusively consider stationary solutions, i.e. we always assume a timelike isometry. In this case, one can consider a timelike dimensional reduction to three dimensions and dualise all vector fields to scalars [32], to obtain an effective euclidean sigma model describing stationary black hole backgrounds. The resulting equations of motion are still rather complicated, so that it is common to consider special linear systems that solve the full equations of motion, but do not provide a full list of possible solutions. There are two such systems known, namely the composite non-BPS system [11] and the almost-BPS system [14], which together account for a representative majority of the explicitly known multi-centre solutions featuring a flat three-dimensional base space. The purpose of this section is to define these systems in terms of four-dimensional, manifestly duality covariant quantities, aiming for a clear description of their general structure. To this end, we make use of the duality symmetries of the three-dimensional theory resulting from the dimensional reduction, through the formalism developed in [33, 13]. We therefore first describe the basics of this effective theory in section 3.1, followed by a discussion of the method of nilpotent orbits in section 3.2. We then present the derivation of the first order flow equations for the composite non-BPS system and the almost-BPS system in sections 3.3 and 3.4 respectively. Note that, while the derivation of the almostBPS system has not appeared before, our section 3.3 is essentially a review of the derivation of the same system in [13], which we include for completeness.", "pages": [ 22 ] }, { "title": "3.1 The three-dimensional non-linear sigma model", "content": "In order to describe stationary asymptotically flat extremal black holes, we introduce the standard Ansatz for the metric in terms of a scale function U ( x ) and the Kaluza-Klein one-form ω ( x ) (with spatial components only), which are both required to asymptote to zero at spatial infinity. Here and henceforth, all quantities are independent of time, so that all scalars and forms are defined on the flat three-dimensional base. The n v +1 gauge fields of the theory F I = dA I for I = 0 , . . . n v include the graviphoton and the vector multiplet gauge fields. Together with their magnetic duals, these can be arranged in a symplectic vector, F , as in (A.2), which transforms linearly under electric/magnetic duality. For a background as in (3.1), the appropriate decomposition of the gauge fields takes the form and accordingly for the field strengths where we defined the gauge field scalars ζ , arising as the time component of the gauge fields, and the one-forms w describing the charges. Here, F is defined as the spatial component of the field strength, which is not closed but satisfies according to (A.15), which can be written as Note that this first order equation determines the ζ in terms of the vector fields w and the scalars. Upon dimensional reduction over the time direction, the set of moduli t i parametrising the coset space (2.7) are extended to include the scaling factor U and the scalar dual to the angular momentum ω in (3.1), as well as the fields ζ in (3.3), which altogether parametrize the para-quaternionic symmetric space 10 This defines the so-called c ∗ -map, which can be related to the standard c -map [34] by analytic continuation. The three-dimensional symmetry group Lie algebra g 3 decomposes as where the weights refer to the eigenvalues under the adjoint action of the gl 1 generator. The grade one generators in l (1) 4 are associated to the gauge invariance with respect to a constant shift of the scalars ζ , and accordingly the grade two generator corresponds to the shift of integration constant defining the scalar dual to ω . As discussed in more detail in [13], the equations of motion for the scalar fields parametrising the symmetric space M 3 are expressed in terms of the corresponding Maurer-Cartan form Here, v ∈ G 3 is a coset representative describing the scalar fields, while P is the coset component of the Maurer-Cartan form, defining the scalar momenta. Similarly, B is the sl 2 ⊕ g 4 component defining the pulled back spin connection. In components, the scalar momenta are defined as where we introduce some shorthand notations, based on the central charges in (A.11), that will be used for the remainder of the section. At this stage it is important to introduce some properties of the g 3 algebra. The components of an element of the Lie algebra g 3 in the coset component g 3 /circleminus ( sl 2 ⊕ g 4 ), as given in (3.9) above, correspond to U (1) × U (1) × K 4 irreducible representations as the two complex parameters w and Z and the two complex vectors ¯ Z a , Σ a transform in the C n v representation of K 4 , according to the decomposition (2.12), the same as the scalar field momenta e a i dt i in four dimensions. As one can check explicitly, the quadratic trace invariant which defines the SL (2) × G 4 invariant norm is equivalent to the effective Lagrangian in the background (3.1). The sl 2 algebra is realised on these components as where ρ and λ are a real and a complex parameter respectively, parametrising the sl 2 group. Similarly, the action of g 4 can be written as using the parameters defined below (2.12), as Note that the g 4 action defined by these equations corresponds to the divisor group in (3.6), rather than the original g 4 as given in four dimensions in (2.13), the two being related by a conjugation in g 3 . Finally, we give the components of the sl 2 ⊕ g 4 component of the Maurer-Cartan form, B , along sl 2 and along g 4 where G a b ( B ) defines the k 4 valued traceless 11 component of the pulled back spin connection on M 4 and Q is the pulled back Kahler connection (A.8).", "pages": [ 22, 23, 24 ] }, { "title": "3.2 Nilpotent orbits", "content": "The basic observation for constructing black hole solutions in four dimensions using the three-dimensional Euclidean theory describing stationary solutions is that regular stationary solutions of N = 2 supergravity with a flat three-dimensional base metric can be described by a three-dimensional momentum P that is nilpotent as a Lie algebra element. This implies in particular that P can be written in terms of the basis element e α of a nilpotent subalgebra of g 3 . Such a subalgebra is always associated to a semi-simple element 12 h of sl 2 ⊕ g 4 such that where n defines the maximal possible eigenvalue of ad h in g 3 . This implies for instance the equation which defines a first order constraint on the components of P . In order for (3.17) to be consistent with the equations of motion and the Bianchi identity, the covariant derivative of the generator h must satisfy These equations are satisfied if d B h also lies in the nilpotent algebra defined by h , or equivalently if one imposes that Note that this is not necessarily the most general solution for the generator h , as it might be possible to construct special solutions, for which e.g. there are preferred spacetime directions described by a nonzero derivative. Considering this type of solutions can be understood as the natural generalisation of the BPS black hole solutions in [7, 6] to include supersymmetric string solutions. These are not accounted for by the standard black hole ansatz, but are are allowed by the requirement of preserved supersymmetry. Similarly, we will only consider the generic situation in (3.19), which should be satisfied for all composite black hole solutions with a flat three-dimensional base. Without loss of generality, one can always choose the generators h ∈ sl 2 ⊕ g 4 such that only its components λ and Λ a do not vanish, as these are the only generators with a positive Cartan norm 13 . Equation (3.19) can then be viewed as first order equations for these auxiliary components, which can be solved to determine their evolution in space in terms of the physical fields. With this information at hand, equation (3.17) then defines first order equations for the physical fields, which contain the auxiliary components λ and Λ a and determine w and Σ a of (3.9) in terms of e U Z ( /starF ) and e U Z a ( /starF ), plus some possible constraints on the latter if the dimension of the coset component of the nilpotent algebra defined by h is strictly less than 2 n v +2. Given the definitions in the previous section, it is possible to make (3.17)-(3.19) explicit, in order to determine the auxiliary components and the first order flow directly. A simple example is given by the BPS system, which is characterised by an element of h ∗ ∈ sl 2 , i.e. Λ a ( h ∗ ) = 0. It follows that its only nonvanishing component is λ ( h ∗ ) = e iα , where this phase is identified with the phase α that defines the covariantly constant spinors as in [6]. The action of this generator follows from (3.11) as while the relevant eigenvalue equation (3.17) is simply The last relation clearly imposes a linear relation between the derivatives of the fourdimensional scalars and the gauge fields upon using (3.20). The phase e iα is determined by (3.19), which can be shown to reduce to where in the second equality we used the first of (3.20) and (3.21). These equations are easily seen to be equivalent to the BPS system of [7, 6] and we refer to [33] for a more detailed analysis of the eigenvalue equation (3.16) for this generator, leading to the system of equations describing multi-centre BPS black holes. Similarly, non-BPS solutions with vanishing central charge at the horizons are described in a similar fashion, with λ = 0 and a normalised rank one Λ a ( i.e. c abc Λ b Λ c = 0 and Λ a ¯ Λ a = 1) [33]. Other examples of such constructions using nilpotent orbits have been used to obtain the systems describing respectively single centre non-BPS black holes with a non-vanishing central charge at the horizon and the composite non-BPS multi-centre system [13]. In the next two sections we discuss in some detail the construction of the two multi-centre non-BPS systems, the composite non-BPS and the almost-BPS, which form the basis of this paper.", "pages": [ 25, 26 ] }, { "title": "3.3 Composite non-BPS flows", "content": "The composite non-BPS system [11, 12, 13], describes configurations of interacting nonBPS centres and corresponds to an eigenvalue equation as in (3.16), where the relevant element h C belongs to g 4 . As the scalar momentum P must be of positive grade with respect to h C (given that p α > 0), we consider the relevant decomposition of the coset component, parametrising 2 ⊗ R 2 n v +2 through (3.9), in terms of grades with respect to an element h C ∈ g 4 . The relevant graded decomposition is in this case for g 4 itself and for the coset component, i.e. P ∈ ( 2 ⊗ R n v ) (1) ⊕ 2 (3) . We can choose h C to be Hermitian ( i.e. to lie in g 4 /circleminus ( u (1) ⊕ k 4 )), so that it is realised for where Ω a describes a very small vector, as in (2.18). Equivalently, Ω a is in the U (1) × K 4 orbit of the Jordan algebra identity. More explicitly, one finds the following action on the coset component which is to be identified with the elements of the coset component, through (3.16) for p α = 1 and p α = 3, according to (3.24). Considering a general linear combination of the grade one and three solutions as in [13], one can express w and Σ a in terms of Z and Z a as which are the explicit first order relations for the scalar momenta, upon using the definitions in (3.9). These flow equations contain the auxiliary components N [Ω] and Ω a of h C , which satisfy (2.18) and define a very small vector R of unit mass through The flow equations for these auxiliary fields are given by (3.19), which in this system reduces to in view the fact that d B h C lies in g 4 and is inert under g 5 by definition, and the decomposition in (3.23). Using the explicit form of B in (3.13)-(3.15) and the first order flow (3.27), one computes the components of d B h as where we explicitly separated the terms depending on the derivative of the vector R . It is now straightforward (though cumbersome) to evaluate (3.29), imposing the above relations and which can be shown using (2.11). The result one finds is that (3.29) is identically satisfied for (3.30), if and only if the combination of the first two terms in Λ a ( d B h ) vanishes. This condition implies [13] that there exists a constant symplectic vector ˆ R , since where the second equation follows from the first by use of standard special geometry identities. It follows that the generator h C is in this case determined by a constant very small projective vector ˆ R and the scalar fields are such that One can now return to (3.27), which becomes a first order flow equation for the scalars dU , /stardω and dt i in terms of the gauge fields and the constant vector ˆ R . Further details and the characterisation of solutions to these equations are given in section 4. A special case of this system, namely the single centre subclass, was discussed in detail in [13]. Indeed, one expects a system describing multi-centre black hole solutions to contain a consistent subsystem describing single centre black holes. In the case at hand, one can restrict the above equations (3.27) to this subsystem by imposing that the scalar momentum is also of positive grade with respect to the generator 1 2 ( h C + h ∗ ) where h C and h ∗ are the generators that define the composite system through (3.26) and the BPS system through (3.20) respectively. The result one obtains is that the single centre nonBPS momenta satisfy (3.27), for Z and Z a constrained to satisfy exactly the same phase dependent projection (2.59) above, which represents a constraint on the physical degrees of freedom that is necessary to reduce to single centre solutions. One can interpret this constraint in terms of the decomposition in (2.29), upon identifying the very small vector ˆ R in (3.33) with the vector R in section 2.2. It follows that (3.34) implies that only the grade ( -1) and (+3) components of the gauge fields (and thus charges) are allowed. We refrain from discussing the single centre system, as we will deal with the multi-centre case in what follows. However, we will sometimes make use of structures present already in the single centre case, referring to [13] for an in depth discussion.", "pages": [ 26, 27, 28 ] }, { "title": "3.4 Almost-BPS flows", "content": "We now turn to the derivation of the first order flow equations for the almost-BPS system [14] of multi-centre black hole solutions. Our treatment is purely in terms of four/three dimensional quantities, but we still refer to this system as almost-BPS, as we will find that it contains the original almost-BPS equations, derived from the BPS conditions of five dimensional supergravity. However, the five dimensional system is only defined in a fixed frame and does not allow for generic charges at the centres, while we will present a manifestly duality covariant form of these equations, as we did for the composite non-BPS system in the previous section. In this sense, the almost-BPS system below is larger, as it contains the one in five dimensions, as well as all the charge configurations that can be obtained from it by dualities. Our covariant system also represents the general form of the almost-BPS system of [11], where a frame was chosen to match with the five dimensional results. Closely related first order flow equations have recently been derived using similar methods in [21] for the STU model. 14 In this case Ω a reduces to three phases, which altogether with α define equivalently the first order equations. Nevertheless, the authors in [21] use the second order equations of motion to determine the evolution of these phases, whereas we consider the first order equations (3.19), which imply the second order ones by construction. For the almost-BPS system, the eigenvalue equation (3.16) is defined by the generator h A = h C +2 h ∗ , where we use the same generators as defined in (3.26) and (3.20) above. The corresponding graded decomposition of sl 2 ⊕ g 4 is then given by which is exactly the same as the one in (3.23) with respect to the g 4 components. The additional notation R n v and R n v refers to two conjugate representations of the real group g 5 . Similarly, the coset component is decomposed as so that P ∈ ( 1 ⊕ R ¯ n v ) (1) ⊕ ( R n v ) (3) ⊕ 1 (5) , i.e. it lies in the positive grade component. Due to the presence of the generator h ∗ , which by itself defines the BPS system, 15 the first order system reduces to the BPS system (3.21) for most of the components of the field strength F , except for the ones that define the solution for which the expressions of w and Σ a take the opposite sign, where u is a real function. The projection to this component is obtained as for which one easily computes that P 2 = P . One can then verify that the first order system obtained by changing the sign of the components above, given by solves (3.16), as expected. Using the definitions in (3.9), these equations become a first order flow for the four dimensional scalars in terms of the gauge field strengths. However, the auxiliary fields e -iα and Ω a parametrising h A are still arbitrary, we therefore need to impose (3.19) to obtain their dependence on the physical fields. First, the condition on d B h ∗ to be of positive grade is the same as the one for the BPS system in (3.22) and leads to the condition, [33] where the last term is now given in terms of scalars by (3.39) and is therefore more complicated than in (3.22). Analysing the components of (3.19) along g 4 to obtain the remaining auxiliary components, Ω a , one finds that it reduces again to (3.29) in exactly the same way as for the corresponding quantities in the composite non-BPS system. In fact, one can easily check that γ ( d B h A ) and G a b ( d B h A ) are exactly the same as in (3.30) since they only depend on the algebra (2.13). In contrast, manipulating the expression for Λ a ( d B h A ) is sensitive to the particular form for the scalar momenta in terms of the gauge fields. In order to write the resulting equation in a suggestive way, it is useful to draw intuition from the analogous vector in the single centre class 16 , as well as from known multi-centre solutions, which indicate that the magnetic dual of the vector described by Ω a is a more convenient variable. We therefore consider the dual very small vector of mass one, R ∗ , defined as in (2.49), i.e. we impose With this definition, one can use (3.39) to show that As in (3.30), the second bracket in the last expression satisfies (3.29) identically, whereas the first term vanishes upon imposing that R ∗ is related to a constant vector ˆ R ∗ by similar to (3.32) for the composite system. In contrast to (3.30), there are two terms containing the derivative of R ∗ in (3.42), so that one could expect more general solutions than the one shown above. However, one can verify that any solution for R ∗ other than (3.43) is such that it is mutually nonlocal with its derivative dR ∗ , which is not allowed for an everywhere very small vector. This concludes our presentation of the derivation of the first order flow equations for the almost-BPS system. One may now consider solutions to (3.39), which is a first order flow equation for the scalars dU , /stardω and dt i in terms of the gauge fields and the constant vector ˆ R ∗ , upon using (3.41) and (3.43). In section 5 we discuss the real form of these equations, show that they correspond to a linear system and give the characterisation of their solutions in terms of local functions.", "pages": [ 28, 29, 30, 31 ] }, { "title": "4. Composite non-BPS system", "content": "In this section, we present in detail the steps required to characterise solutions of the flow equations for the composite non-BPS system in terms of local functions. The starting point is the solution of the nilpotency condition (3.27), written explicitly as a first order flow system for the four dimensional scalars and the metric degrees of freedom where F is the spatial component of the field strengths defined in (3.3). The vector Ω a is related to the constant very small vector, ˆ R , through (3.28) and (3.32) above. For later reference, we give the inverse relations for the field strengths In order to solve this system, we first construct the electromagnetic potentials and use the resulting structure to simplify the equations in section 4.1. In section 4.2 we exhibit the relevance of the T-dualities introduced in section 2. We then proceed to rewrite these equations as a linear system of differential equations and discuss its integration in terms of local functions in sections 4.3 and 4.4 respectively. The reader interested in applications can find a summary of the final form of the system in section 4.5.", "pages": [ 31 ] }, { "title": "4.1 The electromagnetic potentials", "content": "In order to solve the system (4.1), we construct the gauge field momenta (3.5) using the derivative of R given by (3.32), to obtain where we made extensive use of the special geometry identities in section A. The first term is manifestly a total derivative, whereas the others are along the very small vector R and must therefore combine into the derivative of a single function. This requirement, along with (3.32), leads to the condition where M is an arbitrary function, so that the gauge field momenta take the form with the corresponding central charges given by for later reference. The structure of (4.5) can be used to show that one vector is always trivial, simplifying the system. To see that, we compute whereas taking the imaginary part of (4.1) one gets Finally, using (3.3), one finds that which implies that one vector field is always absent. In terms of the graded decomposition in section 2, the vanishing component is along the very small vector dual to R , given in (2.49), as will be shown shortly. One can now combine (4.6) and (4.9) to disentangle the term proportional to /stardω in the definition of the scalar flow equation, so that (4.1) becomes Applying the same procedure on (4.2), one obtains the inverse relations of (4.10) for the central charges Z ( dw ) and Z a ( dw ). The charge vectors dw can then be straightforwardly constructed with the result where we used the shorthands Note that the above results are in direct correspondence with the ones in [13], where the single centre system was treated.", "pages": [ 31, 32, 33 ] }, { "title": "4.2 Connection to T-dualities", "content": "Given the structure above, it is useful to define a second distinguished very small vector which is mutually nonlocal with ˆ R , as in (2.49)-(2.50) through Here, we identified the function Y in (4.13) with the one in (2.50) and we included an overall rescaling. One can check that 〈 ˆ R, ˆ R ∗ 〉 = 4 and that the central charges of ˆ R ∗ above satisfy (2.18). This new vector is defined in exactly the same way as the second constant vector used in [13] in the single centre case, but is not constant in the full composite non-BPS system. As explained in (2.5)=(2.6), a non-constant very small vector with a constant non-zero symplectic product with ˆ R must be of the form where R ∗ 0 is a constant very small vector satisfying 〈 ˆ R,R ∗ 0 〉 = 4 and exp[T + ] is a T-duality matrix leaving ˆ R invariant. It then follows that the derivative of ˆ R ∗ takes the form which is identically closed by the fact that T-dualities are abelian. The above are consistent with known solutions [11, 12], in which ˆ R ∗ takes the form of a T-duality whose parameters are harmonic functions, acting on a constant vector along p 0 . Within the composite non-BPS system, one can explicitly compute the components of the derivative of ˆ R ∗ using (4.4) and (4.10), as where we indicated that the components in this basis are given by the variation of ˆ R ∗ itself under the T-duality transformations (2.56), as shown in (2.58). In order to obtain this result, one has to identify ˆ R as the grade (+3) very small vector, that is invariant under all T-dualities T + , which we assume henceforth. The explicit values for the parameters are given by where we used the combination for brevity. Note that in the single center class of [13] these expressions were shown to vanish, consistent with the fact that ˆ R ∗ is constant if the T-dualities are rigid. Given the above, we can directly apply all considerations of section 2, since the presence of the two constant very small vectors ˆ R and R ∗ 0 implies that the grading shown in (2.29) is relevant for the integration of the system. It follows that R ∗ 0 is identified as the grade ( -3) very small vector. As discussed in (2.58)-(2.59), a generic T-duality is parametrised by a rank three grade (+1) vector, which we denote by K , so that (4.15)-(4.16) become The rank of K can be verified by checking that the quartic invariant of the vector d ˆ R ∗ in (4.17) is vanishing, i.e. In what follows, we will generally drop the subscript K on T + K for simplicity, since this is the only T-duality appearing throughout. Finally, it is worth commenting on the difference between the dual very small vector in (2.49), which was used to define ˆ R ∗ in (4.14), and the constant vector R ∗ 0 of grade ( -3), that might seem confusing. It is important to realise that (2.49) simply defines a possible dual vector, which is not unique. Since two very small vectors not commuting with ˆ R are related by exactly a finite T-duality leaving ˆ R invariant, one may choose any other vector in that orbit. We will fix this ambiguity by defining the function K to vanish in the asymptotic region, or equivalently R ∗ 0 = ˆ R ∗ | r →∞ .", "pages": [ 33, 34 ] }, { "title": "4.3 The linear system", "content": "One can now use the vector ˆ R ∗ of (4.14) in a way similar to the vector R was used in section 4.1, to project the flow equations and simplify the system. Indeed, taking the inner product with dw one finds where we defined the function V . Note that V is not a harmonic function, since ˆ R ∗ is not a constant vector. Combining this with (3.32), (4.4) and (4.10), we can determine the combination Ω i dt i in terms of V , M and the metric components as We now use all the above information to write the expression (4.11) for dw in a suggestive form. To this end, we use the expression for the derivative of ˆ R ∗ in (4.17), combined with inspiration drawn from the analogous computation performed in [13] for the single centre class. After a long but straightforward computation we obtain Here, we used (2.50) to define a new phase as which will be useful in what follows. One can compute the derivative of e -iα using (3.33) to first obtain from which follows the relation We emphasise that these equations are completely analogous to the ones relevant for the single centre system, but now involve generically non-harmonic functions M and V . Additionally, (4.25) contains the parameters for the particular T-duality appearing in the derivative of ˆ R ∗ in (4.18). These can be rewritten by observing that the action of a T-duality on the symplectic section takes the form Using this relation and (4.28) in (4.25) leads to where we also used the fact that ˆ R is by definition inert under the T-dualities in question. This is the final form of the flow equations in the real basis, where the local T-dualities have parameters given by (4.18) above.", "pages": [ 34, 35 ] }, { "title": "4.4 Integration and local structure", "content": "Due to the presence of the flat connection for the T-dualities, it is not possible to solve the system of equations (4.30) in terms of harmonic functions only. However, the scalar and vector fields can be written as where the vector of functions H is the solution to the non-harmonic equation In order to disentangle the derivatives on H and T + and cast this as a Poisson equation, we introduce the rescaled vector of functions in terms of which we find while (4.33) takes the form It is possible to give a systematic characterisation of the solution for H 0 and exp[T + ] using the following crucial observation. From the expression (4.36) for the vector fields, we compute for the derivative of H 0 that where N a is the combination of scalar momenta defined in (4.19). One can easily show that the central charges in (4.38) satisfy the reality constraint (2.59), i.e. we have This constraint was found to be crucial to describe single centre solutions in [13], where it was analysed in some detail. Using (2.30), one computes that it is equivalent to the constraints These equations are manifestly duality covariant, and in particular T-duality covariant. Therefore, using (4.15) one finds that the vector d H 0 satisfies the constraints In a similar fashion, one can show that the vector of functions H 0 itself satisfies the same constraint, using (4.31), and therefore only half of its components are allowed. Indeed, a vector satisfying this constraint does not contain components of grade (+1) and ( -3), so that in terms of (2.29), one finds and therefore describes only n v + 1 functions instead of the 2 ( n v + 1) one would have a priori, i.e. it lies on a Lagrangian subspace. One of these functions is clearly the function V in (4.23), describing the grade (+3) component of H 0 along the direction of R . The remaining n v functions span the grade ( -1) vector space, according to the decomposition imposed by the T-duality T + , and are undetermined for the moment. Using this decomposition in (4.37), we find the following grade assignments for each term where we used the fact that T + is of grade (+2). Now, since the second term in (4.43) lies in a subspace orthogonal to both the other terms, it is clear that this equation decomposes in two independent equations, as In deriving the second equation we used the fact that applying a T-duality on a generic vector as in (4.42) results in a vector of grade (+1), which cannot vanish for a physical solution 17 unless the T-duality matrix is trivial. We now analyse each of the two equations in turn. First, it is easy to express (4.45) in terms of the vector of T-duality parameters in (4.20), as so that the parameters K are identified as a rank three vector of harmonic functions. Note that K is a priori a generic vector of grade ( -1), i.e. it lies in the same Lagrangian subspace as the vector H 0 above, with the additional restriction of a vanishing component along R , since 〈 ˆ R ∗ , d ˆ R ∗ 〉 = 0 by definition. However, we should note that the constant part of K is not physical and can be absorbed into R ∗ 0 by imposing the boundary condition that K vanishes asymptotically. This choice is useful in the discussion of explicit solutions. We now turn to the Poisson equation (4.44) for the vector H 0 . As shown in (4.43), the source term of is along the unique grade (+3) component, that is along the vector R . We have therefore identified H 0 as a vector lying in a Lagrangian submanifold containing R , whose components along the grade ( -1) directions are n v harmonic functions and the component along the direction of R is a single non-harmonic function, V . One can directly compute the source term by varying the combination in (4.29) and using (4.35), to find which is explicitly proportional to the vector R , up to a real function. We can write this result in a simpler form by taking the inner product of (4.36) with ˆ R ∗ and comparing with (4.23), to obtain where we used (4.15) and (4.34). The Poisson equation for this function can now be found by projecting (4.44) along R ∗ 0 , as Note that, by (4.47), the right hand side of the first equality is nonvanishing, since it is the inner product of R with its magnetic dual. The second equality is a direct consequence of (2.40). We moreover record the following relations for the expressions involving T-duality matrices in (4.29) and (4.47), which are direct consequences of (2.35) and (2.40), in combination with (4.35). These equations, along with (4.22), allow us to evaluate the action of T-duality generators on the various objects relevant to the system. As the action of a finite T-duality is expressed as a finite sum due to the nilpotency of T + , it follows that one may also compute the action of finite T-dualities, as alluded to above. This is especially important when trying to find explicit solutions to this system [26]. The final quantities to be fixed are the angular momentum vector, ω , and the function, M , appearing in the expression for the scalars. In the single centre class, these are both harmonic and are dual to each other. Indeed, in that case the T-duality parameters in (4.18) must vanish and the first of these equations imposes exactly that M and ω are harmonic. In the more general multi-centre case, one has to compute the nontrivial γ + to obtain the analogous equation. This can be done straightforwardly using (4.18), (4.34)-(4.36) and (4.50) to show that One can alternatively obtain the same expression by manipulating the definition of the non-harmonic function M in (4.4) using the above results. The integrability condition on (4.51) leads to the following Poisson equation for M : which can be solved once the grade -1 component of H 0 and K are chosen. This concludes our duality covariant presentation of the composite non-BPS system in terms of the real basis. In the next section, we summarise the final form of the equations to be solved and we comment on some properties of the solutions.", "pages": [ 35, 36, 37, 38, 39 ] }, { "title": "4.5 Summary of results", "content": "In this short section, we summarise all relevant formulae for the composite non-BPS system in the real basis. All relations presented here were shown explicitly in the previous sections and we refer to the discussion there for the details. We find it however useful for future applications to give a self-contained account of the final form of the system. The ansatze for the metric and gauge fields are given in (3.1) and (3.3) in terms of the function e U , the one-form ω and the spatial vector fields dw , while the electromagnetic potentials are fixed by (4.5). The first order flow equation for the composite non-BPS system is given by (4.30), as Here, M , V , are functions to be specified below, while ˆ R and ˆ R ∗ are a constant and a non-constant very small vector respectively, where 〈 ˆ R, ˆ R ∗ 〉 = 4. The non-constant ˆ R ∗ is related to a constant very small vector, R ∗ 0 , by (4.15) which also satisfies 〈 ˆ R,R ∗ 0 〉 = 4. In this and all equations in this section, T + K is a generator of the T-dualities leaving ˆ R invariant, parametrised by a vector of harmonic functions, K . As discussed in section 2, the vector of parameters K lies in the grade ( -1) component of the vector space according to the decomposition implied by the T-duality. It is therefore a three-charge vector satisfying which indeed specifies a vector of n v degrees of freedom. The solutions to the flow equation (4.53) are simplified by introducing a vector, H 0 , of grade ( -1) ⊕ (+3), i.e. satisfying Note that (4.55) is trivially a solution of the first equation, found by setting the grade (+3) component, 〈 R ∗ 0 , H 0 〉 , to vanish. The equations resulting from (4.53) take the form (4.36), where V is now identified with the grade (+3) component of H 0 , as V = 〈H 0 , R ∗ 0 〉 . The compatibility relation for the last relations leads to the field equation for H 0 , as in (4.44) and (4.50) As the right hand side of this relation is only along ˆ R , it follows that all grade ( -1) components of H 0 are harmonic, whereas V is not, leading to (4.49), as by taking the inner product of (4.59) with R ∗ 0 . The final dynamical equation required is the one for the function M in (4.58) and the angular momentum vector ω , both of which are conveniently given by (4.51), as Taking the divergence of this equation, one obtains a Poisson equation for M . These equations can be seen to be equivalent to the known formulation of the composite non-BPS system, as given in a fixed duality frame in [11, 12], by making a choice for the constant vectors ˆ R and R ∗ 0 . In fact, these two papers use different frames for describing the system, both of which can be reached from our general formulation. The frame of [12] is found by choosing where we disregard the (arbitrary) normalisation. Similarly the frame in [11] is found by interchanging the expressions for the two vectors in (4.62), corresponding to an electric/magnetic duality. We close with some comments on the structure of the solutions. First, the physical scalars and the metric scale factor can be obtained by solving (4.57) in the standard way [8]. Since all quantities above are appropriate combinations of the single centre solution in [13], up to overall T-dualities, it is possible to use many of the results given there. For instance, the metric scale factor is given by where we used the fact that the quartic invariant is by definition invariant under all Tdualities. This expression is identical to the corresponding one for the single centre class, despite the fact that the functions M and V are not harmonic in the present context. As in [13], one can simplify the expression for e U as follows. The decomposition of the vector H 0 -1 2 V ˆ R in grades ( -1) ⊕ (+3) implies that I 4 ( H 0 -1 2 V ˆ R ) is linear in V , as any other power of the grade (+3) would vanish identically. In particular we find that so that we can show the equality I 4 ( H 0 -1 2 V ˆ R ) = -I 4 ( H 0 ). It then follows that This equation implies that the n v grade ( -1) harmonic components in H 0 must correspond to a rank three charge ( i.e. a large electric charge in five dimensions), so that (4.65) leads to a non-degenerate metric. Based on (4.65), we conclude that the system above indeed describes the interactions of black holes that are non-supersymmetric in isolation, since I 4 ( H 0 ) must be negative globally and in particular at each centre, for a regular geometry. We refrain from giving an explicit expression for the physical scalars, as these would involve the action of an arbitrary abelian isometry with parameters K on the physical scalars for the single centre class as given explicitly in [13]. Of course, the scalars can be computed from the standard formulae in [8] for any desired solution. The characteristic features of all solutions in this class is that each one of the centres must be of the non-BPS type in isolation, as explained above, and that the charges at all centres must commute with the vector ˆ R and must not commute with the vector R ∗ 0 , by regularity. It is then clear that such solutions do not exist for generic non-BPS charges at the centres. In addition, once an allowed charge configuration is fixed, one cannot have arbitrary values for the moduli at infinity. This is due to the constraint on H 0 , which contains only n v +1 asymptotic constants and the fact that some of these constants, together with the 2 n v parameters in ˆ R and R ∗ 0 , turn out to parametrise charges that are not described by the poles of H 0 , through (4.53) and (4.58). A simple example of this situation is given by the class of two centre solutions for models with n v ≥ 3. In this case, the two non-BPS charges are in fact arbitrary, since one can always find a choice of ˆ R and R ∗ 0 for any pair of non-BPS charges. However, it is possible to show [26] that the asymptotic moduli are constrained to lie on a ( n v +2)-dimensional hypersurface of the 2 n v -dimensional moduli space. Note that this is very different from the multi-centre BPS solutions, where solutions exist a priori everywhere in moduli space and walls of marginal stability arise only when the constrains implied by global regularity are imposed. In the present case however, local solutions seem to exist only in certain hypersurfaces of moduli space, while walls of marginal stability might still arise on these constrained surfaces. Finally, it is worthwhile commenting on the evaluation of T-dualities appearing as matrices in the equations above. As we have shown in several examples, one can avoid introducing explicit matrices, instead computing the action of T-dualities on the relevant vectors by use of the definition (4.46) and equations (4.50). Indeed, we find that it is possible to reduce all required computations to a recursive application of these three relations. As the grading involves only four subspaces, this procedure terminates after at most three steps.", "pages": [ 39, 40, 41 ] }, { "title": "5. Almost-BPS system", "content": "In this section we present in detail the characterisation of solutions to the almost-BPS system of equations, in analogy with the steps taken in the previous section for the composite non-BPS system. While the discussion here is self-contained, we will occasionally refer to section 4, in order to highlight similarities and recycle some results. The starting point is the solution to the nilpotency condition given in (3.39), which we repeat explicitly here where Z = Z ( /starF ), Z a = Z a ( /starF ) are the central charges of the spatial field strengths F in (3.3). The vector Ω a is connected to a constant very small vector, ˆ R ∗ , through (3.41)-(3.43). For later reference, we also give the inverse relations, which read where we used as shorthand the one-form The final required equation is the compatibility equation (3.40), which can be rearranged using (5.2) and (5.3) to obtain Here, we used the function Y defined in the first equality of (2.50) and the fact that it has unit real part. Similarly, one can show using the flow equations above, that µ is also given by the expressions that will be used in due time. In order to solve these equations, we follow a path similar to the last section, by considering the electromagnetic potentials in section 5.1 and using them to simplify the equations. The connection with T-dualities is shown in section 5.2, while sections 5.3 and 5.4 are devoted to the linear system of equations governing this system and its integration in terms of local functions respectively. The reader interested in applications can find a summary of the final form of the system in section 4.5.", "pages": [ 41, 42 ] }, { "title": "5.1 The electromagnetic potentials", "content": "As a first step towards the solution of the system, we decompose the field strength F in the electromagnetic potentials and the vector fields that define the conserved charges. The electromagnetic potentials for this system are computed by their definition (3.5) as where we used (5.4) and the definition (3.41) to rearrange terms. The first term in (5.6) is already a total derivative, so that the last term must combine into the derivative a vector, which is necessarily proportional to R ∗ . Using the relation of R ∗ to a constant vector in (3.43), this requirement leads to the condition where we introduced the a priori arbitrary real function W . The result for the electromagnetic potentials takes the form while the corresponding central charges are We can now construct the vector potentials by the definition which leads to the expression where we use the shorthand in (5.3) and Here, it is worth pointing out that, unlike in the composite non-BPS system (cf. (4.9)), there is no component of the vector fields that is vanishing a priori. Nevertheless, the projection of the vector fields in (5.11) along the available constant vector ˆ R ∗ is still relevant and can be computed as where Y is again as in (2.50). This can be simplified by imposing consistency of the two expressions for µ in (5.5) with (5.7), combined with (3.43), leading to Now, it is simple to show that (5.13) and (5.14) imply that the projection 〈 ˆ R ∗ , dw 〉 is given by a harmonic function, V , defined as while the function W is fixed as where β is an arbitrary constant. Using these results, (5.7) simplifies to which will be used in due time.", "pages": [ 42, 43, 44 ] }, { "title": "5.2 Connection to T-dualities", "content": "We now discuss the relevance of T-dualities for the almost-BPS system, which will be important for the integration of the flow equations, as for the composite non-BPS system in the previous section. In order to exhibit this, we consider the very small vector which is always mutually nonlocal with the constant vector ˆ R ∗ and is not constant in general. As is clear by their definitions these two vectors are related in exactly the same way as the pair of very small vectors, R and R ∗ in (2.17) and (2.49) respectively, up to rescalings. Note that the situation is opposite to the one for the composite non-BPS system, where R is a constant vector up to rescaling, while ˆ R ∗ is not constant. Based on the discussion in section 2, we associate R to the grade (+3) component of the decomposition (2.29), while R ∗ is identified as the corresponding grade ( -3) component. Moreover, one can check that the normalised very small vector has a constant inner product with ˆ R ∗ , namely 〈 ˆ R, ˆ R ∗ 〉 = 4. As explained in (2.5)-(2.6), the condition that ˆ R is a very small vector can be generally written as where R 0 is a constant very small vector and exp[T -] is a T-duality matrix leaving ˆ R ∗ invariant, parametrised by a grade (+1) vector of functions. It follows that the derivative of ˆ R can be expressed as which is closed by the property that T-dualities are abelian, exactly as in the composite non-BPS case. This is consistent with the known formulation of the almost-BPS system in five dimensions [14, 15, 16], which can be written in terms of a T-duality parametrised by harmonic functions, acting on the scalar and vector fields. We can find the relevant T-duality parameters in (5.20) by explicitly computing the derivative of ˆ R , using the flow equations (5.1) for the almost-BPS system. After a lengthy but straightforward computation, we obtain that the derivative of (5.19) is indeed given by where we indicated that the result is given by the variation of ˆ R under the T-duality transformation in (2.62), as shown in (2.63). The values for the one-form generators are given by The expression for γ -can be rewritten using (5.5) in a form similar to the corresponding T-duality parameter for the composite non-BPS system in (4.18), as where we used the definition (2.50) for the function M in terms of the phases e iα and N [Ω]. In this form, it is manifest that setting the T-duality parameters γ -, Q a -to zero, one finds that the angular momentum is given in terms of a harmonic function, M , while the second of (5.23) becomes a reality constraint on the scalar flow, similar to (2.59) in the composite non-BPS system. This restriction therefore leads to the single centre subclass, which is common to both the composite non-BPS and almost-BPS systems. Applying the considerations of section 2 on the system at hand, we recall that a generic T-duality leaving ˆ R ∗ invariant is parametrised by a rank three grade ( -1) vector of parameters, which we denote by K , so that (5.20)-(5.21) become In what follows, we will generally suppress the explicit subscript K from the T-duality generators for simplicity. One checks that K is indeed a rank three vector, i.e. These equations are clearly dual to the corresponding equations (4.22) for the composite non-BPS system. Alternatively, the same property follows from the fact that d T -R 0 is of grade (+1) and the rank of such a vector is at most three. Indeed, one can directly verify that the vector d K , as defined in (5.25), satisfies the constraint (2.64) and (2.65), which are equivalent to the real constraint These equations are explicit realisations of the general situation discussed in section 2. Finally, we stress the difference between the constant very small vector R 0 of grade (+3) and the original vector in (3.28), which was also used to define ˆ R in (5.19). This can seem confusing, especially in view of the fact that we used the constant vector R 0 to define the constraint in (5.28) and ultimately the grading of the vector space. However, as already explained in the analogous situation for the composite non-BPS system, below (4.22), the grading associated to the T-dualities leaving invariant ˆ R ∗ is only defined up to the action of the T-dualities themselves. In this respect, one can chose any constant vector R 0 in this orbit to define the grading. Given this redundancy in the definition, it will be convenient to fix T -to vanish in the asymptotic region, such that R 0 = ˆ R | r →∞ .", "pages": [ 44, 45, 46 ] }, { "title": "5.3 The linear system", "content": "We are now in a position to use the above results to write the system of flow equations in the real basis, in terms of the symplectic section, V , the two very small constant vectors ˆ R ∗ , R 0 , and the relevant T-duality generators. To show this, we insert (5.16) and (5.17) in (5.11) to eliminate the spurious quantities W and µ in favor of V and dV respectively. Moreover, it is useful to note the relation as well as the identity which follows from (5.21), and we remind the reader that the explicit expression for d ˆ R is given by (5.22) with parameters as in (5.23). One can then verify that the expression is equivalent to (5.11) above. This result is manifestly duality covariant in the real basis, as it is written in terms of real symplectic vectors only. In particular, note that (5.31) is completely analogous to the corresponding result (4.30) for the composite non-BPS system, up to the term explicitly proportional to the angular momentum. However, it is simple to shown that this term is unphysical, after considering the electromagnetic potentials as well. These are computed by combining the result (5.8) with (5.16) for the function W , to find which leads to the following expression for the total spatial field strengths Given that the constant β does not appear in the gauge invariant total field strengths, we conclude it corresponds to a residual gauge transformation of the type A → A + β ˆ R ∗ dt and therefore is unphysical. Henceforth we set β = 0 in all relations, for simplicity. With this choice, (5.31) is formally exactly the same as its counterpart in the composite non-BPS system in (4.30), up to changing the relevant T-dualities from those leaving R invariant to those leaving R ∗ invariant. Note however that this choice of β is not the most convenient one for all purposes, as for example in showing that the above equations describe multi-centre BPS solutions as a particular case. It can be shown that this is the case when V is a constant, but one only recovers the standard form of BPS solutions after imposing β = 1 V , as is clear from equation (5.32).", "pages": [ 46, 47 ] }, { "title": "5.4 Integration and local structure", "content": "The presence of the T-duality connection d T -in (5.31) does not allow for a straightforward solution in terms of harmonic functions, but one can follow steps similar to the composite non-BPS system in order to solve the system in terms of local functions. We can write the scalar and vector fields as where the vector of functions H is the solution to the non-harmonic equation This can be simplified and cast as a Poisson equation after introducing a rescaled vector, as which in turn is the solution to the equation This can be formally obtained from (4.37) upon exchange of T-duality transformations. In terms of the new vector, H 0 , the scalar and vector fields are given by which is the final form of the system in the real basis. The solutions to the above system can be characterised using the fact that the components of the vector H 0 are restricted, in the following way. The form (5.38) of the vector fields allows us to compute The crucial observation is that the central charges in (5.39) satisfy the reality constraint (2.59), as which restricts the components of the corresponding vector to lie on a particular Lagrangian subspace [13]. The same constraint holds for the integrated vector H 0 , for which The real form of this relation is the same as for the composite non-BPS system (4.56), which we recall in this section for completeness where we used (5.25) to undo an overall T-duality on all terms in this equation. The constraints (5.42) are exactly dual to (5.28), as they are related by replacing ˆ R ∗ and R 0 . We therefore conclude that H 0 and K lie in opposite Lagrangian subspaces, i.e. the two vectors have no common directions and span 2 n v +1 independent components in the 2( n v +1)-dimensional vector space. To be more precise, the vector K is of grade (+1), as explained in section 5.2, while the vector H 0 and its derivative lie in the Lagrangian subspace composed by grade ( -1) and (+3) components in the decomposition (2.29), as exactly as in (4.42). Applying this to (5.37), the following pattern arises for the various terms where we used the fact that T -lowers the grade of a vector by ( -2). In direct correspondence with (4.43)-(4.45) for the composite non-BPS system, we find that (5.44) decomposes into two equations according to its graded decomposition, as where in the second equation we used the property that no T-duality T -leaves the vector (5.43) invariant. This is true because the grade 3 component of H 0 can be identified as the nowhere vanishing harmonic function V defined in (5.15), as as can be seen by contracting (5.38) by ˆ R ∗ . We now analyse each of the two equations (5.45)-(5.46) in turn. The solution to (5.46) is equivalent to the condition where we used (5.25). It follows that the vector of parameters K is a generic grade (+1) vector of harmonic functions, K , which is of rank three. Note that, in this system, the poles of K represent new independent physical charges, since this vector is by definition linearly independent from H 0 . In fact, the poles of this function are the only relevant information, since one may always absorb the constant part of K into R 0 in (5.25). We now turn to the Poisson equation (5.45) and observe that the source term is of grade ( -1), according to (5.44). It follows that only n v -1 out of the n v components of H 0 are sourced, leading to an equal number of non-harmonic functions. The remaining component is the harmonic function V , already identified in (5.47) above. The source term in (5.45) can be computed explicitly using (2.62), with the result i.e. it is proportional to the vector defined in (5.27). This vector is of grade ( -1) by construction and it can be verified to satisfy the constraint in (5.40). We can now rewrite (5.45) as where we used the definition of K in (5.25) and (5.27). Finally, we present the covariant form for the equation determining the angular momentum and the function M in (2.50) and (5.24). The starting point is the first of (5.23), which upon use of (5.5) can be written as where we also used (5.38) in the second equality. Explicit computation of the last expression using the definition (2.36) leads to the alternative form Taking the divergence and the curl of this equation one obtains the relevant equations for the function M and the angular momentum respectively. The resulting Poisson equation for M reads and can be solved once H 0 and K are specified. Note that upon setting the parameters, K , of the T-dualities to vanish, these equations imply that M is a harmonic function, while ω is the corresponding dual one-form, consistent with the single centre class. This concludes our duality covariant presentation of the almost-BPS system in terms of the real basis. In the next section, we summarise the final form of the equations to be solved and we comment on some of the properties of solutions.", "pages": [ 47, 48, 49 ] }, { "title": "5.5 Summary of results", "content": "In this short section, we summarise the relevant formulae for the almost-BPS system in the real basis. All relations presented here were shown explicitly in the previous sections and we refer to the discussion there for further details. We find it however useful, both for clarity and for future applications, to give a as self-contained as possible account of the final form of the system. The ansatze for the metric and gauge fields are given in (3.1) and (3.3) in terms of the function e U , the one-form ω and the spatial vector fields dw , while the electromagnetic potentials are fixed by (5.32). The first order equation for the almost-BPS system is given by (5.31), as Here, M , V , are functions to be specified below, while ˆ R ∗ and ˆ R are a constant and a non-constant very small vector respectively, where 〈 ˆ R, ˆ R ∗ 〉 = 4. Here, the non-constant ˆ R is related to a constant very small vector, R ∗ 0 , by (5.20) which again satisfies 〈 R 0 , ˆ R ∗ 〉 = 4. In all equations, T -K is a generator of the T-dualities leaving ˆ R ∗ invariant, parametrised by a vector of harmonic functions, K . As discussed in section 2, the vector parameter K lies in the grade (+1) component of the vector space according to the decomposition implied by the T-duality. It is therefore a three-charge vector satisfying which indeed specifies a vector of n v degrees of freedom. The solutions to the flow equation (5.54) are simplified by introducing a vector, H 0 , of grade ( -1) ⊕ (+3), i.e. satisfying Note that (5.56) follows from a similar constraint, obtained by interchanging ˆ R ∗ with R 0 , that projects to the (+1) ⊕ ( -3) component of the vector space. The equations resulting from (5.54) upon use of H 0 in (5.57), take the form (5.38), where V is now identified with the grade (+3) component of H 0 , as V = 〈H 0 , ˆ R ∗ 〉 . The compatibility relation for these relations leads to the field equation for H 0 , as in (5.50) As the right hand side of this relation is a vector of grade ( -1), the corresponding components of H 0 are not harmonic, whereas V is, as can be seen by taking the inner product of (5.60) with ˆ R ∗ where we used (5.56). The final dynamical equation required is the one for the function M in (5.58) and the angular momentum vector ω , both of which are conveniently given by (5.52), as Taking the divergence of this equation, one obtains a Poisson equations for M . The equations above can be seen to be equivalent to the known formulation of the almost-BPS system, as given in five dimensional supergravity [14, 15, 16], by making a choice for the constant vectors R 0 and ˆ R ∗ . Indeed, upon choosing where we disregard the (arbitrary) normalisation, one can show a complete equivalence of the above to the original system in [14]. This particular frame is convenient in that it allows to lift to five dimensional solutions that are locally but not globally supersymmetric. However, our formulation of the almost-BPS system is closed under four dimensional dualities and includes all duals of the system in [14]. More recently, it was shown in [29] that some of the BPS structure is preserved in four dimensions as well, upon reinterpreting the constant vector R ∗ 0 as Fayet-Iliopoulos terms in a gauged theory. We close with some comments on the structure of the solutions. First, the physical scalars and the metric scale factor can be obtained by solving (5.58) in the standard way [8], once H 0 and M are solved for. Since all quantities above are appropriate combinations of the single centre solution in [13], up to overall T-dualities, it is possible to use many of the results given there. For instance, the metric scale factor is given by (4.65), as in exactly the same way as for the composite non-BPS system. However, in this case the situation is richer and more complicated, in view of the fact that the grade ( -1) components of H 0 are not harmonic and its grade 3 component V does not necessarily carry a pole at all centres. Indeed, it turns out that not all black holes described by the almost-BPS system are non-supersymmetric in isolation. On the contrary, the presence of both BPS and non-BPS types of centres, is the distinguishing property of this system, as shown in [14, 15, 16]. Clearly, the fact that the harmonic functions K lie in a subspace independent of the one where H 0 lives is the crucial ingredient that allows for both BPS and non-BPS types of charges to exist simultaneously. As seen in the case of the composite non-BPS system, solutions do not exist for all charge configurations and this holds also in the almost-BPS system. Moreover, it is not possible to obtain arbitrary asymptotic moduli for a given allowed charge configuration, for exactly the same reasons explained in section 4.5. Indeed, (4.53) and (4.58) have exactly the same structure in both cases, so that some of the n v +1 asymptotic constants in H 0 and the parameters of ˆ R and R ∗ 0 will correspond to charges rather than moduli. We once again refer to [26] for more details on the structure of almost-BPS solutions in four dimensions and for explicit examples.", "pages": [ 50, 51, 52 ] }, { "title": "6. Conclusion", "content": "In this paper, we gave a comprehensive treatment of the flow equations describing multicentre under-rotating black holes in N = 2, D = 4 supergravity coupled to vector multiplets with a symmetric scalar manifold. In particular, we considered the non-linear sigma model obtained after timelike dimensional reduction to three dimensions and derived the general, frame independent, flow equations for two systems of multi-centre non-BPS black holes, namely the composite non-BPS and almost-BPS systems. This represents a generalisation of the systems given in specific frames in [11, 14], to systems that are closed under electric/magnetic duality. The resulting structure for the vector fields and scalars in terms of real symplectic vectors turns out to be very similar for both systems. In particular, both systems are described in terms of space-dependent transformations along abelian subgroups of isometries on the scalar target space. In terms of the natural embedding to string theories, these subgroups of the full duality group are conjugate to the so-called spectral flow transformations, that are combinations of Tdualities with gauge transformations on the p -form gauge fields. In this paper, we refer to them simply as T-dualities for brevity. The main distinction between these solutions and the BPS multi-centre solutions, is that the electromagnetic vector fields are not harmonic anymore, but satisfy instead where the functions K are themselves harmonic. Note that the consistency of this equation requires that the generators T ± are indeed abelian, as for T-dualities. It follows from this equation that the poles of the harmonic functions K contribute to the electromagnetic charges in a non-linear way. Despite the interpretation of these functions as parameters of abelian isometries of the scalar manifold, they are not associated to a gauging of the theory. The crucial property that makes a general discussion in terms of covariant objects possible is that the action of general T-dualities can be given explicitly using the quartic invariant, I 4 , of symmetric special Kahler geometry. Indeed, as summarised in sections 4.5 and 5.5, all relevant equations are written in terms of this invariant only, evaluated for the real vectors parametrising the solutions. In this form, these duality covariant systems are not significantly more complicated than the equations given for the composite non-BPS system in [11] and for the almost-BPS system in [14] and can be solved in exactly the same way. The main advantage of the formulation displayed in this paper is that one need not define solutions in a fixed duality frame in terms of generic parameters, and only compute the electromagnetic charges and asymptotic moduli a posteriori, as in the constructions of [15, 16, 11, 12]. In contrast, one can start from any configuration of physical charges satisfying the required criteria associated to each system and construct the corresponding solution, using the results summarised in sections 4.5 and 4.5. This is in particular very useful for studying the domain of stability of these solutions in moduli space. Using this formulation, one can start from a given set of electromagnetic charges consistent with the system ( e.g. they have to all mutually commute with a common very small vector R in the composite non-BPS system), and parametrize the most general very small vectors R and R ∗ satisfying the corresponding constraints. Using the formulation of this paper, one can then determine the most general solution associated to a given charge configuration and define the domain of existence of such solutions in moduli space. As opposed to BPS solutions, the domain of existence of such solutions in moduli space will be restricted to a hypersurface of non-zero co-dimension. The normal directions to the hypersurface are probably not forbidden physically, but rather push us out of the domain where we know how to describe the solution. For example, the BPS solutions within the almost-BPS system only exist on a co-dimension one hypersurface, but only due to the fact that the charges at all centers must be compatible with a single constant vector R ∗ 0 , leading to a subset of all BPS solutions. Nonetheless, one can still wonder if there are walls of marginal stability within the hypersurfaces defined by each of the two systems described in this paper, i.e. whether there are boundaries of the domain of existence of such solutions at finite values of the moduli. We intend to study two-centre configurations in the aim of exhibiting (or not) walls of marginality for non-BPS solutions in a forthcoming publication. From a more general point of view, the unified description of the two known non-BPS systems and its relative simplicity are encouraging for further uncovering the structure of non-BPS solutions in supergravity. In particular, the isometries of the scalar manifold seem to play a crucial role not only in the effective three-dimensional theory, but also in the real formulation in four dimensions. It would be interesting to understand the role of these isometries in the reduction of the equations of motion to first order systems, which is not clear from our treatment in terms of nilpotent orbits. In fact, it is known that higher orbits, describing more complicated systems of non-BPS solutions, exist and one might hope that similar structures as the ones described in this paper appear in those cases as well.", "pages": [ 52, 53 ] }, { "title": "Acknowledgement", "content": "We thank Hermann Nicolai for pointing out to us reference [30]. This work was supported by the French ANR contract 05-BLAN-NT09-573739, the ERC Advanced Grant no. 226371 and the ITN programme PITN-GA-2009-237920. The work of SK was supported in part by the ANR grant 08-JCJC-0001-0, and by the ERC Starting Independent Researcher Grant 240210-String-QCD-BH.", "pages": [ 53 ] }, { "title": "A. N = 2 supergravity and symmetric special Kahler geometry", "content": "The bosonic Lagrangian of N = 2 supergravity coupled to n v vector multiplets reads [35, 36] Here, the F I µν = ∂ µ A I ν -∂ ν A I µ for I = 0 , . . . n v encompass the graviphoton and the gauge fields of the vector multiplets and G µν I are the dual field strengths, defined in terms of the F I µν though the scalar dependent couplings, whose explicit form will not be relevant in what follows. The gauge field equations of motion and Bianchi identities can then be cast as a Bianchi identity on the symplectic vector whose integral over any two-cycle defines the associated electromagnetic charges through The physical scalar fields t i , which parametrize a special Kahler space M 4 of complex dimension n v , only appear in (A.1) through the section, V , of a holomorphic U (1) × Sp (2 n v + 2 , R ) bundle over M 4 . Choosing a basis, this section can be written in components in terms of scalars X I as where F is a holomorphic function of degree two, called the prepotential, which we will always consider to be cubic for completely symmetric c ijk , i = 1 , . . . n v , and we introduced the cubic norm N [ X ]. The section V is subject to the constraint and is uniquely determined by the physical scalar fields t i = X i X 0 up to a local U (1) transformation. The U (1) gauge invariance of (A.1) is ensured by the appearance of the Kahler connection Q µ in the covariant derivative. The Kahler potential on M 4 is defined up to an arbitrary holomorphic function f ( t ) as and we fixed the U (1) gauge invariance in terms of Kahler transformations by requiring that the Kahler connection is determined by the Kahler potential as such that where D i V is the corresponding Kahler covariant derivative on the components of the section. With the prepotential (A.5), the special geometry identities [37] reduce to which are used throughout the main text. We introduce the following notation for any symplectic vector J with the understanding that when the argument is form valued, the operation is applied component wise. For instance, the central charge of the gauge field is for the prepotential (A.5). With these definitions it is possible to introduce a scalar dependent complex basis for symplectic vectors, given by ( V , D i V ), so that any vector J can be expanded as whereas the symplectic inner product can be expressed as Finally, we introduce the notion of complex selfduality of the gauge fields (A.2), which satisfy the identity where J is a scalar dependent complex structure defined as", "pages": [ 54, 55 ] } ]

[ { "title": "Black String Flow", "content": "Roberto Emparan a,b , Marina Mart'ınez b a Instituci'o Catalana de Recerca i Estudis Avan¸cats (ICREA) Passeig Llu'ıs Companys 23, E-08010 Barcelona, Spain b Departament de F'ısica Fonamental and Institut de Ci'encies del Cosmos, Universitat de Barcelona, Mart'ı i Franqu'es 1, E-08028 Barcelona, Spain", "pages": [ 1 ] }, { "title": "Abstract", "content": "We give an exact description of the steady flow of a black string into a planar horizon. The event horizon is out of equilibrium and provides a simple, exact instance of a 'flowing black funnel' in any dimension D ≥ 5. It is also an approximation to a smooth intersection between a black string and a black hole, in the limit in which the black hole is much larger than the black string thickness. The construction extends easily to more general flows, in particular charged flows.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Recent studies of black holes and black branes have sparked an interest in stationary spacetimes admitting event horizons that are not Killing horizons, i.e., the null generators of the horizon are not parallel to the generators of isometric time translations. 1 From a technical viewpoint, the theorems [3, 4, 5] that would forbid this possibility are evaded since the horizons are non-compactly generated. From a physical perspective, such horizons connect two asymptotic regions of infinite extent which have different surface gravities, i.e., different temperatures. They can be regarded as describing a steady heat flow between two infinite heat reservoirs that keep a temperature gradient constant in time. In this article we describe a remarkably simple, exact solution for a 'flowing horizon'. The explicit nature of the construction allows a detailed study of the properties of the flow. Since the spacetime is Ricci-flat, it shows that, in contrast to previous descriptions of horizon flows (motivated by AdS/CFT) [1, 2, 6, 7, 8, 9], a negative cosmological constant is not essential for their existence. In order to motivate the construction, let us first imagine a thin black string, of thickness r bs , that falls vertically into a very large black hole of radius r bh glyph[greatermuch] r bs . 2 The black string has a much larger surface gravity than the black hole. If the string is free-falling into the black hole horizon, then there are no external forces acting on the system and we can expect that the two horizons merge smoothly. 3 This is not a stationary configuration: the black hole is accreting mass from the string that flows into it and therefore must grow in size. However, this effect becomes negligible if we take the limit r bh →∞ keeping r bs fixed, and focus on the region where the two horizons meet. The black hole horizon then becomes an acceleration, Rindler-type infinite horizon, into which the black string flows by falling freely across it. Going to the rest frame of the falling black string, the acceleration horizon disappears: we are left with the spacetime of a static black string. In other words, if we take a static black string and view it from a frame that accelerates along the direction of the string, what we observe is a string in free fall into an acceleration horizon. We will construct the event horizon for such accelerated observers (taking also into account their dragging by the string, as we will see), and show that it interpolates smoothly between the Rindler horizon of a 'cold', infinitely-large black hole, at large distances from the string, and the Killing horizon of the 'hot' black string when far from the acceleration horizon. It is clear that the spacetime has a timelike Killing vector - which defines the rest frame of the string - but the flowing event horizon is not mapped into itself under its action. The horizon of this 'black string flow' is closely similar to the 'black funnels' of [11, 12, 13, 7, 14, 8, 9], where a string-like horizon that in one direction extends towards the AdS boundary, in the other direction smoothly merges with the infinitely extended horizon of an AdS black brane. The two constructions differ in their asymptotics but otherwise describe essentially similar phenomena. Our construction should approximate well the horizon of an AdS funnel much thinner than the AdS radius, in the region where it joins the AdS black brane. Indeed, it should give the universal description of all neutral, non-rotating, thin black funnels over distances sufficiently close to the horizon.", "pages": [ 2, 3 ] }, { "title": "2 Horizon of black string flow", "content": "In the rest frame of the free-falling black string, and in D = n + 4 spacetime dimensions, the metric is with In the absence of the string ( f = 1), the null surfaces t = z + t 0 , with constant t 0 , are (future) acceleration horizons, i.e., event horizons for trajectories of asymptotically uniform acceleration along z . Often this is made manifest by changing to coordinates adapted to accelerating observers, but this is actually not needed, nor is it very practical in the present case. Instead, it is simpler to trace back an appropriate congruence of null rays from null asymptotic infinity. 4 In our case, one condition that we clearly want to be satisfied is that the null rays reach the conventional Rindler horizon far from the string, i.e., where the dot denotes derivative with respect to an affine parameter λ . These turn out to be the main conditions that we will need to impose, with all other initial conditions on the geodesics following naturally. The angles on S n +1 remain fixed for each null geodesic so that the event horizon preserves the symmetry SO ( n +2). The equations for t ( λ ), z ( λ ), r ( λ ) are easy to obtain from and where glyph[epsilon1] and p are two integration constants coming from the isometries generated by ∂ t and ∂ z . Then (2.3) is satisfied by setting Null hypersurface H f . The null hypersurface ruled by outgoing geodesics can now be characterized by the one-form equation i.e., each value of t 0 giving a different null hypersurface that ends at a different value of the null coordinate in future null infinity. Obviously, any of them can serve as our event horizon, differing simply by a translation in t -z . We shall denote the null hypersurface with t 0 = 0 as H f - the flowing, or funnel, horizon. Clearly it is not a stationary horizon: the action of ∂ t changes t 0 and therefore it does not map a hypersurface onto itself but rather onto another one. The explicit form of the integral in r in (2.8) is not particularly illuminating and we give it in the appendix. The surface H f is plotted in fig. 1. We can introduce coordinates x ± adapted to these null surfaces, defined by These are null one-forms normal to the hypersurfaces defined by dx ± = 0. The vectors ∂/∂x ± = ( ∂ t ± ∂ z ) / 2 are instead spacelike vectors tangent to these hypersurfaces, i.e., dx -· ( ∂/∂x + ) = 0 and dx + · ( ∂/∂x -) = 0. Moreover, the hypersurfaces dx + = 0 (resp. dx -= 0) are symmetric under the action of ∂/∂x -(resp. ∂/∂x + ). 0 The geometry (2.1) written in coordinates ( x -, z, r, Ω) takes the form H f is the null surface x -= 0. Taking also its time reversal, namely the null surface x + = 0, we can regard the region x -< 0, x + > 0 that they bound as the Rindler wedge modified by the presence of the black string, see fig. 2. Null geodesic congruence. The null geodesics that rule H f are easily obtained. Using the freedom to rescale λ we set glyph[epsilon1] = p = 1. Then, since we have and t ( λ ) is obtained from (A.1). Here ζ is an integration constant that labels each null ray of the congruence. It corresponds to the value of z for the ray when λ = 0, i.e., when r = r 0 and t →-∞ . We can then take ( λ, ζ ) as the coordinates on the null hypersurface (together with the angles of S n +1 ). If we eliminate them we obtain the hypersurface t ( r, z ) in (2.8). For each value of ζ we have a light ray outgoing in the r direction, which initially hovers just above the black string horizon, until it escapes out to infinity moving in the r and z directions, see fig. 1. Note that r = r 0 is reached at a finite value of the affine parameter, namely λ = 0. This is in fact the same as for null outgoing trajectories outside the horizon in the Schwarzschild geometry: they have glyph[epsilon1] > 0 and reach r = r 0 in the past horizon at a finite affine parameter. In our construction the same happens for the null geodesics in H f . Taking λ < 0 they are extended into the interior of the white hole until they reach at r = 0 the past curvature singularity of the solution. Event horizon and black string drag. H f given by (2.8) is a codimension-1 null hypersurface that extends to asymptotic infinity. It is the future null boundary of a region of spacetime, and it is natural to ask what are the timelike trajectories that have H f as their event horizon. According to eq. (2.12), all light rays on H f move towards r →∞ as the affine parameter grows. Then, any timelike trajectory that remains within bounded values of r will cross H f at a finite time. That is, H f is not an event horizon for observers that remain within a finite range of the black string: they all fall across H f eventually. We interpret this phenomenon physically as a dragging effect. A boosted black string (one that moves at constant velocity) has around itself an ergosurface at a constant radial distance. Observers inside this surface cannot remain static but are dragged along with the string. In our configuration the black string is accelerating, i.e., its velocity grows, and so the ergoregion grows too. We expect then that any trajectory that remains within a finite distance from the string will be dragged along with it and eventually cross the acceleration horizon, thus moving to the future of H f . An observer who wants to avoid crossing H f must not only accelerate in the z direction, but it must also move out towards r →∞ . These are the observers that have H f as their event horizon. 5 It is straightforward to extend this analysis to the class of observers whose motion is confined inside the wedge in fig. 2 by considering trajectories which are time-reversal invariant around t = 0. Funnel geometry. Consider a cross-section of this horizon at constant t (or equivalently, at constant x + ). From (2.9) and (2.10), the metric induced on it is This geometry describes an infinite funnel: at r → ∞ it becomes flat space, while near r = r 0 , where f vanishes linearly in r , we find an infinite throat with the geometry R × S n +1 , with sphere radius r 0 . 6 We can describe this surface as the curve √ which we represent in fig. 3. This illustrates clearly the idea that the black string and the Rindler horizon merge smoothly into a funnel-shaped horizon.", "pages": [ 3, 4, 5, 6, 7, 8 ] }, { "title": "3 Out-of-equilibrium flow", "content": "The vector is an affine generator of the null geodesic congruence. It is convenient to consider the following non-affine generator of the future event horizon, This is normalized in such a way that near and far from the black string we recover the generators of the black string horizon and of the acceleration horizon. Often the Rindler horizon generator is taken to be the boost vector z∂ t + t∂ z , which at t = z becomes t ( ∂ t + ∂ z ). However, this is not adequate for us: the boost vector gives a finite, dimensionless surface gravity κ = 1, to the Rindler horizon. This is the acceleration of observers at unit proper distance from the horizon, and not the surface gravity in the infinite radius limit of a black hole, which is zero. The surface gravity κ ( glyph[lscript] ) of glyph[lscript] is defined as the non-affinity factor of the geodesics, Since λ is an affine parameter we easily find that This surface gravity decreases monotonically from its asymptotic value at the black string horizon at r = r 0 , where κ ( glyph[lscript] ) → n/ (2 r 0 ), down to κ ( glyph[lscript] ) → 0 at large r where the horizon approximates the planar Rindler horizon. Since the horizon is out of equilibrium, we can expect that its expansion be positive. In order to compute it, consider the geometry of sections at constant λ , Here we use the coordinate ζ on the surface, instead of r as in (2.13), since ∂/∂r does not commute with d/dλ and therefore is not a good coordinate for the congruence. The area element on this surface is (where ω n +1 is the area element of S n +1 ) and therefore the expansion of d/dλ is This is indeed positive, so the area of the horizon grows to the future. It is also monotonically decreasing, vanishing at r →∞ . If we consider the expansion associated to glyph[lscript] we get This is again positive, but now it vanishes both at r = r 0 and at r → ∞ . Thus the flow of the vector glyph[lscript] reflects the property that this event horizon interpolates between two asymptotic horizons, each of which is asymptotically in equilibrium at a different temperature. Finally, note that not only is the horizon out of global equilibrium, i.e., has non-constant κ , but it is also away from local equilibrium. By this we mean that the gradient r 0 ∂ z ln κ ( glyph[lscript] ) becomes large in the region r glyph[greatermuch] r 0 . Then the surface gravity at a section of the horizon at constant z , with radius r ( λ ), is not well approximated by the surface gravity of a black string with that horizon radius - as should be clear from fig. 3. As a consequence, the flowing horizon cannot be described in the effective hydrodynamic theory for black strings [17, 18].", "pages": [ 8, 9, 10 ] }, { "title": "4 Charged flows", "content": "Our previous analysis can be easily extended to more general static black string metrics of the form where all the metric functions are assumed positive outside the black string horizon and tend to 1 at r →∞ . By a suitable choice of the radial coordinate we could set H = 1, or instead R = T . Each choice has its virtues, so we shall keep this radial gauge freedom. Solutions with T = Z are qualitatively different from those with T < Z (and when T > Z there are no null geodesics with glyph[epsilon1]/p = 1). When T = Z the string worldsheet is Lorentz-invariant and the notion of the string falling along its length is not well defined. The black string horizon does not merge with the Rindler horizon, but instead the two just intersect. This can be easily seen by performing the conventional change to Rindler coordinates, t = ρ sinh η , z = ρ cosh η : the black string horizon at r = r 0 , where T ( r 0 ) = 0, and the Rindler horizon at ρ = 0, form two intersecting null surfaces, both with zero expansion. Later we discuss a relevant instance of this. Henceforth we restrict ourselves to T < Z . The flowing event horizon is characterized by and in terms of the coordinates ( λ, ζ ) on the congruence, where ζ labels different null rays and λ the affine parameter along them, we have The metric on a constantt section (or constant λ ) of this horizon is For the non-affine null geodesic generator the surface gravity is This vanishes as r → ∞ , while close to T = 0 it reproduces the surface gravity of the event horizon of the black string, The expansion is The first term inside the brackets is due to the expansion along the string direction, while the latter two correspond to the spherical expansion in the radial direction. The last factor comes from glyph[lscript] µ ∂ µ r . A natural class of solutions to study are charged strings, in particular electrically charged ones. The qualitative properties differ depending on whether the charge is string-charge, i.e., the strings are electric sources of a 2-form potential B µν , or 0-brane charge, which sources a Maxwell 1-form potential A µ . String charge. Configurations with string charge are of interest for several reasons. The neutral black string flow of previous sections is unstable, since the spacetime (2.1) is itself unstable [19]. However, string-charged black strings that are sufficiently close to extremality, but not necessarily extremal, are stable. An interesting instance are black strings with fundamental string charge, i.e., black F-strings. Above extremality the horizon can be regarded as the gravitational description of a thermal spectrum of excitations on a stack of fundamental strings. The 'F-string flow' horizon then describes, in gravitational terms, the flow of these excitations down a very large black hole that the string intersects. Even if the F-string charge allows to tune down the temperature of the black string, only at extremality can it be in thermal equilibrium with the infinitely large black hole. This extremal limit has Lorentz-invariance along z , with T = Z , so in this case there is actually no flow. Above extremality the string excitations are at a higher temperature than the black hole, and the system appears to differ from those in which the string excitations are in thermal equilibrium with a finite-temperature horizon (as studied in a worldsheet approach, e.g., [20, 21, 22, 23], or in the blackfold approach [24, 25]). In our construction, when the black string is not extremal it is not mining energy from the black hole, but rather dumping it. 0-brane charge. 0-brane charge on a black string breaks Lorentz symmetry on the worldsheet at any temperature, including at extremality. The horizons of these strings can then merge smoothly with the Rindler horizon and there is always a non-zero flow. For extremal string flows the surface gravity associated to glyph[lscript] is zero only at the black string, where (4.7) vanishes, and at the Rindler horizon at r →∞ . Inbetween them, the surface gravity is generically non-zero, as is also the expansion θ ( glyph[lscript] ) . So these are always out-of-equilibrium configurations. One may wonder what drives the flow when both its endpoints are at zero temperature. It is easy to see that it is driven by a gradient of the electric potential, i.e., an electric field along the horizon. The charge on the string is in free fall across the acceleration horizon. On the event horizon, this phenomenon is a charge current from the black string to the planar horizon, driven by an electric field. This field on the event horizon is the projection (pullback) of the spacetime electric field that the static black string creates. Clearly this field points in the direction of increasing r , and thus, on the event horizon, it points from the black string towards the planar horizon. It may be interesting to understand better these charge flows. In particular the appearance of a temperature on the horizon of the extremal string flow, in which the asymptotic endpoints are at zero temperature, is suggestive of resistive (Joule) dissipation of the electric current on the horizon. Finally, 0-brane charge does not prevent the instability of the black string, as this charge can be redistributed along the horizon. However, the addition of string charge can make these solutions stable, even supersymmetric. In the latter case, the flowing horizon is not parallel to the timelike vector associated to the Killing spinors, and therefore need not be an extremal horizon.", "pages": [ 10, 11, 12, 13 ] }, { "title": "5 Outlook", "content": "The black string flow studied above approximates a system where a very thin black string smoothly pierces a black hole. A similar-looking configuration has been found in the late-time evolution of the black string instability [26] - including a flow from the string that makes the black hole grow. It would be interesting to study in more detail the geometry in the latter case, near the region where the 'black hole' and 'black string' meet, to see if it conforms to the flowing horizon we have constructed. Black funnels in AdS can be interpreted holographically in dual terms as a flow of Hawking radiation in the boundary theory, emitted from a black hole through a thermal radiation fluid that extends to infinity [1, 7]. For our flowing geometries, a similar interpretation is also possible - although only to some extent, since the quantum degrees of freedom of the dual radiation are not known. In order to understand how this works, consider first the Rindler horizon, without the string. If we impose Dirichlet boundary conditions on a timelike surface S at a fixed, finite proper distance from the horizon, then the gravitational dynamics of the system can be described in terms of a dual thermal 'Rindler fluid' on S [27]. If we introduce the black string, then there will be a black hole horizon on S where it intersects the black string. The dual description, in terms of the quasilocal stress-energy tensor on S , will then exhibit a flow of the Rindler fluid qualitatively similar to that in AdS. Note also that in this Rindler-fluid set up, the C-metric yields an exact droplet solution in a four-dimensional bulk. The construction is like that of a black hole on a thin, planar domain wall in [28]. The method we have employed of finding non-equilibrium acceleration horizons in stationary black hole spacetimes can be extended to other situations of interest, for instance: (i) rotating black strings, to yield rotating string flows; (ii) black strings in AdS, to obtain black funnels in (hyperbolic) AdS black branes; (iii) Schwarzschild black holes, to find the event horizon in the final plunge of extreme-mass-ratio black hole collisions. We plan to report on these systems elsewhere.", "pages": [ 13 ] }, { "title": "Acknowledgments", "content": "We gratefully acknowledge conversations with Pau Figueras, Veronika Hubeny, and Don Marolf. Work supported by MEC FPA2010-20807-C02-02, AGAUR 2009-SGR168 and CPAN CSD2007-00042 Consolider-Ingenio 2010.", "pages": [ 14 ] }, { "title": "A Explicit integration of the event horizon", "content": "Setting for simplicity r 0 = 1, the event horizon (2.8) is the surface When n = 2 we find The expression simplifies for other values of n , e.g., At large values of r the surface tends to so for larger n the horizon asymptotes more rapidly to t = z . In fact for n = 1 , 2, the limit of r →∞ at fixed z or fixed t does not tend to t = z (although it is always the case that dt → dz at r →∞ ). The interpretation is that, as might be expected, low-codimension black string flows spread much more in the transverse directions than higher-codimension flows. Nevertheless, the spatial geometry of the horizon (2.13) is asymptotically flat in all dimensions.", "pages": [ 14 ] } ]

[ { "title": "ABSTRACT", "content": "Prepared for submission to JHEP", "pages": [ 1 ] }, { "title": "Borun D. Chowdhury", "content": "Department of Physics, Arizona State University, Tempe, Arizona 85287, USA Abstract: There are two evidences for information loss during black hole evaporation: (i) a pure state evolves to a mixed state and (ii) the map from the initial state to final state is non-invertible. Any proposed resolution of the information paradox must address both these issues. The firewall argument focuses only on the first and this leads to order one deviations from the Unruh vacuum for maximally entangled black holes. The nature of the argument does not extend to black holes in pure states. It was shown by Avery, Puhm and the author that requiring the initial state to final state map to be invertible mandates structure at the horizon even for pure states. The proof works if black holes can be formed in generic states and in this paper we show that this is indeed the case. We also demonstrate how models proposed by Susskind, Papadodimas et al. and Maldacena et al. end up making the initial to final state map non-invertible and thus make the horizon 'cool' at the cost of unitarity. Keywords: Fuzzballs, Firewalls, Black Holes, Information Loss Paradox", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "There are two results in Hawking's black hole evaporation analysis [1] which serve as evidence for loss of unitarity: Result 1. A pure initial state evolves to a mixed state, Result 2. The map from the initial state to the final state is many to one and thus non-invertible. Both of these originate from the assumption that the horizon of a black hole is in the Unruh vacuum, independent of the state of the matter that formed the black hole. Any proposed resolution of the information paradox must address/fix both these issues. 1 Note that in order to fix Result 1, the entropy of the outside radiation has to start decreasing no later than when half of the entropy of black hole has evaporated away [2, 3]. In Ref [4] Mathur demonstrated the stability of Hawking's result. The essence of his proof is: Statement 1. A pure state at the horizon implies increase in the entanglement entropy of radiation at each step of the evaporation. In fact, this is only the leading order result and Mathur actually showed that small deviations from purity at the horizon still imply an increase in the entanglement entropy of the radiation at each step of the evaporation. This statement implies Result 1. Since the Unruh vacuum is a pure state, this implies that small corrections to the Unruh vacuum at the horizon cannot fix Result 1. Mathur used this result to argue for structure at the horizon, a defining characteristic of fuzzballs. Recently Almheiri et. al. (AMPS) [5] observed that the contrapositive of the above statement: Statement 2. A pure final state implies a mixed state at the horizon after half of the entropy of the black hole has evaporated (Page time) implies that, assuming purity of final state, a black hole maximally entangled with its radiation cannot have its horizon in the Unruh vacuum state. Claiming that any other state would not allow free infall, AMPS dubbed the state at the horizon a firewall. 2 The qualifier about when the horizon has to become a mixed state, comes from the result of Ref. [2, 3] mentioned above. This has resulted in a discussion about 'young' (pure state) and 'old' (maximally entangled with radiation or some other system) black holes. While AMPS expressed a belief that firewalls must exist before Page time, the nature of the proof in [4, 5] (trying to address only Result 1) does not allow one to conclusively argue that. 3 In [5] and most of the discussion following it, Result 2 has not been considered at all except in Ref. [6] to the best of our knowledge. In [6] the author, along with Avery and Puhm, took a different approach than Ref. [4]. We argued that purity of final state is necessary but not sufficient for unitarity. We showed that if one considers fixing Result 2 also, i.e. demand that the initial to final state map be invertible, then there has to be an order one deviation from the Unruh vacuum at the horizon at every step of evaporation ( i.e. even before Page time/for pure states). 4 Naively, there seems to be a caveat to our proof. We show that for generic states in the e S BH sized Hilbert space of the black hole, the horizon cannot be in any fiducial state in general and the Unruh vacuum in particular. However, if the actual microstates of the black hole are special in that they live in a smaller dimensional sub Hilbert space then our result above does not apply for the following reason. One may still conjecture some special dynamics for special states such that the horizon is predominantly in the Unruh vacuum before Page time [10]. In fact, allowing for some amount of non-locality (see for example [11]), special non-local dynamics for special states may allow the horizon to be predominantly in the Unruh vacuum at all times. One reason to suspect such special states for black holes would be that 't Hooft [12] showed that the information content of black holes formed from stellar collapse is S 3 / 4 BH which is parametrically smaller that S BH . However, properties of black holes should not depend on how they are formed and whether or not they are in special states unless there is a fundamental obstruction to making black holes in non-special states. In this paper we show that it is possible to adiabatically collapse radiation in an arbitrary state living in a Hilbert space of size e S BH , to form a black holes in any of the e S BH states. While this claim was made by Zurek in [13], it turns out to not work for Schwarzschild black holes. Nevertheless, it does work for the D1-D5-P black string. The difference comes from the sign of the specific heat in the two cases. While we demonstrate the possibility of adiabatic collapse explicitly for the D1-D5-P black string, we expect it to work for any big black hole in AdS. After all, in the context of AdS/CFT the claim just amounts to saying we can excite the CFT into any one of the microscopic states accounting for the microcanonical entropy. There doesn't seem to be any fundamental obstruction to it. Having established this, our proof of [6] is on a firm footing. We then analyze some recent attempts to do away with fuzzballs/firewalls [10, 11, 14]. We find that the common theme in all of them is to find a way to oppose Statement 2. This is done in various ways (e.g. the so called A = C models postulate the inside and the radiation are not independent systems). However, Statement 2 is closely related to Result 1. Whatever model one proposes, Result 2 still needs to be taken care of to ensure unitarity. Concretely, just by establishing that the 'horizon' (however a given model defines it) is pure does not mean it is in the Unruh vacuum. We find that one then has two ways to interpret these models: Thus the main result of the paper is that a smooth, information free horizon in a fiducial state (Unruh vacuum) generically leads to information loss. Thus, we conclude that if unitarity is to be preserved then the horizon of a black hole cannot be in the Unruh vacuum at any time during the evolution for generic states. Since a horizon cannot exist but in an Unruh vacuum state, our result shows that, as far as one can trust qubit models, quantum gravity microstates have to end in a (possibly complicated Planck scale) cap outside the horizon i.e. fuzzballs. We go on to discuss how analytic continuation and thermofield doubling are related and why one should not trust the results they provide for 'behind the horizon physics'. We also propose a solution to the puzzle raised by Sen that if quantum gravity effects become important at horizon scale, why does semi-classical gravity work so well for black hole microstate counting. While this paper was being written, Ref. [15] discussing structure at horizons for pure states using different arguments appeared on the arXiv. It also criticizes the claims of [11, 14] based on arguments independent of those in this paper. The plan of the paper is as follows. In Section 2 we briefly review the information loss paradox, the stability of Hawking's result [4] and the firewall argument [5]. In Section 3 we discuss how black holes may be formed in any of the e S BH states by adiabatically compressing radiation in a box. In Section 4 we give a intuitive explanation of the proof of [6] and explain how certain models proposed to do away with fuzzballs/firewalls either do not work generically, or lead to loss of unitarity depending on the interpretation. We end with conclusions and discussions in Section 5.", "pages": [ 2, 3, 4, 5 ] }, { "title": "2.1 Leading order Hawking result", "content": "In [1] Hawking showed that a black hole formed from the collapse of a pure state evaporates away into radiation which is in a thermal state. The final radiation being in a mixed state and the map from initial to final state being non-invertible both imply a breakdown of unitarity, a problem otherwise known as the information paradox. Hawking's process can be understood in terms of stretching of 'nice slices' (see [16]). When such slices in the vacuum state (Unruh vacuum) stretch, the stretching produces particle pairs at intervals ( t ∼ M ) which were shown in [17] to be given by We have adopted the notation where hatted quantities are for the Hilbert space inside the black hole and unhatted ones are for the Hilbert space outside. C is a normalization constant and γ is an order one number. One can truncate this to the first two terms to make the process analyzable using qubits. We view the black hole as a system and the outside as another system and the horizon as the interface of these two systems. In these terms, Hawking's result implies that a state of the black hole described by n qubits evolves to This process terminates after n steps because of energy conservation. Each emission carries away some energy leading to the black hole eventually evaporating away. 5 It is easy to see that the von Neumann entropy of the radiation outside starts at zero and grows by log 2 at each step going upto n log 2. Since we started with a pure state, one of the requirements for unitarity is that the entropy goes back to zero after n steps. However, one may think that this is a leading order process and there could be small corrections to it. These could originate from non-perturbative effects, for instance, making them of the order of e -GM 2 . The large number of states e GM 2 could offset this smallness and restore unitarity. In the qubit models the number of states is 2 n and so the small corrections would then be ∼ 2 -n . Page conjectured in [2] (and this was proven in [3]) that for typical states in a Hilbert space H of dimension D H , the von-Neuman entropy of a subsystem A of dimension D A increases from zero with the dimension of D A till D 2 A = D H and then starts to decrease, going back to zero when D A = D H . So any small corrections would have to make the entropy of the radiation turn around after half the black hole has evaporated ( n/ 2 steps) and bring it down to zero in the end ( n steps) as shown in Figure 1a. This would be one of the requirements for unitarity.", "pages": [ 5, 6 ] }, { "title": "2.2 Small corrections to Hawking's process and purity of final state", "content": "Note that the state at the interface is a two qubit system which, in principle, could be in a four dimensional Hilbert space spanned by the Bell states but, according to Hawking's result, is in a one dimensional Hilbert space spanned by | ϕ 1 〉 . Consider the state of the system when m qubits of radiation have already been emitted. In the computational basis the state of the system may then be written as Mathur introduced small corrections to the Hawking's process in [4] by changing the evolution (2.4) to where note that ˆ P 1 and ˆ P 2 act only on the hatted qubits and the superscripts in | ϕ i 〉 denote the location of the newly created qubit pair for hatted and unhatted qubits respectively. He showed that as long as where the expectation value in (2.8) is taken with respect to the state | BH + rad m 〉 , the entropy of the radiation keeps increasing at every step, never turning around to make the final state pure as shown in Figure 1b. The condition (2.8) serves to keep the corrections to the leading process (2.4) small. Avery [18] generalized this to include the other two directions in the space of Hilbert space of the two qubits (2.5) at the intersection completing Mathur's proof. Simply stated, small corrections to the Hawking's process which respect effective field theory outside the black hole ( ˆ P s do not act on the unhatted qubits) lead to an ever increasing von-Neumann entropy of the radiation outside and thus have no hope of restoring unitarity. Since Hawking's process arose because of the state at the horizon being in the Unruh vacuum, the above result can be restated as Small corrections to the Unruh vacuum at the horizon maintain an ever increasing entropy of the radiation outside as shown in Figure 1b, thus precluding purity of the final state and therefore unitarity. Recently, Almheiri et. al. [5] (AMPS) argued for the contrapositive of the above statement Requiring the entropy of the radiation to go down after Page time, in accordance with Figure 1a, implies large deviations from the Unruh vacuum at the horizon after Page time , to argue that an infalling observer would see drastic consequences on trying to fall through the horizon of an 'old' black hole. They dubbed the horizon of an old black hole a firewall. 6 Note that 'old' black hole means a black hole entangled with some other system. While this system is the Hawking radiation in this case, it could be an arbitrary system. In the case of big black holes in AdS/CFT, the other system may be thought of as a system coupled to the CFT which sucks out the Hawking radiation leading to mixed state [9, 20].", "pages": [ 6, 7, 8 ] }, { "title": "2.3 Possibility of structure at horizon before Page time/ for pure states", "content": "While one would expect that any structure at the horizon would be independent of how 'old' the black hole is, or more precisely whether the black hole is in a pure state or maximally mixed state, the nature of the argument in [4, 9, 18] depends on just the purity of the final state of the radiation and thus the non-smoothness of the horizon can only be inferred after Page time. In [6] the author, along with Avery and Puhm, argued that purity of final state is only one of the requirements of unitarity. Using other requirements for unitarity - linearity, norm preservation and invertibility - we argued that the horizon cannot be smooth even before Page time, i.e. for black holes in pure states. Our proof depends on the assumption that the initial black hole state is not special i.e. one can start in any of the e S BH states. This is certainly not true for black holes formed from stellar collapse [12]. In such cases the Hilbert space of pre-collapse configurations are much smaller than the Hilbert space of the black hole formed after collapse. Thus, such states are special in a sense. If black holes always formed in some special states, it is possible to speculate that the dynamics of black hole evaporation may be fine tuned to act on such special states in such as way so as to delay any significant deviation from the Unruh vacuum at the horizon till after Page time(see [10] for instance). In fact, one may even speculate non-local fine tuned dynamics to act on special states to keep the horizon predominantly in the Unruh vacuum. However, in the next section we will see that there is no fundamental obstruction in creating black holes in generic states. This will put the proof of [6] on a firm footing.", "pages": [ 8 ] }, { "title": "3 The storage capacity of a black hole and its efficient utilization", "content": "Black hole entropy is huge compared to ordinary matter we encounter everyday. In particular it is given by the area of the black hole in Planck units Entropy is a measure of information and therefore of storage capacity. A natural question is whether all this storage space can be used. In other words, can a black hole be formed in any of the e S BH states (or superpositions thereof) or is the actual storage capacity considerably limited (like an old 5 1/4' floppy disk with bad sectors)? Black holes are usually thought of as formed from collapse of stars so the information content of a black hole formed from collapse is at best the amount of information in the pre-collapsed star. This amount was estimated in [12]. The most probable state of matter of energy E in a ball of volume V is a gas at temperature T with the equation of state The entropy and therefore the logarithm of the number of possible pre-collapse configurations of the star is The constants c 1 , c 2 above and c 3 below are order one numbers. For the star to exist, it has to be bigger than its own Schwarzschild radius Thus, we find the entropy of the star to be Since the number of states is exponential in the entropy, it seems that while the black hole could in principle have existed in many different states, it actually exists in far fewer ones, at least for black holes formed by collapse of stars. When a state in a Hilbert space gets mapped to a bigger Hilbert space, a corollary of the no-cloning theorem is that the map has to be of the form where | Φ 〉 is a fiducial state independent of | ψ 〉 [18]. In time this tensor product nature of the state will change as the state gets scrambled but it does not change the fact that the map is from a smaller Hilbert space to a bigger one and so these states are special in a sense. This has implications for the information paradox and the recent discussion about fuzzballs/firewalls. In terms of qubit models, the above discussion amounts to saying that the black hole is a n -qubit system and n 3 / 4 qubits contain some information of the state but the rest are in a fiducial state. For large n , the number of qubits carrying any actual information is a small fraction and one may imagine the black hole state to be more or less independent of the pre-collapse state. Then one may conjecture special dynamics for these small set of special states which allow a drama free experience at the horizon. So, it is very important to ask if black holes are limited to a exponentially suppressed number of total states available or they can be formed in any one of the e S BH states.", "pages": [ 9, 10 ] }, { "title": "3.1 Schwarzschild black hole", "content": "First we begin by looking at Schwarzschild black holes in 1 + 3 dimensions and ask if we can arrange things such that In [13], Zurek showed that while the process of free streaming Hawking radiation produces more entropy in the radiation than is lost by the black hole 7 , the black hole may be evaporated in an adiabatic fashion so that the entropy gained by the radiation is equal to that lost by the black hole. The idea is to put a black hole in a perfectly reflecting box so that it is in equilibrium with its own radiation. The walls of the box can be connected to a piston so that it can be adiabatically expanded with no net entropy production. Once all, or at least a major fraction, of the energy has been converted into radiation one can tweak the state of the radiation to produce any one of the e S BH states and adiabatically collapse the tweaked state again to form a black hole in any desired state. He concluded that the states of the black hole correspond to pre-collapse configurations. This process is depicted in Figure 2. Let us revisit this problem. We take a system consisting of radiation of energy E rad and a black hole of energy E BH in a box of volume V . The entropies of the individual subsystems are and the entropy of the total system is the sum of the entropies of the subsystems, S = S rad + S BH . The most stable configuration would be the one which maximized the entropy subject to the constraint E = E rad + E BH being constant. These amount to [21], We can rewrite the total entropy and the condition for stable equilibrium (3.10) in terms of the common temperature (3.9) and respectively. Now we immediately see a problem. In order for the gas to have more entropy than the black hole we need which is inconsistent with the stability condition. So, contrary to the claims of Zurek, we can not adiabatically evaporate a Schwarzschild black hole sufficiently to confirm that the entropy of the black hole is a measure of pre-collapse configurations. It is easy to see what the source of the above problem is. The black hole has a negative specific heat and when we expand the box adiabatically, at some point the black hole will become small enough so that the total specific heat will become negative. Then the black hole will not remain in stable equilibrium with its radiation. While it seemed bound to happen at some point, it was not obvious a priori that this will happen before the radiation can have more entropy than the black hole. However, just because this procedure does not work for Schwarzschild black holes, does not mean it will not work for other black holes as we will see next.", "pages": [ 10, 11 ] }, { "title": "3.2 D1-D5 black string", "content": "The D1-D5 system has been very useful in the study of black holes. We will study this system for its storage capacity. One of the key differences of the near extremal D1-D5P black string compared to the Schwarzschild black hole is that it has positive specific heat. Thus we may bypass the problem we faced in the preceding section. We give a brief description of this system sufficient for our purposes. For a more in-depth review of the system see [22] for example. We compactify type IIB on string theory on S 1 × T 4 with the four torus being string scale and the radius of S 1 being R . We wrap n 1 D1 branes on S 1 and n 5 D5 branes on S 1 × T 4 . We focus on the limit R glyph[greatermuch] gl s where D-branes are a lot heavier than momentum along S 1 and any extra energy goes into exciting equal amounts of momentum and antimomentum along S 1 . The metric and other fields of this system may be found in full detail in [23]. For our purposes it is sufficient to consider the system when there is no rotation in the non-compact direction and no net momentum. This simplified metric may be found in [20]. The energy, entropy and temperature in terms of the quantized charges are The above expression for the entropy can be obtained from the gravity as well as field theory [24]. The system will behave differently for high and low temperatures but as we will see we need only worry about high temperatures. At high temperatures ( T rad R glyph[greatermuch] 1) the box is effectively 5 + 1 dimensional and the energy and entropy of radiation is given by where c 4 and c 5 are some order one constants. This makes the total entropy The condition for equilibrium is that the subsystems have the same temperature This equilibrium is always stable as so we do not expect problems like we had with the Schwarzschild black hole in adiabatically evaporating the black string. As we increase the volume of the surrounding box adiabatically, the temperature will decrease. There is evidence from the weakly coupled D1-D5 system at the orbifold point that around TR ∼ 1 there will be a phase transition coming from the long string sector converting into shorter strings [25]. Further, when TR ∼ 1 the surrounding box becomes 4 + 1 dimensional so the radiation system also undergoes a phase transition. Fortunately we will not have to worry about these subtleties since by that time the radiation will have much more entropy than the black string as we will see. In order to see this we first write the entropy in terms of the common temperature We want to start with a high temperature configuration such that most of the entropy is in the black string. The entropy is then We want to adiabatically expand the box till the temperature is order one T f R ∼ 1. The entropy is the same as before but in terms of the final volume is Equating (3.20) and (3.21) and using RT i glyph[greatermuch] 1 we find so most of the entropy is in the radiation by the time TR ∼ 1. One can now change the state of the radiation and adiabatically pump it back into the D1-D5 system producing it in any of the e S BS states. We expect this to work for all black holes with positive specific heat. Specifically, it should work for all big black holes in AdS. After all, in that case all we are saying is that the dual CFT can be put in any one of the e S BH states accounting for its microcanonical entropy and there does not seem to be any fundamental obstruction to this.", "pages": [ 11, 12, 13 ] }, { "title": "4 The cost of cool horizons", "content": "In the previous section we established that it is indeed possible to make black holes in any one of e S BH states. This process will be slow but that should not in any fundamental way affect the properties of the black hole. In this section we will be talking about such black holes in arbitrary states. We will review an earlier result [6] which showed that such black holes cannot have Unruh vacuum at the horizon even before Page time. We will then end by demonstrating how some of the arguments against fuzzballs/firewalls do not refute the results of Ref. [6].", "pages": [ 13 ] }, { "title": "4.1 Unitarity requires order one corrections at each step of emission", "content": "Since the argument about small corrections made by Mathur and Avery was based on purity (or lack thereof) of the final state of the radiation from a black hole, the contrapositive of the statement could only be made after Page time. This has resulted in a lot of discussion about 'young' vs. 'old' black holes. Expecting this to be a red herring, the author along with Avery and Puhm revisited the problem [6]. Instead of focussing merely on purity of final radiation, which is necessary but not sufficient for unitarity, we analyzed other requirements - linearity, norm-preservation and invertibility. Either by appealing to finiteness of the black hole Hilbert space (which appears violated in the nice slice description but is necessary in a unitary theory) or by appealing to energy conservation one can argue that the process of radiation from an n -qubit system terminates after n steps (see footnote 5). Thus, all the information of n qubits have to be mapped onto the outside Hilbert space in n steps. This allows very little wiggle room. At each step one qubit worth of information has to be removed from the inside Hilbert space and mapped to the outside Hilbert space. The proof of this rather intuitive idea is technical and we will not repeat it here. The interested reader can find it in Ref. [6]. Fortunately, the result can be stated quite simply: The typical state of the interface of two interacting systems, at least one of which is local (the outside), can not preferentially be in a subspace of the the Hilbert space spanned by (2.5). Instead, the typical state necessarily explores the full four dimensional Hilbert space. Our result is stronger than the results of [4, 18] and proves that there has to be an order one correction to the Unruh vacuum at the horizon even for young black holes. At this point one may choose to say that a firewall has to be there from the beginning but since the main point of the fuzzball proposal is that there is structure at the horizon precluding universal infall, our result strongly supports the fuzzball conjecture. In fact, it is not obvious how one should see firewalls as distinct from fuzzballs (see also [9]). The distinction seems to originate from confusing the fuzzball proposal with fuzzball complementarity in the original AMPS paper. The fuzzball proposal argues for structure at the horizon which would change the evolution of typical ( E ∼ T BH ) quanta by order one and is agnostic about the infall of non-typical quanta. This proposal is primarily a constructive one with many string theoretic microstates of gravity having been systematically found over the last decade(see [22, 26-29] for reviews). The latter, on the other hands, is a relatively recent proposal which states that non-typical infalling quanta may still experience a universal behavior described by free infall. In this article we will not say much about fuzzball complementarity as our goal is to argue that structure at the horizon - fuzzballs - are present from very early on, possibly being delayed till scrambling time. For a recent discussion on fuzzball complementarity, especially in relation to AMPS, see [30]. In this paper we shall refer to order one deviation from the Unruh vacuum as fuzzballs mainly and occasionally as firewalls when describing responses to AMPS.", "pages": [ 13, 14 ] }, { "title": "4.2 Why some 'resolutions' of our result/fuzzballs/firewalls do not work", "content": "The results of [6] are in direct contradiction with some papers claiming to resolve some or all issues with fuzzballs/firewalls so we will address three such models out of the burgeoning literature. There are two results in [1] which show violation of unitarity during black hole evaporation: Thus, even if one proposes a model, with or without a mechanism to bypass Hawking's pair production, that ends up giving a final state which is pure, it still needs to be checked for invertibility of the map from the initial to the final state. We will see that the proposed resolutions that we discuss, end up making the horizon smooth at the cost of unitarity. The proofs of Mathur [4], Avery [18] and AMPS [5] only address the first issue above and boil down to using strong subadditivity on three systems. Following the notation of these three papers they are A - the early radiation, B - the outgoing hawking quantum and C - the infalling hawking quantum. We will not repeat the proofs here but the punch line of the argument forwarded by AMPS is that purity of the final state implies the horizon cannot be pure after Page time glyph[negationslash] In particular, this implies the horizon cannot be in the Unruh vacuum which is pure glyph[negationslash] where recall that in the qubit models | ϕ 1 〉 is the Unruh vacuum (2.2). A lot of responses to AMPS have focussed on finding a way to make the BC system pure in such a way that B is maximally entangled with C . The problem with this approach is that the Unruh vacuum is not the only state with this property. 8 In particular this property is shared by all the four Bell states (2.5) and only one of them, | ϕ 1 〉 , plays the role of the Unruh vacuum (2.2). The others are an order one deviation and cannot allow free infall. More explicitly, In fact, the most general two qubit state with S BC = 0 and with B and C maximally entangled is of the form where 0 ≤ θ ≤ π, 0 ≤ φ, χ < 2 π . We will refer to these as generalized Bell pairs. The fact that S BC = 0 does not imply that | BC 〉 = | ϕ 1 〉 and instead only implies that | BC 〉 is one of the generalized Bell pairs (4.4) will be the central theme in the problems with the models we discuss below. 9", "pages": [ 14, 15 ] }, { "title": "Model 1 - Distillable entanglement model", "content": "In [10] Susskind explains that if we take a Hilbert space of size 2 n and divide it into system A of size 2 m and B of size 2 n -m , any typical state | ψ 〉 on A ∪ B may be written as for n, m, n -m glyph[greatermuch] 1, where U A , U B are unitary operations on A and B respectively. This follows from the result of Page [2, 3]. What has been done here is that the entanglement between system A and system B has been first expressed in terms of Bell pairs (2.5) (or more precisely in terms generalized Bell pairs (4.4)) and then these have been rotated into the first Bell pair | ϕ 1 〉 . A count of how much two systems are entangled is the distillable entanglement. However, that entanglement generically is not in the form of | ϕ 1 〉 and to convert all other generalized Bell pairs to | ϕ 1 〉 requires a state dependent transformation. One can then posit that the dynamics of the black hole is such that it takes a typical state and acts on it with ( U A ⊗ U B ) -1 for equal sized systems so that m = n/ 2. Then before Page time quanta from system A are emitted in such a way that the interface is always in the state This will allow free infall. However, after Page time this process has to stop as now quanta from system B will be emitted and there can be no free fall. Thus, in [10] Susskind argued that firewalls can be avoided before Page time but not after. One problem with this kind of argument, as mentioned above, is that ( U A ⊗ U B ) -1 is state dependent A worse problem is the fact that it is a non-invertible map. Recall that the information problem stemmed from a many to one non-invertible map in the first place. If this is the model of black hole evaporation then the question is how would Bob standing outside the black hole collecting all the radiation coming from the black hole construct the original state | ψ 〉 . A system which can perform the map (4.6) would have to be at least as big as the black hole and according to the no-hiding theorem [31], would have absorbed all the information. After the black hole evaporates, Bob has access to the radiation but it carries no information. Bob would need access to the system which bleached out the black hole to produce the map (4.6) but there is no such system left. So in summary the problem with this model is that it restores purity but still looses unitarity. In fact a model equivalent to this was explicitly constructed in [6] to show how it does not restore unitarity even though the final radiation is pure.", "pages": [ 15, 16 ] }, { "title": "Model 2 - 'A=C' models", "content": "Another argument voiced by many, based in no small extent on the original idea of black hole complementarity, is that the inside Hawking quantum C should not be thought of as distinct from the early radiation quanta A . The claim is that this invalidates a key assumption in the proofs of [4, 5, 18]. Papadodimas and Raju tried to make this argument concrete by proposing a model based on this idea in [11]. They envision the following dynamics for infall. When Alice encounters a qubit B , it is maximally entangled with the rest of the qubits when the full state is a typical state in the Haar measure sense. However, the entanglement entropy is log 2 so B is entangled with one non-locally spread out qubit C such that S BC = 0. Papadodimas and Raju posit infall dynamics such that it is this non-local C which Alice encounters right after B . They claim that this will make Alice's experience fuzzball/firewall free both before and after Page time. The problem with this is of course that S BC = 0 is not enough to say the state is the Unruh vacuum as explained in the beginning of this section and mentioned explicitly in (4.2) and (4.3). To say that the observer always encounters BC such that it is in the state | ϕ 1 〉 implies that the non-local dynamics involves the bleaching operation which converts all generalized Bell pair states (4.4) to the Unruh vacuum | ϕ 1 〉 . Thus we again have a non-invertible map like in the previous model and the cost we pay for smooth horizons is information loss.", "pages": [ 16, 17 ] }, { "title": "Model 3 - 'ER=EPR' model", "content": "A variant of the firewall argument may be made by replacing the Hawking radiation by an arbitrary heat bath which is entangled with the CFT [9, 20, 32]. A similar idea was recently discussed by Maldacena and Susskind in [14] but with the opposite conclusion. To begin, note that Maldacena [33] has proposed that a system of two CFTs which are entangled in a particular way is dual to the eternal AdS black hole. Maldacena and Susskind argue that it is possible for Alice to collect all the radiation of an old black hole and collapse it to form another black hole. While these two black holes will be maximally entangled they might not be in the state (4.10). In this case the horizons will not be 'cool'. This is the analogue of our (4.2) and (4.3). However, Maldacena and Susskind claim that if Alice has an extremely powerful quantum computer she can have it do a quantum operation to put the two black holes in the state (4.10) leading to 'cool' horizons. While we agree with the above, we would like to add something to this. Of course, if Alice has access to the original black hole and radiation, she can have her extremely powerful quantum computer do a quantum bleaching operation on the total system to put it in the state (4.10). However, by unitarity such an operation would require the quantum computer to extract all the information of the original state onto a storage device [31]. If this has to work for generic states the size of the storage device has to be at least as big as e S BH . This is shown in Figure 3a. So we see that in order to convert the original black hole and radiation into the eternal black hole state, Alice has to have access to another storage device as big as the black hole to save the information of the original state into. However, for the information loss problem we are not interested in what Alice can do but what quantum gravity does. If instead of Alice, quantum gravity has to do this quantum bleaching operation on generic states of the black hole, then what is this extra storage space? This is shown in Figure 3b. If such a storage space does not exist (or is not accessible to asymptotic observer Bob), as seems to be the case then we are back with information loss. 10", "pages": [ 17, 18 ] }, { "title": "4.3 Relation to earlier work refuting A = C models", "content": "It was observed by Bousso earlier in [34] that A = C models for generic maximally entangled states require a many to one map. 11 This was later also mentioned in [8] where he noted that there seems to be problems with linearity because of this. While our work in this section has some overlap with the technical findings of Bousso's work, but our paper goes beyond this. Though he claims that for many to one maps there seem to be problems with linearity, this misses a very important physical implication of non-invertibility. As we discussed in detail above, such non-invertible maps are quantum operations which bleach a given Hilbert space. Such bleaching operations, if they are to preserve unitarity, need to transfer the information of the original state on the Hilbert space to some other Hilbert space (see Figure 3). While one can certainly imagine such operations being performed by Alice with access to a huge storage, in quantum gravity such a storage is missing and any model which proposes such maps are basically advocating information loss. Of course if one advocates that information is lost, then there is no reason to even assume the final state is pure and Hawking's original model is fine.", "pages": [ 18, 19 ] }, { "title": "5 Conclusion and discussion", "content": "Information loss paradox cannot be solved by simply proposing a model that makes the final state of the radiation pure. The map from the initial to the final state has to be invertible. We had demonstrated in [6] that the latter requirement implies order one deviations from the Unruh vacuum during the entire evaporation of a black hole. A loophole in our proof could be found if black holes only forms in special states. If such were the case, one could postulate special dynamics for special states which could keep the horizon predominantly in the Unruh vacuum. However, in this paper we have established that it is indeed possible to make black holes in any one of the e S BH states. In light of this, models of [10, 11, 14] may be interpreted in two ways: (i) they only allow a fuzz/fire free horizon for special states, (ii) they work on all states to allow fuzz/fire free horizons but at the cost of unitarity. Neither of these is a pleasing option. The problems with the first option are justifying the fine tuned dynamics for black holes formed from astrophysical collapse and treating black holes formed by fast and adiabatic collapse differently. Occam's razor would suggest that there is fuzzy structure at the horizon throughout a black hole's lifetime. The problem with option two is of course that we are back where we started - information loss. We are of the opinion that the results of this paper (see also [32]) strongly support the idea that the horizon of black holes are not smooth and instead the geometries end in fuzzy-stringy states outside the horizon i.e. fuzzballs. See [22, 26-29] for some reviews of this proposal. We end with a few general comments on why the black hole picture seems to work in so many respects despite being so problematic as far as unitarity is concerned.", "pages": [ 19 ] }, { "title": "5.1 Analytic continuation and thermo-field doubling", "content": "The technique of thermo-field doubling (TFD) [35] is used as a calculational tool in thermal field theory. If one has a density matrix in a thermal state of system 'Right' one can formally purify this mixed state by doubling the Hilbert space by adding system 'Left' which is identical to the system 'Right' and writing the purified state as Note that one recovers (5.1) from (5.2) by tracing over the system 'Left'. This procedure has an intimate relationship with analytic continuation. The eternal Schwarzschild black hole is obtained by maximally analytically continuing the Schwarzschild solution on the right, beyond the horizon. Israel observed that the left side of such a black hole is a TFD of the right side [36]. In fact this result also applies to the two Rindler wedges obtained by Rindler decomposition of Minkowski space. Using ideas from AdS/CFT, Maldacena proposed that the eternal AdS black hole is dual to a system of two CFTs in a TFD state [33]. However, (5.2) is not the only purification of (5.1). For simplicity we discuss these ideas using qubits. Let us look at the density matrix The TFD method would tell us that the purified state is We certainly recover (5.3) by tracing over the hatted Hilbert space in (5.6). However, we also recover the (5.3) by tracing over the hatted Hilbert space in any of the following among a continuum of other possibilities (4.4). In fact, we can also purify (5.3) as with n arbitrary. How do we understand this ambiguity in 'purification'? The point is that any mixed state will be mixed with something (its environment/ancillia/heat bath) and without access to the rest of the system, it is simply not possible to predict what the full state is. The TFD technique assumes that the system is mirrored in the heat bath. 12 For instance, let us assume an accelerating Bob sees the thermal state (5.1). While a purification of the kind (5.2) corresponds to analytic continuation of Bob's wedge giving Minkowski vacuum, another possible purification contains a Rindler (accelerating) elephant in the left wedge. These scenarios are shown in Figure 4. Being accelerated, Bob will see a horizon in addition to the thermal state (5.1) as shown in Figure 4a. If he uses analytic continuation or TFD purification he would expect the full state to be the Minkowski vacuum state (5.2) as shown in Figure 4b. However, suppose the actual state is in fact the one with a Rindler elephant in the left wedge, as shown in Figure 4c. Bob would not know that and will tell Alice that she may safely fall through the horizon. When Alice actually does so, she may get hit by a firewall of water splashed by the Rindler elephant in Figure 4d. The moral of the story is that by just having access to the thermal density matrix (5.1) Bob is not justified in saying the full state is the thermo-field double state. Alternately, Bob is not justified in doing an analytic continuation.", "pages": [ 19, 20 ] }, { "title": "5.2 Why does black hole counting work or what about the successes of semiclassical gravity?", "content": "The discussion above tells us that if unitarity is to be preserved, quantum gravity effects become important at horizon scale. Unitarity dictates that there is no hairless horizon and no interior of a black hole. Of course without these two, the phrase black hole does not make sense and the above discussion tells us that the true microstate of quantum gravity are fuzzballs which end in a quantum-fuzzy-stringy mess before the horizon. However, horizon scale is parametrically larger than Planck scale where one would naively have expected quantum gravity to become important. One question which has been asked in the context of fuzzballs is - if this is the case how does one explain success of semi-classical gravity like black hole state counting [37]? Here we will attempt to answer this question. There are two ways to measure the entropy of a system (among others) - in the microcanonical ensemble and in the canonical ensemble. In the microcanonical ensemble one actually counts the number of microstates consistent with the given macroscopic charges. In the canonical ensemble on the other hand, the entropy is given by In the canonical ensemble the density matrix is thermal ρ = e -βH from which we get where ˆ ρ = ρ Trρ is the normalized density matrix. So the entropy in the canonical ensemble is the von-Neuman entropy. Thus, we see that the canonical ensemble entropy measures how much the system is entangled with the heat bath . One may then think that the states counted in the microcanonical ensemble end up getting entangled with the heat bath in the canonical ensemble in the thermodynamic limit. Now, suppose we want to ask how many fuzzballs exists for a certain energy. For specificity we consider asymptotically AdS fuzzballs which are supposed to be dual to the CFT microstates. We can either count them one by one 13 or we can couple the fuzzballs to a heat bath and count entanglement. It turns out that for gravitational thermal density matrices there is an easy way to count entanglement. From the discussion above we learnt that a thermal state may be purified in many different ways. The TFD purification is the simplest in many ways as it models the heat bath as another copy of the system. If we take the TFD purification of a thermal ensemble of fuzzballs, Van Raamsdonk argued in Ref. [7, 32, 45] that we get the eternal AdS black hole. For mysterious reasons, the Bekenstein-Hawking entropy of the eternal AdS black hole counts the entanglement entropy of one CFT with its thermofield double [33]. Thus, the Bekenstein-Hawking entropy is a measure of entanglement of a thermal ensemble of fuzzballs with its heat bath. To summarize, the Bekenstein Hawking entropy counts the entanglement entropy of a thermal ensemble of fuzzballs with its heat bath. If we had chosen to purify the thermal ensemble to a state other than the TFD we would not have had a geometric interpretation and would not have been able to obtain the answer so elegantly. However, that operation would not have effected the entanglement entropy. Also note that while the TFD purification makes entropy counting easier, it does not actually tell us the properties of the heat bath and the interaction of the ensemble of fuzzballs with it.", "pages": [ 21, 22 ] }, { "title": "Acknowledgements", "content": "I would like to thank Steve Avery, Bartek Czech, Samir Mathur, Ashoke Sen and Erik Verlinde for helpful discussions.", "pages": [ 22 ] } ]

[ { "title": "Radiatively induced symmetry breaking and the conformally coupled magnetic monopole in AdS space", "content": "1 , 2 Ariel Edery ∗ , 3 Noah Graham † 1 Department of Physics, Bishop's University, 2600 College Street, Sherbrooke, QC J1M 1Z7 2 Kavli Institute for Theoretical Physics, University of California, Kohn Hall, Santa Barbara, CA 93106 U.S.A. 3 Department of Physics, Middlebury College, Middlebury, VT 05753", "pages": [ 1 ] }, { "title": "Abstract", "content": "We implement quantum corrections for a magnetic monopole in a classically conformally invariant theory containing gravity. This yields the trace (conformal) anomaly and introduces a length scale in a natural fashion via the process of renormalization. We evaluate the one-loop effective potential and extract the vacuum expectation value (VEV) from it; spontaneous symmetry breaking is radiatively induced. The VEV is set at the renormalization scale M and we exchange the dimensionless scalar coupling constant for the dimensionful VEV via dimensional transmutation. The asymptotic (background) spacetime is anti-de Sitter (AdS) and its Ricci scalar is determined entirely by the VEV. We obtain analytical asymptotic solutions to the coupled set of equations governing gravitational, gauge and scalar fields that yield the magnetic monopole in an AdS spacetime.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The t'Hooft Polyakov magnetic monopole [1, 2] is a toplogical soliton solution of non-Abelian gauge theory. When the gauge fields are coupled to scalar fields with the typical symmetry breaking potential V ( φ ) = λ 4 ( φ 4 -a 2 φ 2 ), the SU (2) gauge symmetry gets broken down to the residual U (1) gauge of the electromgnetic field leading to a magnetic monopole. The spontaneous breaking of gauge symmetry leaves the 10 parameter Poincar'e group intact. A few years ago, it was shown that one can add the gravitational field to this picture and enlarge the symmetry group to the 15 parameter conformal group SO(2,4) by conformally coupling the scalar fields to the metric [3]. The original gravity sector does not include an Einstein-Hilbert term because that term is not conformally invariant; instead, one uses the conformally invariant Weyl squared term. A cosmological constant is also forbidden by conformal invariance. In this massless model, the usual a 2 φ 2 term is replaced by the conformal coupling term -Rφ 2 / 6, where R is the Ricci scalar. The vacuum expectation value (VEV) is now expressed in terms of R . The theory is classically scale invariant, and one must therefore choose a VEV value by hand (or alternatively specify the asymptotic value of the Ricci scalar). Magnetic monopole solutions in a background anti-de Sitter (AdS) space were then found numerically [3]. The static sherically symmetric solution at large distances (outside the monopole core) is SchwarzschildAdS. It should be stressed that the solution in [3] is not a black hole (BH) solution since the core is non-singular and static. By definition, a BH has a horizon and there exits a region in the interior which is non-stationary, where all Killing vectors are spacelike [4] (extremal BHs are the exception since their interiors can be static or stationary, i.e. they possess a timelike Killing vector [4]). In this work, we introduce a length scale by considering the effects of quantum fluctuations of scalar fields at one loop in a classical gravitational and gauge field background. We calculate the contribution of the trace anomaly through the Schwinger-Dewitt coefficient a 2 ( x ) of the theory. To determine the vacuum expectation value (VEV), we derive the effective potential at one loop for a massless theory with a triplet of scalar fields and Rφ 2 / 6 and λφ 4 / 4! interactions. The coupling constant λ is a running coupling constant which is defined at a renormalization scale M . We use this fact to trade the dimensionless λ for the dimensionful VEV of the theory, through dimensional transmutation [5]. The asymptotic (background) spacetime is AdS and its Ricci scalar is completely determined by the VEV. The effective potential is also used to evaluate the composite operator [ E ] that enters the quantum-corrected equations of motion. Finally, we derive the equations of motion governing the gravitational, gauge and scalar fields and solve them analytically in the asymptotic region to obtain the relation between the Ricci scalar and the VEV. Though conformal invariance is presented here in the context of the magnetic monopole, there is a more general interest in scale and conformal symmetry as a possible fundamental principle in physics and cosmology, as discussed for example in [6]. As described there, the classical action of the standard model is already consistent with global scale symmetry if the Higgs mass is dropped. Furthermore, on cosmic scales, the nearly scale-invariant spectrum of primordial fluctuations seems to demand an explanation based on a fundamental symmetry in nature. In short, the authors argue that scale and conformal symmetry may be the clue to fitting observations on very small and very large scales in a coherent theory (see [6] for more details and references). The conformal anomaly has also attracted interest as an approach to the cosmological constant problem [7]. The argument put forward is that the cosmological constant problem may arise from the infrared sector of the effective theory of gravity where the conformal anomaly is expected to be relevant.", "pages": [ 2, 3 ] }, { "title": "2 Quantum corrections", "content": "Quantum vacuum fluctuations break explicitly the global scale symmetry and hence the entire conformal symmetry. The energy-momentum tensor develops a nonzero trace that can be evaluated via the Schwinger-Dewitt coefficient a 2 ( x ) in four dimensions. This is the well known trace or conformal anomaly [8, 9]. The effects due to vacuum fluctuations of matter fields can be included via an effective action W . It is convenient to split W into two parts: a divergent part W div and a renormalized finite part W ren = W -W div . W div is due to high-frequency fluctuations and is therefore local and state-independent [8]. It is expressed as an integral over a 2 ( x ), which can be calculated via a set of 'curvatures' [10]. W div is incorporated into the original action S by adding the appropriate counterterms. We label this S ren , the renormalized local part of the one loop effective action. W ren is the finite state-dependent nonlocal part of the effective action due to long wavelength fluctuations that sample the entire geometry. For example, the finite part of the vacuum expectation value of the energy-momentum tensor, 〈 T µν 〉 , that will appear on the right hand side of the gravitational equations of motion, can be viewed as a Casimir effect which also arises from long wavelength quantum fluctuations [11].", "pages": [ 3 ] }, { "title": "2.1 Schwinger-Dewitt coefficient for the monopole action and trace anomaly", "content": "The conformally invariant action that leads to a magnetic monopole coupled to gravity contains a metric g µν , a triplet of scalar fields φ a and nonabelian gauge fields A a µ . It is given by [3] where C µνστ is the Weyl tensor, F a µν is the gauge field strength defined by ∇ µ A a ν -∇ ν A a µ + ε a bc A b µ A c ν , and D µ φ a is the covariant derivative defined by ∇ µ φ a + ε a bc A b µ φ c . The square of the Weyl tensor is denoted as C 2 and F 2 ≡ F a µν F µν a , ( Dφ ) 2 ≡ D µ φ a D µ φ a and φ 2 ≡ φ a φ a . The Ricci-Levi-Civita three-dimensional coefficients ε abc are totally antisymmetric. Raising or lowering of the internal (latin) indices does not change the sign so that φ a = φ a and implicit summation is assumed throughout e.g. φ a φ a = φ 2 1 + φ 2 2 + φ 2 3 . The action (2.1) is invariant under the conformal transformations g µν → Ω 2 ( x ) g µν and φ a → φ a / Ω( x ) where Ω( x ) is called the conformal factor, an arbitrary positive smooth function. We consider the vacuum fluctuations of the triplet φ a in a gravitational and gauge field classical background. The Euclidean effective action is given by W = 1 2 tr ( ln ˆ H ) where ˆ H is the hessian of the Euclidean version of the action (2.1). It takes the following form [10] The above operator is a 3 × 3 matrix in the internal vector space of the triplet φ a and ˆ 1 is the identity matrix. Here ˆ P arises from the self interaction term λ ( φ a φ a ) 2 / 4!, 1 6 R ˆ 1 arises from the conformally coupled scalar field term 1 6 Rφ a φ a and g µν D µ D ν arises from the covariant derivative term D µ φ a D µ φ a . The gravitational and nonabelian gauge fields are treated classically. One identifies three 'curvatures' for the operator (2.2) that enter into the calculation of the effective action [10]. These are the Riemann curvature associated with g µν , the commutator curvature ˆ R µν associated with the covariant derivative D µ and defined by and the potential ˆ P which is its own curvature. The divergent part W div of the Euclidean effective action can be expressed entirely in terms of these curvatures as [10] We now evaluate the curvatures ˆ P and ˆ R µν for the Euclidean version of the action (2.1). The matrix elements of the potential ˆ P are To obtain ˆ R µν , we first evaluate the double covariant derivative, The commutator is then given by where the contracted epsilon identity ε jk/lscript ε jmn = δ km δ /lscript n -δ kn δ /lscriptm was used. Comparing (2.8) with (2.3), we obtain the commutator curvature The trace of the operators that appear in ˆ a 2 ( x, x ), evaluated using the formulas (2.6) and (2.9) for ˆ P and ˆ R µν respectively, are The quantity of interest that appears in the integrand of W div is where the factor of 11 / 3 in front of λ 2 stems from having a triplet of scalar fields (it is equal to 3 for a single scalar field). To cancel the divergent part W div , we simply add the appropriate counterterm to the original action (2.1). The renormalized local part of the action at one loop is simply The total derivatives /square R and /square ( φ a φ a ) appearing in (2.11) are not included in the above action as they lead to boundary terms that have no bearing on the equations of motion. The terms C µνστ C µνστ and R µνστ R µνστ do not appear in (2.12) since they can be eliminated in favor of R µν R µν and R 2 using the equality C µνστ C µνστ = R µνστ R µνστ -2 R µν R µν + R 2 / 3 and the fact that the integral of R µνστ R µνστ -4 R µν R µν + R 2 , the Gauss-Bonnet integral, is a topological invariant (the Euler number) in four dimensions. The quantities e , λ , α , and β are renormalized constants. They are running coupling constants governed by a renormalization group equation which we do not need to state here for our purposes (for details see [12, 13]). At one loop, the constants α and β are related by α = -β/ 3 and the sum of the first two terms in (2.12) are conformally invariant. At higher loops, non-conformally invariant terms like R 2 are generated [12, 13] and the two constants become independent. Our calculations will be at one loop, so we assume the relation α = -β/ 3.", "pages": [ 3, 4, 5 ] }, { "title": "2.2 Anomalous trace", "content": "To find the trace, we first note that under a conformal tranformation the inverse metric transforms according to g µν ( x ) → e 2 σ ( x ) g µν ( x ) and the scalar field transforms according to φ a ( x ) → e σ ( x ) φ a ( x ) where σ ( x ) is introduced for convenience and is related to the conformal factor Ω( x ) (previously introduced) by e -σ ( x ) ≡ Ω( x ). This yields the functional relation The functional variation of the one-loop effective action with respect to σ ( x ) is proportional to the Scwinger-Dewitt coefficient a 2 ( x ) [10] and using (2.13) this yields The vacuum expectation value of the energy-momentum tensor is defined as The renormalized composite operator [ E ] can be obtained readily once the effective potential has been calculated. From (2.15), its trace is given by where and", "pages": [ 6 ] }, { "title": "2.2.1 AdS space", "content": "As already discussed, the magnetic monopole solution is obtained when the spacetime is asymptotically anti-de Sitter space. This is a maximally symmetric spacetime with constant Ricci scalar R (positive in our notation). It is the submanifold obtained by embedding a hyperboloid in a flat five-dimensional spacetime of signature(+,+,-,-,-). The universal covering space of AdS space has the topology of R 4 and can be represented by the following metric: where ( r, θ, φ ) cover the usual range of spherical coordinates ( r ≥ 0, 0 ≤ θ ≤ π , 0 ≤ φ< 2 π ) and -∞ The symmetry group (isometry group) of AdS space is the ten parameter group SO(2,3). The only maximally form invariant rank two tensor under this group is the metric g µν (times a constant) so that the expectation value of the energy-momentum tensor in AdS space can be expressed in terms of its trace: 〈 T µν 〉 0 = 1 4 g µν 〈 T µ µ 〉 0 (2.21) where the zero subscript means that the quantity is evaluated asymptotically in AdS space, the vacuum spacetime. 3 The effective potential for a massless theory with Rφ 2 and λφ 4 interactions In this section we obtain the one loop effective potential by summing all the one loop oneparticle irreducible (1PI) Feynman diagrams in the presence of λφ 4 / 24 and -Rφ 2 / 6 interactions. The coupling constant λ is defined at a renormalization scale M . We choose the renormalization scale M to coincide with the VEV, the minimum of the effective potential. This in turn fixes the value of λ ; the dimensionless constant λ is traded for the dimensionful VEV through dimensional transmutation. The expectation value of the composite operator [ E ], which appears both in the trace anomaly and in the equations of motion for the scalar field, can be readily obtained from the effective potential. Note that at one loop the effective potential and the expectation value of composite operators do not generate new geometrical curvature terms; they are generated starting only at two loops [12, 13]. In particular, for the calculation of the effective potential, the √ -g factor plays no role at one loop. The calculation proceeds in the same manner as in flat space, though of course the Ricci scalar R is non-zero and acts as a vertex for the Rφ 2 interaction. The (classical) potential is given by U = λ 4! φ 4 -1 6 Rφ 2 . (3.22) We have a triplet of scalar fields and without loss of generality we take the vacuum expectation value to lie along the third component φ 3 . For φ 3 loops, U generates two vertices: -R/ 3 and λφ 2 3 / 2. They can be combined into a single vertex given by the second derivative U '' ( φ 3 ) = -R/ 3+ λφ 2 3 / 2. The vacuum expectation value of φ 1 and φ 2 are zero but they can still fluctuate in loops. The vertex for both is U '' ( φ 1 ) = U '' ( φ 2 ) = -R/ 3 + λφ 2 3 / 6 which is equivalent to replacing the coupling constant λ by λ/ 3 in the previous case. There are three sets of one loop 1PI Feynman diagrams; these are depicted in Fig.1. Let the classical field φ c ( x ) be defined as the vacuum expectation value of φ 3 in the presence of some external source J ( x ) φ c ( x ) = 〈 0 | φ 3 ( x ) | 0 〉 〈 0 | 0 〉 ∣ ∣ ∣ J . (3.23) ∣ where J appears in the action in the usual fashion via the source term Jφ . The effective potential is obtained by summing all the diagrams in Fig.1. Note that the propagator is massless. For the first set of diagrams, the one-loop contribution yields V = i ∫ d 4 k (2 π ) 4 ∞ ∑ n =1 1 2 n [ -R/ 3 + λφ 2 c / 2 k 2 + i/epsilon1 ] n = 1 16 π 2 ∫ Λ 0 k 3 E ln [ 1 + -R/ 3 + λφ 2 c / 2 k 2 E ] (3.24) where k E is the Euclidean momenta and Λ is a momentum cut-off. The integral in (3.24) can be readily evaluated but we do not write it out explicitly here. The other two sets of Feynman diagrams can be evaluated by simply replacing λ by λ/ 3 in (3.24). The one loop contribution to the potential is then V 1 = V +2 V [ λ → λ/ 3] . (3.25) As it stands, the expression V 1 is divergent in the infinite Λ limit. This is handled in the usual fashion by adding the necessary counterterms and then imposing the appropriate renormalization conditions. The total potential is given by V tot = λ 4! φ 4 c -1 6 Rφ 2 c + V 1 + Aφ 2 c + Bφ 4 c (3.26) where the last two terms are the counterterms. The constants A and B are determined via the renormalization conditions λ = d 4 V tot dφ 4 c ∣ ∣ ∣ ∣ φ c = M and -R 3 = d 2 V tot dφ 2 c ∣ ∣ ∣ ∣ φ c =0 . (3.27) The renormalization scale M sets the scale for the theory. Substituting A and B back into (3.26), taking the infinite Λ limit and then collecting terms into compact expressions, we obtain the one loop effective potential V eff = λ 4! φ 4 c -1 6 Rφ 2 c + 1 1152 π 2 [ 18 ( -R 3 + λφ 2 c 2 ) 2 ln [ 2 R -3 λφ 2 c 2 R ] +36 ( -R 3 + λφ 2 c 6 ) 2 ln [ 2 R -λφ 2 c 2 R ] +5 Rλφ 2 c + λ 2 φ 4 c ( 9 2 ln [ 2 R 2 R -3 λM 2 ] +ln [ 2 R 2 R -λM 2 ]) ] +∆ (3.28) where ∆ = λ 2 φ 4 c ( -1584 R 4 +11904 M 2 R 3 λ -20360 M 4 R 2 λ 2 +12480 M 6 Rλ 3 -2475 M 8 λ 4 ) 13824 π 2 ( -2 R + M 2 λ ) 2 ( -2 R +3 M 2 λ ) 2 . (3.29) Let φ c = v be the vacuum expectation value (VEV). It takes on this value in the asymptotic (background) spacetime, which is AdS space. We will see later, in section 4.1, that solving the equations of motion asymptotically yields the relation (4.59) between R 0 , the Ricci scalar of AdS space, and the VEV: R 0 = √ 110 λv 2 . The VEV occurs at the minimum of the effective potential in the AdS background spacetime, where dV eff dφ c ∣ ∣ ∣ ∣ φ c = v R = √ 110 λv 2 = 0 . (3.30) We set the arbitrary scale M to be equal to the VEV i.e. φ c = v = M . Equation (3.30) then yields a numerical value of λ = C/D = 2519 . 926, where the exact expressions for C and D are √ C = 288 π 2 (197355230 110 -1144879587) and D = 299795296 √ 110 -3734763307+ (12 ln(2 √ 110 -1)+18ln(2 √ 110 -3) -15ln(440))( -96294467 √ 110+524292560) . (3.31) The ratio of the one loop correction to the tree (classical) result for the potential can be readily calculated to be -0 . 504. Such ratios are typical of one loop corrections in massless theories (e.g. in massless λφ 4 theory in flat space with a single scalar field the ratio is close to -1 [14]). We discuss in the conclusions how adding gauge field fluctuations can effect this scenario. We started with a classical massless theory, a conformally invariant theory with λφ 4 and Rφ 2 interactions. After including one loop quantum corrections, the dimensionelss parameter λ has been traded for the dimensionful VEV. An important result is that the Ricci scalar of AdS space is now completely determined by the value of the VEV. 3.1 Composite operator [E] The composite operator [E(y)] is defined via (2.18). It appears in the trace but also as a quantum correction to the equations of motion for the scalar fields (see section 4 below). Inserting this operator into an n -point Green's function Γ n ( x 1 , ..., x n ) and integrating over all y yields n times the same Green's function [13]. The Feynman diagrams are therefore identical to those used to evaluate the effective potential V eff , namely those of Fig.1 (the only difference is that the symmetry factor is multiplied by n ). The upshot is that [ E ] can be obtained by taking the negative of the derivative of the one loop part of the effective potential (3.28), V loop = V eff -U , and then multiplying it by φ c , [ E ] = -φ c dV loop dφ c = -1 1152 π 2 [ 36 ( -R 3 + λφ 2 c 2 ) ln [ 2 R -3 λφ 2 c 2 R ] λφ 2 c -108 ( -R 3 + λφ 2 c 2 ) 2 λφ 2 c 2 R -3 λφ 2 c +24 ( -R 3 + λφ 2 c 6 ) ln [ 2 R -λφ 2 c 2 R ] λφ 2 c -72 ( -R 3 + λφ 2 c 6 ) 2 λφ 2 c 2 R -λφ 2 c +10 Rλφ 2 c + λ 2 φ 4 c ( 18 ln [ 2 R 2 R -3 λM 2 ] +4ln [ 2 R 2 R -λM 2 ]) ] -4∆ . (3.32) 4 Equations of motion for the magnetic monopole The quantum-corrected equations of motion for the metric, scalar and gauge fields are derived in appendix A and are given by equations (A.2), (A.3), and (A.4) respectively. For the magnetic monopole, we seek static spherically symmetric solutions where the spatial symmetry (isometry) and gauge symmetry are both SO(3). These can be viewed as the lowest energy or ground state solution [3]. The metric, scalar triplet, and non-abelian gauge fields take on the Figure 1: One loop Feynman diagrams for a triplet of scalar fields. The two vertices corresponding to -Rφ 2 / 6 and λφ 4 / 24 interactions, which are replaced by a single vertex (black dot). The last two sets of diagrams, where the components φ 2 and φ 1 run around the loop, have a vertex with 1 / 3 the coupling constant λ of the first set of diagrams. The propagator is massless. Figure 1: One loop Feynman diagrams for a triplet of scalar fields. The two vertices corresponding to -Rφ 2 / 6 and λφ 4 / 24 interactions, which are replaced by a single vertex (black dot). The last two sets of diagrams, where the components φ 2 and φ 1 run around the loop, have a vertex with 1 / 3 the coupling constant λ of the first set of diagrams. The propagator is massless. following spherically symmetric form [3]: metric: ds 2 = N ( r ) dt 2 -ψ ( r ) dr 2 -r 2 ( dθ 2 +sin 2 ( θ ) dφ 2 ) (4.33) scalar: φ a ( r ) = f ( r ) r a r = f ( r )[sin θ sin ϕ, sin θ cos ϕ, cos θ ] (4.34) gauge: A µ a = q ( r ) ξ µ a where ξ µ a are the Killing vectors for SO(3), namely ξ µ 1 = [0 , 0 , cos ϕ, -sin ϕ cot θ ] , ξ µ 2 = [0 , 0 , -sin ϕ, -cos ϕ cot θ ] and ξ µ 3 = [0 , 0 , 0 , 1] . (4.35) It will be convenient to work with a ( r ) ≡ 1+ r 2 q ( r ) instead of q ( r ). There are four functions of r to determine: the 'metric' fields N ( r ) and ψ ( r ), the 'gauge field' a ( r ) and the 'scalar' field f ( r ). It is convenient to obtain the equations of motion by direct variation of these functions. The Lagrangian corresponding to S ren is given by L = 4 π ∫ ∞ 0 dr √ N ψ r 2 ( αR 2 + β R µν R µν -1 4 e 2 F a µν F µν a + D µ φ a D µ φ a + 1 6 Rφ a φ a -λ 4! ( φ a φ a ) 2 ) = 4 π ∫ ∞ 0 L dr . (4.36) The quantities that appear in (4.36) evaluated using Eqs. 4.33-4.35 are R = -2 ψ ' r ψ 2 -N ' 2 2 ψN 2 + N '' ψN -N ' ψ ' 2 N ψ 2 -2 r 2 + 2 r 2 ψ + 2 N ' r N ψ (4.37) R µν R µν 3 ) 2 +( R 4 ) 2 ˜ R 2 ≡ ψR rr = ψ ' r ψ 2 + N ' 2 4 N 2 ψ -N '' ψ 2 ψN + N ' ψ ' 4 N ψ 2 (4.39) = ˜ R j ˜ R j ≡ ( ˜ R 1 ) 2 +( ˜ R 2 ) 2 +( ˜ R ˜ where ˜ R 1 ≡ N R tt = -ψ ' N ' 4 Nψ 2 -N ' 2 4 ψ N 2 + N '' 2 ψN + N ' r N ψ (4.38) ˜ R 3 ≡ r 2 R θθ = ψ ' 2 r ψ 2 + 1 r 2 -1 r 2 ψ -N ' 2 r N ψ (4.40) ˜ R 4 ≡ r 2 sin 2 θ R φφ = ˜ R 3 (4.41) F a µν F µν a = 4 a ' 2 r 2 ψ + 2( a 2 -1) 2 r 4 (4.42) D µ φ a D µ φ a = -f ' 2 ψ -2 a 2 f 2 r 2 (4.43) a f 2 2 r Rφ φ a = 2 ψ 2 2 4 r ψ ' N N ' ψr +2 N '' ψr N ψ ' N ' r N 4 ψ N +4 ψN 2 +4 N N ' r ψ } (4.44) N { - - - - ( φ a φ a ) 2 = f 4 and √ -g = √ N ψ r 2 sin θ . (4.45) 2 2 2 2 2 2 2 Variation with respect to the metric function N ( r ) yields ∂ L ∂N -∂ ∂r ∂ L ∂N ' + ∂ 2 ∂r 2 ∂ L ∂N '' = √ -g 2sin θ 〈 T tt 〉 The equation of motion for N is √ Nψr 2 2 N ( αR 2 + β ˜ R j ˜ R j ) + √ Nψr 2 ( 2 αR ∂R ∂N +2 β ˜ R j ∂ ˜ R j ∂N ) -[ √ Nψr 2 ( 2 αR ∂R ∂N ' +2 β ˜ R j ∂ ˜ R j ∂N ' ) ] ' + [ √ Nψr 2 ( 2 αR ∂R ∂N '' +2 β ˜ R j ∂ ˜ R j ∂N '' ) ] '' + 1 6 √ N ψ ( f 2 -f 2 ψ -f 2 r ψ ' ψ -r 2 ff ' ψ ' ψ +4 rff ' +2 r 2 ff '' ) -λr 2 f 4 ψ 48 √ Nψ -( ( a 2 -1) 2 ψ +2 r 2 a ' 2 ) 4 e 2 r 2 √ Nψ -a 2 f 2 ψ √ Nψ -r 2 f ' 2 6 √ Nψ = √ N ψ r 2 2 〈 T tt 〉 (4.46) where the Ricci scalar R is given by (4.37), ˜ R j = ˜ R j is given by Eqs.4.38-4.41 and implicit summation over j = 1 , 2 , 3 , 4 is assumed. Variation with respect to the metric function ψ ( r ) yields the equation ∂ L ∂ψ -∂ ∂r ∂ L ∂ψ ' + ∂ 2 ∂r 2 ∂ L ∂ψ '' = -√ -g 2sin θ 〈 T rr 〉 (4.47) and we obtain √ Nψr 2 2 ψ ( αR 2 + β ˜ R j ˜ R j ) + √ Nψr 2 ( 2 αR ∂R ∂ψ +2 β ˜ R j ∂ ˜ R j ∂ψ ) -[ √ Nψr 2 ( 2 αR ∂R ∂ψ ' +2 β ˜ R j ∂ ˜ R j ∂ψ ' ) ] ' + [ √ Nψr 2 ( 2 αR ∂R ∂ψ '' +2 β ˜ R j ∂ ˜ R j ∂ψ '' ) ] '' -λr 2 f 4 N 48 √ Nψ -N ( a 2 -1) 2 4 e 2 R r 2 √ Nψ + N 2 a ' 2 2 e 2 ( Nψ ) 3 / 2 + N 2 r 2 f ' 2 2( Nψ ) 3 / 2 + rff ' ( 4 N 2 + rN ' N ) 6( Nψ ) 3 / 2 + f 2 ( N 2 -N 2 ψ -6 a 2 N 2 ψ + rNN ' ) 6( Nψ ) 3 / 2 = -√ N ψ r 2 2 〈 T rr 〉 . (4.48) Lagrange's equations for the gauge field a is given by ∂ L ∂a -∂ ∂r ∂ L ∂a ' = 0 (4.49) which yields the equation of motion 2 a ( a 2 -1) + 4 ae 2 f 2 r 2 -2 ψ a '' r 2 + a ' r 2 ψ ( ψ ' ψ -N ' N ) = 0 . (4.50) Lagrange's equations for the scalar field f is given by ∂ L ∂f -∂ ∂r ∂ L ∂f ' = -√ Nψr 2 f [ E ] (4.51) which yields the equation of motion -4 a 2 f 2 r 2 + f 2 R 3 -λ 6 f 4 + 2 f f '' ψ + f f ' ψ ( 4 r + N ' N -ψ ' ψ ) = -[ E ] (4.52) where [ E ] is given by (3.32) and R is the Ricci scalar given by (4.37). The above equations of motion are for static field configurations. Therefore the higher derivative terms in the metric field equations (4.46) and (4.48) pose no issues as they are spatial not time derivatives. Higher spatial derivatives appear in many branches of physics e.g. in physical acoustics the wave equation is modified by a term with four spatial derivatives when the bending stiffness of a vibrating string is included. 4.1 Relation between Ricci scalar of AdS space and the VEV: asymptotic analytical solution We now solve the equations of motion (4.46),(4.48), (4.50) and (4.52) analytically in the asymptotic region to show that the Ricci scalar of AdS space is determined entirely by the VEV. The vacuum expectation value of the energy momentum tensor in AdS space is given by (2.21) and the non-zero components are 〈 T tt 〉 = 1 4 1 1 + k r 2 〈 T µ µ 〉 0 ; 〈 T rr 〉 = -1 4 (1 + k r 2 ) 〈 T µ µ 〉 0 〈 T θθ 〉 = -1 4 r 2 〈 T µ µ 〉 0 ; 〈 T φφ 〉 = -1 4 r 2 sin 2 θ 〈 T µ µ 〉 0 (4.53) where the AdS metric (2.20) was used. The trace 〈 T µ µ 〉 is given by 〈 T µ µ 〉 = a 2 ( x ) 16 π 2 -[ E ] = 1 16 π 2 { 1 60 ( R µνστ R µνστ -R µν R µν + /square R ) -1 6 F 2 + 11 3 λ 2 4! φ 4 -5 36 λ /square [ φ 2 ] } -[ E ] , (4.54) where (2.17) and (2.19) were used and [ E ] is given by (3.32). We now evaluate (4.54) in AdS space, the asymptotic spacetime. Asymptotically, F a µν → 0, /square [ φ 2 ] → 0, R ρσµν = k ( g ρµ g σν - R µν = 3 k g µν so that R = 12 k , /square R = 0 and R µνστ R µνστ -R µν R µν = -12 k 2 . Substituting these values into (4.54), we obtain 〈 T µ µ 〉 0 = -1 80 k 2 π 2 + 11 1152 λ 2 π 2 v 4 -[ E ] 0 (4.55) where [ E ] 0 is [ E ] evaluated in asymptotic AdS space. The boundary conditions for the magnetic monopole [3] are that asymptotically, as r → ∞ , the spacetime is AdS where N → 1+ kr 2 , ψ → 1 / (1+ k r 2 ), f → v , r f ' → 0 ( f ' drops off faster than 1 /r ), r 2 f '' → 0, a → 0 and a ' → 0. Both v and k are positive constants. Substituting these boundary conditions into the gravity equation (4.46) (or (4.48)) and using (4.55) yields the following relation v 2 2 -λv 4 48 k = 1 8 k ( -1 80 k 2 π 2 + 11 1152 λ 2 π 2 v 4 -[ E ] 0 ) . (4.56) We can eliminate [ E ] 0 above by solving the scalar equation (4.51) asymptotically. This yields 4 k v 2 -λv 4 6 = -[ E ] 0 (4.57) where we used that R = R 0 = 12 k in AdS space. Substituting the above into (4.56) yields the solution so that k = √ 110 12 λv 2 (4.58) R 0 = √ 110 λv 2 . (4.59) The Ricci scalar of AdS space is therefore determined solely by the VEV since the value of λ is known (it is no longer a free parameter having been traded for the dimensionful VEV). Substituting (4.58) into (4.57), yields [ E ] 0 = -λv 4 6 ( 2 √ 110 -1). This agrees with expression (3.32) asymptotically i.e. after substituting R = √ 110 λv 2 and φ c = v = M . The remaining unbroken U(1) is associated with F µν 3 and the magnetic field is defined via F ij 3 = /epsilon1 ijk B k . Asymptotically, a ( r ) → 0 and the function q ( r ) appearing in (4.35) approaches -1 /r 2 . It is easy to verify that one obtains a radial magnetic field that varies as 1 /r 2 at large distances, corresponding to a magnetic monopole. 5 Conclusions In previous work [3], a magnetic monopole solution in AdS space was obtained without introducing explicitly a mass term. In that calculation, spontaneous symmetry breaking (SSB) of gauge symnmetry responsible for the magnetic monopole occurred via gravitation itself through the coupling term Rφ 2 in a conformally invariant action. This works as long as a length scale is introduced by hand because classically there is no length scale. In the present work, we introduced a renormalization scale into the massless theory by considering quantum corrections. Symmetry breaking was radiatively induced ' a la Coleman-Weinberg [5], albeit in a more complicated massless theory containing gravity where the one loop effective potential must take into account the Rφ 2 interaction in addition to the usual λφ 4 . The dimensionless λ , defined at the renormalizaton scale M , was traded for the dimensionful VEV and the Ricci scalar of the background AdS spacetime was determined entirely by the VEV. Though we discussed the quantum corrections of a classical conformal invariant theory in the context of the magnetic monopole, the techniques and results presented here could potentially have wider consequences. For example, in a recent article [6], scale and conformal symmetry are presented as fundamental principles for physics and cosmology. The authors have a model containing a Higgs field H , a dilaton field φ , standard model fields as well as gravity. The authors point out that in a conformally invariant theory there is no mechanism at the classical level to set the scale of φ 0 , the minimum of the dilaton field φ . They then mention that quantum corrections may alleviate this problem in a fashion that is reminiscent to [5]; our calculation provides a concrete implementation of this proposal. Our work can now be naturally extended in a few ways. First, one can add gauge field fluctuations in the calculation of the effective potential. Then the coupling contant λ can be expressed in terms the electromagnetic coupling constant e as in [5]. In that scenario, the two parameters in the theory become e and the dimensionful VEV. Second, we used the symmetry of AdS space to solve for 〈 T µν 〉 0 , the asymptotic value of 〈 T µν 〉 . This allowed us to solve the equations of motion analytically in the asymptotic regime and to obtain an expression relating the Ricci scalar of AdS space to the VEV. The interior spacetime obeys spherical symmetry but not the symmetry of AdS space. Therefore, the finite and nonlocal part of 〈 T µν 〉 for the interior would require a more elaborate calculation. One could then obtain numerical solutions of the interior. It is of interest to see how these numerical solutions containing quantum corrections in an AdS background compare with those obtained in the classical context of General Relativity (GR)[17]. Third, we worked with a background AdS spacetime because the Ricci scalar of AdS space had the right sign for classical SSB [3]. However, the VEV here is obtained from the quantum-corrected effective potential. In the presence of quantum corrections, de Sitter (dS) space could well be a viable background spacetime. The Ricci scalar of the background AdS space was determined solely by the VEV, which should apply to dS space as well. Such solutions could be of greater cosmological interest. Acknowledgments AE acknowledges support from an NSERC discovery grant. He thanks KITP for a four week stay during the summer of 2012 where part of this work was completed. This research was, supported in part by the National Science Foundation under Grant No. NSF PHY11-25915. NG was supported in part by the National Science Foundation (NSF) through grant PHY1213456. References [1] G. 't Hooft, Nucl. Phys. B 79 , 276, (1974). [2] A.M. Polyakov, JETP Lett. 20 , 194 (1974). [3] A. Edery, L. Fabbri and M. B. Paranjape, Class. Quant. Grav. 23 , 6409 (2006) [hep-th/0603131]. [4] A. Edery and B. Constantineau, Class. Quant. Grav. 28 , 045003 (2011)[arXiv:1010.5844]. [5] S. Coleman and E. Weinberg, Phys. Rev. D 7 , 1888 (1973). [6] I. Bars, P. Steinhardt and N. Turok, [arXiv:10307.1848]. [7] E.C. Thomas, F.R. Urban and A. R. Zhitnitsky, JHEP 08 ,043 (2009) [arXiv:0904.3779] [8] V.F. Mukhanov and S. Winitzki, Introduction to Quantum Effects in Gravity , Cambridge University Press, (2007). [9] N.D. Birrell and P.C. W. Davies, Quantum Fields in Curved Space , Cambridge University Press, (1982). [10] A. O. Barvisnsky, Yu. V. Gusev, G.A. Vilkovisky and V.V. Zhytnikov, Nucl. Phys. B 439 , 561 (1995)[hep-th/9404187]. [11] V.M. Mostepanenko and N.N. Trunov, The Casimir Effect And Its Applications , Oxford University Press, (1997), M. Bordag , U. Mohideen and V.M. Mostepanenko, Phys. Rep. 353 , 1 (2001), A. Zee, Quantum Field Theory in a Nutshell , Princeton University Press, (2003), A. Edery, J. Phys. A: Math. Gen. 39 , 685 (2006) [math-ph/0510056], A. Edery, Phys. Rev. D 75 , 105012 (2007) [hep-th/0610173], A. Edery, J. Stat. Mech. P06007 (2006) [hep-th/0510238]. [12] L.S. Brown and J.C. Collins, Ann. Phys. 130 , 215 (1980). [13] S.J. Hathrell, Ann. Phys. 139 , 136 (1982). [14] T. Cheng and L. Li, Gauge Theory of Elementary Particle Physics , Oxford University Press, (1984). [15] M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory , Westview Press, (1995). [16] S. Caroll, Spacetime and Geometry: An Introduction to General Relativity , Benjamin Cummings, (2003). [17] P. Breitenlohner, P. Forg' a cs and D. Maison, Nucl. Phys. B 383 , 357 (1992); K. Lee, V. P. Nair and E. J. Weinberg, Phys. Rev. D 45 , 2751 (1992); K. Lee, V. P. Nair and E. J. Weinberg, Phys. Rev. Lett. 68 , 1100 (1992); M.E. Ortiz, Phys. Rev. D 45 , 2586 (1992); H. Hollmann, Phys. Lett. B 338 , 181 (1994). A Equations of motion with quantum corrections The equations of motion for the metric, non-abelian gauge fields, and scalar triplet are obtained by variation of the total action S = S ren + W ren with respect to each field. For the metric we obtain 2 √ -g δ δg µν S ren = -〈 T µν 〉 (A.1) where we used (2.21). With S ren given by (2.12) the metric field equations are where αH µν + β K µν + M µν = -〈 T µν 〉 (A.2) H µν ≡ 2 √ -g δ δg µν ∫ R 2 √ -g d 4 x = 4 ∇ µ ∇ ν R -4 g µν /square R -g µν R 2 +4 RR µν K µν ≡ 2 √ -g δ δg µν ∫ R αβ R αβ √ -g d 4 x =4 ∇ ν ∇ α R α µ -2 /square R µν -g µν /square R +4 R α µ R αν -g µν R αβ R αβ . M µν ≡ 2 √ -g δ δg µν ∫ { -1 4 e 2 R F 2 +( Dφ ) 2 + 1 6 Rφ 2 -λ 2 R φ 4 } √ -g d 4 x = 1 4 e 2 R g µν F 2 -1 e 2 R F a µβ F β ν a +2 D µ φ a D ν φ a -g µν ( ( Dφ ) 2 -λ 2 R φ 4 ) + 1 3 ( g µν ∇ α ∇ α φ 2 -∇ µ ∇ ν φ 2 +( R µν -1 2 g µν R ) φ 2 ) . For the scalar field, we have 2 √ -g δ δφ a ( S ren + W ren ) = 0, which yields the equation ( 2 D µ D µ φ a -( R 3 -4 λ 2 R φ 2 ) φ a ) φ a = 1 √ -g φ a δW ren δφ a = [ E ] . (A.3) For the gauge field we simply have 2 √ -g δ δA ν S ren = 0, since 2 √ -g δ δA ν W ren is zero when evaluated in the vacuum state where F a µν → 0 and A a µ is a pure gauge which we set to zero. The equation of motion for the gauge field is then ∇ µ F µν a + ε bc a A µ b F µν c = 2 e R ε bc a D v φ b φ c . (A.4) The symmetry group (isometry group) of AdS space is the ten parameter group SO(2,3). The only maximally form invariant rank two tensor under this group is the metric g µν (times a constant) so that the expectation value of the energy-momentum tensor in AdS space can be expressed in terms of its trace: where the zero subscript means that the quantity is evaluated asymptotically in AdS space, the vacuum spacetime.", "pages": [ 7 ] }, { "title": "3 The effective potential for a massless theory with Rφ 2 and λφ 4 interactions", "content": "In this section we obtain the one loop effective potential by summing all the one loop oneparticle irreducible (1PI) Feynman diagrams in the presence of λφ 4 / 24 and -Rφ 2 / 6 interactions. The coupling constant λ is defined at a renormalization scale M . We choose the renormalization scale M to coincide with the VEV, the minimum of the effective potential. This in turn fixes the value of λ ; the dimensionless constant λ is traded for the dimensionful VEV through dimensional transmutation. The expectation value of the composite operator [ E ], which appears both in the trace anomaly and in the equations of motion for the scalar field, can be readily obtained from the effective potential. Note that at one loop the effective potential and the expectation value of composite operators do not generate new geometrical curvature terms; they are generated starting only at two loops [12, 13]. In particular, for the calculation of the effective potential, the √ -g factor plays no role at one loop. The calculation proceeds in the same manner as in flat space, though of course the Ricci scalar R is non-zero and acts as a vertex for the Rφ 2 interaction. The (classical) potential is given by We have a triplet of scalar fields and without loss of generality we take the vacuum expectation value to lie along the third component φ 3 . For φ 3 loops, U generates two vertices: -R/ 3 and λφ 2 3 / 2. They can be combined into a single vertex given by the second derivative U '' ( φ 3 ) = -R/ 3+ λφ 2 3 / 2. The vacuum expectation value of φ 1 and φ 2 are zero but they can still fluctuate in loops. The vertex for both is U '' ( φ 1 ) = U '' ( φ 2 ) = -R/ 3 + λφ 2 3 / 6 which is equivalent to replacing the coupling constant λ by λ/ 3 in the previous case. There are three sets of one loop 1PI Feynman diagrams; these are depicted in Fig.1. Let the classical field φ c ( x ) be defined as the vacuum expectation value of φ 3 in the presence of some external source J ( x ) ∣ where J appears in the action in the usual fashion via the source term Jφ . The effective potential is obtained by summing all the diagrams in Fig.1. Note that the propagator is massless. For the first set of diagrams, the one-loop contribution yields where k E is the Euclidean momenta and Λ is a momentum cut-off. The integral in (3.24) can be readily evaluated but we do not write it out explicitly here. The other two sets of Feynman diagrams can be evaluated by simply replacing λ by λ/ 3 in (3.24). The one loop contribution to the potential is then As it stands, the expression V 1 is divergent in the infinite Λ limit. This is handled in the usual fashion by adding the necessary counterterms and then imposing the appropriate renormalization conditions. The total potential is given by where the last two terms are the counterterms. The constants A and B are determined via the renormalization conditions The renormalization scale M sets the scale for the theory. Substituting A and B back into (3.26), taking the infinite Λ limit and then collecting terms into compact expressions, we obtain the one loop effective potential where Let φ c = v be the vacuum expectation value (VEV). It takes on this value in the asymptotic (background) spacetime, which is AdS space. We will see later, in section 4.1, that solving the equations of motion asymptotically yields the relation (4.59) between R 0 , the Ricci scalar of AdS space, and the VEV: R 0 = √ 110 λv 2 . The VEV occurs at the minimum of the effective potential in the AdS background spacetime, where We set the arbitrary scale M to be equal to the VEV i.e. φ c = v = M . Equation (3.30) then yields a numerical value of λ = C/D = 2519 . 926, where the exact expressions for C and D are The ratio of the one loop correction to the tree (classical) result for the potential can be readily calculated to be -0 . 504. Such ratios are typical of one loop corrections in massless theories (e.g. in massless λφ 4 theory in flat space with a single scalar field the ratio is close to -1 [14]). We discuss in the conclusions how adding gauge field fluctuations can effect this scenario. We started with a classical massless theory, a conformally invariant theory with λφ 4 and Rφ 2 interactions. After including one loop quantum corrections, the dimensionelss parameter λ has been traded for the dimensionful VEV. An important result is that the Ricci scalar of AdS space is now completely determined by the value of the VEV.", "pages": [ 7, 8, 9, 10 ] }, { "title": "3.1 Composite operator [E]", "content": "The composite operator [E(y)] is defined via (2.18). It appears in the trace but also as a quantum correction to the equations of motion for the scalar fields (see section 4 below). Inserting this operator into an n -point Green's function Γ n ( x 1 , ..., x n ) and integrating over all y yields n times the same Green's function [13]. The Feynman diagrams are therefore identical to those used to evaluate the effective potential V eff , namely those of Fig.1 (the only difference is that the symmetry factor is multiplied by n ). The upshot is that [ E ] can be obtained by taking the negative of the derivative of the one loop part of the effective potential (3.28), V loop = V eff -U , and then multiplying it by φ c ,", "pages": [ 10 ] }, { "title": "4 Equations of motion for the magnetic monopole", "content": "The quantum-corrected equations of motion for the metric, scalar and gauge fields are derived in appendix A and are given by equations (A.2), (A.3), and (A.4) respectively. For the magnetic monopole, we seek static spherically symmetric solutions where the spatial symmetry (isometry) and gauge symmetry are both SO(3). These can be viewed as the lowest energy or ground state solution [3]. The metric, scalar triplet, and non-abelian gauge fields take on the following spherically symmetric form [3]: gauge: A µ a = q ( r ) ξ µ a where ξ µ a are the Killing vectors for SO(3), namely It will be convenient to work with a ( r ) ≡ 1+ r 2 q ( r ) instead of q ( r ). There are four functions of r to determine: the 'metric' fields N ( r ) and ψ ( r ), the 'gauge field' a ( r ) and the 'scalar' field f ( r ). It is convenient to obtain the equations of motion by direct variation of these functions. The Lagrangian corresponding to S ren is given by The quantities that appear in (4.36) evaluated using Eqs. 4.33-4.35 are R µν R µν 3 ) 2 +( R 4 ) 2 a f 2 2 r Rφ φ a = 2 ψ 2 2 4 r ψ ' N N ' ψr +2 N '' ψr N ψ ' N ' r N 4 ψ N N { - - - - 2 2 2 2 2 2 2 Variation with respect to the metric function N ( r ) yields The equation of motion for N is where the Ricci scalar R is given by (4.37), ˜ R j = ˜ R j is given by Eqs.4.38-4.41 and implicit summation over j = 1 , 2 , 3 , 4 is assumed. Variation with respect to the metric function ψ ( r ) yields the equation and we obtain Lagrange's equations for the gauge field a is given by which yields the equation of motion Lagrange's equations for the scalar field f is given by which yields the equation of motion where [ E ] is given by (3.32) and R is the Ricci scalar given by (4.37). The above equations of motion are for static field configurations. Therefore the higher derivative terms in the metric field equations (4.46) and (4.48) pose no issues as they are spatial not time derivatives. Higher spatial derivatives appear in many branches of physics e.g. in physical acoustics the wave equation is modified by a term with four spatial derivatives when the bending stiffness of a vibrating string is included.", "pages": [ 10, 12, 13, 14 ] }, { "title": "4.1 Relation between Ricci scalar of AdS space and the VEV: asymptotic analytical solution", "content": "We now solve the equations of motion (4.46),(4.48), (4.50) and (4.52) analytically in the asymptotic region to show that the Ricci scalar of AdS space is determined entirely by the VEV. The vacuum expectation value of the energy momentum tensor in AdS space is given by (2.21) and the non-zero components are where the AdS metric (2.20) was used. The trace 〈 T µ µ 〉 is given by where (2.17) and (2.19) were used and [ E ] is given by (3.32). We now evaluate (4.54) in AdS space, the asymptotic spacetime. Asymptotically, F a µν → 0, /square [ φ 2 ] → 0, R ρσµν = k ( g ρµ g σν - R µν = 3 k g µν so that R = 12 k , /square R = 0 and R µνστ R µνστ -R µν R µν = -12 k 2 . Substituting these values into (4.54), we obtain where [ E ] 0 is [ E ] evaluated in asymptotic AdS space. The boundary conditions for the magnetic monopole [3] are that asymptotically, as r → ∞ , the spacetime is AdS where N → 1+ kr 2 , ψ → 1 / (1+ k r 2 ), f → v , r f ' → 0 ( f ' drops off faster than 1 /r ), r 2 f '' → 0, a → 0 and a ' → 0. Both v and k are positive constants. Substituting these boundary conditions into the gravity equation (4.46) (or (4.48)) and using (4.55) yields the following relation We can eliminate [ E ] 0 above by solving the scalar equation (4.51) asymptotically. This yields where we used that R = R 0 = 12 k in AdS space. Substituting the above into (4.56) yields the solution so that The Ricci scalar of AdS space is therefore determined solely by the VEV since the value of λ is known (it is no longer a free parameter having been traded for the dimensionful VEV). Substituting (4.58) into (4.57), yields [ E ] 0 = -λv 4 6 ( 2 √ 110 -1). This agrees with expression (3.32) asymptotically i.e. after substituting R = √ 110 λv 2 and φ c = v = M . The remaining unbroken U(1) is associated with F µν 3 and the magnetic field is defined via F ij 3 = /epsilon1 ijk B k . Asymptotically, a ( r ) → 0 and the function q ( r ) appearing in (4.35) approaches -1 /r 2 . It is easy to verify that one obtains a radial magnetic field that varies as 1 /r 2 at large distances, corresponding to a magnetic monopole.", "pages": [ 14, 15 ] }, { "title": "5 Conclusions", "content": "In previous work [3], a magnetic monopole solution in AdS space was obtained without introducing explicitly a mass term. In that calculation, spontaneous symmetry breaking (SSB) of gauge symnmetry responsible for the magnetic monopole occurred via gravitation itself through the coupling term Rφ 2 in a conformally invariant action. This works as long as a length scale is introduced by hand because classically there is no length scale. In the present work, we introduced a renormalization scale into the massless theory by considering quantum corrections. Symmetry breaking was radiatively induced ' a la Coleman-Weinberg [5], albeit in a more complicated massless theory containing gravity where the one loop effective potential must take into account the Rφ 2 interaction in addition to the usual λφ 4 . The dimensionless λ , defined at the renormalizaton scale M , was traded for the dimensionful VEV and the Ricci scalar of the background AdS spacetime was determined entirely by the VEV. Though we discussed the quantum corrections of a classical conformal invariant theory in the context of the magnetic monopole, the techniques and results presented here could potentially have wider consequences. For example, in a recent article [6], scale and conformal symmetry are presented as fundamental principles for physics and cosmology. The authors have a model containing a Higgs field H , a dilaton field φ , standard model fields as well as gravity. The authors point out that in a conformally invariant theory there is no mechanism at the classical level to set the scale of φ 0 , the minimum of the dilaton field φ . They then mention that quantum corrections may alleviate this problem in a fashion that is reminiscent to [5]; our calculation provides a concrete implementation of this proposal. Our work can now be naturally extended in a few ways. First, one can add gauge field fluctuations in the calculation of the effective potential. Then the coupling contant λ can be expressed in terms the electromagnetic coupling constant e as in [5]. In that scenario, the two parameters in the theory become e and the dimensionful VEV. Second, we used the symmetry of AdS space to solve for 〈 T µν 〉 0 , the asymptotic value of 〈 T µν 〉 . This allowed us to solve the equations of motion analytically in the asymptotic regime and to obtain an expression relating the Ricci scalar of AdS space to the VEV. The interior spacetime obeys spherical symmetry but not the symmetry of AdS space. Therefore, the finite and nonlocal part of 〈 T µν 〉 for the interior would require a more elaborate calculation. One could then obtain numerical solutions of the interior. It is of interest to see how these numerical solutions containing quantum corrections in an AdS background compare with those obtained in the classical context of General Relativity (GR)[17]. Third, we worked with a background AdS spacetime because the Ricci scalar of AdS space had the right sign for classical SSB [3]. However, the VEV here is obtained from the quantum-corrected effective potential. In the presence of quantum corrections, de Sitter (dS) space could well be a viable background spacetime. The Ricci scalar of the background AdS space was determined solely by the VEV, which should apply to dS space as well. Such solutions could be of greater cosmological interest.", "pages": [ 15, 16 ] }, { "title": "Acknowledgments", "content": "AE acknowledges support from an NSERC discovery grant. He thanks KITP for a four week stay during the summer of 2012 where part of this work was completed. This research was, supported in part by the National Science Foundation under Grant No. NSF PHY11-25915. NG was supported in part by the National Science Foundation (NSF) through grant PHY1213456.", "pages": [ 17 ] }, { "title": "A Equations of motion with quantum corrections", "content": "The equations of motion for the metric, non-abelian gauge fields, and scalar triplet are obtained by variation of the total action S = S ren + W ren with respect to each field. For the metric we obtain where we used (2.21). With S ren given by (2.12) the metric field equations are where For the scalar field, we have 2 √ -g δ δφ a ( S ren + W ren ) = 0, which yields the equation For the gauge field we simply have 2 √ -g δ δA ν S ren = 0, since 2 √ -g δ δA ν W ren is zero when evaluated in the vacuum state where F a µν → 0 and A a µ is a pure gauge which we set to zero. The equation of motion for the gauge field is then", "pages": [ 18, 19 ] } ]

[ { "title": "ABSTRACT", "content": "We study quantum cosmological models for certain classes of bang/crunch singularities, using the duality between expanding bubbles in AdS with a FRW interior cosmology and perturbed CFTs on de Sitter space-time. It is pointed out that horizon complementarity in the AdS bulk geometries is realized as a conformal transformation in the dual deformed CFT. The quantum version of this map is described in full detail in a toy model involving conformal quantum mechanics. In this system the complementarity map acts as an exact duality between eternal and apocalyptic Hamiltonian evolutions. We calculate the commutation relation between the Hamiltonians corresponding to the different frames. It vanishes only on scale invariant states.", "pages": [ 1 ] }, { "title": "Conformal Complementarity Maps", "content": "Jos'e L.F. Barb'on † and Eliezer Rabinovici /star † Instituto de F´ısica Te´orica IFT UAM/CSIC C/Nicolas Cabrera 13. UAM, Cantoblanco 28049. Madrid, Spain [email protected] /star Racah Institute of Physics, The Hebrew University Jerusalem 91904, Israel and Theory Group, Physics Department, CERN CH 1211, Geneva 23. Switzerland [email protected]", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "There are quite a few cases in quantum field theory where the same set of data can be described in several ways. The two dimensional Ising model and electric magnetic duality are examples. In string theory a very large set of such relations was uncovered over the years. Backgrounds which have for example different metric, topology, number of small or large dimensions and singularity, commutativity and associativity structures were identified. The AdS/CFT type relations are in this class. In an effort to come to grips with the special challenges presented by black hole physics a concept named Black Hole Complementarity was put forward [1]. Sets of observables defined outside and inside the horizon, while not commuting among the sets, were each supposed to give a full description of the system. The consequences of such a suggestion are still being processed (cf. [2] and its wake). In a previous paper we have been brought to suggest a relation which touches all these types of dualities [3]. The systems discussed, a certain type of crunching AdS spacetimes, have a cosmological horizon separating those observers who meet the crunch in finite proper time from those who get to live for an infinite proper time. The situation is thus similar to a black hole of infinite entropy. It was claimed in [3] that the exterior physics can be described, via AdS/CFT tools, by a specific class of non-singular timeindependent QFTs living on a time-dependent de Sitter (dS) world volume, whereas the horizon interior could be described by a time-dependent QFT living on a static Einstein universe. The two holographic descriptions are related by a conformal transformation, which becomes equivalent to a complementarity map for this system. The conformal complementarity relates the 'eternal' Hamiltonian evolution of dS space-time to a finite-time interval of the Einstein universe, which we call 'apocalyptic'. This property is visible in the short-distance description of the QFT, and can be studied with effective Lagrangian methods, something we address in sections 2 and 3. Furthermore, if the system is simplified by 'dimensional reduction' to the conformal quantum mechanics of de Alfaro, Fubini and Furlan [4], the complementarity transformation becomes explicitly expressible for any wave function in the Hilbert space. We analyze in sections 4 and 5 the details of this d = 1 system, including the mapping of observables on both sides of the duality. Section 6 is devoted to a formal extension of the eternal/apocalyptic duality to arbitrary QFTs and we end with a discussion of conceptual puzzles and open questions in section 7, where we also succumb to the temptation to relate these ideas to some features of our universe.", "pages": [ 2 ] }, { "title": "2. A Simple Model Of Cosmological Complementarity", "content": "A relation between horizon complementarity and conformal symmetry is inherent in AdS/CFT as a result of basic rules of the correspondence. An AdS d +1 space-time does not define a canonical metric on the d -dimensional boundary but rather defines a boundary conformal structure, i.e. a conformal class of d -metrics. Conformal maps between these metrics extend naturally as bulk diffeomorphisms, whose global properties produce some degree of ambiguity in the precise rules by which a given abstractly defined CFT codifies the bulk geometry. To appreciate the point, let us consider the AdS d +1 global manifold with metric where lengths are measured in units of the AdS radius of curvature. According to the standard AdS/CFT rules [5], we may regard (2.1) as the vacuum state of a dual CFT on the Einstein manifold E d = R × S d -1 with metric Small perturbations of (2.1) quantized on a low-energy effective field theory approximation can be regarded as low-lying excitations of the CFT on (2.2) and can be described by a Hamiltonian picture for all values of the Einstein-frame time variable t ∈ R . Alternatively, we could have started with a different presentation of the AdS d +1 spacetime, with a metric which we denote 'the bubble': made out of a de Sitter foliation of AdS. Taking the ρ → ∞ limit and rescaling by the divergent factor of sinh 2 ( ρ ) we have a different conformal boundary metric: given by the global de Sitter manifold. Thus, we can also regard the version of AdS given by (2.3) as the bulk dual of the CFT on the dS d global manifold with metric (2.4) (cf. [6]). Not surprisingly, the two boundary metrics are conformally related by a Weyl rescaling and a time diffeomorphism: a map that should be unitarily represented in the Hilbert space and operator algebra of the abstractly defined CFT. On the other hand, the 'bubble' version of AdS given by (2.3), with coordinate domains -∞ < τ < ∞ and 0 ≤ ρ < ∞ , only covers a proper subset of the whole global AdS manifold (2.1): while the r -slices generate the whole AdS bulk, the ρ slices only cover the causal diamond subtended by the t ∈ [ -π 2 , π 2 ] interval of E-time and bounded by the null surfaces ρ = 0, τ = ±∞ . This raises the question of how the two CFT descriptions can be unitarily equivalent while one of the bulk duals is strictly contained into the other. It turns out that the two bulk formulations are truly equivalent, in the sense that each one of them contains all the information needed to reconstruct the other [7,8,9]. The key fact making this equivalence possible is the existence of a common initial value surface in both bulk domains. As shown in Figure 2, the Hamiltonian development of the dS-foliated patch shares an initial-value surface with the E-foliation of the global AdS manifold, namely the τ = t = 0 surface. Therefore, any perturbative bulk state defined on an arbitrary t = constant surface may be unitarily 'pulled-back' to the t = 0 initial-value surface, which coincides with the τ = 0 initial surface of the dS time slices. This operation is performed with the evolution operator generated by the t -Hamiltonian, the generator of translations in the foliation by t = constant hyper-surfaces, ˜ H ∼ i∂ t . Once the state is 3 defined at τ = 0, we may 'push-forward' this state to any τ = constant surface in the dS patch, acting with the τ -Hamiltonian H ∼ i∂ τ .", "pages": [ 3, 4, 5 ] }, { "title": "2.1. Extracting UV Data", "content": "The 'pull-back/push-forward' method described here (to follow the terminology of [10]), provides a simple operational definition of 'horizon complementarity' in a very concrete example. As it stands, the construction applies to perturbative states around the vacuum AdS manifold. In seeking generalizations, it is natural to look at the asymptotic (UV) data, whose non-perturbative CFT interpretation is most straightforward. In this vein, we look for the effect on the AdS boundary of the alternative Hamiltonian foliations in the bulk and pick a natural map between τ = constant and t = constant surfaces to represent the complementarity. Directly matching the fixedt and fixedτ surfaces at the AdS boundary (cf. Figure 3) provides such a natural map, determining a particular time-diffeomorphism which we shall denote t τ . We can find its explicit form using the common SO ( d ) symmetry of (2.1) and (2.3) to set d Ω d -1 = 0 in both metrics. Introducing coordinates s = tan -1 ( r ) -π/ 2 and u ± = 1 2 ( t ± s ) we obtain the metric of the AdS 2 section of (2.1): If instead we define v ± = 1 2 ( τ ± η ) with we find a metric for the d Ω d -1 = 0 section of (2.3). By direct inspection, we can check that (2.6) and (2.7) are related by the transformation on their domain of overlap. This includes the AdS boundary, defined by u ± = t/ 2 and v ± = τ/ 2. On this boundary, the diffeomorphism (2.8) reduces to the sought-for time-map: or, equivalently cos( t τ ) = 1 / cosh( τ ). The result is of course consistent with the conformal map between boundary dS d and E d metrics (2.5). Associating unitary evolution operators to the two time foliations we may write for the unitary map between the two states at fixed t or fixed τ respectively. Our particular matching of time foliations, given by the diffeomorphism t τ , allows us to interpret (2.10) as the unitary implementation of the conformal map Ω between E d and dS d , i.e. Notice that U Ω acts on the Hilbert space at a given value of the either time parameter 1 , sending a dS-frame state | Ψ 〉 τ at dS time τ into an E-frame state | ˜ Ψ 〉 t with t = t τ . The singularity of this map at t = ± π/ 2 does not translate into a physical singularity for perturbative states around the AdS vacuum. Those states are perfectly smooth in the E-frame and may be continued for all values of t ∈ R . The crucial issue of whether this smoothness is expected for more general states will be addressed in the next section. The relation (2.10) was motivated by the geometry of the Hamiltonian flows in the AdS geometry, and the evolution operators could be constructed in the low energy theory of the bulk, describing perturbative states around the AdS vacuum manifold. However, the resulting operators are parametrized by time variables that make sense in the exact CFT, so it is natural to promote (2.10) as a non-perturbative definition of the 'complementarity map'.", "pages": [ 5, 6, 7 ] }, { "title": "2.2. (In)Completeness", "content": "The same method can be implemented for the case of the Poincar'e patch, defined for t ' ∈ R and y > 0. The physics on this patch is codified by the Minkowski version of the CFT, i.e. picking a conformal boundary with metric R × R d -1 , and the unitary complementarity map between (2.12) and (2.1) can be constructed as in (2.11), using the common t ' = t = 0 initial value surface. An interesting example where this method does not work in a naive fashion is provided by the hyperbolic foliation of AdS: where the radial coordinate is defined in the domain ¯ r > 1 and the boundary metric is taken to be R × H d -1 , the second factor being a ( d -1)-dimensional hyperboloid. This time, the ¯ t = 0 surface does not cover the whole t = 0 surface of the global manifold. The null surface ¯ r = 1 is a horizon of a particular black hole solution with hyperbolic horizon geometry and Hawking temperature T = 1 / 2 π , which suggests that a situation similar to that of the eternal AdS black hole is at play [11]. Indeed, one can cover the complete initial value surface with the ¯ t = 0 section of two hyperbolic patches of the form (2.13), each one of them dual to the CFT living on R × H d -1 . The global AdS background is dual to the CFT on the S d -1 vacuum, which can be regarded as an entangled state of the Hilbert spaces supported on each hemisphere of S d -1 . Since a ( d -1)-dimensional hemisphere is conformal to H d -1 , let U denote the unitary operator implementing the map on the CFT Hilbert space. The global vacuum can be written then as an entangled state with data on two copies of the hyperbolic CFTs (cf. [12]): where E hyp is an energy eigenvalue of the CFT quantized on the H d -1 spatial manifold 2 (the sum in (2.14) is symbolic, since the spectrum of hyperbolic energies is continuous). Operators on a single copy see the AdS vacuum as a mixed thermal state with the temperature T = 1 / 2 π of the hyperbolic AdS black hole. 3 An important comment regarding (2.14) is that, while the S d -1 vacuum state | VAC S d -1 〉 of the CFT should map smoothly to the global AdS geometry, the same cannot be said of each individual eigenstate of the hyperbolic Hamiltonian | E hyp 〉 . As emphasized in [12], such states are expected to harbor bulk singularities (akin to 'firewalls' [2]) on the horizon of the hyperbolic patch. In the next section we shall add a simple classical argument in favor of this interpretation. Complementarity maps from a left-right symmetric slicing in hyperbolic time ¯ t to some global E-frame slice, t , can be specified by operators of the form In this expression, the first factor pulls the fixed-¯ t state in the product hyperbolic CFT back into the ¯ t = 0 slices, undoes the conformal map back to each left-right hemispheres and finally it pushes the full S d -1 state forward in E-frame time t . Notice, however, that U C defined in (2.15) makes use of the two copies of the CFT on disjoint hyperboloids, and the resulting operator does not have a straightforward interpretation as a unitary representation of a conformal map in the full CFT defined on R × S d -1 . We note that the d = 1 case has interesting peculiarities. The union of the back-toback hyperbolic patches of AdS 1+1 coincides with the bubble patch. Their boundary is dS 1 , consisting of two disconnected lines, each one representing one static dS patch (cf. Figure 4). The map between Poincar'e and global frames also simplifies. We compute here for future use the associated boundary time diffeomorphism. Let the Poincar'e patch of AdS 1+1 be represented by the metric with y ≥ 0 and t ' ∈ R , covering a proper subset of the global AdS 1+1 whose metric we write as with x ∈ R and t ∈ R , the right and left boundaries corresponding to the limits x →±∞ respectively. As indicated in Figure 4, a natural time-diffeomorphism t t ' is induced on the boundary metrics by the matching of time slices at the R boundary x = y = + ∞ . To find this boundary diffeomorphism we begin by transforming (2.16) by the change of variables ζ ± = t ' ± 1 /y , leading to which in turn may be transformed into the global version (2.6) under the further redefinition ζ ± = tan( v ± ). Evaluating the chain of coordinate changes at the R boundary, we find for the required time-diffeomorphism. It is tempting to promote the picture of complementarity maps outlined in this section to conformal maps in CFT 1 , i.e. a model of conformal quantum mechanics which would encode the physics of AdS 1+1 spaces. On the other hand, the d = 1 version of the AdS/CFT correspondence is rich with subtleties (cf. for instance [14,15]) which makes it a rather special case. Despite these caveats, we will find in the coming sections that many aspects of the complementarity maps discussed here do find analogs in the simplest models of conformal quantum mechanics.", "pages": [ 8, 9, 10, 11 ] }, { "title": "3. Singular Maps Versus Singular States", "content": "We have argued that a version of horizon complementarity for perturbative bulk states around the global AdS vacuum can be analyzed in terms of conformal maps between the E d and dS d versions of the dual CFT. This conformal rescaling, which we refer to as the EdS map, sends the whole Hamiltonian development of the dS manifold into a compact domain of Einstein-frame time. We refer to this situation as the 'eternal/apocalyptic duality'. Accordingly, we speak of the 'eternal Hamiltonian', dual to the dS time variable, τ , and the 'apocalyptic Hamiltonian', dual to the E-frame time variable, t . The conformal transformation U Ω is singular at the endpoints of apocalyptic time t = t /star = ± π/ 2, but the physics of perturbative states around AdS is smooth, as the E-frame Hamiltonian acts smoothly on those states for | t | > π/ 2. It is possible to envisage states without such a smooth continuation, for which the apocalyptic time development is truly singular in a physical sense. Let us consider a classical state with the properties of a codimension-one brane, supported on a fixed ρ trajectory in (2.3). Such a state is stationary with respect to the τ -Hamiltonian, but it is accelerating, asymptotic to a null surface, in the E-frame of the global AdS geometry. Therefore it requires an infinite supply of t -energy, and its t -time evolution is not expected to be smooth for ∆ t > π . An example of this behavior is given by a O ( d, 1)-invariant configuration similar to a Coleman-de Luccia (CdL) bubble, which expands exponentially in an ambient AdS space and produces a crunch as in Figure 5. 4 Brane-like states producing crunch singularities are a rather more interesting arena where ideas of complementarity can be probed. Since the whole space-time crunches, they behave in some sense as infinite-entropy limits of black holes -even the boundary of AdS 'crunches' in finite global time. Local observables associated to constantρ trajectories are analogous to 'exterior' black hole observables, whereas local observables associated to constantr trajectories are analogous to 'infalling' observables. Unlike the black hole case, we can identify infalling 'observers' even on the AdS boundary, so that the complementarity map must be visible in the deep UV data of the CFT. It is precisely the conformal transformation between 'eternal' and 'apocalyptic' Hamiltonian flows what provides this 'UV remnant' of the complementarity map, visible in the microscopic formulation of the CFT. In what follows, we study the transformation between eternal and apocalyptic Hamiltonians from various points of view, starting with a Landau-Ginzburg description of the codimension-one brane states.", "pages": [ 11, 12 ] }, { "title": "3.1. Effective Landau-Ginzburg Models", "content": "An approximate description of O ( d, 1)-invariant brane states can be achieved by defining a radial collective coordinate φ which can be regarded as a field degree of freedom in the CFT. Assuming that this world-volume field is weakly coupled, it can be assigned a canonical mass dimension. A brane situated at ρ = ρ M can be expressed by arranging the effective dynamics such that the collective field φ obtains an expectation value where M is the mass scale associated to the fixed radial position ρ = ρ M . According to the IR/UV relation of AdS/CFT we have (cf. [3]) a relation which is valid provided d > 2 and M /greatermuch 1 in units of the dS curvature radius, two conditions that we assume to be valid throughout this section. The simplest effective dynamics supporting such a classical condensate on dS is given by the effective (long wavelength) Landau-Ginzburg (LG) action where the effective potential can be written as The first term and the marginality of the operator appearing in the second term are dictated by conformal invariance 5 , including the conformal curvature coupling with The non-linear terms in (3.4) correspond to a marginal operator of mass dimension d and a relevant operator of dimension ∆ < d , whose coupling introduces the conformal symmetrybreaking scale M . The factor ε = ± 1 controls the sign of the relevant operator, and we must require λ > 0 for global stability. In general, there may be many relevant operators and a host of irrelevant operators correcting (3.3), but the simplified form of (3.4) will suffice for our qualitative discussion. Taking λ = O (1) and M /greatermuch 1 we can find condensates of the form (3.1) provided ε = -1. In fact, we get both a stable condensate and an unstable one, as shown in Figure 6. The unstable condensate was interpreted in [18,3] as a sphaleron configuration which all the properties of a CdL bounce in the bulk. Interestingly, this configuration is present even for the globally unstable model with no relevant operator, M = 0 and λ < 0. Such models were studied extensively in [19,20,21,22,18,3] as holographic duals of crunch singularities. It was recognized in [21,23,3] that the stable condensates in globally well-defined models are perfectly suited to the AdS/CFT embedding of space-times with crunch singularities. The classical LG description of condensate states on dS should be accurate when the scale of the condensate is much larger than the dS temperature, i.e. M /greatermuch 1 in our notation. In the opposite limit, M /lessmuch 1, the effective LG theory should receive large quantum corrections. On the other hand, this is the limit where classical gravity descriptions in the bulk admit a linearized approximation (cf. the appendix of [21]), the result being O ( d, 1)-invariant geometries with very small bubbles and the same crunching behavior as in Figure 4 The conformal complementarity (EdS) map (2.11) becomes particularly simple in the classical approximation to the LG dynamics (3.3). Given the conformal map between the two frames (2.5), an extension to the full effective LG field dynamics is achieved by postulating the conformal transformation of the basic field variable, as dictated by its scaling dimension: ˜ φ ( t ) = Ω( t ) d -2 2 φ ( τ t ) . (3 . 5) This transformation sends the dS-invariant condensate 〈 φ 〉 M ∼ M d -2 2 into the t -dependent E-frame configuration which is a solution of the E-frame system with an effective potential featuring an explicit time-dependent coupling of the relevant operator. This coupling causes the total energy, as well as the kinetic and potential energies of the state ˜ φ ( t ) to blow up at the 'bang-crunch' times t /star = ± π/ 2, showing that the singularities at the 'apocalyptic' times are physical in terms of the E-frame variables. The E-frame Hamiltonian is itself singular at t = t /star , so that the t time evolution cannot possibly continue smoothly beyond the apocalyptic times. We conclude that a particular class of O ( d, 1)-invariant states in dS-frame variables are seen as a singular (crunching) states in the E-frame, as a result of a singular driving term in the E-frame Hamiltonian. By inverting (3.5) we can study how an E-frame t -stationary condensate looks when analyzed in dS-frame variables. Such states have U (1) × O ( d ) symmetry and have the form 〈 ˜ φ 〉 ˜ M ∝ ˜ M d -2 2 (notice that we now take ∂ t ˜ M = 0). This t -static configuration is a solution of the static E-frame potential The corresponding dS-frame field is which vanishes exponentially in global dS time for d > 2. After appropriately normalizing the O (1) proportionality constant in (3.9), this solution is driven by the dS-frame LG 'potential' which now features a negative-definite, τ -dependent relevant operator turning-off as | τ | → ∞ . The value of the LG potential evaluated on the solution (3.9) also redshifts to zero as | τ | → ∞ . We thus conclude that the U (1) × O ( d )-invariant condensates on the E-frame 'dilute away' when analyzed in dS-frame variables. Broadly speaking, we can identify two qualitatively different types of states. One natural class is given by those states which are completely smooth in the E-frame and can be continued through all t ∈ R with a time-independent non-singular E-frame Hamiltonian. We refer to these as smooth states. When analyzed in the eternal frame, their distinctive feature is the 'diluting' nature as | τ | → ∞ . A second class of states is given by those which are asymptotically τ -stationary in the eternal frame, but distinct from the trivial CFT vacuum on dS. The natural way of engineering such states is to deform the CFT by a relevant operator and consider stationary states looking like condensates induced by the new relevant interactions. These states, while completely regular in the eternal dS frame, are singular in the E-frame and therefore called crunch states. It should be clear that the smooth and crunch states do not share the same phase (or Hilbert) space. They actually occur in different systems, in the sense that they need different Hamiltonians to be supported as stationary states. If we fix, say the dS frame, crunch states need a non-trivial dS-invariant relevant deformation to be turned on, while smooth states already exist in dS systems whose Hamiltonian has no such deformation turned on. We have chosen to discuss the conformal map which rises naturally from the diffeomorphisms discussed in section 2. It maps a non compact region into a compact one independent of the dynamics brought about by the specific Hamiltonian involved. This more universal approach required us to disentangle the singularity inherent in such a transformation from a possible dynamical one. One could have chosen a conformal transformation akin to a unitary gauge in gauge theories. It would be ab initio useful when there is a physical singularity to be exposed in one frame, like the unitary gauge is useful in the BEH phase. The transformation will be defined on the fields (cf. equation (3.5)) in such a way that Ω is multiplied by the product of the expectation value of the scalar field φ in the dS frame and the Hubble scale. This product vanishes in the cases when there is no condensate and thus renders the transformation to be ill defined in those cases.", "pages": [ 12, 13, 14, 15, 16 ] }, { "title": "3.2. Classical Firewalls", "content": "It is interesting to inquire to what extent this description generalizes to other versions of the conformal frame duality studied here, such as the examples of AdS foliations related to CFTs on flat or hyperbolic space-times. Let K k represent the standard constant-curvature manifold in d -1 dimensions, with k = 0 , ± 1 controlling the sign of the Ricci curvature, i.e. K 0 = R d -1 , K 1 = S d -1 and K -1 = H d -1 . We can describe the global, Poincar'e and hyperbolic patches of AdS at once with the family of metrics: The k = 1 case with r 1 ≥ 0 is the standard metric of the global AdS manifold (2.1). The case k = 0 with r 0 ≥ 0 gives the Poincar'e patch (2.12) of AdS, and finally k = -1 with | r -1 | ≥ 1 returns the two mirror hyperbolic patches given by (2.13). The time variables t k in (3.11) define natural Hamiltonian flows for CFTs on R × K k . In the notation of the previous section, we have t = t 1 , t ' = t 0 and ¯ t = t -1 . It is interesting to inquire about the fate of condensate states with the symmetries of R × K k , corresponding to brane-like states defined by r k = constant in (3.11). In particular, one can consider condensates on R × R d -1 with Poincar'e invariance ISO ( d -1 , 1) and condensates on R × H d -1 with symmetry U (1) × O ( d -2 , 1). 6 The crucial property making the k = 0 and k = -1 cases special is the non-compact nature of the spatial section K k . Unlike the EdS map studied so far, this implies that the conformal transformation to the E-frame: has singularities even on the t = 0 spatial section, at those points on S d -1 where the infinite boundary of K k is mapped. In particular, a maximally symmetric condensate on R × K k of the form is mapped to an E-frame field with nontrivial space-time profile, and sharing the singularities of the Weyl function Ω( x ). The configuration (3.14) solves the E-frame effective equations of motion with a relevant perturbation The physical interpretation in the E-frame is that of an inhomogeneous injection of energy with sharp divergences at the singularities of the conformal map. This happens for k = 0 at a single point on S d -1 , whereas the infinite injection of energy occurs in the k = -1 case along the complete equatorial S d -2 which separates S d -1 into two hemispheres. It follows that singularities of 'firewall' type are expected in the global bulk description of such states, in agreement with the philosophy expressed in [24]. The price we pay for the ability to use a classical set up is the need to realize the state in a CFT perturbed by a large relevant operator, but the take-away message ends up being the same. The behavior of homogeneous condensate states described in this section should admit a natural extension for small perturbations around these states, such as finite-particle excitations. On the other hand, it would be interesting to generalize the present purely classical description to the full quantum theory. The presence of strongly time-dependent couplings makes the problem challenging. Fortunately, a number of structural properties of the complementarity maps can be studied in a simplified quantum mechanical model, where time-dependent couplings can be studied at considerable ease.", "pages": [ 17, 18 ] }, { "title": "4. Conformal Quantum Mechanics", "content": "In order exhibit these ideas in an explicit quantum framework we can study the quantum mechanical version of the Landau-Ginzburg models associated to conformal complementarity maps. A natural construction arises as the d → 1 limit of the above, in which we replace the classical d -dimensional conformal dynamics of the LG collective degree of freedom by its d = 1 analog. It turns out that this simple procedure is somewhat nontrivial, since the different frames will be found to retain some characteristic features in the d = 1 system. The basic building block is given by the de Alfaro-Fubini-Furlan (AFF) Conformal Quantum Mechanics (CQM) with Hamiltonian [4] for one LG-type degree of freedom φ with canonical momentum π . The conformal group acts on the Hilbert space of this theory as an SL (2, R ) algebra generated by the Hamiltonian (4.1), the dilatation operator D = 1 2 { φ, π } and the special-conformal generator C = 1 2 φ 2 , with commutation relations which follow from the basic canonical Heisenberg algebra [ φ, π ] = i . The AFF Hamiltonian is classically bounded-below for repulsive potentials with λ > 0. Even when the potential becomes attractive, it remains well defined at the quantum level as long as λ > -1 / 4. The spectrum is still well defined for λ > -1 / 4 when the system is quantized on L 2 ( R + ), i.e. on wave-functions Ψ[ φ ] with inner product and vanishing boundary condition at the origin, lim φ → 0 Ψ[ φ ] = 0. More specifically, the Hamiltonian has a positive-definite continuous spectrum 7 with delta-function normalization for -1 / 4 < λ . A discrete spectrum can be obtained by placing the system on a 'harmonic trap', i.e. by adding a harmonic potential term with some frequency ω , where C = 1 2 φ 2 is the generator of special conformal transformations. The main advantage of this IR regularization is the preservation of a nice SL (2 , R ) action on the spectrum, since the Hamiltonian is still linear in the SL (2 , R ) generators. This leads in particular to an equally-spaced discrete spectrum for the trapped Hamiltonian H ω . The trapped models are analogous to the higher-dimensional conformal field theories defined on spheres, with a gapped spectrum, i.e. the model referred above as the E-frame CFT. The analogy can be sharpened by doing 'dimensional reduction', namely taking the d → 1 limit of (3.4). The conformal mass term does survive this limit. The curvature's vanishing is compensated by the behavior of the conformal coupling ξ d , the product leading to a finite result. Explicitly, one finds for the LG model on X k = R × K k and for the LG model on dS d : It is interesting that we get the same tachyonic 'anti-trapping' frequency for the d → 1 limits of the hyperbolic and dS theories. This result is natural given the interpretation of the LG models as world-volume descriptions of codimension-one branes on AdS, since we have seen in section 2 that hyperbolic and 'bubble' patches of AdS are identical for d = 1. The complete LG action can be derived following the logic of [25]. We can drop a particle probe of mass m in AdS 1+1 and analyze its near-boundary, slow-motion dynamics in each of the relevant patches: in the notation of (3.11). The particle action reads and takes the form of a CQM system with parameters ω 2 k = k/ 4 and λ = 2 m 2 : in the limit r k /greatermuch 1 and | dr k /dt k | /lessmuch 1, where we have used the field redefinition 8 Although the probe-brane derivation is very transparent, its logical relation to a welldefined AdS 2 /CFT 1 duality is still far from clear. The asymptotic boundary conditions in AdS 2 are very sensitive to back-reaction from any finite-energy perturbation [14] and the most likely interpretation of the AdS 2 /CFT 1 correspondence involves a large Hilbert space with exactly degenerate states on the CFT side [15], whose precise relation to AFF-like models is an open problem. We shall not deal with such subtleties in this paper, our aim being more modest. Namely we use the AFF model as a quantum arena to study the eternal/apocalyptic map, while at the same time offering a tentative bulk interpretation of the results.", "pages": [ 18, 19, 20, 21 ] }, { "title": "4.1. Deformations And Bound States", "content": "We may thus consider three different versions of the CQM model. The standard AFF model with ω 2 = 0 (no trapping) will be regarded as the analog of the M-frame, whereas the model with positive trapping ω 2 = 1 / 4 corresponds to the E-frame. Finally, the model with tachyonic anti-trapping ω 2 = -1 / 4 will be interpreted as the dS-frame (or equivalently the hyperbolic frame). More generally, we can deform the AFF model (either trapped or untrapped) by adding a relevant operator contributing to the potential energy as with ∆ < 1 (the trapping harmonic term being the particular case ∆ = -1). We notice that positive relevant deformations with ε > 0 and ∆ < 0 behave qualitatively like the trapping term (4.2), in the sense that they remove all the largeφ 'scattering states' near zero energy. Hence, we interpret the models with such a strongly relevant deformation as completely gapped. For ∆ = -1 we have the strict harmonic trapping, analogous to the E-frame CFT. For ∆ < -1 they present a steeper wall, mimicking a confining potential with a gap proportional to M as M /greatermuch 1. Since the complete largeφ region is removed from the spectrum, we suggest to interpret such 'confining' models as analogous to a sharp wall where AdS 2 is terminated, as in a 'bubble of nothing' [26,27]. On the other hand, relevant deformations in the window 0 < ∆ < 1 are very mild at large values of φ , preserving the continuum of largeφ scattering states near zero energy. Therefore, we interpret these deformations as leaving behind a sort of 'IR CFT fixed point', such as the effective field theory describing the IR behavior of a system where spontaneous symmetry breaking has occurred. In particular, for ε < 0 and large M we find localized classical ground states at 〈 φ 〉 ∼ 1 / √ M which we may identify as 'condensate' states (cf. figure 7). Such states are analogous to codimension-one brane states propagating in AdS 2 . The ∆ = 1 case is the marginal deformation. Interestingly, a negative ε = -1 deformation does not automatically imply a global instability of the model, reminiscent of the CdL solutions discussed in the classical models above. The reason is the improved quantum stability 9 which sets the critical value of the effective coupling at λ critical = -1 / 4, a phenomenon analogous to the limited tolerance of tachyons in AdS [28]. The AFF model admits exact solutions for the condensate states for the particular case of a ∆ = 1 / 2 deformation, since the resulting induced potential (4.7) is a standard Coulomb interaction. It follows that a spectrum of bound states can be constructed as the radial Hydrogen wave functions continued to real values of the angular momentum, i.e. as (hypergeometric) solutions of with discrete spectrum of energies The notion of condensate states is inherently semiclassical in the particular case of the dS-frame Hamiltonian, which we write here explicitly, perturbed by a negative ∆ = -1 operator. The condensate state induced by the last term is necessarily metastable (cf. Figure 8). If this metastable well is deepened by going to large M , the decay probability to the quadratic runaway region is of order exp( -a M 2 / 3 ) for some constant a . The bulk interpretation is that of a probe particle which can tunnel out of an accelerating fixedρ trajectory, into the low radius region of AdS 2 . Any such probe that tunnels back to the interior of AdS fails to reach the boundary with infinite Eframe energy, and thus the crunch is prevented. We will return to this intriguing question in section 6.", "pages": [ 21, 22, 23 ] }, { "title": "5. The CQM Complementarity Map", "content": "We now study the 'conformal complementarity' in the CQM model. For d = 1, it reduces to the conformal transformation induced by the time-diffeomorphism acting as a map between 'eternal' time evolution, τ ∈ R , and 'apocalyptic' time evolution, t ∈ [ -t /star , t /star ]. At the classical level we seek the appropriate Weyl function Ω( t ) which maps the E-frame version of CQM: with ˜ ω 2 = 1 / 4, into the two canonical models of 'eternal' type: where ω 2 = 0 for the M-frame CQM and ω 2 = -1 / 4 for the dS-frame CQM (we use the same time variable for both eternal models for simplicity of notation). The answer is obtained by direct substitution of the conformal rescaling φ ( τ ) = ˜ φ ( t ) √ Ω( t ) into the actions. We find the required behavior up to a boundary term: where 10 provided the Weyl function satisfies a relation that we may interpret as an 'anomalous' transformation law for the frequencies. It is useful to define ˜ for future use, as a measure of such anomalous scaling behavior. In this notation, (5.5) reads Plugging into (5.5) the actual values of the frequencies, we find two solutions of the non-linear differential equation which, not surprisingly, exactly match the time diffeomorphisms (2.9) and (2.19) found in the context of purely geometrical considerations in AdS 1+1 . We have the EM map between the E-frame and M-frame systems, i.e. between the trapped and ordinary AFF models: The second solution is the standard EdS map, between the trapped and tachyonic versions of the AFF model: A useful parametrisation of the two Weyl functions at once is where α = 1 for the EdS map and α = 2 for the EM map. Notice that the singularities of Ω( t ) occur at t = ± t /star with t /star = απ/ 2, in agreement with the geometrical features of AdS 1+1 Penrose diagrams, showing that the Minkowski patch covers a larger portion of the AdS boundary as compared to the dS (hyperbolic) patch (cf. Figure 4). A relevant operator deformation of the form in the eternal frame transforms into an analogous term in the apocalyptic frame, where the mass parameters are related by Notice that the map between (5.11) and (5.12) works also for time-dependent mass parameters, and (5.13) implies that either M or ˜ M must be time-dependent in one of two frames.", "pages": [ 23, 24, 25 ] }, { "title": "5.1. Quantum Map", "content": "The field redefinition between the eternal and apocalyptic frames is generalized to a full quantum map by a correspondence between wave functions given by the explicit transformations and its inverse In these expressions, τ ( t ) and its inverse give the appropriate time diffeomorphism transforming the eternal and apocalyptic frames. The first factor in (5.14) and (5.15) is a Jacobian accounting for the correct normalization of both wave functions and the phase is the result of the boundary term in time (5.3). It can be checked explicitly that this map sends solutions of the apocalyptic Schrodinger equation into solutions of the eternal Schrodinger equation and viceversa, where the two dual Hamiltonians are defined as with The quantum map (5.14) and (5.15) is formally a canonical transformation a change of variables that holds in quantum averages up to some anomalous terms coming form the boundary terms. We have the basic rules which allow us to formulate the general map of observables for general polynomial functions of the canonical operators. In average values defined by we have ˜ ˜ These two equations can be used to extract information about the behavior of any observable. A consequence of the quantum map defined above is the complementarity of time evolutions in the respective 'eternal' and 'apocalyptic' frames. In particular, we can explicitly check the non-commutativity of the time evolution operators, by directly showing that the respective Hamiltonians fail to commute, even at the initial surface where Ω = 1. We can compute [ H, ˜ H ] by expressing, say ˜ H in eternal-frame variables as Hence we have the identity from which we find the commutator of the Hamiltonian operators in terms of the function A defined in (5.6) and (5.7), and the dilation operator D = 1 2 { φ, π } = 1 2 { ˜ φ, ˜ π } = ˜ D . (5 . 25) Since A /negationslash = 0 even for Ω = 1, the two Hamiltonians do not commute in general. Note that they do commute on those states which are annihilated by the generator of the scale transformations. This interpretation makes contact with the fact that the vacuum AdS manifold realizes the boundary conformal group as an isometry group, showing that the bulk geometry is actually codifying the quantum state of the dual CFT. It is interesting to notice that the scale-invariant wave function, satisfying D Ψ 0 = 0, is given by Ψ 0 ( φ ) = φ -1 / 2 and is not normalizable, i.e. it does not sit in the Hilbert space of bound states. Still, its norm is less divergent than that of a plane wave.", "pages": [ 26, 27, 28 ] }, { "title": "5.2. States And Observables", "content": "The quantum map (5.14) and its inverse (5.15) are completely general, valid for any state with arbitrary wave-function. Both versions of the complementarity map have a Weyl function which smoothly tends to the identity near the origin of times, Ω = 1 for t = τ = 0, and blows up at the 'ends of time', with a pole-like behavior Ω( t ) ∼ ( t -t /star ) -α near t = ± t /star = ± απ/ 2. In order to further fix the intuition about the meanings of the quantum map, we can consider a smooth τ -static wave function in the eternal quantum mechanics, with width Γ and centered around φ 0 . Its dual to the apocalyptic frame has a narrowing width ˜ Γ( t ) = Γ / √ Ω( t ) as t →± απ/ 2, with its center migrating to the origin as ˜ φ 0 ( t ) = φ 0 / √ Ω( t ), while at the same time the phase oscillates wildly. Therefore, the ˜ Ψ wave function is infinitely squeezed into the UV region (small φ ) as we approach the 'apocalypse'. ˜ To be more precise, let us focus on the operator map (5.22) for the particular cases of interest. We shall adopt a terminology rooted in the behavior of the state in the apocalyptic frame (E-frame) as diagnosed by the average values of polynomials in the canonical operators or natural observables such as the kinetic, potential and total energy. States with smooth E-frame behavior at t = ± t /star can be continued beyond the 'apocalyptic' times ± t /star and will be termed smooth ( S ), while those with divergent matrix elements will be denoted as singular . Among the singular states, we shall refer to crunches ( C ) when the E-frame potential energy plummets to minus infinity: ˜ Conversely, starting with t -static wave function with fixed width ˜ Γ and centered at ˜ φ 0 in the E-frame system, it corresponds to an eternal wave function slipping into the deep IR (large φ ), trailing the peak at φ 0 ( τ ) = ˜ φ 0 √ Ω( τ ), and widening at a rate of order Γ( τ ) = Γ √ Ω( τ ). Singular states with the opposite-sign divergence ˜ will be called bubbles ( B ) as analogs of 'bubbles of nothing'. The most interesting among singular states are those that look stationary in the eternal frame, i.e. with a finite | τ | → ∞ limit of 〈 V ( φ ) 〉 , such as the 'condensate states' considered in section 5. Any such state with a non-zero value of the eternal potential energy has an apocalyptic potential energy diverging as Ω( t ) ∼ ( t -t /star ) -α , the hallmark of a singular state. The anomalous transformation terms do affect the scaling of the kinetic energy: ˜ For a condensate-type state in the eternal frame, the three terms in this equation scale as ( t -t /star ) -α , ( t -t /star ) -1 and ( t -t /star ) α -2 respectively. For either the EM or the EdS map, there is always a singular term for generic values of the eternal frame averages, confirming that the eternally stationary state is a singular state in the apocalyptic frame. The anomalous terms (second and third on the right hand side of (5.26)) are subdominant for the EM model ( α = 2), but have the same scaling as the first term in the EdS case ( α = 1). 11 Starting with a smooth state in the E-frame, with finite and generic values of apocalyptic observables at t = t /star , the corresponding large-time behavior in the eternal frame follows from the inverse transformations. The potential energy vanishes as 〈 V ( φ ) 〉 ∼ Ω -1 → 0 as | τ | → ∞ for any value of α (recall this potential energy excludes the purely quadratic trapping term, to be considered below). We say that the state dilutes away in the eternal frame. The inverse of the (5.26) relation ˜ ˜ ˜ implies a similar diluting behavior for the ordinary scaling term of the kinetic energy. The anomalous terms depending on derivatives of Ω have a potentially interesting behavior, since they scale as ( t -t /star ) α -1 and ( t -t /star ) α -2 respectively. Hence, for α = 1 (EdS map) we do get a divergent contribution to the kinetic energy in the | τ | → ∞ limit. We can understand this behavior by recalling that the EdS map relates the trapped CQM in the E-frame to the anti-trapped (i.e. tachyonic) CQM in the dS-frame. This is characteristic of the d = 1 case and implies that smooth states look as falling down a harmonic cliff in the eternal frame. The potential energy coming from the trapping also diverges as ˜ canceling the kinetic-energy infinity coming from the last term in (5.27). Hence, the total energy does stay finite in the eternal runaway state. Even for α = 2, i.e. the EM map, the anomalous terms produce an interesting behavior, since the last one yields an asymptotically constant value of the potential energy. This is to be understood as a state running away to large values of φ , with asymptotically constant kinetic energy, i.e. a standard scattering state in the AFF quantum mechanical model. We conclude that the quantum complementarity map sends smooth E-frame states (which do not look 'apocalyptic' in this frame) into states which run away towards large φ values in the eternal frame, with finite total energy but diverging kinetic and potential components in the particular case of the EdS map. Starting with a stationary state in the eternal frame, modeled as a 'condensate' in the terminology of section 5, the apocalyptic description carried by the ˜ φ, ˜ π operator algebra, sees it as a singular state. We say it is a crunch when the divergent potential energy is negative, and a bubble of nothing when it diverges to positive infinity.", "pages": [ 28, 29, 30 ] }, { "title": "5.3. Evanescent Crunches?", "content": "As previously indicated, the EdS map has peculiar properties related to the tachyonic character of the dS-frame Hamiltonian. Recall that the EdS map sends the standard trapped CQM (E-frame system) with Hamiltonian into the tachyonic CQM (dS-frame system) with Hamiltonian The largeφ instability of the eternal-frame CQM implies that the standard negative deformations of the dS Hamiltonian with constant M and 0 < ∆ < 1, fail to induce a absolutely stable condensate at 〈 φ 〉 ∼ M -1 / 2 with large M . In fact, the resulting states are only metastable, with a decay width of order Γ ∼ M exp( -aM 2 / 3 ) for some O (1) constant a . Since the eternality of the condensate is related to the crunchy character of the state in the apocalyptic frame, it is interesting to inquire whether this metastability, inducing a finite life-time for the condensate, is capable of regularizing the crunch singularity. We can approximate the very-large τ wave-function of such states as where Ψ cond is a normalized state which solves the Schrodinger equation in the large M limit and represents the condensate in the absence of the tachyonic instability, and Ψ run is a state representing the runaway down the inverted harmonic potential after tunneling through the barrier. We can define this running state as the eternal dual from some generic finite-energy state in the E-frame system. Upon transforming this wave function to the E-frame system, the ˜ Ψ cond component has the characteristic crunchy behavior we mentioned above, whereas ˜ Ψ run is a smooth state in the apocalyptic frame. The amplitude of the crunchy component does vanish in the t → π/ 2 limit as Since we have seen characteristic observables to diverge as inverse powers of t -t /star , the contribution of ˜ Ψ cond to expectation values is of order where b is some positive constant of O (1) whose detailed value depends on the particular observable being evaluated. In the semiclassical limit where this description of the tunneling is valid, we have Γ ∼ M exp( -a M 2 / 3 ) /lessmuch 1, so that Γ /lessmuch b and the quantum depletion of the condensate is not fast enough to turn off the crunchy behavior of the state. On the other hand, if the depletion rate should become of O (1), the exponent in (5.31) could change sign and the corresponding expectation value be smoothed out. We see that the potential for a quantum-mechanical smoothing of the crunch exists, by considering 'condensates' with sufficiently fast decay rate. Formally, this situation can be engineered by tuning M /lessmuch 1 in units of the background curvature. While the notion of 'condensate' is not well defined in such a limit, it is worth mentioning that such states do exist in the dual AdS description (cf. the model discussed in the appendix of [21]) and they have the same qualitative behavior as the more obvious crunch states described here. 12", "pages": [ 30, 31, 32 ] }, { "title": "6. Generalized Duality Between Eternity and Apocalypse", "content": "The description of conformal complementarity maps as quantum canonical transformations in CQM can be formally extended to higher-dimensional field theories. Consider two conformally related d -dimensional Riemannian manifolds X and ˜ X with Weyl rescaling function Ω( x ). Let us define a LG model on X with classical action The relevant perturbations depend on mass scales M i . This model can be rewritten as a perturbed LG model on ˜ X with action ˜ plus some boundary terms. The new potential reads in terms of rescaled point-dependent mass scales ˜ M i = Ω M i which now become 'sourceterms', and with the basic field redefinition ˜ φ = Ω d -2 2 φ . The boundary terms are defined as where /epsilon1 = -1 for a space-like boundary component and /epsilon1 = +1 for a time-like boundary component and ∇ is the covariant derivative on X . ˜ ˜ For the particular case of a Weyl rescaling function which is only dependent on time, and a compact spatial section K , we can regard the X manifold as a cosmology while ˜ X is a static cylinder with base K : ˜ If Ω maps a finite interval t ∈ [ -t /star , t /star ] into the real line τ ∈ R we can use the terminology that has become standard along this paper and regard X as the 'eternal frame' and ˜ X as the 'apocalyptic frame'. Then, we only have space-like boundaries at t = ± t /star , so that ∆ ˜ K = ˜ K ( t /star ) -˜ K ( -t /star ), with The boundary term (6.4) plays no significant role at the classical level, but does feature in the quantum treatment. The formal construction of the states and the canonical map parallels the previous formalism explained in section 5.2, except that Schrodinger-picture wavefunctionals replace wave-functions and canonical momenta are defined in terms of functional derivatives in the usual formal fashion, π ( x ) = -iδ/δφ ( x ) and ˜ π ( x ) = -iδ/δ ˜ φ ( x ). This leads to analogous quantum maps generalizing (5.22): The Hamiltonian complementarity presented in (5.24) generalizes as well. The eternalframe Hamiltonian can be written in the form and its apocalyptic counterpart: Explicit calculation then shows that where ˜ D = 1 2 ∫ K { ˜ φ, ˜ π } and the anomalous term generalizes to 13 Although these relations are derived by simple canonical manipulations, we expect them to hold in all generality for general field theories, showing that the Hamiltonian complementarity induced by conformal mappings trivializes when acting on scale-invariant states, annihilated by the dilation operator. It is natural to interpret this result as underlying the fact that the global AdS vacuum is invariant under the action of the dilation operator, represented in the bulk as an isometry. On the other hand, for any state with an intrinsic scale, we expect a non-trivial quantum complementarity between the two Hamiltonian evolutions.", "pages": [ 32, 33, 34 ] }, { "title": "7. Discussion", "content": "In this paper we have analyzed aspects of the general idea, going back at least to [7,9], that complementarity maps can be realized as conformal transformations (or more general field-redefinitions) in holographic models. A particularly simple example of this program was proposed in [3], in terms of condensate states on perturbed CFTs defined on dS space-time. A conformal map to the same CFT on the Einstein universe (E d ), but now perturbed by a time-dependent coupling, serves as the 'infalling' frame in the sense of horizon complementarity. We have singled out the change of time variables, from an eternal history in dS d , to a finite or 'apocalyptic' one in E d , as an 'UV remnant' of the complementarity map, which can be studied using conventional Lagrangian methods. We have done so at the level of classical Landau-Ginzburg models of brane-like states, extending the analysis already presented in [3]. A full quantum analysis is possible for the d = 1 version, conformal quantum mechanics, which retains some qualitative properties akin to a AdS 2 dual, despite the fact that no actual reconstruction of bulk dynamics is available. In this case, the quantum map is a canonical transformation which rescales the canonical operator basis by a time-dependent factor. Once identified, the construction can be formally extended to higher dimensions. The existence of two operator algebras: {O} for the 'exterior' observables and { ˜ O} for the 'infalling' observables, is similar to the case of electric/magnetic duality or T-duality in string theory. In order to sharpen the analogy, we must enrich the models by allowing the coexistence of both 'windings' and 'momenta'. So far we have considered extremely simple states by way of example, such as either dS condensates or E-frame stationaries. In order to realize the quantum complementarity map in a more physical fashion we must introduce an appropriate 'measuring apparatus' for each operator algebra. Consider for instance a dS condensate state. Any physical system constructed from stationary states around the condensate ground state will only measure the eternal properties of the dS state. On the other hand, a physical system whose physical size is 'comoving' with the Hubble expansion, will be able to 'measure' a crunch in the t time variable. It is tempting to us to use our own universe as an example (in the limit G N → 0 with fixed Hubble constant). The Higgs condensate has fixed size in units of the Hubble constant, but a comoving 'observer', anchored on the 'realm of the nebulae', will measure the { ˜ O} operator algebra. At large τ -times such an observer is necessarily made of 'neurons' separated by super-horizon distances, so that its workings appear completely non-local to an observer furnished with the {O} operator algebra. Conversely, in its own frame the { ˜ O} observer will see the {O} observer as a shrinking entity whose own Hamiltonian ramps up the eigenfrequencies to produce the illusion of eternity in the face of an impending crunch. The main limitation of these considerations is the absence of an actual reconstruction of operators with approximate bulk locality, in the spirit of [7,29,30,31,32]. In this sense, we have strived to characterize horizon complementarity in the absence of an actual 'horizon', using only deep UV data. The dichotomy between local and strongly non-local observables in the CFT should then become even more drastic when translated to reconstructed bulk operators.", "pages": [ 34, 35 ] }, { "title": "Acknowledgements:", "content": "The work J.L.F. Barbon was partially supported by MEC and FEDER under a grants FPA2009-07908 and FPA2012-32828, the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042), Comunidad Aut'onoma de Madrid under grant HEPHACOS S2009/ESP-1473 and the spanish MINECO Centro de Excelencia Severo Ochoa Program under grant SEV-2012-0249. The work of E. Rabinovici is partially supported by the American-Israeli Bi-National Science Foundation, the Israel Science Foundation Center of Excellence and the I Core Program of the Planning and Budgeting Committee and The Israel Science Foundation 'The Quantum Universe'.", "pages": [ 35, 36 ] }, { "title": "References", "content": "0709 , 089 (2007) [arXiv:hep-th/0703220]. J. A. Hutasoit, S. P. Kumar and J. Rafferty, 'Real time response on dS 3 : the Topological AdS Black Hole and the Bubble,' JHEP 0904 , 063 (2009) [arXiv:0902.1658 [hep-th]].", "pages": [ 38 ] } ]

[ { "title": "Chemical potentials in three-dimensional higher spin anti-de Sitter gravity", "content": "Marc Henneaux 1 , 2 , Alfredo Pérez 2 , David Tempo 2 , Ricardo Troncoso 2 , 3 ∗ 1 Physique théorique et mathématique and International Solvay Institutes, Université Libre de Bruxelles, Campus Plaine C.P.231, B-1050 Bruxelles, Belgium. 2 Centro de Estudios Científicos (CECs), Casilla 1469, Valdivia, Chile and 3 Universidad Andrés Bello, Av. República 440, Santiago, Chile.", "pages": [ 1 ] }, { "title": "Abstract", "content": "We indicate how to introduce chemical potentials for higher spin charges in higher spin anti-de Sitter gravity in a manner that manifestly preserves the original asymptotic W -symmetry. This is done by switching on a non-vanishing component of the connection along the temporal (thermal) circles. We first recall the procedure in the pure gravity case (no higher spin) where the only 'chemical potentials' are the temperature and the chemical potential associated with the angular momentum. We then generalize to the higher spin case. We find that there is no tension with the W N or W ∞ asymptotic algebra, which is obviously unchanged by the introduction of the chemical potentials. Our argument is not perturbative in the chemical potentials.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Higher spin gauge theories in 3 spacetime dimensions [1-4], which provide a useful laboratory for understanding higher spin gauge theories in 4 and higher dimensions [5-7], have attracted recently a considerable amount of interest. One reason for this surge of activity is the rich asymptotic structure displayed by the theory at infinity, where the W -algebras, or their supersymmetric extensions in the graded case, emerge as asymptotic symmetry algebras [8-10]. This opens the door to an investigation of holography with the powerful tools of two-dimensional conformal field theory and representation theory of W -algebras [11-14]. As it is by now well known, anti-de Sitter gravity in 3 dimensions is described by a sl (2 , R ) ⊕ sl (2 , R ) gauge theory [15, 16]. It was recognized in [17] that the conditions expressing that the gravitational field approaches at infinity the anti-de Sitter solution give, in the Chern-Simons formulation, the conditions implementing the familiar Hamiltonian reduction of the sl (2 , R ) -current algebra to the Virasoro algebra [18-21]. This yields the gauge theory derivation of the asymptotic Virasoro algebra and central charge first obtained in [22] in the metric formulation. The remarkable fact that the geometrical anti-de Sitter boundary conditions implement the algebraic Hamiltonian reduction remains valid for simple and extended supergravities [23, 24] and for higher spin gauge theories [8-10]. Recently, exact black hole solutions supporting a non trivial higher spin field have been obtained [25-27] (see also [28]). However, in spite of the simplicity of these black hole solutions, a suitable characterization of their global charges and their entropy is a subject which is not free of controversy, because there are tensions between various approaches, which give different results (see [29] for a lucid discussion). We show in this paper that this tension is somewhat artificial because it results from a non standard incorporation of the chemical potentials that obscures the asymptotics and hence the correct definition of the charges. Once the chemical potentials are properly introduced along the lines indicated below, there is no difficulty with the asymptotics. Our approach uses the Hamiltonian formalism, which provides a particularly transparent analysis. It is not perturbative. Our paper is organized as follows. In the next section, we recall how the chemical potentials are introduced in the metric formulation of pure gravity in three dimensions and then translate the results in Chern-Simons terms. We find that the chemical potentials ap- pear through the temporal components of the connection (along the thermal circles). This is in perfect agreement with experience from four-dimensional gravity where the chemical potential for the electric charge is well known to be associated with the zeroth component of the electromagnetic vector potential in the Reissner-Nordström solution. In Section III we extend the analysis to include higher spin charges and their chemical potentials. The approach makes it obvious that non-vanishing chemical potentials do not change the asymptotical properties because these potentials enter only the Lagrange multipliers. Finally, we give comments and conclusions in Section IV. We display a black hole solution that fulfills our conditions. In a subsequent paper [30, 31], we shall further discuss the asymptotics and the thermodynamics of this solution. It should be stressed that our method agrees with the discussion of [32, 33].", "pages": [ 2, 3 ] }, { "title": "II. CHEMICAL POTENTIALS IN THE CHERN-SIMONS FORMULATION OF PURE ANTI-DE SITTER GRAVITY", "content": "To begin with, we start with a discussion on how the temperature and the chemical potential for the angular momentum enter in the sl (2 , R ) ⊕ sl (2 , R ) formulation of gravity. This simpler case illuminates the central points. Similar considerations may be found in [34].", "pages": [ 3 ] }, { "title": "A. Metric formulation", "content": "In the usual formulation of black hole thermodynamics, the temperature and the chemical potential for the angular momentum do not enter the metric of the black hole explicitly. They appear indirectly through the identifications involving the imaginary time and the angle, which must be made to avoid a singularity at the horizon in the Euclidean section. This means that the range of the coordinates is not fixed but varies from one solution to another. It is useful to have a description in which the range of the coordinates is fixed once and for all. This can be achieved by redefining the time coordinates t → λt ' and θ → θ ' = θ + ωt , where λ and ω are chosen such that t ' and θ ' have a constant range. This induces a non trivial lapse and shift in the three-dimensional black hole solution [35, 36], which reads (dropping primes on coordinates), with If one chooses the coordinates t and θ such that N ∞ = 1 , N θ ∞ = 0 , then the ranges of the identifications in t and θ depend on the solution. If one wants fixed ranges, one must therefore allow for N ∞ and N θ ∞ to vary. We impose that on the Euclidean section t ∼ t +2 πl and θ ∼ θ +2 π (always). The variables N ∞ and N θ ∞ are clearly related to the temperature and the chemical potential for the angular momentum and will for this reason be called 'the chemical potentials'. [We use quotation marks here because the temperature stands on a special footing but nevertherless it is convenient in what follows to include it among the standard chemical potentials.] We shall from now on deal with the grand canonical ensemble, where the chemical potentials are held fixed to arbitrary values. The appropriate variational principle has then N ∞ and N θ ∞ fixed. One finds the value of the conjugate variables, namely the mass M and the angular momentum J on-shell, by requiring the absence of singularity in the Euclidean section at the horizon, which imposes in particular that N θ = -J 2 r 2 N ∞ + N θ ∞ should vanish at the horizon. /negationslash /negationslash When N ∞ = 1 and N θ ∞ = 0 , the metric does not fulfill at infinity the boundary conditions of [22], which, from the present perspective, would correspond to fixed β = 1 2 πl and zero chemical potential for the angular momentum. However, it is very easy to translate these boundary conditions to generic values of the chemical potentials, just like it is very easy to translate the asymptotic flat boundary conditions written in cartesian coordinates to spherical coordinates through the appropriate coordinate transformation. The asymptotic symmetry is of course the same. When the chemical potentials are introduced, one should not talk about a relaxation of the boundary conditions, but rather of a (straightforward in this case) extension of the formalism to cover different values of the (held fixed) chemical potentials. The only case where the metric is not asymptotically AdS is when N ∞ = 0 , which corresponds to the infinite temperature limit and to a degenerate metric ( det g = 0 ). We shall not consider this case in this paper.", "pages": [ 3, 4 ] }, { "title": "B. Connection formulation", "content": "How do the chemical potentials enter the Chern-Simons connection? We claim that they appear as additional contributions to the thermal circles around the horizon ( dt contributions to the connection), explicitly (after the r -dependent gauge transformation of [17] has been performed to eliminate the r -dependence to leading order): (asymptotically) where ν ± are constants and called the chemical potentials of the ChernSimons formulation 1 . Indeed, with constant L ± 's, the metric corresponding to (3) is (1) with and", "pages": [ 5 ] }, { "title": "C. Asymptotic Analysis", "content": "We now show that the introduction of the chemical potentials does not modify the Virasoro asymptotics. This is in fact direct, and physically mandatory, but we provide an explicit argument since some confusion arose in the spin-3 case. The discussion is most transparent in the Hamiltonian formalism. On a slice t = const, say the initial slice t = 0 , the connection is asymptotically given by The gauge transformations that preserve this form of the connection are asymptotically parametrized by a gauge parameter that takes the form where ε ± are at this stage arbitrary functions of θ and also of the slice under consideration, i.e., t , since one can make independent gauge transformations that preserve (7) on each slice. Here, prime denotes the derivative with respect to θ . The motion from one slice to the next is a gauge transformation parametrized by the Lagrange multiplier a ± 0 associated with the Chern-Simons Gauss constraint. To preserve the asymptotic form (7), a ± 0 should be of the form (8). The choice of the Lagrange multiplier which is made when the chemical potentials are not switched on is simply ε ± = 1 , so that a ± 0 = ± a ± θ . The equations of motion imply that the fields are chiral with ± chiralities, and asymptotically given by (3) with ν ± = 0 . The choice of Lagrange multipliers which is made when the chemical potentials are switched on is ε ± = 1 + ν ± yielding now (3) with ν ± non zero when one integrates the equations. It is immediate, by very construction, that: To close this subsection, we note that the introduction of the chemical potentials through the temporal components of the connection (i.e., the components along the thermal circles) is in fact familiar from the thermodynamics of Reissner-Nordström black holes, where the chemical potential for the electric charge is introduced through the temporal component A 0 of the electromagnetic vector potential. A 0 is the Lagrange multiplier for Gauss' law. This procedure actually guarantees the interpretation of the Lagrange multipliers as chemical potentials. Indeed, quite generally, the Lagrange multipliers λ A enter the action as -∫ d d xλ A G A -λ A ∞ q A where G A are the constraints (which vanish on-shell) and q A the charges, which are given by the surface terms at infinity that must accompany the bulk constraints to make the variational principle well defined [37]. On shell, this sum reduces to -λ A ∞ q A .", "pages": [ 5, 6 ] }, { "title": "D. Some comments", "content": "We thus see that it is rather straightforward to handle the chemical potentials in the Chern-Simons formulation. It has been proposed in the literature to include them not along the thermal circle, but along the conjugate null directions, as Now, this is more than just a choice of the Lagrange multiplier. One sees that this choice has also the effect of modifying the phase space variables a ± θ as which is not any more of the requested asymptotic form (7). However, even though not of the requested asymptotic form, one can bring the spatial connection to it by redefinitions, so that the 'penalty' paid is not very high: the formulas need only direct, although somewhat awkward, adjustments. As we shall see, this is not the case for the higher spin charges, where this different approach destroys the asymptotics.", "pages": [ 7 ] }, { "title": "III. HIGHER SPIN CHEMICAL POTENTIALS", "content": "We now turn to the higher spin case. For definite, we consider the theory based on sl (3 , R ) ⊕ sl (3 , R ) , which contains a spin 3 field in addition to the spin 2 one, and which illustrates the main points.", "pages": [ 7 ] }, { "title": "A. Direct approach", "content": "The asymptotic form of the connections when the chemical potentials are not included in them read [8, 9], after the r -dependent gauge transformation of [17] has been performed, So, on a slice t = const, the spatial connection (which contains the reduced phase space variables) takes the asymptotic form The gauge transformations that leave this asymptotic form invariant are [8, 9] where ε ± , χ ± are arbitrary functions of θ (on each slide). Furthermore, the functions L ± and W ± of the canonical variables appearing in the asymptotic form of the connection transform as [8, 9] leading to the W 3 algebra (with central charge). As we explained above, the Lagrange multipliers a ± 0 must preserve the boundary conditions and so must be of the form (12). Now, in the case where the chemical potentials are not incorporated in the connection [8, 9], one simply takes ε ± = 1 , χ ± = 0 and one finds that on-shell, the connections are chiral and take the asymptotic form (11). If one wants to include the chemical potentials in the connection, one should, as explained above, allow for extra terms in a ± 0 . The temporal component of the connection reads now where ν ± and µ ± are constants and are the respective chemical potentials for the spin-2 part and the spin-3 part. The connection is therefore Note that this is not really a relaxed set of boundary conditions since the spatial part of the connection is unchanged. Only the Lagrange parameters are modified. This is an extension of the formalism that incorporates the chemical potentials. Just as in the pure gravity case, it is obvious that this choice is such that:", "pages": [ 7, 8 ] }, { "title": "B. Comments", "content": "If one were to introduce the chemical potentials through extra non-vanishing components of the connection not along the thermal circles but along the conjugate timelike directions, one would run into serious difficulties. Indeed, if one were to impose asymptotically one would modify the spatial connection in a way incompatible with the W 3 symmetry since the terms proportional to the chemical potentials µ ± for the spin-3 charge enter a ± θ multiplied by Lie algebra generators that are not highest (lowest) weight states and hence are not compatible with the asymptotic conditions (11) implementing the Hamiltonian reduction of the sl (3) current algebra to the W 3 algebra, The 'offending terms' are absent when the chemical potentials µ ± are zero (although rescalings are still needed in that case), but present otherwise. In fact, when the chemical potentials µ ± are non zero, the full asymptotic asymptotic symmetry at infinity, i.e., the set of all gauge transformations preserving the asymptotic form of a θ is the algebra W 2 3 corresponding to the other non trivial embedding of sl (2 , R ) into sl (3 , R ) [30, 31]. In the enveloping algebra of W 2 3 , one may try to pick out a W 3 algebra, for instance by requiring analyticity in µ . Perturbative efforts in that direction may be found in [38] where the equations have been analyzed to finite order O ( µ 4 ) .", "pages": [ 9 ] }, { "title": "IV. CONCLUSIONS", "content": "We have shown in this paper how to incorporate the chemical potentials associated with higher spin charges in higher spin three-dimensional gravity. Although we considered only the case of sl (3 , R ) , it is clear that our method extends straightforward to any sl ( N, R ) (with any non trivial embedding of sl (2 , R ) and even to infinite-dimensional higher-spin algebras). As here, the asymptotic symmetry algebra is obviously unchanged when the chemical potentials are switched on. The method is straightforward because the chemical potentials enter only the temporal components of the connection, which are Lagrange multipliers. The canonical variables and hence the canonical generators of the symmetry at infinity - are unaffected. In that sense, the boundary conditions with chemical potentials included are not true relaxations of the original boundary conditions. True relaxations would be relaxations on the behavior of the canonical variables and not just the Lagrange multipliers. The Hamiltonian formalism makes the analysis particularly transparent and direct. The argument is non-perturbative and exact in the chemical potentials. As we pointed out, the introduction of the chemical potentials through the temporal (i.e., along the thermal circles) components of the connection is in fact familiar from the thermodynamics of Reissner-Nordström black holes. The thermodynamical significance of the Lagrange multipliers as chemical potentials is immediate given the structure of the action, where the constraints and the accompanying charges (given by surface integrals at infinity) are multiplied by the Lagrange multipliers. It is very satisfying that what works in four dimensions also works in three. In a future paper [30, 31], we shall provide more insight on the analysis by investigating the static black hole solution endowed with a spin3 -field where L , W , µ , ν are integration constants. We shall confirm through the study of this specific ' W 3 -black hole\" that there is no tension between the holographic and canonical approaches. The analysis of the thermodynamics and the conformal properties is direct, because the coefficients L and W in the connection really mean what they are, namely, the generators of the W 3 algebra, without needing translation through a dictionary. We shall also investigate the rotating solution, which is given by (16) with L ± , W ± , ν ± and µ ± fixed to constants, which we write by displaying explicitly the spatial and temporal components as well as the analog ' W (2) 3 -black holes\". Perhaps the following question can be asked as a final note. We have seen that the chemical potentials ν ± of the gravitational sector can be absorbed through a definition t → αt , θ → θ + ωt of the coordinates, at the price of having new ranges for the new coordinates. It would be interesting to see if analogous absorptions of the higher spin chemical potentials could take place by suitable redefinitions in the yet-to-be-found geometry incorporating the higher spin fields. Note added: In the interesting paper [39], devoted to the higher spin black hole solutions of [25], it is advocated that the chemical potentials should be defined in that case in terms of the components of the connection along the thermal circles. This work relies on the approach of [38], where the W 3 -symmetry was proved to be present (perturbatively, to order O ( µ 4 ) ). We are grateful to the authors of [39] for drawing our attention on their work.", "pages": [ 10, 11 ] }, { "title": "Acknowledgments", "content": "We thank C. Martínez and C. Troessaert for helpful discussions. M.H. thanks the Alexander von Humboldt Foundation for a Humboldt Research Award. The work of M.H. is partially supported by the ERC through the 'SyDuGraM' Advanced Grant, by IISN - Belgium (conventions 4.4511.06 and 4.4514.08) and by the 'Communauté Française de Belgique' through the ARC program. The work of A.P., D.T. and R.T. is partially funded by the Fondecyt grants N · 1130658, 1121031, 3110122, 3110141. The Centro de Estudios Científicos (CECS) is funded by the Chilean Government through the Centers of Excellence Base Financing Program of Conicyt. spin gravity: A review,' J. Phys. A 46 , 214001 (2013) [arXiv:1208.5182 [hep-th]].", "pages": [ 11, 12, 14 ] } ]

[ { "title": "Jan Pieter van der Schaar ∗ and I-Sheng Yang †", "content": "IOP and GRAPPA, Universiteit van Amsterdam, Science Park 904, 1090 GL Amsterdam, Netherlands Abstract: We clarify under what conditions slow-roll inflation can continue almost undisturbed, while briefly evolving through a (semi-classically) metastable false vacuum. Furthermore, we look at potential signatures in the primordial power spectrum that could point towards the existence of traversed metastable false vacua. Interestingly, the theoretical constraints for the existence of traversable metastable vacua imply that Planck should be able to detect the resulting features in the primordial power spectrum. In other words, if Planck does not see features this immediately implies the non-existence of metastable false vacua rolled through during the inflationary epoch.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "One pending observation in cosmology is the existence of a metastable vacuum different from our own. Up to now the possibilities discussed in the literature are limited to primordial events that took place before, and set the initial conditions of, slow-roll inflation. For instance bubble nucleation resulting in detectable levels of negative curvature [1-4] or bubble collisions leading to specific patterns in the Cosmic Microwave Background sky [5,6]. As a consequence a long enough period of slow-roll inflation will erase all memory of these events and leave us with nothing to observe. A possibility that we would like to consider here is to have a metastable vacuum during slow-roll inflation. At first this sounds quite contradictory. Indeed, slow-roll inflation is an attractor solution involving a slowly changing state that eventually comes to and end because the slow change can no longer be maintained. A false vacuum by definition implies an attractor solution that is trapped in a non-changing state. Na¨ıvely, one might think these two attractor solutions should be mutually exclusive. In other words, a slow-roll inflationary trajectory should not be able to 'pass through' a metastable false vacuum without being trapped there. We will show that the above intuition is incorrect. First of all we should note that this is not new. After all there are a large number of slow-roll inflationary models and it should not come as a surprise that some of them, in some parameter range, do contain one or more metastable false vacua. In particular there has been a lot of work over recent years on adding specific features to simple slow-roll models and study the observable outcome [7, 8]. Although the stand-alone properties of these features in the potential are typically not of main concern and therefore less studied, in some parameter range they actually do correspond to metastable false vacua, as we will point out. We will use the simplest possible model - slow-roll with a canonical scalar field to illustrate the dynamics of rolling through a metastable false vacuum. As one should expect vacuum stability requires a 'big' barrier in the potential. At the same time, the barrier cannot be too 'big' to disrupt the slow-roll process. By carefully defining what turn out to be different standards of 'big' in this simplest scenario, we show that it is possible to satisfy both conditions at the same time. Our basic example allows us to derive roughly the necessary conditions for traversing a stable vacuum during slow-roll inflation. Specifically we identify under what conditions a stable vacuum only fractionally disturbs the slow-roll process by a factor of 10 -1 ∼ 10 -2 . The leading order instability of the false vacuum will then be through Coleman-deLuccia tunneling, suppressed by e -10 ∼ e -100 . Including the observational constraints, this happens to be the parameter range that is consistent with the WMAP7 [9, 10] results and can be probed by Planck [11]. In general, a feature corresponding to a traversable false vacuum will result in an oscillating pattern on the power spectrum and the bispectrum [7,12]. Notably slow-roll disturbances smaller than a factor of 10 -1 ∼ 10 -2 will not only be unobservable by Planck, but at the same time result in an unstable 'vacuum'. This suggests that unlike many other properties of slow-roll inflationary models, passing through a metastable false vacuum feature can actually be ruled out. On the other hand, when a feature would be detected by Planck one will be hard-pressed to uniquely identify it as due to a metastable false vacuum. For that one would need to know the details of the shape and the phase of the oscillation, which is notoriously difficult. Our basic result automatically generalizes to multi-field inflationary models as long as the slow-roll process follows the gradient flow and the effective field along the gradient flow has a canonical kinetic term. Although this includes a significant portion of the available models, there are some notable exceptions. It excludes slow-roll models that are not driven by gradient flow [13,14], models with effective DBI kinetic terms [15] and models that cannot be described by effective scalar fields [16]. Even so, we would like to stress that this mechanism implies additional opportunities, not just restricted to the period of slow-roll inflation, to observe the existence of metastable false vacua within our observable cosmological history. This includes for instance the reheating process and many possibly many other interesting scenarios waiting to be further investigated. The structure of this paper is as follows. In Sec.2, we discuss some existing models that can include a false vacuum, or that are in some ways relevant to such a possibility. In Sec.3, we introduce and analyze the basic single field inflationary model and derive the necessary conditions to slowly roll through a metastable false vacuum. In Sec.4, we briefly discuss the observable consequences and point out that a metastable false vacuum can be consistent with the WMAP7 and recent Planck data. Even better, if Planck does not see anything, traversable metastable false vacua are ruled out. Finally in Sec.5, we summarize, discuss generalizations and point towards interesting future directions.", "pages": [ 2, 3, 4 ] }, { "title": "2. Existing Models", "content": "This section is effectly a short review of related slow-roll models. Readers familiar with the topic can choose to skip it and proceed to our main analysis in Section3. To start out let us have a look at some known models that could possibly feature the presence of metastable false vacua during inflation. An obvious set of slow-roll theories to consider are Chain-Inflation models [17], described by (variants of) the following potential For m 4 c ∆ φ c /greatermuch k c , it contains many false vacua roughly at φ = N ∆ φ c . The essential idea is that a vacuum will rapidly tunnel to the next one by nucleating many bubbles. These bubbles percolate and the entire spacetime transits into the next vacuum state. Repeating this process is comparable to slow-roll evolution, as shown in Fig.1. However, in order for the repeated percolation to take place, the typical lifetime of these false vacua should be at most one Hubble time 1 . As such they do not qualify as metastable false vacua. However, there are two ways to modify this model that will allow the presence of metastable false vacua. The first modification comes from better understanding the classical transitions whenever domain walls collide [21-24]. For the potential given by Eq. (2.1), percolating bubbles will not just stop at the next vacuum. The domain walls will cross each other and automatically proceed down the chain [19], as shown in Fig.2. This implies that even though the initial false vacuum has to be short-lived to allow the first generation of bubbles to percolate, subsequently all other false vacua can have exponentially long lifetimes. Once started, the classical crossing of subsequent domain walls walls automatically results in an effective phase of slow-roll inflation. Recently this mechanism was further improved by introducing compact extra dimensions [25-27]. Essentially a periodic identification in Fig.2 will mean that an initial bubble will collide with itself in the compact dimensions. As a consequence one bubble is enough to start the entire process and all of the false vacua can have long lifetimes. Obviously these models are quite interesting, but for our purposes here they are unnecessarily contrived. For one they rely on the special mechanism of classical domain wall transitions, whereas we would like to show that passing through a metastable false vacuum does not require a special mechanism and is quite a generic possibility. Another modification, or parameter regime, in Chain-Inflation models is to consider small m c in Eq. (2.1), such that This makes the potential monotonically decreasing, so obviously tuning k c can lead to a standard slow-roll model with small periodic disturbances. This type of potential is in fact motivated by monodromy inflation and the periodic feature can lead to interesting signatures [28]. Most of the work in this direction assumes Eq. (2.2), but a priori that is not a necessary restriction. If we increase m c slightly such that V is no longer monotonically decreasing, it does not mean that the slow-roll evolution is suddenly significantly disturbed. Indeed, interestingly we can keep increasing m c until the false vacua become metastable, while at the same time keeping the slow-roll process minimally disturbed. In order to demonstrate that a metastable false vacuum does not necessarily ruin slow-roll evolution, we clearly do not have to consider a potential with many minima as in Eq. (2.1). It can simply be understood by studying the dynamics of a slow-roll potential featuring one false vacuum. This connects nicely to studies of potentials with step features, which in some cases might correspond to a false vacuum [7, 12]. In the next section we will show that the condition for a minimum to correspond to a metastable false vacuum is directly related to how much it disturbs the slow-roll process. As a corollary we conclude that slow-roll evolution in inflationary models with step features, or in models of monodromy inflation, can be maintained even in parameter ranges where the false vacua are metastable.", "pages": [ 4, 5, 6 ] }, { "title": "3. Rolling Through a Vacuum", "content": "In this section we will derive the conditions for (almost) undisturbed slow-roll evolution while passing through a metastable false vacuum. To keep things as general as possible we study the simplest example with just one canonical scalar field. Let us consider the following potential, that includes a potentially metastable false vacuum in the field range between -φ f < φ < 3 φ f , As shown in Fig.3, there is a local minimum at φ = 0. The barrier is described by two parameters, its height and its width, which in our case can be identified with Moreover, this potential has the convenient property that at φ = -φ f and φ = 3 φ f , the false vacuum feature connects smoothly to the unperturbed slope ∣ Clearly, this particular set-up should be well-suited to study the (slow-roll) dynamics in the presence of a false vacuum. By construction the false vacuum feature stops disturbing slow-roll when the slope of the potential returns to the original value k just before it entered the feature. As it turns out the dynamics is most conveniently parametrized by two dimensionless parameters, α and β , that we will shortly define. The parameter α controls the stability of the false vacuum, with α > 1 corresponding to metastability. The parameter β controls the disturbance of the slow-roll dynamics, with β /greatermuch 1 corresponding to undisturbed slow-roll evolution.", "pages": [ 6, 7 ] }, { "title": "3.1 Disturbed slow-roll evolution", "content": "Let us assume slow-roll evolution just before the scalar field enters the false vacuum feature. Just before entering the feature we identified the slope of the potential as k and demanding that the first slow-roll condition is satisfied we obtain where M 2 p ≡ 1 8 πG and V i is the value of the potential just before entering the false vacuum feature. We will require that the false vacuum feature only slightly disturbs the slow-roll process. In order for this to be possible, the range in field space ∼ φ f of the barrier must be smaller than the field range corresponding to a single e-fold. Using that k ≈ -3 H ˙ φ in the slow-roll limit, this means Assuming this condition is satisfied, passing through this feature will take less than one Hubble time and the slow-roll phase should not be significantly disturbed. Correspondingly the Hubble friction is not draining a significant amount of energy and we can roughly estimate the amount of energy change caused by the false vacuum feature For this to be a small perturbation, as assumed, it should be much smaller than the typical kinetic energy during slow-roll evolution, i.e. So far, this has been completely general, but now we want to make use of the fact the for the potential under consideration the slope k and the field value φ f are related by Eq. (3.4). Plugging this relation into Eq.(3.8) we arrive at the final condition for undisturbed slow-roll evolution in the presence of a false vacuum, introducing the dimensionless parameter β , One can easily check that this final result is indeed self-consistent with the starting assumption, in the sense that large β implies that the field range φ f is traversed in far less than a single e-fold. To be precise, defining the field range traveled in a single e-fold as ∆ φ 1 , one derives that In other words the fraction of Hubble time spend moving through the false vacuum is measured by β -2 , which for β /greatermuch 1 is much smaller than 1. A numerical solution for /epsilon1 in Fig.4 confirms our analytical estimation. Note that the second slow-roll parameter, η = ˙ /epsilon1/ ( H/epsilon1 ) is strongly disturbed, for a short duration. That is why the spectrum of small fluctuation can be significantly affected while the overall slow-roll process is not.", "pages": [ 7, 8, 9 ] }, { "title": "3.2 Vacuum Stability", "content": "Now that we have derived the condition for a traversable false vacuum, let us next investigate the criteria for (meta-) stability. For the local minimum at φ = 0 to be a metastable false vacuum, we need to make sure the following two non-perturbative processes are exponentially suppressed: To find the probability for the false vacuum to decay via a Hawking-Moss instanton we compute, as usual, the Euclidean action, giving This result of course provides us with the exponent of the decay rate In order for the Hawking-Moss decay rate to be exponentially suppressed, the following condition therefore needs to be satisfied We will soon see that the above condition can be considered redundant, i.e. is automatically satisfied when both the Coleman-deLuccia decay is exponentially suppressed and the false vacuum does not significantly interrupt the slow-roll evolution. Let us now turn to the Coleman-de Luccia decay rate. To start we first estimate the domain wall tension, Note however that in this case we should consider tunneling to a point on the potential with a non-zero slope instead of another minimum, implying that the thin-wall approximation is invalid. In other words, the bubble size roughly equals the thickness of the wall m -1 . If we would nevertheless estimate the tunneling rate using the standard thin-wall formula, we obtain A more accurate thick-wall numerical calculation 3 of the instanton solution reveals an additional factor of ∼ 300. With the above result we now arrive at the second dimensionless parameter α , which should satisfy the following condition to exponentially suppress Coleman-deLuccia false vacuum decay Combining conditions Eq. (3.18) and Eq.(3.9), which guarantees undisturbed slowroll evolution, we note that the condition for exponentially suppressed Hawking-Moss decay of the false vacuum Eq. (3.14) is automatically satisfied and therefore redundant As a corollary the leading order instability of the metastable false vacuum which minimally disturbing the slow-roll process is always through Coleman-deLuccia tunneling, which is governed by Eq. (3.18). To summarize: in order for a false vacuum feature to be metastable and at the same time not disturbing slow-roll inflation requires that the two dimensionless parameters α ≡ φ f m and β ≡ m H both be large. This corresponds to inflationary potentials with false vacua described by the following 'hierarchy' of scales φ f > m > H . Such a hierarchy can easily be constructed in general, but the magnitude of primordial density perturbations does impose an additional constraint, as we will discuss next.", "pages": [ 9, 10, 11 ] }, { "title": "4. Observational constraints", "content": "So far we have been treating α and β as two free parameters, but arbitrarily tuning them will have observational consequences. This is because they implicitly control H and /epsilon1 . In particular, the observed magnitude of the primordial density perturbation implies that Combining this expression with with Eq. (3.4), (3.9) and (3.18), one sees that As a consequence arbitrarily large values for both of α and β are excluded. There remains some room however to satisfy both conditions, for instance for a β between 10 and 50 we can still slow-roll through a metastable vacuum α /greaterorsimilar 1. Disturbing slow-roll even less, i.e. for β /greaterorsimilar 50, the false vacuum necessarily has to be unstable in order to satisfy the observational constraint on the primordial density perturbation. Note that the primordial density perturbation also constrains a combination of the inflationary scale H/M p and the first slow-roll parameter /epsilon1 , since δρ ρ = 3 H 10 πM p √ 2 /epsilon1 = 10 -5 . Comparing this to Eq. (4.2) one readily sees that the parameter constraints are independent, implying that the condition to slow-roll through a metastable false vacuum does can in principle be satisfied for arbitrary inflationary scales H/M p . We can next ask what kind of signature a traversed metastable false vacuum leaves on the CMB sky. Fortunately there exists a large body of work on the observational signatures of isolated features on the slow-roll potential. A particular type of feature that is very similar to a metastable false vacuum is the step feature [7,12]. Translating the natural parameters describing a step feature (width and relative height) to the specific toy-model false vacuum studied here one finds The original analysis applied to both negative as well as positive relative heights of the step, whereas a false vacuum can of course only be compared to a rising step feature. As it turns out the width of the step feature in field space is not directly relevant, instead the meaningful parameter is the fraction of Hubble time the field spends in this field range, which as we saw in Eq. (3.10) is given by β -2 . Looking at the above parameter identification this means that large β , besides ensuring slow-roll evolution through the false vacuum feature, should be similar to a sharp step feature. This is exactly the interesting regime studied in [12], so we can apply their results directly. According to [12], a sharp step feature leads to oscillating behavior in the power spectrum. The relative step height determines the amplitude and the WMAP7 constraint roughly equals [9] So we conclude that β /greaterorsimilar 10 does not conflict with the WMAP7 observational bounds on step features. In our specific toy-model the false vacuum feature can only affect the observables by disturbing the slow-roll process. Presumably a detailed power- and bispectrum analysis of the recent Planck data [11] will improve this bound significantly. As a consequence a false vacuum feature with β ∼ 10 has a good chance to be observed. Most interestingly, Planck will not be able to see β /greaterorsimilar 100 as estimated in [12]. This is a common situation in the literature of slow-roll features-the proposed feature can still exist even if unseen by Planck. However our case is entirely different. Due to the constraint from ( δρ/ρ ), Eq. (4.2), larger values β /greaterorsimilar 100 fail the metastability condition ( α /greaterorsimilar 1) for the false vacuum feature. Therefore 'not being seen' by Planck actually rules out the existence of a slow-roll traversed metastable false vacuum, at least in single canonical field models which are favored by recent Planck data. If an oscillating feature in the power spectrum is observed, would it be possible to explicitly link it to a metastable false vacuum feature? Obviously this will be difficult, since we have utilized the observational bounds as derived for generic sharp step features. Most likely detection might lead to a decent estimate on the amplitude of the oscillating feature, which leaves a large degeneracy among all possible feature shapes. A metastable false vacuum is just one possibility. To really distinguish among different (sharp) features, one would need to accurately observe all details of the oscillating pattern, like the phase and the specific shape, which might be a tall order for some time to come.", "pages": [ 11, 12, 13 ] }, { "title": "5. Discussion", "content": "In this article we explicitly determined the conditions that need to be satisfied in order for a false vacuum feature in a slow-roll potential to be 1) slow-roll traversable and 2) metastable. For this to be possible the scale φ f corresponding to the width of the false vacuum feature has to be larger than the scale m corresponding to the mass, or curvature, of the false vacuum potential, which has to be bigger than the scale of slow-roll inflation H , i.e. φ f > m /greatermuch H . In general this hierarchy of scales can easily be satisfied, but the magnitude of primordial density perturbations ( δρ/ρ ) ∼ 10 -5 severely restricts the parameter space. Interestingly the remaining parameter range, expressed in terms of α and β , can be significantly tightened, and perhaps even excluded, by looking for oscillating signatures in the power spectrum as measured by Planck. So the existence of traversed metastable false vacua during slow-roll inflation is ruled out if Planck does not detect any oscillating features in the power spectrum. It would be interesting to further study the detailed observational signatures of a rolled through metastable false vacuum feature. In particular it is of interest to see whether and how a metastable false vacuum can be distinguished from other types of (sharp) features. This might in particular involve a careful analysis of higher order statistical observables of the primordial density fluctuations. The biggest degeneracy might be between a rising step (slowing down slow-roll) that could be a metastable false vacuum and a lowering step (speeding up slow-roll) that is clearly not a candidate for a metastable false vacuum. According to [12] the difference in the observational signal is only in the initial phase of the oscillation in the power- (or bi-) spectrum. Clearly the recent Planck data set should be analyzed in detail to either rule out the existence of slow-roll traversed metastable false vacua, or in case of a feature detection, try to ascertain its properties as accurately as possible. Let us make some final remarks regarding the 'coincidence' of Planck being able to rule out the existence of traversable metastable false vacua. Our current observational capability of course should not have a fundamental physical meaning. However Eq. (4.2) provides an intriguing relation between the false vacuum stability, traversability, and the value of density perturbation. In a universe where ( δρ/ρ ) is larger, for example 10 -1 ∼ 10 -2 , the presence of a metastable false vacuum would necessarily trap the inflaton. A trapped universe can later tunnel out. If it tunnels in the φ direction, the previous e-foldings are lost and the effective duration of inflation is reduced. If it tunnels out in other directions, most likely there will be no inflation at all [1,31]. Both cases lead to a smaller chance for the standard cosmology and a bigger chance for an empty universe. This relation might be useful in the multiverse framework [32]. Our value of ( δρ/ρ ) is notoriously difficult to come by as an anthropic prediction. Holding all other cosmological parameters fixed, a larger value of ( δρ/ρ ) increases the physical density of entropy/baryon/observers. This implies a na¨ıve runaway problem [33, 34]-Our universe should have a bigger ( δρ/ρ ) if we are anthropically selected from a multiverse. Our finding points to a possible solution to that problem. If for some reason the vacuum-like features are common along a slow-roll potential, then models with larger ( δρ/ρ ) are less stable in the sense that they are prone to be trapped by false vacua and produce less observers. This implies an anthropic disadvantage for larger ( δρ/ρ ) and maybe a natural cutoff for the runaway problem.", "pages": [ 13, 14 ] }, { "title": "Acknowledgemements", "content": "We would like to thank Raphael Bousso, Ben Freivogel, Matthew Kleban, Eugene Lim, and Gary Shiu for helpful discussions. This work is supported in part by the research program of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organization for Scientific Research (NWO).", "pages": [ 14 ] } ]

[ { "title": "ABSTRACT", "content": "We estimate the power spectra of CMB temperature anisotropy in localized regions on the sky using the WMAP 7-year data. Here, we report that the north hat and the south hat regions at the high Galactic latitude ( | b | ≥ 30 · ) show anomaly in the power spectrum amplitude around the third peak, which is statistically significant up to 3 σ . We try to figure out the cause of the observed anomaly by analyzing the low Galactic latitude ( | b | < 30 · ) regions where the galaxy contamination is expected to be stronger, and regions that are weakly or strongly dominated by the WMAP instrument noise. We also consider the possible effect of unresolved radio point sources. We found another but less statistically significant anomaly in the low Galactic latitude north and south regions whose behavior is opposite to the one at the high latitude. Our analysis shows that the observed north-south anomaly at high latitude becomes weaker on the regions with high number of observations (weak instrument noise), suggesting that the anomaly is significant at sky regions that are dominated by the WMAP instrument noise. We have checked that the observed north-south anomaly has weak dependences on the bin-width used in the power spectrum estimation and the Galactic latitude cut. We have also discussed the possibility that the detected anomaly may hinge on the particular choice of the multipole bin around the third peak. We anticipate that the issue of whether the anomaly is intrinsic one or due to the WMAP instrument noise will be resolved by the forthcoming Planck data. Key words : cosmology: cosmic microwave background - cosmology: observations - methods: data analysis", "pages": [ 1 ] }, { "title": "Local anomalies around the third peak in the CMB angular power spectrum of the WMAP 7-year data", "content": "Kyeong Yeon Ko 1 , 3 , Chan-Gyung Park 2 , and Jai-chan Hwang 3 1 Korea Astronomy and Space Science Institute, Daejeon, Korea 2 Division of Science Education and Institute of Fusion Science, Chonbuk National University, Jeonju, Korea 3 Department of Astronomy and Atmospheric Sciences, Kyungpook National University, Daegu, Korea E-mail : [email protected], [email protected], [email protected] (Received February 1, 2013; Accepted March 21, 2013)", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The cosmic microwave background radiation (hereafter CMB) provides us with a wealth of information on the history of the universe. The thermal black body nature of the CMB energy spectrum is now considered as the firm evidence of the hot big bang scenario for the beginning of the universe (Alpher & Herman 1948; Dicke et al. 1965; Penzias & Wilson 1965). The existence of large-scale structure in the universe also implies that there were primordial density perturbations as the seeds for structure formation. It was expected that these inhomogeneities would have the imprint on the CMB as the minute temperature fluctuations (anisotropy) (Sachs & Wolfe 1967; Peebles & Yu 1970; Bond & Efstathiou 1987). The CMB anisotropy was discovered by the Cosmic Background Explorer (COBE) Differential Microwave Radiometers experiment (Smoot et al. 1992) and has been confirmed by many ground-based and balloon-borne experiments (see Hu & Dodelson 2002; Scott & Smoot 2006 for reviews and references therein). Recently, the Wilkinson Microwave Anisotropy Probe (WMAP) has opened a new window to the precision cosmology by measuring the CMB temperature anisotropy and polarization with high resolution and sensitivity (Bennett et al. 2003; Jarosik et al. 2011). For every data release, the WMAP team presented their estimation of the angular power spectra for temperature and polarization anisotropy (Hinshaw et al. 2003, 2007; Page et al. 2003, 2007; Nolta et al. 2009; Larson et al. 2011). By comparing the measured CMBpower spectra with the theoretical prediction, the WMAP team determined the cosmological parameters with a few % precision (Spergel et al. 2003, 2007; Komatsu et al. 2009, 2011), and found that the observed CMB fluctuations are consistent with predictions of the concordance ΛCDM model with scale-invariant adiabatic fluctuations generated during the inflationary epoch (Spergel et al. 2003; Peiris et al. 2003; Komatsu et al. 2011). Recent ground-based and balloon-borne experiments that have performed the CMB power spectrum measurement and the cosmological parameter estimation include the South Pole Telescope (SPT) (Keisler et al. 2011), the QUaD experiment (Brown et al. 2009), Arcminute Cosmology Bolometer Array Receiver (ACBAR) (Reichardt et al. 2009), the Cosmic Background Imager (CBI) (Mason et al. 2003), the Atacama Cosmology Telescope (ACT) (Fowler et al. 2010), the Degree Angular Scale Interferometer (DASI) (Carlstrom et al. 2003), BOOMERANG (Jones et al. 2006), Archeops (Benoˆıt et al. 2003), and MAXIMA (Lee et al. 2001). In the CMB data analysis the two-point statistics such as the correlation function and the power spectrum has been widely used. In particular, the relation between the CMB angular power spectrum and the cosmological physics is well understood, and the tight constraints on the cosmological parameters can be directly obtained by comparing the measured power spectrum with the theoretical prediction. Therefore, the accurate estimation of the angular power spectrum from the observed CMB maps is the essential step. The efficient techniques to measure the angular power spectrum from the CMB temperature fluctuations with incomplete sky coverage have been constantly developed (e.g., G'orski 1994; Tegmark 1997; Bond et al. 1998; Oh et al. 1999; Szapudi et al. 2001; Wandelt et al. 2001; Hansen et al. 2002; Hivon et al. 2002; Mortlock et al. 2002; Hinshaw et al. 2003; Wandelt et al. 2003; Chon et al. 2004; Efstathiou 2004; Eriksen et al. 2004a; Wandelt et al. 2004; Brown et al. 2005; Polenta et al. 2005; Fay et al. 2008; Dahlen & Simons 2008; Das et al. 2009; Mitra et al. 2009; Ansari et al. 2010; Chiang & Chen 2012). Until now, the analysis of the WMAP CMB data has been made using the whole sky area except for strongly contaminated regions (Hinshaw et al. 2003, 2007; Nolta et al. 2009; Larson et al. 2011; Saha et al. 2006, 2008; Souradeep et al. 2006; Eriksen et al. 2007a; Samal et al. 2010; Basak & Delabrouille 2012). On the other hand, there was only a small number of studies on the power spectrum measurement on the partial regions of the sky (e.g., Eriksen et al. 2004b; Hansen et al. 2004a,b; Ansari et al. 2010; Yoho et al. 2011; Chiang & Chen 2012). Especially, Yoho et al. (2011) detected degree-scale anomaly around the first acoustic peak of the CMB angular power spectrum measured on a small patch of the north ecliptic sky. In this work, we measure the angular power spectra from the WMAP 7-year temperature anisotropy data set on some specified regions of the sky. We found that at high Galactic latitude regions there is the north-south anomaly or asymmetry in the power amplitude around the third peak of the angular power spectrum, which is the main result of this paper. Our result differs from the wellknown hemispherical asymmetry in the angular power spectrum and the genus topology at large angular scales (Park 2004; Eriksen et al. 2004b,c, 2007b; Hansen et al. 2004b; Bernui 2008; Hansen et al. 2009; Hoftuft et al. 2009). This paper is organized as follows. In Sec. 2, we describe the method of how to measure the angular power spectrum. The application of the method to the WMAP 7-year data and the comparison with the WMAP team's measurement are shown in Sec. 3. In Sec. 4, we measure the power spectra on various sky regions defined with some criteria, and present our discovery of the north-south anomaly around the third peak of the angular power spectrum at high Galactic latitude regions. We also discuss the potential origin of such a north-south anomaly. Section 5 is the conclusion of this work. Throughout this work we have used the HEALPix software (G'orski et al. 1999, 2005) to measure the pseudo angular power spectra and to generate the WMAP mock data sets.", "pages": [ 1, 2 ] }, { "title": "2. Angular power spectrum estimation method", "content": "In this section we briefly review how the angular power spectrum is measured from the observed CMB temperature anisotropy maps. Throughout this work we do not consider the CMB polarization. In the ideal situation where the temperature fluctuations on the whole sky are observed, we can expand the temperature distribution T ( n ) in terms of spherical harmonics as where n denotes the angular position on the sky, Y lm ( n ) is the spherical harmonic basis function, and the monopole ( l = 0) and dipole ( l = 1) components have been neglected. From the orthogonality condition of the spherical harmonics, the coefficients a lm can be obtained from where d Ω denotes the differential solid angle on the sky. If the temperature anisotropy is statistically isotropic, then the variance of the harmonic coefficients a lm is independent of m and the angular power spectrum C l is given as the ensemble average of two-point product of the harmonic coefficients, /negationslash where the bracket represents the ensemble average and δ ll ' is the Kronecker delta symbol ( δ ll ' = 1 for l = l ' , and δ ll ' = 0 for l = l ' ). For a Gaussian temperature distribution, it is known that all the statistical information is included in the two-point statistics like the angular power spectrum C l . For the single realization of temperature fluctuations on the sky, the ensemble average is replaced with the simple average of independent modes belonging to the same multipole l and the angular power spectrum becomes with uncertainty due to the cosmic variance In the practical situation, however, we usually exclude some portion of the sky area where the contamination due to the Galactic emission and the strong radio point sources is expected to affect the statistics of the CMB anisotropy significantly. The limited angular resolution, the instrument noise, and the finite pixels of the processed map also prevent us from using the formulas, Eqs. (1)-(5), which are valid only in the ideal situation. The incomplete sky coverage with an arbitrary geometry and the limited instrumental performance break the orthogonality relation between the spherical harmonic functions. Thus, measuring angular power spectrum from the realistic observational data is more complicated. There are two popular ways of measuring the angular power spectrum from the observed CMB data with incomplete sky coverage. One is the maximum likelihood estimation method based on the Bayesian theorem (Bond et al. 1998; Tegmark 1997), and the other is the pseudo power spectrum method that applies the Fourier or harmonic transformation directly (Hivon et al. 2002). For the latter, Hivon et al. (2002) developed the so called MASTER (Monte Carlo Apodised Spherical Transform Estimator) method which estimates the angular power spectrum from the high resolution CMB map data. For mega-pixelized data sets like those from WMAP or Planck (Planck Collaboration et al. 2011a), the MASTER method is faster and more efficient than the maximum likelihood method. We refer to the pseudo angular power spectrum as ˜ C l to distinguish from the true angular power spectrum C l which we try to estimate. Two quantities differ from each other due to the incomplete sky coverage, the limited angular resolution and pixel size, and the instrument noise, and they are related by (Hansen et al. 2002; Hivon et al. 2002) where B l is the beam transfer function, F l is the pixel transfer function due to finite pixel size of the CMB map, 〈 ˜ N l 〉 is the average pseudo noise power spectrum due to the instrument noise, and M ll ' is the mode-mode coupling matrix that incorporates all the effect due to the incomplete sky coverage. The mode-mode coupling matrix is written as (7) where W l is the power spectrum of the window function for incomplete sky coverage (see Sec. 3.2), and the last factor on the right-hand side enclosed with the parenthesis represents the Wigner 3j symbol. We use the Fortran language and the Mathematica software ∗ to calculate the Wigner 3j symbols; see also Hivon et al. (2002) and Brown et al. (2005) for the numerical computation of the Wigner 3j symbols. To reduce the statistical variance of the measured angular power spectrum due to the cosmic variance and the instrument noise, we usually average powers within the appropriate l bins. According to Hivon et al. (2002), the binning operator is defined as The reciprocal operator is defined as where b is the bin index. For ease of comparison, we use exactly the same l -binning adopted by the WMAP teams ( b runs from 1 to 45) (Larson et al. 2011). The true power spectrum can be estimated from where is a kernel matrix that describes the mode-mode couplings due to the survey geometry, the finite resolution and pixel size, and the binning process. Ideally the average pseudo noise power spectrum 〈 ˜ N l 〉 can be estimated by the Monte Carlo simulation that mimics the instrument noise. However, because of the statistical fluctuations in the noise power spectra itself, it is rather difficult to obtain an accurate estimation of the power contribution due to the instrument noise from the single observed data. In our analysis we evade this problem by measuring the cross-power spectra between different channel data sets, in the same way as adopted by the WMAP team (see the next section).", "pages": [ 2, 3 ] }, { "title": "3.1 WMAP 7-year data", "content": "The WMAP mission was designed to make the full sky CMB temperature and polarization maps with high accuracy, precision, and reliability (Bennett et al. 2003). The WMAP instrument has 10 differencing assemblies (DAs) spanning five frequency bands from 23 to 94 GHz: one DA each at 23 GHz (K1) and 33 GHz (Ka1), two DAs each at 41 GHz (Q1, Q2) and 61 GHz (V1, V2), and four DAs at 94 GHz (W1, W2, W3, W4), with 0 . · 82, 0 . · 62, 0 . · 49, 0 . · 33, and 0 . · 21 FWHM beam widths, respectively. In this work we use the foreground-reduced WMAP 7-year temperature fluctuation maps that are prepared in the HEALPix format with N side = 512 (resolution 9; r9). The total number of pixels of each map is 12 × N 2 side = 3 , 145 , 728. Following the WMAP team's analysis, we use only V and W band maps to reduce any possible contamination due to the Galactic foregrounds. We use the WMAP beam transfer function ( B l ) and the number of observations ( N obs ) for each DA. All the data sets used in this work are available on the NASA's Legacy Archive for Microwave Background Data Analysis (LAMBDA) † .", "pages": [ 3, 4 ] }, { "title": "3.2 Weighting schemes", "content": "The window function assigns a weight to the temperature fluctuation on each pixel before the harmonic transformation is performed. In our analysis the window function maps have been produced based on two weighting schemes, the uniform weighting scheme and the inverse-noise weighting scheme (Hinshaw et al. 2003). The WMAP team provides mask maps which assigns unity to pixels that are used in the analysis and zero to pixels that are excluded to avoid the foreground contamination. We use the KQ85 mask map with resolution 9 that includes about 78.3% of the whole sky area. The mask map excludes the Galactic plane region with the strong Galactic emission and the circular areas with 0 . · 6 radius centered on the strong radio point sources. The uniform weighting scheme uses the mask map as the window function directly, where M ( p ) denotes a mask map on a pixel p on the sky. The power spectrum at high l region is usually dominated by the instrument noise. In the WMAP data the noise level on each pixel is modeled by where N obs ( p ) is the number of observations on the pixel p and σ 0 is the global noise level ( σ 0 = 3 . 319, 2 . 955, 5 . 906, 6 . 572, 6 . 941, 6 . 778 mK for V1, V2, W1, W2, W3, W4 DAs, respectively). To reduce the effect of the instrument noise, the inverse-noise weighting scheme is defined as the product of the mask map and the map of number of observations, The window function at 94 GHz W4 frequency channel (DA) is shown in Fig. 1, together with a histogram of the number of observations. Although the WMAP team applies the uniform weighting scheme for l < 600 and the inverse-noise weighting scheme for l > 600 (Larson et al. 2011), in this work we simply apply the single weighting scheme over the whole range of l considered (2 ≤ l ≤ 1200) during the power spectrum estimation and compare the results based on the two weighting schemes.", "pages": [ 4 ] }, { "title": "3.3 Angular power spectrum measured from the WMAP data", "content": "In the WMAP team's data analysis, the power spectrum at low multipoles ( l ≤ 32) was measured by a Blackwell-Rao estimator that is applied to a chain of Gibbs samples obtained from the foreground-cleaned CMBmapwhile the power spectrum at high multipoles (32 < l ≤ 1200) by the MASTER pseudo power spectrum estimation method (Larson et al. 2011). Throughout this work, the angular power spectrum over the whole l range (2 ≤ l ≤ 1200) is measured based on the pseudo power spectrum estimation method because our primary attention is paid on the high l region where the peaks are located. Thus, as shown below our results at low multipoles are somewhat different from the WMAP team's results. The WMAP V and W bands have two (V1 and V2) and four (W1, W2, W3, and W4) separate frequency channels (DAs). Using the Anafast program in the HEALPix package, we measured 15 pseudo cross power spectra ( ˜ C l ) for different channel combinations (V1W1, V1W2, and so on) based on the uniform and the inverse-noise weighting schemes. During the production of each pseudo cross power spectrum, Anafast program estimates two separate sets of harmonic coefficients for each DA using the formula Eq. (2) where T ( n ) is now replaced with W ( n ) T ( n ) and W ( n ) is the window function defined in Eqs. (10) and (12) for the corresponding DA. We set the maximum multipole as l = 1200. For each channel combination, we calculate the mode-mode coupling matrix M ll ' and the kernel matrix K bb ' using the beam transfer, the pixel transfer functions, and the same l -binning as defined by the WMAP team (Larson et al. 2011). For the power spectrum of window function in Eq. (7) we use the cross power spectrum of window functions for the corresponding DA combination, which is obtained from the Anafast program by inserting the window functions at two different frequency channels as the input data. The squared beam transfer function B 2 l appearing in Eq. (6) is also modified into the product of beam transfer functions from the two different frequency channels. The final binned cross power spectrum ˆ C b for each channel combination is obtained from Eq. (8) neglecting the term for the average noise power spectrum 〈 ˜ N l 〉 . Since the properties of the instrument noise for each channel are independent, the process of crosscorrelation statistically suppresses the contribution of the instrument noise to the power spectrum estimation. This technique has an advantage that our power spectrum estimator is not biased by noise if the noise in the two independent channels is uncorrelated; the crosscorrelation between the uncorrelated noise vanishes statistically. Thus, the measured cross-power spectra are independent of instrument noise from the individual channels in every practical sense (see Hinshaw et al. 2003). Note that the self-combination of each frequency channel data (like V1V1, V2V2, W1W1) is not considered during the analysis because it is essentially needed to subtract the contribution of noise power from the measured auto-power spectrum and it is a difficult task to precisely estimate the noise power from the single observed map (see the discussion at the end of Sec. 2). For each binned cross power spectrum we subtract the contribution due to the unresolved radio point sources expected in the WMAP temperature anisotropy maps by using the information presented in Nolta et al. (2009) and Larson et al. (2011). The amplitude of the binned power spectrum due to the unresolved point sources can be written as Here S i l is the point-source spectral function given by where ν k and ν k ' denote the individual frequency channel belonging to i -th channel combination, r ( ν k ) is a conversion factor from antenna to thermodynamic temperature, ν Q = 40 . 7 GHz is the Q-band central frequency, 10 3 A ps = 9 . 0 ± 0 . 7 µ K 2 , and β = -2 . 09 (see also Huffenberger et al. 2006; Souradeep et al. 2006). The subindex l is dummy because the spectral function does not depend on it. To combine these cross power spectra into the single angular power spectrum, in fact we need the Monte Carlo simulation data sets with the similar properties of the WMAP observational data. Using the Synfast program of HEALPix and assuming the concordance flat ΛCDM model with parameters, Ω b h 2 = 0 . 02260, Ω c h 2 = 0 . 1123, Ω Λ = 0 . 728, n s = 0 . 963, τ = 0 . 087, ∆ 2 R ( k = 0 . 002 Mpc -1 ) = 2 . 441 × 10 -9 (Table 14 of Komatsu et al. 2011), we have made 1000 mock data sets that mimic the WMAP beam resolution and instrument noise for each channel. We use the CAMB software to obtain the theoretical model power spectrum (Lewis et al. 2000). We assumed that the temperature fluctuations and instrument noise follow the Gaussian distribution. In the simulation data sets we do not consider any effect due to the residual foreground contamination. We have analyzed the one thousand WMAP simulation mock data sets in the same way as the real data set is analyzed. Before combining the cross power spectra into the single power spectrum, each cross power spectrum has been averaged within l -bins specified to reduce the statistical fluctuations due to the cosmic variance and the instrument noise. We use the same l -bins defined by the WMAP 7-year data analysis (Larson et al. 2011). Thus, in a case when we measure the power spectrum from the WMAP data with a sky fraction much smaller than the area enclosed by KQ85 mask, the measured powers at low and high l regions become uncertain and fluctuate significantly with strong crosscorrelations between adjacent bins (see below). We average the 15 binned cross power spectra into the single angular power spectrum by applying the combining algorithm which is similar to that used in the WMAP first year data analysis (Hinshaw et al. 2003). For each l -bin, we construct a covariance matrix defined as where i and j denote the cross combination of the WMAP frequency channels (V1V2, V1W1, W1W3, and so on), b is the binning index ( b = 1,. . . ,45), C i b is the measured cross power spectrum for channel combination i , ¯ C i b is the ensemble averaged cross power spectrum for the same channel combination expected in the concordance ΛCDM model (we have used one thousand WMAP mock data sets to estimate the ensemble averaged quantities), and is the covariance matrix between errors expected to be caused during the subtraction of powers due to unresolved point sources. We use σ 2 src = ( δA ps ) 2 = (0 . 0007 µ K 2 sr) 2 . The final combined angular power spectrum is obtained by To estimate the uncertainty for the combined power spectrum ˆ C b , we obtain one thousand combined power spectra ( ˆ C sim b ) together with their average and standard deviation at each bin. The standard deviation measured at each bin is used as the uncertainty (or error bar) for the measured combined power spectrum. By comparing the theoretical model power spectrum with the average power spectrum obtained from the simulation data sets, we can check whether our power spectrum measurement algorithm works correctly or not. The left panel of Fig. 2 shows the angular power spectra of the WMAP 7-year temperature maps measured with the uniform (red) and the inverse-noise (blue dots with error bars) weighting schemes. We have used the sky area defined by the KQ85 mask that was adopted in the WMAP team's temperature analysis. For a comparison, the WMAP team's measurement has been shown together as grey dots with error bars (Larson et al. 2011). Our estimations of the angular power spectrum in both the uniform and the inversenoise weighting schemes are consistent with the WMAP team's estimation. We note that the amplitude of the power spectrum at the 41th bin (851 ≤ l ≤ 900) is slightly larger than the WMAP team's estimation, but they are statistically consistent with each other. Although statistically consistent, our result is a bit different from the WMAP team result at low l region because the WMAP team applied the different estimation method (Blackwell-Rao estimator based on Gibbs sampling) at this low l region while we simply used the pseudo power spectrum estimation method over the whole l range. The right panel of Fig. 2 shows the averaged angular power spectra measured from the 1000 mock data sets that mimic the WMAP instrument performance based on the concordance ΛCDM model. The averaged power spectra based on both the uniform and the inverse-noise weighting schemes coincide with each other and restore the assumed model power spectrum (green curve) within measurement uncertainties, which demonstrates that our measuring algorithm works correctly. However, our algorithm gives a slightly positive bias with respect to the true value at the last two bins at the highest multipoles, where the uncertainty due to the finite size of the pixels is significantly large. Comparing the results based on the two weighting schemes, we can see that in a case of the inverse-noise weight scheme the size of error bars is bigger (smaller) at lower (higher) multipoles.", "pages": [ 4, 5, 6, 7 ] }, { "title": "4. Power spectra of various local areas on the sky", "content": "Here we present the angular power spectra measured on various local regions on the sky defined based on the simple criteria such as the Galactic latitude or the number-of-observations cuts. To define a partial sky region, we basically use the KQ85 mask map and the number of observations at the W band (W4 DA). By putting a limit on the Galactic latitude or the number of observations, we have obtained several local regions with different characters. Figure 3 summarizes the mask maps (or the window functions in the uniform weighting scheme) that are used in our power spectrum measurements. We estimate the angular power spectra on each local area using the corresponding mask map as a window function. Especially, the north hat ( b ≥ 30 · ; b is the Galactic latitude) and the south hat ( b ≤ -30 · ) regions defined by the simple Galactic latitude cut show a difference in the power amplitude around the third peak of the angular power spectrum, which can be interpreted as the anomaly effect (see Fig. 4 below). To assess the statistical significance of the observed anomaly, we compare the difference of power spectrum amplitudes with the prediction of the fiducial flat ΛCDM model. Then, we search for the origin of the observed anomaly by considering possibilities due to the residual Galaxy contamination, the WMAP instrument noise, and unresolved point sources.", "pages": [ 7 ] }, { "title": "4.1 North hat and south hat", "content": "Figure 4 displays the angular power spectra measured on the north hat and the south hat regions. Each power spectrum has been estimated in both the uniform and the inverse-noise weighting schemes separately. The overall features in the measured angular power spectra are similar to the power spectrum measured on the whole sky region with KQ85 mask (grey curves). However, the amplitudes of the power spectrum around the third peak show opposite deviations in the south hat and the north hat. We can see such a difference more dramatically in the middle panels of Fig. 4, where the difference of powers between the south hat (SH) and north hat (NH) are displayed for an ease of comparison. We have also estimated the average and the variance of differences between SH and NH powers from the one thousand WMAP mock data sets, which are shown as dark green dots with 1 σ (dark grey) and 3 σ (light grey) error bars. Since we have assumed the homogeneity and isotropy of the universe during the production of the WMAP mock data sets, the average of differences is expected to be zero over the whole l range. We note that in the third peak (that corresponds to the 40th bin; 801 ≤ l ≤ 850) the difference between SH and NH regions is statistically significant because it is located around the 3 σ deviating from the zero value. The histogram of differ- etween SH and NH powers in the 40th bin measured from the ΛCDM based WMAP mock data sets and the value measured from the WMAP data (vertical red dashed line) also confirms this result (bottom panels of Fig. 4).", "pages": [ 7, 8 ] }, { "title": "4.2 Low latitude area", "content": "The foreground contamination at high Galactic latitude regions like the north hat and the south hat is generally expected to be small. Although the foreground model has been subtracted from the observed temperature fluctuations and the sky region that are significantly contaminated by the Galactic emission has been excluded by the KQ85 mask, we can expect that the residual foreground emission at low latitude regions may be stronger than high latitude regions such as the north hat and the south hat. We estimated angular power spectra on two separate regions at the low Galactic latitude defined as low latitude north (0 · < b < 30 · ) and south ( -30 · < b < 0 · ) regions, where the Galactic plane region has been excluded by the KQ85 mask. The results are presented in Fig. 5 with a similar format as in Fig. 4. The angular power spectra measured on the sky regions that are expected to be contaminated by the residual Galactic foreground emission show different features from those seen in the cases of the north and south hats. The behavior of power spectrum amplitudes around the third peak is opposite to the case of north versus south hat regions. The power on the low latitude north region is larger than that on the south region, and the difference between the north and the south regions is more significant at the 41th bin (851 ≤ l ≤ 900). The difference is again statistically significant with a 3 σ deviation from the zero value (for uniform weighting scheme), which is another anomaly found in this work. Although not shown here, if we consider the whole northern and southern hemispheres as two localized regions, then the anomalies at high and low Galactic latitudes are compensated and not noticeable. Furthermore, there are two more noticeable differences between the north and the south regions at the second bin ( l = 3) and at the last bin (1101 ≤ l ≤ 1200). For the latter, the measured difference exceeds 3 σ from the zero value for the inversenoise weighting scheme (see also Table 1 below). Therefore, based on this comparison, it seems that the northsouth anomaly around the third peak of the angular power spectrum observed on the high Galactic latitude north and south hat regions is not due to any possible residual Galactic foreground emission.", "pages": [ 8, 9 ] }, { "title": "4.3 Instrument noise", "content": "Although the WMAP satellite probed the CMB temperature fluctuations on the whole sky, its scanning strategy is somewhat inhomogeneous such that the regions near the ecliptic poles were probed many times as compared to the regions near the ecliptic plane. Thus, it is expected that the CMB anisotropy is strongly affected by the instrument noise in the ecliptic plane region. To quantify such an effect due to the WMAP instrument noise, we use the number of observations, N obs ( p ), at the WMAP W4 frequency channel, which is shown in Fig. 1. Defining regions with high and low instrument noise by simply putting threshold limits on the number of observations results in the geometrical sky regions whose boundary is not smooth and the pixels near the boundary are not contiguous with many small islands out- side the primary region, which prevents us to obtain the unbiased estimation of the power spectrum. To avoid this problem, we have used the Smoothing program of the HEALPix software to smooth the map of number-of-observations at the W4 frequency channel with a Gaussian filter of 1 · FWHM. Then, we define the north hat (or the south hat) regions with high instrument noise by selecting pixels with N obs ( p ) < 2500 on the smoothed number-of-observations map. We also exclude the island areas that are not included in the primary big regions. Similarly, to define the regions with low instrument noise, we set N obs ( p ) > 2035 on the same map. The threshold value for the number of observations at the region of high (low) instrument noise has been set to obtain the sky area of about 15% (13%) of the whole sky for N obs ( p ) < 2500 ( N obs ( p ) > 2035). For ecliptic plane regions with high instrument noise (i.e., with low number of observations), we choose the threshold value to include the larger sky area for the purpose of reducing the statistical variance in the power spectrum measurement. The sky regions defined in this way are shown in Fig. 3 ( e )-( h ). We have measured angular power spectra on the ecliptic plane regions with small number of observations (defined as N obs < 2500) that belong to the north hat or the south hat regions. The results are shown in Fig. 6. The angular power spectra measured on the ecliptic plane regions (that belong to the north hat or the south hat regions) show the behavior that is similar to the case of the whole north hat and south hat regions (see Fig. 4). The difference of power spectrum amplitudes between the south and the north hats still statistically significant, deviating from the zero value up to 3 σ . We have also measured angular power spectra on the ecliptic pole regions with large number of observations (defined as N obs ( p ) > 2035), which are shown in Fig. 7. Here, the power spectra measured on the ecliptic pole regions in the north hat and the south hat are consistent with each other in the l range around the third peak. They are also very consistent with the power spectrum measured on the whole sky with KQ85 mask (grey dots with error bars). Around the third peak, the power spectrum measured on the ecliptic pole regions in the north hat is very similar to the case of the whole north hat region, while the power spectrum measured on the ecliptic pole regions in the south hat has decreased in amplitude compared with the case of the whole south hat region (see Fig. 4). As a result, the observed north-south difference in the 40th bin of the power spectrum is around 1 σ confidence limit and is not statistically significant any more (bottom pan- els of Fig. 7). These results strongly suggest that the north-south anomaly around the third peak likely originates from the unknown systematic effects contained on the sky regions that are affected by strong instrument noise. Based on the one thousand WMAP mock data sets, we count the number of cases where the magnitude of the north-south power difference is larger than the observed value for several chosen local regions. The results are summarized in Table 1 for particular l -bins where the high statistical significance is seen. The listed p -values are probabilities of finding the northsouth (absolute) difference larger than the observed value. Although the statistical significance for the north-south anomaly at the high latitude regions becomes weaker for the inverse-noise weighting scheme, the observed anomaly is still rare in the ΛCDM universe with p = 0 . 7%. Comparing the cases for high and low instrument noise at the high latitude regions strongly suggests that the north-south anomaly around the third peak come from the region dominated by the WMAP instrument noise.", "pages": [ 10, 11, 12 ] }, { "title": "4.4 Point sources", "content": "Recently, the Planck point source catalogue has been publicly available (Planck Collaboration 2011b). We take the Planck Early Release Compact Source Catalogue (ERCSC) at 100 GHz frequency band and exclude the sources at high Galactic latitude ( | b | ≥ 30 · ) whose angular position on the sky is located at the excluded region with M ( p ) = 0 in the KQ85 mask map. The remaining 163 radio point sources at | b | ≥ 30 · are considered as the sources that were newly discovered by the Planck satellite at 100 GHz. By excluding (i.e., by assigning M ( p ) = 0 at) the circular areas centered on these point sources with radius 0 . · 6 in the WMAP team's KQ85 mask map, we have produced a new mask map (or window function) and measured the angular power spectra based on this new window function. Because the number of newly discovered sources are very small, the additional exclusion of point sources almost does not affect the angular power spectra on the north and the south hat regions. Therefore, at the present stage we cannot find any evidence that the north-south anomaly around the third peak of the angular power spectrum originates from the unresolved point sources.", "pages": [ 12 ] }, { "title": "4.5 Dependence of the north-south anomaly on the bin-width and the Galactic latitude cut", "content": "Here we investigate the effect of the bin-width and the Galactic latitude cut on the detected north-south anomaly around the third peak in the angular power spectrum measured from the WMAP 7-year data. Our detection of the north-south anomaly is based on the 40th bin around the third peak which ranges over 851 ≤ l ≤ 900 (∆ l = 50); see Fig. 4. By fixing the bin center at l = 825, we have increased the bin-width by 20% (796 ≤ l ≤ 855; ∆ l = 60) and 50% (788 ≤ l ≤ 862; ∆ l = 75). During the re-binning process, we reduce the total number of bins into 43 and widen the width of adjacent bins appropriately; now our focused bin centered at l = 825 is located at the 39th bin. Based on the new l -binning, we have measured angular power spectra from the WMAP 7-year data and one thousand sets of WMAP mock observations. The results are shown in Fig. 8. In the case of the uniform weighting scheme, the increase of the bin-width increases the statistical significance of the north-south anomaly; there is no occurrence of such an anomaly larger than the observed one among the one thousand WMAP simulations. However, in the case of the inverse-noise weighting scheme, the statistical significance slightly decreases for the 50% increased bin-width (see Table 1 for the summary of statistical significances). We also look into how our conclusion is sensitive to the Galactic latitude cut. We compare the power spectrum measurements for different latitude cuts, | b | = 25 · , 30 · , 35 · , 40 · , and 45 · (Fig. 9; the case of | b | = 30 · is presented in Fig. 4). We notice that the north-south anomaly becomes the most significant in the case of | b | = 35 · cut, and no such an anomaly occurs in both weighting schemes among the one thousand WMAP mock data sets (see Table 1). Furthermore, the statistical significance of this anomaly becomes weaker as the latitude cut higher or lower than | b | = 35 · is applied. For | b | = 25 · (45 · ), the p value is 5 . 9% (12%) in the inverse-noise weighting scheme. Up to the Galactic latitude cut | b | = 40 · , the north-south anomaly is maintained with a high statistical significance (see Table 1). For higher latitude cuts like | b | ≥ 45 · , however, the north-south anomaly in the power spectrum amplitude does not have the statistical significance any more. According to the two results shown above, the observed north-south anomaly has weak dependences on the bin-width used in the power spectrum estimation and the Galactic latitude cut, which implies that the north-south anomaly is the realistic feature present in the WMAP data and supports our primary conclusion.", "pages": [ 12, 13 ] }, { "title": "5. Conclusions", "content": "In this work we have compared the angular power spectra measured on various local sky regions using the WMAP 7-year temperature anisotropy data. The sky regions studied in this work are the low Galactic contamination regions at high Galactic latitude (north hat and south hat), the strong Galactic contamination regions at low Galactic latitude north and south regions, the regions dominated by the WMAP instrument noise (ecliptic plane regions), and the regions of low instrument noise (ecliptic pole regions). We found that the power spectra around the third peak measured on the north hat and the south hat regions show an anomaly which is statistically significant, deviating around 3 σ from the ΛCDM model prediction. We have tried to identify the cause of this anomaly by performing the similar analysis on the low latitude regions and the regions with high or low instrument noise. Curiously, in the low Galactic latitude ( | b | < 30 · ) there appears another but less statistically significant northsouth anomaly around the third peak, whose behavior is opposite to the one seen in the high latitude regions and compensates the anomalies in the whole northern and southern hemispheres. At the present situation, we cannot draw any firm conclusion for the origin of the observed anomaly. However, we found that the observed north-south anomaly maintains with the high statistical significance in the power spectra measured on the regions with high instrument noise, and the anomaly becomes weaker in the power spec- on the regions with low instrument noise. Thus, in our present analysis the observed anomaly is significant on the sky regions that are dominated by the WMAP instrument noise. We have verified that the observed north-south anomaly around the third peak has only weak dependences on the bin-width used in the power spectrum estimation and the Galactic latitude cut, which strengthens our conclusion (see Figs. 8 and 9). The location of the third peak in the angular power spectrum corresponds to the angular scale that approaches the WMAP resolution limit and that is dom- inated by the WMAP instrument noise. Because the Planck satellite probes the CMB anisotropy with higher angular resolution and sensitivity than the WMAP, we expect that the origin of the observed anomaly will be identified in more detail when the Planck data becomes publicly available (Planck Collaboration et al. 2011a). It is possible that our detection of the north-south anomaly may be strongly driven by a posteriori statistics based on the fact that we have computed the power spectra on two different parts of the sky and only then noticed a peculiar discrepancy between the two power amplitudes in one bin around the third peak, neglecting the fact that there could have been a similar discrepancy at any of the other multipole bins (see Zhang & Huterer 2010 for the related discussion). One may argue that we may properly quantify the statistical significance of the detected anomaly from the distribution of maximum north-south differences (in unit of standard deviation) over all multipole bins. The result for north and south hats ( | b | ≥ 30 · ) in the uniform weighting scheme is shown in Fig. 10. In the histogram of one thousand values of the maximum (among 45 bins) north-south power difference in unit of standard deviation, each estimated from the WMAP mock observations, the detected north-south anomaly around the third peak becomes statistically less significant; the probability of finding cases with a deviation larger than the measured value is p = 75 / 1000 = 7 . 5% which is slightly less than 2 σ confidence limit. However, it should be emphasized that the point described above is correct only under the condition that the power spectrum at different multipoles is equally affected by exactly the same CMB physics, Galaxy foreground, and instrument noise properties, which is generally not true in our case of harmonic space. The power spectrum at low multipoles is dominated by the integrated Sachs-Wolfe effect and is more affected by the survey geometry while the power spectrum at higher multipoles is more concerned with the physics of acoustic oscillation and the WMAP instrument noise. For example, it is not justified to consider all multipole bins in quantifying the statistical significance of an anomaly detected in one multipole bin if the anomaly originates from the phenomenon that is influential at the local multipole range. Therefore, our original interpretation of the detected north-south anomaly around the third peak as statistically significant based on the analysis focusing on the corresponding multipole bin is still valid before the origin of the anomaly becomes unveiled. As the Planck data will be soon available, we anticipate that our issue of whether the anomaly is intrinsic one or due to the WMAP instrument noise will be resolved by the Planck data.", "pages": [ 13, 14, 15 ] }, { "title": "ACKNOWLEDGMENTS", "content": "We acknowledge the use of the Legacy Archive for Microwave Background Data Analysis (LAMBDA). Support for LAMBDA is provided by the NASA Office of Space Science. Some of the results in this paper have been derived using the HEALPix and the CAMB softwares. This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-341-C00022).", "pages": [ 15 ] }, { "title": "REFERENCES", "content": "Alpher, R.A. & Herman, R. 1948, Evolution of the Universe, Nature, 162, 774 Ansari, R. & Magneville, C. 2010, Partial CMB maps: bias removal and optimal binning of the angular power spectrum, MNRAS, 405, 1421 Basak, S. & Delabrouille, J. 2012, A needlet internal linear combination analysis of WMAP 7-year data: estimation of CMB temperature map and power spectrum, MNRAS, 419, 1163 Benoˆıt, A., Ade, P., Amblard, A., et al. 2003, The cosmic microwave background anisotropy power spectrum measured by Archeops, A&A, 399, L19 Bennett, C.L., Halpern, M., Hinshaw, G., et al. 2003, First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Preliminary Maps and Basic Results, ApJS, 148, 1 Bernui, A. 2008, Anomalous CMB north-south asymmetry, Phys. Rev. D, 78, 063531 Bond, J.R. & Efstathiou, G. 1987, The statistics of cosmic background radiation fluctuations, MNRAS, 226, 655 Bond, J.R., Jaffe, A.H., & Knox, L. 1998, Estimating the power spectrum of the cosmic microwave background, Phys. Rev. D, 57, 2117 Brown, M.L., Castro, P.G., & Taylor, A.N. 2005, Cosmic microwave background temperature and polarization pseudoC l estimators and covariances, MNRAS, 360, 1262 Brown, M.L., Ade, P., Bock, J., et al. 2009, Improved Measurements of the Temperature and Polarization of the Cosmic Microwave Background from QUaD, ApJ, 705, 978 Carlstrom, J.E., Kovac, J., Leitch, E.M., & Pryke, C. 2003, Status of CMB polarization measurements from DASI and other experiments, New Astron. Rev., 47, 953 Chiang, L.Y. & Chen, F.F. 2012, Direct Measurement of the Angular Power Spectrum of Cosmic Microwave Background Temperature Anisotropies in the WMAP Data, ApJ, 751, 43 Chon, G., Challinor, A., Prunet, S., Hivon, E., & Szapudi, I. 2004, Fast estimation of polarization power spectra using correlation functions, MNRAS, 350, 914 Dahlen, F.A., & Simons, F.J. 2008, Spectral estimation on a sphere in geophysics and cosmology, Geophysical Journal International, 174, 774 Das, S., Hajian, A., & Spergel, D.N. 2009, Efficient power spectrum estimation for high resolution CMB maps, Phys. Rev. D, 79, 083008 Dicke, R.H., Peebles, P.J.E., Roll, P.G., & Wilkinson, D.T. 1965, Cosmic Black-Body Radiation, ApJ, 142, 414 Efstathiou, G. 2004, Myths and truths concerning estimation of power spectra: the case for a hybrid estimator, MNRAS, 349, 603 Eriksen, H.K., O'Dwyer, I.J., Jewell, J.B., et al. 2004a, Power Spectrum Estimation from High-Resolution Maps by Gibbs Sampling, ApJS, 155, 227 Eriksen, H.K., Hansen, F.K., Banday, A.J., G'orski, K.M., & Lilje, P.B. 2004b, Asymmetries in the Cosmic Microwave Background Anisotropy Field, ApJ, 605, 14 Eriksen, H.K., Novikov, D.I., Lilje, P.B., Banday, A.J., & G'orski, K.M. 2004c, Testing for Non-Gaussianity in the Wilkinson Microwave Anisotropy Probe Data: Minkowski Functionals and the Length of the Skeleton, ApJ, 612, 64 Eriksen, H.K., Huey, G., Saha, R., et al. 2007a, A Reanalysis of the 3 Year Wilkinson Microwave Anisotropy Probe Temperature Power Spectrum and Likelihood, ApJ, 656, 641 Eriksen, H.K., Banday, A.J., G'orski, K.M., Hansen, F.K., & Lilje, P.B. 2007b, Hemispherical Power Asymmetry in the Third-Year Wilkinson Microwave Anisotropy Probe Sky Maps, ApJ, 660, L81 Fay, G., Guilloux, F., Betoule, M., et al. 2008, CMB power spectrum estimation using wavelets, Phys. Rev. D, 78, 083013 Fowler, J.W., Acquaviva, V., Ade, P.A.R., et al. 2010, The Atacama Cosmology Telescope: A Measurement of the 600 < /lscript < 8000 Cosmic Microwave Background Power Spectrum at 148 GHz, ApJ, 722, 1148 G'orski, K. M. 1994, ApJ, On determining the spectrum of primordial inhomogeneity from the COBE DMR sky maps: Method, 430, L85 G'orski, K.M., Wandelt, B.D., Hansen, F.K., Hivon, E., & Banday, A.J. 1999, The HEALPix Primer, arXiv:astro-ph/9905275 G'orski, K.M., Hivon, E., Banday, A.J., et al. 2005, HEALPix: A Framework for High-Resolution Discretization and Fast Analysis of Data Distributed on the Sphere, ApJ, 622, 759 Hansen, F.K., G'orski, K.M., & Hivon, E. 2002, Gabor transforms on the sphere with applications to CMB power spectrum estimation, MNRAS, 336, 1304 Hansen, F.K., Banday, A.J., & G'orski, K.M. 2004a, Testing the cosmological principle of isotropy: local power-spectrum estimates of the WMAP data, MNRAS, 354, 641 Hansen, F.K., Balbi, A., Banday, A.J., & G'orski, K.M. 2004b, Cosmological parameters and the WMAP data revisited, MNRAS, 354, 905 Hansen, F.K., Banday, A.J., G'orski, K.M., Eriksen, H.K., & Lilje, P.B. 2009, Power Asymmetry in Cosmic Microwave Background Fluctuations from Full Sky to Sub-Degree Scales: Is the Universe Isotropic?, ApJ, 704, 1448 Hinshaw, G., Spergel, D.N., Verde, L., et al. 2003, First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: The Angular Power Spectrum, ApJS, 148, 135 Hinshaw, G., Nolta, M.R., Bennett, C.L., et al. 2007, Three-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Temperature Analysis, ApJS, 170, 288 Hivon, E., G'orski, K.M., Netterfield, C.B., et al. 2002, MASTER of the Cosmic Microwave Background Anisotropy Power Spectrum: A Fast Method for Statistical Analysis of Large and Complex Cosmic Microwave Background Data Sets, ApJ, 567, 2 Hoftuft, J., Eriksen, H.K., Banday, A.J., et al. 2009, Increasing Evidence for Hemispherical Power Asymmetry in the Five-Year WMAP Data, ApJ, 699, 985 Hu, W. & Dodelson, S. 2002, Cosmic Microwave Background Anisotropies, ARA&A, 40, 171 Huffenberger, K.M., Eriksen, H.K., & Hansen, F.K. 2006, Point-Source Power in 3 Year Wilkinson Microwave Anisotropy Probe Data, ApJ, 651, L81 Jarosik, N., Bennett, C.L., Dunkley, J., et al. 2011, Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Sky Maps, Systematic Errors, and Basic Results, ApJS, 192, 14 Jones, W.C., Ade, P.A.R., Bock, J.J., et al. 2006, A Measurement of the Angular Power Spectrum of the CMB Temperature Anisotropy from the 2003 Flight of BOOMERANG, ApJ, 647, 823 Keisler, R., Reichardt, C.L., Aird, K.A., et al. 2011, A Measurement of the Damping Tail of the Cosmic Microwave Background Power Spectrum with the South Pole Telescope, ApJ, 743, 28 Komatsu, E., Dunkley, J., Nolta, M.R., et al. 2009, Five-Year Wilkinson Microwave Anisotropy Probe Observations: Cosmological Interpretation, ApJS, 180, 330 Komatsu, E., Smith, K.M., Dunkley, J., et al. 2011, Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation, ApJS, 192, 18 Larson, D., Dunkley, J., Hinshaw, G., et al. 2011, Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Power Spectra and WMAP-derived Parameters, ApJS, 192, 16 Lee, A.T., Ade, P., Balbi, A., et al. 2001, A High Spatial Resolution Analysis of the MAXIMA-1 Cosmic Microwave Background Anisotropy Data, ApJ, 561, L1 Lewis, A., Challinor, A., & Lasenby, A. 2000, Efficient Computation of Cosmic Microwave Background Anisotropies in Closed Friedmann-Robertson-Walker Models, ApJ, 538, 473 Mason, B.S., Pearson, T.J., Readhead, A.C.S., et al. 2003, The Anisotropy of the Microwave Background to l = 3500: Deep Field Observations with the Cosmic Background Imager, ApJ, 591, 540 Mitra, S., Sengupta, A.S., Ray, S., Saha, R., & Souradeep, T. 2009, Cosmic microwave background power spectrum estimation with non-circular beam and incomplete sky coverage, MNRAS, 394, 1419 Mortlock, D.J., Challinor, A.D., & Hobson, M.P. 2002, Analysis of cosmic microwave background data on an incomplete sky, MNRAS, 330, 405 Nolta, M.R., Dunkley, J., Hill, R.S., et al. 2009, FiveYear Wilkinson Microwave Anisotropy Probe Observations: Angular Power Spectra, ApJS, 180, 296 Oh, S.P., Spergel, D.N., & Hinshaw, G. 1999, An Efficient Technique to Determine the Power Spectrum from Cosmic Microwave Background Sky Maps, ApJ, 510, 551 Page, L., Nolta, M.R., Barnes, C., et al. 2003, First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Interpretation of the TT and TE Angular Power Spectrum Peaks, ApJS, 148, 233 Page, L., Hinshaw, G., Komatsu, E., et al. 2007, Three-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Polarization Analysis, ApJS, 170, 335 Park, C.-G. 2004, Non-Gaussian signatures in the temperature fluctuation observed by the Wilkinson Microwave Anisotropy Probe, MNRAS, 349, 313 Peebles, P.J.E. & J.T. Yu, J.T. 1970, Primeval Adiabatic Perturbation in an Expanding Universe, ApJ, 162, 815 Peiris, H.V., Komatsu, E., Verde, L., et al. 2003, First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Implications For Inflation, ApJS, 148, 213 Penzias A.A. & Wilson R.W. 1965, A Measurement of Excess Antenna Temperature at 4080 Mc/s, ApJ, 142, 419 Planck Collaboration, Ade, P.A.R., Aghanim, N., et al. 2011a, Planck early results. I. The Planck mission, A&A, 536, A1 Planck Collaboration, Ade, P.A.R., Aghanim, N., et al. 2011, Planck early results. VII. The Early Release Compact Source Catalogue, A&A, 536, A7 Polenta, G., Marinucci, D., Balbi, A., et al. 2005, Unbiased estimation of an angular power spectrum, J. Cosmol. Astropart. Phys., 11, 1 Reichardt, C.L., Ade, P.A.R., Bock, J.J., et al. 2009, High-Resolution CMB Power Spectrum from the Complete ACBAR Data Set, ApJ, 694, 1200 Sachs, R.K. & Wolfe, A.M. 1967, Perturbations of a Cosmological Model and Angular Variations of the Microwave Background, ApJ, 147, 73 Saha, R., Jain, P., & Souradeep, T. 2006, A Blind Estimation of the Angular Power Spectrum of CMB Anisotropy from WMAP, ApJ, 645, L89 Saha, R., Prunet, S., Jain, P., & Souradeep, T. 2008, CMB anisotropy power spectrum using linear combinations of WMAP maps, Phys. Rev. D, 78, 023003 Samal, P.K., Saha, R., Delabrouille, J., et al. 2010, Cosmic Microwave Background Polarization and Temperature Power Spectra Estimation Using Linear Combination of WMAP 5 Year Maps, ApJ, 714, 840 Scott, D. & Smoot, G. 2006, Cosmic Microwave Background Mini-Review, arXiv:astro-ph/0601307 Smoot, G.F., Bennett, C.L., Kogut, A., et al. 1992, Structure in the COBE differential microwave radiometer first-year maps, ApJ, 396, L1 Souradeep, T., Saha, R., & Jain, P. 2006, Angular power spectrum of CMB anisotropy from WMAP, New Astron. Rev., 50, 854 Spergel, D.N., Verde, L., Peiris, H.V., et al. 2003, First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters, ApJS, 148, 175 Spergel, D.N., Bean, R., Dor'e, O., et al. 2007, Three-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Implications for Cosmology, ApJS, 170, 377 Szapudi, I., Prunet, S., Pogosyan, D., Szalay, A.S., & Bond, J.R. 2001, Fast Cosmic Microwave Background Analyses via Correlation Functions, ApJ, 548, L115 Tegmark, M. 1997, How to measure CMB power spectra without losing information, Phys. Rev. D, 55, 5895 Wandelt, B.D., Hivon, E., & G'orski, K.M. 2001, Cosmic microwave background anisotropy power spectrum statistics for high precision cosmology, Phys. Rev. D, 64, 083003 Wandelt, B.D. & Hansen, F.K. 2003, Fast, exact CMB power spectrum estimation for a certain class of observational strategies, Phys. Rev. D, 67, 023001 Wandelt, B.D., Larson, D.L., & Lakshminarayanan, A. 2004, Global, exact cosmic microwave background data analysis using Gibbs sampling, Phys. Rev. D, 70, 083511 Yoho, A., Ferrer, F., & Starkman, G.D. 2011, Degreescale anomalies in the CMB: Localizing the first peak dip to a small patch of the north ecliptic sky, Phys. Rev. D, 83, 083525 Zhang, R., & Huterer, D. 2010, Disks in the sky: A reassessment of the WMAP'cold spot'?, Astroparticle Physics, 33, 69", "pages": [ 15, 16, 17, 18 ] } ]

[ { "title": "Jae-Weon Lee ∗", "content": "Department of energy resources development, Jungwon University, 5 dongburi, Goesan-eup, Goesan-gun Chungbuk Korea 367-805 (Dated: December 16, 2018) It is shown that the first law of thermodynamics and the holographic principle applied to an arbitrary large cosmic causal horizon naturally demand the zero cosmological constant and non-zero dynamical dark energy in the form of the holographic dark energy. Semiclassical analysis shows that the holographic dark energy has a parameter d = 1 and an equation of state comparable to current observational data, if the entropy of the horizon saturates the Bekenstein-Hawking bound. This result indicates that quantum field theory should be modified at large scale to explain dark energy. The relations among dark energy, quantum vacuum energy and entropic gravity are also discussed. Type Ia supernova (SN Ia) observations [1, 2], the Sloan Digital Sky Survey (SDSS) [3-6] and cosmic microwave background observations [7] all indicate that the current universe is expanding at an accelerating rate. The expansion can be explained if there is a negative pressure fluid called dark energy of which pressure p DE and energy density ρ DE satisfy w DE ≡ p DE /ρ DE < -1 / 3. Being one of the most important unsolved puzzles in modern physics and cosmology, the dark energy problem consists of three sub-problems [8]; why it is so small, nonzero, and comparable to the critical density at the present. We also need to explain why the cosmological constant Λ is so small or exactly zero. Solving this problem is not an easy task, because quantum field theory (QFT) predicts huge zero point energy that can play a role of Λ. It is very hard to reconcile the great success of QFT at small scales with this failure of QFT to explain dark energy. There are already many works on this problem [9-11], however, the problem seems to be far from a solution. In this paper, it is suggested that if the holographic principle holds for a cosmic causal horizon, the cosmological constant should be exactly zero and there should be holographic dark energy (HDE) consistent with the recent observational data. The holographic principle [12] is a conjecture claiming that all of the information in a region can be described by the physics at the boundary of the region and that the maximal number of degrees of freedom in the region is proportional to its surface area rather than the volume. More specifically, it was conjectured that the Bekenstein-Hawking Entropy is the information bound that a region of space with a surface area A can contain [13]. Based on black hole physics Cohen et al [14] proposed that the total energy in a region can not be larger than that of a black hole of that size. Therefore, if the region has a size r h = O ( H -1 ), the vacuum energy density is bounded as ρ Λ ≤ O ( M 2 P H 2 ), where H = da/adt is the Hubble parameter with the scale factor a , and M P = √ /planckover2pi1 c/ 8 πG is the reduced Planck mass. Interestingly, saturating the bound gives HDE comparable to the observed dark energy density ρ Λ ∼ 10 -10 eV 4 . However, Hsu [15] pointed out that with the Hubble horizon HDE behaves like matter rather than dark energy. Li [16] suggested that using the future event horizon as IR cutoff we can solve this problem. Recently, based on the holographic principle Verlinde [17] and Padmanabahan [18] proposed a remarkable idea linking gravity to entropy, which brings out many follow-up studies [19-30]. Verlinde derived the Newton's equation and the Einstein equation by assuming that energy inside a holographic screen is the equipartition energy E h ∼ T h N for the screen with the temperature T h and the number of bits N . On the other hand, in a series of works [31-34], Lee et al. suggested that the energy of gravitational systems could be explained by considering information loss at causal horizons. For example, we pointed out that a cosmic causal horizon with a radius r ∼ O ( H -1 ) has temperature T h ∼ 1 /r , entropy S h ∼ r 2 and a kind of thermal energy E h ∼ T h S h ∼ r , which can be dark energy [35]. This dark energy, dubbed 'quantum informational dark energy' [36] or 'entanglement dark energy' [31] by the authors, is similar to the entropic dark energy based on the Verlinde's idea [37-40]. It was also suggested that black hole mass and the Einstein equation itself can be derived from the relation dE h = k B T h dS h , that might have a quantum information theoretic origin [32]. Similarities between this theory and Verlinde's theory were investigated in [34, 41]. In this paper we assume that the holographic principle and the following first-law like definition of the horizon energy hold for a cosmic causal horizon such as the cosmic event horizon or the apparent horiozn. This energy could be the equipartition energy [17], energy from Landauer's principle associated with information loss at the horizon [31] or simply the energy defined by the Clausius relation. Inspired by the entropic [17, 42] or quantum information theoretic [31, 34] interpretation of gravity we take the holographic principle and the horizon energy in Eq. (2) as guiding principles for dark energy study. Let us first recall the cosmological constant problem in the context of QFT. The (classical) time independent cosmological constant Λ c appears in the gravity action as Since the energy-momentum tensor T µν for the vacuum fluctuation 〈 T µν 〉 is usually proportional to a spacetime metric (See for example [43]), 〈 T µν 〉 has been regarded as a candidate for the cosmological constant and dark energy. To calculate its expectation value one usually integrates the zero point energy /planckover2pi1 ω/ 2 for each mode of quantum fields in a flat spacetime. Thus, the energy density of the quantum vacuum is approximately given by where k U ∼ M P is a UV-cutoff and k I ∼ 1 /r is an IR-cutoff. Unfortunately, as is well known, for k U ∼ M P , the estimation gives ρ q ∼ M 4 P ∼ 10 109 eV 4 which is too large to explain the observed dark energy density ρ DE ∼ 10 -12 eV 4 . On the other hand, if we subtract this zero point energy to calculate the renormalized vacuum energy for the universe, we obtain ρ q ∼ H 4 , which is too small compared to the observed dark energy. It is often argued that after taking the vacuum expectation of quantum fields, the Friedmann equation gets an additional constant contribution Λ q = ρ q /M 2 P c 2 = 〈 T 00 〉 /M 2 P c 2 from the vacuum quantum fluctuation ρ q in Eq. (4). (Here, k c is the spatial curvature parameter, which we will set zero for simplicity, and ρ m is the matter energy density.) Thus, the total cosmological constant is Λ = Λ c +Λ q , and the total vacuum energy density is given by Without a fine tuning it seems to be almost impossible for two terms to cancel each other to result in the tiny observed upper bound for the cosmological constant. This is the essence of the cosmological constant problem. Then, in the context of QFT, from where could horizon energy ρ h arise? Recall that ρ q in Eq. (4) was estimated in a flat spacetime. However, for a curved spacetime, after a Bogoliubov transformation there appear excited states in addition to the vacuum. The normal ordered quantum vacuum energy (i.e., with the subtraction of the zero point energy) in a curved spacetime with the UV and the IR cutoffs often has a term in the form of ρ h . One can do a volume integral of /planckover2pi1 ω with the Bogoliubov coefficient β k for the quantum fields in a curved spacetime to obtain ρ h . For example, using the result in [47], it was shown in [46] that ρ vac for the de Sitter universe contains an extra term in addition to the usual zero point energy, where β k ∼ H/k . If we choose M P as the UV-cutoff k U , this extra term gives ρ ' vac ∼ M 2 P H 2 ∼ ρ h . Thus, it is possible that ρ h is actually the average quantum fluctuation energy above the zero point vacuum energy of the curved spacetime in the bulk [48]. This dark energy may be also identified to be the energy of cosmic Hawking radiation [46]. However, this calculation still can not explain why we can ignore the zero point energy in the bulk. It seems that there is no plausible way to overcome this difficulty, as long as we rely on the conventional QFT. We need another fundamental ingredient to solve this problem. Alternatively, we can take not the bulk QFT but the holographic principle as a postulate and describe the bulk physics using only the DOF on the horizon. In this holographic context, to estimate the bulk energy density we can treat the quantum fields on the horizon as a collection of oscillators on the spherical surface with a lattice constant of order O ( k -1 U ). Then, to obtain ρ h we have to sum the zero-point energy of the oscillators with frequency ω on the horizon surface rather than those in the bulk. This rough estimation results in the HDE density, because where Σ i represents a summation over the horizon oscillators with the temperature T h , and the number of oscillators are proportional to the horizon area ∼ ( r/k -1 U ) 2 . At the last step we used the equipartition approximation /planckover2pi1 ω ∼ k B T h ∼ 1 /r . Note that this is just an order of magnitude estimation for comparison, and more accurate solution requires a careful calculation with an appropriate horizon. This result indicates that the bulk QFT overestimates the independent DOF in the bulk and the true vacuum energy of the bulk could be the zero point energy of the boundary DOF on the horizon, which is of order of the normal-ordered bulk vacuum energy in the conventional QFT. What gives the small HDE could be the smallness of the number of independent DOF in the bulk. This redundancy of the bulk DOF can explain why we cannot obtain the correct dark energy density by simply calculating the zero point energy of the bulk. In short, QFT is not a complete theory at the cosmic scale. We need to calculate the horizon energy E h as the vacuum energy of the universe without using QFT. Let us consider a causal cosmic horizon with a radius r , having generic holographic entropy and temperature with constants η and /epsilon1 (See Fig. 1). (Note that these quantities contatin /planckover2pi1 and are usually derived by semiclassical calculations.) In this case the universe is similar to a big black hole with an expanding horizon. For the Bekenstein entropy η = π , and the Hawking-Gibbons temperature /epsilon1 = 1 / 2 π . By assuming the first law and integrating dE h on the isothermal surface Σ of the causal horizon with Eqs. (9) and (10), we obtain the horizon energy Then, the energy density due to E h is given by which has the form of the holographic dark energy [16]. Note that this semiclassical derivation of HDE is different from the usual derivation based on UV-IR relations. ρ h here corresponds to the estimation of the surface vacuum energy in Eq. (8). This kind of dark energy was also derived in terms of entanglement energy [31] and quantum entanglement force [41]. From the above equation we immediately obtain a formula for the constant which is the important parameter determining the nature of HDE. If S h saturates the Bekenstein bound and T h is the Hawking-Gibbons temperature /planckover2pi1 c/ 2 πk B r , then η/epsilon1 = 1 / 2 and d = 1. Thus, the holographic principle applied to a cosmic causal horizon naturally leads to HDE with d = 1 [41], which is favored by observations and theories [44, 45]. There are few works on fixing d value. Li found d = 1 by assuming the universe as a classical black hole [16]. On the contrary in this paper d value is obtained by considering the semiclassical quantities. Let us turn to the cosmological constant problem in this context. From the holographic viewpoint, it is very simple to see why the cosmological constant Λ should be zero. If we apply the holographic principle and the definition of the horizon energy (Eq. (11)) to the cosmic horizon, the bulk vacuum energy density ρ vac in Eq. (6) should be smaller than the horizon energy density ρ h in Eq. (12). Since the principle is one of our starting postulates, the principle should hold strictly even for arbitrary large r , and hence, the vacuum energy E Λ proportional to Λ r 3 is problematic. It clearly violates the holographic principle for large r , where vacuum energy is dominant. According to the principle and the first law of thermodynamics with T h ∝ 1 /r , the total horizon energy E h is proportional to r . For r > r c ≡ √ 3 d 2 Λ , E Λ > E h and the holographic principle can be violated. Thus, the principle holds true for arbitrary r only if the cosmological constant Λ is exactly zero. Note that this argument holds for arbitrary small coefficient of r 3 term in E Λ as long as r can increase infinetely. As the universe expands, the inequality E Λ ≤ E h would be violated eventually. For example, if the cosmological constant is the dark energy, constant ρ Λ is about the present critical density ρ c = 3 H 2 M 2 P / 8 π ∝ 1 /t 2 and within a few Hubble times ρ Λ r 3 will exceed ρ h r 3 . Thus, we can say that the holographic principle insists that the cosmological constant is zero (i.e., ω DE = -1). (Here, we have /negationslash excluded an implausible case that Λ c and a constant part of the quantum contribution miraculously cancel each other to result in ρ vac ∼ 1 /r 2 .) This solution to the cosmological constant problem has its own cost. At the large cosmic scale, we have to abandon QFT and accept the holographic principle and the dark energy problem becomes much easier. As long as the principle holds, the argument about the zero cosmological constant would be valid. Since the principle also solves the other subproblems about dark energy, this approach seems to be promising. Furthermore, the solution above is free from some difficulties often encountered by other approaches such as infrared or ultraviolet modifications of gravity, adjusting initial conditions, or dynamical attractor mechanisms (See [49] for example.). They failed to explain both of the early small universe and current large universe and why the QFT vacuum loops or cosmological phase transitions did not curl up the universe. Let us discuss these facts in detail. First, our approach based on the holographic principle suggests that the energy density from the vacuum loop energy for a quantum field with a UV cutoff energy scale M is ρ M = O ( M 2 H 2 ) /lessmuch O ( M 2 P H 2 ) not of O ( M 4 ). Since the Friedmann equation is ρ tot = 3 H 2 M 2 P , the vacuum loop energy is not a dominant contribution to ρ tot unless M /similarequal M P . Second, our approach could also avoid the issue related to the cosmological phase transitions. For example, consider a phase transition of the scalar field field φ with a thermal effective potential V ( φ, T ), showing the transition at the temperature T = T c . Then, in conventional approaches even if ρ Λ was set to be 0 before the transition, during the phase transition the potential could generate temporally the energy difference between the false vacuum and the true vacuum, which is of -O ( T 4 c ) = -O ( M 2 P H 2 ), where H is the Hubble parameter at the transition. If the absolute value of energy matters, this energy difference could act as a negative cosmological constant and make the universe rapidly collapse. However, in our theory there is always positive O ( M 2 P H 2 ) dark energy that could cancel the negative energy term and prevent the collapse. Third, unlike the dynamical attractor theories, our theory does not directly rely on the contributions of matters to energy-momentum tensor and hence we do not need a feedback mechanism adjusting ρ Λ to precision 10 -120 as long as the horizon radius is O ( H -1 ). One can easily see the similarity between our theory and entropic gravity. In entropic gravity the horizon energy is given by the equipartition law E h = NT h / 2, which is essentially equivalent to our dark energy E h = ∫ T h dS , because S ∼ N in general. Following [41] and [40] one can also obtain an entropic force for the dark energy which could be also identified as a 'quantum entanglement force' as in [41], if S h is the entanglement entropy. Let us compare predictions of our theory with observational data. From ρ DE = ρ h and a cosmological energymomentum conservation equation, one can obtain an effective dark energy pressure [16] in the bulk from which one can derive the equation of state. At this point we need to choose a horizon among various cosmological horizons such as an apparent horizon, a Hubble horizon, and a future event horizon. In the simplest case, only the event horizon can result in the accelerating universe [16]. Thus, from now on we assume the case that the causal horizon is the cosmic event horizon. For this case one can find the equation of state for holographic dark energy as a function of the redshift z as shown in Ref. [16]. Fig. 1 compares this prediction with d = 1 to the observational data obtained from the 182 gold SN Ia data, the baryon acoustic oscillation, SDSS, and the 3-year Wilkinson Microwave Anisotropy Probe (WMAP) data. One can also find that the equation of state [5, 16] w 0 = -1 3 ( 1 + 2 √ Ω 0 Λ d ) , and its change rate at the present w 1 with w DE ( a ) /similarequal w 0 + w 1 (1 -a ). Here the current dark energy density parameter Ω 0 Λ /similarequal 0 . 73. If we use an entanglement entropy calculated in [41] for S h , one can obtain d slightly different from 1. It is also straightforward to study the cases with other horizons such as apparent horizons or Hubble horizons. We saw that the predictions of our theory well agree with the recent observational data. Note that although the cosmological constant is most favored by the cosmological observations, the observational data still allow dynamical dark energy models. For d = 1 these equations give w 0 = -0 . 903 and w 1 = 0 . 104. According to WMAP 5-year data [51], w 0 = -1 . 04 ± 0 . 13 and w 1 = 0 . 11 ± 0 . 7. WMAP 7-year data with the baryon acoustic oscillation, SN Ia, and the Hubble constant yields w 0 = -0 . 93 ± 0 . 13 and w 1 = -0 . 41 +0 . 72 -0 . 71 [7]. It was also shown that holographic dark energy models with an inflation with a number of e-folds N e /similarequal 65 can solve the cosmic coincidence problem [16, 52] thanks to a rapid expansion of the event horizon during the inflation. I summarize how the holographic principle and the horizon energy can solve the dark energy problem. In this theory the dark energy density is small due to the holographic principle, comparable to the critical density due to the O (1 /H ) horizon size, and non-zero due to the quantum fluctuation. The vacuum fluctuation energy is not huge but comparable to the observed dark energy, because conventional QFT overestimates the actual independent DOF. The holographic principle and the first law of thermodynamics also demand that the cosmological constant is zero, because the nonzero time independent cosmological constant is inconsistent with them. Compared to previous works on HDE, our work has following new features. First, albeit simple, the dark energy theory in this paper seems to provide us a logically self-consistent explanations to the all subproblems of the dark energy including the cosmological constant problem. Second, the parameter d is obtained using semiclassical parameters such as Hawking temperature incorporating quantum effects to some extent. Third, the relations among HDE, QFT vacuum energy and entropic gravity are studied. Note that our solution is more than a simple transformation of one problem into another one, because the formalism we used here is based not on the conventional QFT but on the holographic principle that could allow a reformulation of gravity and quantum mechanics in terms of thermodynamics as recently suggested by some authors [17, 48, 53]. Therefore, there is interesting possibility that these thermodynamic approaches could open a new route to understanding not only dark energy but also the unification of quantum mechanics and gravity in the future.", "pages": [ 1, 2, 3, 4, 5, 6 ] }, { "title": "ACKNOWLEDGMENTS", "content": "This work was supported in part by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the ministry of Education, Science and Technology (2010-0024761) and the topical research program (2010-T-1) of Asia Pacific Center for Theoretical Physics.", "pages": [ 6 ] } ]

[ { "title": "A new variable in scalar cosmology with exponential potential", "content": "Hyeong-Chan Kim 1, ∗ 1 School of Liberal Arts and Sciences, Korea National University of Transportation, Chungju 380-702, Korea We present a new way describing the solution of the Einstein-scalar field theory with exponential potential V ∝ e √ 6 βφ/M Pl in spatially flat Friedmann-Robertson-Walker space-time. We introduced a new time variable, L , which may vary in [ -1 , 1]. The new time represents the state of the universe clearly because the equation of state at a given time takes the simple form, w = -1 + 2 L 2 . The universe will inflate when | L | < 1 / √ 3. For β ≤ 1, the universe ends with its evolution at L = β . This implies that the equation of state at the end of the universe is nothing but w = -1 + 2 β 2 . For β ≥ 1, the universe ends at L = 1, where the equation of state of the universe is one. On the other hand, the universe always begins with w = 1 at L = ± 1. PACS numbers: 98.80.-k, 98.80.Cq, 04.20.Jb Keywords: exact solution, scalar cosmology The discovery of cosmic acceleration in 1997, which is regarded to be originated from a dark energy, changed the modern cosmology from the basics. In the context of fundamental theories, it is an important problem to understand the origin of the dark energy. The dimensional reduction of higher dimensional M/string theory typically give rise to a scalar fields with exponential potentials coupled to four-dimensional gravity. It is usually argued that these the homogeneous scalar field slowly rolling on the potential may give rise to a dark energy, which is typically called as quintessence [1, 2]. It is interesting to understand whether exponential potentials could describe observational data for the late-time cosmic acceleration. In cosmology, exponential potentials were much investigated in the past and general exact solutions have appeared [3, 4] . In particular, a general solution in four dimensions was obtained in Ref. [5]. Especially, in Ref. [6], it was also shown that in the simplest case of a homogeneous scalar field coupled to an exponential potential can be solved in a direct way in d -dimensions by introducing new variables which decouple the system. In Ref. [7], it was shown an example of cosmology which starts with a decelerating expansion, at some point it experiences a transitory period of acceleration, and it ends with decelerating expansion again. The origin of the acceleration was further clarified in Ref. [8]. As noted in Ref. [6], the explicit general solution will present a clear view in parametric space on the physical origin of the acceleration. We are interested in the universe which is spatially flat, homogeneous, and isotropic, with metric: where a ( t ) is the scale factor. We find exact cosmological solutions of Einstein equation coupled to a scalar field φ with action in standard form, where we set M 2 Pl = 1 / (8 πG ). The dynamics of the scalar field and gravity can be dealt with a pair of equations where overdot and prime denote derivatives with respect to time and the scalar field, respectively, and H = ˙ a/a is the Hubble parameter. We consider the exponential potential, Without loss of generality we set β > 0. In the case of an exponential potentials, the scalar cosmology in four dimensions were investigated [9] and general exact solutions were found [6, 10]. Interesting properties of the cosmological solutions with exponential solutions were also discussed [11-13]. In Refs. [14-18], it was shown that the equations of motion (3) and (4) is equivalent to the ' generating equation ': which is typically called as the Hamilton-Jacobi equation, supplemented by Summarizing, the two coupled differential equations (3) and (4) with respect to time is reduced to one non-linear first order differential equation (6) with respect to the scalar field supplemented by the equation giving the dynamics (7). We solve Eq. (6) directly to obtain the generating function in the cases of the exponential potentials. One may easily guess a specific solution of the generating function since the derivative of the exponential is nothing but an exponential: where a real generating function of this form exists only when | β | < 1. This example was dealt in Ref. [14] and its fixed point properties including contributions from perfect fluid were studied in Ref. [11]. The scalar field and scale factor corresponding to this is given by The whole cosmological solutions with the potential (5) were studied in Ref. [6] by changing the equations into two Riccati equations using a couple of coordinates transformation. The model was also studied in terms of Noether charge method in Ref. [19] and Hamilton-Jacobi method [20]. The model is extended to include a perfect fluid numerically in Ref. [4, 11]. The solution (9) corresponds to a solution approaching to the fixed point ( x, y ) = ( λ/ √ 6 , (1 -λ 2 / 6) 1 / 2 ) in Ref. [11]. 1 Another two fixed points ( ± 1 , 0) in the reference corresponds to V = 0, which is not relevant at the present situation. In this work, we present the whole generating functions by solving Eq. (6) directly. The general solution of Eq. (6) representing expanding universe is where we restrict the range of L to be | L | ≤ 1. For the range of | L | > 1, the generating function describes the potential V ( φ ) = -3 M 2 Pl H 2 0 e √ 6 βφ M Pl , which we are not interested in at the present cosmological situation. φ c is chosen to be a parameter characterizing the maximum possible value of the scalar field during evolution, which will be shown below, and the scalar field evolves as For β = 1, this equation is ill-defined. But, by using β → 1 limit, we get The solution (9) corresponds to the limit φ c →∞ with L = β . The schematic plot for the time evolution of the scalar field is given in Fig. 1. The equation of state parameter of the universe can be expressed in a quite simple form: Because of the simplicity, it becomes easier to understand the behavior of the universe as a function of L . The equation of state becomes that of the cosmological constant at L = 0. The universe inflates if w < -1 / 3, which gives -1 / √ 3 < L < 1 / √ 3. The relation between the cosmological time and the time L is explicitly given in Appendix A. Rather than explicitly describing the whole detail, let us remind the following key results: For β < 1, there are two corresponding universes. One begins at L = -1 ( t = 0) and ends at L = β ( t = ∞ ) and the other begins at L = +1 ( t = 0) and ends at L = β ( t →∞ ). Remember that the time arrow is reversed in the second case. For β ≥ 1, the universe begins at L = -1 ( t = 0) and ends at L = 1 ( t = ∞ ). Noting H = a -1 da/dt , the scale factor, using Eq. (A2) in Appendix A, is given by Integrating, we get the scale factor as a function of L , The metric (1), now, can be written as, These solutions are not new but found in Ref. [6] in a different form. For various specific parameter values, the solutions were also found in Ref. [5]. In the rest of this work, we analyze the behavior of the universe for each parameter space. Let us consider the case with 0 < β < 1 first. Because the scale factor at L = β diverges, there are two independent universes divided by the range of L with -1 ≤ L < β and β < L ≤ 1, in which the time runs in (0 , ∞ ). First consider the universe resides in -1 ≤ L < β . The universe begins at L = -1 with φ → -∞ . The universe is in a singular state because the scalar energy density diverges, which can be seen from Eq. (A1). As the scalar field climbs up the potential, its velocity monotonically decreases due to the friction of the Hubble parameter and the inclination of the potential. The Hubble parameter also decreases until L reaches L = 1 / √ 3. At this time, the universe starts to inflate because the speed of the scalar field is slowed down and the scalar potential is big enough. The scalar field arrive at φ = φ c at time L = 0 and its velocity vanishes there. Thereafter, it turns its direction and starts to decrease. However, note that the scalar velocity vanishes at L = β . Therefore, even if the scalar field decreases continually, its velocity approaches zero in the future. This results in a interesting result: If β ≤ 1 / √ 3, the universe inflates eternally even though the scalar field continually decreases so that the scalar potential goes to zero. On the other hand, if β > 1 / √ 3, the inflation will end at L = 1 / √ 3 and the universe enters into the decelerating expansion. The universe will ends at L = β . Therefore, the equation of state at the end of the universe is simply given by Let us obtain the asymptotic form of the scalar field and the scale factor for L ∼ β . The asymptotic form, in fact, is given by the fixed point solution (9). Then, from Eq. (11), one gets Then, one can integrate Eqs. (10) and (A1) to obtain the same evolutions of the scale factor and the scalar field as in Eq. (9). For β < 1 / √ 3, the attractor denotes the inflationary attractor. On the other hand, for β ≥ 1 / √ 3, it is not an inflationary attractor but is still an attractor approaching a state with its equation of state w = -1 + 2 β 2 . We next consider the case with β ≤ L ≤ 1. The universe begins at L = 1 with φ →∞ and a = 0. At the beginning, the universe starts from a singular state. Even though the scalar field slides down the potential, its speed cannot always increase because the friction due to the Hubble parameter competes with the slope of the potential. The scalar velocity starts to decrease at some φ . At L = 1 / √ 3, its speed becomes small enough to support the inflation of the universe. If β < 1 / √ 3, the universe will go into eternal inflating stage at L = 1 / √ 3. On the other hand, if β > 1 / √ 3, the universe fails to get into the inflating stage but continues the decelerating expansion. The later evolutions can also be approximated by the fixed point solution (9). The cosmological time runs (0 , ∞ ) during -1 ≤ L ≤ 1. The universe begins at L = -1 with φ →-∞ and a = 0. At the beginning, the universe starts from a singular state. As the scalar field climbs up the potential, the velocity (therefore kinetic energy) of the scalar field decreases due to the friction of the Hubble parameter and the slope of the potential. For -1 / √ 3 < L < 1 / √ 3, the kinetic energy is small enough to inflate the universe. The velocity vanishes at φ = φ c , when L = 0. Afterward, the scalar field starts to slide down the potential. However, as seen in Eq. (A1), the scalar velocity vanishes at L = 1. Therefore, it cannot increase at all times but starts to decrease at some field value. In the future, the scalar velocity goes to zero as φ →-∞ . The universe, after the temporal inflation, goes into decelerating expansion. Let us now consider the asymptotic behavior of the solutions in the cosmological time. We first consider the early universe around L ∼ -1. Using Eq. (A4), the scalar field and the scale factor behaves as where φ i = ( β +1) φ c + √ 2 3 M Pl log ( 3 √ 2 2 β +1 2( β -1) (1+ β -1 ) β β -1 ) is a lengthy function of φ c and β and We next consider the later time solution with L ∼ 1. As t →∞ , the scalar field and the scale factor behaves as Comparing a i and a f , we find that the scale factor during the inflationary period is enhanced by Unless β -1 ∼ 10 -90 , this value is not big enough to support the present observational data for inflation. The asymptotic form of the scalar field and the scale factor are explicitly dependent on the choice of φ c and deviates from the fixed point solution (9). In fact, this corresponds to a kinetic attractor, with the equation of state p = ρ corresponding to matter with dominance of kinetic energy with equation of state, w -→ 1 . The solution in this case can be obtained by taking the β → 1 limit in Eq. (15). The evolution of the scalar field was given in Eq. (12). The derivative of the scale factor with respect to L is the same as in Eq. (14) with β = 1. Integrating this equation with Eq. (A6), we get the scale factor, The equation of state is also the same as Eq. (13). As -1 ≤ L ≤ 1, the cosmological time runs (0 , ∞ ). The universe begins from L = -1 with a = 0 and φ = -∞ . Both of the initial scalar velocity and Hubble parameter diverges. Due to the friction from the Hubble parameter, the scalar velocity decreases with time and the scalar field starts to approach φ c . At -1 / √ 3 < L < 1 / √ 3, the scalar velocity becomes small enough to support the inflation. The scalar field takes its maximum value φ c at L = 0 and then bounces back to decrease. /negationslash On the whole, the evolution of the scale factor and the scalar field is completely different from that of β = 1. We now write down the asymptotic form of the solutions for L ∼ -1 using Eq. (A7), On the other hand, around L ∼ 1 using Eq. (A8), the scale factor becomes where H 1 = 3 / 2 × H 0 e √ 3 2 φ c /M Pl . Notice that the limiting behaviors are completely different from the cases of β = 1. This limiting behavior can also be seen in the specific solutions in Ref. [5]. /negationslash In summary, we directly attacked the Hamilton-Jacobi equation for the scalar cosmology with exponential potential V ∝ e √ 6 βφ/M Pl . We have reproduced the known solutions and introduced a new time variable L , which varies in -1 ≤ L ≤ 1. In describing the evolution of the universe, we have found that the new time variable is very convenient to identify the state of the universe at a given time. The most beautiful property of it is that the equation of state of the scalar field is determined by the value very easily, w = -1 + 2 L 2 . From this, we notice that the universe will expands with accelerating rate if -1 / √ 3 < L < 1 / √ 3. The evolution of the universe are characterized by the value of β . For 0 < β < 1, the universe ends at L = β . Therefore, the equation of state at the end of the universe is nothing but w = -1+2 β 2 . For 0 < β ≤ 1 / √ 3 the universe ends with eternal later time inflation irrespective of the initial condition of the scalar field. On the other hand, for β > 1, the universe will ends at L = 1 and the equation of state of the universe will approach to w = 1, implying the kinetic dominance. The temporal inflation during -1 / √ 3 < L < 1 / √ 3 makes the scale factor is increased by the factor [ ( β +1) / ( β -1) ] 1 / 3 . However, this is not enough to explain the cosmic inflation or the present later time accelerating expansion. To see how useful the variable L in describing the scalar cosmology is, more general systems need to be analyzed. For example, in addition to the scalar field one may include a perfect fluid, an axion [21], or a gauge field [22]. If the scalar field plays the role of the dark energy, the value of β should be very close to zero, β /lessmuch 1 for dark energy dominated future. However, as pointed out by Townsend [3], there is a conjecture saying that such spacetime with future event horizon cannot arise from classical compactification of String/M-theory. For partial proof for this conjecture, consult Ref. [23]. If this is true, the value should be restricted to be β ≥ 1 / √ 3 discarding the previous possibility.", "pages": [ 1, 2, 3, 4, 5, 6 ] }, { "title": "Acknowledgement", "content": "HCK was supported in part by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (MEST) (No.2010-0011308).", "pages": [ 6 ] }, { "title": "Appendix A: The cosmological time and L", "content": "From Eq. (7), the time derivative of the scalar field is given by where we have used Eq. (11) to get Now, we obtain the relation between the two times L and t by Note that the arrow of time of the two time is dependent on the sign of β -L . Because the exponent of 1 / ( β -L ) is always larger than 1, we see that the cosmological time go to infinity at L = β . At L = -1, the exponent of 1 / ( L +1) is always smaller than one for positive β . Therefore, the time will take a finite value, which we choose to zero. At L = 1, the exponent of 1 / (1 -L ) is equal to or larger than one if β ≥ 1. Therefore, t →∞ for β ≥ 1 and t takes a finite value for β < 1, which we choose to zero. Explicitly, Eq. (A2) can be integrated in a closed form, where t 0 will be used to set the initial time to be zero. However, it is too complex to use directly. We may simply note that the arrow of the time is the same as the increase of L for L < β and is reversed for L > β and use the asymptotic forms below. Around L →-1, the relation becomes, Around L → 1, Eq. (A3) becomes Now we write down the evolutions in β = 1 case. With β → 1 limit, the generating function becomes The time derivative of the scalar field and L with respect to the cosmological time is given by For L →-1, the second equation of Eq. (A6) becomes For L → 1, the relation with the cosmological time becomes", "pages": [ 6, 7 ] } ]

[ { "title": "CHARACTERISATION AND REPRESENTATION OF NON-DISSIPATIVE ELECTROMAGNETIC MEDIUM WITH A DOUBLE LIGHT CONE", "content": "MATIAS F. DAHL ABSTRACT. We study Maxwell's equations on a 4 -manifold N with a medium that is nondissipative and has a linear and pointwise response. In this setting, the medium can be represented by a suitable ( 2 2 ) -tensor on the 4 -manifold N . Moreover, in each cotangent space on N , the medium defines a Fresnel surface . Essentially, the Fresnel surface is a tensorial analogue of the dispersion equation that describes the response of the medium for signals in the geometric optics limit. For example, in isotropic medium the Fresnel surface is at each point a Lorentz light cone. In a recent paper, I. Lindell, A. Favaro and L. Bergamin introduced a condition that constrains the polarisation for plane waves. In this paper we show (under suitable assumptions) that a slight strengthening of this condition gives a pointwise characterisation of all medium tensors for which the Fresnel surface is the union of two distinct Lorentz null cones. This is for example the behaviour of uniaxial medium like calcite. Moreover, using the representation formulas from Lindell et al. we obtain a closed form representation formula that pointwise parameterises all medium tensors for which the Fresnel surface is the union of two distinct Lorentz null cones. Both the characterisation and the representation formula are tensorial and do not depend on local coordinates.", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "We will study the pre-metric Maxwell's equations, where Maxwell's equations are written on a 4 -manifold N and the electromagnetic medium is described by a suitable antisymmetric ( 2 2 ) -tensor κ on N that pointwise is determined by 36 real parameters. In each cotangent space on N , the electromagnetic medium determines a fourth order polynomial surface called the Fresnel surface that can be seen as a tensorial analogue of the dispersion equation. The Fresnel surface describes the response of the medium to signals in the geometric optics limit [OFR00, Rub02, HO03, PSW09, RRS11]. In this work we will assume that the medium is skewon-free . Then there are only 21 free parameters and such medium models non-dissipative medium. For example, under suitable assumptions the skewon-free assumption will imply that Poynting's theorem holds [HO03, Dah10]. On an orientable manifold one can show that invertible skewon-free ( 2 2 ) -tensors are in one-to-one correspondence with area metric . By an area metric, we here mean a ( 0 4 ) -tensor on N that defines a symmetric non-degenerate inner product for bivectors. Area metrics appear when studying the propagation of a photon in a vacuum with a first order correction from quantum electrodynamics [DH80, SWW10]. The Einstein field equations have also been generalised into equations where the unknown field is an area metric [PSW07]. For further examples, see [PSW09, SWW10]. We know that in isotropic medium like vacuum, the Fresnel surface is a Lorentz null cone at each point in N . That is, Lorentz geometry describes the propagation of light in isotropic medium. Conversely, it was conjectured in 1999 by Y. Obukhov and F. Hehl [OH99, OFR00] that isotropic medium is the only (non-dissipative and axion-free) medium where the Fresnel surface is a Lorentz null cone. This was partially proven already in [OFR00]. However, the full conjecture was only established in [FB11] by A. Favaro and L. Bergamin. For an alternative proof, see [Dah11a] and for further discussions and related results, see [OR02, HO03, LH04, Iti05] and Section 3.2 below. Since the Fresnel surface is a 4 th order polynomial surface, the Fresnel surface can also decompose into the union of two distinct Lorentz null cones. For example, this is the case in uniaxial medium like calcite (CaCO 3 ) [BW99, Section 15.3]. In such medium, the propagation properties of the medium does not only depend on direction, but also on the polarisation of the wave. In uniaxial medium, there are two eigenpolarisations and one null cone for each polarisation. In consequence, there is one Fermat's principle for each polarisation [PSW09]. This is the the source for the physical phenomenon of double refraction. We know that uniaxial medium is an example of medium with two distinct null cones. A natural next task is to understand the structure of all medium tensors with this property. This is the main result in [Dah11b], which gives the complete local description of all non-dissipative medium tensors for which the Fresnel surface is a double light cone (up to suitable assumptions). The importance of this result is that it shows that are three and only three medium classes with this behaviour. Moreover, the theorem gives explicit coordinate expressions for each medium class. The first medium class is a slight generalisation of uniaxial medium. The second class seems to be a new class of mediums. The last class seems to be unphysical; heuristic arguments and preliminary numerical tests suggest that Maxwell's equations are not hyperbolic in that class [Dah11b]. In the below, this result is summarised in Theorem 3.5. The main contribution of this paper is Theorem 5.1. Under suitable assumptions, this theorem gives a tensorial characterisation (condition (ii) in Theorem 5.1) of all non-dissipative medium tensors for which the Fresnel surface is two distinct light cones. In a suitable limit, the condition also reduces to the closure condition κ 2 = -λ Id for a λ > 0 that characterises medium with a single light cone [HO03]. Moreover, in Theorem 5.1 we give a tensorial representation formula (equation (64)) that parameterises all non-dissipative medium tensors with two distinct light cones. Both the characterisation and representation formula are pointwise results. The background and motivation for Theorem 5.1 comes from a recent paper by I. Lindell, A. Favaro and L. Bergamin [LBF12]. In Section 4 we will briefly summarise some of the results from [LBF12]. In this paper, the authors introduces a second order polynomial condition on the medium tensor (equation (54) in the below). Equation (54) is derived from a constraint on polarisation of plane waves, and in [LBF12] it is shown that whenever condition (54) is satisfied (plus some additional assumptions), the Fresnel surface always factorises into two second order surfaces. In Section 4.3 we will further motivate that equation (54) is in fact a general factorisability condition for the Fresnel surface. At first this might seem unexpected since equation (54) was initially derived from a constraint on polarisation, yet it is able to constrain the behaviour of signal speed. However, the explanation is that for electromagnetic waves, polarisation and signal speed are not independent properties but tied together. In Theorem 5.1, condition (ii) is a slight strengthening of equation (54). Also, representation formula (64) in Theorem 5.1 is adapted from [LBF12] and constitute a subclass of generalised Q -medium introduced by I. Lindell and H. Wall'en in [LW02]. A further technical discussion on Theorem 5.1 is given in the end of Section 5. Some of the computations in the paper rely on computer algebra. For further information about the Mathematica notebooks for these computations, please see the author's homepage.", "pages": [ 1, 2 ] }, { "title": "2. PRELIMINARIES", "content": "By a manifold N we mean a second countable topological Hausdorff space that is locally homeomorphic to R n with C ∞ -smooth transition maps. All objects are assumed to be smooth where defined. Let TN and T ∗ N be the tangent and cotangent bundles, respectively. For k ≥ 1 , let Ω k ( N ) be antisymmetric tensor fields with k lower indices (that is, k -forms). Similarly, let Ω k ( N ) be antisymmetric tensor fields with k upper indices. Moreover, let Ω 2 2 ( N ) = Ω 2 ( N ) ⊗ Ω 2 ( N ) . Let also C ∞ ( N ) be the set of scalar functions (that is, ( 0 0 ) -tensors). The Einstein summing convention is used throughout. When writing tensors in local coordinates we assume that the components satisfy the same symmetries as the tensor. /negationslash ˜ where sgn: R → R is the sign function , sgn x = x/ | x | for x = 0 and sgn x = 0 for x = 0 . If locally ˜ ˜ ˜ When the chart is clear from context, we will simply write G = 1 2 G ij dx i ∧ dx j . Similarly, if κ ∈ Ω 2 2 ( N ) then in each chart κ is represented by a κ | U ∈ Ω 2 2 ( U ) and locally for suitable components κ ij rs . Moreover, if κ ij rs and ˜ κ ij rs are components for κ in overlapping charts ( U, x i ) and ( U, x i ) then we obtain the transformation rule ˜ ˜ ˜ Compositions involving twisted tensors are computed in the natural way by composing local tensors. For example, if κ, η ∈ ˜ Ω 2 2 ( N ) their composition defines an element κ · η ∈ Ω 2 2 ( N ) and if κ, η and κ · η are written as in equation (4) then If M is orientable, then twisted tensors coincide with their normal (or untwisted) counterparts. For example, if M is orientable, equation (5) implies that ˜ Ω 2 2 ( N ) = Ω 2 2 ( N ) . There are also other way to define twisted forms. Equation (1) coincides with definition of a pseudo-form in [Fra04]. For a global definition of twisted forms using the orientation bundle, see [AMR01, Supplement 7.2A]. 2.2. Tensor densities. In addition to tensors and twisted tensors, we will need tensor densities and twisted tensor densities. A ( p q ) -tensor density of weight w ∈ Z on a manifold N is determined by components T a 1 ...a p b 1 ··· b q in each chart ( U, x i ) , and on overlapping charts ( U, x i ) and ( ˜ U, ˜ x i ) we have the transformation rule [Spi99], The Levi-Civita permutation symbols are denoted by ε ijkl and ε ijkl . Even if these coincide as combinatorial functions so that ε ijkl = ε ijkl , they are also different as they globally define different objects on a manifold. Namely, if ε ijkl , ε ijkl and ˜ ε ijkl , ˜ ε ijkl are defined on overlapping coordinate charts ( U, x i ) and ( U, x i ) , respectively, then ˜ ˜ ˜ A twisted ( p q ) -tensor density of weight w ∈ Z on N is defined in the same way, but with an additional sgn det ( ∂ ˜ x i ∂x j ) factor in the transformation rule as in equations (3) and (5). ˜ That is, ε ijkl defines a ( 0 4 ) -tensor density of weight -1 on N and ε ijkl defines a ( 4 0 ) -tensor density of weight 1 . For future reference, let us note that where δ i j is the Kronecker delta symbol and brackets [ i 1 . . . i p ] indicate that indices i 1 , . . . , i p are antisymmetrised with scaling 1 /p ! .", "pages": [ 3, 4 ] }, { "title": "2.3. Maxwell's equations on a 4 -manifold. On a 4 -manifold N , the premetric Maxwell's equations read", "content": "where d is the exterior derivative, F ∈ Ω 2 ( N ) , G ∈ ˜ Ω 2 ( N ) , J ∈ ˜ Ω 3 ( N ) and κ ∈ ˜ Ω 2 2 ( N ) . Here, F, G , are called the electromagnetic field variables , J describes the electromagnetic sources, tensor κ models the electromagnetic medium and equation (12) is known as the constitutive equation . In local coordinates, equations (10)-(12) reduce to the usual Maxwell's equations. For a systematic treatment, see [Rub02, HO03]. If locally F = 1 2 F ij dx i ∧ dx j , G = 1 2 G ij dx i ∧ dx j and κ is written as in equation (4) then constitutive equation (12) is equivalent with Thus equation (12) models electromagnetic medium with a linear and pointwise response. Suppose κ ∈ ˜ Ω 2 2 ( N ) and suppose ( U, x i ) is a chart. Then the local representation of κ in equation (4) defines a pointwise linear map Ω 2 ( U ) → Ω 2 ( U ) . In U we can therefore represent κ by a smoothly varying 6 × 6 matrix. To do this, let O be the ordered set of index pairs { 01 , 02 , 03 , 23 , 31 , 12 } , and if J ∈ O , let dx J = dx J 1 ∧ dx J 2 , where J 1 and J 2 are the individual indices for J . Say, if J = 31 then dx J = dx 3 ∧ dx 1 . Then a basis for Ω 2 ( U ) is given by { dx J : J ∈ O } , that is, This choice of basis follows [HO03, Section A.1.10]. By equation (4) it follows that where κ J I = κ J 1 J 2 I 1 I 2 . Let b be the natural bijection b : O → { 1 , . . . , 6 } . Then we identify coefficients { κ J I : I, J ∈ O } for κ with the smoothly varying 6 × 6 matrix P = ( κ J I ) IJ defined as κ J I = P b ( I ) b ( J ) for I, J ∈ O . Suppose P = ( κ J I ) IJ and ˜ P = ( ˜ κ J I ) IJ are smoothly varying 6 × 6 matrices that represent tensor κ in overlapping charts ( U, x i ) and ( U, x i ) . Then equation (5) is equivalent with where ˜ ˜ ˜ ˜ ˜ and ∂ ˜ x J ∂x I is defined similarly by exchanging x and ˜ x . For matrices T = ( ∂x J ∂ ˜ x I ) IJ and S = ( ∂ ˜ x J ∂x I ) IJ , we have T = S -1 , whence equation (5) is further equivalent with the matrix equation ˜ In a chart ( U, x i ) , we define trace κ : U → R and det κ : U → R as the trace and determinant of the pointwise linear map Ω 2 ( U ) → Ω 2 ( U ) . When P is as above it follows that trace κ = trace P and det κ = det P . When these definitions are extended into each chart on N equation (17) shows that trace κ ∈ ˜ C ∞ ( N ) and det κ ∈ C ∞ ( N ) . Moreover, if κ is written as in equation (4), then /negationslash At a point p ∈ N we say that κ is invertible if (det κ ) | p = 0 . If Id is the identity tensor Id ∈ Ω 2 2 ( N ) , then writing Id as in equation (4) gives Id ij rs = δ i r δ j s -δ i s δ j r . For f ∈ ˜ C ∞ ( N ) it follows that trace f Id = 6 f . 2.4. Decomposition of electromagnetic medium. At each point of a 4 -manifold N , an element of ˜ Ω 2 2 ( N ) depends on 36 parameters. Pointwise, such ( 2 2 ) -tensors canonically decompose into three linear subspaces. The motivation for this decomposition is that different components in the decomposition enter in different parts of electromagnetics. See [HO03, Section D.1.3].", "pages": [ 4, 5 ] }, { "title": "Proposition 2.1. Let N be a 4 -manifold, and let", "content": "Then ˜ Ω 2 2 ( N ) = Z ⊕ W ⊕ U, (18) and pointwise, dim Z = 20 , dim W = 15 and dim U = 1 . If we write a κ ∈ ˜ Ω 2 2 ( N ) as κ = (1) κ + (2) κ + (3) κ with (1) κ ∈ Z , (2) κ ∈ W , (3) κ ∈ U , then we say that (1) κ is the principal part , (2) κ is the skewon part , (3) κ is the axion part of κ [HO03]. For a proof of Proposition 2.1 as stated above, see [Dah11a], and for further discussions, see [Rub02, HO03, Fav12]. In ˜ Ω 2 2 ( N ) there is a canonical isomorphism ˜ Ω 2 2 ( N ) → ˜ Ω 2 2 ( N ) known as the Poincar'e isomorphism [Gre78, Fav12]. Let us first give a local definition. If κ ∈ ˜ Ω 2 2 ( N ) on a 4 -manifold N , we define κ as the element κ ∈ Ω 2 2 ( N ) defined as when κ and κ are written as in equation (4). Equations (7)-(8) imply that this assignment defines an element κ ∈ ˜ Ω 2 2 ( N ) . For κ ∈ Ω 2 2 ( N ) we define κ in the same way and we also have a canonical isomorphism Ω 2 2 ( N ) → Ω 2 2 ( N ) . Proposition 2.2. Suppose N is a 4 -manifold and κ ∈ ˜ Ω 2 2 ( N ) . The next proposition collects results for κ . In particular, part (i) states that κ can be interpreted as a formal adjoint of κ with respect to the wedge product for 2 -forms. In consequence, the Poincar'e isomorphism is closely related to the decomposition in Proposition 2.1. For example, κ ∈ ˜ Ω 2 2 ( N ) has only a principal part if and only if κ = κ and trace κ = 0 . For a further discussion, see [Fav12]. Proof. Part (i) follows by writing out both sides in equation (20) in coordinates. Parts (ii) and (iii) follow by part (i) . Part (iv) is a direct computation. For part (v) we have for all u, v ∈ Ω 2 ( N ) , and the claim follows since the right hand side vanishes. /square If ρ is a twisted scalar tensor density of weight 1 on a 4 -manifold N and A,B ∈ Ω 2 ( N ) then we define ρA ⊗ B as the twisted tensor in ˜ Ω 2 2 ( N ) defined as follows. If locally A = 1 2 A ij ∂ ∂x i ∧ ∂ ∂x j and B = 1 2 B ij ∂ ∂x i ∧ ∂ ∂x j then when ρA ⊗ B is written as in equation (4). That ρA ⊗ B transforms as an element in ˜ Ω 2 2 ( N ) follows by equation (7). Similarly when ρ is an untwisted scalar density we define ρA ⊗ B ∈ Ω 2 2 ( N ) by equation (21). For both twisted and untwisted ρ we have identities In Section 4.2 and in the proof of Theorem 5.1 we will need the following lemma. where ρ is a scalar tensor density of weight 1 , A,B ∈ Ω 2 ( N ) and f ∈ ˜ C ∞ ( N ) . Then κ | p = 0 at a point p ∈ N implies that f | p = 0 and ρ | p = 0 or A | p = 0 or B | p = 0 . If κ is written as in equation (4) and A,B are written as above, then equation (26) states that Proof. By restricting the analysis to p and introducing notation A I = A I 1 I 2 and B I = B I 1 I 2 , we obtain Setting I = J and summing implies that ∑ I ∈ O ρA I B I = 0 . Multiplying each equation in (27) by A I B J and ε IJ and summing I, J yields two scalar equations. Eliminating f from these equations gives and the claim follows. /square 2.5. The Fresnel surface. Let κ ∈ ˜ Ω 2 2 ( N ) on a 4 -manifold N . If κ is locally given by equation (4) in coordinates { x i } , let If { ˜ x i } are overlapping coordinates, then equations (5), (7) and (8) imply that components G ijkl 0 satisfy the transformation rule Thus components G ijkl 0 define a twisted ( 4 0 ) -tensor density G 0 on N of weight 1 . The Tamm-Rubilar tensor density [HO03, Rub02] is the symmetric part of G 0 and we denote this twisted tensor density by G . In coordinates, G ijkl = G ( ijkl ) 0 , where parenthesis indicate that indices ijkl are symmetrised with scaling 1 / 4! . If locally ξ = ξ i dx i it follows that G ijkl ξ i ξ j ξ k ξ l = G ijkl 0 ξ i ξ j ξ k ξ l , and we call G ijkl ξ i ξ j ξ k ξ l the Fresnel polynomial . The Fresnel surface at a point p ∈ N is defined as By equation (29), the definition of F p ( κ ) does not depend on local coordinates. Let F ( κ ) = ∐ p ∈ N F p ( κ ) be the disjoint union of all Fresnel surfaces. The Fresnel surface F ( κ ) is a fundamental object when studying wave propagation in Maxwell's equations. Essentially, equation G ijkl ξ i ξ j ξ k ξ l = 0 in equation (30) is a tensorial analogue to the dispersion equation that describes wave propagation in the geometric optics limit. Thus F ( κ ) constrains possible wave speed(s) as a function of direction. In general the Fresnel surface F p ( κ ) is a fourth order polynomial surface in T ∗ p ( N ) , so it can have multiple sheets and singular points [OH04]. There are various ways to derive the Fresnel surface; by studying a propagating weak singularity [OFR00, Rub02, HO03], using a geometric optics [Iti09, Dah11a], or as the characteristic polynomial of the full Maxwell's equations [SWW10]. The tensorial description of the Fresnel surface is due to Y. Obukhov, T. Fukui and G. Rubilar [OFR00].", "pages": [ 6, 7, 8 ] }, { "title": "3. RESULTS FOR SKEWON-FREE MEDIUM", "content": "In this section we collect a number of results for twisted skewon-free tensors that we will need in the proof of Theorem 5.1. Suppose L is an element in Ω 1 ( N ) ⊗ Ω 1 ( N ) on an n -manifold N . Then we can treat L as a pointwise linear map Ω 1 ( N ) → Ω 1 ( N ) . By linear algebra, it follows that around each p ∈ N there are coordinates such that at p , components ( L j i ) ij is a matrix in Jordan normal form. Since there are only finitely many ways an n × n matrix can be decomposed into Jordan blocks, it follows that there are only a finite number of normal forms for L | p . It should be emphasised that the structure of the Jordan normal form is unstable under perturbations of the matrix. Hence, the normal form is in general only valid at one point. The normal form theorem in [SWW10] is essentially an analogous result for skewon-free elements κ in Ω 2 2 ( N ) . The difficulty in proving such a result is easy to understand. The matrix that represents κ at a point is a 6 × 6 matrix. By a linear transformation in R 6 , we can transform this into an Jordan normal form, but such a transformation, a priori has 36 degrees of freedom. On the other hand, for a coordinate transformation on N , the Jacobian only has 16 degrees of freedom. It is therefore not obvious that coordinate transformations have enough degrees of freedom to transform κ into a normal form. See equation (17). For a further discussion, see [SWW10, Dah11c]. The below theorem summarises the normal form theorem in [SWW10] specialised to the setting that we need here. Let us make three comments. First, the below theorem is formulated for twisted κ ∈ ˜ Ω 2 2 ( N ) instead of for area metrics in [SWW10] (which are ordinary tensors) or untwisted κ ∈ Ω 2 2 ( N ) in [Dah11c]. Second, the theorem contains the technical assumption that κ is invertible and the Fresnel surface has no 2 -dimensional subspace. This greatly simplifies the result since it implies that there are only 7 possible normal forms and one does not need any conjugations by Hodge operators. These assumptions will also appear in Theorem 5.1. For a further discussion of these assumptions, see end of Section 5. Third, the reason the normal form theorem is useful can be seen from Proposition 2.1. Namely, in arbitrary coordinates, a skewon-free κ ∈ ˜ Ω 2 2 ( N ) depends on 21 parameters. However, from Theorem 3.1 we see that each normal form depends only on 2 , 4 or 6 parameters. This reduction of parameters will make the computer algebra feasible in Theorem 5.1. The division into metaclasses in [SWW10] is based on the Jordan block structure of the matrix representation of κ at a point. Since this structure is unstable under perturbations, it can be difficult to determine the metaclass both in the numerical case and the symbolic case [LZW97]. Theorem 3.1. Suppose N is a 4 -manifold and κ ∈ ˜ Ω 2 2 ( N ) . If p ∈ N and Then there exists coordinates { x i } 3 i =0 around p such that the 6 × 6 matrix ( κ J I ) IJ that represents κ | p in these coordinates is one of the below matrices: In each matrix the parameters satisfy α 1 , α 2 , . . . ∈ R , β 1 , β 2 , . . . ∈ R \\ { 0 } and sgn β 1 = sgn β 2 = · · · . Proof. Let ( U, x i ) be coordinates around p , and let P = ( κ J I ) IJ be the 6 × 6 -matrix that represents κ at p in these coordinates. By treating U as a manifold with coordinates { x i } 3 i =0 , equation (4) defines a tensor κ ∈ Ω 2 2 ( U ) . Since κ is invertible at p and F p ( κ ) has no 2 -dimensional subspace, the Jordan normal form of P can not have a Jordan block of dimension 2 , . . . , 6 that corresponds to a real eigenvalue of P . For area metrics this is established in Lemma 5.1 in [SWW10]. (Or, for a translation to elements in Ω 2 2 ( U ) , see the proof of Theorem 2.1 in [Dah11b].) In the terminology of [SWW10] and [Dah11b] this implies that κ | p is of Metaclasses I, . . . , VII. Hence Theorem 3.2 in [Dah11c] (the restatement of the normal form theorem in [SWW10]) implies that around p , manifold U has a coordinate chart ( ˜ U, ˜ x i ) such that at p , we have TPT -1 = R, (38) where T = ( ∂x J ∂ ˜ x I ) IJ is as in equation (16) and R is one of the 6 × 6 matrices in equations (31)-(37) for some parameters α 1 , α 2 , . . . ∈ R and β 1 , β 2 , . . . > 0 . Since ( U, x i ) is a chart in N it follows that ( ˜ U, ˜ x i ) is also a chart in N . Multiplying equation (38) by sgn det ( ∂x i ∂ ˜ x j ) and comparing with equation (17) shows that sgn det ( ∂x i ∂ ˜ x j ) R is the matrix that represents κ ∈ ˜ Ω 2 2 ( N ) in coordinates { ˜ x i } 3 i =0 . If sgn det ( ∂x i ∂ ˜ x j ) = 1 or if R is in Metaclasses I, IV, VI, VII, the claim follows. On the other hand, if sgn det ( ∂x i ∂ ˜ x j ) = -1 and R is in Metaclasses II, III, V, it remains to prove that we can change the signs of the 1 -entries in the normal forms by an orientation preserving coordinate transformation. Let { ̂ x i } 3 i =0 be coordinates determined by ̂ x i = J i j ˜ x j for a suitable 4 × 4 matrix J = ( J i j ) ij . For Metaclass III a suitable Jacobian is ( J i j ) ij = diag(1 , -1 , -1 , 1) , and for Metaclass II and V a suitable Jacobian is 3.2. Non-birefringent medium. By a pseudo-Riemann metric on a manifold N we mean a symmetric ( 0 2 ) -tensor g that is non-degenerate. If N is not connected we also assume that g has constant signature. By a Lorentz metric we mean a pseudo-Riemann metric on a 4 -manifold with signature ( -+++) or (+ ---) . Let /sharp be the isomorphisms /sharp : T ∗ N → TN , so that if locally g = g ij dx i ⊗ dx j then /sharp ( α i dx i ) = α i g ij ∂ ∂x j . Using the /sharp -isomorphism we extend g to covectors by setting g ( ξ, η ) = g ( ξ /sharp , η /sharp ) when ξ, η ∈ T ∗ p ( N ) . For a Lorentz metric g the light cone at a point p ∈ N is defined as and analogously to the Fresnel surface we define N ( g ) = ∐ p ∈ N N p ( g ) . If g is a pseudo-Riemann metric on a 4 -manifold N , then the Hodge star operator of g is defined as the ∗ g ∈ ˜ Ω 2 2 ( N ) such that if locally g = g ij dx i ⊗ dx j , and ∗ g is written as in equation (4), then where det g = det g ij and g ij is the ij th entry of ( g ij ) -1 . Then ∗ g has only a principal part. See for example, [HO03, Fav12]. Moreover, if g is a Lorentz metric and κ = ∗ g , we have Equation (40) is the motivation for defining N ( g ) as a subset of the cotangent bundle. Definition 3.2. Suppose N is a 4 -manifold and κ ∈ ˜ Ω 2 2 ( N ) . Then κ is non-birefringent if there exists a Lorentz metric g on N such that equation (40) holds. Thus, in non-birefringent medium, the Fresnel surface F p ( κ ) has only a single sheet, and there is only one signal speed in each direction. In non-birefringent medium it follows that propagation speed can not depend on polarisation. On N = R 4 , a specific example of a nonbirefringent medium is κ = √ /epsilon1 µ ∗ g , where g is the Lorentz metric g = diag( -1 /epsilon1µ , 1 , 1 , 1) on R 4 . Then constitutive equation (12) models standard isotropic medium on R 4 with permittivity /epsilon1 > 0 and µ > 0 . The next theorem gives the complete characterisation of all non-birefringent media with only a only a principal part. Theorem 3.3. Suppose N is a 4 -manifold. If κ ∈ ˜ Ω 2 2 ( N ) satisfies (2) κ = 0 , then the following conditions are equivalent: Implication (i) ⇒ (ii) was conjectured in 1999 by Y. Obukhov and F. Hehl [OH99, OFR00]. Under some additional technical assumptions the implication was already proven in [OFR00]. However, the general case was only established in [FB11] by A. Favaro and L. Bergamin by a case by case analysis using the normal form theorem in [SWW10]. For an alternative proof using a Grobner basis, see [Dah11a] and for similar results, see [LH04, Iti05, RRS11] and Section 3.3 below. Implication (iii) ⇒ (i) is a direct computation. In the setting of electromagnetics, implication (ii) ⇒ (iii) seems to first to have been derived by M. Schonberg [Rub02, Sch71]. For further derivations and discussions, see [HO03, Rub02, OFR00, OH99, Jad79]. When a general κ ∈ ˜ Ω 2 2 ( N ) on a 4 -manifold N satisfies κ 2 = -f Id for a function f ∈ C ∞ ( N ) one says that κ satisfies the closure condition . For physical motivation, see [HO03, Section D.3.1]. For a study of more general closure relations, and in particular, for an analysis when κ might have a skewon part, see [Fav12, LBF12], and Section 4.3 below. Definition 3.4. Suppose N is a 4 -manifold and κ ∈ ˜ Ω 2 2 ( N ) . If p ∈ N we say that the Fresnel surface F p ( κ ) decomposes into a double light cone if there exists Lorentz metrics g + and g -defined in a neighbourhood of p such that and N p ( g + ) = N p ( g -) . /negationslash If g, h are Lorentz metrics, then N p ( g ) ⊂ N p ( h ) implies that at p we have g = Ch for some C ∈ R \\ { 0 } . See for example [Tou65]. Thus, if κ decomposes into a double light cone, then κ is not non-birefringent. Under some assumptions, the next theorem gives the complete pointwise description of all medium tensors with a double light cone. The theorem generalises the result in [Dah11b] to twisted tensors. Theorem 3.5. Suppose N is a 4 -manifold and κ ∈ ˜ Ω 2 2 ( N ) . Furthermore, suppose that at some p ∈ N Then exactly one of the below three possibilities holds: and either α 1 = α 2 or β 1 = β 2 or both inequalities hold. /negationslash /negationslash /negationslash /negationslash Conversely, if κ is defined by one of the above three possibilities, then the Fresnel surface of κ decomposes into a double light cone at p . Proof. For κ ∈ Ω 2 2 ( N ) the result is proven in [Dah11b, Theorem 2.1] (up to a permutation of coordinates in Metaclass I). The generalisation to κ ∈ ˜ Ω 2 2 ( N ) follows by the same argument used to prove Theorem 3.1. The converse direction can be verified by computer algebra using the explicit Lorentz metrics given in [Dah11b]. /square In Theorem 3.5, uniaxial medium is given by Metaclass I when α 1 = α 2 = α 3 = 0 . The main conclusion of the theorem is that there are two (and only two) additional classes of medium where the Fresnel surface decomposes (Metaclasses II and IV). In all three classes, there are explicit formulas for the Lorentz metrics that factorise the Fresnel surface. For a further discussion of these metrics, see [Dah11b]. In Theorem 5.1 we will show that under suitable assumptions every skewon-free medium with a double light cone can be written as in equation (43). This medium class is a special class of generalised Q -medium introduced by I. Lindell and H. Wall'en in [LW02]. For further discussions of this medium class, see [LW04, Fav12, LBF12]. Proposition 3.6. Suppose N is a 4 -manifold, g is a Lorentz metric, ρ is a twisted scalar density of weight 1 , A ∈ Ω 2 ( N ) and C 1 ∈ R \\ { 0 } and C 2 ∈ R . Moreover, suppose κ ∈ Ω 2 2 ( N ) is defined as Then κ is skewon-free the following claims hold pointwise in N : (i) κ is non-birefringent if and only if A = 0 or ρ = 0 . (ii) κ has a double light cone if and only if ρ = 0 , A = 0 and /negationslash Proof. We restrict the analysis to a point p ∈ N , and let { x i } 3 i =0 be coordinates around p such that the Lorentz metric has components g = ± diag( -1 , 1 , 1 , 1) at p . For claim (i) , let us note that the axion component of κ does not influence the Fresnel polynomial. See for example [HO03]. Thus κ is non-birefringent when A = 0 or ρ = 0 . For the converse direction, suppose κ is non-birefringent. Then Theorem 3.3 implies that ( κ -1 6 trace κ Id) 2 = -λ Id for some λ > 0 . Writing out the last equation and solving the associated Grobner basis equations (see [CLO07, Dah11a])shows that A = 0 or ρ = 0 . For claim (ii) , let us write A = 1 2 A ij ∂ ∂x i ∧ ∂ ∂x j . Then the Fresnel polynomial at p is given by where g ij = ( g -1 ) ij and H ij = C 1 g ij -2 ρA ia g ab A bj (see [LW02, LBF12]). Moreover, where E ∈ R is an expression that depends on ρ, C 1 and A . We will not need the explicit expression for E . However, by computer algebra we see that the same E also appears in det H for matrix H = ( H ij ) ij . Then equation (46) yields /negationslash /negationslash If κ has a double light cone, claim (i) implies that A = 0 and ρ = 0 . Moreover, by Proposition 1.5 in [Dah11b] and since polynomials have a unique factorisation into irreducible factors [CLO07, Theorem 5 in Section 3.5], we have det H < 0 and equation (47) implies inequality (44) for det κ . Conversely, if the inequalities in claim (ii) are satisfied, then equation (47) shows that det H < 0 , so g and H both have Lorentz signature at p . To complete the proof we need to show that there is no constant C ∈ R \\{ 0 } such that g ij = CH ij . Since A = 0 and ρ = 0 , this follows by inspecting equations g ii = CH ii for i = 0 , . . . , 3 . /square /negationslash /negationslash", "pages": [ 8, 9, 10, 11, 12, 13, 14 ] }, { "title": "4. DECOMPOSABLE MEDIA", "content": "In this section we first describe the class of decomposable medium introduced in [LBF12]. In particular, in Theorem 4.3 we describe the sufficient conditions derived in [LBF12] that imply that a medium is decomposable. In Theorem 5.1 these conditions will play a key role. In Section 4.3 we will describe some results that suggest that condition (i) in Theorem 4.3 is a general factorisability condition for the Fresnel polynomial. Following [LBF12] we restrict the analysis to R 4 so that we can work with plane waves. 4.1. Plane waves in R 4 . We say that a tensor T on R 4 is constant if there are global coordinates for R 4 where components for T are constant. If we assume that many tensors are constant, we assume that they are constant with respect to the same choice of coordinates. Below we also use notation Ω k ( N, C ) to denote the space of k -forms on a manifold N with possibly complex coefficients. /negationslash /negationslash Suppose κ ∈ Ω 2 2 ( R 4 ) is constant and F, G ∈ Ω 2 ( R 4 ) are defined as where Φ is a function Φ: R 4 → R such that d Φ is constant and non-zero, X,Y ∈ Ω 2 ( R 4 , C ) are constant and not both zero. If F and G solve the sourceless Maxwell's equations we say that F and G is a plane wave . Proposition 4.1. Suppose κ ∈ Ω 2 2 ( R 4 ) is constant and Φ is a function Φ: R 4 → R such that d Φ is constant and non-zero. Moreover, suppose X,Y are constant 2 -forms X,Y ∈ Ω 2 ( R 4 , C ) . If F and G are defined by equations (48) , then the following conditions are equivalent: /negationslash Proof. Let ξ = d Φ . If F and G is a plane wave then ξ = 0 implies that /negationslash The first equation in equation (51) implies that there exists a constant 1 -form α ∈ Ω 1 ( R 4 , C ) such that X = ξ ∧ α . It is clear that α and ξ ∧ α are both non-zero, since otherwise X = Y = 0 . Combining the latter two equations in equation (51) implies that Since this linear equation for α has a non-zero solution, it follows that ξ ∈ F ( κ ) . See for example, [OFR00, Rub02, HO03, Dah11a]. This completes the proof of implication (i) ⇒ (ii) . For the converse implication it suffices to verify that equations (48)-(50) define a solution to Maxwell's equations. /square Definition 4.2. Suppose κ ∈ Ω 2 2 ( R 4 ) is constant. Then we say that κ is decomposable if there exist non-zero and constant A,B ∈ Ω 2 ( R 4 ) such that if F, G is a plane wave solution to Maxwell's equations, then (53) F ( A ) = 0 or F ( B ) = 0 . Theorem 4.3. Suppose κ ∈ Ω 2 2 ( R 4 ) is constant. Furthermore, suppose for constants α, β, γ ∈ R and β, γ are not both zero. Then κ is decomposable (and condition (53) holds for the same A and B as in condition (54) ). Before the proof, let us note that by Lemma 2.3, the right hand side in equation (54) is non-zero if and only if A,B and ρ are all non-zero. Proof. (Following [LBF12].) Suppose condition (54) holds for some α, β, γ, ρ, A, B . Moreover, suppose F, G is an arbitrary plane wave for κ as in equation (48). To prove the claim we need to show that condition (53) holds. Proposition 4.1 implies that Y = κ ( X ) and whence equation (20) implies that Let { x i } 3 i =0 be coordinates for R 4 where all the aforementioned tensors are constant. Then Here, the first equality follows by condition (54) and (55), and the latter equality follows by a computation in coordinates. Since A and B are real, it follows that F ( A ) = 0 or F ( B ) = 0 . /square In Theorem 5.1 we will see that all the medium tensors in Theorem 3.5 are decomposable. In particular, uniaxial medium is decomposable. The next proposition shows that isotropic medium determined by a Hodge star operator is never decomposable. Proposition 4.4. Suppose κ ∈ Ω 2 2 ( R 4 ) is defined as where C 1 ∈ R \\ { 0 } , C 2 ∈ R and g is a constant indefinite pseudo-Riemann metric on R 4 . Then κ is not decomposable. Proof. Let us first assume that g is a Lorentz metric and let { x i } 3 i =0 be coordinates such that g = k diag( -1 , 1 , 1 , 1) for some k ∈ {-1 , 1 } . At 0 ∈ R 4 , it follows that For a contradiction, suppose κ is decomposable. By Proposition 4.1 there exists a non-zero and constant A,B ∈ Ω 2 ( R 4 ) such that for all ξ, α ∈ T ∗ 0 ( R 4 ) that satisfy ξ ∈ F 0 ( κ ) and /negationslash Let G is the subset G ⊂ F 0 ( κ ) \\ { 0 } for which each coordinate belongs to { 0 , 1 , √ 2 , √ 3 } . That is, one can think of G as a discretisation of F 0 ( κ ) in one quadrant of T ∗ 0 ( R 4 ) . In total there are 19 such points, and for each ξ ∈ G , we can find two linearly independent α ∈ T ∗ 0 ( R 4 ) such conditions (57) holds, cf. [Dah11a]. Insisting that equation (56) holds for all such ξ and α gives 19 × 2 = 38 second order polynomial equations for variables in A and B . Computing a Grobner basis for these equations and solving implies that either A = 0 or B = 0 . See [CLO07]. Hence κ is not decomposable. When g has signature ( --++) the claim follows by repeating the above argument. /square 4.3. Factorisability of the Fresnel polynomial. In what follows condition (i) in Theorem 4.3 will play a key role. Let us therefore introduce the following definition. Definition 4.5. If κ ∈ Ω 2 2 ( R 4 ) is constant and satisfies condition (i) in Theorem 4.3, then we say that κ is algebraically decomposable . In [LBF12], I. Lindell, L. Bergamin and A. Favaro showed that if κ is algebraically decomposable (plus some additional assumptions), then the Fresnel polynomial of κ always factorises into the product of two quadratic forms. In this section we summarise this result in Theorem 4.6. Moreover, we will see that for algebraically decomposable medium, the Fresnel polynomial seems to factorise even when the additional assumptions in Theorem 4.6 are not satisfied. These results suggest (but do not prove) that the definition of algebraically decomposable medium might be a sufficient condition for the Fresnel polynomial to factorise. Let us first note that the class of algebraically decomposable media contains a number medium classes as special cases. If κ is purely skewon, then κ + κ = 0 and κ is algebraically decomposable. Also, if κ satisfies the mixed closure condition κ · κ = λ Id [LBF12, Fav12], then κ is algebraically decomposable. If κ has no skewon part, then κ = κ and the definition of algebraically decomposable medium simplifies. Thus, if κ has no skewon part and if κ is a self-dual medium (so that α Id + βκ + γκ 2 = 0 ) [Lin08], then κ is algebraically decomposable. In particular, skewon-free medium that satisfies the closure condition κ 2 = λ Id [HO03] is algebraically decomposable. Equation (54) that defines algebraically decomposable medium is a nonlinear equation in κ . Suppose { x i } 3 i =0 are coordinates for R 4 , P ∈ R 6 × 6 is the matrix P = ( κ J I ) IJ that represents κ and A,B ∈ R 6 are the column vectors A = ( A I ) I and B = ( B I ) I that represent bivectors A and B with components as in Section 2.4. Then equation (54) reads /negationslash where A t is the matrix transpose and E ∈ R 6 × 6 is the matrix E = ( ε IJ ) IJ . Numerically, E = ( 0 I I 0 ) , where 0 and I are the zero and identity 3 × 3 matrices. When γ = 0 , equation (58) is structurally similar to an algebraic Riccati equation [GLR05]. The next theorem summarises the factorisation result from [LBF12], but restated in the present setting. Theorem 4.6. If κ ∈ Ω 2 2 ( R 4 ) is algebraically decomposable and α, β, γ, ρ, A, B in equation (54) satisfy one of the below conditions: /negationslash /negationslash Then the Fresnel polynomial of κ factorises into the product of two quadratic forms. Let us note that equation (59) is a non-linear equation for D . A priori , the equation has real solutions, complex solutions, or no solutions for D . For a discussion of the last possibility, see below. Pointwise trace( ρD ⊗ D ) = 0 holds if and only if D ∧ D = 0 or ρ = 0 . Let us outline the argument in [LBF12] used to prove Theorem 4.6. Suppose Ω 2 2 ( R 4 ) is algebraically decomposable. If assumption (i) holds, then by rescaling we may assume that β = 1 . Then, since κ + κ = 2( (1) κ + (3) κ ) , it follows that /negationslash for some σ ∈ Ω 2 2 ( R 4 ) with only a skewon part. This gives an explicit representation formula for all κ that satisfy condition (54) with γ = 0 . Computing the Fresnel polynomial for κ shows that it factorises into two quadratic forms. On the other hand, when assumption (ii) holds, then Theorem 4.7 in the below shows that equation (54) transforms into η · η = λ Id for some λ = 0 by a transformation similar to completing the square. Thus, to understand the structure of algebraically decomposable medium that satisfy assumption (ii) , we only need to understand the simpler equation η · η = λ Id with λ = 0 . In [LBF12] the latter equation is solved (see also [Fav12]) using two explicit representation formulas similar to equation (60). Using these representation formulas, the Fresnel polynomial can again be computed, and in both cases it factorises into a product of quadratic forms. /negationslash The next theorem from [LBF12] describes the transformation property of equation (54) used in the proof of Theorem 4.6. The proof is a direct computation using identities (22)(25). For a general discussion of transformation properties for the matrix algebraic Riccati equation, see [CPL10, LR12]. /negationslash Theorem 4.7. Suppose κ ∈ Ω 2 2 ( R 4 ) is algebraically decomposable such that equation (54) holds with γ = 0 . If, moreover, there exists a D ∈ Ω 2 ( R 4 ) such that equation (59) holds, then η ∈ Ω 2 2 ( R 4 ) defined as satisfies /negationslash /negationslash Suppose κ is algebraically decomposable such that equation (54) holds with γ = 0 and β 2 -αγ = 0 . Now we can not use Theorem 4.6 do decise whether the Fresnel polynomial factorises. However, by computer algebra we can find explicit examples of medium tensors with the above properties. Preliminary computer algebra experiments using such expressions suggest that the Fresnel polynomial always seems to factorise when the above assumptions are met. However, the factorisation seems be qualitatively different. Condition β 2 -αγ = 0 seems to imply a linear factor in the Fresnel polynomial. For example, the Fresnel polynomial can factorise into the product of irreducible 1 st and 3 rd order polynomials. On the other hand, suppose κ is algebraically decomposable such that equation (54) holds with γ = 0 , β 2 -αγ = 0 and equation (59) has no real solution for D . Now we can neither use Theorem 4.6 do decise whether the Fresnel polynomial factorises, but we may again construct explicit examples of medium tensors with the above properties. Using these expressions, preliminary computer algebra experiments suggest that the Fresnel polynomial also seems to factorise in this case. In conclusion, these initial observations together with Theorem 4.6 suggest that the definition of algebraically decomposable medium could be a sufficient condition for the Fresnel polynomial to factorise. Lastly, let us note that algebraic Riccati equations, and more generally, quadratic matrix equations, appear in a number of fields. In view of Theorem 4.6 and equation (58), it is, however, interesting to note that quadratic matrix equations appear in the study of polynomial factorisation in one variable [BG05]. Differential Riccati equations also appear in the problem of factoring linear partial differential operators of second and third order [GS04]. /negationslash", "pages": [ 14, 15, 16, 17, 18 ] }, { "title": "5. CHARACTERISATION AND REPRESENTATION OF MEDIA", "content": "WITH A DOUBLE LIGHT CONE Theorem 5.1. Suppose N is a 4 -manifold, and κ ∈ ˜ Ω 2 2 ( N ) is skewon-free and invertible at a point p ∈ N . Then the following conditions are equivalent: /negationslash and κ satisfies inequality (44) at p . As described in the introduction, the above theorem is the main result of this paper. A discussion of the theorem is postponed to the end of this section. In the Theorem 5.1 we will use the computer algebra technique of Grobner bases [CLO07] to eliminate variables from polynomial equations. This technique was also used in [Dah11b]. Let C [ u 1 , . . . , u N ] the ring of complex coefficient polynomials C N → C in variables u 1 , . . . , u N . For polynomials r 1 , . . . , r k ∈ C [ u 1 , . . . , u N ] , let be the the ideal generated by r 1 , . . . , r k . Suppose V ⊂ C N is the solution set to polynomial equations p 1 = 0 , . . . , p M = 0 where p i ∈ C [ u 1 , . . . , u N ] . If I is the ideal generated by p 1 , . . . , p M , the elimination ideals are the ideals defined as Thus, if ( u 1 , . . . , u N ) ∈ V then by [CLO07, Proposition 9, Section 2.5] it follows that p ( u k +1 , . . . , u N ) = 0 for any p ∈ I k , and I k contain polynomial consequences of the original equations that only depend on variables u k +1 , . . . , u N . Using Grobner basis, one can explicitly compute I k [CLO07, Theorem 2 in Section 3.1]. In the below proof this has been done with the built-in Mathematica routine 'GroebnerBasis' . The same technique of eliminating variables was also a key part of the proof of Theorem 3.5 in [Dah11b]. Proof. Let us first prove implication (i) ⇒ (ii) . By [Dah11b, Proposition 1.3] condition (i) implies that F p ( κ ) has no two dimensional subspace. By Theorem 3.5 we only need to check three medium classes. Metaclass I. If κ | p is in Metaclass I, then κ can be written as in equation (31) with conditions on the parameters given by Theorem 3.5. Suppose α 1 = α 2 . Then Theorem 3.5 implies that β 1 = β 2 . Let ρ = 1 2 ( β 2 2 -β 2 1 ) , µ = -α 1 , λ = β 2 2 . Moreover, let A and B be bivectors defined as A = 1 2 A ij ∂ ∂x i ∧ ∂ ∂x j and similarly for B , with coefficients /negationslash /negationslash where subdiagonal terms are determined by antisymmetry. For these parameters, computer algebra shows that equation (63) holds. On the other hand, if α 1 = α 2 , suitable parameters are and where Bivector B is defined by the same formula as for A , but by replacing √ σ with -√ σ . Metaclass II. If κ | p is in Metaclass II, then κ can be written as in equation (32) with conditions on the parameters given by Theorem 3.5. Suitable parameters are ρ = β 1 / 2 , µ = -α 1 , λ = β 2 1 and /negationslash Metaclass IV. If κ | p is of Metaclass IV, then κ can be written as in equation (34) with conditions on the parameters given by Theorem 3.5. If α 1 = α 3 , then suitable parameters are and where and B is defined as in Metaclass I. On the other hand, if α 1 = α 3 , then suitable parameters are ρ = 1 2 ( β 2 1 + α 2 4 ) , µ = -α 3 , λ = β 2 1 and This completes the proof of implication (i) ⇒ (ii) . For the converse implication (ii) ⇒ (i) , suppose that κ satisfies the conditions in (ii) . By Theorem 3.1 we may assume that there are coordinates { x i } 3 i =0 around p such that at p , tensor κ is given by one of the matrices in equations (31)-(37) for some parameters as in Theorem 3.1. Let us consider each of the seven cases separately. Metaclass I. If κ | p is in Metaclass I, then there are coordinates { x i } 3 i =0 around p such that κ is given by equation (31). By scaling A and B we may assume that ρ | p = 1 . Moreover, writing out equation (63) and eliminating variables in A and B using a Grobner basis (see above) yields equations that only involve λ, µ and the parameters in κ . The rest of the argument is divided into three subcases: /negationslash It follows that α 1 , α 2 , α 3 can not be all distinct, and by a coordinate change, we may assume that α 2 = α 3 . If α 1 = α 2 = α 3 , equation (68) implies that µ = -α 1 . Then equation (31) implies that κ = -β 1 ∗ g + α 1 Id at p , where g is the Hodge star operator for the locally defined Lorentz metric g = diag( -1 , 1 , 1 , 1) . Then equation (63) implies that ρ ( A ⊗ B + B ⊗ A ) = 0 . Since this contradicts Lemma 2.3, we have α 1 = α 2 and κ has a double light cone at p by Theorem 3.5. /negationslash Case 2. If exactly two of β 1 , β 2 , β 3 coincide, then after a coordinate change we may assume that β 1 = β 2 = β 3 . Then the Grobner basis equations imply that either λ = β 2 1 or λ = β 2 2 . If λ = β 2 1 , the Grobner basis equations imply that α 1 = α 2 = α 3 and β 1 = β 2 = β 3 . We may therefore assume that λ = β 2 2 . Then the Grobner basis equations imply that µ = -α 2 = -α 3 , and κ has a double light cone at p by Theorem 3.5. Case 3. If all β 1 , β 2 , β 3 are all distinct, then the Grobner basis equations imply that These equations imply that we must have λ = β 2 i and µ = -α i for some i ∈ { 1 , 2 , 3 } . If i = 1 the Grobner basis equations imply that α 1 = α 2 = α 3 and β 1 = β 2 . This contradicts the assumption that all β i are distinct. Similarly, i = 2 and i = 3 lead to contradictions, and Case 3 is not possible. Metaclass II. If κ | p is in Metaclass II, there are coordinates { x i } 3 i =0 around p such that κ is given by equation (32). Writing out equation (63) and eliminating variables as in Metaclass I gives equations that only involve variables λ, µ and the variables in κ . Solving these equations give and κ has a double light cone at p by Theorem 3.5. Metaclass III. If κ | p is in Metaclass III, there are coordinates { x i } 3 i =0 around p such that κ is given by equation (33). Eliminating variables as in Metaclass I implies that β 1 = 0 . Thus κ | p can not be in Metaclass III. /negationslash Metaclass IV. If κ | p is in Metaclass IV, there are coordinates { x i } 3 i =0 around p such that κ is given by equation (34). We have α 4 = 0 since otherwise span { dx 1 | p , dx 2 | p } ⊂ F p ( κ ) . Moreover, since κ is invertible at p it follows that α 2 3 = α 2 4 . Writing out equation (63), eliminating variables as in Metaclass I, and solving implies that /negationslash and κ has a double light cone at p by Theorem 3.5. /negationslash Metaclass VI. If κ | p is in Metaclass VI, there are coordinates { x i } 3 i =0 around p such that κ is given by equation (36). Eliminating variables as in Metaclass I implies that Metaclass V. If κ | p is in Metaclass V, there are coordinates { x i } 3 i =0 around p such that κ is given by equation (35). We may assume that α 3 = 0 , since otherwise span { dx i | p } 3 i =1 ⊂ F p ( κ ) . Eliminating variables as in Metaclass I, and solving implies the contradiction λ + α 2 3 = 0 . Since λ > 0 it follows that κ | p can not be in Metaclass V. Metaclass VII. If κ | p is in Metaclass VII, there are coordinates { x i } 3 i =0 around p such that κ is given by equation (37). Eliminating variables as in Metaclass I and solving implies that Since λ > 0 , it follows that κ | p can not be in Metaclass VI. Since λ > 0 , it follows that κ | p can not be in Metaclass VII. This completes the proof of implication (ii) ⇒ (i) . Implication (iii) ⇒ (i) is a restatement of Proposition 3.6. To prove implication (i) ⇒ (iii) we proceed as in implication (i) ⇒ (ii) and by Theorem 3.5 we only need to check three medium classes. Also, by Proposition 3.6 we do not need to prove inequality (44) since it follows form the other conditions in (iii) when (i) holds. Metaclass I. If κ | p is in Metaclass I, there are coordinates { x i } 3 i =0 around p such that κ is given by equation (31) with conditions on the parameters given by Theorem 3.5. Suppose α 1 = α 2 . Let C 1 = -β 2 2 Ψ √ | det g | , C 2 = α 2 , Ψ = β 2 2 β 1 and in coordinates { x i } , let ρ be defined by ρ = ( β 2 2 -β 1 2 ) / (2 β 1 ) . Then equation (64) holds when A = 1 2 B ij ∂ ∂x i ∧ ∂ ∂x j when coefficients B ij are as in equation (65) and g is the Lorentz metric g = g ij dx i ⊗ dx j with coefficients Onthe other hand, suppose α 1 = α 2 . Let Ψ be one of the two roots to the quadratic equation /negationslash where D 3 is defined as in [Dah11b, Theorem 2.1 (i) ] Since sgn β 1 = sgn β 2 , the discriminant of equation (72) is strictly positive. Thus Ψ ∈ R \\ { 0 } and sgn Ψ = sgn β 1 . Let Ξ ∈ R be defined as /negationslash /negationslash Since α 1 = α 2 we see that Ψ = β 2 2 β 1 is not a solution to equation (72) whence Ξ = 0 . Let C 1 , C 2 be as in the α 1 = α 2 case and let ρ = sgn Ξ . Then equation (64) holds when g is the Lorentz metric given by equation (71) and A = 1 2 A ij ∂ ∂x i ∧ ∂ ∂x j is given by i 3 Metaclass II. If κ | p is in Metaclass II, there are coordinates { x } i =0 around p such that κ is given by equation (32) with conditions on the parameters given by Theorem 3.5. Let C 1 = -1 β 1 √ | det g | , C 2 = α 1 and ρ = 1 / 2 . Then equation (64) holds when A = 1 2 A ij ∂ ∂x i ∧ ∂ ∂x j is as in equation (66) and g is the Lorentz metric g = g ij dx i ⊗ dx j with coefficients Metaclass IV. If κ | p is in Metaclass IV, there are coordinates { x i } 3 i =0 around p such that κ is given by equation (34) with conditions on the parameters given by Theorem 3.5. Suppose α 1 = α 3 . Let C 1 = β 1 Ψ √ | det g | , C 2 = α 1 , Ψ = α 4 /β 1 and ρ = ( α 2 4 + β 2 1 ) / (2 α 4 ) . Then equation (64) holds when A = 1 2 B ij ∂ ∂x i ∧ ∂ ∂x j when B ij are as in equation (67) and g is the Lorentz metric g = g ij dx i ⊗ dx j with coefficients /negationslash Onthe other hand, suppose α 1 = α 3 . Let Ψ be one of the two roots to the quadratic equation 2 , where (see [Dah11b, Theorem 2.1 (iii) ]), /negationslash /negationslash Then Ψ ∈ R \\{ 0 } and since α 1 = α 3 equation (75) implies that Ψ = α 4 β 1 . Thus Ξ ∈ R \\{ 0 } when Let C 1 , C 2 be as in the α 1 = α 3 case and let ρ = sgn Ξ . Then equation (64) holds when g is the Lorentz metric in equation (74) and A is the bivector A = 1 2 A ij ∂ ∂x i ∧ ∂ ∂x j with coefficients This completes the proof of implication (i) ⇒ (iii) . /square Let us first emphasise that the conditions in Theorem 5.1 are written analogously to the conditions in Theorem 3.3. In each theorem, condition (i) is the dynamical description of the medium, condition (ii) is a characterisation of the medium and condition (iii) is a general representation formula. Let us also emphasise that in suitable limits, condition (63) in Theorem 5.1 reduces to the closure condition κ 2 = -λ Id in Theorem 3.3, and representation formula (64) in Theorem 5.1 reduces to κ = f ∗ g in Theorem 3.3. Let us also emphasise that in both theorems, all conditions are tensorial, and do not depend on coordinate expressions. A difference between the theorems is that Theorem 3.3 is a global result, while Theorem 5.1 is a pointwise result. All the mediums in Theorem 5.1 satisfy the technical assumptions in Theorem 4.6 with either D = A or D = B when A and B are as in equation (63). As described in the introduction, condition (ii) in Theorem 5.1 is a slight strengthening of the conditions derived in [LBF12] (see Theorem 4.3 in the above). Representation formula (64) in Theorem 5.1 is also adapted from [LBF12]. For constant medium tensors on R 4 , Theorem 5.1 implies that if κ is invertible, skewon-free and has a double light cone, then κ is algebraically decomposable, and hence decomposable by [LBF12] (see Theorem 4.3). In this setting, Theorem 5.1 explicitly shows that the behaviour of signal-speed imposes a constraint on the behaviour of polarisation. This can be seen as somewhat unexpected. However, the explanation is that polarisation and signal speeds are not independent for a propagating wave, but constrained by equation (52). For a further discussion, see [Dah11a]. It is also instructive to note that condition (63) is a second order polynomial constraint on the coefficients in κ , but the definition of a double light cone involves the Fresnel surface, which is a constraint involving third order polynomials of the coefficients in κ . The same phenomenon appears in equivalence (i) ⇔ (ii) in Theorem 3.3. Part of condition (ii) is condition (a) , that states that the Fresnel surface of κ contains no two dimensional subspace. Let us describe five results where this condition also appears. First, if the Fresnel surface of a κ ∈ ˜ Ω 2 2 ( N ) can be written as F p ( κ ) = { ξ ∈ T ∗ p ( N ) : ( g ( ξ, ξ )) 2 = 0 } for a pseudo-Riemann metric g , then condition (a) is satisfied if and only if g has signature ( - -++) . This follows by a result of J. Montaldi [Mon07]. For example, if g = diag( -1 , -1 , 1 , 1) , then F p ( κ ) contains the 2 -dimensional subspace span { ∂ ∂x 0 + ∂ ∂x 3 , ∂ ∂x 1 + ∂ ∂x 2 } . Second, one can prove that condition (a) is always satisfied if κ decomposes into a double light cone (Proposition 1.3 in [Dah11b]). Third, in matter dynamics systems, condition (a) can be motivated by the behaviour of energy [RRS11]. In the terminology of [RRS11], condition (a) can be replaced by the stronger condition that κ is bihyperbolic . Fourth, condition (a) also appears in the study of the well posedness of Maxwell's equations as an initial value problem [SWW10]. Lastly, in the normal form representation of skewon-free medium tensors in [SWW10], condition (a) simplifies the representation since the condition excludes all but the first 7 coordinate representations. See [SWW10] and Section 3.1 in the above. When equivalence holds in Theorem 5.1, there does not seem to be a simple relation between parameters C 1 , C 2 , ρ, A, g in equation (64) and parameters µ, λ, ρ, A, B in equation (63). However, if equation (64) holds for an A such that A ∧ A = 0 (that is, A is decomposable or simple [Coh05, p. 185]), then equation (63) holds for parameters Using a Grobner basis argument one can show that the tensor κ defined by equation (31) when β 1 = β 2 = β 3 = 1 , α 1 = 1 and α 2 = α 3 = 2 is invertible and has a double light cone. However, it can not be written as in equation (64) for an A such that A ∧ A = 0 . Acknowledgements. This work has been supported by the Academy of Finland (project 13132527) and by the Institute of Mathematics at Aalto University. I would like to thank Luzi Bergamin, Alberto Favaro and Ismo Lindell for useful discussions on this topic.", "pages": [ 19, 20, 21, 22, 23, 24, 25 ] } ]

[ { "title": "INVARIANT CLASSIFICATION OF VACUUM PP-WAVES", "content": "R. MILSON, A. COLEY, D. MCNUTT Abstract. We solve the equivalence problem for vacuum PP-wave spacetimes by employing the Karlhede algorithm. Our main result is a suite of Cartan invariants that allows for the complete invariant classification of the vacuum pp-waves. In particular, we derive the invariant characterization of the G 2 and G 3 sub-classes in terms of these invariants. It is known [5] that the invariant classification of vacuum pp-waves requires at most the fourth order covariant derivative of the curvature tensor, but no specific examples requiring the fourth order were known. Using our comprehensive classification, we prove that the q ≤ 4 bound is sharp and explicitly describe all such maximal order solutions.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "In general relativity, identical spacetimes are often given in different coordinate systems, thereby disguising the diffeomorphic equivalence of the underlying metrics. It is consequently of fundamental importance to have an invariant procedure for deciding the question of metric equivalence. One approach to this problem is to utilize scalar curvature invariants, obtained as full contractions of the curvature tensor and its covariant derivatives [2]. However, a particularly intriguing situation arises when we consider pp-waves, space-times that admit a covariantly constant null vector field [8, Chapter 24].. Some time ago it was observed that all curvature invariants of a pp-wave spacetime vanish [15]. Subsequently all space-times with the VSI property (vanishing scalar invariants) and the more general CSI property (constant scalar invariants) were classified [13, 3]. It is now known that either a spacetime is uniquely determined by its scalar curvature invariants, or is a degenerate Kundt spacetime [2, 4]; the VSI and CSI solutions belong to this more general class. To invariantly classify the degenerate Kundt spacetimes, and pp-waves in particular, one must therefore use the Karlhede algorithm [7] [8, Chapter 9.2], which is the Cartan equivalence method [1] adapted to the case of 4-dimensional Lorentzian manifolds. The invariant classification proceeds by reducing the 6-dimensional Lorentz frame freedom by normalizing the curvature tensor R and its covariant derivatives, R q . The unnormalized components of R q are called Cartan invariants . We define the IC (invariant classification) order of a given metric to be the highest order q required for deciding the equivalence problem for that metric. An upper bound on the IC order is often referred to as the Karlhede bound . Set t -1 = 0 and d -1 = 6 (the dimension of the Lorentz group). At each order q ≥ 0, let 0 ≤ t q -1 ≤ t q denote the number of functionally independent Cartan invariants and let 6 ≥ d q -1 ≥ d q denote the dimension of the joint isotropy group of the normalized R,R 1 , . . . , R q . The algorithm terminates as soon as t q -1 = t q and d q -1 = d q . A value of d q = 0 means that there exists an invariant tetrad. If t q < 4, then Killing vectors are present. The dimension of the isometry group is 4 -t q + d q . Henceforth, we will refer to the sequence ( t 0 , t 1 , . . . , t q ) as the invariant count . In this paper, we focus on a particularly simple class of VSI spacetimes: the vacuum pp-waves, whose metric has the simple form shown in equation (9) below. The symmetry classes for pp-waves were initially classified by Kundt and Ehlers [6] [8, Table 24.2] for vacuum solutions, and subsequently extended by Sippel and Goenner [16] to the general case. The Karlhede bound for pp-waves was investigated in [5] and [9] where q ≤ 4 was established; however, it was not known whether this bound is sharp, or if it could be lowered further. Despite the fact that these metrics have a very simple form, depending on just one parametric function f ( ζ, u ) (see equation (9) below), the present paper is the first to present a complete invariant classification for vacuum pp-waves, and to establish the sharpness of the q ≤ 4 bound. glyph[negationslash] All vacuum pp-waves have at least one Killing vector. Kundt and Ehlers identified 3 classes of G 2 solutions, 4 classes of G 3 solutions, a universal form for the G 5 solutions, and two types of homogeneous G 6 solutions. Below, we exhibit explicit Cartan invariants that distinguish the various special sub-classes in an invariant fashion. glyph[negationslash] The G 1 , G 2 , G 3 solutions ( α = 0) and the G 5 , G 6 solutions ( α = 0) form two distinct solution branches; here α is a fundamental 1st order invariant which will be defined precisely in Section 2. The classification of the α = 0 class is summarized in Figure 1. The numbers in the solution labels refer to the invariant count with the initial 0 and any trailing 3 omitted. Thus, solution form AP 123 refers to a metric with an invariant count of (0 , 1 , 2 , 3 , 3) while AP 122 refers to a G 2 solution with an invariant count of (0 , 1 , 2 , 2). The G 1 solutions have three independent invariants and thus their label indices end with a 3. For the same reason, the indices of the G 2 solutions end with a 2 while the indices of the G 3 , G 5 solutions end with a 1. glyph[negationslash] From the point of view of invariant classification there are 4 classes of generic G 2 solutions. We label these A 22 , B 22 , C 22 , L 22 and summarize their invariant classification in Table 1 (the Cartan invariants in the third column will be defined in Section 3.) Kundt-Ehlers described forms B 22 and L 22 . Their third G 2 form is glyph[negationslash] where F is a holomorphic function and k a real constant. The k parameter is not essential, and if k = 0 can be normalized to k → 1 by means of a coordinate transformation. In terms of the present terminology, the Kundt-Ehlers solutions of type (1) belong to class C 22 in the the case of k = 1, and and to class A 22 if k = 0. glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] One benefit of the invariant classification is a clear description of the mechanism of specialization of the G 1 → G 2 → G 3 solutions. In order to understand the G 1 → G 2 specialization one first has to understand the invariant mechanism by which the solution forms in Table 1 arise. To that end, we show in Proposition 3.3 that all of vacuum pp-wave solutions of interest can be reduced to the following form where F is a holomorphic functions and where g i = g i ( u ) , i = 1 , 2 , 3 are complex valued functions of one variable. This general ansatz, which we name A ∗∗ 23 , bifurcates into a number of more specialized forms, which are summarized in Table 2 of the Appendix. Roughly speaking, there are 6 solution forms, which we label by A,B,C, P,E,L and by numerical indices that describe the invariant count. An asterisk denotes a generic precursor of a more specialized solution. The labels P,E,L refer to, respectively, solutions of power, exponential and logarithmic type. Roughly speaking, the Kundt-Ehlers G 2 solution forms are appropriate specializations of the A,B,C and L solution forms. The G 1 → G 2 specialization can be understood via the notion of a 'precursor solution'. This is a G 1 solution that is mild generalization of a corresponding G 2 solution. For example the precursor of the B 22 solution is the B 23 solution where g = g ( u ) is an arbitrary complex valued function of one variable. Precursors of the other G 2 solutions have an analogous form. The invariant conditions that define the various precursor classes are listed in Table 4 of the Appendix. In each case, the specialization to a G 2 involves the loss of the gζ term, or equivalently, the vanishing of a certain higher order invariant. As we show below, a vacuum pp-wave has no zeroth order invariants [8], and generically two independent first order invariants, α, α ∗ . In order to understand the G 2 → G 3 specialization it is necessary to understand the sub-class of solutions for which t 1 = 1; i.e, metrics for which the invariants α and α ∗ are functionally dependent. We refer to such solutions as belonging to the (0,1) class and devote Section 4 to their analysis. Thus, the specialization to the G 3 solutions follows the following path: where the middle step consists of type (0,1) G 2 solutions; summarized in Table 7 of the Appendix. Another consequence of our analysis is a firm determination of the Karlhede bound for vacuum pp-waves. It turns that q ≤ 4 is the sharp bound. Theorem 1.1. There exist vacuum pp-wave spacetimes with an IC order q = 4 . Every such metric belongs to one of the four classes exhibited in Table 6. Note that metrics that require 4th order invariants for invariant classification necessarily have a (0,1,2,3,3) as their invariant count. glyph[negationslash] glyph[negationslash] The rest of the paper is organized as follows. Section 2 is an introductory description of the Karlhede algorithm as it applies to the class of vacuum pp-wave metrics. In particular, this section describes the fundamental bifurcation into the generic α = 0 class and the specialized α = 0 subclass. The invariant classification of the former consists of 8 sub-class types shown in Figure 2. Section 3 introduces the various Cartan invariants necessary for the generic classification and derives the A,B,C,P,E,L solution forms in an invariant manner. Section 4 deals with the type (0,1) solutions in the α = 0 class. Section 5 classifies the G 2 -precursor solutions. Section 7 derives and classifies the G 1 metrics having maximal IC order; the proof of Theorem 1.1 is given here. Sections 3, 4, 5, 7, when taken together, constitute the invariant classification of the G 1 solutions; the specialization diagram for the various G 1 sub-classes is presented in Figure 4 of the Appendix. Sections 6 and 8 deal with the invariant classification of the G 2 and G 3 solutions, respectively. The glyph[negationslash] α = 0 branch consists of G 5 and G 6 solutions. There is a generic G 5 solution that specializes into two distinct classes of homogeneous G 6 solutions, as per Figure 3. This branch of the classification is discussed in Section 9 and summarized in Table 9. Remark: the invariant analysis in Section 8 brings to light a minor classification mistake found in line 6 of [8, Table 24.2]. This line describes a G 3 class which is listed as BL 11 in our Table 8. Kundt-Ehlers give the solution as au -2 ln ζ with a a real constant. This is incorrect; the leading coefficient should be an arbitrary complex number.", "pages": [ 1, 2, 3, 4, 5 ] }, { "title": "2. Vacuum pp-wave spacetimes", "content": "Throughout, we use the four-dimensional Newman-Penrose formalism [12] adapted to a complex, null-tetrad ( e a ) = ( m a , m ∗ a , glyph[lscript] a , n a ) = ( δ, δ ∗ , D, ∆). These vectors satisfy with all other cross-products zero. Equivalently, letting θ 1 , . . . , θ 4 denote the dual coframe, the metric is given by The connection 1-form and the the curvature 2-form are defined, respectively by The connection components are labeled by the 12 Newman-Penrose scalars: The curvature components are labelled by the Ricci scalar Λ = ¯ Λ, traceless Ricci components Φ AB = ¯ Φ BA , A,B = 0 , 1 , 2, and Weyl components Ψ C , C = 0 , . . . , 4: where θ ab = θ a ∧ θ b . A pp-wave is a space-time admitting a covariantly constant null vector field. this entails Such space-times are necessarily Petrov type N or type O and belong to the Kundt class [8, Sect. 24.5]. A vacuum pp-wave that isn't flat-space is necessarily type N: glyph[negationslash] Applying a boost and a spatial rotation we normalize the tetrad by setting Ψ 4 → 1. Therefore, there are no 0th order Cartan invariants. The remaining frame freedom consists of the 2-dimensional group of null rotations. The above constraints can be integrated to yield the following class of exact solutions [8, Section 24.5]: where f = f ( ζ, u ) is analytic in ζ . The above form is preserved by the following class of transformations: The Bianchi identities [8, (7.32c) (7.32d)] impose: Using the notation of [5], the non-vanishing 1st order components are: The transformation law for these components is [8, (7.7c)] glyph[negationslash] where z is a complex valued scalar. Therefore, α is a 1st order Cartan invariant and the invariant classification divides into two cases: α = 0 and α = 0. In the first case, γ is an invariant, while in the 2nd case, we fix the tetrad by normalizing γ → 0. We consider these two cases in more detail. glyph[negationslash] Proposition 2.1. Suppose that α = 0 . Then, d p = 0 for p ≥ 1 . The possible values of the invariant count sequence are: The first 3 possibilities describe a G 1 , the next 2 possibilities are a G 2 , and the last possibility is a G 3 . The Cartan invariants are generated by and their complex conjugates, where the above spin coefficients are calculated relative to the normalized Ψ 4 → 1 , γ → 0 tetrad. Proposition 2.2. Suppose that α = 0 . Then d p = 2 for all p . The possible values of the invariant count sequence are The first possibility describes a G 5 . The second possibility describes a G 6 (homogeneous space). The Cartan invariants are generated by and their complex conjugates, calculated relative to a tetrad normalized by Ψ 4 → 1 . In the following sections we will show that each of these cases describes a welldefined class of solutions, and go on to derive a the canonical forms for the metric in each case. glyph[negationslash] We now turn to the proof of Proposition 2.1, which concerns the α = 0 case. The NP equations [8, (7.21f) (7.21o)] imply the additional constraints The non-vanishing 2nd order curvature components are [5, (4.2a)-(4.2t)]: Therefore, the independent 2nd order Cartan invariants are µ, ν, δ ∗ α and the corresponding complex conjugates. The commutator relations are The NP-equations imply the following relations amongst the invariants: Higher order relations follow in a straight-forward manner from these and from the commutator relations. Fixing Ψ 4 → 1 reduces the isotropy to null rotation. Fixing γ → 0 eliminates this frame freedom. Therefore, the isotropy is trivial. Equation (21) implies that α is not constant. All invariants are annihilated by D . Therefore, there are either 3, 2, or 1 independent Cartan invariants. The conclusions of Proposition 2.1 now follow directly from the Karlhede algorithm. Next we present the proof of Proposition 2.2, which treats the α = 0 class. As was mentioned above, the 1st order Cartan invariants are generated by The Newman-Penrose equations [8, (7.21f) (7.21o) (7.21r)] imply There is only one non-zero 2nd order curvature component, namely The operator transformation law for null rotations is [8, (7.7a)] Therefore, by (28), ∆ n γ is well-defined, despite the fact that no canonical choice of ∆ exists and is invariant with respect to null rotations. By [8, (7.6a)-(7.6d)] all commutators are spanned by δ, δ ∗ , D . This implies that Therefore there are two possibilities. Either γ is a constant, in which case we have a homogeneous G 6 ; or γ is the unique independent invariant, in which case we have a G 5 . This concludes the proof of Proposition 2.2.", "pages": [ 5, 6, 7, 8 ] }, { "title": "3. The G 1 solutions", "content": "glyph[negationslash] In this section, we derive solutions for certain key G 1 sub-classes. We assume that α = 0 for the remainder of this section. The solutions are summarized in Table 2 and 3. In the tables, F = F ( z ) is an analytic function; g = g ( u ) is complex-valued function of u ; h = h ( u ) is a real-valued function of u ; and k is a real constant. The meaning of g 1 , g 2 , h 1 , h 2 , k 1 , k 2 are analogous. glyph[negationslash] In the preceding section we established that α = 0 solutions admit an invariant tetrad characterized by the normalizations Let ω 1 , ω 2 = ( ω 1 ) ∗ , ω 3 = ( ω 3 ) ∗ , ω 4 = ( ω 4 ) ∗ denote the coframe dual to δ, δ ∗ , ∆ , D . We introduce the following key invariants. glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] Even though the γ → 0 normalization is the most obvious way to select an invariant tetrad, an equally useful normalization is ˆ ∆ α → 0. The reason is that a Killing vector V necessarily annihilates all invariants, and hence it will turn out to be useful to work in a frame where ˆ ∆ is a linear combination of Killing vectors. For a given vector field V let us write where glyph[negationslash] The following proposition shows that if AA ∗ = 1, then the normalization ˆ ∆ α → 0 selects a well-defined invariant tetrad. Proposition 3.1. Suppose that AA ∗ = 1 . Then, every vector field that satisfies glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] has the form V = a ˆ ∆+ bD, a = 0 . If AA ∗ = 1 , but B = 0 , then (42) does not have a solution. If AA ∗ = 1 and B = 0 , then there is a 1-parameter family of solutions to (42) . Proof. The null-rotation transformation law for ∆ is [8, 7.7 (c)], Hence, by (20)-(22) and (30) we seek a scalar ˆ z such that If AA ∗ = 1, the solution is glyph[negationslash] If AA ∗ = 0, then the system has rank 1. In this case the system is consistent if and only if Next, we establish some key relations for these invariants and certain other scalars that will prove useful in our calculations. Proposition 3.2. Suppose that α = 0 . If the normalization (29) holds then glyph[negationslash] where glyph[negationslash] We also have Furthermore, if Q = Q ( a, u ) then We begin by deriving some a key classes of G 1 solutions; all the various solutions discussed in this paper are subclasses of these general categories. Proposition 3.3. Suppose that α = 0 . The following conditions are equivalent: glyph[negationslash] glyph[negationslash] where F = F ( z ) is an analytic function such that F ''' ( z ) = 0 and where g i = g i ( u ) , i = 1 , 2 , 3 are complex-valued such that g 1 , g 2 = 0 . Furthermore, δM = 0 if and only if g 1 = g -2 2 ; i.e., glyph[negationslash] Hence, Hence, by (61), condition (62) is equivalent to The latter condition is equivalent to (68). glyph[negationslash] glyph[negationslash] glyph[square] Proposition 3.4. Suppose that B 1 = 0 and AA ∗ = 1 . Then the following are equivalent: (i) B 2 /B 1 = k , is a real constant and (ii) f ( ζ, u ) = F ( h i k ζ ) h 2 + gζ . In addition M = 0 if and only if g 1 = g 2 = 1 ; i.e., Proof. Our first claim is that (63) is equivalent to the following conditions: Note that since α = 0, by (46), we must have L = 0. We now consider two cases. glyph[negationslash] glyph[negationslash] First, let us consider the case of δM = 0. Note that in this case (62) holds trivially. Also, in this case, L ua = 0, and hence without loss of generality, g 5 = 0. The case of M = 0 is true if and only if L u = 0. Here g 5 = 0 and g 1 = g 2 = 1. glyph[negationslash] Let us now consider the generic case where δM = 0. In this case, (62) can be restated as Observe that Hence, (62) is equivalent to Next, we observe that Proof. Let C = 1 + i k so that condition (i) is equivalent to or Im( B/C ) = 0. Suppose that (i) holds. By (31), Hence, assuming (69) and by (47) (50), By Proposition 3.2, the above is both real and holomorphic in a , and hence independent of a . Hence, glyph[negationslash] glyph[negationslash] where h 1 = h 1 ( u ) = 0 is real. Conversely, (71) with h 1 = 0 implies condition (i). Hence, glyph[negationslash] Proposition 3.5. Suppose that A = 1 . Then, the following are equivalent: (i) B 1 = 0 , and (ii) f ( ζ, u ) = F (e i h ζ ) + gζ . Proof. By (31) (47) (50), condition (i) is equivalent to glyph[negationslash] where L a = -2, by assumption. Hence, where h 1 = h 1 ( u ) is real. Hence, Proposition 3.6. The following are equivalent: (i) AA ∗ = 1 , and (ii) L = Pa + g whereb g = g ( u ) and P = P ( u ) such that PP ∗ + P + P ∗ = 0 . Proof. By (50), A ∗ is holomorphic in a . Hence, if AA ∗ = 1 then, A must be independent of a ; i.e., L a = P = P ( u ). Since A ∗ = 1 /A = -1 -L a , condition (ii) follows. glyph[square] glyph[negationslash] Proposition 3.7. Suppose that A 2 = 1 . The following are equivalent: (i) AA ∗ = 1 and (ii) f ( ζ, u ) = (e g 1 ζ ) i h + g 2 ζ Proof. By Proposition 3.6, condition (i) is equivalent to where, by assumption, P = 0 , -2. Hence Re(1 /P ) = -1 / 2, whence glyph[negationslash] where h = h ( u ) = 0 is real. Hence, glyph[negationslash] Hence (ii) follows with Proposition 3.8. The following are equivalent: (i) A = -1 and (ii) f ( ζ, u ) = exp( g 1 ζ ) + g 2 ζ Proof. By the Lemma, condition (i) can be restated as L = g . Hence, Proposition 3.9. The following are equivalent: (i) A = 1 and (ii) f ( ζ, u ) = g 1 log ζ + g 2 ζ . Proof. By the Lemma, condition (i) can be restated as L = -2 a + g . Hence, Note that if A = 1, then B = µ -µ ∗ . Hence, if A = 1, then B 1 = 0 automatically. Above we showed that α, α ∗ generate the 1st order invariants. Generically, these are independent and hence, generically, the invariant count is (0 , 2). However, an important subclass occurs for which d α ∧ d α ∗ = 0. We will refer to these as the (0 , 1) solutions. The next two Propositions characterize the (0,1) solutions in terms of invariants. glyph[negationslash] Proposition 4.1. If µ = 0 , then d α ∧ d α ∗ = 0 if and only if B = 0 . In this case, the condition AA ∗ = 1 follows automatically. If µ = 0 , then d α ∧ d α ∗ = 0 if and only if AA ∗ = 1 . Proof. By (21) (22), glyph[negationslash] Hence, the condition d α ∧ d α ∗ = 0 is equivalent to the conjunction of AA ∗ = 1 and B = 0. However, if µ = 0 and B = 0, then A = µ ∗ /µ , and hence AA ∗ = 1 automatically. Therefore, if µ = 0, then the condition B = 0 suffices. On the other hand, if µ = 0, then B = 0, and therefore the condition AA ∗ = 1 suffices. glyph[square] glyph[negationslash] Proposition 4.2. Suppose that B = 0 and AA ∗ = 1 . Then, necessarily A is a constant and δM = 0 . Proof. By Proposition 3.6, L = Pa + g where P = P ( u ) , g = g ( u ). Since B = 0, we have Taking the derivative with respect to a gives L ua = 0. Hence, A must be a constant. Furthermore, by (48) (60), as was to be shown. Lemma 4.3. Suppose that B = 0 and AA ∗ = 1 . If A = 1 , then glyph[negationslash] where k is a real constant, and h = h ( u ) is real. If A = 1 , then Proof. By Propositions 3.6, 4.2, L = Pa + g where g = g ( u ) and P = -( A +1) /A is a constant. Hence, equation (72) can be restated as glyph[negationslash] If A = 1, we multiply both sides by A/ ( A -1) to obtain Re( A/ ( A -1) g ' ( u )) = 0. This gives us (73). If A = 1, then (74) follows immediately. glyph[square] Proposition 4.4. A type (0 , 1) solution belongs to one of the classes shown in Table 3. Proof. By Proposition 4.1, B = 0 and AA ∗ = 1. We proceed by cases. Suppose that A 2 = 1. By Proposition 3.7, glyph[negationslash] Since (1 + P/ 2) = ( A -1) / (2 A ), we must have g 1 = k +i h by Lemma 4.3 . This gives form P 13 . Next, consider the case A = -1. Here, L = k +i h . By Proposition 3.8 we arrive at form E 13 . Finally, if A = 1, then (74) and Proposition 3.9 give form L 13 . glyph[square] glyph[square]", "pages": [ 8, 9, 10, 11, 12, 13, 14 ] }, { "title": "5. The G 2 precursors", "content": "glyph[negationslash] As above, we assume that α = 0 and that δ, δ ∗ , ∆ , D is a tetrad normalized so that Ψ 4 → 1 and γ → 0. In this section we classify the solutions that satisfy the following definition. Definition 5.1. We say that a vacuum pp-wave metric is a G 2 -precursor if there exists a vector field V = V 1 δ + V 2 δ ∗ + V 3 ∆+ V 4 D such that glyph[negationslash] glyph[negationslash] A Killing vector annihilates all invariant scalars and invariant differential forms [14, Ch. 8-10]. Thus, the 'precursor' terminology reflects the fact that (75) is a necessary, but not sufficient condition, for the existence of a Killing vector independent from D = ∂ v . The requisite propositions and proofs are presented below. The resulting classification of precursor solutions is summarized in Tables 4 and 5. Proposition 5.2. Let V = V 1 δ + V 2 δ ∗ + V 3 ∆+ V 4 D be a vector field. Relation L V ω 1 = L V ω 3 = 0 holds if and only if C = α ∗ V 1 is a constant, while V 3 satisfies Proof. By (20) - (22) and (30), By (51) and the definition of C , We also have the following identity: The desired equivalence follows immediately. Proposition 5.3. If B = 0 , then (75) is equivalent to the the conjunction of (84) and the condition Proof. Note the following structure equations, which are dual to the commutator relations (16)-(19) If (75) holds, then glyph[negationslash] glyph[square] because α, α ∗ , µ are the structure functions in (86) (87). But, if 3 functions on a 4-dimensional manifold are annihilated by 2 independent vector fields, then they must be functionally dependent. Therefore (84) holds. By Proposition 3.1, where ˆ ∆ is defined as per (37), and a, b are some functions. By Proposition 5.2, are constants. Hence, by (33), is a constant. Conversely, suppose that (84) and (85) hold. By assumption, (89) holds for some constant C . Hence, C/X = C ∗ /X ∗ is real. Set V = C/X ˆ ∆. This is a real vector field such that, by construction, L V α = L V α ∗ = 0. Since α, α ∗ , µ are functionally dependent, we also have L V µ = 0. By relation (80), L V ω 1 = 0. By (86) and (83), glyph[negationslash] Since L V ω 3 is real and µ = 0 by assumption, it follows that L V ω 3 = 0. glyph[square]", "pages": [ 15, 16 ] }, { "title": "Remark: Observe that", "content": "glyph[negationslash] Hence, if B 1 = 0, then condition (85) can be conveniently expressed as B 2 /B 1 = k where k is a real constant. glyph[negationslash] We now show that type (0 , 2) precursor solutions belong to the 4 classes shown in Table 4. Proposition 5.4 characterize the precursor solutions for which V 3 = 0. Proposition 5.5 characterizes precursor solutions for which V 1 = 0. This leaves the case where both V 1 , V 3 are non-zero. Since we are considering type (0 , 2) solutions, we exclude the possibility that B = 0. The possibility that B = 0 but AA ∗ = 1 is excluded by Proposition 3.1. The remaining possibilities can be divided into the case B 1 = 0 and the case B 1 = 0. Proposition 5.6 deals with the former and 5.7 with the latter. glyph[negationslash] Proposition 5.4. There exists a vector field V such that glyph[negationslash] if and only if A = 1 . Proof. Suppose that (90) holds. By (78), C + C ∗ = 0, and hence C = α ∗ V 1 is imaginary. Hence, by (76), C + C ∗ A = 0, which means that A = 1. Conversely, if A = 1, then in order for (76) - (78) to hold, it suffices to set V 1 = i /α ∗ , V 3 = 0. glyph[square] Proposition 5.5. There exists a vector field V such that glyph[negationslash] if and only if µ = 0 . Proof. Suppose that (91) holds. Hence, by (79) , Therefore, µ = 0. To prove the converse, it suffices to take V 3 = e a + a ∗ . Relations (77) and (78) follow by (60) (61). glyph[square] glyph[negationslash] glyph[negationslash] Proposition 5.6. Suppose B 1 = 0 , AA ∗ = 1 . The following are equivalent: (i) condition (75) holds; (ii) B 2 /B 1 = k, ∆ X 1 = 2 X 2 1 ; (iii) f ( ζ, u ) = F ( u -i k ζ ) u -2 + gζ . Proof. Suppose that (i) holds. Since V 3 is real, by Proposition 5.2, Since B = 0, we have µ = 0 also. Hence, C = 0, by (76). Hence, glyph[negationslash] glyph[negationslash] glyph[negationslash] Hence, C = 1 + i k , without loss of generality, and B 2 /B 1 = k . Furthermore, since X/X ∗ = B/B ∗ , we have Therefore, (ii) follows by (78). Next, we show that (ii) implies (iii). By Proposition 3.4, f ( ζ, u ) = F ( h i k ζ ) h 2 + gζ belongs to class B ∗ 23 . In the proof of Proposition 3.4, we showed that where h 1 = h 1 ( u ) is real. Hence, by (61) In the last step we can omit the constant of integration because of transformation freedom (12). Therefore Following the steps in the proof of Proposition 3.4 gives h = u -1 , which specializes solution form B ∗ 23 to form B 23 . Finally we show that (iii) implies (i). For this, it suffices to set V 1 = C/α ∗ where C = 1 + i k and to set Conditions (77) (78) follow by (60) (61). glyph[square] Proposition 5.7. Suppose that B 1 = 0 , µ = 0 , AA ∗ = 1 . The following are equivalent: (i) condition (75) holds; (ii) ∆ X 2 = 0 ; (iii) f ( ζ, u ) = F (e i u ζ ) + gζ . glyph[negationslash] glyph[negationslash] glyph[negationslash] Proof. Let us show that (i) implies (ii). As above, C = α ∗ V 1 = 0 is a constant such that Im( B/C ) = 0. Since B 1 = 0 we have C = i without loss of generality. Hence, V 3 = 1 /X 2 and ∆ X 2 = 0 by (78). Next, we show that (ii) implies (iii). By assumption, f ( ζ, u ) = F (e i h ζ ) + gζ belongs to class C ∗ 23 . Since B = i B 2 we have by Hence, by Proposition 3.5 where h 1 ( u ) = -2 h ' ( u ) = 0 is real. Since ∆ X 2 = 0, we infer that h 1 is a constant. Hence, without loss of generality, h ( u ) = u . glyph[negationslash] Finally we show that (iii) implies (i). For this, it suffices to set V 1 = C/α ∗ where C = i and to set We now classify the type (0 , 1) precursor solutions. glyph[negationslash] glyph[negationslash] Proposition 5.8. Suppose that B = 0 , A = 1 , µ = 0 . Then (75) holds if and only if Proof. Suppose that (75) holds. By Proposition 4.2, A is a constant. Hence, (93) follows by (76) (78). Conversely, suppose that (93) holds. By Proposition 5.2, we seek a constant C such that and such that the above V 3 satisfies (77) and (78). First, observe that A ∗ = 1 /A and µ ∗ = Aµ . Hence, Therefore, V 3 is well-defined for any choice of C . By Proposition 4.2, δ ( αµ ) = 0. Hence Hence, (77) is satisfied for all choices of C . We now turn to condition (78). By (47) and (73) of Lemma 4.3 where h = h ( u ) is real. Hence, by (61), is a real constant. If k = 0, then condition (78) can be satisfied by taking C = i. If k = 0, (78) is satisfied by taking C = A/ ( A -1) + i /k . With this choice, glyph[negationslash] Proposition 5.9. The type (0,1) precursor solutions belong to one of the classes shown in Table 5. glyph[negationslash] Proof. By Proposition 5.4 the B = 0 , A = 1 solutions are automatically precursor solutions with V 3 = 0 , V 1 = 0. We now classify all precursor solutions that admit a vector field that satisfies (75) with V 3 = 0. We consider two cases: µ = 0 and µ = 0. Suppose the former. By the above Lemma, a precursor solution is characterized by the condition ∆ 2 (1 /µ ) = 0, which is equivalent to glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] where h is the parameter in solution forms P 13 , E 13 , L 13 . This gives us four classes of solutions. Class BP 13 corresponds to the case A = -1 and k = 0. In this case, the solution of (95), without loss of generality, is h = 1 k log u . Class CP 13 corresponds to A = -1 and k = 0. Here, without loss of generality, the solution to (95) is h = u . Similarly, the condition A = -1 gives solution classes BE 13 and CE 13 . Finally, consider the case of A = 1. Here µ ∗ = µ . Hence, by Proposition 3.9 L = -2 a +i k + h where h is real. By Lemma 5.2 we require that glyph[negationslash] glyph[negationslash] Since δ ( αµ ) = 0, we automatically have δ (1 /µ ) = α ∗ µ ; condition (77) is automatically satisfied. Hence, a necessary and sufficient condition for a precursor solution is ∆(1 /µ ) = -1, or equivalently ∆ µ = µ 2 . This is equivalent to h '' ( u ) = h ' ( u ) 2 , which, by employing the freedom (12), gives us h ( u ) = -log u . Employing the integration steps in Proposition 3.9, this gives us f = Cu -2 log ζ + gζ , which is solution form BL 13 . Next, suppose that µ = 0 , AA ∗ = 1. Here it suffice to specialize one of the Table 3 solutions. For classes P 13 and E 13 we set h → 0. For the logarithmic solution L 13 we set h → k , where the latter is a constant. glyph[square]", "pages": [ 16, 17, 18, 19 ] }, { "title": "6. The G 2 solutions", "content": "In this section we characterize and classify the vacuum pp-waves with two independent Killing vectors. Since a Killing vector annihilates the invariant 1-forms ω 1 , . . . , ω 4 , every G 2 solution is a specialization of the precursor metrics discussed in the preceding Section. We first present the invariant characterization of the generic, type (0 , 2 , 2) solutions, and then present the characterization of the type (0 , 1 , 2 , 2) solutions. We then pass to a detailed classification, the results of which are displayed in Tables 1 and 7. Proposition 6.1. A type (0,2,2) G 2 solution is characterized by (75) and glyph[negationslash] Proof. By Proposition 2.1, the 2nd order Cartan invariants are generated by A,µ,ν . Suppose that there exists a Killing vector V independent from D . Condition (75) follows by definition. Since Killing vectors annihilate invariants, there are at most two functionally independent invariants. Hence, (96) must hold. Conversely, suppose that (75) and (96) hold. Dependence of µ follows by Proposition 5.3. Furthermore, where V is the vector field in (75). By (20) - (22) and (30), Hence L A = 0. Therefore, the invariant count is (0 , 2 , The type (0 , 1 , 2 , 2) solutions split into two branches, depending on whether or not µ is independent of α . We consider each branch in turn. glyph[negationslash] Proposition 6.2. Suppose that d α ∧ d α ∗ = 0 but that d α ∧ d µ = 0 . Then a G 2 solution is characterized by the condition Proof. If V is a Killing vector then L V ν = 0. In a G 2 solution there are two such independent vector field, which means that α, µ, ν must be functionally dependent. Let us prove the converse. We will show that the invariant count is (0 , 1 , 2 , 2), which signifies a G 2 solution by the Karlhede algorithm. By Proposition 2.1, the secondorder invariants are generated by µ, A, ν and their complex conjugates. Suppose that glyph[negationslash] By Propositions 4.1 and 4.2, Hence, all second order invariants depend on α, µ . The third order invariants are generated by δ ∗ A,δµ, ∆ µ, ∆ ν , and their complex conjugates. Since A is a constant and ν is a function of α, µ , and since relation (27) holds, it suffices to show that δµ depend on α, µ . By Proposition 4.2 and by (21), glyph[negationslash] Proposition 6.3. Suppose that d α ∧ d α ∗ = d α ∧ d µ = 0 , but that d α ∧ d ν = 0 . Then a G 2 solution is characterized by the conditions glyph[negationslash] Lemma 6.4. Suppose that B = 0 , µ = 0 . The following are equivalent: (i) d α ∧ d µ = 0 and(ii) A = 1 , ∆ µ = µ 2 . Proof. By assumption, µ ∗ = Aµ . By Proposition 4.2, relation (98) holds. Hence, glyph[negationslash] Proof of Proposition 6.3. By Propositions 4.1 4.2, A is a constant. Hence, using the reasoning in the proof of Proposition 6.2 above, ν ∗ , ∆ µ, ∆ ν generate the second and third-order invariants. If µ = 0, then by Lemma 6.4, ∆ µ is a function of µ , which itself is a function of α . If µ = 0, then afortiori ∆ µ = 0. That means that ν ∗ , ∆ ν generate all second and third-order invariants. Therefore, (99) (100) suffice for a G 2 solution. glyph[square] We now classify the (0 , 2 , 2) solutions. Throughout, V denotes the 2nd Killing vector independent from D . The G 2 solutions can be further subdivided according to whether V 3 = 0 or V 3 = 0. glyph[negationslash] By Proposition 5.4, the (0,2) precursor with V 3 = 0 is of class L 22 . The remaining (0,2) precursors are B 23 , C 23 , A 23 . As we show below, the specialization from the precursor class to the G 2 class is governed by the vanishing of the Y and Υ invariants, which are defined in (34) and (36), respectively. Proposition 6.5. Suppose that f ( ζ, u ) = F ( u -i k ζ ) u -2 + gu -2 -i k ζ, k = 0 belongs to the B 23 precursor class. The following are equivalent: (i) d α ∧ d α ∗ ∧ d ν = 0 , (ii) ˆ Υ = 0 , (iii) g ' ( u ) = 0 . glyph[negationslash] Proof. By Proposition 5.3, V = X -1 1 ˆ ∆ annihilates ω 1 , ω 2 , ω 3 , α, µ . Above, we already noted that L V A = 0. By (31) (33), L V X = 0 also. Let ˆ ν be the invariant defined in (35). By Proposition 5.2 and (26) (27) , where ˆ ν is the invariant defined by (35). This proves the equivalence of (i) and (ii). A direct calculation shows that This proves the equivalence of (ii) and (iii). glyph[square] Remark 1: If g ' ( u ) = 0, then by (10) we can absorb the g ( u ) u -2 -i k ζ term into the F ( u -i k ζ ) u -2 term. Remark 2: the invariant ˆ ν can be calculated directly by employing the tetrad that respects the normalization ˆ ∆ α = 0. The null rotation that sends ∆ → ˆ ∆ maps ν → ˆ ν . Proposition 6.6. Suppose that f ( ζ, u ) = F (e i u ζ ) + g e i u ζ, belongs to the C 23 precursor class. The following are equivalent: (i) d α ∧ d α ∗ ∧ d ν = 0 , (ii) ˆ Υ = 0 , (iii) g ' ( u ) = 0 . Proof. The proof is similar to the argument employed in Proposition 6.5 above. The formulas that differ are Proposition 6.7. Suppose that f ( ζ, u ) = g 1 log ζ + g 2 ζ belongs to the logarithmic L 23 precursor class. The following are equivalent: (i) d α ∧ d α ∗ ∧ d ν = 0 , (ii) Y = 0 , (iii) g 2 = 0 . Proof. By Proposition 5.4, V = Im( α -1 δ ∗ ) annihilates ω 1 , ω 3 , α, µ . Hence, condition (i) is equivalent to L V ν = 0. We have This proves the equivalence of (i) and (ii). A direct calculation shows that This proves the equivalence of (ii) and (iii). glyph[square] Proposition 6.8. Suppose that f ( ζ, u ) = F ( ζ ) + gζ belongs to the A 23 precursor class. The following are equivalent: (i) d α ∧ d α ∗ ∧ d ν = 0 , (ii) ∆ ν = 0 , (iii) g ' ( u ) = 0 . Proof. By Proposition 5.5 a multiple of ∆ annihilates ω 1 , ω 3 , α, α ∗ . Hence (i) is equivalent to (ii). A direct calculation shows that This proves the equivalence of (ii) and (iii). Note that if g ' ( u ) = 0, then we can absorb the gζ term into the F ( ζ ) term. glyph[negationslash] glyph[negationslash] glyph[negationslash] We now classify the G 2 solutions of type (0 , 1 , 2 , 2). By definition, these are specializations of the type (0 , 1) precursors. The latter solutions fall into three groups: (i) V 3 = 0, (ii) V 3 = 0 and d α ∧ d µ = 0, (iii) V 3 = 0 and d α ∧ d µ = 0, where V is the vector field that satisfies (75). Case (i) is class L 23 . The specialization to a G 2 solution is described, mutatis mutandi, by Proposition 6.7 above. Case (ii) consists of classes L 13 , AP 13 , AE 13 , AE 13 . The specialization to G 2 solutions is described by Propositions in 7.2, 7.4, 7.6, 7.8 of the following section. Case (iii) consists of classes BP 13 , CP 13 , BE 13 , CE 13 . By Proposition 6.2, the specialization to a G 2 solution is characterized by the condition d α ∧ d µ ∧ d ν = 0. The following Proposition analyzes this condition. The key invariant here is ˜ Υ, as defined by (40). Lemma 6.9. Suppose that B = 0 and AA ∗ = 1 , A = 1 . Then glyph[negationslash] with ˜ ∆ defined as in (41) . Proof. Since M = αµ , no generality is lost if replace d µ with d M . By Proposition 4.2, δM = 0. By (24) By (31), µ ∗ = Aµ . Hence, by (21) (22) we seek the kernel of the following matrix: By Proposition 4.1 d α ∧ d α ∗ = 0; hence, the above matrix has rank 2. Since A ∗ = 1 /A , the kernel is invariant under complex conjugation. Therefore, since A = 1, a basis for the kernel is D and glyph[negationslash] Proposition 6.10. Suppose that f ( ζ, u ) = ( k 0 z ) i k 1 u -2 + gu -2 z , or f ( ζ, u ) = exp( z ) + gz where z = u -i k ζ or z = e i u ζ ; i.e., f ( ζ, u ) belongs to one of the following classes: BP 13 , CP 13 , BE 13 , CE 13 . Then, the following are equivalent: (i) d α ∧ d µ ∧ d ν = 0 , (ii) ˜ Υ = 0 , (iii) g ' ( u ) = 0 . glyph[negationslash] Proof. By assumption, B = 0 , AA ∗ = 1 , A = 1. Hence, there exists a V such that condition (75) holds. Since L V α = L V µ = 0, by Lemma 6.9 V is a multiple of ˜ ∆. Hence, ˜ X/ ˜ X ∗ = C/C ∗ where C = α ∗ V 1 , and hence V = C/ ˜ X ˜ ∆. In the proof of Proposition 5.8 we showed that ∆(1 /µ ) is a constant. It follows that L V ∆ µ = 0 and hence L V ˜ X = 0 also. Therefore, the desired condition is equivalent to ˜ ∆(˜ ν/ ˜ X ) = 0 glyph[square] where ˜ ν is the invariant defined in (39). By (26), (27) (77), (78) This proves the equivalence of (i) and (ii). A direct calculation shows that Remark 1: If g ' ( u ) = 0, then by (10) we can absorb the the 2nd term in f ( ζ, u ) into the first term. Remark 2: the invariant ˜ ν can be calculated directly by employing a null-rotated tetrad that sends ∆ → ˜ ∆ and ν → ˜ ν .", "pages": [ 19, 20, 21, 22, 23 ] }, { "title": "7. The maximal IC order class.", "content": "This section is devoted to the proof of Theorem 1.1; we exhibit and classify all vacuum pp-wave solutions with a (0 , 1 , 2 , 3) invariant count. The (0 , 1) class is defined by the condition d α ∧ d α ∗ = 0. If α, µ are independent, then the (0 , 1 , 2) condition requires that ν, ν ∗ be functions of α, µ . However, by Proposition 6.2, this forces a G 2 solution, and therefore can be excluded from the (0 , 1 , 2 , 3) classification. Thus, we have narrowed the search for (0 , 1 , 2 , 3) solutions to the following class: glyph[negationslash] The middle condition forces some restrictions. By Lemma 6.4, the analysis divides into two cases: B = 0 , A = 1 , ∆ µ = µ 2 , µ = 0 and µ = 0 , AA ∗ = 1. The former possibility specifies class BL 13 ; the latter classes AP 13 , AE 13 , AL 13 . We begin by describing the specialization from class BL 13 to class BL 123 . The Y invariant employed below is defined in (34). Proposition 7.1. Suppose that f ( ζ, u ) = Cu -2 log ζ + gζ belongs to class BL 13 . The following are equivalent: (i) d α ∧ d ν ∧ d ν ∗ = 0 , (ii) ∆log( Y Y ∗ ) = 4 µ , (iii) g = ku -2 e i h , where k is a real constant and h = h ( u ) is real. Proof. Our assumption implies Hence, by (26) (27) This proves the equivalence of (i) and (ii). Writing g = e h 1 +i h 2 , a direct calculation shows that Therefore, (ii) is equivalent to ' 1 uh ( u ) = - 2 , which is equivalent to (iii). glyph[square] We now prove that generically the above solution is (0,1,2,3), and in the process derive the condition for specialization to a G 2 solution. Proposition 7.2. Suppose that f ( ζ, u ) = u -2 ( C log ζ + k e i h ζ ) belongs to class BL 123 . The following are equivalent: (i) d α ∧ d ν ∧ d∆ ν = 0 , (ii) ∆( α ∆log Y ) = 0 , (iii) e i h = u i k 1 , where k 1 is a real constant. Proof. All of the relations given in the proof of Proposition 7.1 hold. Furthermore, by (16) - (19) Thus, a direct calculation shows that Since αµ is a constant, the factor on the right can be written as This proves the equivalence of (i) and (ii). Furthermore, a direct calculation gives We now consider the case of µ = 0 , AA ∗ = 1. Proposition 7.3. Suppose that f ( ζ, u ) = ( k 0 ζ ) 2i k 1 + gζ belongs to the AP 13 class. The following are equivalent: (i) d α ∧ d ν ∧ d ν ∗ = 0 , (ii) (iii) g = k 2 e i h (1 -2i k 1 ) where k 2 is a real constant and h = h ( u ) is real. Proof. Our assumption and Proposition 4.2 imply that µ = 0 and that A is a constant satisfying AA ∗ = 1. Hence, by (26) and (27) This proves the equivalence of (i) and (ii). Writing a direct calculation shows that where C = C ( k 0 , k 1 ) is a constant. This proves the equivalence of (ii) and (iii). glyph[square] We now prove that generically the above solution is (0,1,2,3), and in the process derive the condition for specialization to a G 2 solution. Proposition 7.4. Suppose that f ( ζ, u ) = ( k 0 ζ ) 2i k 1 + k 2 e i h (1+2i k 1 ) ζ belongs to class AP 123 . The following are equivalent: (i) d α ∧ d ν ∧ d∆ ν = 0 , (ii) ∆ 2 Y 1 -A A -3 = 0 , (iii) f ( ζ, u ) = ( k 0 ζ ) 2i k 1 + Cu -2 -i k 1 ζ , where C is a complex constant. Proof. All of the relations given in the proof of Proposition 7.3 hold. Furthermore, by (16) - (19) , From there, a direct calculation shows that This proves the equivalence of (i) and (ii). Furthermore, a direct calculation gives where ˜ C is a complex constant. This proves the equivalence of (ii) and (iii). glyph[square] Finally, we consider the AE and the AL classes. Propositions 7.5 and 7.7 derive the form of the (0 , 1 , 2) solutions for the cases A = -1 and A = 1, respectively. Propositions 7.6 and 7.8 prove that these solutions are generically of type (0 , 1 , 2 , 3) and derive the condition for the specialization to the corresponding (0 , 1 , 2 , 2) G 2 solution. Mutatis mutandi, these Propositions are proved in the same way as Propositions 7.3 and 7.4 above. Proposition 7.5. Suppose that f ( ζ, u ) = exp( kζ ) + gζ belongs to the AE 13 class. The following are equivalent: (i) d α ∧ d ν ∧ d ν ∗ = 0 , (ii) ∆log( Y/Y ∗ ) = 0 , (iii) g = e i k 1 e h where k 1 is a real constant and h = h ( u ) is real. Proposition 7.6. Suppose that f ( ζ, u ) = exp( k 0 ζ ) + e i k 1 e h ζ belongs to the AE 123 class. The following are equivalent: (i) d α ∧ d ν ∧ d∆ ν = 0 , (ii) ∆ 2 Y -1 / 2 = 0 , (iii) e h = Cu -2 where C is a complex constant. Proposition 7.7. Suppose that f ( ζ, u ) = e i k log ζ + gζ belongs to the AL 13 class. The following are equivalent: (i) d α ∧ d ν ∧ d ν ∗ = 0 , (ii) ∆log( Y Y ∗ ) = 0 , (iii) g = k 2 e i h where k 2 is a real constant and h = h ( u ) is real. Proposition 7.8. Suppose that f ( ζ, u ) = e i k 0 log ζ + k 1 e i h ζ belongs to the AL 123 class. The following are equivalent: (i) d α ∧ d ν ∧ d∆ ν = 0 , (ii) ∆ 2 log Y = 0 , (iii) e i h = u i k 2 where k 2 is a real constant. Finally we remark that a suitable change of variable (10) (13) allows for two equivalent representation for solution classes AP 122 , AE 122 , AL 122 : It follows that classes AP 122 , AE 122 are specializations of the generic G 2 solution B 22 , while AL 122 is a specialization of C 22 . In this section we classify the G 3 solutions. The invariant count is (0 , 1 , 1) and hence these solutions are characterized by α = 0 and glyph[negationslash] glyph[negationslash] The condition d α ∧ d A = 0 is redundant, because by Propositions 4.1 and 4.2, a G 3 solution satisfies B = 0 , AA ∗ = 1 , d A = 0. By Lemma 6.4 there are two branches: (i) B = 0 , A = 1 , ∆ µ = µ 2 , µ = 0; and (ii) µ = 0 , AA ∗ = 1. By Propositions 7.1, 7.3, 7.5, 7.7 the condition d α ∧ d ν ∧ d ν ∗ = 0, which is weaker than d α ∧ d ν = 0, specializes these two branches to (0 , 1 , 2 , 3) solutions. Therefore the G 3 solutions arise as the following sequence of specializations: Therefore, to classify the G 3 solutions it suffices to begin with the classes BL 13 , AP 13 , AE 13 , AL 13 and impose the specialization is d α ∧ d ν = 0. Proposition 8.1. Suppose that f ( ζ, u ) = Cu -2 log ζ + gu -2 ζ belongs to class BL 13 . The following are equivalent: (i) d α ∧ d ν = 0 , (ii) Y = 0 , (iii) g = 0 . Proof. Using the relations from the proof of Proposition 7.1, we have This proves the equivalence of (i) and (ii). A direct calculation shows that This proves the equivalence of (ii) and (iii). glyph[square] Proposition 8.2. Suppose that f ( ζ, u ) = ( k 0 ζ ) 2i k 1 + gζ belongs to the AP 13 class. The following are equivalent: (i) d α ∧ d ν = 0 , (ii) Y = 0 , (iii) g = 0 . Proof. Using the relations from the proof of Proposition 7.3, we have This proves the equivalence of (i) and (ii). A direct calculation shows that α ∗ Y ∗ = Cζ 1 -2i k 1 g, The proof of the following two Propositions uses the same argument as above. One merely specializes A →-1 and A → 1, respectively. Proposition 8.3. Suppose that f ( ζ, u ) = exp( kζ ) + gζ belongs to the AE 13 class. The following are equivalent: (i) d α ∧ d ν = 0 , (ii) Y = 0 , (iii) g = 0 . Proposition 8.4. Suppose that f ( ζ, u ) = e i k log ζ + gζ belongs to the AL 13 class. The following are equivalent: (i) d α ∧ d ν = 0 , (ii) Y = 0 , (iii) g = 0 . In this section we derive and classify the metric forms in the α = 0 class. By Proposition 2.2 the corresponding solutions are either G 5 or G 6 . Proposition 9.1. The following are equivalent: (i) α = 0 and (ii) f ( ζ, u ) = g 2 ζ 2 + g 1 ζ + g 0 , where as usual g i = g i ( u ) , i = 0 , 1 , 2 denote complex valued functions of one variable. Proof. A direct calculation shows that where Note that a form-preserving transformation (10) - (13) can be used to set g 1 , g 0 → 0. Hence, without loss of generality a solution in the α = 0 class has the form f ( ζ, u ) = gζ 2 , where g = 0. glyph[negationslash] It will be convenient to set g = e 4 A , where A = A ( u ) is complex valued. A direct calculation then shows that We are now in a position to derive and classify the homogeneous G 6 solutions. Such solutions are characterized by the condition ∆ γ = 0, which ensures that the fundamental Cartan invariant γ is a constant. At this point the G 6 classification bifurcates, depending on the value of A u . We consider the generic case in Proposition 9.2, and the singular case in Proposition 9.3. The classification is summarized in Table 9. glyph[negationslash] Proposition 9.2. Suppose that f ( ζ, u ) = e 4 A ζ 2 , ∆ γ = 0 , and glyph[Rfractur] γ = 0 . Then, without loss of generality, glyph[negationslash] Proof. If ∆ γ = 0, then γ is a constant. By assumption, A u = 0 , and so γ/γ ∗ is also a constant. It will therefore be convenient to write where both k is a real constant and h = h ( u ) is real. A direct calculation now gives which implies glyph[negationslash] where k 1 = 0 is a real constant. Substituting into (113) gives which means that k, k 1 are essential constants, while k 2 can be gauged away. Applying the change of variables (12) gives the desired solution form. glyph[square] Proposition 9.3. Suppose that f ( ζ, u ) = e 4 A ζ 2 , ∆ γ = 0 , and glyph[Rfractur] γ = 0 . Then, without loss of generality, where k 0 is a real constant. Proof. The super-singular case of γ = 0 corresponds to A u = k = 0. From now on, we suppose that γ is a non-zero imaginary constant. It follows that where k is some real constant. The desired conclusion follows immediately. glyph[square]", "pages": [ 23, 24, 25, 26, 27, 28 ] }, { "title": "10. Conclusions", "content": "In our search for those vacuum PP-wave spacetimes in which the fourth-order covariant derivatives of the curvature tensor are required to classify them entirely, we have produced an approach to invariantly classifying the vacuum PP-wave spacetimes. Our approach is based on Cartan invariants and the Karlhede algorithm and is necessitated by the fact that a the class of vacuum PP-waves has vanishing scalar invariants [2]. Our classification is finer than the analysis of each spacetime's isometry group alone. The summary of this invariant approach to classification is given in tables 1 - 8 with specialization relations summarized in Figures 1 and 4. For any spacetime, the classification begins with the fact that the components of the curvature tensor and its covariant derivatives produce all of the invariants required. The Karlhede algorithm provides an algorithmic approach to determining the lowest order, q , of covariant differentiation needed to classify the space, canonical forms for the components of the curvature tensor and the number of functionally independent invariants, ( t 0 , t 1 , . . . , t q ) arising from the collection of all components of the curvature tensor and its covariant derivatives up to order q . For vacuum pp-waves we have demonstrated that q ≤ 4 and have classified all solutions that attain an IC order of 4. Table 6 summarizes the maximal order solutions. By characterizing the G 2 and G 3 solutions in terms of invariant conditions, the invariant approach also sheds light on the origin of the additional Killing vectors. Another remarkable finding is the fact that the maximal order solutions of Table 6 are direct precursors of the G 3 solutions first discovered by Kundt and Ehlers. In terms of the metric form, the mechanism of specialization is the disappearance of an additive term; e.g., Outside of the invariant classification of spacetimes, the study of the invariant structure of the Riemann tensor and its covariant derivatives reveal the interconnection between spacetimes with less symmetry and their more symmetric counterparts and how these arise as specialization of the classifying manifold. Furthermore by imposing conditions on the Cartan invariants we produced definite examples of spacetime with little or no symmetry. This is particularly relevant for the PP-wave spacetimes as before our work little was known about those spacetimes admitting D = ∂ v as the sole Killing vector. The approach used to invariantly classify the PP-waves is not limited to this class alone. One may repeat the process for the other half of the plane-fronted waves, the Kundt waves [10]. Together these spacetimes constitute the entirety of all Petrov type N VSI spacetimes: the class of spacetimes where all scalar curvature invariants vanish. These spacetimes are a special case of the CSI spacetimes , where all scalar curvature invariants are constant, and so the Karlhede algorithm is the only approach to invariantly classifying these spaces. Future research direction involve the extension of the invariant classification to all VSI space-times, and even the full class of Kundt-degenerate spacetimes. The question of the physical and phenomenological interpretation of the classifying invariants is also unresolved, although some steps in this direction are ongoing [11].", "pages": [ 28, 29 ] }, { "title": "11. Acknowledgements", "content": "The authors would like to thank Georgios Papadopoulos for useful discussions. The research of RM and AC is supported, in part, by NSERC discovery grants.", "pages": [ 29 ] }, { "title": "Appendix A. Tables of exact solutions", "content": "Tables 2 and 3 summarize the exact solutions derived in Section 3. Tables 4 and 5 summarize the precursor solutions derived Section in 5. Tables 1 and 7 give the G 2 solutions. glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash]", "pages": [ 29, 30 ] }, { "title": "References", "content": "glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] Dept. Mathematics and Statistics, Dalhousie U., Halifax, Nova Scotia B3H 4R2, Canada E-mail address : [email protected], [email protected], [email protected]", "pages": [ 30, 31, 32 ] } ]

[ { "title": "All the solutions of the form M 2 × W Σ d -2 for Lovelock gravity in vacuum in the Chern-Simons case", "content": "Julio Oliva Instituto de Ciencias F'ısicas y Matem'aticas, Universidad Austral de Chile, Valdivia, Chile ∗ and Universidad de Buenos Aires, FCEN-UBA, Ciudad Universitaria, Pabell'on I, 1428, Buenos Aires, Argentina.", "pages": [ 1 ] }, { "title": "Abstract", "content": "In this note we classify a certain family of solutions of Lovelock gravity in the Chern-Simons (CS) case, in arbitrary (odd) dimension, d ≥ 5. The spacetime is characterized by admitting a metric that is a warped product of a two-dimensional spacetime M 2 and an (a priori) arbitrary Euclidean manifold Σ d -2 of dimension d -2. We show that the solutions are naturally classified in terms of the equations that restrict Σ d -2 . According to the strength of such constraints we found the following branches in which Σ d -2 has to fulfill: a Lovelock equation with a single vacuum (Euclidean Lovelock Chern-Simons in dimension d -2), a single scalar equation that is the trace of an Euclidean Lovelock CS equation in dimension d -2, or finally a degenerate case in which Σ d -2 is not restricted at all. We show that all the cases have some degeneracy in the sense that the metric functions are not completely fixed by the field equations. This result extends the static five-dimensional case previously discussed in Phys.Rev. D76 (2007) 064038, and it shows that in the CS case, the inclusion of higher powers in the curvature does not introduce new branches of solutions in Lovelock gravity. Finally we comment on how the inclusion of a non-vanishing torsion may modify this analysis.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Gravity in higher dimensions has proved to be an interesting arena to test how generic are the notions gained in four dimensional gravitational physics. Even in higher dimensional General Relativity (GR), properties as uniqueness and stability of solutions in vacuum may depart completely from their four-dimensional counterpart (for a recent summary of the state of the art see [1]). Maintaining the second order character of the field equations in higher dimensions, it is possible to consider a more general setup than the one defined by Einstein's gravity, since as proved by Lovelock in [2] the most general parity-even Lagrangian in arbitrary dimension d , that gives second order field equations for the metric is given by an arbitrary linear combination of the dimensional continuations of all the lower dimensional Euler densities. This gives rise to the so-called Lovelock gravity, the simplest case after GR being the Einstein-Gauss-Bonnet (EGB) gravity. In this theory, in addition to the EinsteinHilbert and cosmological terms, one includes a term which is quadratic in the curvature and gives non-trivial field equations in dimensions grater than four. This quadratic combination is very precise, in such a way that the possible higher derivative terms cancel each other and one gets second order field equations. Since the field equations come from a diffeomorphism invariant action, their divergence vanishes identically. To find exact and analytic solutions of these theories is a non-trivial problem when one departs from spherical symmetry 1 . For example, a problem that is solved in a very simple manner in GR, corresponds to finding the most general solution of the form where Σ d -2 is an arbitrary Euclidean manifold of dimension d -2. Einstein equations plus a cosmological constant in vacuum imply that the metric functions do not depend on t , and are given by where µ is an arbitrary integration constant and Σ d -2 must be an Einstein manifold fulfilling the equation Here ˜ R ij is the Ricci tensor of Σ d -2 and ˜ g ij its metric [3]. Solving exactly the same problem in Lovelock gravity is more complicated. For example, in the EGB theory for the static case, the work [8] solves this problem in arbitrary dimension finding a rich set of causal structures. For arbitrary values of the coupling constants of the theory, the analysis done in [8] reduces to the done previously reported in [9], where it was proved that if one assumes Σ d -2 to be Einstein, then one can show that it must also obey a quadratic restriction on the Weyl tensor which includes a new parameter θ . That parameter appears in the lapse function and even more, it modifies the asymptotic behavior of the metric (see also [10]). For arbitrary Σ, beyond the EGB case not much is known. The static solution in the spherically symmetric case was found in [11]. When Σ d -2 is a constant curvature manifold, a Birkhoff's theorem was proved in [12] (see also [13]). Reference [12] also shows that Birkhoff's theorem is not valid when the coupling constants are fixed in a precise way and some degeneracies may appear since in such cases, some of the metric functions are not determined by the field equations (for some particular cases, this was previously observed in reference [14]). Lovelock theory, being a gravity theory with higher powers in the curvature, could have more than one maximally symmetric solution, and the mentioned degeneracies appear precisely at the regions in the space of couplings in which some of these vacua coincide 2 (for some static black hole solutions, with constant curvature horizons in this case see [16]). It would be interesting therefore to classify all the solutions of the form (1) in higher curvature Lovelock theories. In this work we focus on the odd-dimensional case, when the highest possible power of the curvature is present in the Lagrangian and all the vacua coincide. This theory is known as Lovelock-Chern-Simons (LCS) theory (for a recent review see [17]). The action for a general Lovelock theory can be written as where κ and α p are arbitrary (dimensionfull) coupling constants, ε a 1 ...a d is the Lorentz invariant Levi-Civita tensor, R ab := dω ab + ω ac ω b c is the curvature two-form written in terms Lorentz connection one-form ω ab , and e a is the vielbein. [ x ] stands for the integer part of x . Wedge exterior product between differential forms is understood. Finally, latin indices { a i , b i } run from 0 to d -1. The term with p = 0 in (5), corresponds to a volume term that gives the contribution of the cosmological constant, for p = 1 one gets the Einstein-Hilbert term, while for p = 2 the Lagrangian reduces to the Gauss-Bonnet term. As mentioned before, here we will focus on the case d = 2 n +1 and the coefficients α p are given by where l 2 is the squared curvature radius of the unique ( AdS ) maximally symmetric solution. For simplicity we will focus on the case l 2 > 0, nevertheless the de Sitter case is trivially obtained by analytically continuing l → il , while the flat limit (up to some subtleties that will be mentioned when necessary) can be obtained by taking l →∞ . When torsion vanishes, the field equations coming from (5) with the couplings given by", "pages": [ 2, 3, 4 ] }, { "title": "(6) can be written as", "content": "where we have defined the concircular curvature two-form as ¯ R ab := R ab + 1 l 2 e a e b . In terms of tensors, if we use the generalized Kronecker delta of strength one denoted by δ α 1 ...α p β 1 ...β p , by defining the concircular curvature tensor ¯ R αβ γδ = R αβ γδ + 1 l 2 δ αβ γδ , the field equations (7) read In the next section we will prove that all the solutions of the form (1), for the field equation (7) (or equivalently (8)) fall into one of the following three different classes: Case I: The manifold Σ d -2 is arbitrary and the metric reads where µ is an integration constant. /negationslash Case II: For ξ = 0, if the manifold Σ d -2 satisfies the following (scalar) restriction where ˜ R ij is the curvature two-form intrinsically defined on Σ d -2 and the indices { i, j } run on Σ d -2 , then the metric reads with c 1 ( t ) and c 2 ( t ) arbitrary integration functions. In the flat limit ( l → ∞ ) the metric reduces to In the case ξ = 0 (which does not exist in the limit l → ∞ ) the restriction on Σ d -2 is obtained by setting ξ = 0 in (10) and the metric reads where again c 1 ( t ) and c 2 ( t ) are arbitrary integration functions. Note that in all of these cases, by redefining the time coordinate, one can gauge away one of the two integration functions, but not both simultaneously. and the metric reads with f ( t, r ) an arbitrary function. This result extends the static five dimensional case previously analyzed in [18]. In Case I, we see that the manifold Σ d -2 is arbitrary, i.e. it is not fixed by the field equations. In Case II, the manifold Σ d -2 is fixed by a single scalar equation which, even after using diffeomorphism invariance, in general it is not enough to determine a metric on it. Finally in Case III, we see that the lapse function f 2 ( t, r ) is left arbitrary by the field equations. Therefore we conclude that, in the previously mentioned sense, all the cases have some degeneracy.", "pages": [ 4, 5, 6 ] }, { "title": "II. PROOF OF THE CLASSIFICATION", "content": "To develop the proof of the classification it is useful to have the components of the curvature two-form with respect to some basis for the metric (1). If we define the components of the vielbein as where ˜ e i is the vielbein intrinsically defined on Σ d -2 , then the nontrivial components of the concircular curvature two-form ¯ R ab read where ˜ R ij is the curvature two-form intrinsically defined on Σ d -2 and A , B , C , F , H and J are functions of t and r defined by Primes denote derivation with respect to r while dots derivation with respect to t . There are three kinds of equations depending on whether the free index in (7) goes along the time direction, radial direction or along the manifold Σ d -2 , which respectively reduce to After introducing explicitly in these equations the components of the concircular curvature two-form (17)-(20), we get the following three equations where we have defined ε i 1 ...i 2 n -1 := ε 01 i 1 ...i 2 n -1 . Considering the combinations e 0 G 0 + e 1 G 1 = 0 and e 1 G 0 + e 0 G 1 = 0 one respectively gets This immediately splits the analysis in two cases defined by the (would be) constraint on Σ d -2 If (23) doesn't hold, then we need to impose F = B and H = C , the former implies that g ( t, r ) = g ( r ), while the later implies f ( t, r ) = S ( t ) g ( r ). The function S ( t ) can be set to 1 without lost of generality by means of a redefinition of the time coordinate. Therefore in this branch (i.e. provided (23) doesn't hold), we have that (21) and (22) imply f ( t, r ) = g ( t, r ) = f ( r ) = g ( r ). If (23) holds, then G 0 = 0 = G 1 without imposing any restriction on the function f and g at the moment. Note that the quantities with tilde on top depend only on the coordinates in Σ d -2 , while the combination Jr 2 , could depend on both t and r . At the moment this is not relevant since equations (21) and (22) are factorized in any case, but later we will see that the consistency of equation (23) strongly constraints the metric functions. If we consider now equation G 0 = 0, in the case in which (23) doesn't hold and therefore f 2 ( r ) = g 2 ( r ) then we can see that H identically vanishes, while the vanishing of the function F implies that g 2 = r 2 l 2 -µ , where µ is an integration constant. As mentioned, in this branch we also have f 2 = r 2 l 2 -µ (since f 2 = g 2 ), and therefore one can see by direct evaluation that A identically vanishes also. Therefore H = F = A = 0 and then equation G i = 0 is also trivially satisfied without imposing any restriction on Σ d -2 . This concludes the proof of Case I outlined in the introduction. On the other hand if (23) holds (as mentioned before) G 0 and G 1 vanish identically and then at the moment, the functions f ( t, r ) and g ( t, r ) are not restricted. Before continuing to equation G i = 0, let us go back to the problem of the consistency of equation (23). Considering the derivative of this equation with respect to the t and r , we respectively obtain therefore ( Jr 2 ) ' = ( Jr 2 ) · = 0 and consequently Jr 2 = -ξ with ξ a constant, or the second term in both (24) and (25) vanishes, implying a new scalar restriction on Σ d -2 that contains terms of order n -2 in the curvature and might also depend on t, r , therefore its compatibility must be analyzed as well. The first case ( Jr 2 = -ξ ), implies g 2 ( t, r ) = g 2 ( r ) = r 2 l 2 + ξ where ξ is an integration constant. Note also that when n = 2, we are forced to set ( Jr 2 ) ' = ( Jr 2 ) · = 0 otherwise the volume element on Σ d -2 would vanish. On the other hand (for n > 2), if we assume ( Jr 2 ) ' and ( Jr 2 ) · to be nonvanishing we can divide these factors obtaining the new mentioned scalar restriction on Σ d -2 , which reads Again, we must consider the consistency of this equation by taking its derivative with respect to the parameters r and t . This respectively gives If n = 3 we are forced again to set ( Jr 2 ) ' = ( Jr 2 ) · = 0 (otherwise the volume element of Σ d -2 should vanish) which fixes g 2 = r 2 l 2 + ξ , while for n > 3 we can consider ( Jr 2 ) ' and ( Jr 2 ) · to be nonvanishing and divide by these expressions, therefore obtaining another scalar restriction on Σ d -2 , which this time, includes powers of the curvature of order n -3. Repeating this procedure n -1 times one eventually gets /negationslash /negationslash and if the expressions ( Jr 2 ) ' and ( Jr 2 ) · are nonvanishing then we would have that the volume form of Σ d -2 must vanish, arriving to a contradiction. We have proved then that in the case F = B and H = C , the consistency of equation (23) implies that ( Jr 2 ) ' = ( Jr 2 ) · = 0 which in turn implies that g 2 = r 2 l 2 + ξ with ξ an integration constant, and consequently (23) reads which now depends only on the coordinates of Σ d -2 . The remaining structure comes from the analysis of equation G i = 0. Note that g 2 = r 2 l 2 + ξ further implies H = 0 = F , therefore G j reduces to /negationslash If ξ = 0, the equation A = 0 allows to integrate f ( t, r ), which in this case reads while for ξ = 0 it integrates as The latter case is not defined in the flat limit ( l →∞ ) while in such a limit, when g 2 = ξ , the equation A = 0 gives the following expression for f : In all of these expressions c 1 ( t ) and c 2 ( t ) are arbitrary integration functions and note that one of them can be gauged away by a redefinition of the time coordinate. /negationslash Summarizing, in this branch we have that if ξ = 0, the metric reads which in the limit l →∞ takes the form while for ξ = 0 we have where c 1 ( t ) and c 2 ( t ) are arbitrary integration functions and Σ d -2 fulfills in both cases, the same scalar equation (31). Note that here the constant ξ appears in the restriction on Σ d -2 and can be scaled to ± 1 when it is non-vanishing. This ends the proof of Case 2 outlined in the introduction. /negationslash If A = 0, equation (32) implies a tensor restriction on Σ d -2 , which naturally, is stronger than its trace given by (31). When this tensor restriction holds, the metric reads with Σ d -2 constrained by and the function f ( t, r ) is arbitrary. This concludes the proof of Case 3 outlined in the introduction. This tensor restriction corresponds to an Euclidean Lovelock CS equation in dimension d -2 = 2 n -1. This concludes the proof of the classification.", "pages": [ 6, 7, 8, 9, 10, 11 ] }, { "title": "On the causal structures", "content": "For the Case 2 and Case 3, the ( t, r )-part of the metrics obtained depend on arbitrary functions of the time coordinate, therefore the causal structure of this spacetimes is not fixed. Note that this dependence cannot be gauged away completely by a diffeomorphism. Nevertheless, a few comments on the causal structures are in order in all of the three cases when the integration functions are chosen to be constants, i.e. c 1 ( t ) = c 1 and c 2 ( t ) = c 2 . In Case I, the solution describes a black hole. This solution reduces to the one found in [19]. In such case also, its thermodynamics and causal structure coincide with that of the three-dimensional Ba˜nados-Teitelboim-Zanelli (BTZ) black hole [20] where, for generic values of µ , the causal structure singularity at r = 0 of the three-dimensional case is now replaced by a curvature singularity as can be seen by evaluating, for example, the Ricci scalar. In Case II, the metric might describe the traversable wormhole found in [21], which is asymptotically AdS at both asymptotic regions. This is the case when ξ = -1 and | c 2 lc 1 | < 1, which can be seen directly by performing the change of coordinates r = l cosh ρ and allowing the coordinate ρ to go from -∞ to + ∞ . In this case, the metric reduces to where ρ 0 = -tanh -1 ( c 2 lc 1 ) and we have properly rescaled the time coordinate. The conditions under which the propagation of a scalar field on this background is stable, was studied in [22], and some holographic properties of strings attached to the boundaries have been explored in [23]. For ξ = 0 the metric reduces to /negationslash When c 1 = 0 this spacetime is asymptotically locally AdS, while if c 1 = 0, the ( t, r )-part of the metric reduces to a Lifshitz geometry (geometry with an anisotropic scaling symmetry), with a dynamic exponent equals to z = -1. Since in Case III the lapse function is arbitrary, the causal structure is also undefined even in the static case.", "pages": [ 11, 12 ] }, { "title": "Does torsion help removing the degeneracy?", "content": "The field equations coming from the variation with respect to the spin connection in Lovelock theory, do not necessarily imply that torsion should vanish (for some explicit solutions see e.g. [24]). For example in five dimensions, in first order formalism for the Lovelock CS case, the field equations coming from the variation with respect to the vielbein and the spin connection are respectively given by where we have introduced the torsion two-form T e := De e := 1 2 e e α T α µν dx µ ∧ dx ν . Therefore choosing the Levi-Civita connection is ad-hoc. Then it is natural to wonder whether the equations coming from the torsion may help removing the degeneracy. Posing the question in a different manner one could ask : is there a non-degenerate branch of solutions of (39)(40) in which the vielbein and the spin connection are compatible with the local isometries of Σ d -2 ?. It is clear that there are particular cases in which the torsion may not be vanishing and anyway the system is degenerated since, if for example we choose the (non-Riemannian) curvature to be constant R ab = -1 l 2 e a e b , then the torsion is left completely arbitrary by the field equations. Note also that, since this theory has an extra symmetry that mixes the spin connection and the vielbein (see [17]), the arbitrariness in the torsion can be transformed into an arbitrariness of the line element constructed out from the corresponding vielbein. A thorough analysis with the inclusion of torsion will be presented elsewhere [25].", "pages": [ 12, 13 ] }, { "title": "Further comments", "content": "As studied for the static quadratic case in [8], when one considers Lovelock theories that do not belong to the subclass of Lovelock CS, but nevertheless the couplings are related in such a way that there is a unique vacuum, there are also sectors in which some of the metric functions are arbitrary. Therefore this phenomenon seems to be more related to the fact of having degenerate maximally symmetric solution than with the appearance of an extra symmetry. In such non-Lovelock CS theories, as well as in the Lovelock CS ones, this degeneracy allows to have interesting causal structures as solutions (see e.g. [26]). Nevertheless in the former cases, there are more restrictions on Σ d -2 , which on one hand can be thought of as helping to remove the degeneracy, while in the other hand could be not compatible beyond the constant curvature case. A simple set of geometries beyond constant curvature manifolds (or their products) are product of the homogenous three dimensional Thurston geometries, which have been recently found to provide simple examples of transverse sections of hairy black holes for some Lovelock theories in even dimensions [27]. In the context of compactifications of Lovelock CS theories, involving metrics that are products of constant curvature spaces, the degenerate behavior is also present as it was proved in reference [28] back in the early 90's. The inclusion of matter fields seems to help removing the mentioned degeneracies (see for example references [29]). If one departs from the underlying ( A ) dS symmetry group, static spherically symmetric solutions of gravitational CS theories with matter fields, have also been recently considered in [30]. In this reference, the authors considered a Chern-Simons theory evaluated on a Lie algebra that is obtained by performing what the authors called an S -expansion procedure [31] from the AdS algebra and a particular semigroup S , which provides an approach to obtain GR in odd-dimensions from a CS theory. It would be interesting to study further the properties of these theories and to integrate them in the general ansatz (1) classifying the possible non-degenerate sectors.", "pages": [ 13 ] }, { "title": "IV. ACKNOWLEDGMENTS", "content": "We thank Andr'es Anabal'on, Fabrizio Canfora, Francisco Correa, Gustavo Dotti, Sourya Ray and Steven Willison for many useful conversations. We thank also the support of Becas Chile Postdoctorales, CONICYT, 2012 and FONDECYT grant 11090281.", "pages": [ 14 ] } ]

[ { "title": "Isotropic universe with almost scale-invariant fourth-order gravity", "content": "Hans-J¨urgen Schmidt and Douglas Singleton May 13, 2013 Institut fur Mathematik, Universitat Potsdam, Germany 1 Am Neuen Palais 10, D-14469 Potsdam, [email protected]", "pages": [ 1 ] }, { "title": "Abstract", "content": "We study a class of isotropic cosmologies in fourth-order gravity with Lagrangians of the form L = f ( R )+ k ( G ) where R and G are the Ricci and Gauss-Bonnet scalars respectively. A general discussion is given on the conditions under which this gravitational Lagrangian is scale-invariant or almost scale-invariant. We then apply this general background to the specific case L = αR 2 + β G ln G with constants α, β . We find closed form cosmological solutions for this case. One interesting feature of this choice of f ( R ) and k ( G ) is that for very small negative value of the parameter β the Lagrangian L = R 2 / 3+ βG ln G leads to the replacement of the exact de Sitter solution coming from L = R 2 (which is a local attractor) to an exact, power-law inflation solution a ( t ) = t p = t -3 /β which is also a local attractor. This shows how one can modify the dynamics from de Sitter to power-law inflation by the addition of an G ln G -term. Keywords: Gauss-Bonnet term, Friedmann space-time, scale-invariant gravity, fourth-order gravity AMS-Classification: 83D05, 83F05, 83C15, 53Z05, 85A40", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "From the huge class of theories of gravitation which can be considered for describing and explaining the early evolution of the Universe, it is the subclass of scale-invariant ones which plays a prominent role. The reason for this prominence is that almost all physical theories and their resulting cosmologies have some limiting regime which is free from any scales. In the present paper we investigate a class of cosmologies which include not only scale-invariant theories but also almost scale-invariant theories. In section 2 we will present concrete definitions and results related to the several variants of '(almost) scale-invariant theories'. The Lagrangians which we consider are of the form where R is the curvature scalar and is the Gauss-Bonnet scalar. By placing restrictions on the form of the functions f ( R ) and k ( G ) we will obtain theories which are scale-invariant and almost scale-invariant. We will focus on cosmological solutions to these scaleinvariant and almost scale-invariant Lagrangians - in particular spatially flat Friedmann space-times. We obtain general features for these almost scaleinvariant cosmologies, and for certain cases we are able to completely integrate the resulting Friedmann-like equations to confirm older results and to obtain new exact, closed form solutions in terms of the Friedmann metric scale factor, a ( t ). Before moving to the detailed calculations we give a brief review of work in this area which has some connection with the present paper. Cosmological models where the action depends on the Gauss-Bonnet scalar G are discussed in [1], [2], [3], and [4]. In [5], exact solutions for k ( G ) = G β are given which have an ideal fluid source, a power law scale factor, a ( t ) = t p , with p depending on β and the equation of state of the fluid; a ( t ) is the related cosmic scale factor. Further papers on this topic are [6], [7], [8], [9], [10], [11], [12], [13], and [14]. In [15], the ΛCDM epoch reconstruction from F ( R, G ) and modified Gauss-Bonnet gravities is presented. In this work models with Lagrangians R + k ( G ), or more general f ( R )+ k ( G ), and also R + ξ ( φ ) G + φ ,i φ ,i are discussed especially for the spatially flat Friedmann models. The paper [16] investigates ΛCDM cosmological models, using Lagrangians of the form L = k ( G ), and L = R + k ( G ). For the pure k ( G ) gravity, and a spatially flat Friedmann model with scale factor a ( t ) where t is synchronized time, the following results are obtained: A de Sitter space-time with Hubble parameter h = ˙ a a > 0 has G = 24 h 4 . This de Sitter solution is a vacuum solution if the condition Gdk/dG = k ( G ) is fulfilled. The exact power-law solution of the form a ( t ) = t p exists if the following condition is fulfilled: i.e. if k ( G ) = G (1 -p ) / 4 , or more completely the Euler-type eq. (1.2) has solutions k ( G ) = c 1 G + c 2 G (1 -p ) / 4 with constants c 1 and c 2 - compare with eqs. (59) and (60) of [16]. The term c 1 G is a divergence, and so does not contribute to the field equation, so, seemingly, only power-law Lagrangians k ( G ) = G (1 -p ) / 4 produce the exact solution a ( t ) = t p . However, this is not the complete truth: If one takes the example p = -3, then the solutions of eq. (1.2) become k ( G ) = c 1 G + c 2 G ln G , a case not mentioned in [16]. So, besides powers of G , also G ln G leads to exact solutions a ( t ) = t p . Further recent papers on k ( G ) gravity are [17], [18], [19], and [20]. In [21], the Lagrangian is discussed, and cosmological closed-form solutions are given, including the just mentioned exact solution a ( t ) = t p with p = -3. In [22], the anomalous velocity curve of spiral galaxies is modelled by Lagrangians of type L ( R, G, ✷ G ), where ✷ denotes the D'Alembertian, especially in the form In a first approximation, one can assume ˜ G ≈ G , so this Lagrangian has similarity with that one from eq. (1.3). In [23], the stability of power-law solutions in cosmology is discussed for L = G , which gives non-trivial results for space-time dimension exceeding 4 only. In [24], solutions for L = R + √ G with a Friedmann scale factor of power-law form, i.e. a ( t ) = t p are given. In [25], the Lagrangian L = R n + βG n/ 2 is investigated. Further Lovelock models along the line of [25] are given in [26], whereas in [27] the case k ( G ) = G n + βG ln G is discussed. In [28], the stability of the cosmological solutions with matter in f ( R, G ) gravity is discussed, with special emphasis on the stability of the de Sitter solution, and with Lagrangians of the type R + R n G m . Analogous models for f ( R )-gravity can be found in [29], where the case L = R 3 / 2 is related to Mach's principle. In [30] an exact solution for L = R 2 is given. Further models are discussed in [31], [32], [33], [34], [35], and [36]. In [37], the stability of models within theories of type L = R + R m + R n with n < 0 < m is discussed, and exact power-law solutions are obtained. Further papers on this topic are [38], [39], [40], [41], and [42]. In [43], the case f ( R ) = R 3 / 2 is studied. In [44], the following strict result is shown: Exact power-law cosmic expansion in f ( R ) gravity models with perfect fluid as source is possible for f ( R ) = R n only. Newer models of this kind can be found in [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68], [69], [70], [71], [72], [73], [74], [75], [76], [77], [78], [79], [80], and [81]. The conformal Weyl theory, especially the value of the perihelion advance in this theory, has been discussed in [82], [83], [84], [85], [86], [87], [88], [89], [90], and [91]. For theories in lower-dimensional space-times see e.g. [92], [93], and [94]. Our motivation for considering Lagrangians of the form given in eq. (1.1) is as follows: We study the cosmological aspects of a specific version of F ( R, G ) gravity which is scale-invariant in the sense that in the absence of matter no fundamental length exists within that theory. One can contrast this with R ± l 2 R 2 theories which have the fundamental length l . In connection with this we discuss and clarify that there are slightly different notions of scale-invariance, and we carefully distinguish between them. We do not insist on second-order field equations, 2 so also non-linear dependences of the Lagrangian on G are included with the result, that the field equations are of fourth-order in general. Similarly, we do not motivate our research by string theory, 3 but rather we want to present possible models for the observed evolution of the universe which includes both inflationary phase at early times and the present acceleration (normally attributed to some fluid/field generically termed 'dark energy') without the need to introduce additional matter fields. Thus our motivation is as follows: First the leading principle is that in the first approximation, the Einstein-Hilbert Lagrangian R is the right one for weak fields. Second, a non-linear addition to the Einstein-Hilbert Lagrangian depending on R , especially of the form R 2 or R 2 ln R , gives the desired early time inflationary behaviour, see e.g. the early papers [31] on this topic. Third, the further addition of a term non-linear in G to the Lagrangian was proposed in [37], [3] and others as a possible alternative for dark energy, which is the generic term for the substance postulated to drive the current accelerated expansion of the Universe. To make the task tractable of finding which of these various modified gravity theories can give the observed late time acceleration the authors of [6] developed 'the reconstruction program for the number of modified gravities including scalar-tensor theory, f ( R ), F ( G ) and string-inspired, scalar-Gauss-Bonnet gravity. The known (classical) universe expansion history is used for the explicit and successful reconstruction of some versions (of special form or with specific potentials) from all above modified gravities.' The paper is organized as follows: As already said, in section 2 we give a general discussion of scale-invariance and almost scale-invariance. This general discussion motivates our special choice for the Lagrangian. Section 3 gives a brief, self-contained review of relevant formulas concerning the Gauss-Bonnet scalar and Gauss-Bonnet gravity. This is done since k ( G ) models are much less known that f ( R ) models. Section 4 gives our main new results which follow from the almost scaleinvariant Lagrangian of the form eq. (1.5). Section 5 summarizes and gives conclusions about the results presented in this paper.", "pages": [ 2, 3, 4, 5 ] }, { "title": "2 Notions of (almost) scale-invariance", "content": "We first give the exact definitions of what we mean by an (almost) scaleinvariant gravitational action or gravitational Lagrangian. A theory of gravitation with a geometric Lagrangian L = L ( g ij , ∂ k ) is defined by a scalar L which depends on the metric and its partial derivatives up to arbitrary order. The signature of the metric is ( -+ . . . +) and g = det g ij . Within this section, we assume the dimension of space-time to be D ≥ 2. Then the gravitational action I is defined by A scale-transformation, also called a homothetic transformation, is a conformal transformation with a constant conformal factor. In another context, scale-transformations can also be interpreted as transformations that change the applied length unit. The transformed metric, ˜ g ij , is defined as where c is an arbitrary constant. For the inverted metric one gets The Christoffel affinity Γ i jk , the Ricci tensor R ij and the Riemann tensor R i jkl do not change under the scale-transformation given in eq. (2.2), and also all covariant derivatives of the Ricci and the Riemann tensor are homothetically invariant. However R , G and g are changed under the transformation of eq. (2.2.) as follows: Definition: The action (2.1) is called scale-invariant, if ˜ I = I according to eq. (2.2). It is called almost scale-invariant, if the difference ˜ I -I is a topological invariant. The Lagrangian L is called scale-invariant if there exists a constant m , such that Finally, the Lagrangian L is called almost scale-invariant, if the difference ˜ L -e mc L is a divergence. Of course, the sum of a scale-invariant action and of an arbitrary topological invariant is always an almost scale-invariant action. Likewise, the sum of a scale-invariant Lagrangian and a divergence is always an almost scaleinvariant Lagrangian. At first glance one might be tempted to conclude that the converse should be true, that an almost scale-invariant action can always be written as the sum of a scale-invariant action plus a topological invariant and that an almost scale-invariant Lagrangian can always be written as the sum of a scale-invariant Lagrangian plus a divergence. However, as we will show below, there exist non-trivial examples of almost scale-invariant actions which cannot be represented in the form of such a sum. The following relations between these four notions of scale-invariance exist: If L is scale-invariant, then with eq. (2.3) we get so for m = -D , a scale-invariant Lagrangian gives rise to a scale-invariant action. Likewise for m = -D , an almost scale-invariant Lagrangian gives rise to an almost scale-invariant action, because the space-time integral of a divergence represents a topological invariant. Let us now take the example L = f ( R ) with an arbitrary but sufficiently smooth function f and ask, under which circumstances, this leads to scaleinvariance. We have to distinguish two cases: D = 2 and D > 2. For D = 2, the scalar R represents a divergence, whereas for D > 2, no function of R has such a property. Let us start with the more tractable case D > 2. As no function of R gives a divergence, the notions of scale-invariance and almost scale-invariance coincide. For L = f ( R ) to be a scale-invariant Lagrangian there must exist an m such that the following relationship holds using eqs. (2.2), (2.3), and (2.4). With f ' denoting the derivative of f with respect to its argument we get from eq. (2.5) by applying d/dc Putting c = 0 into this equation we get a differential equation for f ( R ): which is solved by with integration constant c 1 . As expected, just the powers of R lead to scale-invariant Lagrangians. The corresponding action I turns out to be scale-invariant for m = -D only, i.e. L = R D/ 2 leads to a scale-invariant action, for D = 4 this is the celebrated L = R 2 . Let us now turn to the less trivial case D = 2, where R represents a divergence. We look for the set of all almost scale-invariant Lagrangians. For a Lagrangian of the form L = f ( R ) to be almost scale-invariant requires that there exists an m such that where v depends on c only to ensure that v · R is a divergence for every c . Applying d/dc we now get Inserting c = 0 and abbreviating v ' (0) by c 2 we get in place of eq. (2.6) now We divide by R , apply d/dR and get which is solved by with integration constants c 3 and c 4 . This is just what one expected from the beginning: The divergence c 3 R added to the power-law term c 4 R -m/ 2 , i.e. the added divergence term v ( c ) · R in eq. (2.8) leads to the extra divergence term c 3 R in eq. (2.11). However, eq. (2.10) possesses a further solution besides eq. (2.11): For m = -2 eq. (2.10) is solved by The result of eq. (2.12) was already noted in [92]: Besides what one would have expected, the action I = ∫ R ln R √ -gd 2 x turns out to be almost scaleinvariant. Now we perform the analogous analysis for the Lagrangian L = k ( G ). For dimension D ≤ 3, G vanishes, so this case is not interesting. For dimension D ≥ 5, no function of G is a divergence, so we get the expected result: scaleinvariance and almost scale-invariance coincide. Every power of G leads to a scale-invariant Lagrangian, and the action I = ∫ G D/ 4 √ -gd D x is scaleinvariant. So, D = 4 remains the only interesting case. Here, G represents a divergence, and we ask for the set of all almost scale-invariant Lagrangians. The condition that the Lagrangian L = k ( G ) be almost scale-invariant means that there exists an m such that Applying d/dc we now get Inserting c = 0 and abbreviating v ' (0) by c 2 we get We divide by G , apply d/dG and get which is solved by with constants c 3 and c 4 . This is just what one expects from the beginning: The divergence c 3 G added to the power-law term c 4 G -m/ 4 . However, eq. (2.15) possesses one further solution besides eq. (2.16): For m = -4 one gets The result in eq. (2.17), see eq. (1.3), was already noted in [21]: Besides what one would have expected, the action I = ∫ G ln G √ -gd 4 x turns out to be almost scale-invariant. An important property, valid not only for scale-invariant but also for almost scale-invariant Lagrangians is the following: If g ij is a vacuum solution and ˜ g ij is homothetically related to g ij , then ˜ g ij is also a vacuum solution. For the Lagrangians of type L = f ( R ) + k ( G ) and dimension D = 4, only L = αR 2 leads to a scale-invariant action, and only L = αR 2 + βG ln G leads to an almost scale-invariant action. This is a strong argument for a further detailed study of the gravitational Lagrangian Of course, the term γC ijkl C ijkl - see [90] - by itself has a scale-invariant action, but we did not consider it in this paper, as it has no influence on the field equation within the Friedmann models. The terms Λ and R are added here since in the weak field limit such terms appear effectively. Einstein's theory of general relativity has a scale-invariant Lagrangian, but only if the cosmological term is absent 4 , but not a scale-invariant action. In closing this section we note that one can construct scale-invariant Lagrangians which do not have the form L = f ( R ) + k ( G ). One example is with a constant n and an arbitrary (transcendental) function f .", "pages": [ 5, 6, 7, 8, 9, 10 ] }, { "title": "3 On the Gauss-Bonnet scalar", "content": "The field equations for the Lagrangian L = f ( R ) are given by, see for example eq. (2.27) of [42], Since the case L = f ( R ) has been widely studied we will not go into further details here but simply refer the interested reader to the overview [42]. The case when L = k ( G ) is much less known than the case L = f ( R ) so we give some further details here. For the spatially flat Friedmann metric (given below in eq. (4.1)) the Gauss-Bonnet scalar G becomes where h is the Hubble parameter h = ˙ a/a and γ = ˙ h/h 2 . For the Lagrangian k ( G ) with k G = dk/dG , the corresponding vacuum field equation is given in eq. (3.3) of [21] as See eqs. (A4), (A5) of [21], specialized to the space-time dimension n = 4: C ijkl is the Weyl tensor and we define C 2 = C ijkl C ijkl and With this notation, the first (unnumbered) equation of the appendix of [21] reads Inserting eqs. (3.4) and (3.6) into eq. (3.5) we get Further it holds that and With these notations we can rewrite eq. (3.3) as Inserting k ( G ) = G n into eq. (3.10) we get with k G = nG n -1 Inserting k ( G ) = G · ln G into eq. (3.10) we get with k G = 1 + ln G", "pages": [ 10, 11 ] }, { "title": "4 Cosmological solutions for αR 2 + βG ln G", "content": "In this section we use the background developed in the previous sections to give a general study of spatially flat Friedmann space-times for almost scaleinvariant Lagrangians. In particular we focus one the case L = αR 2 + βG ln G which the analysis of section 2 pointed out as an important and unique case. We start by setting up our system and notation. First, the cosmological metric we use is the spatially flat Friedmann space-time given as with positive cosmic scale factor a ( t ). The dot denotes d/dt , h = ˙ a/a is the Hubble parameter, and R = 6(2 h 2 + ˙ h ) is the curvature scalar. 5 Without loss of generality we assume h ≥ 0. If this is not the case then it is always possible to invert the time direction so as to get h ≥ 0. If h appears in the denominator, this automatically includes the additional assumption, that h = 0. 6 It proves useful to define the function which shall be used to replace ˙ h in subsequent formulas. In terms of γ we get R = 6 h 2 (2 + γ ). The deceleration parameter (i.e. q = -aa/ (˙ a ) 2 ) is related to γ via q = -1 -γ . Sometimes it proves useful to use τ = ln a as an alternative time coordinate. With a dash denoting d/dτ , we get with ˙ τ = h the following formula: We now give some results which will be useful in dealing with the almost scale-invariant Lagangians of the form given in eq. (1.5). First we note that, /negationslash 6 This is not a real restriction, as a constant function a ( t ) is the trivial Minkowski spacetime with h ≡ 0, and solutions, where h ( t ) = 0 at isolated points t only, can be matched by pieces with h = 0. In other words: if h ( t ) = 0 at isolated points t then these are always connected to regions where h ( t ) = 0. /negationslash /negationslash assuming a spatially flat Friedmann metric of the form given in eq. (4.1), that the vacuum field equation for a Lagrangian of the form L = F ( G,R ), where F is a function of R and G , is (see eq. (15) of reference [5]) where F G = ∂F/∂G and F R = ∂F/∂R . Eq. (4.4) is the 00-component of the vacuum field equation for the Lagrangian L = F ( G,R ). All other components of the vacuum field equation are fulfilled if eq. (4.4) is valid. For the almost scale-invariant Lagrangian from eq. (1.5) L = αR 2 + βG ln G we get from eq. (4.4) We look for solutions of eq. (4.5) with αβ = 0. After some lengthy but straightforward calculations it turned out that for the vacuum equation (4.4) following from eq. (1.5) and restricting to the spatially flat Friedmann space-time eq. (4.1), no cosmic bounce and no cosmic recollapse is possible; the proof was done by inserting a Taylor expansion for a ( t ) into the field equation (4.4) and to show, that regular local extrema of this function do not exist. This result is not very surprising, as one knows this property to be valid already for both of the ingredients of eq. (1.5), i.e. for L = R 2 and L = G ln G . This fact simplifies the calculations as h ( t ) cannot change its sign, and we do not need do match pieces of different sign of h together. 7 /negationslash /negationslash In a second step, we look for constant values γ = 0 related to the scale factor a ( t ) = t p representing the self-similar solutions. 8 To this end, we insert γ = -1 /p into eq. (4.5). Without loss of generality we assume α = 1 / 3 which transforms eq. (4.5) to This equation can be solved for β by As a first estimate we can see the following: The leading term in the limit p →∞ of eq. (4.6) is 0 = -6 /p -2 β . If we insert β = -3 /p into eq. (4.6), then we get qualitatively the following result: A very small negative value of the parameter β in the Lagrangian leads to the replacement of the exact de Sitter solution from L = R 2 , which is a local attractor, to a PLI (power-law inflation) exact solution a ( t ) = t p = t -3 /β which also represents a local attractor. This shows how one can modify the dynamics from de Sitter to power-law inflation by the addition of the G ln G -term. Let us now go more into the details of the system. The limit β →∞ in eqs. (4.7)/(4.8) is essentially the limit to the Lagrangian L = G ln G from (4.8) and is related to the limit p → -3 in eq. (4.7), see the discussion near eq. (1.3) for this case. At some points, the Cauchy problem fails to be well-posed: For R = 0, i.e. γ = -2, the field equation following from L = R 2 has the property that it is fulfilled by every space-time having R ≡ 0. Thus, the fourth-order field equation (3.1) possessing 10 independent components in the general case, now degenerates to one single second order scalar field equation, namely R = 0. 9 Further, for G ≤ 0, the Lagrangian G ln G is not immediately defined; while this might be compensated by writing G ln | G | instead for G < 0, there remains to be a mild singularity at G = 0, i.e. for γ = -1. However, we are in the lucky circumstances that for those cosmological models where this model may play a role, i.e. near de Sitter and near to PLI behaviour, we have G > 0 anyhow. Therefore we restrict the following discussion to the region γ > -1. With 3 α = 1, eq. (4.5) can be rewritten as follows: Here we meet the third case that the Cauchy problem is not well-posed: namely at points where 1 + β + γ = 0. 10 We chose to restrict to that region where 1 + β + γ > 0. For those cases, where γ ' = 0, eq. (4.9) can be solved for β by Analysing this eq. (4.10) one can see that for a given value β , zero, one or two related values γ exist. The range of values β , where no γ exists, is the interval In the third step we see that eq. (4.9) being of first order can be analysed qualitatively to see the asymptotic behaviour of the solutions: both for the past singularity as for the future expansion, the solutions tend to a solution with scale factor a ( t ) = t p . We count the degrees of freedom as follows: For a general solution of a fourth-order field equation and one free unknown, in this case a ( t ), one expects 4 initial values, namely a (0) and the first three derivatives of a ( t ) at t = 0. The remaining information is contained in the field equations. However the 00-component of the field equation is a constraint, reducing the order by one. Thus the general solution is expected to have 3 free constants - one is a t -translation, the second is multiplication of t by some factor, the third is multiplication of a ( t ) by a constant factor. In this sense, a general solution can be given even if it has no free parameter. 11 Let us sum the details: For every fixed β > 0, exactly three solutions exist. They can be described as follows: Find that value γ ( β ) which has 0 < γ ( β ) < 1 / 3 and solves eq. (4.10). Then the solution with γ = γ ( β ) is just the self-similar solution a ( t ) = t p with p = -1 /γ < -3 discussed already earlier, see eq. (4.7). 12 The second solution has γ > γ ( β ) and γ ' < 0 throughout, starts from an initial singularity with γ → ∞ and attracts the self-similar solution a ( t ) = t p which, of course, reaches a → ∞ in a finite time t . The third solution has -1 < γ < γ ( β ) and γ ' > 0 throughout, starts from an initial singularity of type a ( t ) = t and also attracts the self-similar solution a ( t ) = t p for a →∞ . For -0 . 45 ≈ ( √ 7 -4) / 3 ≤ β < 0 we have essentially the same behaviour 12 Of course, for p < 0, writing simply a ( t ) = t p may be misleading, because then h < 0 appears; but we believe that it is sufficiently clear that for p < 0, writing simply a ( t ) = t p means a ( t ) = ( t 0 -t ) p with t < t 0 . as in the previous case: One self-similar solution a ( t ) = t p , but now with p > 0, and two other ones, both having a ( t ) = t p as attractor for t → ∞ . For the remaining negative values of β the instabilities become more serious, and the behaviour of the solution has several different types of singularities. We conclude that probably the physically sensible range of our model is in the interval β ≥ ( √ 7 -4) / 3. In closing this section we show that it is possible to completely integrate eq. (4.5) for the special case when α = 0; i.e. when L = βG ln G one can integrate the system completely to the point where one has an explicit form for the scale factor a ( t ). This special case can be thought of as the limit where β >> α so that system is dominated by the Gauss-Bonnet logarithmic term. In this case taking α = 0 eq. (4.5) becomes γ ' = Aγ 2 + Bγ + C where the prime indicates differentiation with respect to τ and where A, B, C are However we will carry through the calculation until almost the end using general A, B, C since in this way the analysis can also be applied to the special Lagrangians L = R 2 n and L = G n which also lead to an equation for γ of the form γ ' = Aγ 2 + Bγ + C . Integrating this equation for γ yields where D = √ -B 2 +4 AC . Inverting eq. (4.11) gives Now taking into account the form of A, B, C from eq. (4.11) we find that D = √ -B 2 +4 AC is imaginary. Thus in eq. (4.13) we replace D with iD 1 where D 1 = √ B 2 -4 AC and taking into account that tan( iD 1 ) = i tanh( D 1 ) we find that eq. (4.13) becomes Next, we use this γ ( τ ) from eq. (4.14) to solve the equation for h = ˙ a/a (where the overdot is differentiation with respect to t ). First we note that We have used h = dτ/dt in arriving at the final result. Now we integrate eq. (4.15) for γ ( τ ) from eq. (4.14) which yields or solving for h ( τ ) gives Now using h ( t ) = ˙ a ( t ) /a ( t ) for the left hand side and τ = ln[ a ( t )] in the right hand side of eq. (4.17) gives where in the last line we have inserted the specific values of A, B, C, D 1 for this cases when L = G ln G i.e. A = -3, B = -2, C = 1 and D 1 = 4. Finally integrating eq. (4.18) gives where k is an integration constant and 2 F 1 ( a, b ; c ; z ) is the hypergeometric function. One should now solve eq. (4.19) for a ( t ) to obtain the scale factor as a function of t . Or a simpler method - given the presence of 2 F 1 in eq. (4.19) - is to plot t versus a and then flip the graph about the line t = a thus graphically giving a ( t ). If one does this then one finds that a ( t ) is an exponentially growing function of t , thus having a term like G ln G might be a way to obtain an early inflationary stage to the Universe.", "pages": [ 12, 13, 14, 15, 16, 17 ] }, { "title": "5 Discussion", "content": "In section 2 we had introduced the notions of scale-invariant and almost scale-invariant Lagrangians. For the set of Lagrangians L = f ( R ) + k ( G ) we found out that L is scale-invariant if with constants α , β , and n . This is a closed set of functions. If we add the divergence γG with constant γ , we arrive trivially at the following set of almost scale-invariant Lagrangians Of course, both Lagrangians give rise to the same set of field equations, but the 4-parameter set eq. (5.2) fails to be a closed set of functions. To see this, we take the limit n → 1 in eq. (5.2) while α remains constant and β = -γ = 1 / ( n -1). We get In arriving at this result we have defined n -1 = ε and G = e z and used Result: Besides the trivial almost scale-invariant Lagrangians, as defined by eq. (5.2), the class of almost scale-invariant Lagrangians include also the Lagrangians of the form 13 Of course the inclusion of matter will change the behaviour of the cosmological solutions discussed in this paper, but in the early stages of the Universe, with matter in the form of dust or radiation, the behaviour of the solutions above will only be marginally modified by the presence of the matter. The behaviour described in this paper will thus essentially correctly describe the dynamics. Finally, let us reformulate one of the key results of this work given in section 4 which is closely related to analogous calculations done for higher dimensions in [23]: For p > 0, the Lagrangian has the spatially flat Friedmann with scale factor a ( t ) = t p as exact vacuum solution. For large values p this is a local attractor solution and it represents a model for power-law inflation. /negationslash", "pages": [ 17, 18 ] }, { "title": "Acknowledgements", "content": "Useful comments by S. Deser, Q. Exirifard, A. de la Cruz-Dombriz and S. Odintsov are gratefully acknowledged. DS acknowledges the support of a DAAD (Deutscher Akademischer Austauschdienst) grant to do research at Universitat Potsdam.", "pages": [ 19 ] } ]

[ { "title": "Deformations of Polyhedra and Polygons by the Unitary Group", "content": "Etera R. Livine 1, 2, ∗ 1 Laboratoire de Physique, ENS Lyon, CNRS-UMR 5672, 46 All'ee d'Italie, Lyon 69007, France 2 Perimeter Institute, 31 Caroline St N, Waterloo ON, Canada N2L 2Y5 (Dated: June 27, 2018) We introduce the set of framed (convex) polyhedra with N faces as the symplectic quotient C 2 N // SU(2). A framed polyhedron is then parametrized by N spinors living in C 2 satisfying suitable closure constraints and defines a usual convex polyhedron plus extra U(1) phases attached to each face. We show that there is a natural action of the unitary group U( N ) on this phase space, which changes the shape of faces and allows to map any (framed) polyhedron onto any other with the same total (boundary) area. This identifies the space of framed polyhedra to the Grassmannian space U( N ) / (SU(2) × U( N -2)). We show how to write averages of geometrical observables (polynomials in the faces' area and the angles between them) over the ensemble of polyhedra (distributed uniformly with respect to the Haar measure on U( N )) as polynomial integrals over the unitary group and we provide a few methods to compute these integrals systematically. We also use the Itzykson-Zuber formula from matrix models as the generating function for these averages and correlations. In the quantum case, a canonical quantization of the framed polyhedron phase space leads to the Hilbert space of SU(2) intertwiners (or, in other words, SU(2)-invariant states in tensor products of irreducible representations). The total boundary area as well as the individual face areas are quantized as half-integers (spins), and the Hilbert spaces for fixed total area form irreducible representations of U( N ). We define semi-classical coherent intertwiner states peaked on classical framed polyhedra and transforming consistently under U( N ) transformations. And we show how the U( N ) character formula for unitary transformations is to be considered as an extension of the Itzykson-Zuber to the quantum level and generates the traces of all polynomial observables over the Hilbert space of intertwiners. We finally apply the same formalism to two dimensions and show that classical (convex) polygons can be described in a similar fashion trading the unitary group for the orthogonal group. We conclude with a discussion of the possible (deformation) dynamics that one can define on the space of polygons or polyhedra. This work is a priori useful in the context of discrete geometry but it should hopefully also be relevant to (loop) quantum gravity in 2+1 and 3+1 dimensions when the quantum geometry is defined in terms of gluing of (quantized) polygons and polyhedra.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Inspired by loop quantum gravity [1], and more particularly the spinorial formalism [2-4] and the structures of twisted geometry [5], we discuss the phase space of polyhedra in three dimensions and its quantization, which serves as basic building of the kinematical states of discrete geometry. More precisely, following [7], we show that the Grassmannian space U( N ) / (U( N -2) × SU(2)) is the space of framed (convex) polyhedra with N faces up to 3d rotations. The framing consists in the additional information of a U(1) phase per face. This provides an extension of the Kapovich-Milson phase space [8] for polyhedra with fixed number of faces and fixed areas for each face. Indeed, we describe the Grassmannian as the symplectic quotient C 2 N // SU(2), which provides canonical complex variables for the Poisson bracket. This construction allows a natural U( N ) action on the space of polyhedra, which has two main features. First, U( N ) transformations act non-trivially on polyhedra and change the area and shape of each individual face. Second, this action is cyclic: it allows to go between any two polyhedra with fixed total area (sum of the areas of the faces) and in particular to generate any polyhedron from the totally squeezed polyhedron with only two non-trivial faces. Upon quantization, the framed polyhedron phase space leads to the Hilbert space of SU(2) intertwiners, which is interpreted as the space of quantum polyhedra. We perform a canonical quantization from the complex variables of C 2 N // SU(2) and all the classical features are automatically exported to the quantum level. Each face carries now a irreducible representation of SU(2), i.e. a half-integer spin j , which defines the area of the face. Intertwiners are then SU(2)-invariant states in the tensor product of these irreducible representations. These intertwiners are the basic building block of the spin network states of quantum geometry in loop quantum gravity. The U( N ) action on the space of intertwiners changes the spins of the faces and each Hilbert space for fixed total area (sum of the spins) defines an irreducible representation of the unitary group U( N ), as shown in [7]. Once again, the U( N ) action is cyclic and allows to generate the whole Hilbert space from the action of U( N ) transformation on the highest weight vector. This construction provides coherent intertwiner states peaked on classical polyhedra, as used in [9]. At the classical level, we will use the U( N ) structure of the space of polyhedra to compute the averages of polynomial observables over the ensemble of polyhedra distributed along the uniform Haar measure. We will underline a phenomenon of concentration of measure, which peaks random polyhedra on spherical configurations for large number of faces N . Furthermore, we will show how to use the Itzykson-Zuber formula from matrix models [10] as a generating functional for these averages. It computes the integral over U( N ) of the exponential of the matrix elements of a unitary matrix tensor its complex conjugate. At the quantum level, we will show that the character formula, giving the trace of unitary transformations either over the standard basis or the coherent intertwiner basis, provides an extension of the Itzykson-Zuber formula. It allows in principle to generate the expectation values of all polynomial observables (and thus their spectrum). The plan of this paper goes as follows. In section II, we define and describe the phase space of framed polyhedra, its parameterization in terms of spinor variables and the action of U( N ) transformations. In section III, we show how to compute the averages and correlations of polynomial observables using group integrals over U( N ) and we discuss the Itzykson-Zuber integral as generating function. In section IV, we discuss the quantum case, with the Hilbert space of SU(2) intertwiners, coherent states and the character formula. In section V, we investigate the lower-dimensional analog of polygons (in two dimensions), we show that the unitary group is replaced by the orthogonal group and that the Grassmannian O( N ) / (O( N -2) × SO(2)) defines the phase space for framed polygons. We then discuss the issue of gluing such polygons together into a consistent 2d cellular decomposition, as a toy model for the gluing of framed polyhedra into 3d discrete manifolds. These constructions are relevant to quantum gravity in 2+1 and 3+1 dimensions, especially to discrete approaches based on a description of the geometry using glued polygons and polyhedra such as loop quantum gravity (and dynamical triangulations). The goal is to clarify how to parametrize the set of polygons/polyhedra and their deformations, and to introduce mathematical tools to compute the average and correlations of observables over the ensemble of polygons/polyhedra at the classical level and then the spectrum and expectation values of geometrical operators on the space of quantum polygons/polyhedra at the quantum level. In this context, we hope that this work will be useful to the study of the dynamics of (loop) quantum gravity, especially in its formulation in terms of spinor networks and twisted geometries, but it should also be relevant to the study of the structure of discrete geometries and cellular decompositions.", "pages": [ 1, 2 ] }, { "title": "A. A Quick Review of the Kapovich-Milson Phase Space", "content": "Let us consider N vectors /vector V i in R 3 that satisfy a closure condition, that their sum vanishes: By a theorem due to Minkowski, these determine a unique convex polyhedron with N faces, such that the /vector V i 's are the (outward) normal vectors to the faces, that is the faces have area V i = | /vector V i | ∈ R + and unit normal ̂ v i = /vector V i / | /vector V i | ∈ S 2 . The reconstruction of the polyhedron is not trivial and the shape of the faces depend non-trivially on the set of chosen vectors. The interested reader can find details on the reconstruction algorithm in [11]. The space of polyhedra P N = { ( /vectorv i ) | ∑ N i =1 /vector V i = 0 } has dimension (3 N -3), and if we consider the set of equivalence classes under 3d rotations we get the space P 0 N with dimension (3 N -6). Generally, these spaces do not have an even dimension and are not symplectic manifolds. However, if we fix the areas V i of all N faces, we get the Kapovich-Milson phase space [8]: Definition II.1. Let us consider the space product of N 2-spheres for fixed V i 's: This is a symplectic manifold provided with the Poisson structure on each of the N spheres (scaled by their radii): Then the closure conditions ∑ i V a i = ∑ i V i v a i = 0 form a first class constraint system, that generates global SO(3) rotations on the set of the N vectors /vector V i , or equivalently on the set of the N unit vectors ̂ v i . This defines by symplectic reduction the Kapovich-Milson phase space for convex polyhedra with N faces and fixed face areas V i : This manifold has dimension (2 N -6) . Instead of removing N degrees of freedom from the space of polyhedra P r N by fixing the individual face areas and thus obtaining the manifold P { V i } N with even dimension (3 N -6) -N and carrying a symplectic structure, we will now add N degrees of freedom to embed P 0 N into a larger phase space of framed polyhedra P z N with even dimension (3 N -6) + N . These extra degrees of freedom are angles (or U(1) phases) canonically conjugate to the face areas. They allow to work in a phase space where the areas can vary and have a dynamics. This is a necessary structure when studying the dynamics of loop quantum gravity, where areas and spins do change under time evolution (and space-time diffeomorphisms). We achieve this below by using the spinorial representation of the su (2) algebra 'a la Schwinger as prescribed in [2-5, 7, 9].", "pages": [ 2, 3 ] }, { "title": "B. Spinor Phase Space for Framed Polyhedra", "content": "We will now replace the data of N vectors in R 3 by N spinors. We call a spinor a complex 2-vector z ∈ C 2 for which we will use a bra-ket notation: It lives in the fundamental 2-dimensional representation of SU(2), with the obvious scalar product 〈 w | z 〉 = ¯ w 0 z 0 +¯ w 1 z 1 . We also introduce its dual spinor using the structure map of SU(2): Following [7, 9], we consider sets of N spinors satisfying a closure constraint: Definition II.2. Let us consider the space C 2 N of N spinors z i ∈ C 2 endowed with the canonical symplectic structure { z A i , ¯ z B j } = -i δ AB δ ij with the indices A,B = 0 , 1 . We impose the closure constraints that the 2 × 2 matrix X ≡ ∑ i | z i 〉〈 z i | is proportional to the identity: where the three matrix σ a =1 , 2 , 3 are the Pauli matrices generating SU(2) . These three real constraints are first class and generate the SU(2) action on the N spinors: We define the phase space of framed polyhedra with N faces as the symplectic quotient P z N ≡ C 2 N // SU(2) , that is as the set of collections of N spinors satisfying the closure constraints and up to SU(2) transformations. A simple counting gives that P z N is a (4 N -6)-dimensional manifold, which corresponds to the dimension (3 N -6) of the space P 0 N of N -faced polyhedra (up to 3d rotations) plus N degrees of freedom. More precisely, we introduce the mapping from spinors to 3-vectors: with V = | /vector V | = 〈 z | z 〉 . This mapping is obviously not one-to-one and is actually invariant under the multiplication of the spinor by an arbitrary phase, | z 〉 → e iθ | z 〉 . The inverse mapping is given by [2, 12]: This provides a bijection C 2 ∼ R 3 × U(1). One checks that we have the same Poisson brackets for the vectors as earlier, { V a , V b } = 2 /epsilon1 abc V c , inherited from the canonical bracket on the spinor variables. Using this mapping, we send a collection of N spinors onto a collection of N vectors. The closure constraint then read as before: This defines a convex polyhedron with N faces with areas given by the norm squared of the spinors | V i | = 〈 z i | z i 〉 , with a total area A = 2 λ ≡ ∑ i | V i | = ∑ i 〈 z i | z i 〉 overall. This mapping provides a bijection P z N ∼ P 0 N × U(1) N between our space of framed polyhedra defined in terms of spinors and the space P 0 N of polyhedra with N faces up to 3d rotations times N phases attached to each face [7, 9]. This construction provides a larger phase space where the areas of the faces can vary dynamically. Moreover the spinors are crucial in defining the action of the unitary group U( N ) on the (framed) polyhedra as we will see in the next sections. Finally, we conclude this section by introducing the complex variable ζ = z 1 /z 0 ∈ C for a spinor | z 〉 . This variable ζ commutes with the norm V and parameterizes the 2-sphere defined by the 3-vector /vector V as | z 〉 varies while keeping the radius V = 〈 z | z 〉 fixed: The symplectic structure on the 2-sphere then simply reads in terms of this complex parameter: This variable is specially interesting when studying the Kapovich-Milson phase space for fixed individual face ares, when the phase space is parametrized by these complex variables ζ i =1 ..N constrained by the closure condition.", "pages": [ 3, 4 ] }, { "title": "C. Closing Open Polyhedra and the SL(2 , C ) Action", "content": "A first interesting remark is that the use of spinors provide a natural way to close opened configurations into actual polyhedra. As pointed out in [9, 12, 13], this is achieved through a SL(2 , C ) transformation on the spinors. Indeed, starting with an arbitrary set of (not all vanishing) N spinors | z i 〉 , a priori not satisfying the closure constraints, that is such that the matrix X = ∑ i | z i 〉〈 z i | is not proportional to the identity. Then X is a positive Hermitian operator, it can be diagonalized and written as: where g ∈ SU(2) is unitary, ∆ is a diagonal 2 × 2 matrix with positive entries, ρ = det X = det∆ is positive, Λ = g √ ∆ /ρ 1 4 ∈ SL(2 , C ). Then we act with Λ -1 ∈ SL(2 , C ) on the spinors to get a closed configuration: These new spinors | ˜ z i 〉 trivially satisfy the closure constraints: and thus define a (framed) polyhedron with face areas ˜ V i = 〈 ˜ z i | ˜ z i 〉 and total area: 2 ˜ λ = ∑ i ˜ V i = T r ˜ X = 2 ρ, (12) ρ 2 = det X = 1 2 [ (T rX ) 2 -T rX 2 ] = 1 4 [ (T rX ) 2 -T r ( X/vectorσ ) · T r ( X/vectorσ ) ] = 1 4 [ (2 λ ) 2 -| /vector C| 2 ] . This new total area 2 ˜ λ is always smaller than the initial one 2 λ and obviously coincides when the original spinors already satisfy the closure condition /vector C = 0. It is useful to get a closer at the geometry of this procedure. Starting with the N vectors /vector V i with a non-vanishing sum /vector C /negationslash = 0, we perform a SU(2) transformation g on the spinors | z i 〉 such that the corresponding § 0(3) rotation sends the vector /vector C onto the z -axis: where ˆ e z is the unit basis vector along z -axis. Up to this 3d rotation, we can start directly with such a configuration with /vector C collinear with the z -axis. Then writing the components of the matrix X in terms of the spinors give equations corresponding to the total area and the components of the closure vector: We now defines the rescaled spinors: or explicitly: Second, the rescaling matrix Λ is in SL(2 , C ) and is actually a boost along the z -direction. And we understand the overall SL(2 , C ) transformation from the original arbitrary spinors | z i 〉 to the new closed spinors | ˜ z i 〉 as a rotation to the z -axis followed by a rescaling of the first and second component of the spinors with inverse factor so that the sum of their modulus square match. First, the new spinors | ˜ z i 〉 satisfy the balance equation ∑ i | ˜ z 0 i | 2 = ∑ i | ˜ z 1 i | 2 and the orthogonality equation ∑ ¯ ˜ z 0 i ˜ z 1 i = 0, and thus satisfy the closure condition. They define a closed polyhedron with total area 2 ˜ λ = ∑ i | ˜ z 0 i | 2 + | ˜ z 1 i | 2 = √ 4 λ 2 -| /vector C| 2 .", "pages": [ 4, 5 ] }, { "title": "D. Invariant Parametrization and Cross-Ratios", "content": "The spinors z A i do not commute with the closure constraints /vector C = 0 and are thus not invariant under SU(2) transformations. The first question is to identify SU(2)-invariant observables, which can then be used to parameterize framed polyhedra in the phase space P z N . Natural observables are given by the scalar products between spinors and their dual: These scalar products commute with the closure constraints, and are thus invariant under SU(2) transformations of the spinors, These are the basic variables for the U( N ) formalism for SU(2) intertwiners as developed for loop quantum gravity in [7, 9, 12-14]. From that perspective, the most useful feature is that these variables form a closed algebra under the Poisson bracket, This algebra will get quantized exactly and will provide the basic operators acting on the Hilbert space of intertwiners. The usual vector scalar products /vector V i · /vector V j , measuring the angles between two faces, are easily expressed in terms of these variables, One can write all observables probing the geometric of the polyhedra in terms of E 's or F 's. We can then use these variables to parameterize the space of (framed) polyhedra. On the one hand, the E 's are most particularly relevant because they generate U( N ) transformations compatible with the closure conditions on the spinors. We will use this to define the action of the unitary group U( N ) on polyhedra in the next section. On the other hand, the F 's are holomorphic and offer a enlightening parametrization of the framed polyhedron phase space as we explain below. Moreover, they are crucial in defining coherent intertwiners [9] and in deriving the holomorhic/anti-holomorphic splitting of the simplicity (second class) constraints in loop quantum gravity [12, 15]. The F 's are specially interesting because they are not only invariant under global SU(2) transformations but they are also invariant under global SL(2 , C ) transformations, as it is easy to check: Thus the action of closing an arbitrary set of spinors into a (framed) polyhedron, as described in the previous section, will leave the F 's invariant. We can go further and show that the F 's entirely determine the orbit under SL(2 , C ) in the space of unconstrained spinors C 2 N : Lemma II.3. Considering two sets of spinors | z i 〉 and | w i 〉 such that [ z i | z j 〉 = [ w i | w j 〉 for all indices i, j , and further assuming that there exists a couple of indices k, l such that [ z k | z l 〉 /negationslash = 0 , then there exists a matrix Λ ∈ SL(2 , C ) that maps one onto the other: Proof. Let us first remark that the following identity on 2 × 2 matrices is true, taking into account that [ z k | z l 〉 /negationslash = 0: Indeed, [ z k | z l 〉 /negationslash = 0 implies that | z k 〉 and | z l 〉 are not colinear and span the whole two-dimensional spinor space. Then the previous operator leaves invariant | z k 〉 and | z l 〉 and is thus equal to the identity. Let us now consider the matrix: One checks that its determinant is equal to one, det Λ = 1 2 ((T r Λ) 2 -T r Λ 2 ) = 1, so that Λ ∈ SL(2 , C ). Finally, using the equality of the F -observables for both sets of spinors, we have: Applying this to k, l = 1 , 2, we get 1 : This means that we can obtain all the F ij from the two ( N -2)-dimensional complex vectors F i 1 and F i 2 (for i ≥ 3) plus the scale factor F 12 . This minimal data is defined in terms of 2( N -2) + 1 complex parameters thus (4 N -6) real parameters as expected. Furthermore each SL(2 , C )-orbit has a unique intersection with the space of framed polyhedra P z N . This is a restatement of the isomorphism C 2 N / SL(2 , C ) ∼ C 2 N // SU(2), where SL(2 , C ) is understood as the complexification of SU(2). This is similar to the analysis performed in [16] but the present setting is slightly more general (and actually simpler) since the authors were looking at the Kapovich-Milson phase spaces (at fixed individual face areas). We formalize this as follows: Proposition II.4. Considering two sets of spinors | z i 〉 and | w i 〉 satisfying the closure constraints, and such that [ z i | z j 〉 = [ w i | w j 〉 for all indices i, j , then they are related by a global SU(2) transformation that maps one set of spinors onto the other: Proof. To start with, assuming the closure constraints on the spinors z i , one can get the total area 2 λ = ∑ i V i = ∑ i 〈 z i | z i 〉 from the F 's: Thus the total area associated to both sets of spinors z i and w i are equal. If λ vanishes, then both sets of spinors vanish and are trivially related by an arbitrary SU(2) transformation. Else λ does not vanish and there automatically exists at least a couple of indices ( k, l ) such that F ( z ) kl = [ z k | z l 〉 does not vanish, so that we can apply the previous lemma ensuring that both sets of spinors are related by a SL(2 , C ) transformation. Then the work is to show that this SL(2 , C ) transformation is actually unitary and lays in SU(2). First, we show that all the scalar products are equal, by inserting the closure constraint: Then we fix one index k and consider the SU(2) group element mapping w k to z k : And we check that it actually maps each w i to the corresponding z i : As a result, the key point is that the SL(2 , C ) invariant observables F ij entirely determine a unique (framed) polyhedron. A na¨ıve puzzle is that there are N ( N -1) / 2 such observables F ij , thus giving N ( N -1) real parameters, while the space of framed polyhedra is of dimension (4 N -6). This points to the fact that the F 's variables are not independent and satisfy the Plucker relations (which can be directly checked from their explicit definition in terms of the spinors): This is illustrated by the fact that one can send by a SL(2 , C ) transformation 2 an arbitrary set of spinors z i on a new set of spinors such that the first two spinors are collinear to the complex vectors (1 , 0) and (0 , 1):", "pages": [ 5, 6, 7, 8 ] }, { "title": "E. The Cyclic U( N ) Action", "content": "We now come to the key tool of this paper: U( N ) transformations acting on framed polyhedra with N faces. Following [9, 12 ? ], we introduce the natural action of the U( N ) group on collections of N spinors in C 2 N : The key point is that this action commutes with the closure constraints: Thus this induces an action of unitary group on the space P z N of framed polyhedra. Moreover, taking the trace of the previous equation, we check that this action leaves invariant the total area of the polyhedron: Notice that this action does not simply act on the 3-vectors /vector V i but also involves the individual phases of each spinor. Therefore we truly need the spinors and one can not simply define a U( N )-action on the space of polyhedra P N . At the infinitesimal level, this action is generated by the scalar products 3 between the spinors [2, 9, 12]: where e iα ∈ U( N ) if the matrix α is Hermitian. As we said in the previous section, these generators commute with the closure constraints generating the SU(2) transformations on the spinors, { /vector C , E ij } = 0, confirming that U( N ) transformations commute with the SU(2) action. Finally, we look at their Poisson bracket (16) and check that they form the expected u ( N ) Lie algebra. This parametrizes the polyhedron in terms of the N face areas plus ( N -3) complex cross-ratios, which gives the correct dimension, N + 2( N -3) = (3 N -6). These cross-ratios can be almost translated in the F -variables [18], which hints towards an explicit link between the two considered SL(2 , C ) actions: The SL(2 , C ) action, the fibration of the phase space in terms of SL(2 , C )-orbits and the parametrization in terms of cross-ration turned out powerful when constructing coherent intertwiner states and studying the integration measure over them [16-18]. In particular, it hints towards a link between coherent intertwiner states and conformal field theory. The possible reformulation of our spinor phase space in terms of conformal field theory is postponed to future investigation. We further check that these generators commute with the total area of the polyhedron 2 λ = ∑ i 〈 z i | z i 〉 , thus confirming that the total area is invariant under U( N ) transformations. The key feature of this U( N )-action on the space of framed polyhedron is that the action is cyclic. Indeed, we can reach any configuration up to a global scale factor from the completely degenerate and flat configuration by an arbitrary U( N ) transformation. More precisely, we introduce the trivial reference point: which obviously satisfies the closure constraints, ∑ i | Ω i 〉〈 Ω i | = I . The corresponding 3-vectors are the unit vector in the z -direction, /vector V 1 = ˆ e z , its opposite /vector V 2 = -/vector V 1 = -ˆ e z , and vanishing vectors /vector V 3 = .. /vector V N = 0, thus giving a completely-flat configuration defining a degenerate polyhedron. Acting with an arbitrary U( N ) transformation on this configuration gives: Reversely, considering from an arbitrary collection of N spinors { z i , i = 1 ..N } satisfying the closure constraints, we can rescale it so that it is of the form above: This works because the closure constraints are equivalent to the fact that the first and second components of the spinors form two orthogonal complex N -vectors with equal norms: exactly the same as the first two columns, ( √ λU k 1 ) k and ( √ λU k 2 ), of a unitary matrix U ∈ U( N ) re-scaled by √ λ . Moreover, the stabilizer group of the completely flat polyhedron clearly is U( N -2). Thus the set of collections of N spinors satisfying the closure constraint is identified to the quotient U( N ) / U( N -2). Further quotienting by the action of SU(2) (to get equivalence classes of poyhedra under 3d rotations), this lead us to the following proposition as hinted in [7, 9]: Proposition II.5. We have an action of the unitary group U( N ) on the space P z N of framed polyhedron with N faces. This leads to an isomorphism between P z N = C 2 N // SU(2) and the Grassmannian space U( N ) / (SU(2) × U( N -2)) . In particular, we have the equivalence for a set of spinors z i ∈ C 2 N : Before moving on to the next part of the paper, we would like to re-visit this U( N ) structure of the space of polyhedra from the point of view of the SU(2)-invariant observables. The definition of the spinors | z k 〉 = √ λ | ( U Ω) k 〉 calculation gives for an anti-symmetric matrix β : Contrarily to the U( N ) transformations, these do not leave invariant the total area of the polyhedron, as one can check directly from the Poisson brackets of the F 's and ¯ F 's with 2 λ = ∑ k 〈 z k | z k 〉 : in terms of the unitary matrix U ∈ U( N ) implies the diagonalization of the observables E ij and F ij as N × N matrices: where the off-diagonal components vanish and t U = ¯ U -1 . These definitions of E and F in terms of the matrix U are invariant under transformations U → UG with G ∈ SU(2) × U( N -2). We can also deduce the existence of the unitary matrix U directly from the E 's or F 's. Indeed, first considering the Hermitian matrix of the scalar products E ij = 〈 z i | z j 〉 for a closed configuration of spinors | z i 〉 , the matrix E satisfies a simple polynomial identity: Reversely, this equality is obviously enough to guarantee the existence of U (as already stated in [2]): Result II.6. Considering a N × N Hermitian matrix E satisfying E 2 = T rE 2 E for some λ ∈ R ∗ + , it is diagonalizable with λ as its single non-vanishing and doubly-degenerate eigenvalue: One can also start from the matrix of observables F ij : Result II.7. Considering a non-vanishing N × N matrix F satisfying the Plucker relations (27) , it is automatically antisymmetric and of the following form: Proof. We specialize the Plucker relations to a doublet of indices i, j and the fixed indices k, l = 1 , 2 as before: This means that F is antisymmetric and furthermore that it is of rank 2 (if it is non-vanishing). F being a complex antisymmetric matrix, one can diagonalize it as F = U Σ t U , with U ∈ U( N ) and Σ of the following type: The λ k 's are a priori complex. Since F is of rank 2, there is a single non-vanishing block with λ 1 ∈ C . One can then absorb its phase in the definition of the unitary matrix U and keep its modulus as λ ∈ R + . In the next part III, we will use this reformulation of (framed) polyhedra in terms of unitary matrices to compute systematically the averages and correlations between the normal vectors defining polyhedra and characterizing their shape.", "pages": [ 8, 9, 10 ] }, { "title": "III. COMPUTING AVERAGES THROUGH INTEGRALS ON U( N )", "content": "Now considering the ensemble of (framed) polyhedra provided with the uniform measure or equivalently the U( N ) Haar measure, we study the averages and correlations of polynomial observables and aim at characterizing the shape of a typical polyhedron. In particular, we show how to formulate the averages of polynomial observables in the normal vector /vector V i as integrals over the unitary group U( N ) and how to use the Itzykson-Zuber integral as a generating function for these.", "pages": [ 11 ] }, { "title": "A. Counting Polyhedra: Entropy", "content": "We start by computing the volume of the space of polyhedra with N faces for a fixed total area. This corresponds to computing the entropy for a simplified model of the black hole horizon in loop quantum gravity [20, 21]. When quantized, this model reproduces the loop gravity's entropy calculation through counting the dimensions of SU(2) intertwiner spaces (see [22] and [23] for reviews and detail on the description of the quantum states of a black hole horizon as SU(2) intertwiners). One defines the density of framed polyhedra with N faces and fixed area 2 λ as the following straightforward integral over spinor variables constrained by a total area condition and the closure conditions: where the 8 π -factor is an arbitrary choice of normalization. Integrating over the phases of the spinors, one can perform the change of variables from the z k ∈ C 2 to the vectors /vector V k ∈ R 3 . The change of measure is straightforward to perform [12, 13] and one obtain the density of polyhedra as previously defined in [21]: The most direct way to compute this integral is to Fourier-transform the δ -distribution 4 . One then gets: where the 8 π -factor had been chosen so that ρ 2 [ λ ] = 1 for polyhedra with N = 2 faces. One can find the details of this calculation in appendix A4. The method is actually useful for defining a partition function over the ensemble of polyhedra and computing the averages of polynomial observables by differentiation as outlined in [21]. Another method also shown in [21] is to Fourier-transform the δ -distribution while keeping the spinor variables. One then gets Gaussian integrals which can be easily handled. We will not use this method here. Instead, we would like to highlight the fact that the space of framed polyhedra is isomorphic to the Grassmaniann space U( N ) / U( N -2) × SU(2), which allows for a more geometric interpretation for the volume of P z N . valid for arbitrary values of /epsilon1 ∈ R + . Indeed, keeping the spinor variables in the definition (35) of the density ρ N [ λ ], we write explicitly the total fixed area and closure constraints in terms of the real and imaginary parts of the spinor variables, z A k = x A k + iy A k : This means that we have two unit vectors of dimension 2 N , ( x 0 k , y 0 k ) and ( x 1 k , y 1 k ), both on the (2 N -1)-dimensional sphere S 2 N -1 . The second vector ( x 1 k , y 1 k ) is actually orthogonal to the first vector ( x 0 k , y 0 k ) but also to the vector ( y 0 k , -x 0 k ) itself orthogonal to the former vector. This means that this second vector ( x 1 k , y 1 k ) actually lives on a (2 N -3)-dimensional sphere S 2 N -3 still with unit radius. This leads to a simple geometric interpretation of the density of polyhedra with N faces and fixed total area as the product of the volumes of the spheres S 2 N -1 and S 2 N -3 : where the factor π/ 4 adjusts the over-all normalization of the integrals. From the point of view of unitary groups, the situation is clear: we are computing the volume of the coset U( N ) / U( N -2), which can be decomposed as U( N ) / U( N -1) × U( N -1) / U( N -2), which is isomorphic to the product of the two spheres S 2 N -1 × S 2 N -3 . Below, we will analyze the average of polynomial observables over the ensemble of polyhedra and we will fully use for this purpose the U( N ) structure. In practice, we will normalize all the results by the overall volume of the space of polyhedra at fixed total area by simply using the normalized Haar measure on the unitary group U( N ).", "pages": [ 11, 12 ] }, { "title": "B. Probing the Average Geometry of a Polyhedron and Fluctuations", "content": "We would like to characterize a typical polyhedron drawn at random from the ensemble with the Haar measure on U( N ). To this purpose, we compute the averages of the normal vectors and their correlations. Using the explicit expression of the spinors and vectors in terms of the unitary matrix U ∈ U( N ) as given earlier by (32), the averages of product of the norms V k or vector components V a k can all be re-cast as polynomial integrals over U( N ) of the type: where the number of U 's and of its complex conjugate ¯ U 's must match else the integral vanishes. Here we focus on the explicit computation of these integrals up to the 4rth order, using the basic recoupling theory of U( N ) representations, in order to probe the average geometry and uncertainty of the polyhedra. Below, we will give the generic behavior of the polynomial integrals in section III C and discuss how such integrals can be generated from the Itzykson-Zuber formula in section III D. Starting with quadratic integrals, we compute the average norm of each normal vector: using the orthogonality of the matrix elements of a U( N ) group element in the fundamental N -dimensional representation. This was expected since the total area is 2 λ , which is shared isotropically among the N normal vectors. Beside this, the average of each of the vector components 〈 V a k 〉 vanishes. The next step is to compute the quartic integrals 〈 V 2 k 〉 and 〈 V a k V b l 〉 . This is done using the explicit formula (computed by decomposing the tensor product U ⊗ ¯ U as the matrix elements of the group element U in the trivial and adjoint representations): Applying this to the average squared-norm and correlations between vector components, straightforward calculations give: /negationslash First, this allows to compute the spread of a face area: which means that the probability distribution of the area of a face remains fuzzy even as the number of faces grows. Second, looking at the correlation 〈 V i V j 〉 between the areas of two distinct faces, we can check that ∑ i,j 〈 V i V j 〉 = 4 λ 2 as expected from the fixed total area constraint ∑ i V i = 2 λ . Moreover, we check that the area of faces becomes more and more decoupled as the number of faces grows: Third, we introduce another set of observables Θ ab characterizing the shape of a polyhedron: which vanishes if the normal vectors are distributed spherically, but will be non-vanishing as soon as we deviate from the isotropic distribution (e.g. if the shape of the polyhedron is more ellipsoidal than spherical). Here, we easily check that: Instead of using U( N )-integrals, one could instead compute brutally these averages and correlations as integrals over the normal vectors together with the closure constraints and fixed area constraint. We give the explicit method in appendix A and we recover the formulas above. But we have further computed the mean value 〈 Θ ab Θ cd 〉 in order to get the standard deviation from the spherical configuration: This means that the probability distribution over the ensemble of polyhedra is highly peaked about the spherical configuration. To get a simpler single indicator, we can compute the average of T r Θ 2 . Classically, T r Θ 2 has direct expression in terms of the vector scalar products: which gives: It is always null or positive, T r Θ 2 ≥ 0, and measures somehow the shape of the polyhedron. It is maximal when the polyhedron is flat and gets smaller as the polyhedron becomes more and more spherical. For instance, it vanishes for a cube ( N = 6 faces). We get its average by contracting the indices in the formula above: This can be compared to the concentration of measure on the sphere S 2 N -1 ∼ U( N ) / U( N -1) induced by the Haar measure on U( N ): the uniform measure concentrates very strongly about any equator as N grows large (see e.g. [24] for a description of this phenomenon, focusing on its application to the entanglement of random states). We very probably have a similar concentration of measure on the coset U( N ) / U( N -2). We will have a closer look at this later in section III E. It is interesting to compare these averages to the one of an ensemble of normal vectors without the closure constraints: We use the similar brute-force method by Fourier-transforming the δ -distribution, as done in [21], with /epsilon1 ∈ R + : This allows us to compute this volume: Thinking in terms of spinors, this correspond to the (properly normalized) volume of a (4 N -1)-dimensional sphere. Using the same techniques as given in appendix A of differentiating with respect to the momentum conjugated to the vectors /vector V k , we have computed the averages and correlations of the vector components, which we note with the subscript (0) to distinguish them from the average over the space of polyhedra: /negationslash At leading order in N , we find the same average 〈 V i 〉 and spread 〈 V 2 i 〉 for the individual face areas. Here, we can easily go further and compute exactly all the averages 〈 V n i 〉 for an individual face area. Indeed: with Furthermore, the closure condition is obviously satisfied in average 〈 ∑ i V a i 〉 (0) = 0, but it now has a on-trivial spread: This is due to the vanishing of the correlation between components of two distinct vectors i and j . Indeed the main difference between the ensembles satisfying or not the closure constraints is in the correlations between normal vectors. For an individual vector, it does not change the leading order (in N ) of the averages of the powers of the area 〈 V n i 〉 (though the exact full expression does change), as we will check later in section III D. Going further, we easily check that 〈 Θ ab 〉 (0) = 0 and that the ensemble is also peaked on spherically symmetric sets of vectors. We nevertheless expect a deviation for the averages 〈 Θ ab Θ cd 〉 (0) but we haven't checked this explicitly. Up to now we have looked explicitly at integrals up to order 4 in the normal vectors (up to order 8 in the spinors). Using the U( N ) framework, it is possible to compute generic formulas for all polynomial integrals over the unitary group and thus compute at least at leading order all polynomial averages over the ensemble of (framed) polyhedra, as we will see in the next section. This is much more powerful than the method of differentiating the partition function.", "pages": [ 12, 13, 14, 15 ] }, { "title": "C. Polynomial Averages at Leading Order", "content": "Using the interplay between the irreducible representations of U( N ) and of the permutation group S n , [ ? ] give a systematic formula for polynomial integrals over U( N ): where the sum is over permutations σ and τ . The factor is given explicitly as where the sum is over partitions Λ /turnstileleft n of the integer n , χ Λ is the corresponding character of the permutation group S n , and s Λ ,N ( x 1 , .., x N ) is the corresponding Schur function, with in particular s Λ ,N (1) = s Λ ,N (1 , .., 1) the dimension of the irreducible representation of U( N ) associated with Λ. Furthermore, [25] goes further and uses combinatorics to provide an asymptotic formula for the symbol Wg at large N : in terms of the cycle decomposition of the permutation σ = c 1 . . . c K . For a generic permutation | σ | is the minimal number of transpositions needed to write σ . For a cycle, | c | is simply the length of the cycle minus one. C c is the c -th Catalan number: in terms of binomial coefficients. Large N corresponds geometrically to a very large number of faces and thus at a refinement limit for the polyhedra. This will likely be very useful to understand the large N limit of the distribution of polyhedra and thus study their continuous limit. This result was used in [25] to study the large N limit of the Itzykson-Zuber formula, or more precisely of its derivative, when the normalized traces N -1 T rX k and N -1 T rY k converge at large N (for all k 's). We will investigate below how the Itzykson-Zuber formula can actually be used as the generating function for these polynomial integrals over U( N ). Applying this formula to the product of vector components V a i , the indices j 's and l 's in the integral (61) will all be equal to 1 or 2 and contracted with Pauli matrices σ a lj . The indices i 's and k 's correspond to the index of the vectors between 1 and N . The permutation σ have to match the i 's with the k 's, thus does not mix between different vectors, while the permutation τ have to match the j 's with the l 's and can mix terms corresponding to different vectors. Then one has to compute the traces of product of Pauli matrices corresponding to the cycles of the permutation τ . Thus in theory it is possible to compute systematically the average of any polynomial observables using this formula. In practice, this can become tedious. Nevertheless, the structure of the formula is rather simple (in terms of the permutations σ and τ ) and one could study in a straightforward manner the averages of the powers of an interesting observable (e.g. the individual face area or the volume of the polyhedron) if one wanted. An equivalent formula but worded differently can be found in [26], related to the evaluation of the twirling operator in quantum information and used in the context of the convergence to equilibrium under a random Hamiltonian. Considering the Hilbert space of ⊗ n k =1 C N , on which the unitary operators U ⊗ n act. We consider the representation of the permutation group S n defined by swapping subsystems: where { e i } i =1 ..N forms a basis of C N . Following the notations of [26], we write V σ as short for the operator D ( N ) ( σ ). It is easy to compute the character of this representation D ( N ) : χ D ( N ) ( σ ) = N /lscript ( σ ) , where /lscript ( σ ) is the number of cycles in the cycle decomposition of the permutation σ . Then defining the twirling operator T n ( · ) = ∫ dU U ⊗ n ( · ) U ⊗ n † , we have for any two operators A,B acting on ( C N ) ⊗ n : and the vectors a σ ≡ T r A VV σ -1 and the same for B . The proof can be found in [26]. The matrix M has a simple form: and the whole issue is to invert this matrix, which leads to the same result as presented above when applied to operators A and B taken in the standard basis. More details on the structure and possible computation of M -1 can be found in [26] for the interested reader.", "pages": [ 15, 16 ] }, { "title": "D. Itzykson-Zuber Formula as Generating Function", "content": "The Itzykson-Zuber formula allows to compute the integral over U( N ) of the exponential of matrix elements of U and ¯ U . Based on the localization of integrals, it first appeared in relation to matrix models and two-dimensional quantum gravity [10] and be computed explicitly using the Harish-Chandra formula (e.g. [25]). It goes as follows. Let us consider two N × N matrices X and Y and let ( x i ) and ( y i ) be their respective eigenvalues. We call ∆( X ) = ∏ i ∫ U( N ) dU e iθ T r ( Y U † XU ) = det ( e iθx j y k ) 1 ≤ j,k ≤ N ∆( X )∆( Y ) ( iθ ) -N ( N -1) 2 . (66) Choosing appropriate matrices X and Y , this Itzykson-Zuber formula can be seen as the generating function for all the correlations between the normal vectors over our polyhedron ensemble. In our case, let us give an example with the observable V i and its powers. We have: V i = 〈 z i | z i 〉 = λ ( U i 1 ¯ U i 1 + U i 2 ¯ U i 2 ) = T r ( Y U † XU ) , with Y jk = ( δ j 1 δ k 1 + δ j 2 δ k 2 ) and X ( i ) jk = δ ji δ ki . The matrix Y is fixed and implements the reduction from U( N ) to our space of polyhedron U( N ) / U( N -2). The matrix X selects the considered observables. Then the mean value 〈 exp( iθV i ) 〉 is a Itzykson-Zuber integral: 〈 e iθV i 〉 = c ∫ U( N ) dU e iθ T r ( Y U † XU ) = 1 + ∞ ∑ n =1 ( N -1)! ( n + N -1)! ( n +1)( iθ λ ) n , (67) where c is a normalization constant such that 〈 1 〉 = 1 for θ = 0. The trick to derive this formula is to regularize the Itzykson-Zuber formula by shifting slightly all the eigenvalues of X and Y to ensure that they are different and then to send these regulators to 0 at the end. Then this result gives us directly all the mean values 〈 ( V i ) n 〉 , without having to suitably differentiate the density of state ρ N [ λ ] as in section III B or compute the polynomial U( N ) integrals as in section III C: 〈 V n 〉 = λ n ( n +1)!( N -1)! ( N + n -1)! , (68) which matches our expressions already derived for 〈 V 〉 and 〈 V 2 〉 . We can compare them to the free model without closure constraints as introduced earlier in section III B, which had the following averages (59): 〈 V n 〉 (0) = (2 λ ) n ( n +1)!(2 N -1)! (2 N + n -1)! . First, we notice that these are different (though similar), showing that the two models are clearly distinct and have a different probability distribution for the individual face areas. Second, as claimed earlier, the two expressions nevertheless match at large N for a fixed power n : 〈 V n 〉 ∼ N /greatermuch 1 λ n ( n +1)! N n ∼ 〈 V n 〉 (0) . We can go further and get the formula for the fixed matrix Y but for arbitrary matrix X . We perturb around the actual eigenvalues of Y as y 1 = 1 + /epsilon1 1 , y 2 = 1 + /epsilon1 2 and y k ≥ 3 = /epsilon1 k . Both numerator and denominator of the ItzyksonZuber vanish as all the /epsilon1 i are set to 0. We can nevertheless suitably differentiate both numerator and denominator (using L'Hˆopital rule) until we reach non-vanishing values, here ∂ ( N -2) /epsilon1 N ∂ ( N -3) /epsilon1 N -1 ..∂ /epsilon1 3 ∂ /epsilon1 2 . This leads to for N ≥ 4: The numerator is a modified Vandermonde determinant (but vanishes when θ = 0) while the denominator comes from differentiating the original Vandermonde determinant ∆( Y ) (it is also the determinant of the ( N -2) × ( N -2) matrix whose matrix elements are given by m ij = ∏ i k =1 ( k + j )). This provides a direct formula for the observables ∑ i x i V i for a diagonal matrix X : det ( e iθx j y k ) 1 ≤ j,k ≤ N ∆( Y ) -→ /epsilon1 i → 0 i N ( N +1) 2 θ 3( N -3)+1 ∑ σ /epsilon1 [ σ ] x N -2 σ (1) x N -3 σ (2) ..x σ ( N -2) e iθx σ ( N -1) x σ ( N ) e iθx σ ( N ) ( N -1)! ∏ N -3 k =1 k ! . (69) θ ∑ i x i V i = ( θλ ) T r ( Y U † XU ) for X = ( x 1 , ..x N ) . When the matrix X is arbitrary and not diagonal, its off-diagonal components allows us to probe the correlations between the various spinors z i : ( θλ ) T r ( Y U † XU ) = θ ∑ ij X ij 〈 z i | z j 〉 . Then the Itzykson-Zuber integral can be understood as the generating function for the averages and correlations of the spinor scalar products. From these and taking into account that the vector scalar product is related to the spinor scalar product, |〈 z i | z j 〉| 2 = V i V j + /vector V i · /vector V j , we can extract in principle all the averages and correlations of the SU(2)invariant polynomials in the vector components /vector V a i . It would be interesting to apply these techniques to computing the averages of the powers of the (squared) volume observable, in order to get a better idea of the typical shape of polyhedra, but also because the exact spectrum of the (squared) volume operator at the quantum level is still an open issue. Thus we have seen how the Itzykson-Zuber integral over U( N ) expressed in terms of Vandermonde determinants can be considered as the generating function for the averages of all polynomial observables in the polyhedra's normal vectors. These averages are extracted by suitable differentiating of this Itzykson-Zuber formula. An interesting point is whether the Itzykson-Zuber integrant e iθ T rY U † XU for the fixed considered Y but arbitrary X can have a physical or geometrical relevance, for instance when investigating some (random) dynamics on the space of (framed) polyhedra. We leave this for future investigation. = k ≥ 2 E. Explicit U( N ) Parametrization and Haar Measure We now turn to another method to compute these integrals over U( N ) using an explicit parametrization of the unitary matrices and the corresponding recursive formula for the Haar measure on U( N ) [27]. The goal is to draw a unitary matrix at random with respect to the Haar measure, or more precisely to draw at random its two first columns, that is two ortogonal complex N -vectors of unit norm. The details of the parametrization and construction for the whole unitary matrix can be found in [27]. Here, we will only detail the parametrization of the two first columns and thus of the spinors defining the polyhedra with N faces. The parametrization is best defined recursively. We start with the case N = 2. Two arbitrary orthogonal complex 2-vectors of unit norm can be written as: v (2) = ( e iθ 1 cos α 2 e iθ 2 sin α 2 ) , w (2) = e iφ 2 ( -e iθ 1 sin α 2 e iθ 2 cos α 2 ) , (70) where the phases θ 1 , θ 2 and φ 2 live in [0 , 2 π ] while the rotation angle α 2 's range is [0 , π 2 ]. The normalized Haar measure then reads: dµ 2 = 1 N 2 sin( α 2 ) cos( α 2 ) dα 2 dθ 1 dθ 2 dφ 2 , with N 2 = 1 2 (2 π ) 3 . (71) The components of the two spinors are read directly from these complex vectors: z i = √ λ ( v (2) i w (2) i ) , z 1 = e iθ 1 √ λ ( cos α 2 -e iφ 2 sin α 2 ) , z 2 = e iθ 2 √ λ ( sin α 2 e iφ 2 cos α 2 ) . This provides a parametrization of a unitary matrix in U(2) as expected. Then we can define the two complex vectors v ( N ) and w ( N ) recursively from v ( N -1) and w ( N -1) as: v ( N ) = ( cos α N v ( N -1) e iθ N sin α N ) , w ( N ) = ( cos β N w ( N -1) 0 ) + e iφ N ( -sin α N sin β N v ( N -1) e iθ N cos α N sin β N ) , (72) where we have added four new parameters, θ N , φ N ∈ [0 , 2 π ] and α N , β N ∈ [0 , π 2 ]. The normalized Haar measure is now: dµ N = 1 N N dθ 1 N ∏ k =2 sin α k cos 2 k -3 α k dα k dθ k dφ k N ∏ k =3 sin β k cos 2 k -5 β k dβ k , (73) (2 π ) - 2 N - 1 1) with N n ∏ ∏ We can read the components of the N spinors directly from these two complex vectors, up to the global scale factor √ λ . In total, we have parametrized our spinors using (4 N -4) angles α k , β k , θ k , φ k plus λ . These are (4 N -3) parameters, exactly the dimension of the space of N spinors satisfying the closure constraints. If we want to further gauge fix the SU(2) invariance, we can fix the direction of the last vector /vector V N . In terms of the components of the last spinor, z N = e iθ N (sin α N , e iφ N cos α N sin β N ), this amounts to fixing φ N = α N = β N = 0. Fixing these three parameters, this provides an explicit parametrization of the (4 N -6)-dimensional space P z N of framed polyhedra up to 3d rotations. 2( k If we consider the first vector v ( N ) , we can give its full expression: v ( N ) =         e iθ 1 cos α 2 cos α 3 .. cos α N e iθ 2 sin α 2 cos α 3 .. cos α N e iθ 3 sin α 3 .. cos α N . . . e iθ N cos α N         with dµ ( v ( N ) ) ∝ N ∏ i dθ i N ∏ k =2 sin α k cos 2 k -3 α k dα k . (74) This gives actually a random vector on the complex unit sphere in C N , distributed uniformly with respect to the Haar measure on U( N ). It is well known that there is a phenomenon of concentration of measure on the complex k ≥ 3 ) 2( k - 2) . sphere as N grows, e.g. [24]. More precisely, the integral over the complex sphere is almost equal to the simpler integral over the equator of the sphere (for α N = 0). This is due to the specific shape of the Haar measure in this parametrization, which gets concentrated to the equator as N grows large. In the context of quantum information (and quantum computing), this concentration of measure is often used to argue that arbitrary states are maximally entangled between subsystems as the dimensions of the Hilbert spaces grows large, e.g. [24, 28]. Here we are drawing a second complex vector w ( N ) , which is orthogonal to the first one. It would be interesting to investigate whether there is a similar phenomenon of concentration of measure and what would be its geometrical interpretation on the space of (framed) polyhedra. We postpone such analysis to future investigation. Nevertheless, this explicit parametrization does provide a very useful tool in order to compute the average of any polynomial observable over the space of polyhedra as an explicit trigonometric integral. IV. DEFORMING QUANTUM POLYHEDRA This section is dedicated to the study of the quantum case: we quantize the space of framed polyhedra into the Hilbert space of SU(2) intertwiners interpreted as quantum polyhedra, following the previous work done in [2, 9, 12, 13]. We will see that the Hilbert space of quantum polyhedra has the same structure as the classical set of framed polyhedra. We have indeed a cyclic action of the U( N ) transformations on quantum polyhedra with fixed total boundary area and we can construct coherent polyhedron state labeled by the classical framed polyhedra (up to 3d rotations). Finally, we will give two ways to write the trace of geometrical operators: either using the U( N ) character formula, which is interpreted as the quantum counterpart of the Itzykson-Zuber integral formula or using the coherent states and having an integral over 'fuzzy' polyhedra. A. Quantizing Polyhedra into Intertwiners We canonically quantize the space of spinors C 2 N by promoting the components of the spinors and their complex conjugate to harmonic oscillators: { z A i , ¯ z B j } = -iδ AB δ ij -→ [ a A i , a B j † ] = δ AB δ ij , (75) where we have taken the convention /planckover2pi1 = 1. As shown and used in [2, 9, 12, 13] (see also [3, 4, 29]), the closure constraints C a generating the SU(2) action on spinors, the U( N ) generators E ij and the SU(2)-invariant observables F ij are all quantized without ambiguity and their algebra at the quantum level is without any anomaly. We consistently choose the normal ordering, keeping the annihilation operators a 0 , 1 to the right and the creation operators a 0 , 1 † to the left. For details, the interested reader can refer to those references. We will nevertheless give here a quick summary of the main structures, relevant to our main point, that is the U( N ) action on SU(2) intertwiners. For the sake of completeness, we give the expressions of the basic operators, which are all quadratic in the harmonic oscillators. When there can be no confusion, we will not distinguish the classical quantity from the quantum operator, else we will put a hatˆon the quantum operator. For the SU(2) generators, we have C a = ∑ i V a i with: V a i = ∑ A,B σ AB a a A † i a B i , V z i = ( a 0 † i a 0 i -a 1 † i a 1 i ) , V + i = a 0 † i a 1 i , V -i = a 1 † i a 0 i . (76) These form on each face i the Schwinger representation of the su (2) algebra in terms of two harmonic oscillators. We also introduce the operator giving the total energy of the oscillators living on the face i as the quantization of the norm of the normal vector V i : V i = ∑ A a A † i a A i , [ V i , V a i ] = 0 . (77) As well-known, this SU(2) representation is reducible and irreducible components are obtained by diagonalizing the Casimir operator V i , whose eigenvalues are twice the spin living on that face, 2 j i ∈ N . This is interpreted as usual as the quantization of the individual face areas. We similarly quantize the spinor scalar products: E ij = a 0 † i a 0 j + a 1 † i a 1 j , F ij = a 0 i a 1 j -a 1 i a 0 j , F † ij = a 0 † i a 1 † j -a 1 † i a 0 † j . (78) It is straightforward to compute the commutators of these operators and check that they give the same results as their Poisson brackets. In particular, the E 's and F 's commute with the closure constraint operators C a and thus are SU(2)-invariant. Moreover the E ij form a closed u ( N ) algebra. Using the definition in terms of harmonic oscillators, the Casimir of this u ( N ) algebra is easily related to the total area [7]: ∑ i,j E † ij E ij = E ( E + N -1) , E = N ∑ i E ii = ∑ i V i . (79) Looking at the Hilbert spaces, we start with 2 N copies of the Hilbert space of a single harmonic oscillator, ( H HO ⊗ H HO ) ⊗ N = L 2 ( C 2 N ). Each couple ( H HO ⊗ H HO ) can be decomposed in irreducible representations of SU(2) with arbitrary spin j ∈ N / 2 (given by half the total number of quanta of the oscillators). Then we impose a SU(2)invariance by requiring that the closure constraint operators C a = ∑ i V a i vanish on the states. This is exactly the Hilbert space of SU(2) intertwiners between N irreducible representations: H ( N ) = Inv SU(2) [ ( H HO ⊗H HO ) ⊗ N ] = Inv SU(2) [ N ⊗ i ⊕ j i ∈ N / 2 V j i ] = ⊕ { j i } i =1 ..N Inv SU(2) [ N ⊗ i V j i ] , (80) where we write V j for the irreducible SU(2)-representation of spin j . On this Hilbert space of intertwiners, we have a U( N ) action generated by the E ij . Since the corresponding u ( N )-Casimir ∑ i,j E † ij E ij is determined in terms of the total area operator E whose value is simply the sum of twice the spins ∑ N i (2 j i ), we can simply decompose the space H ( N ) in irreducible components by fixing the value of the total area: H ( N ) = ⊕ J ∈ N R J N , R J N = ⊕ ∑ N j i = J Inv SU(2) [ N ⊗ i V j i ] , (81) where each Hilbert space R J carries an irreducible representation of U( N ), as shown in [2, 7, 9]. The corresponding Young tableaux is given by two horizon lines of equal length J . The corresponding highest weight vector | ψ J 〉 corresponds to a bivalent intertwiner, which is the quantum equivalent of the completely squeezed polyhedron in the classical case: E 11 | ψ J 〉 = J | ψ J 〉 , E 22 | ψ J 〉 = J | ψ J 〉 , ∀ k ≥ 3 , E kk | ψ J 〉 = 0 , E i>j | ψ J 〉 = 0 , (82) where the E ii = V i are the generators of the Cartan subalgebra. In particular, we notice that this highest weight vector is invariant under U( N -2), which corresponds to the expectation that the classical space of framed polyhedra is isomorphic to the Grassmanniann space U( N ) / (U( N -2) × SU(2)). The dimension of each of these irreducible U( N )-representations can be computed using the hook formula. This gives: d N [ J ] = dim R J N = 1 J +1 ( N + J -1 J ) ( N + J -2 J ) . (83) This is the total number of SU(2)-intertwiners for a fixed number of faces N and fixed total area 2 J = ∑ i 2 j i . It is the quantum counterpart of the density of states ρ N [ λ ], which gives the volume of the space of framed polyhedra with N faces and total area 2 λ . Indeed, looking at the large area limit while keeping N fixed, gives: d N [ J ] ∼ J →∞ J 2 N -4 ( N -1)!( N -2)! + NJ 2 N -5 ( N -1)!( N -3)! + . . . , (84) which fits at leading order in J with ρ N [ λ ], as given by (37), for λ = J . Notice that all the terms have the same order in N . Therefore, this limit can be considered carefully. To be more rigorous, one should put the /planckover2pi1 -factors back in the quantum expression, then this is the limit where the Planck area unit is sent to 0, while keeping the total area fixed. Then this amounts to sending the sum of the spin to ∞ , thus giving the wanted result. To summarize the structures, the vector operators V i acts on each subspace V j i living on each face and generate the SU(2)-action on those subspaces. The SU(2)-invariant operators E ij act on each subspace R J N , defined as the space of SU(2) intertwiners for fixed sum of the spins J = ∑ i j i , and they generate a U( N )-action on each of these subspaces. Finally the F ij and F † ij operators respectively act as annihilation and creation operators on the full space of intertwiners H ( N ) allowing to respectively decrease and increase the total area J . These SU(2)-intertwiners are the quantum counterpart of the classical (framed) polyhedra. They are also the basic building blocks of the spin network states of quantum (space) geometry in loop quantum gravity [1]. That identification of intertwiners as quantum polyhedra is the key to the geometrical interpretation of spin network as discrete geometries constructed as (quantum) polyhedra glued together. this identification will be made even clearer below when dealing with coherent intertwiner states peaked on classical framed polyhedra. B. Beyond Intertwiners: non-Closed Quantum Polyhedra Considering the tensor product of N representations of SU(2), one for to each face of the polyhedron, we have imposed up to now the closure constraint and thus required invariance of our tensor product states under SU(2). We can relax this condition and characterize states that recouple to a fixed overall spin J different from 0. This corresponds to the classical case where the closure constraints are broken and the sum of the normal vectors do not vanish but the closure vector has a fixed norm. We are now working on another subspace of ( H HO ⊗H HO ) ⊗ N = L 2 ( C 2 N ), such that the value of the SU(2)-Casimir given as the norm squared of the closure constraint operators C 2 is fixed to J ( J +1): H ( N ) J = Cov J SU(2) [ ( H HO ⊗H HO ) ⊗ N ] = ⊕ { j i } i =1 ..N Cov J SU(2) [ N ⊗ i V j i ] = ⊕ { j i } i =1 ..N Inv SU(2) [ V J ⊗ N ⊗ i V j i ] . (85) This is actually equivalent to having intertwiners, i.e SU(2)-invariant states, between the N original irreducible representations V j i and an extra one V J . We still have the U( N )-action on this Hilbert space H ( N ) J and we can decompose it into U( N ) irreducible representations: H ( N ) J = ∑ J ⊕ ∑ N j i = J Cov J SU(2) [ N ⊗ i V j i ] = ∑ J ⊕ ∑ N j i = J Inv SU(2) [ V J ⊗ N ⊗ i V j i ] , (86) where the total area J is of the same parity as the overall spin J (i.e half-integer or integer depending on J ) and necessarily larger or equal to J . Each of the subspaces at fixed J carries an irreducible representation of U( N ). Its highest weight vector is defined by the (unique) trivalent intertwiner between SU(2)-representations of spins J + J 2 , J -J 2 and J , i.e the values of the Cartan subalgebra generators on it are [7]: E 11 | ψ J J 〉 = ( J + J ) | ψ J J 〉 , E 22 | ψ J J 〉 = ( J -J ) | ψ J J 〉 , ∀ k ≥ 3 , E kk | ψ J J 〉 = 0 . Thus the corresponding Young tableaux contains two horizontal lines of respective lengths ( J + J ) and ( J -J ) and the dimensions of the representations are [7]: d N [ J, J ] = dim R J, J N = dim ∑ J ⊕ ∑ N j i = J Cov J SU(2) [ N ⊗ i V j i ] = 2 J +1 J + J +1 ( N + J + J 1 J + J ) ( N + J -J 2 J -J ) . (87) It is fairly easy to check that summing over all possible values of J ≤ J , we recover the full Hilbert space of intertwiners for ( N +1) faces and fixed total area J : d N +1 [ J ] = ∑ J≤ J d N [ J, J ] . (88) This could be proved directly either by recombining the binomial coefficients or by recursion. Finally, it would be interesting to investigate whether there is a similar procedure to 'close' non-invariant configuration as in the classical case, where we could apply a SL(2 , C ) transformations on an arbitrary non-closed set of spinors in order to map it into a closed set of spinors defining an actual framed polyhedron. We postpone to future investigation the thorough study of the existence on a SL(2 , C )-action on the space of intertwiners and of its properties. C. Probing the shape of Intertwiners Similarly to the classical case, we now would like to compute the traces of geometrical operators on the Hilbert space of SU(2)-intertwiners at fixed number N of faces and fixed total area J = ∑ i j i . We can already deduce some averages from the fixed area condition J = ∑ i j i and the formula for the dimensions of the intertwiner spaces d N [ J ]. We obviously have: 〈 2 j i 〉 = 〈 V i 〉 ≡ 1 d N [ J ] T r H ( N ) V i = 2 J N , (89) which is also equal to the classical average (42). We can also single out explicitly one face/leg of the intertwiner. Then using the dimension of the space of tensor product states for a fixed external spin (or intertwiners with one fixed spin) given in the previous section, we compute: 〈 V 2 i 〉 = 〈 4 j i ( j i +1) 〉 = 1 d N [ J ] ∑ j ≥ J 2 4 j ( j +1) d N -1 [ J, j ] = 6 J ( J + N ) N ( N +1) = 6 J 2 N ( N +1) + 6 J N +1 , . (90) We see that the first term in 〈 V 2 i 〉 fits exactly the classical average (44). The second term is the quantum correction, and is sub-leading in the classical limit defined by taking large J at fixed N . Playing around with the binomial coefficients, one can show the somewhat surprising formula giving the traces of arbitrary powers of the norm: 〈 2 j i (2 j i +1) .. (2 j i + n ) 〉 = 1 d N [ J ] ∑ j ≥ J 2 2 j (2 j +1) .. (2 j + n ) d N -1 [ J, j ] = J (( m +2) J +2 N + m -2) ( m +1)! ( N + J + m -2)! ( N + J -1)! ( N -1)! ( N + m )! (91) = J (( m +2) J +2 N + m -2) ( m +1)! ( N + J + m -2 m -1 ) ( N + m m -1 ) , from which we can recover the traces 〈 V i 〉 and 〈 V 2 i 〉 . We can square the fixed area condition and deduce the correlation between spins i = k : /negationslash 〈( ∑ i V i ) 2 〉 = N 〈 V 2 i 〉 + N ( N -1) 〈 V i V k 〉 i = k = (2 J ) 2 ⇒ 〈 V i V k 〉 i = k = 〈 4 j i j k 〉 = J 2 2(2 N -1) ( N -1) N ( N +1) -6 J ( N -1)( N +1) . (92) /negationslash Similarly, using the closure constraint operator, or in other words the SU(2)-invariance, we can compute: /negationslash /negationslash 〈( ∑ i /vector V i ) 2 〉 = N 〈 V 2 i 〉 + N ( N -1) 〈 /vector V i · /vector V k 〉 i = k = 0 ⇒ 〈 /vector V i · /vector V k 〉 i = k = -6 J ( J + N ) ( N -1) N ( N +1) . (93) If we want to go further and compute traces of operators involving the values of the spins on three or more legs and thus probing the fine structure of the intertwiners, we would have to compute the dimensions of the intertwiner subspaces with fixed spins. Instead of doing this by hand, we can do this consistently using the full U( N )-character formula, which computes the trace of U( N ) transformations instead of simply the dimension which gives the trace of the identity. This the method outlined in [7] and we show here that it should be considered as a generalization to the quantum case of the Itzykson-Zuber formula used as generating function for averages over the ensemble of classical polyhedra. More precisely, the character of the U( N ) representation, of highest weight [ l 1 , ..l N ], computes the trace of a diagonalized unitary transformation U = ( e iθ 1 , .., e iθ N ) as a Schur polynomial: χ [ l i ] ( e iθ i ) = det( e iθ j ( l i + N -i ) ) ij det( e iθ j ( N -i ) ) ij . (94) /negationslash Here, the highest weight is given by l 1 = l 2 = J and this formula defines directly the generating functions for the spin expectation values (or equivalently the V i = 2 j i ): 〈 e i ∑ k θ k E k 〉 = χ [ J,J, 0 ,.. ] ( e iθ k ) d N [ J ] = 1 d N [ J ] det( e iθ j ( J ( δ i 1 + δ i 2 )+ N -i ) ) ij det( e iθ j ( N -i ) ) ij , (95) where the normalization should be such that the expectation value of 1 is 1 (when θ i = 0). The determinant at the denominator is exactly a Vandermonde determinant, while the numerator is a slight modification. This formula contains all the traces of polynomials in the spins j i 's. If we extend the formula to non-diagonal U( N ) transformations (which we can diagonalize of course), we can generate the traces of all scalar products and powers in the basic vectors /vector V i . As in the classical case, extracting these traces requires a careful differentiation of this generating function. It would be interesting if these traces of U( N ) transformations could themselves be physically/geometrically relevant, for instance in the study of the dynamics of polyhedra and intertwiners. D. Interpolating between Classical and Quantum Polyhedra: Coherent Intertwiner States To better understand the link between intertwined states and classical polyhedra, we can build coherent intertwined states peaked on classical framed polyhedra following [9, 12, 13]. Following the conventions of [12, 13], one defines: Definition IV.1. Given a set of spinors z i ∈ C 2 N , we define the coherent intertwiner state | J, { z i }〉 in R J N using the SU(2) creation operators F † : | J, { z i }〉 = 1 √ J !( J +1)!   1 2 ∑ ij [ z i | z j 〉 F † ij   J | 0 〉 = 1 √ J !( J +1)!   1 2 ∑ ij [ z i | z j 〉 ( a 0 † i a 1 † j -a 1 † i a 0 † j )   J | 0 〉 . (96) The scalar products [ z i | z j 〉 are invariant under SU(2) rotations, so the intertwiner states are labeled by the orbits of spinors under global SU(2) transformations. Moreover, these scalar products are also invariant under global SL(2 , C ) transformations, which map arbitrary sets of spinors to spinors satisfying the closure constraints. Thus the coherent intertwiner states are truly labeled by orbits of spinors under global SL(2 , C ) transformations, that is points in the space of framed polyhedra (up to 3d rotations) P z N = C 2 N / SL(2 , C ) = C 2 N // SU(2) as we have seen in section II D. The main results established in [9], and revisited in [12, 13], are two key properties of these intertwiner coherent states: their formulation as group averaging of the tensor product of standard SU(2) coherent states, which establishes their geometrical interpretation as semi-classical polyhedron states, and then their coherence under the action of U( N ). Or more precisely, · Decomposition on SU(2) coherent states: 1 √ J !( J +1)! | J, { z i }〉 = ∑ J = ∑ i j i 1 ∏ i (2 j i )! ∫ SU(2) dg ⊗ i g | j i , z i 〉 , (97) where we group average the tensor product of individual SU(2) coherent states living on each face and defined as: | j, z 〉 = ( z 0 a 0 † + z 1 a 1 † ) 2 j √ (2 j )! | 0 〉 . g | j, z 〉 = | j, gz 〉 , 1 √ 〈 z | z 〉 2 j | j, z 〉 = g ( z ) | j, j 〉 , with g ( z ) = 1 √ 〈 z | z 〉 ( z 0 -¯ z 1 z 1 ¯ z 0 ) . (98) These states living in V j are coherent under the action of SU(2) and thus can all be generated from the highest weight vector | j, j 〉 by acting with SU(2) transformations (up to a norm factor): Finally, they are peaked on the classical vectors /vector V ( z ) = 〈 z | /vectorσ | z 〉 : 〈 j, z | ˆ V a | j, z 〉 〈 j, z | j, z 〉 = 2 j /vector V V , (99) where the expectation value vector has the same direction as /vector V but is normalized to 2 j in term of the spin carried by the state. The group average states for fixed individual spins j i were introduced earlier in [30]. Written as such, the coherent intertwiner states | J, { z i }〉 truly represent the quantized version of a classical framed polyhedron defined as a set of N vectors or spinors up to SU(2) transformations. · Coherence under U( N ) transformations: The action of U( N ) transformations, generated by the operators ˆ E ij at the quantum level, on the coherent intertwiner states amounts to the classical U( N )-action on the set of spinors labeling the state: ˆ U | J, z i 〉 = | J, ( Uz ) i 〉 , U = e ih , ˆ U = e i ∑ kl h kl ˆ E kl , (100) for an arbitrary Hermitian matrix h . This ensures that the behavior of coherent intertwiner states is just the same as classical framed polyhedra. For instance, one can generate all coherent intertwines by acting with U( N ) transformations on the bivalent intertwiner, just the same way as we could generate all (closed) framed polyhedra by acting with U( N ) transformations on the totally squeezed configuration with only two non-trivial faces. This is the key property allowing us to take the trace over the Hilbert space of intertwines by an integral over the unitarity group U( N ), similarly to the classical case. This is explained below in details. Taking into account that the Hilbert space R J N of intertwiners for fixed total sum of the spins is an irreducible representation of U( N ), one can write the identity of that space as an integral over U( N ) acting on a fixed state, say the bivalent intertwiner on the legs 1 and 2, which is exactly the integral over the coherent intertwiner states: I J N = 1 J !( J +1)! ∫ C 2 N ∏ i e -〈 z i | z i 〉 d 4 z i π 2 | J, { z i }〉〈 J, { z i }| . (101) A rigorous proof can be found in [9], and then a simpler proof in [12, 13]. Basically, this comes from writing the identity on the larger Hilbert space H ( N ) in terms of the usual coherent states for the harmonic oscillators and then projecting down on the subspace with fixed total number of quanta J . We also compute the scalar product between two coherent intertwiners [9]: 〈 J, { z i }| J, { w i }〉 = ( det ∑ i | w i 〉〈 z i | ) J =   1 2 ∑ i,j [ w i | w j 〉〈 z j | z i ]   J . (102) For a single set of spinors, this also gives the norm of the coherent intertwiner state: 〈 J, { z i }| J, { z i }〉 =   1 2 ∑ i,j | F ij | 2   J = 1 2 2 J   ( ∑ i V i ) 2 -∣ ∣ ∣ ∣ ∣ ∑ i /vector V i ∣ ∣ ∣ ∣ ∣ 2   J . (103) We clearly see that the norm is maximized when the closure vector vanishes, /vector C = ∑ i /vector V i = 0, and that we have a closed set of spinors thus corresponding to a true polyhedron. This is similarly to the result obtained in [30], which studied the saddle point approximation to the group averaging for fixed spins j i and show that a stationary point exists only if the vectors satisfy the closure constraints. Combining these two formula, we can take the trace of the identity on R J N and recover the dimension of this Hilbert space, which we can express either as a Gaussian integral over the spinors z i or as an integral over the vectors /vector V i : d N [ J ] = T r I J N = 1 J !( J +1)! ∫ C 2 N ∏ i e -〈 z i | z i 〉 d 4 z i π 2 ( det ∑ i | z i 〉〈 z i | ) J = 1 2 2 J J !( J +1)! ∫ R 3 N ∏ i e -V i d 3 /vector V i 4 πV i   ( ∑ i V i ) 2 -∣ ∣ ∣ ∣ ∣ ∑ i /vector V i ∣ ∣ ∣ ∣ ∣ 2   J . (104) It is possible to check this formula directly by performing the integral over the vectors /vector V i as done in [21]. This provides a interpretation of the space of intertwiners as almost-closed polyhedra, or fuzzy polyhedra, which are more and more peaked on true framed polyhedra (satisfying the closure constraints) as the total area J grows. This fits with our earlier claim that the dimension of the Hilbert space R J N behaves at leading order as the classical density of framed polyhedra as the total spin J grows large for a fixed number of faces N . We can know further the compute the trace of any unitary transformation, which provides an integral formula for the U( N )-character given in terms of modified Vandermonde determinant in the previous section: χ [ J,J, 0 ,.. ] ( ˆ U ) = T r J N ˆ U = 1 J !( J +1)! ∫ C 2 N ∏ i e -〈 z i | z i 〉 d 4 z i π 2 〈 J, z i | J, ( Uz ) i 〉 = 1 J !( J +1)! ∫ C 2 N ∏ i e -〈 z i | z i 〉 d 4 z i π 2 ( det ∑ i | ( Uz ) i 〉〈 z i | ) J , (105) with U = e ih and ˆ U = e i ∑ kl h kl ˆ E kl in terms of a Hermitian matrix h as above. In general, the determinant is a little messy: det ∑ i | ( Uz ) i 〉〈 z i | = 1 2 ∑ ijkl U ik U jl F kl ¯ F ij , from which we can compute the trace of ˆ U as Gaussian integral in the spinor variables. It is however simpler when looking at diagonal unitary transformations U = e i ∑ k θ k E k , which act as multiplication by individual phases on each spinor: 〈 J, { z i }| e i ∑ k θ k E k | J, { z i }〉 = 〈 J, { z i }| J, { e iθ i z i }〉 =   1 2 ∑ ij e i ( θ i + θ j ) | F ij | 2   J = 1 2 2 J   ( ∑ i e iθ i V i ) 2 -∣ ∣ ∣ ∣ ∣ ∑ i e iθ i /vector V i ∣ ∣ ∣ ∣ ∣ 2   J , from which we can compute the trace of e i ∑ k θ k E k as an integral over the spinors or the vectors. These traces are to be considered as the generating function for the traces of every polynomial operators in the E 's or F 's or V 's. Beyond this, it would be interesting to investigate if these unitary transformations can be seen as implementing the dynamics of the intertwiners for fixed boundary area (for instance, in the context of quantum black holes, see e.g. [21]), in which case these traces could be provided with a direct physical interpretation. On the other hand, we could also apply our machinery to other geometric operators. For example, an open issue is still to compute the exact spectrum of the (squared) volume operator (see nevertheless e.g. [31]) and we could attempt to compute the traces of the powers of this operator, from which we could reconstruct its spectrum. V. ORTHOGONAL GROUP ACTION ON POLYGONS We will now look at structures in one dimension less and study the space of polygons. We will see that we can similarly define a phase space of framed polygons, also the frame now on each edge will be reduced to a sign ± . Similarly to the case of polyhedra, we will characterize the space of framed polygons for fixed boundary perimeter as a representation of the orthogonal group O( N ) instead of the unitary group. Working in 2d instead of 3d will also us to be more explicit in the reconstruction of the geometrical structure especially on the issue of gluing polygons together to form a two-dimensional discrete manifold. A. Phase Space for Polygons The phase space structure for polygons is much simpler than for polyhedra. Instead of using spinors attached to each face of the polyhedron, we will attach a single complex variable to each edge of the polygon. Let us thus start with { z i } ∈ C N with i = 1 ..N for polygons with N edges and postulate the following canonical Poisson bracket: { z j , ¯ z k } = -iδ jk . (106) This corresponds to a set of N oscillators. Decomposing the complex variables in real and imaginary parts, z j = R j + iI j , these variables look like the real version of the spinors used in the 3d case for polyhedra: z j = R j + iI j ∈ C -→ ( R j I j ) ∈ R 2 . In these real variables, the canonical bracket reads { R j , I k } = 1 2 δ jk . We define a closure constraint to ensure that the complex variables correspond the normal vectors to the edges of a true closed polygon: C = ∑ j z 2 j . (107) The normal vectors are 2-dimensional and correspond to the square of the complex variables: z 2 j = ( R 2 j -I 2 j ) + 2 iR j I j -→ /vectorn j = ( R 2 j -I 2 j 2 R j I j ) ∈ R 2 . (108) As we will see in details in a following section V C, this ensures that we can reconstruct a unique convex polygon, such that these normal vectors are orthogonal to the polygon's edges and their norm give the length of the corresponding edge. Then the perimeter of the polygon is given by the total energy of the oscillators: E = ∑ j | z j | 2 . (109) The real part of the closure constraint generates the multiplication by a global U(1) phase to all the complex variables: e iθ { ∑ k ( R 2 k -I 2 k ) , ·} ( R j I j ) = ( cos θ sin θ -sin θ cos θ ) ( R j I j ) , e iθ { ∑ k ( R 2 k -I 2 k ) , ·} z j = e iθ z j , (110) with θ ∈ R and e iθ ∈ U(1). The closure constraint C = 0 is clearly invariant under this global phase transformation. On the other hand, the imaginary part of C generates global inverse re-scaling of the real and imaginary parts: e -η { ∑ k 2 R k I k , ·} ( R j I j ) = ( e η 0 0 e -η ) ( R j I j ) , (111) with η ∈ R . The closure constraint C = 0 is not invariant under such transformations. On the contrary, we can use them map any set of complex variables onto a closed set satisfying C = 0. Indeed, starting with an arbitrary value of ∑ j z 2 j , we first set the phase of this complex number by multiplication by a global phase. It is then purely real. Second, we set its real part to 0 by the inverse re-scaling which allows to balance the sum of the squares of the real parts and imaginary parts, k R 2 k = k I 2 k . ∑ ∑ Combining these two type of transformations generate the SL(2 , R ) group. As we have just shown, these transformations allow to map any complex N -vector ( z k ) k =1 ..N ∈ C N onto one satisfying the closure constraint C = 0. This is the equivalent of the SL(2 , C ) transformations allowing to map arbitrary sets of N spinors onto a closed framed polyhedron. B. The Orthogonal Group Action As before, we have the obvious U( N )-action on C N now generated by E jk = ¯ z j z k : z i -→ ( Uz ) i = ∑ j U ij z j . (112) It allows to go from Ω = (1 , 0 , .., 0) to an arbitrary vector in C N up to a global scale factor: Ω -→ ( U Ω) i = U i 1 , (113) which means that we are working on the unit complex sphere U ( N ) / U( N -1) ∼ S C N -1 ∼ S R 2 N -1 . This could be an interesting testing ground for the case of the polyhedra since we know well the phenomenon of concentration of measure on the coset U ( N ) / U( N -1) as N grows to infinity. As expected, the U( N ) action leaves invariant the perimeter E : { E ij , E} = 0 . (114) On the other hand, it does not commute with the closure constraint. However, we can introduce a linear combination of the u ( N ) generators that does: { E ij , C} = iz j z i = 0 , { e ij , C} = 0 with e ij ≡ -i ( E ij -E ji ) = -i (¯ z i z j -z i ¯ z j ) . (115) /negationslash These form a o ( N ) algebra and actually generate the following action of O( N ) on the complex N -vector: z i -→ ( Oz ) i = ∑ j O ij z j . (116) It leaves invariant the perimeter and the closure constraint: ∑ i | z i | 2 -→ ∑ jk ∑ i O ij ¯ z j O ik z k = ∑ jk δ jk ¯ z j z k = ∑ i | z i | 2 , ∑ i z 2 i -→ ∑ jk ∑ i O ij z j O ik z k = ∑ jk δ jk z j z k = ∑ i z 2 i . It is interesting that this action is cyclic on the set of vectors satisfying the closure constraint. Indeed starting with the vector ω = (1 , i, 0 , .. 0) with 'unit' perimeter, E = 2, and trivially satisfying the closure constraint, we perform an orthogonal transformation O , with O ij ∈ R and t OO = I : ω i -→ ( Oω ) i = O i 1 + iO i 2 . (117) Thus the orthogonal matrix gives the real and imaginary parts of the complex variables. Reciprocally, starting with a N -vector with unit perimeter E = 2, we write both the fixed perimeter and closure constraints in terms of the real and imaginary parts of the complex coordinates, z i = R i + iI i : ∑ i z 2 i = 0 = ∑ i ( R 2 i -I 2 i ) + 2 i ∑ i R i I i , (118) ∑ i | z i | 2 = 0 = ∑ i ( R 2 i + I 2 i ) , (119) which mean that the real N -dimensional vectors R i and I i are orthonormal, and thus can be identified as the first two columns of an orthogonal matrix, R i = O i 1 and I i = O i 2 . At the end of the day, we will be able to describe averages on the ensemble of polygons as integrals over the orthogonal group. We will go further in this direction, although we can compute similarly to the unitary group case polynomial integrals and a Itzykson-Zuber formula over O( N ). Instead we will focus on the geometrical interpretation of this phase space. C. Reconstructing Polygons Let us describe how we actually go from our complex N -dimensional vector z i satisfying the closure constraint to a real closed polygon (embedded in the flat plane). We would like to interpret the complex variable z i as defining the normal to an edge of the polygon. More precisely, we identify the 2-vector /vectorn i ∈ R 2 normal to the edge i to z 2 i ∈ C = R 2 with the edge length given by the modulus square l i = | z i | 2 . The crucial step of the reconstruction is that we need to (re)-order the edges according to the angle of the normal vector /vectorn i , or equivalently to the phase of z 2 i , so that the angles taken between 0 and 2 π grows with the edge label i . Starting arbitrarily the position /vectorv 1 of the first vertex of the polygon, say on the positive real axis for the sake of simplicity, we reconstruct the next vertex positions /vectorv i from /vectorn i = ( /vectorv i +1 -/vectorv i ) ∧ ˆ e z , or equivalently ( /vectorv i +1 -/vectorv i ) = /vectorn i ∧ ˆ e z , where ˆ e z is the axis orthogonal to the plane. The closure constraint ∑ /vectorn i = 0, equivalent to ∑ z 2 i = 0, ensures that this procedure defines an actual polygon, with /vectorv N +1 = /vectorv 1 . Then we would like to first check that our polygon is convex, i.e. that the angle between two consecutive displacement vectors, or equivalently two consecutive normal vectors, is always at most 180 degrees. Mathematically, this translates to ˆ e z · [( /vectorv i +1 -/vectorv i ) ∧ ( /vectorv i +1 -/vectorv i -1 )] ≥ 0 for all i 's, or equivalently ˆ e z · ( /vectorn i ∧ /vectorn i -1 ) ≥ 0. Since we have ordered all the normals with growing angles, this convexity condition is automatically fulfilled, else the closure constraint can not be satisfied. This concludes he reconstruction procedure for the polygon, which is significantly simpler than for the polyhedron (see e.g. [11]). An interesting feature of our phase space construction is that the normal vectors, and thus the actual geometric polygon, is invariant under the change of sign of individual complex variables z i → -z i . This sign is nevertheless relevant when looking at the action of the orthogonal group on the polygons, i.e two sets of complex variables differing by signs but defining the same polygon will have different images under an orthogonal transformation. This sign 'ambiguity' is the equivalent of the phase of the spinor variables for the polyhedra. Then we similarly introduce the notion of 'signed' polygons, corresponding to 'framed' polyhedra in the 3d case. We expect this sign to be relevant when gluing the polygons together, just as the spinor phases played an essential role when gluing (framed) polyhedra into twisted geometries (encoding the Ashtekar-Barbero connection along the edge [5]). Let us look a bit more into this in the next section. D. Deforming and Gluing Polygons Similarly to the spinor networks introduced as the classical phase space underlying the spin network of loop quantum gravity on a fixed graph [2, 3, 5, 6] and interpreted as twisted geometries, we would like to introduce its two-dimensional equivalent, corresponding to gluing polygons along a given graph. Let us consider an abstract (oriented) closed graph Γ. Around each vertex v of the graph, we will consider one complex variable z v l for each link l attached to v . Reciprocally, for each link l of the graph, we will have two complex variables z s,t l for the two vertices bounding l , for its source and target vertices v = s, t ( l ). We assume the canonical Poisson bracket for each complex variable, plus one closure constraint at each vertex, plus one length matching constraint on each link: { z v l , ¯ z v l } = -i, ∀ v, ∑ l /owner v ( z v l ) 2 = 0 , ∀ l, | z s l | 2 = | z t l | 2 . (120) Geometrically, we have one polygon dual to its vertex. These polygons are then glued together edge by edge along each link of the graph. We call this a 'complex network', where complex stands for the complex variables used on each edge instead of spinors. Let us emphasize that although each polygon is constructed in a fixed plane as a purely two-dimensional object, the glued polygons are not to be thought as in the same plane. Indeed, we can think of each polygon as in its own tangent plane to the overall 2d discrete manifold, with its normal vectors defined in that tangent plane. This can be considered as a toy model for the gluing of polyhedra in 3d and the study of the deformation and dynamics of twisted geometries. There is no shape matching problem as in 3d, where we have an area matching between polyhedra ensuring that the two faces to be glued have the same area but not necessarily the same shape. But we have nevertheless the issue of reconstructing globally the dual cellular complex (i.e the 'triangulation'). A first look easily shows the problems. Let us start with a cellular complex for a 2d manifold, as a set of flat convex polygons glued together, and we consider the graph defined as its dual 1-skeleton and the corresponding complex network living on it encoding the geometrical data of the polygons. If we start modifying the normal vectors around the vertices, still making sure of not changing the closure constraints and the length matching constraints, we can change the angles of the normal vectors around each vertex and nothing a priori ensures that the ordering of the edges remains consistent to the original one and defines the same cellular complex as before. This seems to imply that deforming the complex network can induce a global change of the dual cellular complex (definition of the points dual to the faces/loops of the network) and probably of its topology. An alternative would be to fix the ordering of the edges around each vertex and not modify it while deforming the angles and norms of the normal vectors, thus allowing for the reconstruction of non-convex polygons. We face the same issue(s) in 3d considering the deformations of glued polyhedra and it would probably enlightening to explore the various possibilities and solve these problems in the 2d case studying the dynamics of glued polygons. From this perspective, we plan to report in a separate paper the analysis of the dynamics of these complex networks and the issue of deforming the gluing of polygons. This should involve introducing some Hamiltonian constraints imposing some flatness conditions on the glued polygons and studying the dynamics of the 2d geometry induced by these constraints. Particular care should taken in understanding the role (if any) of the sign ambiguity between the complex variables z 's and the normal vectors /vectorn 's. VI. OUTLOOK: MATRIX MODELS FOR DYNAMICAL POLYHEDRA We would like to finish this paper on the possibility of defining and studying the dynamics of framed polyhedra in the U( N ) framework presented here. We first would like to define the kinetic term, encoding the dynamical degrees of freedom and their Poisson bracket. As the spinor variables have canonical brackets, it is natural to postulate the straightforward kinetic term for them, as assumed in [2]. Then keeping in mind the definition of the spinors in terms of the U( N ) matrix U and the total boundary area 2 λ , z A i = λU iA for the face index i = 1 ..N and the spinor index A = 1 , 2, we can express the kinetic term entirely in terms of the unitary matrix: S kin = ∫ dt -i ∑ k 〈 z k | ∂ t z k 〉 = ∫ dt -iλ T rY U † ∂ t U = ∫ dt + iλ T rUY ∂ t U † , with Y = ( I 2 0 N -2 ) . (121) We then have to define a Hamiltonian and potential. We can not require U( N ) invariance as we would naturally do when dealing with matrix models else our model would collapse to a pure isotropic behavior independent of the unitary matrix U and described only by the dynamics of the total boundary area λ . If we want some dynamics deforming the shape of the polyhedron, a natural possibility 5 to explore is to introduce an external source given for example as a Hermitian matrix X with a non-trivial potential and define the full action in terms of the unitary matrix U : S [ U ] = ∫ dt ( -iλ T rY U † ∂ t U -λ T rY U † XU + V [ X ] ) . (122) The equations of motion are straightforward to compute: λUY U † = ∂ X V, ( i∂ t U + XU ) Y = 0 , (123) with the equation of motion for λ being trivial. The potential V [ X ] should not be taken U( N )-invariant, else the theory would be invariant under the action of the unitary group and thus trivial (with only the global area being dynamical). We should investigate how to choose a physically-relevant potential, for example in relation to cosmological minisuperspaces in quantum gravity (e.g. [32]). This would model the evolution of a given polyhedron, within a twisted geometry, coupled to some external excitation a priori taking into account the interaction of the polyhedron with the rest of the geometry. At the quantum level, this would model the dynamics of an intertwiner with the outside geometry in the context of loop quantum gravity. It would thus be interesting to solve these equations and see the various behavior of the evolution of U in terms of the choice of potential V [ X ]. From this perspective, it would seem possible to model the dynamics of a (framed) polyhedron as a matrix model. It would be interesting to see if the tools of matrix models can be relevant in our framework, especially in the large N limit when we would consider the refinement limit of our polyhedra which we expect to describe some continuous 2d surface (topologically equivalent to a 2-sphere). Conclusion To summarize, we started by explaining how to extend the set of (convex) polyhedra with N faces to a set of framed polyhedra by attaching the extra data of a U(1) phase to each face. This allows to see the set of framed polyhedra (up to 3d rotations) as the symplectic quotient C 2 N // SU(2), defined as the set of collections of N spinors satisfying closure constraints and up to SU(2) transformations. Discussing the various parametrization of this space, we showed that this symplectic manifold is equal to the quotient C 2 N / SL(2 , C ), where we can use a SL(2 , C ) transformation to map any collection of spinors onto one satisfying the closure constraints and thus defining a true geometric polyhedron. Furthermore, following the original work of [7, 9], the space of framed polyhedra can be identified to the Grassmaniann space U( N ) / (SU(2) × U( N -2)) with a natural action of the unitary group U( N ) on framed polyhedra. It is important to emphasize that there is no U( N ) action on polyhedra and that the extra phase attached to each face is essential to the construction. These U( N ) transformations allow to generate any framed polyhedra from the totally squeezed configuration with only two non-trivial faces, and thus allow to go between any two framed polyhedra with equal total boundary area. Such transformations could be instrumental in the study of geometric properties of polyhedron, especially in order to consistently explore the space of polyhedra (either analytically or numerically). Using this U ( N ) structure, we have shown how to compute the average value of geometrical observables, such as polynomials in the area of the faces and the angles between their planes (or normal vectors), can be computed as integrals over the unitary group. We have reviewed various formalisms allowing to compute consistently these polynomial integrals over U( N ). Moreover, we have discussed how the Itzykson-Zuber integral can be used as a generating function for these averages. In short, this formula from matrix models contains all the information about the distribution of polyhedra and their shape with respect to the uniform Haar measure on U( N ). Moving on to the quantum level, we have explained how all the classical features are upgraded automatically upon a canonical quantization of the framed polyhedra phase space. This leads to the Hilbert space of SU(2)-intertwiners (or equivalently SU(2)-invariant states) and one can define semi-classical intertwiner states, that transform coherently under the action of U( N ) and that are peaked on classical framed polyhedra. Then one can compute the trace of polynomial observables. Furthermore, similarly to the classical case, one can use the character formula for U( N ) group elements as a generating function for these polynomial traces and as a extension of the Itzykson-Zuber formula to the quantum case. We provide two different expressions for the U( N )-character, either as a quotient of generalized Vandermonde determinants or as an Gaussian integral over almost-closed configurations of spinors (using the coherent intertwiner formalism). We also showed how we can describe polygons in a similar fashion, trading the unitary group for the orthogonal group and defining a phase space of 'signed' polygons as the Grassmanniann space O( N ) / (SO(2) × O( N -2)). All the same techniques presented for the unitary group and polyhedra can then be straightforwardly translated to polygons. This lower-dimensional toy models allow to discuss more explicitly the geometrical reconstruction of polygons, which is simpler than for polyhedra, and we plan to investigate in the future the details of gluing these polygons together and the definition of consistent dynamics on the resulting 2d discrete manifolds. Quantizing the system, we would then obtain the dynamics of quantum surfaces. The present formalism might also turn out useful in discrete geometry, outside of the realm of quantum gravity, when studying polygons from a purely mathematical point of view. For example, it might be applicable to issues like the search for the largest small polygons, e.g. [33], or other similar problems of geometry. To conclude, we would like to mention a few directions that can be explored based on the present work: · After having understood in details all the kinematics on the space of polyhedra (and polygons), we should move to the study of dynamics along the outline shortly discussed earlier in section VI. In the context of loop quantum gravity, this would mean looking at the dynamics of a fundamental chunk of volume, either at the classical level with a polyhedron or at the quantum level with an intertwiner. In the present framework, it would be most natural to study a deformation dynamics, at fixed number of faces N and fixed total boundary area λ , with the shape of the polyhedron entirely encoded in the unitary matrix U (up to 3d rotations SU(2) and action of the stabilizer group U( N -2)). It could first be interesting to check what would a free evolution on U( N ), of the type U [ τ ] = exp( iτh ) for a fixed Hermitian matrix h , would produce in terms of polyhedra. Then we could deform such an evolution with a non-trivial potential. A second step would be to include a non-trivial dynamics for the total area and number of faces, using the F -operators, to account for an expansion or contraction of the polyhedron. From a physical perspective, it would be interesting to relate such dynamics to cosmological mini-superspace models (as attempted in [32]) or to quantum black hole models. · Here we have developed techniques to compute consistently the average or trace of polynomial observables. We have focused on the area observable, which is well-understood. It would be interesting to apply these methods to a less-understood operator, for instance the volume operator. Indeed the (squared) volume operator is cubic in the normal vectors (or equivalently in the su (2) generators at the quantum level) and determining its full spectrum is a yet-unsolved problem despite great progress [31]. There have been a few very interesting approaches to this issue and hopefully we could get some extra information from the U( N ) approach presented here. · For now, we have focused on a single polyhedron and then a single intertwiner at the quantum level. The next step is to generalize this to bounded regions of twisted geometries, i.e. to look at a bunch of polyhedra glued together and study their algebra of bulk and boundary deformations. This would be relevant for coarse-graining spin networks in loop quantum gravity and investigate the continuum limit of the theory (or at least, define it more rigorously at the kinematical level). Moreover these deformations should somehow be related to the action of diffeomorphisms on the twisted geometries and spin network states. By a quick glance at the corresponding structures, it appears that it will be possible to describe boundary deformations again by U( N ) transformations, but in different representations than used in the case of the single polyhedron. In this context, it seems plausible to be able to describe the boundary dynamics of spin network states as some matrix models, which at a purely speculative level would open a possibility to a link to a conformal field theory description of the boundary of loop quantum gravity (maybe along the CFT description intertwiners already hinted in [17]). Acknowledgment E.L. would like to acknowledge Mehdi Assanioussi for his collaboration on the calculations of entropy and polynomial integrals over the unitary group, as part of his Masters thesis research project at the Laboratoire de Physique ENS Lyon, 'Simple models for black holes in Loop Quantum Gravity' (June 2012, Aix-Marseille university, France). Appendix A: Computing the density of polyhedra and correlations We use the method of [21] to compute the density of polyhedra with N faces and with fixed total area 2 λ and the various averages over the ensemble of such polyhedra. We introduce the following (generating) function: ρ N [ λ ] ≡ 8 π ∫ N ∏ i d 3 /vector V i 4 πV i δ ( ∑ k V k -2 λ ) δ (3) ( ∑ k /vector V k ) . (A1) Following [21], we Fourier-transform both sets of constraints and perform the integrals over the normal vectors /vector V k (assuming /epsilon1 > 0) : ρ N [ λ ] = 8 π e 2 /epsilon1λ ∫ dq 2 π ∫ d 3 /vectoru (2 π ) 3 ∫ N ∏ i d 3 /vector V i 4 πV i e -/epsilon1V i e iqV i e i/vectoru · ∑ i /vector V i = 8 π e 2 /epsilon1λ ∫ dq 2 π ∫ d 3 /vectoru (2 π ) 3 I ( q, /vectoru ) N (A2) with I ( q, /vectoru ) = ∫ d 3 /vector V 4 πV e -/epsilon1V e iqV e i/vectoru · /vector V . The kernel I ( q, /vectoru ) converges due to the regulator /epsilon1 > 0. We first integrate over the angular part of /vector V and then over its norm: I ( q, /vectoru ) = ∫ + ∞ 0 V 2 dV V e -/epsilon1V e iqV ∫ S 2 d 2 ˆ V 4 π e i/vectoru · /vector V = ∫ + ∞ 0 V dV e -/epsilon1V e iqV sin uV uV = 1 u 2 -( q + i/epsilon1 ) 2 . (A3) This is exactly as the Feynman propagator in quantum field theory where /vectoru plays the role of the momentum and q the role of the mass. We then perform the integrals over /vectoru and finally over q : ρ N [ λ ] = 8 π e 2 /epsilon1λ ∫ R dq 2 π e -2 iqλ ∫ + ∞ 0 4 πu 2 du (2 π ) 3 1 ( u 2 -( q + i/epsilon1 ) 2 ) N = e 2 /epsilon1λ ∫ R dq 2 π e -2 iqλ (2 N -4)! ( N -1)!( N -2)! ( /epsilon1 -iq ) 3 -2 N 2 2 N -4 = λ 2 N -4 ( N -1)!( N -2)! . (A4) And we recover the formula (37) for the volume of the space of (framed) polyhedra with N faces and fixed total area. To extract the average of the norm of a normal vector, or its powers, we can differentiate with respect to q . For instance, we have: 〈 V i 〉 = 8 π e 2 /epsilon1λ ρ N ∫ d 3 /vectoru (2 π ) 3 dq 2 π e -2 iqλ I ( q, /vectoru ) N -1 ˜ I ( q, /vectoru ) , (A5) with a modified kernel taking into account the insertion of the observable V i , ˜ I ( q, /vectoru ) = ∫ d 3 /vector V 4 πV V e -/epsilon1V e iqV e i/vectoru · /vector V = -i∂ q I ( q, /vectoru ) = 2( /epsilon1 -iq ) ( u 2 -( q + i/epsilon1 ) 2 ) 2 , which allows to get the expected result: 〈 V i 〉 = 1 ρ N 2 λ 2 N -3 N !( N -2)! = 2 λ N . (A6) /negationslash One can check as a consistency check that ∑ i,j 〈 V a i V b j 〉 = 0 as expected from the closure constraints. Furthermore, one interesting observable is the second moment Θ ab = ∑ i V a i V b i -1 3 δ ab V i V i , which characterizes the shape of the polyhedra and its deviation from the isotropic spherical distribution. From the result above, we have the trivial average 〈 Θ ab 〉 = 0. However, using the technique presented here, one can compute the width(s) or uncertainty of the distribution of the tensor Θ ab around its vanishing mean value. This is rather lengthy calculation although straightforward, which we do not detail here. The final result is: One can go further and compute averages of higher powers of the norm V by differentiating more. For instance, in order to compute 〈 V 2 〉 , one insert the modified kernel [21]: ˜ ˜ I ( q, /vectoru ) = ∫ d 3 /vector V 4 πV V 2 e -/epsilon1V e iqV e i/vectoru · /vector V = -∂ 2 q I ( q, /vectoru ) = -2 ( u 2 -( q + i/epsilon1 ) 2 ) 2 + 8( /epsilon1 -iq ) 2 ( u 2 -( q + i/epsilon1 ) 2 ) 3 . Using this technique, one can also compute the correlations between the norm of two different faces, by modifying the product of kernels in the integral from I N to I N -2 ˜ I 2 . Similarly, one differentiate with respect to the components of the vector /vectoru in order to insert observables depending on the components of the normal vectors /vector V . For instance, we can compute the correlations 〈 V a i V b j 〉 as: ρ ab i ≡ 〈 V a i V b i 〉 = 8 π e 2 /epsilon1λ ρ N ∫ d 3 /vectoru (2 π ) 3 dq 2 π e -2 iqλ I ( q, /vectoru ) N -1 ( -∂ u b ∂ u a I ( q, /vectoru )) = 2 δ ab λ 2 N ( N +1) , (A7) ρ ab i = j ≡ 〈 V a i V b j 〉 = 8 π e 2 /epsilon1λ ρ N ∫ d 3 /vectoru (2 π ) 3 dq 2 π e -2 iqλ I ( q, /vectoru ) N -2 ( -i∂ u b I ( q, /vectoru ))( -i∂ u a I ( q, /vectoru )) = -2 δ ab λ 2 N ( N 2 -1) . (A8) 〈 Θ ab Θ cd 〉 = λ 4 4 ( 4( N 2 + N -2) δ ab δ cd -6( N -1)( δ ac δ bd + δ ad δ bc ) ) 3( N -1) N ( N +1)( N +2)( N +3) . (A9) [1] C. Rovelli, Zakopane lectures on loop gravity , arXiv:1102.3660; P. Don & S. Speziale, Introductory lectures to loop quantum gravity , arXiv:1007.0402; H. Sahlmann, Loop quantum gravity - a short review , arXiv:1001.4188; M. Gaul and C. Rovelli, Loop Quantum Gravity and the Meaning of Diffeomorphism Invariance , Lect.Notes Phys. 541 (2000) 277-324 [arXiv:gr-qc/9910079] [2] E.F. Borja, L. Freidel, I. Garay, E.R. Livine, U(N) tools for Loop Quantum Gravity: The Return of the Spinor , Class.Quant.Grav.28 (2011) 055005 [arXiv:1010.5451] [3] E.R. Livine and J. Tambornino, Spinor Representation for Loop Quantum Gravity , J. Math. Phys. 53, 012503 (2012) [arXiv:1105.3385]; E.R. Livine and J. Tambornino, Loop gravity in terms of spinors , Proceedings of Loops '11 Conference (Madrid, 2011) [arXiv:1109.3572] [4] E.R. Livine and J. Tambornino, Holonomy Operator and Quantization Ambiguities on Spinor Space , to appear in Phys.Rev.D 2013 [arXiv:1302.7142] [5] L. Freidel and S. Speziale, Twisted geometries: A geometric parametrisation of SU(2) phase space , Phys.Rev.D82 5 (2010) 084040 [arXiv:1001.2748] [6] L. Freidel and S. Speziale, From twistors to twisted geometries , Phys.Rev.D82 (2010) 084041 [arXiv:1006.0199] [7] L. Freidel and E.R. Livine, The Fine Structure of SU(2) Intertwiners from U(N) Representations , J.Math.Phys.51 (2010) 082502 [arXiv:0911.3553] [8] M. Kapovich and J.J.Milson, The Symplectic Geometry of Polygons in Euclidean space , [9] L. Freidel and E.R. Livine, U(N) Coherent States for Loop Quantum Gravity , J. Math. Phys. 52 (2011) 052502 [arXiv:1005.2090] [10] C. Itzykson and J.B. Zuber, The planar approximation II. , J. Math. Phys. 21 (1980) 411 [11] E. Bianchi, P. Dona and S. Speziale, Polyhedra in loop quantum gravity , Phys.Rev.D83 (2011) 044035 [arXiv:1009.3402] [12] M. Dupuis and E.R. Livine, Holomorphic Simplicity Constraints for 4d Spinfoam Models , Class. Quant. Grav. 28 (2011) 215022 [arXiv:1104.3683] [13] M. Dupuis and E.R. Livine, Revisiting the Simplicity Constraints and Coherent Intertwiners , Class.Quant.Grav.28 (2011) 085001 [arXiv:1006.5666] [14] F. Girelli and E.R. Livine, Reconstructing Quantum Geometry from Quantum Information: Spin Networks as Harmonic Oscillators , Class.Quant.Grav. 22 (2005) 3295-3314 [arXiv:gr-qc/0501075] [15] M. Dupuis, L. Freidel, E.R. Livine and S. Speziale, Holomorphic Lorentzian Simplicity Constraints , J. Math. Phys. 53 (2012) 032502 [arXiv:1107.5274] [16] F. Conrady and L. Freidel, Quantum geometry from phase space reduction , J.Math.Phys.50 (2009) 123510 [arXiv:0902.0351] [17] L. Freidel, K. Krasnov and E.R. Livine, Holomorphic Factorization for a Quantum Tetrahedron , Commun.Math.Phys.297 (2010) 45-93 [arXiv:0905.3627] [18] V. Bonzom and E.R. Livine, Generating Functions for Coherent Intertwiners , Class. Quantum Grav. 30 (2013) 055018 [arXiv:1205.5677] [19] E. Bianchi and H.M. Haggard, Discreteness of the volume of space from Bohr-Sommerfeld quantization , Phys.Rev.Lett.107 (2011) 011301 [arXiv:1102.5439]; E. Bianchi and H.M. Haggard, Bohr-Sommerfeld Quantization of Space , arXiv:1208.2228 [20] E. Bianchi, Black Hole Entropy, Loop Gravity, and Polymer Physics , Class.Quant.Grav.28 (2011) 114006 [arXiv:1011.5628] [21] E.R. Livine and D.Terno, Entropy in the Classical and Quantum Polymer Black Hole Models , Class. Quantum Grav. 29 (2012) 224012 [arXiv:1205.5733] [22] J. Engle, K. Noui, A. Perez and D. Pranzetti, Black hole entropy from an SU(2)-invariant formulation of Type I isolated horizons , arXiv:1006.0634; J. Engle, K. Noui, A. Perez and D. Pranzetti, The SU(2) Black Hole entropy revisited , JHEP 1105 (2011) 016 [ arXiv:1103.2723] [23] R. Kaul, Entropy of Quantum Black Holes , SIGMA 8 (2012) 005 [arXiv:1201.6102]; J. Diaz-Polo and D. Pranzetti, Isolated Horizons and Black Hole Entropy in Loop Quantum Gravity , SIGMA 8 (2012) 048 [arXiv:1112.0291] [24] P. Hayden, D.W. Leung and A. Winter, Aspects of generic entanglement , Comm. Math. Phys. 265 (2006) 1, 95-117 [arXiv:quant-ph/0407049] [25] B. Collins, Moments and Cumulants of Polynomial random variables on unitary groups, the Itzykson-Zuber integral and free probability , Int. Math. Res. Not. (17) (2003) 953-982 [math-ph/0205010] [26] F.G.S.L. Brand˜ao, P. C'wiklin'ski, M. Horodecki, P. Horodecki, J. Korbicz, and M. Mozrzymas, Convergence to equilibrium under a random Hamiltonian , Phys. Rev. E 86 (2012) 031101 [arXiv:1108.2985] [27] C. Spengler, M. Huber and B.C. Hiesmayr, A composite parameterization of unitary groups, density matrices and subspaces , J. Phys. A: Math. Theor. 43 (2010) 385306 [arXiv:1004.5252]; C. Spengler, M. Huber and B.C. Hiesmayr, Composite parameterization and Haar measure for all unitary and special unitary groups , J. Math. Phys. 53 (2012) 013501 [ arXiv:1103.3408] [28] C. Spengler, M. Huber, A. Gabriel and B.C. Hiesmayr, Examining the dimensionality of genuine multipartite entanglement , Quantum Inf. Process. 12 (2013) 269 [arXiv:1106.5664]; , C. Spengler, M. Huber, S. Brierley, T. Adaktylos and B.C. Hiesmayr, Entanglement detection via mutually unbiased bases Phys. Rev. A 86 (2012) 022311 [arXiv:1202.5058] [29] V. Bonzom and E.R. Livine, A new Hamiltonian for the Topological BF phase with spinor networks , J. Math. Phys. 53 (2012) 072201 [arXiv:1110.3272] [30] E.R. Livine and S. Speziale, A new spinfoam vertex for quantum gravity , Phys.Rev.D76 (2007) 084028 [arXiv:0705.0674] [31] J. Brunnemann and T. Thiemann, Simplification of the Spectral Analysis of the Volume Operator in Loop Quantum Gravity , Class.Quant.Grav. 23 (2006) 1289-1346 [arXiv:gr-qc/0405060]; Discreteness of the volume of space from Bohr-Sommerfeld quanti- K.A. Meissner, Eigenvalues of the volume operator in loop quantum gravity , Class.Quant.Grav. 23 (2006) 617-626 [arXiv:gr-qc/0509049] E. Bianchi and H. Haggard, zation , hys.Rev.Lett.107 (2011) 011301 [arXiv:1102.5439] [32] E.R. Livine and M. Martin-Benito, Classical Setting and Effective Dynamics for Spinfoam Cosmology , Class. Quantum Grav. 30 (2013) 035006 [arXiv:1111.2867] [33] C. Audeta, P. Hansenb, F. Messined and J. Xionge, The Largest Small Octagon , Journal of Combinatorial Theory, Series A, 98-1 (2002) 46?59, http://dx.doi.org/10.1006/jcta.2001.3225 Choosing appropriate matrices X and Y , this Itzykson-Zuber formula can be seen as the generating function for all the correlations between the normal vectors over our polyhedron ensemble. In our case, let us give an example with the observable V i and its powers. We have: The matrix Y is fixed and implements the reduction from U( N ) to our space of polyhedron U( N ) / U( N -2). The matrix X selects the considered observables. Then the mean value 〈 exp( iθV i ) 〉 is a Itzykson-Zuber integral: where c is a normalization constant such that 〈 1 〉 = 1 for θ = 0. The trick to derive this formula is to regularize the Itzykson-Zuber formula by shifting slightly all the eigenvalues of X and Y to ensure that they are different and then to send these regulators to 0 at the end. Then this result gives us directly all the mean values 〈 ( V i ) n 〉 , without having to suitably differentiate the density of state ρ N [ λ ] as in section III B or compute the polynomial U( N ) integrals as in section III C: which matches our expressions already derived for 〈 V 〉 and 〈 V 2 〉 . We can compare them to the free model without closure constraints as introduced earlier in section III B, which had the following averages (59): First, we notice that these are different (though similar), showing that the two models are clearly distinct and have a different probability distribution for the individual face areas. Second, as claimed earlier, the two expressions nevertheless match at large N for a fixed power n : We can go further and get the formula for the fixed matrix Y but for arbitrary matrix X . We perturb around the actual eigenvalues of Y as y 1 = 1 + /epsilon1 1 , y 2 = 1 + /epsilon1 2 and y k ≥ 3 = /epsilon1 k . Both numerator and denominator of the ItzyksonZuber vanish as all the /epsilon1 i are set to 0. We can nevertheless suitably differentiate both numerator and denominator (using L'Hˆopital rule) until we reach non-vanishing values, here ∂ ( N -2) /epsilon1 N ∂ ( N -3) /epsilon1 N -1 ..∂ /epsilon1 3 ∂ /epsilon1 2 . This leads to for N ≥ 4: The numerator is a modified Vandermonde determinant (but vanishes when θ = 0) while the denominator comes from differentiating the original Vandermonde determinant ∆( Y ) (it is also the determinant of the ( N -2) × ( N -2) matrix whose matrix elements are given by m ij = ∏ i k =1 ( k + j )). This provides a direct formula for the observables ∑ i x i V i for a diagonal matrix X : When the matrix X is arbitrary and not diagonal, its off-diagonal components allows us to probe the correlations between the various spinors z i : Then the Itzykson-Zuber integral can be understood as the generating function for the averages and correlations of the spinor scalar products. From these and taking into account that the vector scalar product is related to the spinor scalar product, |〈 z i | z j 〉| 2 = V i V j + /vector V i · /vector V j , we can extract in principle all the averages and correlations of the SU(2)invariant polynomials in the vector components /vector V a i . It would be interesting to apply these techniques to computing the averages of the powers of the (squared) volume observable, in order to get a better idea of the typical shape of polyhedra, but also because the exact spectrum of the (squared) volume operator at the quantum level is still an open issue. Thus we have seen how the Itzykson-Zuber integral over U( N ) expressed in terms of Vandermonde determinants can be considered as the generating function for the averages of all polynomial observables in the polyhedra's normal vectors. These averages are extracted by suitable differentiating of this Itzykson-Zuber formula. An interesting point is whether the Itzykson-Zuber integrant e iθ T rY U † XU for the fixed considered Y but arbitrary X can have a physical or geometrical relevance, for instance when investigating some (random) dynamics on the space of (framed) polyhedra. We leave this for future investigation. = k ≥ 2", "pages": [ 16, 17, 18 ] }, { "title": "E. Explicit U( N ) Parametrization and Haar Measure", "content": "We now turn to another method to compute these integrals over U( N ) using an explicit parametrization of the unitary matrices and the corresponding recursive formula for the Haar measure on U( N ) [27]. The goal is to draw a unitary matrix at random with respect to the Haar measure, or more precisely to draw at random its two first columns, that is two ortogonal complex N -vectors of unit norm. The details of the parametrization and construction for the whole unitary matrix can be found in [27]. Here, we will only detail the parametrization of the two first columns and thus of the spinors defining the polyhedra with N faces. The parametrization is best defined recursively. We start with the case N = 2. Two arbitrary orthogonal complex 2-vectors of unit norm can be written as: where the phases θ 1 , θ 2 and φ 2 live in [0 , 2 π ] while the rotation angle α 2 's range is [0 , π 2 ]. The normalized Haar measure then reads: The components of the two spinors are read directly from these complex vectors: This provides a parametrization of a unitary matrix in U(2) as expected. Then we can define the two complex vectors v ( N ) and w ( N ) recursively from v ( N -1) and w ( N -1) as: where we have added four new parameters, θ N , φ N ∈ [0 , 2 π ] and α N , β N ∈ [0 , π 2 ]. The normalized Haar measure is now: (2 π ) - 2 N - 1 1) with N n ∏ ∏ We can read the components of the N spinors directly from these two complex vectors, up to the global scale factor √ λ . In total, we have parametrized our spinors using (4 N -4) angles α k , β k , θ k , φ k plus λ . These are (4 N -3) parameters, exactly the dimension of the space of N spinors satisfying the closure constraints. If we want to further gauge fix the SU(2) invariance, we can fix the direction of the last vector /vector V N . In terms of the components of the last spinor, z N = e iθ N (sin α N , e iφ N cos α N sin β N ), this amounts to fixing φ N = α N = β N = 0. Fixing these three parameters, this provides an explicit parametrization of the (4 N -6)-dimensional space P z N of framed polyhedra up to 3d rotations. 2( k If we consider the first vector v ( N ) , we can give its full expression: This gives actually a random vector on the complex unit sphere in C N , distributed uniformly with respect to the Haar measure on U( N ). It is well known that there is a phenomenon of concentration of measure on the complex k ≥ 3 ) 2( k - 2) . sphere as N grows, e.g. [24]. More precisely, the integral over the complex sphere is almost equal to the simpler integral over the equator of the sphere (for α N = 0). This is due to the specific shape of the Haar measure in this parametrization, which gets concentrated to the equator as N grows large. In the context of quantum information (and quantum computing), this concentration of measure is often used to argue that arbitrary states are maximally entangled between subsystems as the dimensions of the Hilbert spaces grows large, e.g. [24, 28]. Here we are drawing a second complex vector w ( N ) , which is orthogonal to the first one. It would be interesting to investigate whether there is a similar phenomenon of concentration of measure and what would be its geometrical interpretation on the space of (framed) polyhedra. We postpone such analysis to future investigation. Nevertheless, this explicit parametrization does provide a very useful tool in order to compute the average of any polynomial observable over the space of polyhedra as an explicit trigonometric integral.", "pages": [ 18, 19 ] }, { "title": "IV. DEFORMING QUANTUM POLYHEDRA", "content": "This section is dedicated to the study of the quantum case: we quantize the space of framed polyhedra into the Hilbert space of SU(2) intertwiners interpreted as quantum polyhedra, following the previous work done in [2, 9, 12, 13]. We will see that the Hilbert space of quantum polyhedra has the same structure as the classical set of framed polyhedra. We have indeed a cyclic action of the U( N ) transformations on quantum polyhedra with fixed total boundary area and we can construct coherent polyhedron state labeled by the classical framed polyhedra (up to 3d rotations). Finally, we will give two ways to write the trace of geometrical operators: either using the U( N ) character formula, which is interpreted as the quantum counterpart of the Itzykson-Zuber integral formula or using the coherent states and having an integral over 'fuzzy' polyhedra.", "pages": [ 19 ] }, { "title": "A. Quantizing Polyhedra into Intertwiners", "content": "We canonically quantize the space of spinors C 2 N by promoting the components of the spinors and their complex conjugate to harmonic oscillators: where we have taken the convention /planckover2pi1 = 1. As shown and used in [2, 9, 12, 13] (see also [3, 4, 29]), the closure constraints C a generating the SU(2) action on spinors, the U( N ) generators E ij and the SU(2)-invariant observables F ij are all quantized without ambiguity and their algebra at the quantum level is without any anomaly. We consistently choose the normal ordering, keeping the annihilation operators a 0 , 1 to the right and the creation operators a 0 , 1 † to the left. For details, the interested reader can refer to those references. We will nevertheless give here a quick summary of the main structures, relevant to our main point, that is the U( N ) action on SU(2) intertwiners. For the sake of completeness, we give the expressions of the basic operators, which are all quadratic in the harmonic oscillators. When there can be no confusion, we will not distinguish the classical quantity from the quantum operator, else we will put a hatˆon the quantum operator. For the SU(2) generators, we have C a = ∑ i V a i with: These form on each face i the Schwinger representation of the su (2) algebra in terms of two harmonic oscillators. We also introduce the operator giving the total energy of the oscillators living on the face i as the quantization of the norm of the normal vector V i : As well-known, this SU(2) representation is reducible and irreducible components are obtained by diagonalizing the Casimir operator V i , whose eigenvalues are twice the spin living on that face, 2 j i ∈ N . This is interpreted as usual as the quantization of the individual face areas. We similarly quantize the spinor scalar products: It is straightforward to compute the commutators of these operators and check that they give the same results as their Poisson brackets. In particular, the E 's and F 's commute with the closure constraint operators C a and thus are SU(2)-invariant. Moreover the E ij form a closed u ( N ) algebra. Using the definition in terms of harmonic oscillators, the Casimir of this u ( N ) algebra is easily related to the total area [7]: Looking at the Hilbert spaces, we start with 2 N copies of the Hilbert space of a single harmonic oscillator, ( H HO ⊗ H HO ) ⊗ N = L 2 ( C 2 N ). Each couple ( H HO ⊗ H HO ) can be decomposed in irreducible representations of SU(2) with arbitrary spin j ∈ N / 2 (given by half the total number of quanta of the oscillators). Then we impose a SU(2)invariance by requiring that the closure constraint operators C a = ∑ i V a i vanish on the states. This is exactly the Hilbert space of SU(2) intertwiners between N irreducible representations: where we write V j for the irreducible SU(2)-representation of spin j . On this Hilbert space of intertwiners, we have a U( N ) action generated by the E ij . Since the corresponding u ( N )-Casimir ∑ i,j E † ij E ij is determined in terms of the total area operator E whose value is simply the sum of twice the spins ∑ N i (2 j i ), we can simply decompose the space H ( N ) in irreducible components by fixing the value of the total area: where each Hilbert space R J carries an irreducible representation of U( N ), as shown in [2, 7, 9]. The corresponding Young tableaux is given by two horizon lines of equal length J . The corresponding highest weight vector | ψ J 〉 corresponds to a bivalent intertwiner, which is the quantum equivalent of the completely squeezed polyhedron in the classical case: where the E ii = V i are the generators of the Cartan subalgebra. In particular, we notice that this highest weight vector is invariant under U( N -2), which corresponds to the expectation that the classical space of framed polyhedra is isomorphic to the Grassmanniann space U( N ) / (U( N -2) × SU(2)). The dimension of each of these irreducible U( N )-representations can be computed using the hook formula. This gives: This is the total number of SU(2)-intertwiners for a fixed number of faces N and fixed total area 2 J = ∑ i 2 j i . It is the quantum counterpart of the density of states ρ N [ λ ], which gives the volume of the space of framed polyhedra with N faces and total area 2 λ . Indeed, looking at the large area limit while keeping N fixed, gives: which fits at leading order in J with ρ N [ λ ], as given by (37), for λ = J . Notice that all the terms have the same order in N . Therefore, this limit can be considered carefully. To be more rigorous, one should put the /planckover2pi1 -factors back in the quantum expression, then this is the limit where the Planck area unit is sent to 0, while keeping the total area fixed. Then this amounts to sending the sum of the spin to ∞ , thus giving the wanted result. To summarize the structures, the vector operators V i acts on each subspace V j i living on each face and generate the SU(2)-action on those subspaces. The SU(2)-invariant operators E ij act on each subspace R J N , defined as the space of SU(2) intertwiners for fixed sum of the spins J = ∑ i j i , and they generate a U( N )-action on each of these subspaces. Finally the F ij and F † ij operators respectively act as annihilation and creation operators on the full space of intertwiners H ( N ) allowing to respectively decrease and increase the total area J . These SU(2)-intertwiners are the quantum counterpart of the classical (framed) polyhedra. They are also the basic building blocks of the spin network states of quantum (space) geometry in loop quantum gravity [1]. That identification of intertwiners as quantum polyhedra is the key to the geometrical interpretation of spin network as discrete geometries constructed as (quantum) polyhedra glued together. this identification will be made even clearer below when dealing with coherent intertwiner states peaked on classical framed polyhedra.", "pages": [ 19, 20, 21 ] }, { "title": "B. Beyond Intertwiners: non-Closed Quantum Polyhedra", "content": "Considering the tensor product of N representations of SU(2), one for to each face of the polyhedron, we have imposed up to now the closure constraint and thus required invariance of our tensor product states under SU(2). We can relax this condition and characterize states that recouple to a fixed overall spin J different from 0. This corresponds to the classical case where the closure constraints are broken and the sum of the normal vectors do not vanish but the closure vector has a fixed norm. We are now working on another subspace of ( H HO ⊗H HO ) ⊗ N = L 2 ( C 2 N ), such that the value of the SU(2)-Casimir given as the norm squared of the closure constraint operators C 2 is fixed to J ( J +1): This is actually equivalent to having intertwiners, i.e SU(2)-invariant states, between the N original irreducible representations V j i and an extra one V J . We still have the U( N )-action on this Hilbert space H ( N ) J and we can decompose it into U( N ) irreducible representations: where the total area J is of the same parity as the overall spin J (i.e half-integer or integer depending on J ) and necessarily larger or equal to J . Each of the subspaces at fixed J carries an irreducible representation of U( N ). Its highest weight vector is defined by the (unique) trivalent intertwiner between SU(2)-representations of spins J + J 2 , J -J 2 and J , i.e the values of the Cartan subalgebra generators on it are [7]: Thus the corresponding Young tableaux contains two horizontal lines of respective lengths ( J + J ) and ( J -J ) and the dimensions of the representations are [7]: It is fairly easy to check that summing over all possible values of J ≤ J , we recover the full Hilbert space of intertwiners for ( N +1) faces and fixed total area J : This could be proved directly either by recombining the binomial coefficients or by recursion. Finally, it would be interesting to investigate whether there is a similar procedure to 'close' non-invariant configuration as in the classical case, where we could apply a SL(2 , C ) transformations on an arbitrary non-closed set of spinors in order to map it into a closed set of spinors defining an actual framed polyhedron. We postpone to future investigation the thorough study of the existence on a SL(2 , C )-action on the space of intertwiners and of its properties.", "pages": [ 21 ] }, { "title": "C. Probing the shape of Intertwiners", "content": "Similarly to the classical case, we now would like to compute the traces of geometrical operators on the Hilbert space of SU(2)-intertwiners at fixed number N of faces and fixed total area J = ∑ i j i . We can already deduce some averages from the fixed area condition J = ∑ i j i and the formula for the dimensions of the intertwiner spaces d N [ J ]. We obviously have: which is also equal to the classical average (42). We can also single out explicitly one face/leg of the intertwiner. Then using the dimension of the space of tensor product states for a fixed external spin (or intertwiners with one fixed spin) given in the previous section, we compute: We see that the first term in 〈 V 2 i 〉 fits exactly the classical average (44). The second term is the quantum correction, and is sub-leading in the classical limit defined by taking large J at fixed N . Playing around with the binomial coefficients, one can show the somewhat surprising formula giving the traces of arbitrary powers of the norm: from which we can recover the traces 〈 V i 〉 and 〈 V 2 i 〉 . We can square the fixed area condition and deduce the correlation between spins i = k : /negationslash /negationslash Similarly, using the closure constraint operator, or in other words the SU(2)-invariance, we can compute: /negationslash /negationslash If we want to go further and compute traces of operators involving the values of the spins on three or more legs and thus probing the fine structure of the intertwiners, we would have to compute the dimensions of the intertwiner subspaces with fixed spins. Instead of doing this by hand, we can do this consistently using the full U( N )-character formula, which computes the trace of U( N ) transformations instead of simply the dimension which gives the trace of the identity. This the method outlined in [7] and we show here that it should be considered as a generalization to the quantum case of the Itzykson-Zuber formula used as generating function for averages over the ensemble of classical polyhedra. More precisely, the character of the U( N ) representation, of highest weight [ l 1 , ..l N ], computes the trace of a diagonalized unitary transformation U = ( e iθ 1 , .., e iθ N ) as a Schur polynomial: /negationslash Here, the highest weight is given by l 1 = l 2 = J and this formula defines directly the generating functions for the spin expectation values (or equivalently the V i = 2 j i ): where the normalization should be such that the expectation value of 1 is 1 (when θ i = 0). The determinant at the denominator is exactly a Vandermonde determinant, while the numerator is a slight modification. This formula contains all the traces of polynomials in the spins j i 's. If we extend the formula to non-diagonal U( N ) transformations (which we can diagonalize of course), we can generate the traces of all scalar products and powers in the basic vectors /vector V i . As in the classical case, extracting these traces requires a careful differentiation of this generating function. It would be interesting if these traces of U( N ) transformations could themselves be physically/geometrically relevant, for instance in the study of the dynamics of polyhedra and intertwiners.", "pages": [ 22, 23 ] }, { "title": "D. Interpolating between Classical and Quantum Polyhedra: Coherent Intertwiner States", "content": "To better understand the link between intertwined states and classical polyhedra, we can build coherent intertwined states peaked on classical framed polyhedra following [9, 12, 13]. Following the conventions of [12, 13], one defines: Definition IV.1. Given a set of spinors z i ∈ C 2 N , we define the coherent intertwiner state | J, { z i }〉 in R J N using the SU(2) creation operators F † : The scalar products [ z i | z j 〉 are invariant under SU(2) rotations, so the intertwiner states are labeled by the orbits of spinors under global SU(2) transformations. Moreover, these scalar products are also invariant under global SL(2 , C ) transformations, which map arbitrary sets of spinors to spinors satisfying the closure constraints. Thus the coherent intertwiner states are truly labeled by orbits of spinors under global SL(2 , C ) transformations, that is points in the space of framed polyhedra (up to 3d rotations) P z N = C 2 N / SL(2 , C ) = C 2 N // SU(2) as we have seen in section II D. The main results established in [9], and revisited in [12, 13], are two key properties of these intertwiner coherent states: their formulation as group averaging of the tensor product of standard SU(2) coherent states, which establishes their geometrical interpretation as semi-classical polyhedron states, and then their coherence under the action of U( N ). Or more precisely,", "pages": [ 23 ] }, { "title": "· Decomposition on SU(2) coherent states:", "content": "where we group average the tensor product of individual SU(2) coherent states living on each face and defined as: These states living in V j are coherent under the action of SU(2) and thus can all be generated from the highest weight vector | j, j 〉 by acting with SU(2) transformations (up to a norm factor): Finally, they are peaked on the classical vectors /vector V ( z ) = 〈 z | /vectorσ | z 〉 : where the expectation value vector has the same direction as /vector V but is normalized to 2 j in term of the spin carried by the state. The group average states for fixed individual spins j i were introduced earlier in [30]. Written as such, the coherent intertwiner states | J, { z i }〉 truly represent the quantized version of a classical framed polyhedron defined as a set of N vectors or spinors up to SU(2) transformations.", "pages": [ 23, 24 ] }, { "title": "· Coherence under U( N ) transformations:", "content": "The action of U( N ) transformations, generated by the operators ˆ E ij at the quantum level, on the coherent intertwiner states amounts to the classical U( N )-action on the set of spinors labeling the state: for an arbitrary Hermitian matrix h . This ensures that the behavior of coherent intertwiner states is just the same as classical framed polyhedra. For instance, one can generate all coherent intertwines by acting with U( N ) transformations on the bivalent intertwiner, just the same way as we could generate all (closed) framed polyhedra by acting with U( N ) transformations on the totally squeezed configuration with only two non-trivial faces. This is the key property allowing us to take the trace over the Hilbert space of intertwines by an integral over the unitarity group U( N ), similarly to the classical case. This is explained below in details. Taking into account that the Hilbert space R J N of intertwiners for fixed total sum of the spins is an irreducible representation of U( N ), one can write the identity of that space as an integral over U( N ) acting on a fixed state, say the bivalent intertwiner on the legs 1 and 2, which is exactly the integral over the coherent intertwiner states: A rigorous proof can be found in [9], and then a simpler proof in [12, 13]. Basically, this comes from writing the identity on the larger Hilbert space H ( N ) in terms of the usual coherent states for the harmonic oscillators and then projecting down on the subspace with fixed total number of quanta J . We also compute the scalar product between two coherent intertwiners [9]: For a single set of spinors, this also gives the norm of the coherent intertwiner state: We clearly see that the norm is maximized when the closure vector vanishes, /vector C = ∑ i /vector V i = 0, and that we have a closed set of spinors thus corresponding to a true polyhedron. This is similarly to the result obtained in [30], which studied the saddle point approximation to the group averaging for fixed spins j i and show that a stationary point exists only if the vectors satisfy the closure constraints. Combining these two formula, we can take the trace of the identity on R J N and recover the dimension of this Hilbert space, which we can express either as a Gaussian integral over the spinors z i or as an integral over the vectors /vector V i : It is possible to check this formula directly by performing the integral over the vectors /vector V i as done in [21]. This provides a interpretation of the space of intertwiners as almost-closed polyhedra, or fuzzy polyhedra, which are more and more peaked on true framed polyhedra (satisfying the closure constraints) as the total area J grows. This fits with our earlier claim that the dimension of the Hilbert space R J N behaves at leading order as the classical density of framed polyhedra as the total spin J grows large for a fixed number of faces N . We can know further the compute the trace of any unitary transformation, which provides an integral formula for the U( N )-character given in terms of modified Vandermonde determinant in the previous section: with U = e ih and ˆ U = e i ∑ kl h kl ˆ E kl in terms of a Hermitian matrix h as above. In general, the determinant is a little messy: from which we can compute the trace of ˆ U as Gaussian integral in the spinor variables. It is however simpler when looking at diagonal unitary transformations U = e i ∑ k θ k E k , which act as multiplication by individual phases on each spinor: These traces are to be considered as the generating function for the traces of every polynomial operators in the E 's or F 's or V 's. Beyond this, it would be interesting to investigate if these unitary transformations can be seen as implementing the dynamics of the intertwiners for fixed boundary area (for instance, in the context of quantum black holes, see e.g. [21]), in which case these traces could be provided with a direct physical interpretation. On the other hand, we could also apply our machinery to other geometric operators. For example, an open issue is still to compute the exact spectrum of the (squared) volume operator (see nevertheless e.g. [31]) and we could attempt to compute the traces of the powers of this operator, from which we could reconstruct its spectrum.", "pages": [ 24, 25 ] }, { "title": "V. ORTHOGONAL GROUP ACTION ON POLYGONS", "content": "We will now look at structures in one dimension less and study the space of polygons. We will see that we can similarly define a phase space of framed polygons, also the frame now on each edge will be reduced to a sign ± . Similarly to the case of polyhedra, we will characterize the space of framed polygons for fixed boundary perimeter as a representation of the orthogonal group O( N ) instead of the unitary group. Working in 2d instead of 3d will also us to be more explicit in the reconstruction of the geometrical structure especially on the issue of gluing polygons together to form a two-dimensional discrete manifold.", "pages": [ 25 ] }, { "title": "A. Phase Space for Polygons", "content": "The phase space structure for polygons is much simpler than for polyhedra. Instead of using spinors attached to each face of the polyhedron, we will attach a single complex variable to each edge of the polygon. Let us thus start with { z i } ∈ C N with i = 1 ..N for polygons with N edges and postulate the following canonical Poisson bracket: This corresponds to a set of N oscillators. Decomposing the complex variables in real and imaginary parts, z j = R j + iI j , these variables look like the real version of the spinors used in the 3d case for polyhedra: In these real variables, the canonical bracket reads { R j , I k } = 1 2 δ jk . We define a closure constraint to ensure that the complex variables correspond the normal vectors to the edges of a true closed polygon: The normal vectors are 2-dimensional and correspond to the square of the complex variables: As we will see in details in a following section V C, this ensures that we can reconstruct a unique convex polygon, such that these normal vectors are orthogonal to the polygon's edges and their norm give the length of the corresponding edge. Then the perimeter of the polygon is given by the total energy of the oscillators: The real part of the closure constraint generates the multiplication by a global U(1) phase to all the complex variables: with θ ∈ R and e iθ ∈ U(1). The closure constraint C = 0 is clearly invariant under this global phase transformation. On the other hand, the imaginary part of C generates global inverse re-scaling of the real and imaginary parts: with η ∈ R . The closure constraint C = 0 is not invariant under such transformations. On the contrary, we can use them map any set of complex variables onto a closed set satisfying C = 0. Indeed, starting with an arbitrary value of ∑ j z 2 j , we first set the phase of this complex number by multiplication by a global phase. It is then purely real. Second, we set its real part to 0 by the inverse re-scaling which allows to balance the sum of the squares of the real parts and imaginary parts, k R 2 k = k I 2 k . ∑ ∑ Combining these two type of transformations generate the SL(2 , R ) group. As we have just shown, these transformations allow to map any complex N -vector ( z k ) k =1 ..N ∈ C N onto one satisfying the closure constraint C = 0. This is the equivalent of the SL(2 , C ) transformations allowing to map arbitrary sets of N spinors onto a closed framed polyhedron.", "pages": [ 25, 26 ] }, { "title": "B. The Orthogonal Group Action", "content": "As before, we have the obvious U( N )-action on C N now generated by E jk = ¯ z j z k : It allows to go from Ω = (1 , 0 , .., 0) to an arbitrary vector in C N up to a global scale factor: which means that we are working on the unit complex sphere U ( N ) / U( N -1) ∼ S C N -1 ∼ S R 2 N -1 . This could be an interesting testing ground for the case of the polyhedra since we know well the phenomenon of concentration of measure on the coset U ( N ) / U( N -1) as N grows to infinity. As expected, the U( N ) action leaves invariant the perimeter E : On the other hand, it does not commute with the closure constraint. However, we can introduce a linear combination of the u ( N ) generators that does: /negationslash These form a o ( N ) algebra and actually generate the following action of O( N ) on the complex N -vector: It leaves invariant the perimeter and the closure constraint: It is interesting that this action is cyclic on the set of vectors satisfying the closure constraint. Indeed starting with the vector ω = (1 , i, 0 , .. 0) with 'unit' perimeter, E = 2, and trivially satisfying the closure constraint, we perform an orthogonal transformation O , with O ij ∈ R and t OO = I : Thus the orthogonal matrix gives the real and imaginary parts of the complex variables. Reciprocally, starting with a N -vector with unit perimeter E = 2, we write both the fixed perimeter and closure constraints in terms of the real and imaginary parts of the complex coordinates, z i = R i + iI i : which mean that the real N -dimensional vectors R i and I i are orthonormal, and thus can be identified as the first two columns of an orthogonal matrix, R i = O i 1 and I i = O i 2 . At the end of the day, we will be able to describe averages on the ensemble of polygons as integrals over the orthogonal group. We will go further in this direction, although we can compute similarly to the unitary group case polynomial integrals and a Itzykson-Zuber formula over O( N ). Instead we will focus on the geometrical interpretation of this phase space.", "pages": [ 26, 27 ] }, { "title": "C. Reconstructing Polygons", "content": "Let us describe how we actually go from our complex N -dimensional vector z i satisfying the closure constraint to a real closed polygon (embedded in the flat plane). We would like to interpret the complex variable z i as defining the normal to an edge of the polygon. More precisely, we identify the 2-vector /vectorn i ∈ R 2 normal to the edge i to z 2 i ∈ C = R 2 with the edge length given by the modulus square l i = | z i | 2 . The crucial step of the reconstruction is that we need to (re)-order the edges according to the angle of the normal vector /vectorn i , or equivalently to the phase of z 2 i , so that the angles taken between 0 and 2 π grows with the edge label i . Starting arbitrarily the position /vectorv 1 of the first vertex of the polygon, say on the positive real axis for the sake of simplicity, we reconstruct the next vertex positions /vectorv i from /vectorn i = ( /vectorv i +1 -/vectorv i ) ∧ ˆ e z , or equivalently ( /vectorv i +1 -/vectorv i ) = /vectorn i ∧ ˆ e z , where ˆ e z is the axis orthogonal to the plane. The closure constraint ∑ /vectorn i = 0, equivalent to ∑ z 2 i = 0, ensures that this procedure defines an actual polygon, with /vectorv N +1 = /vectorv 1 . Then we would like to first check that our polygon is convex, i.e. that the angle between two consecutive displacement vectors, or equivalently two consecutive normal vectors, is always at most 180 degrees. Mathematically, this translates to ˆ e z · [( /vectorv i +1 -/vectorv i ) ∧ ( /vectorv i +1 -/vectorv i -1 )] ≥ 0 for all i 's, or equivalently ˆ e z · ( /vectorn i ∧ /vectorn i -1 ) ≥ 0. Since we have ordered all the normals with growing angles, this convexity condition is automatically fulfilled, else the closure constraint can not be satisfied. This concludes he reconstruction procedure for the polygon, which is significantly simpler than for the polyhedron (see e.g. [11]). An interesting feature of our phase space construction is that the normal vectors, and thus the actual geometric polygon, is invariant under the change of sign of individual complex variables z i → -z i . This sign is nevertheless relevant when looking at the action of the orthogonal group on the polygons, i.e two sets of complex variables differing by signs but defining the same polygon will have different images under an orthogonal transformation. This sign 'ambiguity' is the equivalent of the phase of the spinor variables for the polyhedra. Then we similarly introduce the notion of 'signed' polygons, corresponding to 'framed' polyhedra in the 3d case. We expect this sign to be relevant when gluing the polygons together, just as the spinor phases played an essential role when gluing (framed) polyhedra into twisted geometries (encoding the Ashtekar-Barbero connection along the edge [5]). Let us look a bit more into this in the next section.", "pages": [ 27, 28 ] }, { "title": "D. Deforming and Gluing Polygons", "content": "Similarly to the spinor networks introduced as the classical phase space underlying the spin network of loop quantum gravity on a fixed graph [2, 3, 5, 6] and interpreted as twisted geometries, we would like to introduce its two-dimensional equivalent, corresponding to gluing polygons along a given graph. Let us consider an abstract (oriented) closed graph Γ. Around each vertex v of the graph, we will consider one complex variable z v l for each link l attached to v . Reciprocally, for each link l of the graph, we will have two complex variables z s,t l for the two vertices bounding l , for its source and target vertices v = s, t ( l ). We assume the canonical Poisson bracket for each complex variable, plus one closure constraint at each vertex, plus one length matching constraint on each link: Geometrically, we have one polygon dual to its vertex. These polygons are then glued together edge by edge along each link of the graph. We call this a 'complex network', where complex stands for the complex variables used on each edge instead of spinors. Let us emphasize that although each polygon is constructed in a fixed plane as a purely two-dimensional object, the glued polygons are not to be thought as in the same plane. Indeed, we can think of each polygon as in its own tangent plane to the overall 2d discrete manifold, with its normal vectors defined in that tangent plane. This can be considered as a toy model for the gluing of polyhedra in 3d and the study of the deformation and dynamics of twisted geometries. There is no shape matching problem as in 3d, where we have an area matching between polyhedra ensuring that the two faces to be glued have the same area but not necessarily the same shape. But we have nevertheless the issue of reconstructing globally the dual cellular complex (i.e the 'triangulation'). A first look easily shows the problems. Let us start with a cellular complex for a 2d manifold, as a set of flat convex polygons glued together, and we consider the graph defined as its dual 1-skeleton and the corresponding complex network living on it encoding the geometrical data of the polygons. If we start modifying the normal vectors around the vertices, still making sure of not changing the closure constraints and the length matching constraints, we can change the angles of the normal vectors around each vertex and nothing a priori ensures that the ordering of the edges remains consistent to the original one and defines the same cellular complex as before. This seems to imply that deforming the complex network can induce a global change of the dual cellular complex (definition of the points dual to the faces/loops of the network) and probably of its topology. An alternative would be to fix the ordering of the edges around each vertex and not modify it while deforming the angles and norms of the normal vectors, thus allowing for the reconstruction of non-convex polygons. We face the same issue(s) in 3d considering the deformations of glued polyhedra and it would probably enlightening to explore the various possibilities and solve these problems in the 2d case studying the dynamics of glued polygons. From this perspective, we plan to report in a separate paper the analysis of the dynamics of these complex networks and the issue of deforming the gluing of polygons. This should involve introducing some Hamiltonian constraints imposing some flatness conditions on the glued polygons and studying the dynamics of the 2d geometry induced by these constraints. Particular care should taken in understanding the role (if any) of the sign ambiguity between the complex variables z 's and the normal vectors /vectorn 's.", "pages": [ 28 ] }, { "title": "VI. OUTLOOK: MATRIX MODELS FOR DYNAMICAL POLYHEDRA", "content": "We would like to finish this paper on the possibility of defining and studying the dynamics of framed polyhedra in the U( N ) framework presented here. We first would like to define the kinetic term, encoding the dynamical degrees of freedom and their Poisson bracket. As the spinor variables have canonical brackets, it is natural to postulate the straightforward kinetic term for them, as assumed in [2]. Then keeping in mind the definition of the spinors in terms of the U( N ) matrix U and the total boundary area 2 λ , z A i = λU iA for the face index i = 1 ..N and the spinor index A = 1 , 2, we can express the kinetic term entirely in terms of the unitary matrix: We then have to define a Hamiltonian and potential. We can not require U( N ) invariance as we would naturally do when dealing with matrix models else our model would collapse to a pure isotropic behavior independent of the unitary matrix U and described only by the dynamics of the total boundary area λ . If we want some dynamics deforming the shape of the polyhedron, a natural possibility 5 to explore is to introduce an external source given for example as a Hermitian matrix X with a non-trivial potential and define the full action in terms of the unitary matrix U : The equations of motion are straightforward to compute: with the equation of motion for λ being trivial. The potential V [ X ] should not be taken U( N )-invariant, else the theory would be invariant under the action of the unitary group and thus trivial (with only the global area being dynamical). We should investigate how to choose a physically-relevant potential, for example in relation to cosmological minisuperspaces in quantum gravity (e.g. [32]). This would model the evolution of a given polyhedron, within a twisted geometry, coupled to some external excitation a priori taking into account the interaction of the polyhedron with the rest of the geometry. At the quantum level, this would model the dynamics of an intertwiner with the outside geometry in the context of loop quantum gravity. It would thus be interesting to solve these equations and see the various behavior of the evolution of U in terms of the choice of potential V [ X ]. From this perspective, it would seem possible to model the dynamics of a (framed) polyhedron as a matrix model. It would be interesting to see if the tools of matrix models can be relevant in our framework, especially in the large N limit when we would consider the refinement limit of our polyhedra which we expect to describe some continuous 2d surface (topologically equivalent to a 2-sphere).", "pages": [ 28, 29 ] }, { "title": "Conclusion", "content": "To summarize, we started by explaining how to extend the set of (convex) polyhedra with N faces to a set of framed polyhedra by attaching the extra data of a U(1) phase to each face. This allows to see the set of framed polyhedra (up to 3d rotations) as the symplectic quotient C 2 N // SU(2), defined as the set of collections of N spinors satisfying closure constraints and up to SU(2) transformations. Discussing the various parametrization of this space, we showed that this symplectic manifold is equal to the quotient C 2 N / SL(2 , C ), where we can use a SL(2 , C ) transformation to map any collection of spinors onto one satisfying the closure constraints and thus defining a true geometric polyhedron. Furthermore, following the original work of [7, 9], the space of framed polyhedra can be identified to the Grassmaniann space U( N ) / (SU(2) × U( N -2)) with a natural action of the unitary group U( N ) on framed polyhedra. It is important to emphasize that there is no U( N ) action on polyhedra and that the extra phase attached to each face is essential to the construction. These U( N ) transformations allow to generate any framed polyhedra from the totally squeezed configuration with only two non-trivial faces, and thus allow to go between any two framed polyhedra with equal total boundary area. Such transformations could be instrumental in the study of geometric properties of polyhedron, especially in order to consistently explore the space of polyhedra (either analytically or numerically). Using this U ( N ) structure, we have shown how to compute the average value of geometrical observables, such as polynomials in the area of the faces and the angles between their planes (or normal vectors), can be computed as integrals over the unitary group. We have reviewed various formalisms allowing to compute consistently these polynomial integrals over U( N ). Moreover, we have discussed how the Itzykson-Zuber integral can be used as a generating function for these averages. In short, this formula from matrix models contains all the information about the distribution of polyhedra and their shape with respect to the uniform Haar measure on U( N ). Moving on to the quantum level, we have explained how all the classical features are upgraded automatically upon a canonical quantization of the framed polyhedra phase space. This leads to the Hilbert space of SU(2)-intertwiners (or equivalently SU(2)-invariant states) and one can define semi-classical intertwiner states, that transform coherently under the action of U( N ) and that are peaked on classical framed polyhedra. Then one can compute the trace of polynomial observables. Furthermore, similarly to the classical case, one can use the character formula for U( N ) group elements as a generating function for these polynomial traces and as a extension of the Itzykson-Zuber formula to the quantum case. We provide two different expressions for the U( N )-character, either as a quotient of generalized Vandermonde determinants or as an Gaussian integral over almost-closed configurations of spinors (using the coherent intertwiner formalism). We also showed how we can describe polygons in a similar fashion, trading the unitary group for the orthogonal group and defining a phase space of 'signed' polygons as the Grassmanniann space O( N ) / (SO(2) × O( N -2)). All the same techniques presented for the unitary group and polyhedra can then be straightforwardly translated to polygons. This lower-dimensional toy models allow to discuss more explicitly the geometrical reconstruction of polygons, which is simpler than for polyhedra, and we plan to investigate in the future the details of gluing these polygons together and the definition of consistent dynamics on the resulting 2d discrete manifolds. Quantizing the system, we would then obtain the dynamics of quantum surfaces. The present formalism might also turn out useful in discrete geometry, outside of the realm of quantum gravity, when studying polygons from a purely mathematical point of view. For example, it might be applicable to issues like the search for the largest small polygons, e.g. [33], or other similar problems of geometry. To conclude, we would like to mention a few directions that can be explored based on the present work:", "pages": [ 29, 30 ] }, { "title": "Acknowledgment", "content": "E.L. would like to acknowledge Mehdi Assanioussi for his collaboration on the calculations of entropy and polynomial integrals over the unitary group, as part of his Masters thesis research project at the Laboratoire de Physique ENS Lyon, 'Simple models for black holes in Loop Quantum Gravity' (June 2012, Aix-Marseille university, France).", "pages": [ 31 ] }, { "title": "Appendix A: Computing the density of polyhedra and correlations", "content": "We use the method of [21] to compute the density of polyhedra with N faces and with fixed total area 2 λ and the various averages over the ensemble of such polyhedra. We introduce the following (generating) function: Following [21], we Fourier-transform both sets of constraints and perform the integrals over the normal vectors /vector V k (assuming /epsilon1 > 0) : The kernel I ( q, /vectoru ) converges due to the regulator /epsilon1 > 0. We first integrate over the angular part of /vector V and then over its norm: This is exactly as the Feynman propagator in quantum field theory where /vectoru plays the role of the momentum and q the role of the mass. We then perform the integrals over /vectoru and finally over q : And we recover the formula (37) for the volume of the space of (framed) polyhedra with N faces and fixed total area. To extract the average of the norm of a normal vector, or its powers, we can differentiate with respect to q . For instance, we have: with a modified kernel taking into account the insertion of the observable V i , which allows to get the expected result: /negationslash One can check as a consistency check that ∑ i,j 〈 V a i V b j 〉 = 0 as expected from the closure constraints. Furthermore, one interesting observable is the second moment Θ ab = ∑ i V a i V b i -1 3 δ ab V i V i , which characterizes the shape of the polyhedra and its deviation from the isotropic spherical distribution. From the result above, we have the trivial average 〈 Θ ab 〉 = 0. However, using the technique presented here, one can compute the width(s) or uncertainty of the distribution of the tensor Θ ab around its vanishing mean value. This is rather lengthy calculation although straightforward, which we do not detail here. The final result is: One can go further and compute averages of higher powers of the norm V by differentiating more. For instance, in order to compute 〈 V 2 〉 , one insert the modified kernel [21]: Using this technique, one can also compute the correlations between the norm of two different faces, by modifying the product of kernels in the integral from I N to I N -2 ˜ I 2 . Similarly, one differentiate with respect to the components of the vector /vectoru in order to insert observables depending on the components of the normal vectors /vector V . For instance, we can compute the correlations 〈 V a i V b j 〉 as:", "pages": [ 31, 32 ] } ]

[ { "title": "Rydberg states of triatomic hydrogen and deuterium", "content": "Jia Wang ∗ , † and Chris H. Greene ∗ , ‡ Department of Physics, University of Connecticut, Storrs, CT 06269, USA, and Department of Physics, Purdue University, West Lafayette, IN 47907, USA E-mail: [email protected]; [email protected]", "pages": [ 1 ] }, { "title": "Abstract", "content": "The triatomic hydrogen ion (H + 3 ) has spurred tremendous interest in astrophysics in recent decades, and Rydberg states of H 3 have also maintained an important role for understanding H + 3 experiments. In a previous study [J. Chem. Phys. 133 , 234302 (2010)], radiative transitions between neutral H 3 Rydberg states were calculated at wavelengths near 7 microns, and could be compared with mid-infrared laser lines observed in hydrogen/rare gas discharges. The present study extends the investigation to wavelengths near 10 - 13 microns. Rydberg states of D 3 are also treated.", "pages": [ 1 ] }, { "title": "Introduction", "content": "Although triatomic hydrogen (H 3 ) and its ion (H + 3 ) are the simplest polyatomic molecules, they have continued to attract intense interest in diverse contexts, ranging from chemistry to astronomy, ever since their discovery. H + 3 plays an important role in astrophysics since it acts as a proton donor in chemical reactions occurring in interstellar clouds. 1,2 Furthermore, this ion also helps to characterize Jupiter's atmosphere from afar. 3,4 H + 3 is the dominant positively charged ion in molecular hydrogen plasmas and was first identified in 1911 by J. J. Thomson with an early form of mass spectrometry. 5 Without a stable electronic excited state and a permanent dipole moment, H + 3 cannot be observed by electronic spectroscopy or rotational spectroscopy. Therefore, an infrared rotation-vibration spectrum is the only means to observe this ion. The first observation was carried out by T. Oka in 1980. 6 By 2012, more than 600 low-lying rovibrational states of H + 3 had been identified. The good agreement achieved between the experimental spectrum and a first-principles calculation provided a benchmark for calculations on other polyatomic molecules such as water. One of the biggest surprises among the properties of this simple ion H + 3 is its dissociative recombination (DR) rate, which is important for understanding observations of H + 3 in diffuse interstellar clouds. 7 Until 2003, the DR process, H + 3 +e -→ H 3 → H 2 +H or H+H+H, has been studied in several different experiments, and had an order of magnitude discrepancy with theoretical expectation at that time. Building on the previous work of Schneider, Orel, and Suzor-Weiner, 8 Kokoouline and Greene showed 9,10 that intermediate Rydberg states of H 3 play an important role in the dissociative recombination. After Rydberg pathways were included in the theoretical description, along with the Jahn-Teller coupling mechanism that excites the vibrational angular momentum mode of the ion, DR theory was able to resolve the discrepancy. Theory and experiment for this fundamental chemical rearrangement process has now progressed to the point that some energy ranges can even be compared at the level of individual resonance features. 11 Jungen and Pratt have independently demonstrated 12 that the overall value of the DR rate coefficient can be accurately determined from a simplified model once the Jahn-Teller capture mechanism is included. Also in 2003, mid-infrared laser lines at wavelengths near 7 microns in laboratory hydrogen/rare gas supersonic plasmas were observed at Berkeley. 13 Interestingly, strong IR emission from several massive star-forming regions is observed in a similar wavelength range of the spectrum. Later, these laser lines in the Berkeley experiments were assigned to transitions between metastable H 3 Rydberg states, as had been suggested by some detailed theoretical calculations. 14 Alasing mechanism was also proposed: the population inversion is generated by recombination of the ubiquitous H + 3 molecular ion with low-energy electrons. Studies of flowing afterglow plasmas by Glosik et al. suggest a three-body 'collision assisted recombination' mechanism, rather than a simple two-body process because of the high (10 14 cm -3 ) He gas density that is present in the supersonic discharge source. 15 More recently, experiments that study lasing in other energy ranges and in systems of other isotopologues such as D 3 in similar experimental conditions have been renewed. This has motivated us to extend our previous studies to this wavelength range at around 10-13 micron and to calculate the properties of lasing transitions between the Rydberg states of H 3 . An extension of our previous study to treat Rydberg states of the other isotopologue D 3 is also presented.", "pages": [ 1, 2, 3 ] }, { "title": "Method", "content": "Our theoretical approach to the Rydberg states of H 3 is based on multi-channel quantum defect theory (MQDT), one of the most successful techniques for treating Rydberg states in ab initio theory. This approach has been detailed in previous work, 14 so it will only be reviewed briefly here. In our studies, the model of studying molecular Rydberg energy levels of H 3 treats the molecule as a Rydberg electron attached to the H + 3 ion. The interactions between the Rydberg electron and the ion are described by body-frame quantum defects (or the equivalent reaction matrix elements ˜ K ) that depend on the nuclear geometry. In the MQDT approach, a rovibrational transformation can be applied to construct the lab-frame K -matrix using the body-frame quantum defect and the rovibrational wave functions. For p -wave Rydberg states, the body-frame quantum defect parameters can be extracted from ab initio electronic potential surfaces. For higher orbital angular momentum states ( l > 1), a long-range multipole potential model is adopted. The rovibrational transformation can be formulated as follows: Here K ii ' is an element of the laboratory-frame K -matrix, which can be used to solve for eigenener- es E of H 3 by solving the following equation, which is the condition to kill exponentially growing components of the wavefunction at ∞ : The laboratory-frame eigenchannels | i 〉 and the body-frame eigenchannels | α 〉 are connected by the unitary transformation matrix U i α = 〈 i | α 〉 , using the rovibrational wave functions of the H + 3 ion core. To calculate these rovibrational wave functions, an accurate potential energy surface of H + 3 is used, 16,17 and the three-body Schrödinger equation is solved within the hyperspherical adiabatic representation. In a recent paper, 14 rovibrational energy levels of H + 3 are calculated and compared with experiment with an accuracy at about 0.2 cm -1 . Observe that Polyansky and Tennyson achieved an accuracy of 0.02 cm -1 using Jacobi coordinates. 18 Their higher accuracy is due to the inclusion of nonadiabatic effects by using different effective reduced masses for vibration and rotation degree of freedom. Because the implementation of their procedure in hyperspherical coordinates is unclear, we have not attempted to reach this higher level of accuracy in the present calculations. However, the permutation symmetry of the rovibrational wave functions can be easily set up in hyperspherical coordinates, which is an important aspect of the rovibrational transformation. Also, the accuracy of the computed Rydberg state energies of H 3 is mainly limited by the accuracy of the body frame quantum defects, which yields uncertainties of typically a few cm -1 . Therefore, the accuracy of the hyperspherical representation is adequate for our present purposes. The hyperspherical coordinates { R , θ , ϕ } used in our approach are of the Smith-Whitten type, 19 which can be defined by the three interparticle distance r 12 , r 23 and r 31 through the relations: Together with the Euler angles α , β and γ , the three-body system can be described in the body- me. Similar to the usual Born-Oppenheimer approximation, the adiabatic approach treats the hyperadius R initially as an adiabatic variable, and diagonalizes the Hamiltonian in all other degrees of freedom (such as the hyperangles, Euler angles and spin degrees of freedom) yielding a set of adiabatic potentials and channel functions. The adiabatic corrections and couplings are later included using the 'slow variable discretization' method. 20,21 One of the advantages of adopting this choice for the hyperspherical coordinates is that the basis functions used to discretize the Hamiltonian with the proper permutation symmetry can be easily constructed as, where u j ( θ ) are a set of fifth-order basis splines which is unaffected by permutations. Here, the rotational part R N + K + m + ( α , β , γ ) is given by, where D N + m + K + are the Wigner D functions of the Euler angles. The phase of the Wigner function is chosen as by Varshalovich et al . 22 N + is the total angular momentum of the ion, K + is the projection of N + onto the laboratory frame's z-axis, and m + is the projection onto the body frame's Z-axis. Φ Γ g I is symmetry-adapted combinations of nuclear-spin functions for three spin half fermions defined as in a previous paper. 10 Γ = { A , E } represent the the symmetry representations, where g I = 0 for Γ = A and g I = ± 1 (ortho) for Γ = E (para). The permutation symmetries for the basis functions chosen for each degree of freedom are shown in 1. Under the condition that m 2 + g I = 3 n for even K + , and m 2 + g I = 3 n + 3 / 2 for odd K + , it is easy to show that the basis function obeys the permutation symmetry required for three identical fermions: and where", "pages": [ 3, 4, 5, 6 ] }, { "title": "Rydberg transitions of H 3 in the 10 - 13 micron range", "content": "The method described in last section has been applied to calculate 3 p and 3 d Rydberg states of H 3 , showing good agreement with experiments. The 4 d → 4 p and 6 d → 5 p Rydberg transitions were used in Ref. 13 to assign mid-infrared laser lines at wavelengths near 7 microns in laboratory hydrogen/rare gas supersonic plasmas. Here, the Rydberg transitions near 10-13 microns are calculated and shown in 1. These transitions are mainly 7 d → 6 p , 6 d → 6 p and 5 d → 6 p Rydberg transitions.", "pages": [ 6 ] }, { "title": "3 p π Rydberg states of D 3", "content": "Using the method developed to calculate the Rydberg state energy levels for H 3 , we have also calculated energy levels for 3 p π Rydberg states of D 3 . The first step is again calculating the rovibrational states of the ion. In this calculation, the ionic potential surface for D + 3 is adopted from calculations done by Cencek et. al., 16,17 while the same quantum defects as in the case of H + 3 are utilized for the Rydberg state calculation. This should be a good approximation since the quantum defects were calculated under the usual Born-Oppenheimer approximation, where the masses of nucleus are assumed to be infinite. Nevertheless, the rovibrational energy levels of D + 3 are calculated using nuclei mass of deuterium. A major difference between the calculations of rovibrational states of H + 3 and D + 3 is the different permutational symmetries for the two species: the deuterium nuclei are bosons while the hydrogen nuclei are fermions. The symmetry-adapted combinations of nuclear-spin functions Φ Γ g I are constructed in the same way as given in by Kokoouline et. al. 10 Here Γ = { A 1 , A 2 , E } represents the symmetry representations of spin permutation group, and g I = 0 for A 1 (ortho) and A 2 (para) symmetry, while g I = ± 1 for E (meta) symmetry, 23 since E representation is two dimensional. The permutation symmetry of these spin functions are tabulated in 2. The total nuclear-molecular function (including other degree of freedom such as rotation and vibration) should obey the permutation symmetry of three boson system, for example, and where S = 1 + P 12 + P 23 + P 31 + P 12 P 31 + P 12 P 23 . Therefore, the basis functions are constructed as, for Γ = A 1 or E , and, for Γ = A 2 , where g I , m 2 and K + satisfies m 2 + g I = 3 n for even K + , and m 2 + g I = 3 n + 3 / 2 for odd K + . Using these numerical basis states having the appropriate permutation symmetry, the rovibrational states of D + 3 are calculated and compared with experimental results 24 in 3. The r.m.s. difference between our calculation and experimental results is about 0 . 11 cm -1 . Using these accurate rovibrational states, a rovibrational frame transformation is applied to calculate the 3p π Rydberg states of D 3 , and compared with experiment results 25 in 4. From this table, the r.m.s. differences between experiment and our calculations are about 6 cm -1 for almost all the results here. This might due to the quantum defect surface are optimal for H 3 , and the accuracy of our result might be improved by simply shifting the quantum defect by a small constant amount.", "pages": [ 6, 8, 9 ] }, { "title": "Acknowledgement", "content": "This work has been supported in part by the U.S. Department of Energy, Office of Science. We thank Rich Saykally and his group for discussions relating to this study.", "pages": [ 9 ] } ]

[ { "title": "E V Bugaev 1 and B V Mangazeev 2", "content": "1 Institute for Nuclear Research of the Russian Academy of Sciences, 7a, 60th October Anniversary prospect, Moscow 117312, Russia 2 Irkutsk State University, 1, Karl Marx Street, Irkutsk 664003, Russia E-mail: [email protected] Abstract. We show that nucleon electromagnetic structure functions of deep inelastic scattering in Regge-Gribov limit (fixed Q -squared, asymptotically large 1/ x and s ) can be well described in the two-component (soft + hard) approach. In the concrete model elaborated by authors, the soft part of the virtual photon-nucleon scattering is given by the vector meson dominance, with taking into account the radial excitations of the rho-meson and nondiagonal transitions in meson-nucleon interactions. The hard part is calculated by using the dipole factorization, i.e., the process is considered as the dissociation of the photon into a qq -pair (the \"color dipole\") and the subsequent interaction of this dipole with the nucleon. The dipole cross section has a Regge-type s -dependence and vanishes in the limit of large transverse sizes of the dipole. We give the brief description of the model and present results of the detailed comparison of model predictions with experimental data for electromagnetic structure functions of the nucleon.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "During last 10-15 years deep inelastic scattering (DIS) at small x and not very small Q 2 ( Q 2 >1-2 GeV 2 ) has been modeled, mostly, by perturbative QCD (PQCD). Nonperturbative theory, by definition, works, partly at least, in strong coupling regime and takes into account confinement effects. These effects are especially important at small and moderate Q 2 , but are often ignored or incorporated purely phenomenologically. It is so, in particular, in QCD-inspired theories of a Pomeron where the description is based on the two-gluon exchange diagrams. Clearly, nonperturbative effects in DIS must be taken into account because the qq - pairs (produced, at first stage, by the virtual photon) with transverse sizes r  > 2 GeV -1 are strongly overlapped with hadronic states (mostly, with vector mesons). It appears, however, that corresponding nonperturbative effects are partly masked by effects of gluon saturation (the latter effects are important if Q 2 < 2 s Q , where s Q is the saturation scale, and   2 s Q x ~1.5-2.5 GeV 2 for x ~10 -5 -10 -6 ). But, in general, this masking is not complete because kinematical region where saturation is important does not always coincide with region where confinement is essential. It was stressed recently that, in spite of fact that modern PQCD approaches based, e.g., on Color Glass Condensate theory, are rather successful in describing the DIS data, the contribution of the nonperturbative physics to the F2 structure function at small Q 2 may be non-negligible. It appears [1], in particular, that strong coupling based methods (like AdS/CFT [2]) are able to describe F2 data at small Q 2 rather well. The vector meson dominance (VMD) is the truly nonperturbative approach because it operates with hadrons and, consequently, the confinement effects are automatically built in. The most general formulation [3] of the VMD concept (referred to as the 'Generalized VMD') uses the double mass dispersion relation for the forward Compton scattering amplitude A γp : Here, M and M' are the invariant masses of the incoming and outgoing vector mesons, ρ(s,M 2 ,M' 2 ) is the spectral function. The field-theoretical basis of this formula goes back to Sakurai's idea of treating the ρ-meson as gauge boson and to hidden-gauge theories, i.e., theories, in which the vector meson is a gauge boson of the hidden local symmetry (HLS). In last several years a lot of papers appeared, in which holographic models of QCD and, moreover, holographic models of hadrons are elaborated. These models are extra-dimensional, essentially nonperturbative (in particular, color confinement is built in), based on AdS/QCD correspondence. Vector meson dominance and hidden local symmetry are natural consequences of these models. Important (for us) predictions of holographic QCD are as follows: The last two points show that one can, in some sense, say about the 'return of vector dominance': the 'old' vector dominance with the lowest V(1) = ρ is replaced everywhere by a 'new', extended, vector dominance with an infinite tower of vector mesons.", "pages": [ 1, 2 ] }, { "title": "2. Outline of the model", "content": "It is well known that the consistency of the spectral representation (1) with the approximate Bjorken scaling requires, in the approaches based on VMD, rather unnatural (from point of view of hadron physics) strong cancellations between amplitudes of the diagonal and nondiagonal, Vp  V'p, transitions. It had been shown in our previous works [4-7] that such cancellations are not effective. More exactly, there is no motivation for essential cancellations if it is assumed that the vector mesons of VMD models are similar to vector mesons with known properties. On the other hand, if it is assumed that these states are qq -systems with definite mass, qq M , rather than vector mesons, the essential cancellations between diagonal and off-diagonal transitions become possible, due to a general feature of quantum field theory that fermion and antifermion couple with opposite sign (the well-known example is the case of two-gluon exchange). Therefore, in some modern versions of VMD there are no vector mesons at all, only qq -pairs , i.e., these models are not hadronic. As a way out of this situation we proposed the two-component model of DIS at small x (the idea had been suggested in [5]): interactions of the qq - pair (produced by the virtual photon) with the target nucleon can be described by PQCD if the transverse size r  of the pair is small and by VMD if r  is of the order of typical hadronic size. Correspondingly, in this model the structure functions of DIS have two components: the 'soft' component described by VMD and the 'hard' one described by PQCD. Naturally, VMD is used in its 'aligned jet' version [8], i.e., the configurations are selected, in which the q and q , produced by virtual photon, are aligned along the beam direction and, as a consequence, the transverse distance between q and q , on arrival at the target nucleon, is of the order of hadronic size. Amplitudes of nondiagonal transitions (Vp-V'p) are calculated in the two-gluon exchange approximation, and the vector mesons are treated as bound states of quark and antiquark. The wave functions of these bound states are obtained from the solution of Bethe-Salpeter equation with an effective input kernel having the long-range confining part. For a description of the perturbative (hard) component we use the colour dipole model with the dipole cross section having a Regge type s -dependence. For the latter we use the parameterization suggested by Forshaw, Kerley and Shaw [9, 10] (omitting the soft part of it). For dipoles with small transverse size the dipole cross section has a behavior ~   2 2 H r r s  as r goes to zero. We use the FKS formula for calculations of nucleon structure functions without any modification except the region of extremely small x ( x <10 ­5 ). Namely, we assumed that νH slightly increases with a decrease of x ( νH =3.27, νH =4, νH =5 at, correspondingly, x =10 ­5 , x =10 ­7 , x =10 ­9 ). Note, at the end of this section, that similar two-component models of DIS (using the VMD approach in its simplest form, without radial excitations and nondiagonal transitions) had been developed in [11, 12].", "pages": [ 2, 3 ] }, { "title": "3. Main results of the calculations", "content": "The main results of our calculations are presented in figures 1-4. Figure 1 shows an energy dependence of the photoabsorption cross section for the real photon. We assumed that the soft part of   p s   can be parameterized by the Regge type formula ( s in GeV 2 ,   p s   in microbarns). At small s the soft part of   p s   completely dominates. For the hard component of   p s   the Q 2 =0 limit of FKS formula is used, with the assumption that square of the mass of the quark is equal to 0.08 GeV 2 . In figure 3 the comparison of our predictions with data on longitudinal structure function FL is shown. The soft part of FL strongly depends on the assumptions on the ratio of longitudinal to transverse vector meson absorption cross sections, / L T    , which is the parameter of VMD model (due to an absence of experimental data at large energies). We assumed that ξ goes to 1 at s goes to infinity (according with the s -channel helicity conservation) and parameterized the ξ ( s ) dependence by the formula       1/4 1 0.9exp / 10000 s s     , where s in GeV 2 . The comparison with scarce data in 2 Figure 4 shows the x -dependences of F2 for fixed values of Q 2 in the region of very small x , and rather small Q 2 (the region, which is important for experiments with cosmic rays). One can see that F2 slowly increases with a decrease of x , while a relative contribution of the soft component decreases (although even at x~10 -9 and Q 2 ~1 GeV 2 this contribution is not too small, ~40 %). figure 3 shows that the model overestimates FL , especially at large Q . X", "pages": [ 3, 5 ] }, { "title": "4. Conclusions", "content": "We demonstrated that the two-component approach to the theoretical description of diffractive DIS, in which the nonperturbative part is given by the VMD in its modern form (suggested by the AdS/CFT correspondence) is successful in description of experimental data at small x ( x <0.08) and Q 2 ≤10 GeV 2 . This may be enough for using in cosmic rays experiments if one uses muons from the steeply falling cosmic ray muon spectrum. At Q 2 >10 GeV 2 the perturbative part of F2 becomes dominant (for x ~10 -4 ) in our calculations. Note, once more, that we use for the perturbative part the hard Pomeron piece of colour dipole cross section from the works [9,10] (without any modification, in the region x >10 -6 , where the most of data exists). We showed also that in the region of extremely small x , smaller than 10 -6 , and at Q 2 around 1 GeV 2 (this region is especially important for applications in cosmic ray physics) the satisfactory description of structure functions of DIS can be obtained within a framework of the two-component (PQCD + VMD) approach (and, moreover, both components are equally essential).", "pages": [ 5 ] } ]

[ { "title": "Yury Kirienko and Leonid Afremov", "content": "Department of Theoretical and Experimental Physics, School of Natural Sciences, Far-Eastern Federal University, 8, Sukhanova str., Vladivostok, Russia E-mail: [email protected] Abstract. We propose a method for modeling the magnetic properties of nanomaterials with different structures. The method is based on the Ising model and the approximation of the random field interaction. It is shown that in this approximation, the magnetization of the nanocrystal depends only on the number of nearest neighbors of the lattice atoms and the values of exchange integrals between them. This gives a good algorithmic problem of calculating the magnetization of any nano-object, whether it is ultrathin film or nanoparticle of any shape and structure, managing only a rule of selection of nearest neighbors. By setting different values of exchange integrals, it is easy to describe ferromagnets, antiferromagnets, and ferrimagnets in a unified formalism. Having obtained the magnetization curve of the sample it is possible to find the Curie temperature as a function of, for example, the thickness of ultrathin film. Afterwards one can obtain the numerical values for critical exponents of the phase transition 'ferromagnet - paramagnet'. Good agreement between the results of calculations and the experimental data proves the correctness of the method.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Creation of magnetic materials with predetermined properties - the task of unquestionable importance. While the manipulation of individual atoms becomes a routine, nanotechnologies need appropriate calculation methods, which would allow prototyping magnetic properties of materials without heavy calculations on supercomputers. The method proposed below satisfies such requirements. Moreover, it allows calculus of the properties of magnets of any type (ferromagnets, antiferromagnets, ferrimagnets) to be provided using a uniform formalism. Another feature of nano-objects is the existence of size effects at their scale. Obviously, such effects are different in two-dimensional films and three-dimensional particles. Furthermore, it is common to use different methods of modeling for nanoparticles of different shape. But if to consider the theory with short-range interaction, the global geometry of the material should not play any role. In this case, there must be a simulation method that is insensitive to the shape of the simulated sample as a whole. We offer in this paper such a method.", "pages": [ 1 ] }, { "title": "2. General model of magnetic material", "content": "We assume that magnetic atoms are distributed over N sites of the sample with probability p . According to [1], the distribution function for random interaction fields H on a particle located at the origin can be defined as: where δ ( x -x 0 ) - Dirac delta function, h k = h k ( m k , r k ) - field created by atoms with magnetic moments m k located at coordinates r k , r k 0 - the coordinates of lattice sites, F ( m k ) -the distribution function for the magnetic moments, which in the approximation of Ising model for a ferromagnets can be represented as follows: Here θ k - the angle between m k and OZ -axis, α k and β k - relative probabilities of the spin orientation along and against OZ -axis ( θ k = 0 and θ k = π , respectively); m 0 - magnitude of magnetic moment of a magnetic atom. Probabilities α k and β k hold normalization condition α k + β k = 1. In the approximation of nearest neighbors and the direct exchange interaction between magnetic atoms, the equation (1) can be represented as: where k j is a set of nearest neighbors of the magnetic atom numbered as j , z = dim k j - its coordination number; C n z ( k j ) - subset of n atoms of the total number of z nearest neighbors of j th atom; L (Ω) is a binomial set of permutations of an arbitrary set Ω with the amount of elements equal to 2 dimΩ . Introducing symmetric notation α -n ≡ 1 -α n we have got ω l j = ∏ ν = z ( ν =1; l ν ∈ k j ) α ± l v and M l j = ∑ n ∈ k j ± m n = m 0 ∑ n ∈ k j ±| 2 α n -1 | . Finally, J j is the constant of exchange interaction (exchange integral). Using the expression for the distribution function of interaction fields (3), one can obtain equations that determine average relative magnetic moments at each site of lattice: Expression (4) allows to investigate the dependence of total magnetic moment of the sample M = ∑ N j =1 µ j at the temperature T and concentration p , as well as to determine the dependence on the number N of atoms of the temperature of phase transition and the percolation threshold. System of N equations with N unknowns (4) can be solved numerically using Newton's method 1 .", "pages": [ 1, 2 ] }, { "title": "3. Modeling of magnetic materials with different crystalline structure", "content": "Equation (4), despite the complicated form, has written in the algorithmically convenient form. This form allows to simulate the magnetic properties of materials, based only on the knowledge of the crystalline structure of the sample and the numerical values of the exchange integrals. Crucial part of this method is the rule of selection of nearest neighbors for each atom (the way of constructing of the set k j ). By changing this rule of selection we can easily adjust our model to different physical systems. We also can simulate antiferromagnetic and ferrimagnetic materials by reversing signs and values of exchange integrals of individual atoms. Consider two examples of applying the method described above.", "pages": [ 2 ] }, { "title": "3.1. Ultrafine particles", "content": "Consider a nanoparticle of N atoms. Then (4) is a system of N independent equations with N unknowns. In some cases, when p = 1, the symmetry of the particle reduces the number of unknowns, but in general all variables are different. For example, for cubic-shaped nanoprticle with n atoms on the edge and simple cubic lattice there is a system of n 3 non-algebraic equations with n 3 unknowns. Even relatively small cubic particle with 10 atoms on the edge contains 10 3 atoms, which makes calculation non-trivial and significantly reduces its accuracy.", "pages": [ 3 ] }, { "title": "3.2. Thin films", "content": "Now consider ultrathin film that is composed of N infinite monolayers. In a simple case when all atoms from the same layer are equal, (4) turns into a system of N equations with N unknowns. Study of size effects in films is much easier than in the particles, because it is sufficient to simulate a film of 10-15 layers to achieve the bulk properties. (See fig. 1.) General equation that determines the average relative magnetic moment µ n in n -th monolayer is given by (4). Replacing in expressions for ω glyph[lscript] and M glyph[lscript] all α ± n on their average values 〈 α ± n 〉 = (1 ± µ n ) / 2 and substituting (3) into (4), one can obtain the equations that determine µ n in each monolayer: where z n,n is the number of nearest neighbors in the n -th layer, z n -1 ,n is the number of nearest neighbors of the atom in ( n -1)-th layer, located in the n -th layer; i nn = J nn m n /J 11 m 1 , i n -1 ,n = J n -1 ,n m n -1 /J 11 m 1 , i n,n +1 = J n,n +1 m n +1 /J 11 m 1 , t = k T/J 11 m 1 . Using (5), one can study the dependence of the average magnetic moment of the film on its temperature and thickness [2], and, as a consequence, the dependence of Curie temperature on the thickness (see fig. 2).", "pages": [ 3 ] }, { "title": "4. Comparison with experimental data", "content": "Comparison of the results of modeling with experimental data was carried out directly and indirectly.", "pages": [ 4 ] }, { "title": "4.1. Direct comparison: relative Curie temperature change", "content": "In [3], the dependence of Curie temperature T C ( N ) on the size N of cubic-shaped particle was calculated. And then the magnitude of the relative change of the Curie temperature ε ( N ) = T C ( N ) /T C ( N →∞ ) was obtained. Comparison with experimental data [4] revealed the validity of the modeling method.", "pages": [ 4 ] }, { "title": "4.2. Indirect comparison: critical exponents", "content": "It is possible to obtain the value of critical exponent ν of spin-spin correlation for phase transition 'ferromagnet - paramagnet' by appropriate approximation of ε ( N ) defined above. Critical exponents for ultrathin films of different crystalline structures were obtained in [2]. It was shown that the value of ν for three-dimensional Ising model is independent from the type of the lattice. The numerical value of ν is close to the value obtained from RG-calculations [5].", "pages": [ 4 ] }, { "title": "5. Conclusion", "content": "Offered method of simulating allows to substitute the real magnetic material with the relatively simple model, where the key role is played by the coordination number and the exchange integrals. All the magnetic geometry lies in these two integral characteristics. By setting only the rules of choice of nearest neighbors and values of corresponding exchange integrals, we make the modeling procedure very effective.", "pages": [ 4 ] }, { "title": "Acknowledgments", "content": "The work was supported by grant of Scientific Fund of Far Eastern Federal University (FEFU) №12-07-13000-FEFU_a.", "pages": [ 4 ] } ]

[ { "title": "Low energy physics and properties of extra space", "content": "S.G. Rubin National Research Nuclear University 'MEPhI' , E-mail: [email protected]", "pages": [ 1 ] }, { "title": "Abstract", "content": "The mechanism of low energy physics formation in the framework of multidimensional gravity is discussed. It is shown that a wide set of parameters of a primary theory could lead to the observable Universe. Quantum fluctuations of extra space metric and its consequent classical evolution play an important role in this process.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "'A theory of everything' (TOE) is usually considered as a final goal. It means an existence of some universal lagrangian containing fields, symmetries and parameters (coupling constants) that should describe all phenomena in the Universe. But even if the goal was achieved main questions listed above remain unanswered: One may conclude that a TOE can not be a real final theory due to its internal reasons. The alternative approach - multiverse - is also known. In this case it is supposed that many theories (universes) could be realized in Nature and we accidentally live in some space domain (Universe) described by one of the theory. This way is partly declared by the string theory. Main objection to this approach can be expressed shortly in one phrase: 'If everything is possible, it is not a science'. Let me suggest some example as a counterargument. Indeed, it would be strange if somebody would explain the observed Earth mass just saying that a lot of planets with different masses do exist. Does it mean that any scientific researches in this direction are meaningless? Not at all. Instead we develop the theory of planet formation and some other scientific directions like the geology, the geophysics.", "pages": [ 1 ] }, { "title": "2 Formation of low energy physics", "content": "Our aim is to propose the approach that connects observable physical parameters and initial parameters of nonlinear gravity acting in D -dimensional. It will be shown that a universe similar to ours can be formed for a wide range of initial parameters. In case of success, answers to the questions itemized above would be quite evident. The main tools are multidimensional gravity and quantum theory. The latter is needed to justify a quantum creation of manifolds with various metrics. For our purposes we need only the same fact of a nonzero probability of any metric formation in a small space region. The essence of the approach is as follows. Suppose that a sufficient number of extra dimensions does exist. A metric of a whole space M 0 is chosen in the form Here M 4 is our 4-dim space and M is an extra space. It is supposed that every 4-dim lagrangian, e.g. the Standard Model lagrangian is deduced from a primary one that acts in the whole space M 0 . Let this lagrangian contains initial parameters a in . The observed values of the parameters a obs of the Standard Model are their functions Here g M is a metric tensor of the extra space. A form of the function f is a separate problem which was discussed partly in [1]. The internal metric g M is usually fixed by low energy physics what leads to strict connection between two sets a in and a obs . Thus, in spite of many interesting applications have been performed, see e.g. [3, 4], the problem of origin of the same parameters a in remains. Let us make a next step and suppose that the geometry has the form where K is an another extra space with a metric g K . Formula (2) is transformed into the two-step expression where b in is a new set of initial parameters. For the first sight nothing has been changed. In fact we have obtain additional freedom - an internal metric g K may be varied freely. The set a in can be obtained from a variety of sets b in by fitting the metric g K . We will refer to the set a in as a secondary set and to the set b in as a primary one. Our nearest goal is to elaborate some production mechanism of various metrics g K . It seems a trivial task because quantum fluctuations claimed produce any form of metric. The problem becomes much less simple if one takes into account a classical motion of an extra space metric g K just after its nucleation. As a result, a set of final metrics g K ( t →∞ ) appears to be quite limited [1]. This strongly reduces a number of possibilities for the metric variation in (4). As was shown in [1] the metrics could evolve classically to some stationary configuration g stat ( y ) = const that does not depend on an initial metric. This way gives rise a large but limited number of an initial sets a in of parameters that lead to the universe similar to ours. The problem of fine tuning would be hardly solved. Another wide class of theories could be obtained if classical equations for the metric g K have finite-dimensional attractors. In this case a behavior of the metric at large times depends on an arbitrary initial metric and form a continuous set. Thus we come to infinite set of effective theories. So far we have implicitly assumed that the structure of the space is T × M D 0 -1 , i.e. we do not distinguish an extra space and the observable 3-dim space of our Universe. In fact, the huge difference in their sizes has to be explained. As was shown in [2], the reason of their difference is laid in initial conditions. If an initial metric satisfies the conditions where R M , R K are the Ricci scalar of the main space and the extra space correspondingly, their following evolution is different. Relaxation time in the extra space K is much smaller so that an evolution of metric of the main space M proceeds at (almost) stationary extra metric. Initial conditions dictate a shape of the extra space while the latter influences a dynamics of the main space.", "pages": [ 2, 3 ] }, { "title": "3 Explicit form of secondary parameters", "content": "To illustrate the above, consider a toy model with the specific pure gravitational action in a D 0 -dim space Here R M 0 is the Ricci scalar and { b } = b 1 , b 2 , ...b N . We follow only those geometries that represent a direct product and satisfy the inequality written in (5). A classical motion of the metric g K as a function in the extra space M is of interest and we will omit any mentioning about our 3-dim space. Metric where g ab ( t, y ) is the metric in M , γ 1 , 2; ij ( z ) are positively defined internal metrics of the extra space K = K 1 × K 2 and e 2 β 1 , 2 ( t ) are scaling factors (see [5, 3]) of the spaces K 1 , 2 . Also, D = dimM,d = dimK . Evolution of the extra space metric is governed by classical equations for the functions β 1 , 2 ( t ). As was shown in [2], the effective action for the field β has the form As was shown in [6] the lines φ ∗ 1 , 2 = 0 are attractors of the classical equations for the fields φ 1 , 2 . We are interested in solutions of the form φ 1 → 0; 0 < φ 2 < ∞ . Under these assumptions the Ricci scalar is easily obtained. After some algebra with formulas (13) and (6) the reduced action can be obtained where the intermediate set of parameters a is connected to a set of primary parameters b g M , g K are determinants of the metric tensors of spaces M and K . This formula may be considered as the connection between the primary set b and the metric g K of extra space K . Variety of the sets b can be obtained by continuous variation the metric g K at fixed intermediate parameters a . This illustrates the hypothesis: In the framework of multidimensional gravity, the observable set of effective parameter values could be obtained by a continuous set of initial parameter values. Formula (15) indicates that all physical parameters vary with time. Their modern values represent the asymptotes at time tends to infinity. In this paper we discuss the idea of 'inverse landscape'. By this is meant that quantum fluctuations of extra space metric and their subsequent classical evolution could lead to observable values of physical parameters in wide range of initial parameter values of a primary lagrangian. In this case the fine tuning problem seems solvable. The study was supported by The Ministry of education and science of Russian Federation, project 14.A18.21.0789.", "pages": [ 3, 4 ] } ]

[ { "title": "Veronica Dexheimer", "content": "UFSC, Florianopolis, BR Gettysburg College, Gettysburg, USA E-mail: [email protected]", "pages": [ 1 ] }, { "title": "Rodrigo Negreiros", "content": "Instituto de Fisica, UFF, Niteroi, BR", "pages": [ 1 ] }, { "title": "Stefan Schramm", "content": "FIAS - Johann Wolfgang Goethe University, Frankfurt, DE Abstract. We model magnetars as hybrid stars, which have a core of quark matter surrounded by hadronic matter. For this purpose, we use an extended version of the SU(3) non-linear realization of the sigma model in which the degrees of freedom change naturally from hadrons to quarks as the temperature/density increases. The presence of a variable magnetic field allows us to study in detail the influence of Landau quantization and the anomalous magnetic moment on the particle population of the star, more precisely on particles with different spin projections. This allows us to calculate the polarization of the system throughout different phases of the star, hadronic, quark and also a mixed phase.", "pages": [ 1 ] }, { "title": "1. Introduction and Model Description", "content": "Magnetars are compact stars that have extremely high magnetic fields. The highest magnetic field observed on the surface of a star is on the order of 10 15 G. The highest possible magnetic field in the center of stars, on the other hand, can only be estimated through models, even when applying the virial theorem. Some results indicate limiting magnetic fields ranging between B = 10 18 -10 20 G [1-8]. Here we assume that magnetars are not necessarily composed of hadronic matter and describe them using a model that contains both hadronic and quark degrees of freedom. There is no first principles solution for QCD in both the hadronic and quark phases at finite temperature and density. In order to model the deconfinement transition, we use a model that agrees with nuclear saturation properties and reproduces reasonable hyperon optical potentials in the low energy limit and incorporates known QCD properties in the high energy limit. In order to constrain the model at intermediate energies, we compare our model predictions with lattice results such as: a first order phase transition and a pressure functional P(T) similar to Ref. [9] at µ = 0 for pure gauge theory, a crossover at vanishing chemical potential with a transition temperature determined as the peak of the change of the chiral condensate and the Polyakov loop, and the location of the critical end-point [10, 11]. B B The Lagrangian density of the non-linear sigma model (shown in Ref. [12]) represents the interactions between baryons (and quarks) and vector and scalar mesons, the self interactions of scalar and vector mesons and includes an explicit chiral symmetry breaking term which is responsible for producing the masses of the pseudo-scalar mesons. The mesons are treated as classical fields within the mean-field approximation. Finite-temperature calculations include a heat bath of hadronic and quark quasiparticles within the grand canonical ensemble. Within our approach, the hadrons (whole baryon octet) are replaced by quarks (up, down, strange) at high densities and/or temperatures. This happens as the effective masses of the hadrons increase and the effective masses of the quarks decrease within these limits. The aforementioned model (see Ref. [12] for details) is an extended version of the SU(3) non-linear realization of the sigma model. Changes in the order parameters of the model σ and Φ signal chiral symmetry restoration and quark deconfinement, respectively. The potential for Φ is an extension of the Polyakov loop potential [9] modified to also depend on baryon chemical potential. In this way our model is able to describe the entire QCD phase diagram, even at zero temperature. Fig. 1 shows that the model is in good agreement with lattice QCD constraints and that it reproduces the liquid-gas phase transition for symmetric matter. In this figure we also show results for neutron star matter, which is charge neutral and in chemical equilibrium. The phase transitions at low temperatures and high densities are of first order, whereas at high temperatures and low densities the model exhibits smooth crossovers. The SU(3) non-linear realization of the sigma model and its extension (that also contains quarks) have been successful in reproducing nuclear matter properties [13], heavy ion collision data [14], compact star and proto-neutron star properties [15-17]. As compact stars have temperatures of the order of 1 MeV, we can safely set their temperature to zero. As already mentioned, for star calculations we have to take into account charge neutrality and chemical equilibrium. Here we assume that the surface tension between the hadronic and quark phases is small [18] and allow charge neutrality to be global (only the combination of both phases sum up to zero charge). As a consequence, a mixed phase appears in the star. This can be seen in Fig. 2, which also shows that hyperons are almost completely suppressed by the appearance of the quarks. We include in the model a magnetic field in the z-direction that has varying magnitude. This is a more realistic approach than considering a constant magnetic field throughout the star and can prevent the creation of hydrodynamical instabilities due to pressure anisotropy [19-23]. This happens because, in our approach, the magnetic field only becomes extremely high in the center of the star, where the matter pressure is also high (see Ref. [24] for more details). More precisely, we assume an effective magnetic field B ∗ that increases with chemical potential, running from a surface value B surf = 69 . 25 MeV 2 = 10 15 G (when µ B = 938 MeV) to different central values B c at large values of baryon chemical potential following [17] with a = 2 . 5, b = -4 . 08 × 10 -4 and µ B given in MeV. As can be seen in Fig. 3, the values of the effective magnetic field only approach B c at very high baryon chemical potentials and, in practice, only about 70% of B c can be reached inside stars. The use of an explicit dependence of B on the baryon chemical potential instead of on density was chosen to prevent discontinuities in the magnetic field at the phase transition, where the baryon density is discontinuous. The constants a and b and the form of the B ∗ expression were chosen to reproduce (in the absence of quarks) the effective magnetic field curve as a function of density from Refs. [5, 25, 26].", "pages": [ 1, 2, 3 ] }, { "title": "2. Results and Conclusions", "content": "The magnetic field in the z-direction forces the eigenstates in the x and y directions of charged particles to be quantized into Landau levels. The energy levels of all baryons are further split due to the alignment/anti-alignment of their spins with the magnetic field (anomalous magnetic moment effect, AMM). But even when the AMM is not taken into account, like in the quark phase in our model, only one of the spin projections contributes to the zeroth Landau level, creating a spin projection asymmetry in the system. In this work, we focus on the analysis of magnetic field effects on the chemical composition of the neutron star, the total spin polarization and the magnetization of the system. Studies of magnetic field effects on compact star observables can be found in Refs. [17, 27-29]. The particle population is shown in Fig. 4 when a central magnetic field B c = 5 × 10 5 MeV 2 = 7 . 22 × 10 18 G with AMM is considered. The 'wiggles' in the charged particle densities mark the µ B values, for which their Fermi energies cross the discrete threshold of a Landau level. The charged particle population is enhanced due to B , as their chemical potentials increase. Although the AMM is known to make the EOS stiffer, it does not have a very significant effect in the particle population. This fact can be easily understood in terms of polarization, when, instead of looking at the total particle density (sum of spin up and down particle densities) for each species, we look at the spin up/spin down particle densities separately. In this case some of these particles are enhanced while others are suppressed (see Ref. [30] for details). The total polarization of the system is defined as where Q Bi stands for the baryon number of each species (3 for baryons, 1 for quarks and 0 for leptons). We can see in Fig. 5 that the total polarization of the system (taking into account hadrons and quarks) is larger for larger magnetic fields and increases with chemical potential until the quarks appear. After this point the polarization decreases. This happens because the AMM for the quarks is not included in our calculations, as it is not fully understood for these particles. In this case, the spin asymmetry comes exclusively from the spin asymmetric contribution to the zeroth Landau level. For an example of a complete study of the polarization without the AMM see Ref. [31]. The 'wiggles' in Fig. 5 mark again when the Fermi energies of the charged particles cross the discrete threshold of a Landau level. Finally, we turn our attention to the calculation of the magnetization of the system. It is important to notice that in the case with the AMM, not only the magnetization of the charged particles has to be included, but also the magnetization of the uncharged particles. For details on the calculation of the magnetization including the AMM see Ref. [32]. Fig. 6 shows that the magnetization is larger for larger magnetic fields and it increases with baryon chemical potential. As in Figs. 4 and 5, the 'wiggles' mark when the Fermi energies of the charged particles cross the discrete threshold of a Landau level. The magnetization of the system is a very important quantity as it relates with the pressure anisotropy of the system [19-23]. We have shown in this work some possible effects of strong magnetic fields in hybrid stars. The presence of different hadronic and quark degrees of freedom makes this quark-hadron sigma model an ideal tool for such an analysis in the different possible phases of the star. More specifically, we analyzed the effects of strong chemical potential dependent magnetic fields on particles with different spin projections, their total polarization and magnetization.", "pages": [ 3, 4, 5 ] }, { "title": "References", "content": "[2] Cardall C Y, Prakash M and Lattimer J M 2001 Astrophys.J. 554 322-339 ( Preprint astro-ph/0011148 )", "pages": [ 5 ] } ]

[ { "title": "Alberto Salvio", "content": "Scuola Normale Superiore and INFN, Piazza dei Cavalieri 7, 56126 Pisa, Italy Departamento de F´ısica Te´orica, Universidad Aut´onoma de Madrid and Instituto de F´ısica Te´orica IFT-UAM/CSIC, Cantoblanco, 28049 Madrid, Spain E-mail: [email protected] Abstract. We study the Casimir energy due to bulk loops of matter fields in codimensiontwo brane worlds and discuss how effective field theory methods allow us to use this result to renormalize the bulk and brane operators. In the calculation we explicitly sum over the Kaluza-Klein (KK) states with a new convenient method, which is based on a combined use of zeta function and dimensional regularization. Among the general class of models we consider we include a supersymmetric example, 6D gauged chiral supergravity. Although much of our discussion is more general, we treat in some detail a class of compactifications, where the extra dimensions parametrize a rugby ball shaped space with size stabilized by a bulk magnetic flux. The rugby ball geometry requires two branes, which can host the Standard Model fields and carry both tension and magnetic flux (of the bulk gauge field), the leading terms in a derivative expansion. The brane properties have an impact on the KK spectrum and therefore on the Casimir energy as well as on the renormalization of the brane operators. A very interesting feature is that when the two branes carry exactly the same amount of flux, one half of the bulk supersymmetries survives after the compactification, even if the brane tensions are large. We also discuss the implications of these calculations for the natural value of the cosmological constant when the bulk has two large extra dimensions and the bulk supersymmetry is partially preserved (or completely broken).", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Higher dimensional theories have a variety of motivations in the physics of fundamental interactions. The original one, in the pioneering works of Kaluza and Klein (KK), was the unification of the known forces. This idea survived as the decades passed by through superstring theory, which provides a framework where gravity can be unified with the other interactions and strongly indicates that the space-time contains a number of extra dimensions. Other advantages of higher dimensional setups include the possibility to understand the structure of fermion masses and couplings [1] in the Standard Model. Remarkably, extra dimensions also provide mechanisms to understand technical naturalness problems. For example, the gauge hierarchy problem, which queries why the Fermi scale is so much smaller than the Planck mass, can be addressed in the large extra dimension (LED) scenario or by so called warped geometries. It is then natural to ask whether extra dimensions can help us understand the most serious fine tuning issue we know, the cosmological constant problem [2]. To understand the origin of such problem it is useful to look at the usual Einstein-Hilbert gravity 1 coupled to matter where L matter is the part which depends on the matter fields. This represents the leading gravitational tensor theory in the large wavelength limit. Here κ 4 is the 4D Planck scale and Λ 0 the tree level value of the cosmological constant. From this equation it is clear that the quantum vacuum energy 〈 ρ 〉 produced by the gravitational field g µν and any matter particle contributes to the cosmological constant Λ = Λ 0 + 〈 ρ 〉 . The cosmological constant problem consists in the mismatch between the observed value Λ ∼ (10 -3 eV) 4 and the (relatively) enormous contribution to 〈 ρ 〉 due to the known forms of matter: any Standard Model particle with mass m adds to Λ a quantity of order m 4 and therefore a huge fine tuning of Λ 0 is required to obtain the observed value. Any solution of such problem must involve a modification of gravity at an energy scale roughly of order 10 -3 eV. Indeed we know all non-gravitational forces up to the ∼ 10 TeV scale and the only possible modification should therefore occur in the gravity sector, which is much less constrained by the experiments; if such modification emerges only at some intermediate energy scale E int , between 10 -3 eV and 10 TeV, then the low energy effective theory at energies much smaller than E int would be again the Standard Model plus Einstein's gravitational theory and we would be back to the original problem. The required modification occurs in any model with two large extra dimensions. Let us see why. The LED scenario predicts the following ratio between the 4D Planck mass, M Pl ≡ 1 / ( √ 2 κ 4 ), and its higher dimensional counterpart, M ≡ 1 / ( √ 2 κ ) 2 / ( D -2) , [4] where D is the full space-time dimension and r is the linear size of the D -4 dimensional space volume (that is the length scale of the extra dimensions). In this framework one addresses the gauge hierarchy problem by choosing roughly M ∼ TeV, which for D = 6 gives us a KK scale of order 1 /r ∼ 10 -3 eV (above which gravity is modified). It is important to notice that a true solution of the gauge hierarchy problem requires a mechanism to dynamically stabilize r to this large value; one way to do so is to consider a compactification on a topologically non-trivial space with a magnetic flux (flux stabilization). Also, notice that in order for the extra dimensions to be so large, only gravity (among the known interactions) can propagate in the bulk while the Standard Model fields should be confined on a 3+1 dimensional brane (3-brane henceforth), which in the case of interest here, D = 6, has to be of codimension-two. Of course, the fact that gravity is modified at the required energy scale is not by itself a solution of the cosmological constant problem. We need a further mechanism protecting Λ from large quantum corrections. Supersymmetry is a possible candidate because the fermion and boson contributions to the vacuum energy cancel exactly if supersymmetry is unbroken, or, if supersymmetry is broken at an energy scale m S , produce a net vacuum energy of order m 4 S . A crucial observation now is that the supersymmetry breaking scale in the bulk does not have to be of the same order of that on the brane, which is required to be larger than the TeV scale by the LHC experiments. A supersymmetry breaking brane Lagrangian δL b , which, including quantum corrections, is expected to be of order TeV gives rise to a supersymmetry breaking splitting δm 2 KK in the KK spectrum of order κ 2 δL b /r 2 . Here the factor κ 2 reminds us that the brane physics is communicated to the bulk through gravitational interactions while 1 /r 2 is there for dimensional reasons. To the extent that m 2 S is given by δm 2 KK we therefore obtain a cosmological constant of the correct order of magnitude. To summarize where the question mark reminds us our expectation. To confirm it explicit calculations in a concrete model are therefore needed. The idea that supersymmetric large extra dimensions with codimension-two branes can address the gauge hierarchy and the cosmological constant problem was originally proposed in [5]. We refer to these works for an extended discussion. But the fact that extra dimensions in general can help with the cosmological constant problem is older (see for example Refs. [6]) Since any issue of technical naturalness is a quantum problem we believe it is important to have a sistematic approach to compute quantum corrections in codimension-two brane worlds. Here we discuss quantum corrections due to bulk (massive) states, focusing on the one-loop approximation. The main subject of this article is the Casimir energy produced by integrating out a (massive) bulk field, and how to obtain it from its KK mass spectrum. Much of our discussions is based on Refs. [7, 8, 9, 10, 11], but here we also obtain some original results. The main one is a convenient way to compute Casimir energies from KK spectra, which is based on a combined use of dimensional regularization (for the ultraviolet divergences) and zeta function regularization (for the KK sums). The Casimir energy can then be used to calculate the renormalization group equations (RGEs) of both bulk and brane coefficients in the quantum action. Although some results can be applied to any codimension-two model, we study in some detail a concrete class of compactifications, where the extra dimensions form a rugby ball shaped space and the flux of a gauge field stabilizes their size, r . These solutions are supported by two 3branes having both tension [12] and magnetic flux (the leading terms in a derivative expansion of the brane action) [13]. We discuss a class of models having these solutions, including one with bulk supersymmetry, 6D gauged chiral supergravity [14, 15, 16], which we refer to throughout this paper as our supersymmetric example. As shown in Ref. [11], when the two 3-branes have identical localized fluxes one half of the bulk supersymetries are preserved (if the bulk is supersymmetric), implying, interestingly, that the 4D vacuum energy vanishes in this limit. For rugby ball compactifications, the KK towers of many bulk fields are known from previous calculations [7, 8, 9, 10, 11] and we make use of these results to compute explicitly the Casimir energy and the renormalization of bulk and brane coefficients as a function of the bulk mass m of the field we integrate out. Our computational method is very convenient and confirms the results of Ref. [10]. The final expression for the Casimir energy is a polynomial function in mr of degree 6, where the coefficients depend on the brane tension and the bulk and brane localized fluxes. In the supersymmetric case, all these coefficients vanish for identical localized fluxes or, more generically, they are suppressed by the difference between these fluxes. As shown in [11] the 4D cosmological constant inherits this suppression and can be of the observed size. Let us give the outline of this article. In section 2 we introduce our family of models, including both the bulk and brane actions, and the rugby ball solutions; as a concrete supersymmetric example we define 6D gauged chiral supergravity and reexamine the technical naturalness of the cosmological constant in this specific case. We then review the KK spectra of scalars, fermions and vectors for rugby ball compactifications in section 3. The Casimir energy calculation and the bulk renormalization is performed in section 4, treating in some detail the codimension-two case and, more specifically, the rugby ball solutions. In section 5 we finally discuss the form of the Casimir energy due to loops of specific matter fields: the simplest case of a real scalar and (massive) matter multiplets of 6D supergravity.", "pages": [ 1, 2, 3 ] }, { "title": "2. The class of models", "content": "We focus on a class of models which include, in addition to the metric tensor g MN , a set of gauge fields A a M , scalars φ i and fermions ψ r . The bosonic part of the Lagrangian for these fields is where D M is the gauge-covariant derivatives for the scalars, F a MN is the field strength of A a M , and G ij ( φ ), H ab ( φ ) and V ( φ ) are generic functions of the scalars. This is general enough to describe the linearized dynamics of matter supermultiplets in explicit 6D supergravities. The supersymmetric example we shall refer to throughout this paper is 6D gauged chiral supergravity. However, the general class of models we consider also includes non-supersymmetric theories and the results we shall discuss in the following sections hold for them as well (unless otherwise stated). The supersymmetric field content is given by a supergravity-tensor multiplet ( g MN , B MN , φ , ψ M , χ ) - with metric tensor, anti-symmetric KalbRamond field (with field strength G MNP ), dilaton, gravitino and dilatino - coupled to some gauge multiplets ( A a M , λ ) - with gauge fields and gauginos - and some hypermultiplets (Φ I , Ψ) - each of them having hyperscalars and their chiral hyperini. The fermions are all Weyl spinors and satisfy Γ 7 ψ M = ψ M , Γ 7 χ = -χ , Γ 7 λ = λ and Γ 7 Ψ = -Ψ. We will consider a matter content with gauge group of the form G = ˜ G × U (1) R , where U (1) R is an Abelian R-symmetry, and ˜ G is a generic product of simple groups. The bosonic Lagrangian of 6D gauged chiral supergravity is given by where g is the U (1) R gauge coupling and we have set G MNP = 0 for simplicity. As we have already mentioned a realistic realization of the large extra dimensions idea requires the presence of 3-branes, one of which supports the Standard Model fields. We take the brane action to be [13]. where g b is the gauge coupling of the gauge field corresponding to the second term in (5) (which has to be Abelian in order for that term to be gauge invariant). The ellipses denote other terms involving two or more derivatives (including in principle the Standard Model fields), and T b , A b , B b , C b and so on could depend on the scalars φ i . One can now ask again whether the observed value of the cosmological constant emerges naturally in this framework. While no positive answer is found in non-supersymmetric cases, the supersymmetric model has been proven to have very interesting features [5, 17]. One can compute the cosmological constant in three steps: first, the brane fields are integrated out at the quantum level; second, the classical integration in the bulk is performed; third, the quantum corrections of the bulk integration are computed. The first and second steps always give a vanishing cosmological constant if the dilaton does not couple directly to the branes [5, 18]. Ref. [17] has recently shown that the third step is not dangerous either, at least for representations of the supersymmetry algebra which are massless in the 6D sense. The essence of the argument is that e 2 φ acts as a loop counting quantity (as it can be proved by going to the frame defined by ˆ g MN = e φ g MN ) and is very small, of order 1 / ( r 4 TeV 4 ). The last property can be understood by noticing that the bulk Lagrangian enjoys a classical scale invariance under which e φ g MN does not change; in the large extra dimension setup the value e φ r 2 is therefore expected to be fixed by the field equations to a value of order 1 / TeV 2 . The contents of the following sections are useful tools to address the same question in the presence of massive 6D supermultiplets (see also [19] for a study of bulk UV sensitivity for Ricci-flat geometries, including the gravity sector, but without branes.). Let us conclude this section by discussing the solutions of the models we have considered. The simplest solution preserving 4D Poincar'e invariance and providing a flux stabilization is a ( Minkowski ) 4 × S 2 compactification, where the metric of the two extra dimensions is that of a sphere with radius r , (0 ≤ θ ≤ π and 0 ≤ ϕ < 2 π ). Also, the scalars φ i are constant and the non-vanishing components of the field strength are F mn = f/epsilon1 mn , where f is a constant. If there are some matter fields having a non-trivial charge 2 , q ˜ g , under this background gauge field, then its field strength F mn has to satisfy the following quantization condition [20] where N = 0 , ± 1 , ... is an arbitrary integer. Then f must satisfy In the supersymmetric model, the field equations also imply Since all the parameters of the bulk theory are expected to be of order TeV (to the appropriate power) in the large extra dimensions setup, we notice that the first of these two conditions confirms our general expectation that e φ r 2 is of order 1 / TeV 2 . The second one, combined with Eq. (8), gives us N = ± q ˜ g/g . When the background gauge field is along the U (1) R , and so ˜ g = g , N = ± 1 and this solution preserves half of the supersymmetries of the bulk theory [16]. For other embeddings of the background gauge field supersymmetry is instead completely broken. A simple way to introduce 3-branes in this case is starting from the sphere metric and demanding ϕ to have a modified period, ϕ ∼ ϕ +2 πα , where α is a positive constant [12]. This procedure introduces two conical singularities, one at the north pole and the other one at the south pole, with identical defect angles, δ = 2 π (1 -α ). The resulting internal space is called the rugby ball . The brane action in (5) can support these conical singularities and the near-brane boundary conditions imply [13] that the defect angle is In order for this angle to be positive we will assume that L b ≥ 0 at the background solution. The brane sources also changes the flux quantization condition. This arises because the branes themselves can support a localized flux: If the branes are identical the total flux localized in this way is Φ := ∑ b Φ b , in terms of which the flux-quantization condition is (when the background gauge field has some localized flux) where S 2 ( α ) is the rugby ball. The normalization constant f is then given by where N := ω ( N -Φ) and ω := 1 /α . In the supersymmetric case, the above-mentioned classical scale invariance can be broken by the boundary localized fluxes, the second term in Eq. (5), which gives mass to the corresponding flat direction [13]; another reason why these fluxes are useful is that they allow us to recover supersymmetry, when the background gauge field is along the U (1) R , in a continuous limit [13, 17]. To understand this let us take the rugby ball solution, ˜ g = g and Φ b = 0, then N = ωN and the second condition in Eq. (9) together with (13) implies ωN = ± q , which does not allow you to approach the supersymmetric value ω = 1 continuously. In the absence of Φ b instead we do not find this obstruction: using the second condition of (9) in (13) this time we obtain which, in the case ˜ g = g and N/q = ± 1, gives In the presence of boundary localized fluxes supersymmetry can be broken by an arbitrarily small amount. For this reason in the supersymmetric model we will always take the boundary localized flux in the U (1) R direction, g b = g . Let us conclude this section by mentioning that, for branes carrying both tension and flux, there is, remarkably, a case in which one half of the 6D supersymmetries is preserved; this occurs when the background gauge field is along the U (1) R generator, for which there is exactly the same amount of localized flux on the two branes [11] (i.e. Φ b = Φ / 2 both at the north and south branes). Such property has important implications regarding the smallness of the cosmological constant.", "pages": [ 4, 5, 6 ] }, { "title": "3. Spectrum in codimension-two brane worlds", "content": "We now move to the analysis of the linear perturbations around the 3-brane solutions we have considered. Since these configurations preserve 4D Poncar'e symmetry, such analysis is equivalent to computing the 4D particle spectrum (defined in the usual sense). This will provide us with additional physical information; it is an important computation to study the stability of the background solutions [8, 9] and, of special relevance for this article, it is an intermediate step to determine the quantum corrections to the 4D vacuum energy. Indeed in dimensional regularization the one-loop contribution to the quantum potential (the Casimir energy) due to a generic field with bulk mass m is where the collective index n includes all KK numbers and m n represents the full set of KK masses. Also ( -1) F is 1 for bosons and -1 for fermions. The 4D particle spectrum is therefore an important ingredient to compute V and we will use Eq. (16) to compute the renormalization of bulk and brane couplings in sections 4 and 5. In the presence of two extra dimensions we have two KK numbers, n = j, n ; in the rest of this section we give the form of m 2 jn := λ jn /r 2 for (minimally coupled) scalars, fermions and gauge fields on top of the rugby ball geometry 3 sourced by branes with both tension T b and flux Φ b (the leading terms in a derivative expansion). In the absence of localized fluxes, these m 2 jn were computed 4 in Refs. [7, 25, 8]. Subsequently, their form in the presence of Φ b was derived in [10]. Let us consider first the simple case of a minimally coupled real scalar, satisfying the equation ( D M D M + m 2 ) φ = 0, that is coupled to the background gauge field through a monopole number N and boundary localized fluxes Φ b . In this case the scalar spectrum (in the north patch of the gauge potential) is where and Φ + (Φ -) is the flux localized on the north (south) pole of the rugby ball, where cos θ = +1 ( -1). Moving to fermions, the KK spectrum for a field, satisfying ( / D + m ) ψ = 0, that is charged under the U (1) is (using again the north patch of the gauge potential) ∣ ∣ ∣ ∣ where n 1 / 2 = n -σ/ 2 and σ ∈ {± 1 } corresponds to the 4D helicity of the spinor, of which there are 2 (4) each for a 6D Weyl (Dirac) spinor. Finally let us give the KK spectrum arising from a gauge field, which assume not to be in the Lie algebra direction where the background gauge field lies 5 . There a two cases: the gauge field can either be massless or massive. We begin with the massless case, when the field satisfy the equation g MN D M F NP = 0. In an appropriate gauge (e.g. light-cone gauge [26, 27, 28, 8, 9]) the 6D gauge field can be decomposed into four components, each with a spectrum (once again in the north patch of the gauge potential) given by ∣ ∣ ∣ ∣ where ξ ∈ { 0 , 0 , +1 , -1 } for each of the four components and we assume N = 0 , ± 1 to ensure stability [29, 8]. We observe that two modes have exactly the same spectrum as scalars (i.e. those with ξ = 0), while the other two (with ξ = ± 1) have almost the same spectrum. For massive gauge fields we have a more complicated situation because we need a scalar field Φ that is charged under the gauge field we are studying in order to give mass through the Higgs mechanism. In order to interpret 〈 Φ 〉 /negationslash = 0 as a 6D spontaneous symmetry breaking, we require 〈 Φ 〉 to be constant and to be at the minimum of U . Then, in order to solve the background scalar equation, D M D M Φ = 0, we also demand that 〈 Φ 〉 /negationslash = 0 does not break the U (1) where the background gauge field lies: otherwise it would not be possible to have 〈 Φ 〉 constant, at least in the sphere compactification of interest in this paper [30, 31, 25]. If we choose again the light-cone gauge, it is possible to show that a massive gauge field leads to the 4D spectrum of a massless gauge field, Eq. (19), plus that of a scalar, Eq. (17).", "pages": [ 6, 7, 8 ] }, { "title": "4. Casimir energy and bulk renormalization", "content": "The purpose of this section is to compute the Casimir energy due to bulk loops and to show how, consequently, the bulk and brane couplings renormalize. This is an important step to address any problem of technical naturalness, such as the gauge hierarchy and the cosmological constant problem. We will consider in some detail the case in which the branes are of codimension-two, but some of our results will be valid in more general brane worlds. The renormalized couplings will depend as usual on a renormalization energy µ and we will compute explicitly their RGEs.", "pages": [ 8 ] }, { "title": "4.1. A general technique to compute the Casimir energy", "content": "The starting point of this calculation is the formula for the one-loop contribution of a single field to the quantum potential in Eq. (16). Notice that, modulo terms which are independent of X , we have ln X = -∫ ∞ 0 ( ds/s ) exp( -sX ) and therefore Performing the integral in d d p and rescaling the variable s we obtain Consider first the case Re( d ) < 0 and rewrite the integral in Eq. (21) as We now extend this integral function by analytic continuation to all complex d except the non-positive integers (where the function has simple poles). By using where γ is Euler's constant, one then obtains where the dots represent finite r -independent terms. One possible renormalization scheme (which we will adopt from now on) is to subtract the divergent part in the brackets of Eq. (24). The renormalized potential V r (the Casimir energy) can then be written as where r -1 0 has to be identified as a ultraviolet (UV) scale which can be computed once the UV completion is known and It is important to notice that all we need in order to compute this coefficient is the divergent part of V , as it is clear from Eq. (24). The quantity on the right hand side of Eq. (26) is divergent and has to be regularized. Notice that the exponent d/ 2 can effectively act as a regulator for the sum over n (zeta function regularization [32]): for d = 4 the sum is divergent, but one can (and we will) compute (26) for those d such that the sum is convergent and then consider the analytic continuation of the final result at d = 4.", "pages": [ 8, 9 ] }, { "title": "4.2. The Casimir energy for rugby ball compactifications", "content": "A generic form of the spectrum which covers all the cases encountered here, Eqs. (17), (18) and (19), is and b ± , a and τ are real parameters; the only assumption we make is that b ± and a are independent of j and τ is independent of both j and n . Then the contribution to the C parameter from a single field is where ( -1) F = 1 for a boson and ( -1) F = -1 for a fermion. We can now expand { ... } d/ 2 by using the binomial series to obtain The sum over j can be performed by means of the following representation of the Hurwitz zeta function ζ ( s, c ) (valid for Re( s ) < 0 and Re( c ) > 0) Indeed setting s = d -2 k and c = c n = ( | n + b + | + | n -b -| ) ω/ 2 + a in (30) we have Observing that integrals of the form ∫ ∞ 0 dy y h e -y 1 -e -y can be computed explicitly for Re( h ) > 0, we Taylor-expand ye y ∑ n e -c n y around y = 0, and obtain We now need to take the limit d → 4. The previous expression turns out to be well defined in this limit. Also, because of the Γ( -d/ 2) in the denominator, only a finite number of k and k ' contributes to the sum: k = 0 , 1 , 2 , 3 and k ' = 0 , 1 , 2 , 3 , 4 , 5 , 6. This implies two things: ( i ) C is computable once we know n e -c n y , ( ii ) C has the generic structure Since the coefficient C can be computed from the mere knowledge of the UV divergent part of the Casimir energy, we should expect that this formula can be applied to codimensiontwo compactifications which are more general than the rugby ball one. This is because UV divergences are related to the local structure of the space-time and are therefore insensitive to global properties such as its topology.", "pages": [ 9, 10 ] }, { "title": "4.3. Bulk and brane counterterms and renormalized couplings", "content": "The models we have considered are non-renormalizable and so not all divergences can be reabsorbed in counterterms of the same form as the terms in the classical action. However, one can show in very general terms that the number of counterterms needed is always finite at a given order in perturbation theory. In this subsection we review [10] the renormalization due to bulk loops in codimension-two braneworlds (and in particular for rugby ball compactifications) at the one-loop level. At the end we will therefore obtain a finite number of counterterms and renormalized couplings. To perform this calculation we will use effective field theory methods (see for example [33]). Although only bulk loops 6 are computed here, both bulk ad brane counterterms are needed. There is an important difference between them. The bulk counterterms, unlike the brane ones, do not depend on the brane properties and so they can be computed in the sphere limit, α → 1. 4.3.1. Renormalization of the bulk interactions. To capture all the terms needed to reabsorb the UV divergences we write down the most general local Lagrangian with the chosen field content and set of symmetries, which we organize in a derivative expansion L ct B = L ct B 0 + L ct B 2 + L ct B 4 . After renormalization this generates a corresponding series of renormalized interactions L r B = L r B 0 + L r B 2 + L r B 4 . Focusing on the fields that are non-zero in the background we have ... , (39) where ¯ R 2 ( ¯ R 3 ) is a generic linear combination of terms which are quadratic (cubic) in the curvature, that is where a R + b R + c R = 1 so that ¯ R 2 = R 2 when specialized to the sphere geometry (for which R mnpq R mnpq = 2 R mn R mn = R 2 = 4 /r 4 ). A similar expression is used for ¯ R 3 . Calculations on a sphere can only provide the overall couplings ζ R 2 , ζ R 3 and not the separate parameters such as a R , b R and c R (see however [10] and references therein to know the latter quantities). Evaluating the renormalized action at the background sphere solution and integrating over the extra dimensions gives Therefore λ , ζ R , ζ R 2 , and ζ R 3 can be read off respectively from the r 2 , r 0 , r -2 , and r -4 terms in V r (see Eqs. (25) and (35)), while the ζ A and ζ AR coefficients are identified as the N 2 /r 2 and N 2 /r 4 terms respectively. This implies that integrating a bulk field with mass m gives the following contribution to the RGEs and and so on. The 'sph' in s sph , k i emphasizes that these quantities are evaluated on the sphere, while the superscript ' k ' denotes terms involving k powers of N . The renormalization of the gaugefield terms, ζ A and ζ AR , has been done by looking at the N -dependent divergences produced when a particle with charge q ˜ g runs in the loop. 4.3.2. Renormalization of the brane interactions. In this case we have a dependence on the boundary conditions used near the brane but the result should be independent of the boundary conditions on distant branes. To compute the brane contributions we first subtract the (boundary condition independent) bulk contributions found above. Noticing that the bulk counterterms should be integrated over the volume of the rugby ball, which is 4 παr 2 , we define and use δs tot i = ∑ b δs i ( b ) to extract how the interactions on each individual brane renormalize. This can be done as before, by distinguishing the interactions that depend on the gauge field which is non-zero on the background from those that do not. In the following we understand the label ( b ) in δs i ( b ) to have a simpler notation. Writing the most general local brane Lagrangian organized in a derivative expansion, L r b = L r b 0 + L r b 1 + L r b 2 + L ct b 3 + ... , and dropping terms that vanish at the background, we have and so on, where γ µν := g MN ∂ µ x M ∂ ν x N (with the right-hand side computed at the brane position) is the induced metric on the brane. Evaluating these at the background solution gives the following contribution to the Casimir energy Using Eqs. (25) and (35) then gives the following RGEs where δs k 2 are terms with k powers of N .", "pages": [ 10, 11, 12 ] }, { "title": "5. Casimir energy for rugby balls in explicit cases", "content": "In this section we apply the method of sections 4.1 and 4.2 to compute the Casimir energy, i.e. the coefficients s i , produced by specific matter fields, for rugby ball compactifications sourced by branes with tension and flux. Indeed the s i is all we need to obtain the renormalization of the bulk ad brane couplings, as it is clear from Eqs. (42)-(45) and (53). Moreover, as we shall comment later on, the s i can be used to extract the 4D cosmological constant [10, 11].", "pages": [ 12 ] }, { "title": "5.1. A single real scalar.", "content": "We observe that the method of section 4.2 can be applied in the case of real scalars because the spectrum in (17) has the form (27) with The (renormalized) Casimir energy produced by a real scalar is given by Eq. (25) with C given in Eq. (35) and one obtains the following s i coefficients: where we introduced the notation The method of sections 4.1 and 4.2 can also be applied to fermions and gauge fields as their spectra, Eqs. (18) and (19), are both of the form given in (27). The explicit expressions for the s i coefficients for fermions and (massive) gauge fields can be found in [10].", "pages": [ 12, 13 ] }, { "title": "5.2. Supermultiplets", "content": "Let us now consider supermultiplets of 6D gauged chiral supergravity focusing on the hypermultiplets and the gauge multiplets (for which we further restrict to the case in which the gauge field is zero in the background). The main reason is that the cancellation of gauge and gravitational anomalies typically require hundreds of such supermultiplets [34, 35] and so their contribution is expected to dominate the Casimir energy. For example, the first anomaly free theory of this sort that has been found has a large ( E 6 × E 7 × U (1) R ) gauge symmetry with many (456) hypermultiplets [34, 36]. A massless hypermultiplet consists of four massless scalars (called hyperscalars) and one 6D Weyl fermion, the hyperino. A massless gauge multiplets is made of one gauge field and a 6D Weyl fermion, the gaugino. By contrast, a massive 6D matter multiplet consists of a massive gauge field, a massive Dirac fermion and three scalars, a total of eight bosonic and eight fermionic states. Since this is also the number of degrees of freedom of a gauge plus a hypermultiplet, one expects to form a massive supermultiplet by having the gauge boson from a gauge multiplet 'eat' one of the hyperscalars through the Higgs mechanism. 5.2.1. Non-supersymmetric embedding of the background gauge field. We first consider the case in which the background gauge field is not embedded in the U (1) R . In this case supersymmetry is broken both by the branes and the bulk solution. Since the gauge field whose flux is localized on the branes is the U (1) R gauge field, the spectrum and the Casimir energy as well as the renormalization will not depend on Φ b for this choice of the gauge field embedding. The contribution of a hypermultiplet to the s i coefficients is obtained by summing the result for a 6D Weyl fermion to that produced by four hyperscalars 7 . We obtain where here N is the common monopole number of the hyperscalars and hyperino. For a massless ( m = 0) hypermultiplet the only coefficient which matters is s hm 2 and the corresponding Casimir energy is -s hm 2 ln( r/r 0 ) / (4 πr 2 ) 2 . As far as the gauge multiplet is concerned, we should sum the contribution of a gauge field to that of a 6D Weyl fermion, obtaining for the s i coefficients where now N is the common monopole number of the gauge field and gaugino. For a massless gauge multiplet the only important coefficient is s gm 2 and the corresponding Casimir energy is -s gm 2 ln( r/r 0 ) / (4 πr 2 ) 2 . Therefore, if one has massless supermultiplets only, the total contribution to the Casimir energy is approximately given by -( s hm 2 + s gm 2 ) ln( r/r 0 ) / (4 πr 2 ) 2 . The overall sign depends on the particular anomaly free model [34] that one chooses. For a massive multiplet made of a hypermultiplet and a gauge multiplet we have that the s i coefficients are s mm i = s hm i + s gm i . Therefore, by using the explicit expressions for s hm i and s gm i given before, we obtain mm From this result, and from Eqs. (35) and (25), we note that for mr at most of order 1 the obtained Casimir energy is ∼ ln( r/r 0 ) / (4 πr 2 ) 2 . However, for mr /greatermuch 1, integrating out a massive multiplet gives a dangerously large contribution. We will see that an extra suppression can be obtained when the gauge background is along the R-symmetry generator. 5.2.2. Supersymmetric embedding of the background gauge field. Let us now turn to the case in which the background gauge field is along the U (1) R . Hyper and gauge multiplet contributions to the Casimir energy have been computed in [11] and are quite involved. We therefore refer to this work for explicit expressions associated with hyper, gauge and massive multiplets. One important point we want to emphasize here is that s sph i = 0 for this background gauge field embedding and therefore the renormalization group equation of the bulk Casimir energy vanishes, µ∂ µ V r B = 0. The reason is that, as we mentioned, one half of bulk supersymmetry is not broken in this case. In the particular case of identical boundary localized fluxes (Φ b = Φ / 2) also the branes preserve one half of the 6D supersymmetries (see the discussion at the end of section 2). This implies that for balanced fluxes on the two branes also the brane Casimir energy does not run, µ∂ µ V r b = 0 . When the boundary localized fluxes are unbalanced a non-trivial result arises. However, by continuity the result should be suppressed by the difference of the two localized fluxes, ∆Φ. Also, as a remnant of 6D supersymmetry the coefficient s -1 for massive supermultiplets vanishes. When the bulk mass is such that mr is at most of order one the Casimir energy is ∼ ln( r/r 0 ) / (4 πr 2 ) 2 . When mr /greatermuch 1 the contribution of a massive supermultiplet to the Casimir energy is of order ∆Φ( mr ) 4 ln( r/r 0 ) / (4 πr 2 ) 2 and can be again as small as ∼ ln( r/r 0 ) / (4 πr 2 ) 2 for ∆Φ appropriately small. It is very interesting to notice that the localized flux difference does not receive quantum corrections from loops involving brane localized fields only (when they are not charged under the corresponding gauge symmetry) and therefore taking this difference to be very small does not need to have the usual fine tuning. Let us conclude this section by mentioning that the Casimir energy is not exactly equal to the 4D cosmological constant, but is rather equal to the quantum action evaluated at the classical solution. The backreaction of the brane on the bulk is important in this case and leads to a sizeble correction of the solution. This point is extensively discussed in [11]. After taking into account this effect, however, the prediction for the (most UV sensitive part of the) 4D cosmological constant can remain as small as ln( r/r 0 ) / (4 πr 2 ) 2 .", "pages": [ 13, 14, 15 ] }, { "title": "6. Conclusions and outlook", "content": "We discussed the Casimir energy as well as the renormalization of bulk and brane coefficients in the quantum action produced by integrating out (massive) bulk matter in codimension-two brane worlds. In the calculation we focused on the one-loop approximation and explicitly summed over the KK towers. Much of what we presented here is a review of [7, 8, 9, 10, 11], but we also provided some new results, in particular in the technique to perform the KK sums. Regarding the motivations, as discussed extensively in the introduction, codimension-two brane worlds may provide a framework to solve the gauge hierarchy and, when the bulk is supersymmetric, the cosmological constant problem. These are technical naturalness problems and as such they require sistematic ways of computing quantum corrections. We considered in some detail the rugby ball compactifications in which the size r of the extra dimensions is stabilized by the flux of a bulk field. A class of model having these configurations as solutions has been presented, including a concrete supersymmetric theory, 6D gauged chiral supergravity. The two 3-branes required to support these solutions can not only carry tension, but also a localized flux of the same gauge field which stabilizes the extra dimensions (tensions and localized fluxes are the leading terms in a derivative expansion of the brane Lagrangians, Eq. (5)). As discussed in [11], when the localized fluxes on the two 3-branes are identical one half of the bulk supersymmetries is unbroken, which implies that the Casimir energy and, remarkably, the 4D cosmological constant vanish. For the rugby ball compactifications the KK spectra of many types of bulk fields are known [7, 8, 9, 10, 11] and we reviewed their structure for scalars, fermions and gauge fields (see Eqs. (17), (18) and (19) rescpectively). The explicit form of the spectra allows us to have explicit expressions for the Casimir energy and, consequently, for the renormalization group equations, which depend on the flux and brane tensions and fluxes. The calculation of the Casimir energy starting from the KK spectrum exploits a novel efficient technique (see sections 4.1 and 4.2) which allows us to confirm the results of [10]. The explicit form for the Casimir energy we obtain is a polynomial function of mr of degree six, see eqs. (25) and (35). The coefficients s i of the polynomial are computed explicitly as a function of the brane tensions and bulk and brane fluxes; for example, in the simple case of a bulk scalar they are given in Eqs. (55)-(58). When the bulk particle that is integrated out is massless, or m is at most of order 1 /r , the 4D cosmological constant has the desired order of magnitude regardless of the fact that there is bulk supersymmetry. We show that for large m the final result can be appropriately suppressed if we select the supersymmetric model, the bulk solution preserves one of the supersymetries and localized fluxes have very similar values, such that we are close to a supersymmetric setup. Let us mention some outlook of the results presented here. A possible extension of our work is the inclusion of warped geometries, which represent the most general solutions with 4D maximal symmetry. A first step towards this goal is the codimension-one case, which would be interesting by itself as the Randall-Sundrum model can address the gauge hierarchy problem [37] (bulk fields in the Randall-Sundrum model have been considered in [38]). Through the AdS/CFT correspondence quantum loops in the bulk would correspond to 1 /N c corrections in the CFT side, where N c counts the number of 'colors'. Regarding again holography, we notice that the formalism of Refs. [7, 8, 9, 10, 11] to compute the spectrum for codimension-two brane worlds that we reviewed here can also be used to analyze spectral properties of holographic models in the confined phase [39]: these are obtained from models with one extra dimension (the holographic coordinate) by an additional compactified dimension and scalar, vector and fermion fields in the bulk can have a variety of uses ranging from condensed matter 8 [41] to quantum chromodynamics [42]. Finally, an interesting property of supersymmetry with two large extra dimensions is that it could provide a link between the observed value of the cosmological constant and the scale at which modifications of gravity should occur, 1 /r [43]. It would be interesting to know how gravity gets modified in the concrete supersymmetric model we discussed. In the absence of localized fluxes graviton contributions have been computed in [44], but the role of these fluxes, which are important for the dilaton stabilization, remains an interesting target for future research.", "pages": [ 15, 16 ] }, { "title": "Acknowledgements", "content": "We would like to thank Cliff Burgess, Leo van Nierop, Susha Parameswaran and Matt Williams for collaborations, Hyun-Min Lee for much help trying to diagonalize the supergravity sector in early stages of this work and Riccardo Barbieri, Oriol Pujol'as, Seifallah Randjbar-Daemi and George Thompson for useful discussions. This work was partly supported by the EU ITN 'Unification in the LHC Era', contract PITN-GA-2009-237920 (UNILHC) and by MIUR under contract 2006022501.", "pages": [ 16 ] }, { "title": "References", "content": "[28] Randjbar-Daemi S and Shaposhnikov M 2002 Nucl. Phys. B 645 188 ( Preprint hep-th/0206016).", "pages": [ 17 ] } ]

[ { "title": "Rafael D. Sorkin", "content": "Perimeter Institute, 31 Caroline Street North, Waterloo ON, N2L 2Y5 Canada and Department of Physics, Syracuse University, Syracuse, NY 13244-1130, U.S.A. address for email: [email protected]", "pages": [ 1 ] }, { "title": "Abstract", "content": "If a wave function does not describe microscopic reality then what does? Reformulating quantum mechanics in path-integral terms leads to a notion of 'precluded event' and thence to the proposal that quantal reality differs from classical reality in the same way as a set of worldlines differs from a single worldline. One can then ask, for example, which sets of electron trajectories correspond to a Hydrogen atom in its ground state and how they differ from those of an excited state. We address the analogous questions for simple model that replaces the electron by a particle hopping (in discrete time) on a circular lattice. Keywords and phrases : quantum foundations, histories, path-integral, coevent formulation, anhomomorphic coevents, quantum logic. Should we try to form for ourselves an image of the quantum world? Or must our theories find their meaning solely in assertions about laboratory instruments and their readings? In other words, is it permissible to ask the question, To what 'reality' does the quantum formalism refer? I believe that this question is not only a legitimate one, but it is one that we must ask. If we try to avoid it, we fall into a vicious circle because instruments are made of atoms and not vice versa. More importantly, quantum gravity and, especially, cosmology need to deal with parts of nature where one finds neither observers nor instruments, and to which an 'operational point of view' therefore seems unsuited. Beyond the more or less familiar reasons just adduced to support the claim that the problems of quantum gravity and 'quantum foundations' are intertwined, there's another connection that could be important via the concept of relativistic causality. If the condition that 'physical influences propagate causally' could be given an intrinsic formulation, free of references to external observers, and if the resulting criterion were formulated in terms of histories (as Bell's 'local causality' is, for example), then one should be able to decide whether or not quantum mechanics and quantum field theory satisfied this condition. If they did, then it would make sense to require it also of quantum gravity, and this in turn could be the key to constructing a viable quantum dynamics for causal sets. Perhaps the challenge of 'quantum foundations' is not so urgent for quantum computing, which is concerned more with the manipulation of information than with microscopic reality as such. But even there, it seems possible that a more definite picture of the micro-world could someday lead us to widen our conceptions of measurement and of computation. What I hope to illustrate in this paper is a possible answer to the italicized question posed above, an answer that arose from thinking of quantum mechanics in the language of histories, i.e. the language of the path integral. According to this answer, the microworld is described by something called a coevent , but instead of attempting to define that concept in the general case, I will consider a very simple model of a particle hopping on a lattice, and in that setting, will present a short calculation that I hope will indicate more concretely how the proposal is meant to go. In the particular scheme I will present, reality will be something like a trajectory or worldline of the hopping particle, but instead of being a single trajectory, as in classical physics, it will be a set of trajectories. This particular choice is not necessarily the best one, but it is the simplest and therefore appropriate to illustrate the main idea. Notice that in a coevent formulation, reality is not represented by a wave function ψ . Therefore, although a concept like position will have a straightforward meaning, a concept like momentum or energy will not. But then we have to ask how an electron in a hydrogen atom 'knows' for example, whether the atom is in its ground state or in an excited state. Is there something about the trajectories that carries this information? Hence the title of this paper.", "pages": [ 1, 2, 3 ] }, { "title": "I. A simple unitary model - the n-site hopper", "content": "A simple model allowing one to refer to trajectories and ground/excited states is the ' n -site hopper', by which I mean a particle residing on an n -site periodic lattice, and at each of a discrete succession of moments either staying where it is or jumping to some other site, the respective amplitudes being those given by the simple 'transfer matrix' reproduced below. [1][2] In this 'toy world', nothing exists beyond the hopper itself, and since both space and time are discrete, the possible realities or coevents can be computed with minimal difficulty. In order to describe the hopping amplitudes, let us identify the nodes of the lattice with the elements of Z n , the integers modulo n . Further let x ∈ Z n be the location of the particle at some moment and let x ' be its location at the next moment, and write for brevity exp(2 πiz ) ≡ 1 z . The amplitude to go from x to x ' in a single step is then for n odd, and for n even. For example, for n = 6 and with q = 1 1 / 12 , the (un-normalized) amplitudes to hop by 0, 1, 2 or 3 sites respectively are q 0 = 1, q 1 = q , q 4 , and q 9 = -i . It is not difficult to verify that these amplitudes (more precisely the matrix they comprise) are unitary. Interestingly, they take precisely the form of the propagator of a non-relativistic free particle in one-dimension, suggesting that in a suitable n →∞ limit, this hopper model could provide a fully self-consistent regularization of the path-integral for such a particle. For the 2- and 3-site hoppers, the amplitudes are particularly simple, yielding for n = 3 the matrix and for n = 2 the matrix", "pages": [ 3, 4 ] }, { "title": "II. Review of anhomomorphic coevents and the 'Multiplicative Scheme'", "content": "In the formulation I am advocating, 'nature' is represented in terms of histories , which for present purposes means trajectories of the hopping particle. [3][4][5] Physical reality - a 'possible world' - is then described by a coevent which specifies which events happen and which don't, where an event is by definition a set of histories. /star More formally, a coevent will be a function φ that assigns either 0 or 1 to each event, according as the event doesn't or does happen in the world described by φ . From this perspective, our description of the physical world would be complete if we were able to specify fully the actual coevent φ . The role of 'dynamics', then, is to help us toward a fuller such specification by placing conditions on φ that narrow down the range of possibilities that need to be considered. I will assume that the input to this dynamics takes the form of a path integral (or in the case of the hopper, a path sum). Wave functions ψ , insofar as they play a role at all, will provide provisional initial amplitudes that go into the computation of the path-sum (as an approximate summary of past). But is it possible to base a dynamics on a path-sum alone? If we wish to do so then, plainly, we must construe the latter as something more than a technical device to compute transition amplitudes between some initial wave-function and some final one. Instead we will interpret it as providing for any event A , the quantal measure µ ( A ) of that event. † Once again, I will omit the formal definition [6][7][8], since we will see very soon how concretely to compute µ ( A ) in the case at hand.", "pages": [ 4 ] }, { "title": "The preclusion principle", "content": "It is not hard to convince oneself that in one special case µ ( A ) has the meaning of an ordinary probability, namely when the event A can be described as a possible outcome ('instrument reading') of a given laboratory experiment. In such a case, the analyses of numerous gedankenexperiments over the years have made it plausible (albeit people don't always express it this way) that µ ( A ) coincides with the Born-rule probability of the outcome A . But if this is accepted, then it follows immediately that a zero value of µ ( A ) implies that the corresponding instrument-event almost surely does not occur it is precluded . /flat Extending this conclusion to the case of an arbitrary - macroscopic or microscopic - event, we arrive at a dynamical principle of general applicability: If µ ( A ) = 0 then the event A cannot happen.", "pages": [ 5 ] }, { "title": "Preclusion and primitivity", "content": "The preclusion principle requires of any dynamically viable coevent φ that it deny every event whose quantal measure vanishes: φ ( A ) = 0 if µ ( A ) = 0. We have seen that this principle flows naturally from the path-integral, but by itself it is still rather weak, in the sense that a vast number of coevents can satisfy it even when preclusions abound. In our hopper example, for instance, we will have 27 histories, and consequently 2 2 27 = 2 134217728 coevents in toto. The number of precluded events is also large (2017807), but even so, there remain 2 132199921 coevents which are preclusive in the sense that they satisfy our condition. On the other hand, 2 132199921 is only a tiny fraction of 2 134217728 , so one might feel on the contrary that the preclusion principle is rather strong. Be that as it may, I think the weightiest reason why the preclusion principle cannot stand alone is that it seems incapable of yielding the classical conception of reality-asa-single-history when the measure µ is classical (and possibly 'deterministic'). Thus, preclusion alone would not exclude coevents for which an experiment had more than one macroscopic outcome, or for which no definite outcome at all happened. In other words it would not resolve the 'measurement problem'. To complete the dynamical story, then, we will supplement preclusion with a further principle of 'minimality' or primitivity designed to remedy the deficiencies just cited. How properly to frame such a supplementary condition is a question not yet settled, but the simplest proposal is that of the so-called multiplicative scheme , and this is the one I will adopt for the present analysis.", "pages": [ 5, 6 ] }, { "title": "the multiplicative scheme", "content": "Classical physics identified reality with a single history, but that no longer seems possible quantum mechanically because the characteristic phenomenon of interference produces non-classical patterns of preclusion which seem to demand a modified conception of reality. In place of a single history, the multiplicative scheme describes the physical world by a coevent of the form φ = F ∗ , where F is now a set of histories that reduces to a singleton set only in very special (effectively classical) circumstances. As illustrated in figure 1, the coevent F ∗ assigns 1 ('true') to an event A if and only if A is a superset of F . Thus in the diagram, if φ = F ∗ then φ ( A ) = 1 while φ ( B ) = φ ( C ) = 0 . When φ = F ∗ I will refer to F as the support of φ . Within the multiplicative scheme, a coevent is thus fully determined by its support. /star Now when is a coevent φ = F ∗ 'dynamically viable' within the multiplicative scheme? By assumption it must be preclusive, and this means precisely that its support F must not fall wholly within any precluded event. Beyond this, we require further that F be as small as possible consistent with the preclusivity condition just stated. A coevent (or its support) that fulfills all these conditions I will call primitive preclusive , or for short just primitive . A primitive coevent thus describes a 'possible world', where 'possible' is to be understood relative to the given set of preclusions. One sees immediately that in the absence of any preclusions (other than the empty event itself) a primitive preclusive support will consist solely of a single history, and the same holds whenever the pattern of preclusions is of the type that occurs in either classical deterministic, or classically stochastic theories. Much more than this could be said about the multiplicative scheme and its consequences, [9][10][11] but now I want to focus on a very concrete example and on our specific question: Does the particle know its own energy?", "pages": [ 6, 7 ] }, { "title": "III. Histories and amplitudes for the 3-site hopper", "content": "In simple cases, preclusion is decided by whether the sum of the amplitudes vanishes. This will be true for our example, and it will make it easy to find the primitive supports. To simplify as much as possible, let the hopper take three steps and then stop. And let there be nothing else in the world beside this hopper. Our space of histories then comprises exactly 81 trajectories, depending on where the hopper starts from and where it lands at each of the three subsequent moments. In fact, however, we only need to consider the 27 histories shown in figure 2, because one can prove, as a general feature of the multiplicative scheme, that any primitive support must correspond to a sharp final position. That is, one of the three events, 'the hopper terminates at 0', 'the hopper terminates at 1', or 'the hopper terminates at 2', must happen. Without loss of generality we can suppose it is the first of these events, and this is what the figure illustrates. With the final position fixed at x = 0, it is also very easy to decide whether a given set of histories is precluded: this occurs iff the amplitudes of the constituent histories sum to zero. To work out the pattern of preclusions, it thus suffices to know all the amplitudes. Now given a history γ , its net amplitude is the amplitude it inherits from its starting location, multiplied by the amplitudes of the individual hops, the former being given by what one might call the 'initial wave-function' ψ initial . For ψ initial let us consider two possible choices, ψ 0 and ψ + , which we may call by way of analogy 'ground state' and 'traveling wave'. The amplitudes for these are respectively ( ψ (0) , ψ (1) , ψ (2)) = (1 , 1 , 1) and ( ψ (0) , ψ (1) , ψ (2)) = (1 , ω, ω 2 ), where ω = 1 1 / 3 as before. Clearly both ψ 0 and ψ + are eigenvectors of the 'transfer matrix' defined earlier. (The overall normalization of ψ initial is immaterial since it has no effect on preclusion.) It is now straightforward to compute the amplitudes of our 27 histories in each case. The figure exhibits them for the traveling wave, and those for the ground-state are similar (but even simpler to compute since the dependence on the starting position is absent.) Curiously, the amplitudes per se (though not of course the way they are distributed among the histories) turn out to be exactly the same for both cases. Counting them up, one obtains the following multiset: meaning 12 histories with amplitude ω 2 = ω , 9 with amplitude 1, and 6 with amplitude ω .", "pages": [ 7, 8 ] }, { "title": "IV. Primitive coevents for the ground state and the traveling wave", "content": "Which combinations of the histories illustrated in figure 2 yield primitive coevents? If we think in terms of the amplitudes abstractly, it is easy to answer this question, and the same answer will then apply unchanged to both cases of traveling wave and ground state. An event E will correspond to a set of amplitudes (strictly speaking a multiset), and E will be precluded precisely when the amplitudes sum to zero. Because the only real-linear relation among the complex numbers 1, ω , ω is 1 + ω + ω = 0, this will occur when, and only when, the three amplitudes occur in equal numbers within E . It is then easy to see what are the maximal precluded multisets of amplitudes. They are those consisting of 6 copies each of 1, ω , and ω . Now let F be the support of a preclusive coevent. In order to be preclusive, F must not fall wholly within any precluded event, and this means that it must not be possible to adjoin further amplitudes to those of F such that the resulting multiset sums to zero. Plainly F will be protected in this way iff it contains at least 7 copies of 1 or 7 copies of ω . On the other hand, we also want F to be primitive, meaning minimal among the preclusive supports. Again it is easy to see what this means: it must comprise precisely 7 copies of 1 or 7 copies of ω . In this way, we find a total of ( 12 7 ) + ( 9 7 ) = 828 primitive coevents of the multiplicative form φ = F ∗ , each made up of a total of 7 histories. In the case of the traveling wave, for example, one such set of histories corresponds to the last three patterns in the first row of figure 2. Interestingly, all seven of these trajectories move in the positive (counterclockwise) direction and none of them in the contrary direction. Thus the event, P = 'The particle circulates exclusively in the positive sense', happens in the reality described by this coevent. Happily enough, this sense of circulation corresponds perfectly with the 'phase velocity' of ψ + , but we cannot assert that all of the 828 traveling-wave coevents also affirm this same event P . In order to quantify the tendency toward counterclockwise motion, then, let us associate a 'net circulation' with each coevent, as the total number of 'forward' hops less the total number of 'backward' ones. (So for example the above coevent has a net circulation of 3 × 1+3 × 2+1 × 3 = 12.) Averaging this quantity over all 828 primitive coevents yields an average net circulation of 7 / 23. A tendency toward counterclockwise motion is therefore present but not extremely pronounced - just as one might expect since a lattice of only n = 3 positions can lend no more than a very rudimentary meaning to a derivative like 'd(phase)/d(angle)'. The primitive preclusive coevents for the ground-state are again supported on sets of seven histories, as we have already remarked. By symmetry we cannot expect a favoured sense of circulation in this case, but it is also interesting to ask how 'restless' the particle proves to be. Here, it is natural to compare with 'Bohmian' or 'pilot wave' conceptions of reality, since they also give meaning to the notion of particle-trajectory. As far as I know, the Bohmian 'guidance equation' is limited to continuum space and time, where the nearest analog to our hopper ground-state might be the ground-state of a particle in a box. Since the phase of the Schrodinger wave-function is independent of position in that case, the Bohmian particle does not move at all. Rather than explore its surroundings, it just stays put, wherever it happens to find itself. † It turns out that the coevents of the multiplicative scheme paint a very different picture. The event, 'The hopper never moves' is of course denied by all 828 coevents, while the contrary event that 'The hopper never rests' is affirmed by eight of them. (For such a coevent, all seven of the histories in its support are in constant motion.) Of the other 820 primitive coevents, 28 of them affirm that the hopper either never rests or never moves (6 histories vs. 1), while the remaining 792 primitive supports consist entirely of histories such that the hopper rests once and hops twice. Moreover, for none of the coevents does the event 'the hopper avoids some lattice-site' happen. /flat All in all, very peripatetic indeed. /flat Which of course doesn't mean that the complementary event, 'the hopper visits all three lattice sites' ever happens either. That's why the coevents are 'anhomomorphic': with φ = F ∗ defined as above, it can easily happen that two events A and B are complementary subsets of the space of histories, but both φ ( A ) and φ ( B ) vanish.", "pages": [ 10, 11 ] }, { "title": "V. Does the hopper know its energy?", "content": "If the multiplicative scheme (or any other coevent scheme) is a kind of 'equation of motion for coevents', then a primitive preclusive coevent is a kind of 'solution of the equations of motion'. In celestial mechanics such a solution would be the orbit of a planet. One can deduce the binding energy of the planet from a knowledge of its orbit, but the converse is impossible since the energy is only one among a number of orbital parameters. Now consider some microscopic counterpart of this problem, like a hydrogen atom. By analogy one should not expect to deduce a unique coevent φ from a knowledge of the total energy, but one might wonder whether, conversely, the coevent determines the energy unambiguously. Quantum mechanically, energies are determinate only for eigenfunctions of the Hamiltonian operator, and the nearest analogs for these in our hopper-world are the initialamplitude sets, ψ 0 and ψ + (plus, of course, the parity-reversed set ψ -, together with linear combinations like the 'standing wave', ψ + + ψ -). We are thus led to ask whether the primitive coevents pertaining to ψ 0 overlap with those pertaining to ψ + . To the extent that the answer is negative one can indeed deduce the hopper's energy from a knowledge of the coevent that describes its motion. Based on the above analysis of the primitive coevents, the required calculation is straightforward, and the result is that the degree of overlap is zero. No coevent is common to both ψ 0 and ψ + . Moreover it is possible to distinguish ψ 0 from ψ + in terms of relatively elementary consequences. For example if the hopper never rests (meaning φ ( E ) = 1, where E is the event consisting of the fourth, seventh, and tenth through fifteenth histories shown in figure 2), then φ pertains to ψ = ψ 0 . Similarly, if the hopper moves only counterclockwise, or if it rests exactly once, then ψ = ψ + . In this sense, we can say that the hopper does know its own energy. We can also say that it 'knows its own angular momentum', since a similar comparison reveals that the primitive coevents pertaining to ψ + are disjoint from those pertaining to ψ -. Of course the example we've studied is exceptionally simple, both with respect to the hopping amplitudes and the amplitudes of the 'initial states' ψ initial which we have considered. It would be good to analyze also the case of the standing wave, and more generally to extend the analysis to include longer times, and larger lattice sizes n . (The case of shorter times is also instructive. For only two time-steps, one finds that there do exist primitive coevents common to ψ + and ψ 0 . This strengthens the impression that with the passage of time the hopper would 'know' more and more about ψ initial .) Our hopper-world is exceptionally simple in another way too, that relates to the radical inseparability (or 'interconnectedness') that seems to show up in the quantum world. Were we to include a second system or process in our idealized world, say for example a four-site hopper, the coevents would change. Obviously the global coevents would change, but even those induced for the three-site hopper would in general be different. This at least is a feature of the multiplicative scheme, and it is likely true more generally. It is therefore important to study extensions of our model of this type. Extensions of our model in any of the directions just mentioned would of course be interesting. But even without them, I hope the examples studied in this paper suffice to illustrate how one can start to think about the quantum world without invoking the ideas of either evolving wave-functions or external observers. If these examples are not misleading us, then we can already draw some conclusions of more general validity, concerning first of all the relation between wave-functions and descriptions of reality. A histories-based or 'path-integral' formulation of the sort we have been working with has no use for the Schrodinger equation at a fundamental level. The basic concept of precluded event has a 'spacetime character' and refers directly to the histories and their amplitudes, not to any evolving wave-function. On the other hand, something like a wave-function does enter into the 'initial conditions' needed in setting up the path-integral. In principle one should probably replace these initial amplitudes with cosmological boundary conditions imposed directly on the histories, but in our hopperworld there is no moment earlier than t = 0, and we have assumed instead that in a more complete model, we would be able to condense the effect of the true boundary conditions into a set of three initial amplitudes for the hopper at t = 0. This is the wave-function ψ initial in its role as 'effective summary of the past'. What our hopper model teaches us about ψ initial is that it is far from furnishing a detailed description of physical reality. Rather, the coevent which by definition does furnish such a description is determined by ψ initial only to a very limited extent. Reality possesses far more 'internal structure' than is reflected in a wave-function. This observation in turn resolves an old paradox that has recently been emphasized in a cosmological setting by Daniel Sudarsky [13]. In its terrestrial form the paradox asks how the spherically symmetric wave-function resulting from the decay of a spinless nucleus can be compatible with the fact that the daughter nuclei will be found to be localized in angle if one sets up detectors. How does the symmetry break? But in terms of coevents, there's no problem. Just because the ψ -function is symmetric, that doesn't imply the same of the individual coevents. Indeed, we see exactly such a phenomenon in our hopper model, where ψ initial exhibits perfect rotational symmetry while the individual coevents are completely asymmetric. No single coevent shares the symmetry of ψ initial , but only the ensemble of all 3 × 828 of them taken together. For discussions of these questions over a long period, I would like to thank Fay Dowker, Cohl Furey, Yousef Ghazi-Tabatabai, Stan Gudder, Joe Henson, David Rideout, Sumati Surya, and Petros Wallden. I also thank the Foundational Questions Institute (FQXI) for a grant, and the maintainers of Steel Belt Common Lisp (SBCL), whose software I used at several points in the analysis reported above. Finally, I thank the members of the Raman Research Institute (RRI) for their warm hospitality while parts of this paper were being written. This research was supported in part by NSERC through grant RGPIN-418709-2012. Research at Perimeter Institute for Theoretical Physics is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI.", "pages": [ 12, 13, 14 ] } ]

[ { "title": "Petros Wallden", "content": "E-mail: [email protected], [email protected]", "pages": [ 1 ] }, { "title": "Abstract.", "content": "Understanding quantum theory has been a subject of debate from its birth. Many different formulations and interpretations have been proposed. Here we examine a recent novel formulation, namely the coevents formulation. It is a histories formulation and has as starting point the Feynman path integral and the decoherence functional. The new ontology turns out to be that of a coarse-grained history. We start with a quantum measure defined on the space of histories, and the existence of zero covers rules out single-history as potential reality (the Kochen Specker theorem casted in histories form is a special case of a zero cover). We see that allowing coarse-grained histories as potential realities avoids the previous paradoxes, maintains deductive non-contextual logic (alas non-Boolean) and gives rise to a unique classical domain. Moreover, we can recover the probabilistic predictions of quantum theory with the use of the Cournot's principle. This formulation, being both a realist formulation and based on histories, is well suited conceptually for the purposes of quantum gravity and cosmology.", "pages": [ 1 ] }, { "title": "1. Motivation", "content": "Quantum theory is undoubtedly one of the most successful theories. The understanding and interpretation of quantum theory, has been a subject of debate from the birth of the theory until now. There is no unique, widely accepted interpretation of quantum theory. In the recent years, due to the developments in the field of quantum information and the search of a quantum theory of gravity and cosmology, there is a new wave of interest for the foundations of quantum theory. This contribution, is a review of a novel formulation of quantum theory namely the 'coevents formulation'. The motivation for such a formulation, is twofold. First is the need for a realist interpretation of quantum theory. The standard view of quantum theory, is that of an instrumentalist. Avoids to refer to the actual ontology, and makes assertions only related with some idealised measurements carried out by an external observer. However, particularly for the field of quantum cosmology where one treats the full universe as a quantum system, the need for an interpretation that does not require an external observer becomes vital. In this sense the need for a realist interpretation, becomes more than a philosophical investigation. Second is the need for a formulation that treats space and time in equal footing. In constructing a quantum theory of gravity, one is led to the observation that while space and time are totally different objects in quantum theory, they are more-or-less the same in general relativity. This leads to a tension that has both technical and philosophical difficulties (e.g. see the problem of time [1]). We can claim that better suited for the purpose of quantum gravity should be a formulation of quantum theory that is based on full histories of the system (such as in the Feynman path integral) and not a canonical formulation (that relies on the Hamiltonian and uses one-moment-of-time propositions). The latter may not even be applicable for certain approaches to quantum gravity such as the causal sets. The coevents formulation was introduced by Rafael Sorkin in [2] following his earlier work in [3] (see section 3). It is a realist interpretation, that is based on histories. In particular, the starting point is the Feynman amplitudes for histories, and interpreting those amplitudes in an observer-free, context-independent way is the aim of the formulation. In section 2 we will give the standard histories view for classical physics, state which is the importance of precluded sets and stress which is the problem in adopting a 'naive' realist view in quantum physics. In section 3 we will introduce the coevents formulation, giving two different (but equivalent) views. In section 4 we will explore (briefly) certain developments of the formulation. In particular we will define what a classical domain is, show that there exist a unique, context-independent such domain and point out that this is not the case for the consistent histories approach 1 (section 4.1). We will give an account of how probabilistic predictions arise (section 4.2), and point out the role and the status of the initial state in the formulation (section 4.3). In section 5 we will summarise and conclude.", "pages": [ 1, 2 ] }, { "title": "2. Histories, classical physics and quantum measure", "content": "We are taking a histories view. Here we will review classical physics casted in histories perspective and stressing certain important features that become important in quantum physics.", "pages": [ 2 ] }, { "title": "2.1. General Structure", "content": "In describing physics with histories, there are three important mathematical structures. Each element of this space h ∈ Ω, corresponds to a full description of the system, specifying every detail and property. For example, a fine grained history gives the exact position of the system along with the specification of any internal degree of freedom, for every moment of time. For a single non-relativistic particle, Ω would be the space of all trajectories in the physical space. Along with Ω, we define the space U , which consists of all the subsets A ⊆ Ω and the Boolean algebra associated with them (called events algebra), where the addition is defined as the symmetric difference between subsets A + B := A /triangle B and the multiplication given by the intersection of subsets A · B := A ∩ B . Each subset A ⊆ Ω is called event , as in probability theory. Note, that all physical questions that one can ask, correspond to one of those subsets. If, for example, one wishes to ask 'was the system at the region ∆ at time t ?', it corresponds to the subset A defined as { A : h i ∈ A iff h i ( t ) ∈ ∆ } , i.e. all histories that the system at time t is in the region ∆. An important observation, is that maps φ that are homomorphisms 2 , are in a one-to-one correspondence with single elements h of the space of histories Ω. In particular, maps that are the characteristic function for a particular history h , are homomorphisms between the Boolean event algebra and the Boolean truth values algebra. Moreover, all homomorphisms are of this type. There exists one homomorphic map for each one of the (single) elements h of Ω. Due to the one-to-one correspondence of homomorphic maps and single elements h , we could adopt a dual view, and state that classical reality is a homomorphic map φ h between the event algebra U and the truth values algebra T . To sum up, for classical physics, potential realties can be viewed either as single histories h elements of Ω or as homomorphic maps φ h . The potential realities are further restricted by the dynamics. As we will see later in section 3, the ontology in quantum physics changes, but the existence of a dual picture of reality as an event or as a map is maintained. We have explored, so far, the 'kinematic' part of the theory. We have not mentioned the dynamics (Hamiltonian), or initial condition. In classical deterministic physics, given the initial state and the dynamics, we know all the evolution of the system, and thus the full history. In other words, we know which history h from the space of histories Ω is actually realised. Given the initial state and dynamics, there is only one possible potential reality h , the one that is actually realised. This is the meaning of determinism. Obtaining predictions, becomes trivial, since an event is true if and only if it contains the one realised history. We can define a classical measure µ c on U , such that µ c ( A ) = 1 if h ∈ A and µ c ( A ) = 0 otherwise. This gives the probability that an event A occurs, which is always either one or zero. This simple measure, coincides with the truth valuation φ h , of the single history that is realised, but should not be confused, since the analogy holds only for deterministic classical physics. /negationslash In classical stochastic physics, the picture is different. We are not given which history is actually realised, but given the initial state and dynamics, we can obtain a classical measure µ c (non-trivial this time) on the space U of subsets of Ω. The measure of each event, corresponds to the probability that this event occurs. The measure is no longer related with the valuations φ ∈ M . It is important to note here, that the actual realised reality in stochastic physics, will still be one fine grained history h element of Ω. Since our theory is no longer deterministic, the space of potential such histories has many elements and in particular (for finite histories space) all fine grained histories h i that have non-zero measure, µ c ( h i ) = 0. The role of the measurement in stochastic classical physics, is restricting further the potential realities. For example, before tossing a coin, two outcome were possible, but after performing the measurement and looking the outcome, it is determined wether the outcome was heads or tails. In histories formulation, the latter corresponds to the experimenter restricting the set of possible histories in the universe to those that are compatible with his new observation. To sum up, in stochastic classical physics, the ontology remains the same but the mechanism to obtain predictions changes. We will see more on probabilistic predictions for closed classical or quantum systems, in section 4.2 after having introduced the coevents formulation.", "pages": [ 2, 3, 4 ] }, { "title": "2.2. Precluded sets", "content": "As we have seen, the set of potential realities, both in deterministic and stochastic classical physics, is determined by the classical measure that is defined on Ω, which in its term is fixed by the initial state and dynamics. More specifically, for finite Ω, it is the measure zero sets that determine which are the allowed potential realities/histories 3 . We define precluded event to be an event P ⊂ Ω such that its measure is zero µ c ( P ) = 0. The precluded events do not occur. If an event P is precluded, any subset P 1 of P , should also be ruled out. The latter is a vital condition that needs to be satisfied, if one wishes to make any deductive argument 4 . In classical physics, this condition is guaranteed by the fact that we require the maps/elements of M , to be homomorphisms of the Boolean algebras. Simply requiring the condition µ c ( P ) = 0 ⇒ φ ( P ) = 0 leads directly to φ ( P 1 ) = 0 if P 1 ⊆ P and φ ( P ) = 0. While this discussion seems trivial for classical physics, it will become apparent that it is important for quantum physics. A final interesting remark, concerning precluded sets, is that one can fully reconstruct all probabilities using only the set of precluded events provided that he has n →∞ identical copies of the system. Heuristically, any distribution of outcomes of the identical copies, that differs from the one given by probability, has small chance of occurring, and as the number of copies tends to infinity, this chance of occurring also tends to zero. Technically, let A an event and A n independent copies. There exists a unique number p such that the following event to be precluded ( I A is indicator function) See for example [4]. In this sense, all the content of relative frequency interpretation of probability, is included in the precluded sets. Moreover, the precluded sets can be used to recover some predictions via the Cournot principle (see later), even in the absence of many identically prepared copies. This is one more reason why, we choose to give specific importance to precluded events.", "pages": [ 4, 5 ] }, { "title": "2.3. Quantum measure", "content": "The picture described above, cannot be (fully) carried over to quantum theory. The histories space Ω and its subsets/events U remains the same. The main difference arises, mathematically, from the fact that we no longer have a classical measure on U but rather a quantum measure , which we will shortly define. Given an initial condition and dynamics the quantum measure is totally fixed. The initial condition and the dynamics can be given either in form of some initial condition on a path integral along with an action S , or as an initial wavefunction in a Hilbert space along with a Hamiltonian operator. To define the quantum measure (which was first done by Sorkin in [5]) we need to introduce amplitudes to histories. Starting from Feynman's path integral approach, one can assign an amplitude (complex number) to each history. which depends on the initial state and on the dynamics of the system encoded in the action S . Obtaining the transition amplitudes from ( x 1 , t 1 ) to ( x 2 , t 2 ), is done by summing through all the paths P obeying the initial and final condition, i.e. The mod square of this amplitude is the transition probability. One can extend this to any event A ⊆ Ω and proceed to define a quantum measure µ : where the initial state ρ and a final time δ -function is added. Alternatively, one can take a Hamiltonian view, and obtain the quantum measure in the following way. Define the class operator C A Where U ( t ) is the unitary evolution operator that relates to the Hamiltonian via U ( t ) = exp( -iHt ), and P A i is the subspace that history A lied at time t i . This class operator corresponds to the history that the system is at P A 1 at t 1 and at P A 2 at t 2 etc. The projection operators can be at any subspace of the Hilbert space. The quantum measure is then defined to be For cases where there is a well defined time, such as in non-relativistic quantum mechanics, the two definitions are equivalent and one uses the more convenient one. For finite moments of time histories, the operator expression is usually easier to deal with 5 . Note, that the quantum measure is closely related with the decoherence functional [6], since it arises from the diagonal parts of the latter. Since the quantum measure is a non-negative function that is also normalised ( µ (Ω) = 1), one could be tempted to interpret the quantum measure as probability. However this is not possible, due to interference. The additivity condition of probabilities 6 , is not satisfied: /negationslash However a weaker condition holds that shows that there is no three-paths interference, that cannot be deduced from pairwise interference: In other worlds, the quantum measure is fully determined once we know the quantum measure of single histories and of pairs of histories. The specific expression is given for example in [7]: Recently, experiments [8] have confirmed that indeed the condition of eq. (11) holds in nature. To fully characterise a histories theory, we need the following triplet: The histories space, the events algebra and the quantum measure (Ω , U , µ ). /negationslash A partition P of the histories space Ω, is a collection of events { P 1 , P 2 , · · ·} such that P i ∩ P j = ∅ for all i = j and ⋃ i P i = Ω. It is an exhaustive and exclusive collection. Each event of the partition is called cell . A coarse-graining of a histories theory ( P , U P , µ P ) is defined in the following way:", "pages": [ 5, 6 ] }, { "title": "2.4. The trouble with quantum theory", "content": "It is already clear, that since the quantum measure cannot be understood as probability, the picture of classical stochastic physics presented in the previous section might need modification. However, the main reason why one cannot interpret the quantum measure in the same way as the classical measure, appears from consideration of the precluded events. Note that from now on, when we mention precluded events we mean that they have quantum measure zero. In many cases (see below for examples), one can find a collection of precluded events Z = { P 1 , P 2 , · · ·} , where µ ( P i ) = 0, that also cover the full histories space Ω, i.e. that ⋃ i P i = Ω. We will call such collection of precluded events as a zero cover . Note, that since in general P i ∩ P j = ∅ , this collection does not constitute a partition. /negationslash However, as we mentioned earlier, in order to be able to construct deductive arguments, the truth value of any subset of a precluded set has to be 'False'. Having the full Ω covered with precluded sets, imply that no single history h i ∈ Ω can have a truth value 'True'. This also leads to no homomorphic map φ ∈ M existing. We will first consider a simplified example, with a three-slits interference experiment. Consider a point at the screen, where crossing slit A destructively interferes with crossing through slit B and slit B destructively interferes with slit C, i.e. µ ( { h 1 , h 2 } ) = 0 , µ ( { h 2 , h 3 } ) = 0, however µ ( { h 1 , h 2 , h 3 } ) = 0 (see Figure 1). /negationslash Since the quantum measure for hitting that point on the screen is non-zero, we know that particles do hit the screen at this point some times. In these cases we run into paradox, by assuming that the particle followed a single trajectory. Crossing from any slit, is a subset of an event of quantum measure zero and thus cannot occur no matter what possible hidden variables determine the realised trajectory. However, in this example, one can say that we did not cover all Ω with precluded events, because the particle could have been detected at other points of the screen where the above analysis does not hold. Even though the nature of the problem that arose in this example was essentially the same as that for zero covers, this example is not a proper zero cover. We now give, an example of a proper zero cover for a single qubit with initial state | 0 〉 . We measure it in three-moments-of-time 7 . We assume that the qubit evolves in the following (unitary) way: which can arise for some particular Hamiltonian H and time interval t between measurements. We evolve and measure in the {| 0 〉 , | 1 〉} basis, three times. The amplitudes for different histories are given by the expression using the class operators | ψ final 〉 = C α | 0 〉 = P t 3 UP t 2 UP t 1 U | 0 〉 , with P t i being the projection either | 0 〉〈 0 | or | 1 〉〈 1 | depending on what was measured at the moment t i : The labels indicate outcomes of the {| 0 〉 , | 1 〉} measurements from right to left. For example h 4 is the history that we get the outcome | 1 〉 in the two first measurements and | 0 〉 in the third. This leads to the following zero cover (collection of precluded events that their union is Ω): This example is simple and very general, since with little modifications, it can hold for any Hamiltonian for the qubit (choosing different moments of time). More complicated examples can also be constructed for all systems (initial states, Hamiltonian). The Kochen-Specker theorem [9], when casted in the histories language, is precisely a special case of a zero cover (see [7, 10]). It is the existence of these zero covers, that rules out the possibility of the reality being a single-history and thus rules out hidden variables theories. We have to stress however, that the conclusion holds, provided one accepts as starting point the Feynman path integral as the amplitude for all histories, being fine or coarse grained 8 . One attempt to overcome the zero covers paradox, would be to avoid taking all events with quantum measure zero as precluded, i.e. allow for some events attaining truth value 'True', even if they have quantum measure zero. This runs to, at least two problems. One is that we need a physical/natural criterion for when an event of quantum measure zero is actually precluded and when it is not. The second comes from consistent histories approach, and the observation, that quantum measure zero events, decohere (and thus behave classically) with their complement. Having, a zero cover, leads necessarily either to a context-dependent view or to the need of selecting a preferred classical domain for the consistent histories (see section 4.1). Other attempts involve modifying one of the structures we had in classical physics ( U , T , M ).", "pages": [ 6, 7, 8 ] }, { "title": "3. The coevent formulation", "content": "We are now in position to give the technical definition of the coevents formulation. The approach was pioneered by Sorkin [2, 3] and one can find older reviews in [14, 15]. There are two, essentially equivalent, ways of viewing the coevents formulation. Here we will introduce both, starting with what appears as more intuitive, leaving as second the one that was historically developed first. In particular, the first view, is that the reality (ontology of the theory) is a coarse-grained history, in direct analogy with the picture in classical physics, that reality was a fine-grained history h i . The second view, is the dual one, that considers the reality as being the (generalised) valuation map φ , which is in general anhomomorphic, and it is analogous with the view in classical physics, that reality is a homomorphic map φ h i that is the characteristic function of the history h i .", "pages": [ 9 ] }, { "title": "3.1. The coarse-grained history view", "content": "Instead of viewing as possible realities the fine-grained histories h , elements of Ω, we could allow as potential realities, coarse-grained histories, i.e. (as starting point) any event R element of U . However, it is clear from classical physics, that this would lead to allowing potential realities that are unphysical. We need to impose conditions, that among other things, will reproduce the classical picture for the case of classical measure. There are two such requirements. The set of all non-preclusive events we call it NP . We require that an allowed reality R is a non-preclusive event ( R ∈ NP ). /negationslash This condition is satisfied by potential realties in classical physics. However, for a classical measure, all subsets of a measure zero set, are also measure zero sets. It follows, that in classical physics is sufficient to require that potential realities h i (single-histories events) are not precluded ( µ c ( h i ) = 0). /negationslash For a quantum measure, it is possible to have an event A that is not precluded ( µ ( A ) = 0) but is subset of a precluded event P . As we have seen earlier, we wish to rule out such events from potential realties, and this is done by requiring the allowed coevents to be non-preclusive (rather than being simply not precluded). To this point, we stress again the reason to rule out such events. We wish to recover deductive logic, and we cannot have an event given truth value 'True', while one of its supersets is precluded and thus given the truth value 'False'. This requirement states, that from all the non-preclusive events, we consider as potential realities only those that give the most detailed description. This is called the 'maximum detail' condition. It is precisely this condition, that in the case of a classical measure, restricts the possible realties to be fine-grained histories that are not precluded, and thus agree with the picture developed in the previous section. For a quantum measure, the condition of eq. (16) forces us to consider potential realities that are not single-histories, as we will see in examples below. The potential realities are events that are non-preclusive and of maximum detail and we will call them covents 10 . The set of all coevents, given a quantum measure, is R . In order to find the set of possible coevents, we only need to know which events have zero quantum measure. Let us see, which are the allowed coevents for the examples of zero covers we gave in the previous section. In the three-slit example, the precluded events, were { h 1 , h 2 } and { h 2 , h 3 } . The nonpreclusive events (those not included in precluded events) are { h 1 , h 3 } and { h 1 , h 2 , h 3 } which are the elements of NP . From those, only the event { h 1 , h 3 } satisfies eq.(17) since the other event { h 1 , h 2 , h 3 } has as subset the non-preclusive event { h 1 , h 3 } . We therefore have a single coevent as potential reality and R = {{ h 1 , h 3 }} . In the qubit example with the three-moments-of-time, we have many precluded events, namely: One can find that there are the following six coevents: This is an example where the potential realities, are not single-histories and moreover there are many (six) such potential realties. After having defined the new ontology of the theory, as being that of a coevent, we wish to be able to make assertions. Given one realised coevent, which physical questions/events occur, i.e. take truth value 'True'? The truth value 'True' is given to all events that have the realised coevent R as a subset and the rest take truth value 'False'. However, this might lead us to a truth valuation map φ that is not homomorphic. We will analyse this in the second view of the coevent formulation that follows. Note that the above, analysis holds for finite histories space Ω. Maintaining this view for infinite histories space is possible but needs more further analysis in the definitions in order to be rigorously formulated.", "pages": [ 9, 10 ] }, { "title": "3.2. The anhomomorphic valuation view", "content": "In classical physics, we mentioned a dual view of reality. Instead of considering the singlehistory h as the ontology of the theory, one could consider the homomorphic truth valuation map φ h as the reality. These valuation maps are characteristic function of single histories h , i.e. gives truth value 'True' to any event A that includes the history h , and 'False' to all the other events. The second view of the coevents formulation, is to try and generalise the truth valuations φ 's. It should be done in such a way, that is compatible with the paradoxes arising from using a quantum rather than a classical measure. In particular it should be compatible with the presence of zero covers. The modification that we adopt, is weakening the requirement that the truth valuation maps are homomorphisms between the Boolean algebras of U and T . In particular we still require the maps φ to be multiplicative i.e. obey eq.(1), but no longer require them to be additive. /negationslash We should note, that the first attempt of weakening the homomorphism to allow for the zero covers, was keeping the additive property and dropping the multiplication [3] but it was ruled out by some gedanken experiments that provided counter examples where no such map existed at all. There are also other attempts to weaken the homomorphism. All different attempts were coined 'schemes' and the one we have adopted and present in this paper, is the multiplicative scheme. Other than concrete examples that alternative schemes fail, there are several advantages in adopting the multiplicative scheme: (i) being able to have the coarsegrained view, (ii) being guaranteed that there always exists allowed maps no matter which is the quantum measure, (iii) recovering deductive logic and (iv) having a unique classical domain (we will analyse (ii),(iii) and (iv) later). It can be shown, that all multiplicative maps, are in fact characteristic maps of some (nontrivial) event S We call this event S support of the multiplicative map. Two things to note. First, is that the choice of allowed maps as characteristic maps is very similar with what was done in classical physics with the only difference that we now allow for characteristic maps of events that may consist of many fine grained histories A = { h 1 , h 2 , · · ·} . Second thing to note, is that precisely this property of multiplicative maps allows us to have both views of the coevent formulation. There is a one-to-one correspondence of anhomomorphic (but multiplicative) maps and coarsegrained histories (their supports). Other than requiring the allowed coevents to be multiplicative, we need to impose two further conditions to potential realties, as it was done in the coarse-grained history view. This condition together with the multiplicativity condition, guarantees that an event A being subset of a precluded event P , gets truth value 'False'. This plays the same role as the first requirement in the previous section (non-preclusive). A preclusive multiplicative map is primitive if it is not dominated by any other preclusive multiplicative map. For a classical measure, the primitivity condition, leads to maps that are homomorphisms and thus correspond to characteristic maps of single histories, as expected. The potential realities are (also) called coevents, and are valuation maps that are (i) multiplicative, (ii) preclusive and (iii) primitive. We should make here a comment on the terminology used to avoid confusion. As coevent, we defined both the multiplicative, preclusive and primitive map φ R and the support of this map R which is a non-preclusive, maximum detail event. In other words we called coevents, the potential realities of the coevents formulation, no matter if we adopted the coarse grained view or the anhomomorphic valuation view. In literature, the term coevent was used only for the (in general anhomomorphic) maps. However, when restricting attention in the multiplicative scheme as we are in this paper, the coarse grained history view is equivalent. In this case, even referring to the anhomomorphic map is easier done using its support. Since, we no longer have homomorphism between the events Boolean algebra U and the truth values Boolean algebra T the laws of inference do not follow directly. The main deductive inference rule, is the modus ponens that essentially means if A is 'True' and A implies B , ( A → B ), then B also is 'True'. In [16] it was shown, that modus ponens, is applicable for multiplicative maps. In particular, it was shown that if we require that our valuation maps obey modus ponens, we are necessarily led to multiplicative maps. This is one further motivation for the choice of the multiplicative scheme. It is clear however, that due to the fact that we no longer require the maps to be homomorphisms, some of the structure of the Boolean algebra is lost. In particular, we cannot use proofs by contradiction. There exist events A such that both φ ( A ) = 0 and φ ( ¬ A ) = 0. For example, in the three-slit case (see Figure 1), we have a single allowed coevent { h 1 , h 3 } which is realised. Asking the question 'did the system cross from any of slit A or B' ( A = { h 1 , h 2 } ) gets answer 'No'. However, answer 'No' gets the question 'did it cross slit C' ( ¬ A = { h 3 } ). This type of paradox is natural for the quantum world, but should not persist in classical world (see later). In other words, our anhomomorphic logic, allows for deductive proofs but not for proofs by contradiction which is precisely what happens in intuitionistic logic 11 . The latter is partly the motivation for the field of 'constructive mathematics' where mathematics are reformulated using solely deductive (constructive) proofs. In reality, one of the possible coevents is realised. Given that coevent, we should be able to answer whether any given question/event is realised ('True') or not realised ('False'). As long as the event in question A , either includes the realised coevent R 1 , i.e. R 1 ⊆ A or does not intersect it at all R 1 ∩ A = ∅ , it is easy to see that either φ R 1 ( A ) = 1 or φ R 1 ( ¬ A ) = 1. However, if the event A intersects non-trivially R 1 , then both A and ¬ A take truth value 'False'. Note here that while the rule does not hold in the coevents formulation, and we cannot use proofs by contradiction, the opposite rule holds. Namely, if A is 'True' its complement is indeed 'False'.", "pages": [ 10, 11, 12 ] }, { "title": "3.3. Remarks", "content": "The problem in applying the classical picture of histories in the quantum world arises because we have a quantum rather than a classical measure. The quantum measure allows the existence of zero covers. We therefore can no longer maintain the picture of reality as a single fine-grained history or equivalently as a homomorphic map. We are led to extend the potential realities from single fine-grained histories to coevents (either viewed as a coarse-grained histories or as multiplicative maps). After this generalisation, we are guaranteed to have some potential realities that are compatible with any quantum measure (see [7]). The initial conditions and dynamics fix the quantum measure. From that we obtain the precluded events which determine the allowed coevents. Given the quantum measure, we have, in general, many allowed coevents. In this sense, quantum theory appears as a generalisation of classical stochastic physics rather than generalisation of classical deterministic physics. Measurements are treated in a very similar way as in stochastic classical physics. They are used to rule out some of the potential realities/coevents, alas without altering the system (unlike standard Copenhagen quantum mechanics). The new ontology of quantum theory, is that of a coevent, is not affected by measurements (in the way it does in Copenhagen quantum mechanics) and is not contextual. We can therefore conclude that the coevents formulation is realist. A final point to raise here, concerns the role of logic. As we have seen, we retain deductive proofs. However, the precise form of logic, i.e. when other rules of inference can be applied, depends crucially on the set of possible coevents, which in their turn depend on the particular quantum measure. It therefore appears that the logic of quantum world is no longer fixed, but becomes dynamical. To quote Sorkin 'Logic is to quantum, as geometry is to gravity' [14].", "pages": [ 12, 13 ] }, { "title": "4. Developments", "content": "Having given the definition of the coevent formulation, we will now proceed analysing certain important aspects.", "pages": [ 13 ] }, { "title": "4.1. Classical domain and consistent histories", "content": "A very important question for quantum theory of a closed system is how one recovers, at some level/coarse-graining, classical physics. The finest grained description is not expected to behave classically, and thus some counter-intuitive properties such as the anhomomorphisms described earlier (e.g. end of section 3.2) are natural. We expect, though, that if we coarse-grain sufficiently, the arising structure behaves classically. In our case, the non-classical behaviour is the anhomomorphisms of the valuation maps. We call classical coarse graining one that all the allowed coevents, elements of R give rise to homomorphism between the coarse-grained event algebra and the truth values Boolean algebra. Given a quantum measure µ we have many 'allowed' coevents R i ∈ R . A classical coarse graining C is a coarse-graining (see end of section 2.3) such that all the coevents R i give rise to homomorphism. To do so, all coevents R i should be subsets of some cell of the partition: One of the most important results, is that there exists a unique finest grained classical partition, which we will call principle classical partition . This means that all classical partitions arise as coarse grainings of this unique finest grained partition (see [17] the Appendix). The construction of the principle classical partition is done as follows 12 : /negationslash Any measurement we can carry out with 'classical' apparatuses, will correspond to some coarse-graining of the principle classical partition. Therefore, it is not possible to distinguish between two intersecting coevents with any classical measurement. One could view these 'fat' coevents (union of the intersecting ones), as more physical, or treat this property as a short of 'coevents uncertainty'. The importance of this result, for histories formulations, becomes more apparent if we compare the situation with the consistent histories approach [6]. The consistent histories approach, with the use of the decoherence functional, finds partitions P i = { P i 1 , P i 2 , · · ·} of the histories space Ω, such that there is no interference between pairs of cells of the partition. The induced quantum measure becomes classical 14 . Each of these partitions, is called a consistent set , and one could attempt to view each consistent set as a classical domain. However, there does not exist a single finest grained consistent set that all the others arise from. This leads to properties in one consistent set being incompatible with properties on another, and we are forced to one of the following two solutions. Either with some physical mechanism (not known yet) to select a preferred consistent set, or to adopt a contextual view, that the answers to propositions depend on which consistent set one wishes to refer to (for further details see [11]). We have, so far, examined the meaning, existence and uniqueness of a classical domain. What remains, is to examine particular physical examples and how this classical domain emerges from the dynamics and whether it agrees with our classical intuition (pointers in a lab being classical, etc). This is essentially matter of calculation (see section 7 of [14]) and is subject of ongoing research.", "pages": [ 13, 14 ] }, { "title": "4.2. Probabilities", "content": "The analysis we did so far, concerned the potential realities. However, there is much more to a physical theory, than solely stating which are the possible realities. One needs to be able to make predictions, and in quantum theory, these predictions are in general probabilistic. We should be able, for example, to recover the double slit pattern, if we consider as system a large number of particles crossing the double slit apparatus. However, the concept of probabilistic prediction for a single non-repeatable closed system is problematic already in the classical level. For example a statement such as 'the universe has property A with probability 1/3' cannot be falsified by any experiment. Both results, possessing or not possessing the property, are compatible with the 'prediction'. This fails the criterion for a scientific proposition, which is that there should exist possible outcomes that falsies the assumption (see Popper). In our case, the system is closed and ideally non-repeatable (we construct such a theory to be able to treat the whole universe as a quantum system). We therefore need to understand the meaning of probability and its connection to the physical world, and to do so we have to resort to the founders of probability theory. We can see, that in a non-repeatable system, the only testable predictions, are those that concern things occurring with probability one or zero. We have also seen in section 2.2 that in the limit of infinite identical copies, one can recover probabilities from precluded events 15 . Considering the above, one is lead to the Cournot principle [19] (see also Kolmogorov's view [20] and the references in [17]): In a repeated trial, an event A singled out in advance, of small measure ( ≤ /epsilon1 ), rarely occurs. There exist a stronger formulation of the principle where the 'rarely occurs' is replaced with 'does not occur', but we adopt the one above. First thing to note in this definition, is the term 'small measure'. In the coevents formulation we will call such events approximately precluded . It is clear, that this is a subjective choice, and depending on how small /epsilon1 is, different prediction may arise. Once the theoretician makes the choice of /epsilon1 , the rest follows. Second thing to stress, is the importance of selecting the event A in advance. Before exploring the latter, we will give an example of how Cournot's principle does work. Consider tossing a coin 1000 times, and start with the assumption that the coin is fair. Any sequence of 1000 outcomes heads/tails is a fine grained history h i . Suppose we select the event A , which is the history that has all outcomes to be heads 16 . The probability (with the assumption of a fair coin) is clearly negligible (1 / 2 1000 ). We can thus predict that the event A will not occur, for most reasonable choices of /epsilon1 . However, it is conceivable that such an event does occur. In that very rare occasion, that such event occurs, we would be led to believe that the coin was not fair and thus falsify our initial assummption 17 . This is precisely what is done in actual experimental practise, with the only exception, that if our belief for the assumption is strong, we may convince ourselves to further repeat the experiment if that is possible. In a fundamentally closed system however, such option is not available. We return now to the importance of pre-selecting the event to be tested. All the time in nature, things of arbitrary small probability do occur. In the earlier example, of tossing a coin 1000 times, any sequence of outcomes heads/tails, has extremely small measure and it is identically the same with the outcome '1000 times heads' that we dismissed earlier. However, the difference between the two is the following. In the one case (1000 times heads) we could ask whether this will occur before carrying the experiment. For some other random sequence of outcomes (the one that actually happened), we did not ask in advance. If we had given a sequence of 1000 outcomes in advance, and then this precise sequence did occur when we carried out the experiment, we would be very sceptical and troubled, and likely assume that somehow there was a correlation of our mind and the tossing of the coin. If the number of tosses were greater than 1000, then we would be even more surprised. In general, we would dismiss the assumption of a fully fair coin and attempt to find which was the pattern that gave this series of outcomes (in that example, it would be easy to perform few more measurements to test the hypothesis, but this is not always an option for all physical systems). We therefore see that in the use of the weak Cournot principle, there is a dichotomy between the ontology and the prediction of the theory. There exist things of small measure, that despite being conceivable realities, one predicts that they will not occur. And if they do occur, the theoretician is led to the wrong conclusion that his assumptions were wrong 18 . Note here, that the Cournot principle was supported by many of the founders of probability theory, such as Bernoulli, Cournot, Markov, Borel, Levy, Kolmogorov and many others. Borel stated that Cournot's principle 'is the only law of chance', while Levy stated that 'Cournot's principle is the only connection between probability and the empirical world'. The use of the principle in recent times has fallen out of fashion, mainly due to the dominance of the subjective view of probabilities such as the approach of De Finetti where probabilities are treated as betting strategies. However, both the objective view of probabilities and the use of precluded sets as the essence of prediction, suits very well the coevents formulation. This is because the coevents formulation deals with a single closed system, adopts a realist view and uses the preclusion as the central feature in selecting the potential realities/coevents. In coevents formulation the dichotomy between ontology and predictions persists. In particular, the ontology of the theory is the coevents themselves, while to recover probabilistic predictions one takes the quantum measure and applies the weak Cournot principle as described above. In [18] for example, one can see how a discrete version of the double slit pattern is recovered using the quantum measure and applying the Cournot principle 19 .", "pages": [ 14, 15, 16 ] }, { "title": "4.3. The role and status of the initial state", "content": "In the coevent formulation, the role of the initial state, is that it affects the quantum measure and thus determines both the allowed coevents, and the probabilistic predictions (see previous section). An interesting question to ask, would be whether the initial state characterises the system (universe) in the following way. Given the coevent realised 20 , can we uniquely determine which was the initial state? Or in other words, is a universe with initial state Ψ 1 different from one with Ψ 2 ? The answer to those questions is closely related with the interpretation of the (initial) state, as was analysed by Pusey Barrett and Rudolph in [21]. If one views different states as ontological different entities, the answer is 'yes', while if one views them as statistical distributions, the answer is 'no'. Given an initial state Ψ 1 , we can find the set of allowed coevents R 1 = { R 1 1 , R 1 2 , · · ·} and similarly for initial state Ψ 2 , we have R 2 = { R 2 1 , R 2 2 , · · ·} . If there are no common covents R 1 ∩ R 2 = ∅ for any pair of initial states Ψ 1 , Ψ 2 , then we can say the universe starting with Ψ 1 is different from that with Ψ 2 . Otherwise, it is possible that the actual realised coevent is compatible with two (or more) initial states. For the coevents formulation, in [22] was given evidence 21 that if one looks at sufficiently fine-grained description, the sets of coevents starting from different pure states is disjoint, i.e. the universe of Ψ 1 is different from that of Ψ 2 . It follows that one is at least in principle, able to retrodict uniquely the initial state. While the initial state per se, has no ontological status in histories formulations, it obtains such status indirectly. Any possible reality/coevent, can arise from a unique initial state, and in this sense the initial state becomes a property possessed by the system. This observation has also interesting consequences for the field of (quantum) cosmology.", "pages": [ 16, 17 ] }, { "title": "4.4. Other developments and outlook", "content": "There are several other questions that one can ask in relation with the quantum measure and its interpretation in the coevents formulation. Here we will briefly mention some works that have been done giving references. One interesting thing to analyse, that is closely related with the question asked in the previous section, is which properties of our system can we deduce from the coevents themselves, or else which are the properties possessed by the coevents. One could wonder for example, what is the meaning of energy, for a given coevent, i.e. if our ontology is just a subset of histories. In a toy model of a three-sites hopper Sorkin showed that the system 'does know its own energy' from simply looking the coevents [23], and we can treat the energy of the site-hopper as a property the coevents have. Other works that have been done, is the attempt to formally extend the quantum measure for infinite histories space. The problems that arise can be found in [24] and directions on how to overcome them in [25]. The derivation of a Hilbert space structure starting purely from histories is given in [26] and the concept of quantum integration and its use is developed in (for example) [27]. There are several direction to further explore. One direction is exploring the conceptual issues that could arise from (i) considering composite systems, (ii) time extending the coevents (in other words, the effects that fine-graining the quantum measure has, to the set of allowed coevents) and (iii) considering large systems that classicality should emerge. Technical issues to be explored, are: (1) the extension of quantum measure and rigorously founding the formulation for infinite histories space and (2) finding ways to compute the allowed coevents for complicated systems (for the moment, it is very hard to do so, even for histories space with only few histories).", "pages": [ 17 ] }, { "title": "5. Summary and conclusions", "content": "We presented the coevents formulation of quantum theory. It is a realistic, histories formulation that applies to closed quantum systems and is well suited for quantum gravity. The three-fold structure of applying histories was presented, where we have (i) the space of histories Ω along with the event algebra U , (ii) the space of truth values T and (iii) the space of allowed truth valuation maps M . The latter in classical physics is the homomorphic maps φ from U → T . The role of the (classical) measure, the difference of classical deterministic and stochastic physics and the importance of precluded events were analysed. The replacement of the classical measure with quantum measure led us to the need to alter the picture. This was due to the existence of zero covers which ruled out the single-histories as potential realities. As a solution we introduced the coevents formulation, which can be viewed in two different ways. The new ontology, is either coarse-grained histories (non-trivial events) R i that are nonpreclusive and are of maximum detail or multiplicative maps φ R i (in general anhomomorphic), that are preclusive and primitive. The two pictures are equivalent, since there is a one-to-one correspondence between events and multiplicative maps, and the further conditions are also equivalent. With the coevents as ontology, we avoid the paradoxes described, that arose from the existence of zero covers. This is the case, because there is an existence theorem showing that given a quantum measure, there always exist non-trivial coevents (see [7]). The resulting logic, while it is not Boolean, is deductive (the modus ponens inference rule holds [16]). Proofs by contradiction are not, in general, possible due to the failure of eq. (25). However, the opposite rule given by eq. (26) holds. The coevents formulation, appears as a generalisation of classical stochastic physics, where there are many potential realities. The ontology (coevents) is not affected by measurement and is not contextual. It is in this sense that we can call the formulation realist . Furthermore, it was shown that there exists a unique classical domain (unlike consistent histories), and that probabilistic predictions can be recovered by the use of Cournot's principle. Other recent developments and future directions were stated in section 4.4.", "pages": [ 17, 18 ] }, { "title": "Acknowledgments", "content": "The author is grateful to Rafael Sorkin for introducing and developing the coevents formulation. He also wants to thank him and Fay Dowker, Sumati Surya and Yousef Ghazi-Tabatabai for many discussions. This work is partly supported by COST Action MP1006 'Fundamental Problems in Quantum Physics'.", "pages": [ 18 ] } ]

[ { "title": "Andronikos Paliathanasis", "content": "Abstract. A detailed study of the modified gravity, f(R) models is performed, using that the Noether point symmetries of these models are geometric symmetries of the mini superspace of the theory. It is shown that the requirement that the field equations admit Noether point symmetries selects definite models in a self-consistent way. As an application in Cosmology we consider the Friedman -Robertson-Walker spacetime and show that the only cosmological model which is integrable via Noether point symmetries is the ( R b -2Λ ) c model, which generalizes the Lambda Cosmology. Furthermore using the corresponding Noether integrals we compute the analytic form of the main cosmological functions. Keywords: General Relativity, Modified Gravity, Noether symmetries Pacs - numbers:98.80.-k,95.35.+d,95.36.+x Faculty of Physics, Department of Astronomy-Astrophysics-Mechanics,University of Athens, Panepistemiopolis, Athens 157 83, Greece E-mail: [email protected]", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The recent cosmological data indicate that the universe is spatially flat and has suffered two acceleration phases. An early acceleration phase (inflation), which occurred prior to the radiation dominated era and a recently initiated accelerated expansion. An easy way to explain this expansion is to consider an additional fluid with negative equation of state parameter, usually called dark energy, that dominates the universe at late times. In spite of that, the absence of a fundamental physical theory, regarding the mechanism inducing the cosmic acceleration, has given rise to a plethora of alternative cosmological scenarios. Most of them are based either on the existence of new fields in nature (dark energy) or in some modification of Einstein's general relativity (GR), with the present accelerating stage appearing as a sort of geometric effect ('geometrical' dark energy). The simplest dark energy probe is the cosmological constant Λ (vacuum) leading to the ΛCDM cosmology [1-3]. However, it has been shown that ΛCDM cosmology suffers from two major drawbacks known as the fine tuning problem and the coincidence problem [4]. Besides ΛCDM cosmology, many other candidates have been proposed in the literature, such as timevarying Λ( t ) cosmologies, quintessence, k -essence, tachyons, modifications of gravity, Chaplygin gas and others [5-10]. There are other possibilities to explain the present accelerating stage. For instance, one may consider that the dynamical effects attributed to dark energy can be resembled by the effects of a nonstandard gravity theory. In other words, the present accelerating stage of the universe can be driven only by cold dark matter, under a modification of the nature of gravity. Such a reduction of the so-called dark sector is naturally obtained in the f ( R ) gravity theories [11]. In the original nonstandard gravity models, one modifies the Einstein-Hilbert action with a general function f ( R ) of the Ricci scalar R . The f ( R ) approach is a relative simple but still a fundamental tool used to explain the accelerated expansion of the universe. A pioneering fundamental approach was proposed long ago with f ( R ) = R + mR 2 [12]. Later on, the f ( R ) models were further explored from different points of view in [13-15] and indeed a large number of functional forms of f ( R ) gravity is currently available in the literature [16-21]. The aim of the present work is to investigate which f ( R ) models admit extra Noether point symmetries and use the first integrals of these models to determine analytic solutions of their field equations. The idea to use Noether symmetries in cosmological studies is not new and indeed a lot of attention has been paid in the literature (see [22-37]). The main reasons for the consideration of this hypothesis is that (a) the Noether point symmetries provide integrals, which assist the integrability of the system, (b) is a geometric criterion because the Noether symmetries associated with the geometry of the field equations. A fundamental approach to derive the Noether point symmetries of a given dynamical system moving in a Riemannian space has been proposed in [38]. A similar analysis can be found in [39, 40]. The structure of the paper is as follows. The basic theoretical elements of the problem are presented in section 2, where we also introduce the basic FRW cosmological equations in the framework of f ( R ) models. The geometrical Noether point symmetries and their connections to the f ( R ) models are discussed in sections 5. In section 6 we provide analytical solutions for those f ( R ) models which are Liouville integrable via Noether point symmetries. In section 7 we study the Noether symmetries in spatially non flat f ( R ) cosmological models. Finally, we draw our main conclusions in section 8.", "pages": [ 1, 2 ] }, { "title": "2. Cosmology with a modified gravity", "content": "Consider the modified Einstein-Hilbert action: where L m is the Lagrangian of dust-like ( p m = 0) matter and k 2 = 8 πG . Varying the action with respect to the metric 1 we arrive at where the prime denotes derivative with respect to R , G µ ν is the Einstein tensor and T µ ν is the ordinary energy-momentum tensor of matter. Based on the matter era we treat the expanding universe as a perfect fluid which includes only cold dark matter with comoving observers U µ = δ µ 0 . Thus the energy momentum tensor becomes T µν = ρ m U µ U ν , where ρ m is the energy density of the cosmic fluid. Now, in the context of a flat FRW model the metric is The components of the Einstein tensor are computed to be: Inserting (4) into the modified Einstein's field equations (2), for comoving observers, we derive the modified Friedman's equation The contraction of the Ricci tensor provides the Ricci scalar The Bianchi identity /triangleinv µ T µν = 0 leads to the matter conservation law: whose solution is Note that the over-dot denotes derivative with respect to the cosmic time t and H ≡ ˙ a/a is the Hubble parameter. If we consider f ( R ) = R then the field equations (2) boil down to the Einstein's equations a solution of which is the Einstein de Sitter model. On the other hand, the concordance Λ cosmology is fully recovered for f ( R ) = R -2Λ. From the current analysis it becomes clear that unlike the standard Friedman equations in Einstein's GR the modified equations of motion (5) and (6) are complicated and thus it is difficult to solve them analytically. We would like to stress here that within the context of the metric formalism the above f ( R ) cosmological models must obey simultaneously some strong conditions [42]. These are: (i) f ' > 0 for R ≥ R 0 > 0, where R 0 is the Ricci scalar at the present time. If the final attractor is a de Sitter point we need to have f ' > 0 for R ≥ R 1 > 0, where R 1 is the Ricci scalar at the de Sitter point, (ii) f '' > 0 for R ≥ R 0 > 0, (iii) f ( R ) ≈ R -2Λ for R /greatermuch R 0 and finally (iv) Rf '' Rf '", "pages": [ 2, 3 ] }, { "title": "3. Modified gravity versus symmetries", "content": "In the last decade a large number of experiments have been proposed in order to constrain dark energy and study its evolution. Naturally, in order to establish the evolution of the dark energy ('geometrical' in the current work) equation of state parameter a realistic form of H ( a ) is required while the included free parameters must be constrained through a combination of independent DE probes (for example SNIa, BAOs, CMB etc). However, a weak point here is the fact that the majority of the f ( R ) models appeared in the literature are plagued with no clear physical basis and/or many free parameters. Due to the large number of free parameters many such models could fit the data. The proposed additional criterion of Noether point symmetry requirement is a physically meaning-full geometric ansatz. According to the theory of general relativity, the space-time symmetries (Killing and homothetic vectors) via the Einstein's field equations, are also symmetries of the energy momentum tensor. Due to the fact that the f ( R ) models provide a natural generalization of GR one would expect that the theories of modified gravity must inherit the symmetries of the space-time as the usual gravity (GR) does. Furthermore, besides the geometric symmetries we have to consider the dynamical symmetries, which are the symmetries of the field equations (Lie symmetries). If the field equations are derived from a Lagrangian then there is a special class of Lie symmetries, the Noether symmetries, which lead to conserved currents or, equivalently, to first integrals of the equations of motion. The Noether integrals are used to reduce the order of the field equations or even to solve them. Therefore a sound requirement, which is possible to be made in Lagrangian theories, is that they admit extra Noether symmetries. This assumption is model independent, because it is imposed after the field equations have been derived, therefore it does not lead to conflict with the geometric symmetries while, at the same time, serves the original purpose of a selection rule. Of course, it is possible that a different method could be assumed and select another subset of viable models. However, symmetry has always played a dominant role in Physics and this gives an aesthetic and a physical priority to our proposal. In the Lagrangian context, the main field equations (5) and (6), described in section 2, can be produced by the following Lagrangian: in the space of the variables { a, R } . Using eq.(10) we obtain the Hamiltonian of the current dynamical system or Combining the first equation of motion (5) with eq.(12) we find The latter equation together with ρ m = ρ m 0 a -3 implies that where Ω m = ρ m 0 /ρ cr, 0 , ρ cr, 0 = 3 H 2 0 /k 2 is the critical density at the present time and H 0 is the Hubble constant. We note that the current Lagrangian eq.(10) is time independent implying that the dynamical system is autonomous hence the Hamiltonian E is conserved ( dE dt = 0).", "pages": [ 3, 4 ] }, { "title": "4. Noether symmetries", "content": "Before we proceed we review briefly the basic definitions concerning Lie and Noether point symmetries of systems of second order ordinary differential equations (ODEs) The one point parameter transformation with generator X = ξ ( t, x j ) ∂ t + η i ( t, x ) ∂ i is a Lie point symmetry of the system of ODEs (15) if the following condition is satisfied [39,40] where X [2] is the second prolongation of X defined by the formula Condition (18) is equivalent to the relation where X [1] is the first prolongation of X and A is the Hamiltonian vector field If the system of ODEs results from a first order Lagrangian L = L ( t, x j , ˙ x j ) , then a Lie symmetry X of the system (15) is a Noether symmetry of the Lagrangian if the additional condition is satisfied where g = g ( t, x j ) is the gauge function. To every Noether symmetry there corresponds a first integral (a Noether integral) of the system of equations (15) which is given by the formula: where E H is the Hamiltonian of the Lagrangian The vector field X in the augmented space { t, a, R } is and the first prolongation Having given the basic formula for the Noether symmetries we look for analytic solutions of the dynamical system with Lagrangian (10) with the use of Noether Integrals.", "pages": [ 4, 5 ] }, { "title": "5. Noether symmetries of f ( R ) gravity", "content": "The Noether condition (22) for the Lagrangian (10) is equivalent with the following system of eight equations while the 'potential' is The solution of the system (27)-(34) will determine the Noether symmetries. Since the Lagrangian (10) is in the form L = T ( a, ˙ a, R, ˙ R ) -V ( a, R ), the results of [38] can be used 2 . The kinematic term defines a two dimensional metric in the space of { a, R } with line element The Ricci scalar of the two dimensional metric (35) is computed to be ˆ R = 0 , therefore the space is a flat space 3 with a maximum homothetic algebra. The homothetic algebra of the metric (35) consists of the vectors where K are Killing vectors ( K 2 , 3 are gradients) and H is a gradient Homothetic vector. Therefore applying theorem 2 of [38] we have the following cases: Case 1: If f ( R ) is arbitrary the dynamical system admits as Noether symmetry the X 1 = ∂ t with Noether integral the Hamiltonian E . Case 2: If f ( R ) = R 3 2 the dynamical system admits the extra Noether symmetries with corresponding Noether Integrals the non vanishing commutators of the Noether algebra are Case 3: If f ( R ) = R 7 8 the dynamical system admits the extra Noether symmetries with corresponding Noether Integrals and the non vanishing commutators of the Noether algebra are From the time dependent integrals (43),(44) and the Hamiltonian we construct the ErmakovLewis invariant [43,44] Case 4 : If f ( R ) = ( R -2Λ) 3 2 the dynamical system admits the extra Noether symmetries with corresponding Noether Integrals where m = 2 3 Λ . The non vanishing commutators of the Noether algebra are Case 5 : If f ( R ) = ( R -2Λ) 7 8 the dynamical system admits the extra Noether symmetries From the time dependent integrals (47),(48) we construct the time independent integral ¯ I 23 = ¯ I 2 ¯ I 3 . with corresponding Noether Integrals and the non vanishing commutators of the Noether algebra are From the time dependent integrals (43),(44) and the Hamiltonian we construct the ErmakovLewis invariant [44] Case 6: If f ( R ) = R n (with n = 0 , 1 , 3 2 , 7 8 ) the dynamical system admits the extra Noether symmetry /negationslash with corresponding Noether Integral and the commutator of the Noether algebra is [ X 1 , X 7 ] = 2 X 1 . We note that the Noether subalgebra of case 2, { X 1 , X 2 , X 3 } and the algebra of case 4 { X 1 , ¯ X 2 , ¯ X 3 } is the same Lie algebra but not in the same representation. The same observation applies to the subalgebra of case 3 { X 1 , X 5 , X 6 } and the algebra of case 5 { X 1 , ¯ X 5 , ¯ X 6 } . This connection between the Lie groups is useful because it reveals common features in the dynamic systems, as is the common transformation to the normal coordinates of the systems. For the cosmological viability of the models see [21,45,46]", "pages": [ 5, 6, 7, 8 ] }, { "title": "6. Analytic Solutions", "content": "Using the Noether symmetries and the associated Noether integrals we solve analytically the differential eqs.(5), (6) and (7) for the cases where the dynamical system is Liouville integrable, that is for cases 2-5. Case 6 (i.e. f ( R ) = R n ) is not Liouville integrable via Noether point symmetries, since the Noether integral (55) is time dependent 4 .", "pages": [ 8 ] }, { "title": "6.1. Power law model R with µ =", "content": "µ 3 2 3 In this case the Lagrangian eq.(10) of the f ( R ) = R 2 model is written as Changing the variables from ( a, R ) to ( z, w ) via the relations: the Lagrangian (56) and the Hamiltonian (11) become where V 0 = 1 9 . The equations of motion in the new coordinate system are The Noether integrals (39),(40) in the coordinate system { z, y } are The general solution of the system is: The Hamiltonian constrain gives E = z 1 I ' 1 where z 0 , 1 are constants and the singularity condition results in the constrain µ 7 In this case the Lagrangian eq.(10) is written as 6.2. Power law model R with µ = 8 Changing now the variables from ( a, R ) to ( ρ, σ ) via the relations: The Lagrangian (94) and the Hamiltonian (11) become where V 0 = -1 42 . The Euler-Lagrange equations provide the following equations of motion: The Noether integrals (43), (44) and the Ermakov-Lewis invariant 45 in the coordinate system { u, v } are Using the Ermakov-Lewis Invariant, the Hamiltonian (68) and equation (70) are written: And the analytical solution of the system is where B ( t ) = ( 1 2 2 ρ 2 t + ρ 1 √ Σ ) and ρ 1 , 2 , σ 0 are constants with Hamiltonian constrain E = 1 2 ρ 2 . The singularity constrain gives ( ρ 1 ) 2 = 4Σ In the case Σ = 0 the analytical solution is The singularity constrain gives ρ 1 = 0, then the solution is In contrast with the claim of [47] this model is analytically solvable and there exists models which admit Noether integrals with time dependent gauge functions. 6.3. Λ bc CDM model with ( b, c ) = (1 , 3 2 ) 3 / 2 into eq.(10) we obtain Inserting f ( R ) = ( R -2Λ) Changing now the variables from ( a, R ) to ( x, y ) via the relations: the Lagrangian (82) and the Hamiltonian (11) become where V 0 = 1 9 and ¯ m = 6Λ. The equations of motion, using the Euler-Lagrange equations, in the new coordinate system are The Noether integrals (47),(48) in the coordinate system { x, y } are where ω = √ 2Λ / 3. From these we construct the time independent first integral The constants of integration are further constrained by the condition that at the singularity ( t = 0), the scale factor has to be exactly zero, that is, x (0) = 0. The general solution of the system (86)-(87) is: The Hamiltonian constrain gives E = ω ( x 1 G I 1 -x 2 G I 2 ) where x 1 G, 2 G are constants and the singularity condition results in the constrain At late enough times the solution becomes a 2 ( t ) ∝ e 2 ωt In this case the Lagrangian eq.(10) of the f ( R ) = ( R 2Λ) model is written as 6.4. Λ bc CDM model with ( b, c ) = (1 , 7 8 ) -7 / 8 Changing now the variables from ( a, R ) to ( u, v ) via the relations: The Lagrangian (94) and the Hamiltonian (11) become where ¯ m = -28Λ , V 0 = -1 42 . The Euler-Lagrange equations provide the following equations of motion: The Noether integrals (51),(52) and the Ermakov-Lewis invariant (53) in the coordinate system { u, v } are Using the Ermakov-Lewis Invariant (102), the Hamiltonian (97) and equation (98) are written: where λ = 1 2 √ 2 3 Λ . The solution of (104) has been given by Pinney [49] and it is the following: where u 1 -3 . From the Hamiltonian constrain (103) and the Noether Integrals (100),(101) we find Replacing (105) in the Ermakov-Lewis Invariant (102) and assuming φ = 0 we find: where Then the solution is /negationslash where from the singularity condition x (0) = 0 we have the constrain u 1 + u 2 +2 u 3 = 0 , or At late enough time we find A ( t ) /similarequal A 0 , which implies a 2 ( t ) ∝ e λt . In the case where φ = 0 equations (103),(104) describe the hyperbolic oscillator and the solution is From the Ermakov-Lewis Invariant we have where v 1 is a constant. The analytical solution is", "pages": [ 8, 9, 10, 11, 12, 13 ] }, { "title": "7. Noether symmetries in spatially non flat f ( R ) models", "content": "In this section we study further the Noether symmetries in non flat f ( R ) cosmological models. In the context of a FRW spacetime the Lagrangian of the overall dynamical problem and the Ricci scalar are where K is the spatial curvature. Note that the two dimensional metric is given by eq.(35) while the 'potential' in the Lagrangian takes the form Based on the above equations and using the theoretical formulation presented in section 5, we find that the f ( R ) models which admit non trivial Noether symmetries are the f ( R ) = ( R -2Λ) 3 / 2 , f ( R ) = R 3 / 2 and f ( R ) = R 2 . The Noether symmetries can be found in section 5. In particular, inserting f ( R ) = ( R -2Λ) 3 / 2 into the Lagrangian (113) and changing the variables from ( a, R ) to ( x, y ) [see section 6.3] we find where ¯ K = 3(6 1 / 3 K ). Therefore, the equations of motion are The constant term ¯ K appearing into the first equation of motion is not expected to affect the Noether symmetries (or the integrals of motion). Indeed we find that the corresponding Noether symmetries coincide with those of the spatially flat f ( R ) = ( R -2Λ) 3 / 2 model. However, in the case of K = 0 (or ¯ K = 0) the analytical solution for the x -variable is written as /negationslash /negationslash where x ( t ) is the solution of the flat model K = 0 (see section 6.3). Note that the solution of the y -variable remains unaltered. Similarly, for the case of the f ( R ) = R 3 / 2 model the analytical solution is where z ( t ) is the solution of the spatially flat model (see section 6.1).", "pages": [ 13, 14 ] }, { "title": "8. Conclusion", "content": "In the literature the functional forms of f ( R ) of the modified f ( R ) gravity models are mainly defined on a phenomenological basis. In this article we use the Noether symmetry approach to constrain these models with the aim to utilize the existence of non-trivial Noether symmetries as a selection criterion that can distinguish the f ( R ) models on a more fundamental level. Furthermore the resulting Noether integrals can be used to provide analytic solutions. In the context of f ( R ) models, the system of the modified field equations is equivalent to a two dimensional dynamical system moving in M 2 (mini superspace) under the constraint ¯ E =constant. Following the general methodology of [31,38], we require that the two dimensional system admits extra Noether symmetries. This requirement fixes the f ( R ) function and the analytical solutions are computed. It is interesting that two well known dynamical systems appear: the anharmonic oscillator and the Ermakov-Pinney system. We recall that the field equations of the Λ -cosmology is equivalent with that of the hyperbolic oscillator.", "pages": [ 14 ] }, { "title": "Acknowledgments", "content": "This research was partially funded by the University of Athens Special Account of Research Grants no 10812.", "pages": [ 14 ] }, { "title": "Appendix A. Special solutions for the Power law model R n", "content": "/negationslash The case f ( R ) = R n is not Liouville integrable via Noether point symmetries. The zero order invariant will be used in order to find special solutions. Inserting f ( R ) = R n , ( n = 0 , 1 , 3 2 , 7 8 ) into eq.(10) we obtain and the modified field equations are The Noether symmetry (54) is also and a Lie symmetry, hence we have the zero order invariants Applying the zero order invariants in the field equations (A.2)-(A.4) and in the Noether integral (55) we have the following results. The dynamical system admit a special solution of the form where the constants N, E and I 7 are or or Another special solution is the deSitter solution for n = 2 where I 7 = 0 and the spacetime is empty i.e. E = 0.", "pages": [ 14, 15 ] } ]

[ { "title": "Angelo Tartaglia", "content": "Politecnico, corso Duca degli Abruzzi 24, 10129 Torino, Italy, and INFN E-mail: [email protected] Abstract. A new experiment aimed to the detection of the gravito-magnetic Lense-Thirring effect at the surface of the Earth will be presented; the name of the experiment is GINGER. The proposed technique is based on the behavior of light beams in ring lasers, also known as gyrolasers. A three-dimensional array of ringlasers will be attached to a rigid monument; each ring will have a different orientation in space. Within the space-time of a rotating mass the propagation of light is indeed anisotropic; part of the anisotropy is purely kinematical (Sagnac effect), part is due to the interaction between the gravito-electric field of the source and the kinematical motion of the observer (de Sitter effect), finally there is a contribution from the gravito-magnetic component of the Earth (gravito-magnetic frame dragging or Lense-Thirring effect). In a ring laser a light beam traveling counterclockwise is superposed to another beam traveling in the opposite sense. The anisotropy in the propagation leads to standing waves with slightly different frequencies in the two directions; the final effect is a beat frequency proportional to the size of the instrument and its effective rotation rate in space, including the gravito-magnetic drag. Current laser techniques and the performances of the best existing ring lasers allow at the moment a sensitivity within one order of magnitude of the required accuracy for the detection of gravito-magnetic effects, so that the objective of GINGER is in the range of feasibility and aims to improve the sensitivity of a couple of orders of magnitude with respect to present. The experiment will be underground, probably in the Gran Sasso National Laboratories in Italy, and is based on an international collaboration among four Italian groups, the Technische Universitat Munchen and the University of Canterbury in Christchurch (NZ).", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The theory of relativity, both special (SR) and general (GR), tells us that the propagation of light along closed space contours appears to be anisotropic either when the observer is rotating or when the gravitational field belongs to a rotating mass, or of course when both circumstances are present. The purely kinematical effect had initially also a classical description and is known as the Sagnac effect [1]; it is easy to measure by modern interferometric techniques or using ring-lasers. The other two effects, due to the gravitational field of a rotating mass, arise from the coupling of the gravito-electric part of the field with the rotation of the observer (geodetic or de Sitter or Schiff effect, according to the experimental circumstances) [2] and from the gravitomagnetic component of the field (Lense-Thirring effect) [3]. I shall call the two latter phenomena physical or GR effects. The physical rotation effects have been measured so far by two experiments in space. In both cases the precession induced by the rotation of the earth on the axis of a gyroscope has been used. By this means the Gravity Probe B experiment, stemmed from an idea proposed by Schiff [4], verified the terrestrial geodetic effect with a 0.28% accuracy and the Lense-Thirring drag with a 19% accuracy [5]. Another measurement has been made by Ciufolini using the data from the laser ranging of the LAGEOS satellites, launched for a different purpose. From the precession of the orbits of the LAGEOS the Lense-Thirring effect has been confirmed at 10% accuracy [6]. Another experiment based on the laser ranging technique is presently under way: the LARES satellite has been launched on February 13th 2012 on a dedicated mission and is now collecting data with the purpose of measuring the Lense-Thirring at a 1% accuracy [7]. These experiments are not easy and have the inconveniences of being in space. What I am presenting here is a different opportunity to measure the GR effects of rotation resting on the surface of the planet earth (actually under the surface, in an underground location). The proposal is to use light as a probe of the configuration and properties of spacetime. The device to be used is a ring-laser (actually a three-dimensional array of ring-lasers). The name of the experiment is GINGER and it will be described in the following. The GINGER proposal is the result of a collaboration whose Principal Investigator is Angela Di Virgilio of the Pisa section of the Italian INFN and involving primarily groups from a number of Italian institutions (the universities of Florence, Naples, Padua, Pisa, Turin-Politecnico; various INFN sections) and from the German Technische Universitat Munchen. A consultancy link is also active with a group of the Canterbury University of Christchurch (NZ). The detailed proposal is presented in [8].", "pages": [ 1, 2 ] }, { "title": "2. Propagation of light in the field of a rotating mass", "content": "The line element of the space-time of a steadily rotating mass may in general be written as: where polar coordinates in space are used, to evidence the symmetry. The g µν 's are the elements of the metric tensor and each of them does not depend on t and φ due to the symmetry. The mixed term g 0 φ accounts for the rotation of the source, i.e. the central mass, and is responsible for the so called gravito-magnetic effects; g 0 φ cannot be eliminated by any global coordinate transformation. Had we used different coordinates, we would have had in general three elements g 0 i , with the index i labeling the three space coordinates. The three g 0 i 's may be read as the components of a three-vector -→ h , which can be interpreted as the vector-potential of a gravito-magnetic field -→ B g , i. e. of the rotation depending part of the gravitational field [9]. In conventional three-dimensional notation it is: Such a gravito-magnetic field has the configuration of a dipole. Let us then evaluate the time of flight (TOF) of a light beam constrained by some physical system to travel along a closed space path; the TOF is the proper time interval of a laboratory within which the experiment is performed. The result is obtained from eq. (1) considering a null spacely closed world-line and integrating along it. The only term sensitive to the rotation sense is the one containing g 0 φ that linearly multiplies dφ , which is odd in the angle; that term will be the only one left when subtracting the clockwise from the counter-clockwise travel time. In formulae it is:", "pages": [ 2 ] }, { "title": "3. Ring lasers", "content": "The anisotropy pointed out in the previous section is the basis of the operation of a ring laser. Consider the scheme shown on fig. 1. An active cavity produces two light beams traveling in opposite directions; four mirrors deviate the beams to form a closed square path in space. In order to have a ring the mirrors cannot be less than three but of course could be more. The loop forms in turn a resonant cavity. According to formula (3) there is a time of flight difference between the two beams. Since we suppose to be in a stationary condition we have a pair of standing waves; each of them is formed by an integer number of wavelengths, but, due to the TOF difference (3), the clockwise and counter-clockwise wavelengths are slightly different from one another. In the lowest mode the integer is the same for both beams and the superposition leads to a beat note whose frequency is proportional to the TOF difference [8]: P is the length of the loop and λ is the fiducial wavelength of the laser. The beat frequency can be extracted and read at one of the corners of the ring. In the case of an earth-bound laboratory the relevant metric tensor elements can be approximated to the lowest significant order including the angular momentum of the earth, obtaining: In (5) the label ⊕ designates physical quantities belonging to the planet Earth: the total mass M , the angular momentum J and the angular velocity Ω; ω is the angular velocity of the apparatus and in the following we shall assume that ω = Ω ⊕ ; θ is the colatitude of the laboratory and r is the distance from the center of the earth. Using the above approximation it is possible to write down the expected signal in the form of the beat frequency [8]: A is now the area contoured by the light beams; R is the radius of the earth; the unit vectors ˆ u r , ˆ u θ and ˆ u n are respectively: radial, along the local meridian in the sense of increasing colatitude, perpendicular to the plane of the ring (provided it is in a plane). The whole quantity in front of the square bracket is called the scale factor: the bigger it is the stronger is the signal from the apparatus. Within the square brackets of formula (6) the three terms represent three real or effective angular velocities. The first is the rotation rate of the earth and accounts for the kinematical Sagnac effect; the second term corresponds to the geodetic or de Sitter effect; the third and last term is the gravito-magnetic contribution and accounts for the Lense-Thirring effect. The two physical terms (as they are also called) in (6) are of the same order of magnitude and nine orders of magnitude smaller than the dominant Sagnac term.", "pages": [ 2, 3, 4 ] }, { "title": "3.1. Commercial gyrolasers", "content": "Ring lasers are already in use as 'gyrolasers' for the measurement of rotation rates, for instance in airplanes or in submarines. They are replacing the mechanical gyroscopes formerly used for the same purpose, whence the name of gyrolasers. Commercial gyrolasers are compact objects with sensitivities in the order of ∼ 10 -7 rad/s/ √ Hertz obtained with appropriate values of the scale factor resulting from the use of multiply wound optical fibers. Often one single device is made of three gyrolasers aligned along three mutually perpendicular axes in order to sense the full angular velocity three-vector. Fig. (2) shows an example of commercial gyrolaser.", "pages": [ 4 ] }, { "title": "3.2. G-Pisa", "content": "The accuracy of commercial gyrolasers is not enough to perform scientific experiments and can hardly detect the diurnal rotation of the earth. For scientific purposes more refined and bigger instruments are required. An example is G-Pisa, initially developed to test the rotational stability of the Virgo gravitational interferometer located at Cascina near Pisa in Italy. G-Pisa is a square ring with a 1.35 m long side; it is visible in fig.s 3 and 4, both in its real aspect and in a schematic drawing. The source of light is a He-Ne laser with an adjustable output power of a few tens of nW. The device is mounted on a thick granite table that can be held at various pitches from horizontal to vertical. The sensitivity of G-Pisa is somewhat in between 10 -9 and 10 -10 rad/s/ √ Hertz [10]. It is not enough to sense the GR effects, but permits to measure, besides the rotation of the earth, many interesting motions of the surface of the earth at the laboratory, of geophysical origin. G-Pisa is destined to become a test-bed for technologies in preparation for the future GINGER instrument.", "pages": [ 4, 5 ] }, { "title": "3.3. The Cashmere cavern instruments", "content": "During the years a number of ring lasers for fundamental research have been built by a group of the University of Canterbury in Christchurch, NZ. The first one was C-I [11], others followed up to UG-2 whose interclosed area is 834 m 2 [12]. They were all located underground in the Cashmere cavern, near Christchurch (see fig. 5). Unfortunately the important gain in the scale factor obtained thanks to the size of the apparatus is overridden by the mechanical instabilities due to the length of the arms, so UG-2 did not permit measurements of the GR effects. The laboratory in the Cashmere cavern suffered relevant damages from the severe earthquakes that shook the Christchurch area in September 2010 and February 2011, so that now it is not in operation. For the future however the group in the University of Canterbury is planning to build a new triangular ring laser with 5 m sides, mounted on a rigid support, that promises very good performances.", "pages": [ 5, 6 ] }, { "title": "3.4. G in Wettzell", "content": "The best ring laser in the world is at the moment the G ring at the Geodatisches Observatorium Wettzell in Bavaria. It is a square ring, 4 m side, mounted on a monolithic block made of zerodur, a rigid ceramic material with a very small thermal expansion coefficient.The source of light is a He-Ne laser with 20 nW output power. The laboratory is under an artificial mound, in order to isolate it as far as possible from environmental disturbances; the apparatus is in turn kept within a chamber where pressure and temperature are stabilized; the zerodur table rests on a concrete pillar attached to the natural rock pavement under ground. Fig.s 6 and 7 present a picture and a scheme of the instrument. The sensitivity of G arrives to 4 . 5 × 10 -12 rad/s/ √ Hz and is already close to the value required in order to reveal the GR effects of rotation. Fig. 8 shows an example of data taken by G during thirteen days. The peaks correspond to the diurnal polar motion of the terrestrial axis; each of them has a height of the order of 50 µHz or less, i.e. ∼ 10 -8 times the Sagnac signal above which the peaks rise. The accuracy of G is within one order of magnitude from the expected physical rotation terms. The instrument is an excellent sensor of any sort of geophysical rotations as well as of tiny pushes from environmental interactions [13].", "pages": [ 6 ] }, { "title": "4. GINGER", "content": "As we have seen, G in Wettzell is already close to the sensitivity needed for gravito-magnetic measurements so that an additional effort allowed by present laser technology can make an earth experiment to reveal the Lense-Thirring effect feasible. To this purpose the collaboration presented in the introduction is proposing a new experiment relying on the expertise of all members: GINGER. GINGER will not be a single ring laser, but rather a three-dimensional array of rings. Each ring will be a square loop with a 6 m long side. The minimum number of rings is three, with the three normals oriented along mutually perpendicular directions in space; in this way the three space components of the effective angular velocities would simultaneously be measured. Apossibility is to actually have six loops, coupled in pairs of equal orientation; the consequent redundancy would allow a better control of noise. Fig. 9 shows two possible configurations of GINGER. One is a cube whose faces support pairs of equal square loops; the other is an octahedron, carrying three square loops. The octahedral structure allows a better control of the geometry through the diagonals, along which three Fabry-P'erot interferometers would be placed. In all cases the mirrors will be attached to a concrete 'monument' and the stability of the configuration will be insured more through active control of the position of the mirrors than via passive rigidity of the support. The design power of the light source is 200 nW and the quality factor of the cavity is Q = 3 × 10 12 . The site for GINGER could probably be the underground Gran Sasso national laboratory (LNGS) of the INFN in Italy where the appropriate technological facilities are already available and more than 1000 m of rock above the apparatus will be a most effective screen against surface noise. The purpose is to measure the Lense-Thirring effect with a 1% accuracy after one year of integration.", "pages": [ 6, 7, 8 ] }, { "title": "5. Conclusion", "content": "We have seen how ring lasers can be used for fundamental tests of general relativity on earth, having attained, in recent years, an unprecedented accuracy. A synergy between three groups working on ring lasers in the world has brought to the proposal of a dedicated new experiment named GINGER, whose technological improvements will indeed allow for the measurement of the general relativistic effects originated from the rotation of the mass of the earth. Actually if the accuracy will be good enough it is also possible to arrive to put constraints on a couple of PPN parameters. In fact the PPN form of three physical effective rotations that GINGER could reveal is: The γ and α 1 parameters would account for possible anomalous (from the view point of GR) dependencies of the curvature from the mass of the source and for possible preferred frame effects; the -→ W appearing in formula (7) is a gravitational three-vector potential associated with the presence of a preferred frame. In GR it is γ = 1 and α 1 = 0. GINGER has got a principle approbation from the Italian INFN. The seismic noise of the possible location in the LNGS has been checked at the beginning of 2011 by a German team; the G-Pisa ring will be moved, at the beginning of 2013, to the LNGS in order to characterize the place from the viewpoint of the rotational noise. Meanwhile a new ring approximately the same size as G-pisa will be built in order to serve as a test bed for mirrors stability and control. Summing up GINGER is on a track that will lead it to completion and we are confident that its performances will match up with the promises and expectations, paving the way for a number of future investigations using light as an intrinsically relativistic probe for the structure of space-time.", "pages": [ 8 ] }, { "title": "Acknowledgments", "content": "This presentation has been given on behalf of the whole GINGER collaboration mentioned in the introduction and in particular of the authors of ref. [8].", "pages": [ 8 ] }, { "title": "References", "content": "Ruggiero M L and Tartaglia A 2002 Nuovo Cimento B 117 743-767.", "pages": [ 9 ] } ]

[ { "title": "S. Fiorucci, for the LUX Collaboration", "content": "Brown University, Dept. Of Physics, 182 Hope St, Providence RI 02912 [email protected] Abstract . We report on the design, construction and commissioning of the Large Underground Xenon (LUX) dark matter detector at the Sanford Laboratory in Lead, SD, USA. From its inception in 2007, to its construction at a surface laboratory in lead in 2009-2010, its operation in 2011, and its re-installation 1 mile underground in 2012, we review the relevant achievements already obtained and give an outlook on how LUX will become the most sensitive detector in the field in 2013.", "pages": [ 1 ] }, { "title": "1. The LUX Detector", "content": "The large Underground Xenon (LUX) detector is a dual-phase xenon time projection chamber for the direct detection of dark matter particle interactions. With a total target mass of 370 kg, LUX is currently the largest detector of its kind. Aside from its larger size, the LUX detector possesses a number of particularities which contribute to make it extremely competitive. For more details on all technical aspects of the LUX detector the reader is encouraged to consult the recently published overview paper in Ref. [1]. Of particular note are the following: instrumented as a Cherenkov veto, allowing the rejection of events in coincidence with rare cosmic muons traversing the tank.", "pages": [ 1, 2 ] }, { "title": "2. Commissioning of the LUX detector", "content": "The LUX collaboration was formed in 2007 and fully funded jointly by NSF and DOE in 2008. Its history is intimately linked to that of the new Sanford Underground Research Facility (SURF) in Lead, South Dakota. For the first three years, the lab did not exist as a research facility and the R&D effort for LUX was focused at Case Western Reserve University with the 'LUX 0.1' program [8]. This allowed us to build and operate a full-size prototype of the LUX detector, with an emphasis on cryogenics, gas circulation, light readout and slow-control. It also provided a tremendous learning and training opportunity for the students and postdocs who would eventually build and operate the real detector. By June 2010, a dedicated Surface Laboratory at SURF was completed to host the construction and commissioning of the LUX detector. This facility is as much as possible a 1:1 replica of the underground laboratory, including a class 1,000 clean room and a (smaller) water tank. It allowed the collaboration to fully test most systems while the underground facility was being excavated and outfitted. The LUX detector was assembled from the inside out between 2010 and 2011 within the clean room, while external systems, such as the xenon circulation loop, data acquisition electronics, xenon storage system, and nitrogen distribution, were built and tested just outside. In May 2011 the detector assembly was completed with 30 PMTs installed. A first cryogenic test was performed, successfully bringing the internal temperature down to 190 K and keeping it stable over a week, with the detector filled with argon gas. By October 2011 the LUX detector had been outfitted with the rest of the 122 PMTs and final upgrades to internal and external systems were completed. Between October 2011 and February 2012 LUX undertook 'Run 2', during which the detector was immersed in the water shield, the full 370 kg xenon load was condensed, circulation/purification was started and data were acquired. The results were very satisfying (see section 3 below and Ref. [9]). Only a very limited number of fixes and upgrades to perform before underground re-deployment, were identified. These were corrected between March and July 2012. In June and July 2012, the entire experiment was packed up and prepared for transport underground. The Davis Laboratory, 4850 feet underground, was completed in June 2012. The twostory laboratory space includes the 8 m diameter, 6 m tall water tank, meant to provide shielding for the LUX detector, as well as a clean room, a control room, a liquid nitrogen storage and a water filtration unit. Between July and September 2012 all of LUX was transported and reinstalled in the Davis Laboratory. The water tank was filled with water in late October, and LUX started taking Cherenkov data with the detector at vacuum. On December 10, LUX officially started 'Run 3' by introducing ultra-pure xenon gas into the detector, with the goal to cool down and condense the full 370 kg by the end of January 2013.", "pages": [ 2 ] }, { "title": "3. Performance", "content": "LUX 'Run 2' at the SURF Surface Laboratory achieved a number of remarkable results and demonstrated that the experiment was ready to move underground and start looking for dark matter. For more details the reader is encouraged to consult Ref. [9], but the following points are worth emphasizing:", "pages": [ 2 ] }, { "title": "4. Outlook", "content": "The LUX underground science program will consist of two steps. In the first step, after ~1 month of circulating xenon to improve the purity, we will run some initial calibrations and get an idea of the background levels. We will then run in dark matter search mode for a period of ~60 days. The exact duration will be decided in a large part by the observed background levels, and tuning of the acquisition parameters. This will be a non-blind search, and we expect to see very few electronic recoil events in the energy region of interest (in a 100 kg fiducial volume, before cuts), and no nuclear recoil events by a comfortable margin. The result of this short campaign should already make LUX the most sensitive dark matter detector in the world, and will be available in 2013. In a second step, after a new round of extensive gamma and neutron calibrations, LUX expects to start a new 300 day dark matter search campaign. This new search will be blind and will rely on early data and calibration data to define the cuts and search region. The estimated sensitivity at the end of the experiment is a WIMP-nucleus cross-section smaller than 2 × 10 -46 cm 2 for a WIMP mass of 40 GeV/c 2 . More details on this estimate and what assumptions go into it can be found at the end of Ref. [9].", "pages": [ 3 ] }, { "title": "Acknowledgements", "content": "This work was partially supported by the U.S. Department of Energy (DOE) under award numbers DE-FG02-08ER41549, DE-FG02-91ER40688, DOE, DE-FG02-95ER40917, DE-FG02-91ER40674, DE-FG02-11ER41738, DE-SC0006605, DE-AC02- 05CH11231, DE-AC52-07NA27344, the U.S. National Science Foundation under award numbers PHYS-0750671, PHY-0801536, PHY-1004661, PHY-1102470, PHY-1003660, the Research Corporation grant RA0350, the Center for Ultra-low Background Experiments in the Dakotas (CUBED), and the South Dakota School of Mines and Technology (SDSMT). LIP-Coimbra acknowledges funding from Fundação para a Ciência e Tecnologia (FCT) through the project-grant CERN/FP/123610/2011. We gratefully acknowledge the logistical and technical support and the access to laboratory infrastructure provided to us by the Sanford Underground Research Facility (SURF) and its personnel at Lead, South Dakota.", "pages": [ 3 ] } ]

[ { "title": "Pierre Brun", "content": "CEA, Irfu, Centre de Saclay, F-91191 Gif-sur-Yvette - France E-mail: [email protected] Abstract. The high-energy Universe is potentially a great laboratory for searching new light bosons such as axion-like particles (ALPs). Cosmic sources are indeed the scene of violent phenomena that involve strong magnetic field and/or very long baselines, where the effects of the mixing of photons with ALPs could lead to observable effects. Two examples are archetypal of this fact, that are the Universe opacity to gamma-rays and the imprints of astrophysical magnetic turbulence in the energy spectra of high-energy sources. In the first case, hints for the existence of ALPs can be proposed whereas the second one is used to put constraints on the ALP mass and coupling to photons.", "pages": [ 1 ] }, { "title": "1. Motivations for axion-like particle searches", "content": "The Standard Model of particle physics reproduces data incredibly well [1]. Some of its foundations are however not completely understood, like for example the absence of CP violation in quantum chromodynamics (QCD). The most general QCD lagrangian includes a complex phase term which -if not exactly zero- induces CP violation. The non-observation of even a very small electric dipole moment for the neutron [2] implies that this phase is smaller than 10 -11 . This fact looks unnatural an calls for a explanation. A possible one is given by making this phase a dynamical field which value is driven to zero by the action of its classical potential. This is made possible by the introduction of a new U(1) global symmetry which is spontaneously broken at some scale f (this is the so-called Peccei-Quinn symmetry [3]). A new particle that is called the axion is then predicted as an associated pseudo Nambu-Goldstone boson [4, 5]. In the original idea of Peccei and Quinn, f was of the order of the electroweak scale (EW), inducing a mass of ∼ 100 keV for the axion, which was quickly ruled out (see [6, 7] for details). Then it was proposed that f was much greater than the EW scale, leading to a very light and weakly interacting axion (dubbed 'invisible axion'). Axions are experimentally searched through their coupling to photons, from the Sun [8, 9], assuming they make up the galactic dark matter [10] or with LASERs [11]. Beyond the case of the strong CP problem and axions, axion-like particles (ALPs) appear in many models of physics beyond the Standard Model such as string theory [12, 13, 14] as pseudo Nambu-Goldstone bosons associated to the breaking of U(1) symmetries. The properties of the associated particles are similar to that of axions, but in general their mass and coupling to photons are not related, making the corresponding parameter space larger. For the general ALP case, the interaction term with photons is where g γa is the dimensionful coupling between photons and ALPs, F is the electromagnetic tensor and a is the ALP field. On the right hand-side of Eq. 1, the F ˜ F term is expressed as a scalar product of the photon electric field and the magnetic field, revealing the fact that ALPs can couple to photons in the presence of an external magnetic field. In the present article, the use of natural (astrophysical) environments to search for ALPs is emphasized, like during the propagation of very high energy gamma-rays over cosmological distances, and the effect of astrophysical magnetic turbulence on high-energy photon source spectra. First the conventional view of the problem of the opacity of the Universe to gammarays is presented, with the discussion of a possible indication for an anomalously transparent Universe. Although the possible tensions can be solved in a conventional way, they can also be released by invoking ALPs mixing with photons. Then it is shown that this observable could be used as a signature when one tries to make a discovery, but that some uncertainties prevent from using it to derive robust constraints. It is then shown that constraints can be obtained by considering the effect of magnetic turbulence around the sources and finally some examples of constraints are given.", "pages": [ 1, 2 ] }, { "title": "2.1. The conventional view of the Universe opacity to gamma-rays 10", "content": "Very high energy photons (with ∼ TeV energies) traveling through the intergalactic medium encounter different populations of background radiations. The most numerous type of background photons belong to the Cosmic Microwave Background (CMB) and a second population is the extragalactic background light (EBL). The latter has a double bump structure, that comes from direct starlight and emission re-processed by interstellar dust in the infrared band as sketched on Fig. 1. Direct measurement of the EBL is very difficult because of foregrounds and infrared radiation by the instruments. 1 -1 10 -5 -4 10 -3 10 -2 10 -1 10 1 10 TeV gamma-ray astronomy is sensitive to the EBL density and spectrum as it is responsible for the attenuation of extragalactic source fluxes at high energy. The reason for that is the pair production process γ TeV γ EBL → e + e -, for which the threshold lies at TeV energies in the terrestrial frame. For instance considering E EBL ∼ 0 . 1 eV, the threshold energy satisfying E th E EBL > m 2 e (where m e is the mass of the electron) yields E th ∼ 2 . 6 TeV. In Fig. 1 the typical range of the TeV absorption range is indicated by the horizontal arrow. Because of the pair production process, the highest energy photons have a larger optical depth. Before 2006 it was commonly admitted that is was very unlikely to detect TeV photons from sources above z ∼ 0 . 2. The situation changed after HESS observations of two active galactic nuclei (AGNs) at z = 0 . 186 and z = 0 . 165. As reported in [16], when unfolded from the EBL effect, the intrinsic spectra of these sources were found too hard and in tension with the source models. This was the first indication for a Universe slightly more transparent than expected at high energy. Later, AGNs were observed at redshifts as high as 0.536 by MAGIC [17] and possibly 0.61 by HESS [18]. The Universe is indeed more transparent to gamma-rays that expected. That puzzle has conventional solutions, it could be for instance that spectra are actually harder, this can be realized for instance including hadronic components in the AGN jets or in relativistic shock acceleration models. The tension can be removed as well with a revision of the EBL models, mainly with a lower density as for instance in the model of [19] which is compatible with all observations. TeV observations are now even used to provide not only upper limits on the EBL density but actual measurements [20]. The current situation is illustrated in Fig. 2, extracted from [21, 22]. It represents the energy above which the absorption becomes significant (defined by a optical depth τ ( E ) = 1) as a function of the redshift of the source. The lines correspond to models or lower limits for the EBL density and redshift evolution from [19, 23, 24]. Constraints from the spectral indices of different sources are shown as arrows, and the HESS measurement corresponds to the blue band. One source seems to be in tension with the measurement. However the methods that lead to the constraint and the measurements are different as the constraints rely on spectral slope measurements and the measurement comes from the observation of features in the spectra that can be related to the EBL spectral density. A more unified approach might be necessary to get a definite answer on how strong the tension is. Note also that the Fermi collaboration did the same measurement at higher redshifts and found a good compatibility with EBL models [25]. Some studies however still claim for an anomaly, even with the lower EBL limits from [24]. It is the case in [26], where the authors claim for an anomaly, with the caveat that their claim requires to leave out some error bars.", "pages": [ 2, 3, 4 ] }, { "title": "2.2. How ALPs enter the game", "content": "The lack of opacity of the Universe to gamma-rays gave rise to the idea that ALPs could be responsible for this effect. The basic idea is that if mixing between ALPs and photons occur, the beam could travel in the form of ALPs on a significant fraction of way, not producing pairs, as sketched on Fig. 3. If ALPs are converted back to photons before observations, this could lead to a more transparent Universe. To get an idea of the relevant masses and couplings for the ALPs that are invoked here, let us consider the most simple formalism for describing the photon/ALP mixing. The system propagation is described by a Schrodinger-like equation: where ∆ B = g γa B t / 2 describes the photon/ALP coupling ( B t is the transverse projection of the magnetic field), τ is the optical depth related to EBL absorption and ∆ a = -m 2 a / 2 E accounts for the ALP mass. The fact that the imaginary coefficient only applies to the photon part of the wavefunction leads to the change in the overall transparency. In the case of no absorption, the mixing matrix is diagonalized with a rotation angle θ such that tan 2 θ = -2∆ B / ∆ a . The resolution of the propagation equation in the propagation state basis leads to the probability of transition The overall 1/2 coefficient in P γ → a accounts for the two polarizations of the photon. A critical energy E c appears, that defines the energy scale at which strong mixing occurs. From the expression of E c , with cosmological magnetic fields B ∼ 1 nG, an ALP mass m a ∼ neV and a coupling g γa ∼ 10 -11 GeV -1 , the critical energy lies at the TeV scale. It follows that the type of ALPs that are concerned by the so-called transparency hint will fall in a region of low masses and with couplings larger than those of the corresponding axions. The full treatment of the transparency problem in the presence of ALPs requires a 3 × 3 mixing matrix to account for the two polarization states for the photon, and a description of the magnetic field on the path from the source to the observatory. The extragalactic magnetic field is usually described as a patches of coherent domains of 1 Mpc size. The magnetic field strength is the same in all domains but from one domain to the next its orientation changes in a random way (see the sketch of Fig. 4). It can be shown (see [27]) that for random orientations and a large number N of domains, the transition probability is reduced to where P 0 is the transition probability in one domain. From this expression one would expect to have a 1 / 3 drop in the energy spectrum above E c in the limit NP 0 glyph[greatermuch] 1, and a flux that is boosted at high energy (typically above the pair-production related cutoff) as described in [28]. At least two facts lead to revise the above statements. First, in practice the limit NP 0 glyph[greatermuch] 1 is hardly realized. Second, due to the unknown nature of the magnetic field configuration, the prediction on the transmission has an intrinsic variance. Indeed it can happen that the ALPs do not convert back into photons before reaching the Earth, leading in that case to an even more opaque Universe. This is nicely illustrated in Fig. 5 extracted from [29]. Here the red dot-dashed line corresponds to the conventional opacity in the absence of ALPs and the solid black line is the average prediction with ALPs. It appears that the average transparency is indeed higher than the conventional case at high energy. However, the associated uncertainty on the prediction, in other words the variance related to the randomness of the magnetic field is such that the envelope includes the conventional case. Because of that fact, if observed without ambiguity in the future, such an effect might be seen as an indication for ALP detection but could hardly serve as a firm argument for discovery. Another limitation comes from the use of a very optimistic value for the extragalactic magnetic field. For the ALP effect to significantly affect the opacity, magnetic fields of nG strength with Mpc coherence length have to be present in the intergalactic medium. It is actually possible to generate such magnetic fields from inflation or QCD phase transition for instance, but the required strength is very close to current upper limits. This is illustrated in Fig. 6 (from [30]), where the different observational constraints on large scale magnetic field appear together with predictions from models (orange thin lines). There the red cross corresponds to the typical parameters used in the ALP analyses. It lies in a region that can be seen as finetuned given the size of the still open parameter space. At the moment it seems invoking such a strong magnetic field would be acceptable if the tension in the TeV observations was stronger. Fig. 2: Light, medium and dark grey: known observational bounds on the strength and correla- tion length of EGMF, summarized in the Ref. ( 25 ). The bound from Big Bang Nucleosynthesis marked 'BBN' is from the Ref. ( 2 ). The black hatched region shows the lower bound on the EGMF derived in this paper. Orange hatched regions show the allowed ranges of B,λ B for magnetic fields generated at the epoch of Inflation (horizontal hatching) the electroweak phase transition (dense vertical hatching), QCD phase transition(mediumverticalhatching),epochof recombination (rear vertical hatching) ( 25 ). White ellipses show the range of measured magnetic field strengths and correlation lengths in galaxies and galaxy clusters. 8 A clever way to avoid using intergalactic magnetic fields is to remark that if the source is magnetized or embedded in a cluster, then the mixing to ALPs could occur essentially around the source. Then, the magnetic field of the Milky-Way can serve as a target magnetic field to convert back the ALPs into photons. In that case as well, a strong boost can be expected at high energy, as first proposed in [31] and then again in [32]. This effect is used in [33] to estimate possible lower limits on the g γa coupling, but again assuming the tension is real between observations and models for the transparency. Finally it appears that the transparency observable could be used in the future for indication or discovery, if a clear tension was observed. To do so, one might wait for the next generation of gamma-ray telescopes such as CTA to have a significantly larger sample of sources. The problem of having only a few sources can be circumvented by using an energy band for which detections are numerous. This has been proposed in [34], where the authors remark that if the strong mixing regime is realized, the statistical properties of the observed fluxes from X-ray sources could display features distinctive of ALP effects. In that case, sources would be seen with fluxes reduced by a factor of 1/3 on average. Of course having no access to the absolute intrinsic fluxes, this overall factor is not observable. However because of the random nature of the mixing process in astrophysical magnetic fields, the first and second momentum distribution should have different shapes compared to the conventional case. In [34] the authors claim the observation of anomalous features in the momenta distributions. That result has however shown to be questionable in [35] where the effect is claimed to be caused by outliers. Because in that case the detection would rely on shapes of distributions, it is difficult to infer a constraint without a deeper analysis and this effect is again used to propose a hint. Nevertheless, it illustrates one possible use of the stochastic nature of the mixing in astrophysical environments, which is no more a limitation but becomes a tool for identifying possible ALP effects. In the following, it is shown that a careful study of this randomness can lead to observable effects that are used to set constraints on the ALP parameters.", "pages": [ 4, 5, 6 ] }, { "title": "3.1. Effect of the magnetic turbulence", "content": "One peculiar effect of photon/ALP mixing is the fact that the magnetic field turbulence can directly imprint features in the energy spectra from high-energy sources. The exact spectral shape one gets at the end is unpredictable, but as shown in [36] the statistical properties of the induced irregularities are a prediction of the ALP model. The authors of [37] and [29] already noticed that in principle the observed spectra should be very irregular in case of strong photon/ALP mixing, without considering the use of the irregularity as an observable. To account quantitatively for the irregularity, the 2 polarizations of the photon must be considered, so that the evolution of the system after n domains is given by and k stands for the k th domain, of size s k , P k is the rotation matrix between the interaction eigenstates and the propagation eigenstates and the matrix M k describes the mixing. The indexes k are there to recall that from one magnetic domain to the next, the corresponding parameters change due to the different orientations of the magnetic field ( B t is the projection of the magnetic field on the polarization plane and φ is the angle that projection makes with one of the two photon polarization). m γ = 4 παn e /m e is the effective mass of the photon propagating in a plasma with electron density n e . Examples of spectral oscillation patterns in one domain are given in Fig. 7 for different values of δ = g γa B t s/ 2, s being the size of the coherent domain. When several domains are considered, the spectrum ends up being very irregular as shown in Fig. 8 in the case of an unpolarized beam. For that example, an extragalactic source is considered and the magnetic field is typical of that of a galaxy cluster. The top panel of Fig. 8 is the raw signal and the bottom panel is the same signal smoothed by the energy resolution of HESS ( ∼ 15%). In that case the critical energy is of order 1 TeV and the effective photon mass is negligible. Whereas in the case of the extragalactic magnetic field the naive description of the turbulence might be sufficient (essentially because its properties are very poorly known), galaxy cluster magnetic fields may deserve a better treatment. The magnetic field in that case is modeled by a gaussian field with zero mean and a distribution of modes that is described by a Kolmogorov-like spectrum as in Eq. 7: The corresponding power spectrum is modeled by a function resembling that of Fig. 9. In galaxy clusters, the typical coherence length of the magnetic field is 10 kpc and the strength of the field is 1 to 10 µ G.", "pages": [ 7 ] }, { "title": "3.2. Examples of constraints", "content": "The first example of constraints is from the HESS analysis of PKS 2155-304 [38, 39], which is an AGN located at z = 0 . 116. For that source, both the extragalactic magnetic field and the cluster magnetic field can be considered. In the first case, as previously, one has to assume optimistic values of the magnetic field strength for the irregularity signal to be significant. A galaxy cluster is observed around the source, but no magnetic field measurements are available. So in the case of the galaxy cluster magnetic field, conservative values for the strength and the coherence length are assumed (1 µ G and 10 kpc respectively). As HESS observation ranges from hundreds of GeV to a few TeV, from the expression of the critical energy E c it is straightforward to see that the typical ALP masses that are probed are of the order of 10 -8 eV. In [38, 39], it is shown that the observed energy spectrum does not exhibit strong irregularities. Then an estimator of the irregularity is proposed and numerical simulations are used to exclude sets of parameters that lead to significantly too strong irregular behavior. This exclusion has to be done on a statistical basis as each realization of the magnetic field turbulence is different. The results of the analysis are presented in Fig. 10. The method allows to improve the CAST limits in a limited energy range around 20 neV. Another possibility is to use a source that lies at the center of a well studied galaxy cluster. In that case, the magnetic field properties are derived observationally. This is done by studying the Faraday rotation maps of the polarized radio emission from the cluster (see [40] for a review). These studies allow in principle a determination of the full turbulence power spectrum, yielding the intensity of the magnetic field, its coherence scale and the slope of the turbulence spectrum. A very well studied cluster is Hydra, for which a strong X-ray source is present at the center (Hydra A) [41] . In [42], X-ray data from the Chandra satellite are analyzed in order to derive constraints on ALP parameters. In the case of X-rays from Hydra, the diagonal terms in the matrix of Eq. 6 can be simplified. Indeed the pair production related opacity is irrelevant in the case of X-rays (so τ = 0), and the trace of the matrix is dominated by the effective photon mass for m a glyph[lessorsimilar] 10 -11 eV. So the constraints are expected to extend to arbitrarily low ALP masses below that value. In [42], the irregularity is estimated by performing χ 2 tests when deriving the energy spectrum with a forward folding method. ALP parameters yielding a too high level of irregularity compared to the data are excluded. The corresponding exclusion curve is displayed in Fig. 11. It turns out this analysis improves the previous constraints in that mass range from the non-observation of gamma-rays associated with SN 1987 A [43].", "pages": [ 7, 8, 9 ] }, { "title": "4. Outlook", "content": "The study of the high-energy Universe is potentially a nice way to search for axion-like particles. The problem of the transparency of the Universe to gamma-rays can provide a interesting observable, that requires nevertheless the observation of a large number of TeV sources to be robust. This can be achieved with the next generation of Cherenkov telescopes such as CTA. The photon/ALP mixing in astrophysical sources is intrinsically a stochastic process. That fact makes difficult the use of the transparency observations to derive constraints on the ALP parameters. It is noted however that the turbulence of the astrophysical magnetic fields has the effect of inducing irregularities in the energy spectra of sources. The statistical properties of the induced irregularity can be predicted and are used to set limits on the ALP coupling to photons. Because the method is insensitive to the polarization, these constraints go beyond classic ALPs and apply to both F ˜ F and F 2 types of couplings. Two examples of limits are given in the case of TeV and X-ray observations of high-energy emitting sources inside clusters of galaxies.", "pages": [ 9 ] } ]

[ { "title": "Fractional Action Cosmology: Some Dark Energy Models in Emergent, Logamediate and Intermediate Scenarios of the Universe", "content": "Ujjal Debnath, 1, ∗ Surajit Chattopadhyay, 2, † and Mubasher Jamil 3, ‡ 1 Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah-711 103, India. 2 Department of Computer Application (Mathematics Section), Pailan College of Management and Technology, Bengal Pailan Park, Kolkata-700 104, India. 3 Center for Advanced Mathematics and Physics (CAMP), National University of Sciences and Technology (NUST), H-12, Islamabad, Pakistan.", "pages": [ 1 ] }, { "title": "Abstract", "content": "In the framework of Fractional Action Cosmology, we have reconstructed the scalar potentials and scalar fields, namely, quintessence, phantom, tachyon, k-essence, DBI-essence, Hessence, dilaton field and Yang-Mills field. To get more physical picture of the variation of the scalar field and potential with time, we express scale factor in emergent, logamediate and intermediate scenarios, under which the Universe expands differently.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Fractional action cosmology (FAC) is based on the principles and formalism of the fractional calculus applied to cosmology. The fractional derivative and fractional integrals are the main tools in fractional calculus, where the order of differentiation or integration is not an integer. The fractional calculus is immensely useful in various branches of mathematics, physics and engineering [1]. In doing FAC, one can proceed in two different ways [2]: the first one is quite easy as one has to replace the partial derivatives in the Einstein field equations with the corresponding fractional derivatives; the second technique involves deriving the field equations and geodesic equations from a more fundamental way, namely starting with the principle of least action and replacing the usual integral with a fractional integral. This later technique is more useful in giving extra features of the FAC [3]: Rami introduced the FAC by introducing the fractional time integral, Here Γ( ξ ) = ∫ ∞ 0 t ξ -1 e -t dt is the Gamma function, 0 < ξ ≤ 1, 0 < τ < t , m = constant and ˙ x µ = dx µ dτ . The variation yields an extra term in the field equations which he termed as 'variable gravitational constant G '. Moreover, when the weight function in the fractional time integral is replaced with a sinusoidal function, then the solution of the corresponding field equations yield a variable cosmological constant and an oscillatory scale factor [4] where χ = 0 reduces to the standard action. In [5], the authors extended the previous study by working out with a general weight function: Several examples were studied and cosmological parameters were calculated in there. An interesting feature of FAC is that it yields an expanding Universe whose scale factor goes like power law form or exponential form depending on the choice of the weight function. Hence cosmic acceleration can be modeled in FAC. Reconstruction of potentials has been done by several authors in various cases. Capozziello et al [8] considered scalar-tensor theories and reconstruct their potential and coupling by demanding a background ΛCDM cosmology. In the framework of phantom quintessence cosmology, [9] used the Noether Symmetry Approach to obtain general exact solutions for the cosmological equations.In this paper, we are going to reconstruct the potentials and scalar fields, namely, quintessence, phantom, tachyonic, k-essence, DBI-essence, Hessence, dilaton field and Yang-Mills field. Such reconstructions have been studied previously in other gravitational setups [6]. To get more physical insight into the model, we express scale factor in three useful forms [7] namely emergent, logamediate and intermediate scenarios, under which the Universe expands differently. Such expansion scenarios are consistent with the observations with some restrictions on their parameters [7].", "pages": [ 2, 3 ] }, { "title": "II. FRACTIONAL ACTION COSMOLOGICAL MODEL", "content": "For a FRW spacetime, the line element is where a ( t ) is the scale factor and k (= 0 , ± 1) is the curvature scalar. We consider the Universe contains normal matter and dark energy. From Eq. (Ia), the Einstein equations for the space-time given by equation (1) are [3] where T 1 = t -τ , ρ = ( ρ m + ρ φ ) and p = ( p m + p φ ). Here ρ m and p m are the energy density and pressure of the normal matter connected by the equation of state and ρ φ and p φ are the energy density and pressure due to the dark energy. Now consider there is an interaction between normal matter and dark energy. Dark energy interacting with dark matter is a promising model to alleviate the cosmic coincidence problem. In Ref. [10], the authors studied the signature of such interaction on large scale cosmic microwave background (CMB) temperature anisotropies. Based on the detail analysis in perturbation equations of dark energy and dark matter when they are in interaction, they found that the large scale CMB, especially the late Integrated Sachs Wolfe effect, is a useful tool to measure the coupling between dark sectors. It was deduced that in the 1 σ range, the constrained coupling between dark sectors can solve the coincidence problem. In Ref. [11], a general formalism to study the growth of dark matter perturbations when dark energy perturbations and interactions between dark sectors were presented. They showed that the dynamical stability on the growth of structure depends on the form of coupling between dark sectors. Moreover due to the influence of the interaction, the growth index can differ from the value without interaction by an amount up to the observational sensibility, which provides an opportunity to probe the interaction between dark sectors through future observations on the growth of structure. Due to this interaction, the normal matter and dark energy are not separately conserved. The energy conservation equations for normal matter and dark energy are and where H = ˙ a/a is the Hubble parameter. From equation (5) we have the expression for energy density of matter as where ρ 0 is the integration constant.", "pages": [ 3, 4 ] }, { "title": "III. EMERGENT, LOGAMEDIATE AND INTERMEDIATE SCENARIOS", "content": "· Emergent Scenario : For emergent Universe, the scale factor can be chosen as [26] where a 0 , µ, λ and n are positive constants. (1) a 0 > 0 for the scale factor a to be positive; (2) λ > 0, to avoid any singularity at finite time (big-rip); (3) a > 0 or n > 0 for expanding model of the Universe; (4) a < 0 and n < 0 implies big bang singularity at t = -∞ . So the Hubble parameter and its derivatives are given by Here H and ˙ H are both positive, but H changes sign at T 1 = 1 µ log λ . Thus H, ˙ H and H all tend to zero as t →-∞ . On the other hand as t →∞ the solution gives asymptotically a de Sitter Universe. · Logamediate Scenario : Consider a particular form of Logamediate Scenario, where the form of the scale factor a ( t ) is defined as [7] where Aα > 0 and α > 1. When α = 1, this model reduces to power-law form. The logamediate form is motivated by considering a class of possible cosmological solutions with indefinite expansion which result from imposing weak general conditions on the cosmological model. Barrow has found in their model, the observational ranges of the parameters are as follows: 1 . 5 × 10 -92 ≤ A ≤ 2 . 1 × 10 -2 and 2 ≤ α ≤ 50. The Hubble parameter H = ˙ a a and its derivative become, · Intermediate Scenario : Consider a particular form of Intermediate Scenario, where the scale factor a ( t ) of the Friedmann universe is described as [7], where Bβ > 0, B > 0 and 0 < β < 1. Here the expansion of Universe is faster than Power-Law form, where the scale factor is given as, a ( T 1 ) = T n 1 , where n > 1 is a constant. Also, the expansion of the Universe is slower for Standard de-Sitter Scenario where β = 1. The Hubble parameter H = ˙ a a and its derivative become,", "pages": [ 4, 5 ] }, { "title": "A. Quintessence or Phantom field", "content": "Quintessence is described by an ordinary time dependent and homogeneous scalar field φ which is minimally coupled to gravity, but with a particular potential V ( φ ) that leads to the accelerating Universe. The action for quintessence is given by [27] The energy momentum tensor of the field is: which gives The energy density and pressure of the quintessence scalar field φ are as follows The EoS parameter for the quintessence scalar field is given by For ω φ < -1 / 3, we find that the Universe accelerates when ˙ φ 2 < V ( φ ) . The energy density and the pressure of the quintessence (phantom field) can be represented by the minimally coupled spatially homogeneous and time dependent scalar field φ having positive (negative) kinetic energy term given by and where V ( φ ) is the relevant potential for the scalar field φ , /epsilon1 = +1 represents quintessence while /epsilon1 = -1 refers to phantom field. Scalar field models of phantom energy indicate that it can behave as a long range repulsive force [12]. Moreover the phantom energy has few characteristics different from normal matter, for instance, the energy density ρ ( t ) of the phantom field increases with the expansion of the Universe; it can be used as a source to form and stabilize traversable wormholes [14-17]; the phantom energy can disrupt all gravitationally bound structures i.e from galaxies to black holes [18-23]; it can produce infinite expansion of the Universe in a finite time thus causing the 'big rip' [24]. From above equations, we get and and Fig.1 Fig.2 Fig.3 and and In figures 1, 2 and 3, we have plotted the potentials against the scalar fields for the quintessence and phantom fields in emergent, logamediate and intermediate scenarios of the universe respectively in fractional action cosmology. It has been observed in figure 1 that after gradual decay, the potential starts increasing with scalar field for quintessence as well as phantom field models of dark energy in the emergent scenario of the universe irrespective of its type of curvature. On the contrary, when logamediate scenario is considered, the figure 2 exhibits a continuous decay in the potential V with increase in the scalar field φ . A different behavior is observed in figure 3 that depicts the behavior of the potential V against scalar field φ in the case of intermediate scenario of the universe. The blue lines in this figure show a continuous decay in V with increase in φ for quintessence model. However, the red lines exhibit an increasing pattern of V with scalar field φ .", "pages": [ 5, 6, 7, 8 ] }, { "title": "B. Tachyonic field", "content": "A rolling tachyon has an interesting equation of state whose state parameter smoothly interpolates between -1 and 0 [28]. Thus, tachyon can be realized as a suitable candidate for the inflation at high energy [29] as well as a source of dark energy depending on the form of the tachyon potential [30]. Therefore it becomes meaningful to reconstruct tachyon potential V ( φ ) from some dark energy models. An action for tachyon scalar φ is given by Born-Infeld like action where V ( φ ) is the tachyon potential. Energy-momentum tensor components for tachyon scalar φ are obtained as The energy density ρ φ pressure p φ due to the tachyonic field φ have the expressions where V ( φ ) is the relevant potential for the tachyonic field φ . It is to be seen that p φ ρ φ = -1 + /epsilon1 ˙ φ 2 > -1 or < -1 accordingly as normal tachyon ( /epsilon1 = +1) or phantom tachyon ( /epsilon1 = -1). From above, we get and and and and Fig.5 V In figure 4, the V -φ plot for normal tachyon and phantom tachyon models of dark energy is presented for emergent scenario of the universe. Potential of normal tachyon exhibits decaying pattern. However, it shows increasing pattern for phantom tachyonic field φ . It happens irrespective of the curvature of the universe. In the logamediate scenario (figure 5) the potentials for normal tachyon and phantom tachyon exhibit increasing and decreasing behavior respectively with increase in the scalar field φ . From figure 6 we see a continuous decay in the potential for normal tachyonic field in the intermediate scenario. However, in this scenario, the behavior of the potential varies with the curvature of the universe characterized by interacting phantom tachyonic field. For k = -1 , 1, the potential increases with phantom tachyonic field and for k = 0, it decays after increasing initially.", "pages": [ 8, 9, 10, 11 ] }, { "title": "C. k-essence", "content": "In the kinetically driven scalar field theory, we have non-canonical kinetic energy term with no potential. Scalars, modelling this theory, are popularly known as k-essence . Motivated by Born-Infeld action of String Theory, it was used as a source to explain the mechanism for producing the late time acceleration of the universe. This model is given by the action [31] with ignoring higher order terms of Using the following transformations, φ = ∫ d ˜ φ √ | L ( ˜ φ ) | /K ( ˜ φ ) , X = | L | K ˜ X and V ( φ ) = K 2 / | L | , the action can be rewritten as with From the action, the energy-momentum tensor components can be written as and The energy density and pressure of k-essence scalar field φ are given by and where φ is the scalar field having kinetic energy X = 1 2 ˙ φ 2 and V ( φ ) is the k-essence potential. From above, we get and and and From figures 7, 8 and 9 we see that for interacting k-essence the potential V always decreases with increase in the scalar field φ in all of the three scenarios and it happens for open, closed and flat universes.", "pages": [ 11, 12, 13 ] }, { "title": "D. DBI-essence", "content": "Consider that the dark energy scalar field is a Dirac-Born-Infeld (DBI) scalar field. In this case, the action of the field be written as [33] Fig.9 where T ( φ ) is the warped brane tension and V ( φ ) is the DBI potential. The energy density and pressure of the DBI-essence scalar field are respectively given by and and and where γ is given by Now we consider here particular case γ = constant. In this case, for simplicity, we assume T ( φ ) = T 0 ˙ φ 2 ( T 0 > 1). So we have γ = √ T 0 T 0 -1 . In this case the expressions for φ , T ( φ ) and V ( φ ) are given by and When we consider an interacting DBI-essence dark energy, we get decaying pattern in the V -φ plot for emergent and intermediate scenarios in the figures 10 and 12. However, from figure 11 we see an increasing plot of V -φ for for interacting DBI-essence in the logamediate scenario.", "pages": [ 13, 14, 15, 16 ] }, { "title": "E. Hessence", "content": "Wei et al [32] proposed a novel non-canonical complex scalar field named 'hessence' which plays the role of quintom. In the hessence model the so-called internal motion ˙ θ where θ is the internal degree of freedom of and Fig.10 Fig.11 V Fig.12 hessence plays a phantom like role and the phantom divide transitions is also possible. The Lagrangian density of the hessence is given by The pressure and energy density for the hessence model are given by and with where Q is the total conserved charge, φ is the hessence scalar field and V is the corresponding potential. From above we get, and and and and For interacting hessence dark energy, figure 13 shows increase in the potential with scalar field and figures 14 and 15 show decay in the potential with scalar field. This means the potential for interacting hessence increases in the emergent universe and decays in logamediate and intermediate scenarios.", "pages": [ 16, 17, 18, 19 ] }, { "title": "F. Dilaton Field", "content": "The energy density and pressure of the dilaton dark energy model are given by [27] and where φ is the dilaton scalar field having kinetic energy X = 1 2 ˙ φ 2 , λ is the characteristic length which governs all non-gravitational interactions of the dilaton and C is a positive constant. We get, For interacting dilaton field, the scalar field φ always increases with cosmic time T 1 irrespective of the scenario of the universe we consider. This is displayed in figures 16, 17 and 18 for emergent, logamediate and intermediate scenarios respectively.", "pages": [ 19, 20 ] }, { "title": "G. Yangs-Mills Dark Energy", "content": "Recent studies suggest that Yang-Mills field can be considered as a useful candidate to describe the dark energy as in the normal scalar models the connection of field to particle physics models has not been clear so far and the weak energy condition cannot be violated by the field. In the effective Yang Mills Condensate (YMC) dark energy model, the effective Yang-Mills field Lagrangian is given by [34], where K is the re-normalization scale of dimension of squared mass, F plays the role of the order parameter of the YMC where F is given by, F = -1 2 F a µν F aµν = E 2 -B 2 . The pure electric case we have, B = 0 i.e.F = E 2 . From the above Lagrangian we can derive the energy density and the pressure of the YMC in the flat FRW spacetime as and where y is defined as, We get, When we consider Yang-Mills dark energy, we find that E 2 is always increasing with cosmic time T 1 . This is displayed in figures 19, 20 and 21 for emergent, logamediate and intermediate scenarios respectively.", "pages": [ 20, 21, 22 ] }, { "title": "V. CONCLUSION", "content": "This paper is dedicated to the study of reconstruction of scalar fields and their potentials in a newly developed model of Fractional Action Cosmology by Rami [3]. The fields that we used are quintessence, phantom, tachyonic, k-essence, DBI-essence, Hessence, dilaton field and Yang-Mills field. We assumed that these fields interact with the matter. These fields are various options to model dark energy which is varying in density and pressure, so called variable dark energy. Different field models possess various advantages and disadvantages. The reconstruction of the field potential involves solving the Friedmann equations in the FAC model with the standard energy densities and pressures of the fields, thereby solving for the field and the potential. For simplicity, we expressed these complicated expressions explicitly in time dependent form. We plotted these expressions in various figures throughout the paper. In plotting the figures for various scenarios, we choose the following values: Emergent scenario: ξ = . 6, n = 4, λ = 8, µ = . 4, a 0 = . 7, G = 1 (all DE models); Logamediate: ξ = . 6, α = 3, A = 5, G = 1 (all DE models); Fig.21 Intermediate: ξ = . 6, β = . 4, B = 2, G = 1 (all DE models). Moreover in all cases δ = . 05, w m = . 01. In figures 1 to 3, we show the variations of V against φ in the emergent, logamediate and intermediate scenarios respectively for phantom and quintessence field. In the first two cases, the potential function is a decreasing function of the field. For the quintessence field, the potential is almost constant while for the phantom field, the potential increases for different field values. Figures (4-6) show the variations of V against φ in the emergent, logamediate and intermediate scenarios respectively for the tachyonic field. In figure 4, the V -φ plot for normal tachyon and phantom tachyon models of dark energy is presented for emergent scenario of the universe. Potential of normal tachyon exhibits decaying pattern. However, it shows increasing pattern for phantom tachyonic field φ . It happens irrespective of the curvature of the universe. In the logamediate scenario (figure 5) the potentials for normal tachyon and phantom tachyon exhibit increasing and decreasing behavior respectively with increase in the scalar field φ . From figure 6 we see a continuous decay in the potential for normal tachyonic field in the intermediate scenario. However, in this scenario, the behavior of the potential varies with the curvature of the universe characterized by interacting phantom tachyonic field. For k = -1 , 1, the potential increases with phantom tachyonic field and for k = 0, it decays after increasing initially. Similarly figures (7-9) show the reconstructed potentials for the k-essence field. We have seen that for interacting k-essence the potential V always decreases with increase in the scalar field φ in all of the three scenarios and it happens for open, closed and flat universes. When we consider an interacting DBI-essence dark energy, we get decaying pattern in the V -φ plot for emergent and intermediate scenarios in the figures 10 and 12. However, from figure 11 we see an increasing plot of V -φ for for interacting DBI-essence in the logamediate scenario. For interacting hessence dark energy, figures 13 shows increase in the potential with scalar field and figures 14 and 15 show decay in the potential with scalar field. This means the potential for interacting hessence increases in the emergent universe and decays in logamediate and intermediate scenarios. Figures (16-18) discuss the dilaton field while figures (19-21) show the behavior of the Yang-Mills field in the FAC. For interacting dilaton field, the scalar field φ always increases with cosmic time T 1 irrespective of the scenario of the universe and when we consider Yang-Mills dark energy, we find that E 2 in always increasing with cosmic time T 1 .", "pages": [ 22, 23, 24 ] } ]

[ { "title": "Cosmological particle creation in the lab?", "content": "Ralf Sch¨utzhold 1 , ∗ and William G. Unruh 2 , + 1 Fakult¨at f¨ur Physik, Universit¨at Duisburg-Essen, D-47048 Duisburg, Germany 2 Canadian Institute for Advanced Research Cosmology and Gravity Program Department of Physics and Astronomy, University of British Columbia, Vancouver B.C., V6T 1Z1 Canada ∗ [email protected] , + [email protected] August 1, 2018", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "One of the most striking examples for the production of particles out of the quantum vacuum due to external conditions is cosmological particle creation, which is caused by the expansion or contraction of the Universe. Already in 1939, Schrodinger understood that the cosmic evolution could lead to a mixing of positive and negative frequencies and that this 'would mean production or annihilation of matter, merely by the expansion' [Schrodinger, 1939]. Later this phenomenon was derived via more modern techniques of quantum field theory in curved space-times by Parker [Parker, 1968] (who apparently was not aware of Schrodinger's work) and subsequently has been studied in numerous publications, see, e.g., [Birrell & Davies, 1982; Fulling, 1989; Wald, 1994]. Even though cosmological particle creation typically occurs on extremely large length scales, it is one of the very few examples for such fundamental effects where we actually may have observational evidence: According to the inflationary model of cosmology, the seeds for the anisotropies in the cosmic microwave background (CMB) and basically all large scale structures stem from this effect, see Section 5. In this Chapter, we shall provide a brief discussion of this phenomenon and sketch a possibility for an experimental realization via an analogue in the laboratory.", "pages": [ 2 ] }, { "title": "2 Scattering analogy", "content": "For simplicity, let us consider a massive scalar field Φ in the 1+1 dimensional FriedmannRobertson-Walker metric with scale factor a ( τ ) where τ is the proper (co-moving) time and η the conformal time. The latter co-ordinate is more convenient for our purpose since the wave equation simplifies to In the massless case m = 0, the scalar field is conformally invariant (in 1+1 dimensions) and thus the expansion does only create particles for m > 0. After a spatial Fourier transform, we find that each mode φ k ( η ) behaves like a harmonic oscillator with a timedependent potential with k 2 + a 2 ( η ) m 2 → Ω 2 ( t ) and η → t . There is yet another analogy which might be interesting to notice. If we compare the above equation to a Schrodinger scattering problem in one spatial dimension we find that is has precisely the same form after identifying t ↔ x , φ ( t ) ↔ Ψ( x ), and Ω 2 ( t ) ↔ 2 m [ E -V ( x )]. Note that Ω 2 is always greater than zero in our case - which corresponds to propagation over the barrier E > V ( x ). If Ω 2 were less than zero over some region in time, one would have a barrier penetration (i.e., tunnelling) problem E < V ( x ). With the condition that in the past the field has the form e i Ω in t , in the future the solution would be αe i Ω out t + βe -i Ω out t due to scattering from the region where Ω 2 < 0. This would correspond to particle creation with probability proportional to | β | 2 . However even if Ω 2 > 0 everywhere there will still be some scattering (above the barrier). In order to derive the cosmological particle creation, we can study a positive pseudo-norm solution of Eq. (3) which initially behaves as e -i Ω in t and finally evolves into a mixture of positive and negative pseudo-norm solutions - which is in this case equivalent to positive and negative frequencies αe -i Ω out t + βe + i Ω out t (assuming that Ω is constant asymptotically). In the Schrodinger scattering problem, the initial solution e -i Ω t could be identified with a left-moving wave on the left-hand side of the potential 'barrier' while the final solution αe -i Ω t + βe + i Ω t would then correspond to a mixture of left-moving αe -i Ω t and right-moving βe + i Ω t waves on the right-hand-side. As a consequence, the Bogoliubov coefficients α and β are related to the reflection R and transmission T coefficients via α = 1 /T and β = R/T . In this way, the Bogoliubov relation | α | 2 -| β | 2 = 1 is equivalent to the conservation law | R | 2 + | T | 2 = 1 for the Schrodinger scattering problem. The probability for particle creation can be inferred from the expectation value of the number of final particles in the initial vacuum state which reads 〈 0 in | ˆ n out | 0 in 〉 = | β | 2 .", "pages": [ 2, 3 ] }, { "title": "3 WKB analysis", "content": "In order to actually calculate or estimate the Bogoliubov coefficients, let us re-write Eq. (3) in a first-order form via introducing the phase-space vector u and the matrix M If we define an inner product via we find that the inner product of two solutions u and u ' of Eq. (5) is conserved The split of a solution into positive and negative frequencies (i.e., positive and negative pseudo-norm) corresponds to a decomposition in the instantaneous eigen-basis of the matrix Choosing the usual normalization u ± = (1 , ± i Ω) T / √ 2Ω, we find At each time t , we may expand a given solution u ( t ) of Eq. (5) into the instantaneous eigen-vectors where the pre-factors are now defined as time-dependent Bogoliubov coefficients α ( t ) and β ( t ). It is useful to separate out the oscillatory part with the WKB phase Now we may insert the expansion (10) into the equation of motion (5) and project it with the inner product (6) onto the eigen-vectors u ± which gives due to ( u -| ˙ u + ) = ˙ Ω / (2Ω) and ( u + | ˙ u -) = -˙ Ω / (2Ω) while ( u + | ˙ u + ) = ( u -| ˙ u -) = 0. This equation (12) is still exact and very hard to solve analytically - except in very special cases. It can be solved formally by a iterative integral equation It can be shown that this iteration converges to the exact solution for well-behaved Ω( t ) [Braid, 1970]. Standard perturbation theory would then correspond to cutting off this iteration at a finite order, which can be justified if Ω( t ) changes only very little. For the scalar field in Eq. (2) this perturbative treatment should be applicable in the ultrarelativistic limit, i.e., as long as the mass is much smaller than the wave-number. In many cases, however, another approximation - the WKB method - is more useful. This method can be applied if the rate of change of Ω( t ), e.g., the expansion of the universe, is much slower than the internal frequency Ω( t ) itself. Writing with some dimensionless function f of order one, the WKB limit corresponds to Ω 0 /greatermuch ω . In terms of the reflection coefficient R = β/α mentioned earlier, we get which is known as Riccati equation. Again, this equation is still exact but unfortunately non-linear. Neglecting the quadratic term R 2 would bring us back to perturbation theory. In the WKB-limit, the phase factors e ± 2 iϕ are rapidly oscillating and the magnitude of R can be estimated by going to the complex plane. Re-writing the Riccati equation (15) as we may use an analytic continuation ϕ → ϕ + iχ to see that R becomes exponentially suppressed R ∼ e -2 χ . How strongly it is suppressed depends on the point where the analytic continuation breaks down. Since e ± 2 iϕ is analytic everywhere, this will be determined by the term ln Ω. Typically, the first non-analytic points t ∗ encountered are the zeros of Ω, i.e., where Ω( t ∗ ) = 0. In the case of barrier reflection, these points where Ω = 0, i.e., where V = E , lie on the real axis and correspond to the classical turning points in WKB. In our case, we have scattering above the barrier and thus these points become complex - but are still analogous to the classical turning points in WKB. Consequently, we find 1 If there is more than one turning point, the one with the smallest χ ∗ > 0, i.e., closest to the real axis (in the complex ϕ -plane) dominates. If these multiple turning points have similar χ ∗ > 0, there can be interference effects between the different contributions, see, e.g., [Dumlu & Dunne, 2010].", "pages": [ 3, 4, 5 ] }, { "title": "4 Adiabatic expansion and its breakdown", "content": "Note that we could repeat steps (5) till (12) and expand the solution u ( t ) into the firstorder adiabatic eigen-states instead of the instantaneous eigen-vectors u ± . To this end, let us re-write (12) as The eigen-vectors of the matrix N are the first-order adiabatic eigen-states w ± and the eigen-frequencies N · w ± = ± i Ω ad w ± are renormalized to Assuming α in = 1 and β in = 0, the system stays in the adiabatic eigen-state w + to lowest order in ω/ Ω and we get This adiabatic expansion into powers of ω/ Ω can be continued and gives terms like ˙ Ω 2 / Ω 4 and ¨ Ω / Ω 3 to the next order in ω/ Ω (see below). One should stress that this expansion is not the same as in (13) since it is local - i.e., only contains time-derivatives - while (13) is global - i.e., contains time-integrals. Since all terms of the adiabatic expansion (20) are local, they cannot describe particle creation - which depends on the whole history of Ω( t ). In terms of the adiabatic expansion into powers of ω/ Ω, particle creation is a non-perturbative effect, i.e., it is exponentially suppressed, see Eq. (17) and thus cannot be found be a Taylor expansion into powers of ω/ Ω. For any finite ratio of ω/ Ω, this also means that the adiabatic expansion (into powers of ω/ Ω) must break down at some point. To make this argument more precise, let us re-write Eq. (18) in yet another form In this representation, the eigen-values of N are given by ± i Λ and the eigen-vectors read Decomposing the solution w ( t ) into these eigen-vectors and using ˙ w + = ˙ ξ w -as well as ˙ w -= ˙ ξ w + , we find This is the same form as Eq. (22) if we change Λ and ξ accordingly. Thus, by repeating this procedure, we get the iteration law By this iteration, we go higher and higher up in the adiabatic expansion since ξ n always acquires an additional factor of ω/ Ω. Thus, for ω /lessmuch Ω, the values of ξ n quickly decay with a power-law ξ n = O ([ ω/ Ω] n ) initially. As we go up in this expansion, however, the effective rate of change of ξ n increases. For example, if Ω( t ) has one global maximum (or minimum) and otherwise no structure, the time-derivative ˙ Ω / (2Ω 2 ) = tanh(2 ξ 1 ) has two extremal points and a zero in between. By taking higher and higher time derivatives, more and more extremal points and a zeros arise and thus the effective frequency ω eff n of ξ n ( t ) increases roughly linearly with the number n of iterations ω eff n = O ( nω ). Furthermore, the adiabatically renormalized eigen-values Λ n decrease with each iteration. Thus, after approximately n = O (Ω /ω ) iterations, the effective frequency ω eff n becomes comparable to the internal frequency Λ n . At that point, the adiabatic expansion starts to break down. Estimating the order of magnitude of ξ n at that order gives Since the effective external ω eff n and internal Λ n frequencies are comparable and ξ n is very small, we may just use perturbation theory to estimate β and we get β = O ( ξ n ), i.e., the same exponential suppression as in Eq. (21). If we would continue the iteration beyond that order, the ξ n would start to increase again - which the usual situation in an asymptotic expansion, see Figure 1. Carrying on the iteration too far beyond this point, the ˙ ξ 2 n exceed the Λ 2 n and thus we have barrier penetration instead of propagation over the barrier (as occurs for all orders below this value of n ). In this procedure, it is this barrier penetration which gives the mixing of positive and negative pseudo-norm, and the creation of particles. Were the system to remain as propagation over the barrier for all orders n in this adiabatic expansion, one would have no particle creation.", "pages": [ 6, 7 ] }, { "title": "5 Example: inflation", "content": "As an illustrative example, let us consider a minimally coupled massive scalar field in 3+1 dimensions - which could be the inflaton field (according to our standard model of cosmology). Again, we start with the Friedmann-Robertson-Walker metric (1) with a scale factor a ( τ ) and obtain the equation of motion Rescaling the field φ ( τ, r ) = /Omegainv ( τ )Φ( τ, r ) with /Omegainv ( τ ) = a 3 / 2 ( τ ) and applying a spatial Fourier transform, we obtain the same form as in Eq. (3) In the standard scenario of inflation, the space-time can be described by the de Sitter metric a ( τ ) = exp { Hτ } to a very good approximation, where H is the Hubble parameter. In this case, the effective potential ¨ /Omegainv / /Omegainv just becomes a constant (3 H/ 2) 2 and the frequency Ω( τ ) reads Inserting a ( τ ) = exp { Hτ } , we see that modes with different k -values follow the same evolution - just translated in time. (This fact is related to the scale invariance of the created k spectrum.) Initially, this frequency is dominated by the k 2 term and we have ˙ Ω / Ω = -H which means that we are in the WKB regime ˙ Ω / Ω /lessmuch Ω. However, due to the cosmological red-shift, this k 2 term decreases with time until the other terms become relevant. Then the behavior of the modes depends on the ratio m/H . For m /greatermuch H , the modes remain adiabatic (i.e., stay in the WKB regime) and thus particle creation is exponentially suppressed. If m and H are not very different, but still m > 3 H/ 2 holds, the modes are adiabatic again for large times - but for intermediated times, the WKB expansion breaks down, leading to a moderate particle creation. For m < 3 H/ 2, on the other hand - which is (or was) supposed to be the case during inflation - the frequency Ω( τ ) goes to zero at some time and becomes imaginary afterwards. This means that we get a barrier penetration (tunneling) problem where the modes φ k ( τ ) do not oscillate but evolve exponentially in time φ k ( τ ) ∝ exp {± τ √ 9 H 2 / 4 -m 2 } . Here one should remember that the original field does not grow exponentially due to the re-scaling with the additional factor /Omegainv ( τ ) = a 3 / 2 ( τ ). This behavior persists until the barrier vanishes, i.e., the expansion slows down (at the end of the inflationary period) and thus the effective potential ¨ /Omegainv / /Omegainv drops below the mass term. After that, the modes start oscillating again. However, in view of the barrier penetration (tunneling) over a relatively long time (distance), we get reflection coefficients R which are not small but extremely close to unity R ≈ 1. This means that the Bogoliubov coefficients α and β are huge - i.e., that we have created a tremendous amount of particles out of the initial vacuum fluctuations. According to our understanding, precisely this effect is responsible for the creation of the seeds for all structures in our Universe. Perhaps the most direct signatures of this effect are still visible today in the anisotropies of the cosmic microwave background radiation. An alternative picture of the mode evolution in terms of a damped harmonic oscillator can be obtained from the original field in Eq. (28) Initially, the term e -2 Hτ k 2 dominates and the modes oscillate. Assuming m /lessmuch H (which is related to the slow-roll condition of inflation), the damping term dominates for late times and we get a strongly over-damped oscillator, whose dynamics is basically frozen (like a pendulum in a very sticky liquid). The transition happens when H ∼ ke -Hτ , i.e., when the physical wavelength λ = 2 πe Hτ /k exceeds the de Sitter horizon ∝ 1 /H due to the cosmological expansion e Hτ . After that, crest and trough of a wave lose causal contact and cannot exchange energy any more - that's why the oscillations effectively stops. As a final remark, we stress that this enormous particle creation effect is facilitated by the rapid (here: exponential) expansion and the resulting stretching of wavelengths over many many orders of magnitude (i.e., the extremely large red-shift). Therefore, a final mode with a moderate wavelength originated from waves with extremely short wavelengths initially. Formally, these initial wavelengths could be easily far shorter than the Planck length. However, on these scales one would expect deviations from the theory of quantum fields in classical space-times we used to derive these effects. On the other hand, this problem is not only negative - it might open up the possibility to actually see signatures of new (Planckian) physics in high-precision measurements of the cosmic microwave background radiation, for example.", "pages": [ 8, 9, 10 ] }, { "title": "6 Laboratory analogues", "content": "Apart from the observation evidence in the anisotropies of the cosmic microwave background radiation mentioned above, one may study the phenomenon of cosmological particle creation experimentally by means of suitable laboratory analogues, see, e.g., [Unruh, 1981; Barcel'o, Liberati, & Visser, 2011]. The are two major possibilities to mimic the expansion or contraction of the Universe - a medium at rest with time-dependent properties (such as the propagation speed of the quasi-particles) or an expanding medium. Let us start with the former option and consider linearized and scalar quasi-particles (e.g., sound waves) with low energies and momenta propagating in a spatially homogeneous and isotropic medium. Under these conditions, their dynamics is governed by the low-energy effective action Here we assume positive a 2 and non-negative b 2 and c 2 for stability. The factor a 2 ( t ) can be eliminated by suitable re-scaling of the time co-ordinate. Then, after a spatial Fourier transform, we obtain the same form as in Eq. (3). The quasi-particle excitations φ in such a medium behave in the same way as a scalar field in an expanding or contracting Universe with a possibly time-dependent potential (mass) term ∝ b 2 ( t ) φ 2 . In order to avoid this additional time-dependence of the potential (mass) term, the factors b and c must obey special conditions. For example, Goldstone modes with b = 0 correspond to a massless scalar field in 3+1 dimensions - whereas the case of constant c is analogous to a massive scalar field in 1+1 dimensions. As one would intuitively expect, the expansion or contraction of the Universe can also be mimicked by an expanding or contracting medium. Due to local Galilee invariance, such a medium can also be effectively spatially homogeneous and isotropic as in Eq. (32) when described in terms of co-moving co-ordinates. For a quite detailed list of references, see [Barcel'o, Liberati, & Visser, 2011]. There are basically three major experimental challenges for observing the analogue of cosmological particle creation in the laboratory. First, the initial temperature should be low enough such that the particles are produced due to quantum rather than thermal fluctuations. Second, one must be able to generate a time-dependence (e.g., expansion of the medium) during which the effective action in Eq. (32) remains valid (in some sense) but which is also sufficiently rapid to create particles. Third, one must be able to detect the created particles and to distinguish them from the radiation stemming from other sources. For trapped ions, for example (see, e.g., [Schutzhold et al , 2007]), the first and third point (i.e., cooling and detection) is experimental state of the art, while a sufficiently rapid but still controlled expansion/contraction of the ion trap presents difficulties. For Bose-Einstein condensates (see, e.g., [Barcel'o, Liberati, & Visser, 2011] and references therein), on the other hand, the first and third points are the main obstacles.", "pages": [ 10, 11 ] }, { "title": "Acknowledgments", "content": "The authors benefited from fruitful discussions, especially with R. Parentani, during the SIGRAV Graduate School in Contemporary Relativity and Gravitational Physics , IX Edition 'Analogue Gravity', at the Centro di Cultura Scientifica 'A. Volta', Villa Olmo, in Como (Italy, 2011). R.S. acknowledges support from DFG and the kind hospitality during a visit at the University of British Columbia where part of this research was carried out. W.G.U. thanks the Natural Sciences and Engineering Research Council of Canada and the Canadian Institute for Advanced Research, for research support, and the University of Duisburg-Essen for their hospitality while part of this research was carried out.", "pages": [ 12 ] } ]

[ { "title": "Irrotational, two-dimensional Surface waves in fluids", "content": "William G. Unruh Department of Physics and Astronomy, University of British Columbia, Vancouver, B.C., V6T 1Z1 Canada (Dated: October 31, 2018) The equations for waves on the surface of an irrotational incompressible fluid are derived in the coordinates of the velocity potential/stream function. The low frequency shallow water approximation for these waves is derived for a varying bottom topography. Most importantly, the conserved norm for the surface waves is derived, important for quantisation of these waves and their use in analog models for black holes. PACS: 47.90.+a, 92.60.Dj, 04.80.y.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "One of the most fascinating predictions of Einstein's theory of general relativity is the potential existence of black holes - i.e. space-time regions from which nothing is able to escape. Perhaps no less interesting are their antonyms: white holes which nothing can penetrate. Both are described by solutions of the Einstein equations and are related to each other via time-inversion, see e.g. [1, 2]. It is equally fascinating that some of the predictions for fields in a black hole spacetime can be modelled by waves in a variety of other situations, with the interior of the black hole or white hole horizons can be mimicked by fluid flow which exceeds the velocity of the waves in some regions. One of these is the use of surface waves on a incompressible fluid[3]. One can alter the flow properties of the fluid by placing obstacles into the bottom of a flume (a long tank along which the water flows) to speed up and slow down the fluid over these obstacles. One of the difficulties in the theoretical treatment of such systems is the complicated boundary conditions on the bottom of the tank (where the fluid velocities must be tangential to the bottom) and the top (where the pressure of the fluid must be zero or at a constant atmospheric constant pressure). In fact as we will see the equations for the fluid itself are remarkably simple. The interesting physics arises entirely from those boundary condition. We will be interested in irrotational, incompressible flow. While both are certainly approximations for water flow (the former assumes no turbulence, and no viscosity which would create vorticity at the shear layer along the bottom, while the latter assumes that the velocity of sound in the fluid is far higher than any other velocities in the problem). While this problem has been investigated before[4], this is in general in the three dimensional context (which is more difficult) and using approximations and expansions for the shape of the bottom. I will assume that the fluid flow is a two dimensional flow- ie is uniform across the tank and that the tank maintains a constant width throughout. This is much simpler case than three dimensional flow, which allows the coordinate transformations I use. The usual spatial coordinates are x, y with x being the horizontal direction in which the fluid flows, and y is the vertical direction (parallel to the gravitational acceleration, g , directed in the negative y direction. The Euler-Lagrange equations are where the second equation is the incompressibility condition. In the usual way, if we assume that the flow is irrotational, then And the above equation can be written as where ˜ p is the pressure. Let me define the specific pressure, p = ˜ p ρ In the following I will consider only flows in the x -y directions. Everything is assumed to be independent of z . Consider the vector /vector w = /vectore z × /vectorv . This vector also obeys since nothing depends on z . Thus we can define where ˜ ψ also obeys ∇ 2 ˜ ψ = 0 and where (12) Let me now define a new coordinate system. I could use ˜ φ and ˜ ψ , but I will be interested in fluid flows where the velocity approaches a constant value v x = v 0 , v y = 0 at large distances. I will thus instead use the functions ψ, φ defined by as the new coordinates. This choice will also allow me to take the limit as the velocity v 0 goes to zero, where the potentials ˜ φ, ˜ ψ are undefined. Thus at large distances, φ = x and ψ = y . The spatial metric in the xy coordinates is (where the Einstein summation convention has been used where a repeated index implies summation over that index, and where z 1 = x, z 2 = y ). Do not confuse z i with the horizontal direction z which nothing depends on. The Laplacian is for a general metric function of g ij ( z ) is where g ij are the components of the matrix with is the inverse to the matrix of coefficients g ij and where g is the determinant of the matrix with coefficients g ij . For a reference regarding metrics and the coordinate independent equations see almost any book on General Relativity[5]. In two dimensions, if g ij = f ˜ g ij where f is some function of the coordinates z i , then since g ij = 1 f ˜ g ij and g = det ( g ij ) = f 2 det (˜ g ij ) = f 2 ˜ g , we have ∇ 2 = 1 f ˜ ∇ 2 . Metrics such as g ij and ˜ g ij are said to be confomally related. i j Recalling that the change in the metric components from one coordinate system z to a new system ˆ z are given by where the Einstein summation convention has been used, the components of the upper components of the usual flat space metric in this new ˆ z 1 = φ, ˆ z 2 = ψ coordinate system are Ie, the new metric (the inverse of this upper form metric) in these new coordinates is a conformally flat metric Since in the xy coordinates the metric is flat, this metric is also flat in ψ, φ coordinates, (the curvature is not changed by a coordinate transformation) and the scalar curvature in this new coordinate system is zero. Using the equation and thus for the scalar curvature of a metric (and in two dimensions, the scalar curvature is the only independent component of the curvature) one gets (This is valid as long as v 2 v 2 0 I define is not equal to zero anywhere) The Laplacian is, since the metric in ψ, φ coordinates is conformally flat, just for any scalar function Φ. Since in x, y coordinates, the Laplacian of both the scalar functions x and y are zero, they must also be zero in φ, ψ coordinates ( since the Laplacian is an invariant scalar operator), and, as functions of φ, ψ , we have as the equations of motion obeyed by x and y in these new coordinates. ψ is the stream function, and the vector /vectorv is tangent to the surfaces of constant ψ . /vectorv · /vector ∇ ψ = /vectorv · /vector w = 0. The bottom of the flow must be tangent to the flow vector (no flow can penetrate the bottom), and thus must be a surface of constant ψ , which I will take to be ψ = 0. Similarly, if the flow is stationary, the top of the water, no matter how convoluted, must also lie along a streamline, since a particle of the fluid which is at the top, must flow along the top (the velocity of the particles must be parallel to the top surface). This means that the top of a stationary flow ( but not a time dependent flow) also is at a constant value of ψ which I will label ψ T . We also have Solving for x and y as a function of ψ, φ , which is just solving the Laplacian in terms of ψ, φ , gives us the velocity at all points. The boundary condition along the bottom for these functions must be that the velocity along the bottom be parallel to the bottom. If the bottom has the functional form y = F ( x ) then y ( φ, 0) = F ( x ( φ, 0)). On the top of the flow, we have the boundary condition that p = 0. From the Bernoulli equation for a stationary flow is which, if the flow has constant velocity u over a constant depth bottom of height h far away from the obstacle, gives the equation for the top of the flow we have and Bernoulli's equation is v 2 + gy = const along the top surface of the fluid where p = 0. Soving for ∂ ψ y we get Any function H ( ψφ ) which is a solution of ∂ 2 ψ H + ∂ 2 φ H = 0 can be expanded in exponentials e ikφ . We see immediately that the dependence of these modes of ψ must be in terms of e ± kψ or equivalently in terms of cosh ( kψ ) and sinh ( kψ ) for the ψ dependence. Thus, since y obeys that equation, we have with where Writing this in terms of φ, ψ we have the upper boundary condition of This is a complicated, non-linear, boundary condition. Thus while the equations of motion of x, y are simple (Laplacian equals zero), the physics is all contained in the boudary conditions. If we are given y ( x ) as the equation for the bottom, the solution of the above non-linear boundary value problem is difficult. However if , instead of specifying the lower boundary, one specifies the shape of the upper boundary y ( φ, ψ T ), one can use Bernoulli's equation in these new coordinates to determine the ψ derivative of y . Since Then at the lower boundary, This gives the bottom as a parametric set of functions of φ . In figure 1 we have an example of sub to supercritical flow over an obstacle. calculated as above. Note that the obstacle is a reasonable function y ( x ). where Of course, we are not given y ( φ, 0) but rather y ( φ, 0) = F ( x ( φ, 0)). However one can get rapid convergence by iteration", "pages": [ 1, 2, 3, 4, 5 ] }, { "title": "A. v 0 = 0 limit", "content": "The boundary condition equations are easily solved in the limit as v 0 → 0. The upper boundary condition becomes simply y = h and ∂ φ y = 0. This can be solved (in terms of the unknown lower boundary solutions y ( φ, 0) , x ( φ, 0) by which gives via the above equations the solution y i +1 ( φ, ψ ) and thus For small v 0 , one can get a first order correction for the surface value of y ( φ, ψ T ) by taking Ie, for slow flow over a bottom boundary, the stationary solution for that flow is easy to find.", "pages": [ 5, 6 ] }, { "title": "B. Formal General solution", "content": "The general solution to the equation ˜ ∇ 2 F =0 can be written as If F is real, then g ( φ -iψ ) = ( f ( φ + iψ )) ∗ We then have x ( φ, ψ ) = ˆ x ( φ + iψ ) + ˆ x ∗ ( φ + iψ ) (54) Given the boundary conditions along the bottom, we have This of course still leaves the highly non-linear boundary conditions at the top to solve to find x and y everywhere.", "pages": [ 6 ] }, { "title": "II. FLUCTUATIONS", "content": "Let us assume that we have a background solution to the stationary equation, x 0 ( φ, ψ ) , y 0 ( φ, ψ ), or equivalently, φ 0 ( x, y ) , ψ 0 ( x, y ). We want to find the equations for small perturbations around this background flow. Let us also consider a solution to the full time dependent equations, φ ( x, y, t ) , ψ ( x, y, t ) together with their inverses, x ( φ, ψ, t ) , y ( φ, ψ, t ), such that y ( φ ( x, y, t ) , ψ ( x, y, t ) , t ) = y and x ( φ ( x, y, t ) , ψ ( x, y, t ) , t ) = x . Define the small deviations from the background by (58) (61) Then we have Keeping terms only to first order in ' δ ', we have or (where all velocity components are those in the background flow). Similarly and Along the surface, we therefore have But, The Bernoulli equation is where the first ∂ t is defined as the derivative keeping x, y fixed, not φ, ψ fixed. Here p is the specific pressure. Writing this equation perturbatively, we have where all of the velocities are the values of the background velocities at the location φ, ψ . Ie, v x ( φ, ψ ) = v 0 x ( x 0 ( φ, ψ ) , y 0 ( φ, ψ )). We can now rewrite this equation in terms of δφ = δφ ( x 0 ( φ, ψ ) , y 0 ( φ, ψ )) to get Recalling that ∂ φ v x v 2 = ∂ φ ∂ ψ y 0 = ∂ ψ v 0 v y v 2 and ∂ φ v 0 v y v 2 = -∂ ψ v 0 v x v 2 , we finally get The boundary conditions at the bottom are that δx and δy must be parallel to the bottom, or v x δy -v y δx = 0 which is just At the top, the pressure at the surface must be 0. However the surface is no longer simply ψ = ψ T because of the time dependence of the equations. Let us assume that the surface is defined by Since a particle of the fluid which starts on the surface, remains on the surface, we can define the fluid coordinates η, ζ . Then the velocity of the fluid is Thus, assuming that Ψ is also small (the same order as the other ' δ ' terms), we have On the surface, we have the Bernoulli equation, which to first order is But along the surface ψ = ψ T , the background 1 2 v 2 + gy is constant, so the φ derivative is 0. We have Dividing by G = ∂ ψ ( 1 2 v 2 + gy ) and taking the derivative ˜ ∂ t + v 2 v 0 ∂ φ we get as the equation of motion for the surface wave. δφ and δψ are related by the boundary condition δφ = 0 along the bottom. Since both δφ and δψ obey ∇ 2 δψ = ∇ 2 δφ = 0, we have Furthermore, since so For irrotational time-independent flow, the acceleration of a parcel of fluid is /vectorv · ∇ /vectorv = /vector ∇ ( 1 2 v 2 ) and the orthogonal component of this, the centripetal acceleration is Also g∂ ψ y = g v x v 0 v 2 ≈ gv 0 /v so Gv/v 0 is the effective gravitational field orthogonal to the flow lines (including the centripital acceleration) . However it is important to note that it is the effective force of gravity only at the surface of the fluid, not at the obstacle to the flow along the bottom, that is important for the equations of motion.", "pages": [ 6, 7, 8 ] }, { "title": "III. SHALLOW WATER WAVES", "content": "Since φ, ψ are real functions, the solutions can be written as for some function Z . These functions clearly satisfy the Laplacian equation for, and furthermore also satisfy the differential relations on the derivatives of x, y with respect to φ, psi This gives Ie, Z is a real function of a real arguments. which gives or, to first order in ψ T The equation for the waves then becomes We note that this is not a Hermitian operator acting on δφ . Recall that a Hermitian operator is one such that if we assume that all of the boundary terms in the integration by parts are zero. We can rewrite the equation for δφ by dividing by v 2 as This is a symmetric equation, derivable from an action, This action has the global symmetry δφ → e iµ δφ and thus has the usual Noether current associated with this symmetry. In particular it has the conserved norm", "pages": [ 8, 9 ] }, { "title": "IV. DEEP WATER WAVES", "content": "For deep water waves, we can assume that either Z ( φ + iψ T ) >> Z ( ψ -iφ T ) or Z ( φ + iψ T ) << Z ( ψ -iφ T ). (ie, we assume that as analytic functions, Z goes to zero either in the upper or lower half plane.) Let us also assume it is the first case, and let us define ˆ Z ( φ ) = Z ( φ + iψ T ), and that ˜ ∂ t δφ = iωδφ We then have If we assume that K = i ( ∂ φ ln ( ˆ Z )) is large and negative, such that ˆ Z varies faster than v 2 or G , we have approximately or or The equation in general is Fourier transforming with respect to φ and ψ , and using the fact that δψ = 0 at ψ = 0, the functions δφ, δψ then can be written as since again they obey the Laplacian equal to zero in these variables. Defining B ( k, t ) = A ( k, t ) cosh ( kψ ) this can be written as Thus the equation of the surface waves can be written as Ie, we get the usual tanh dispersion relation for the transition from shallow to deep water waves. This equation is symmetric and real, and thus if δφ is a solution, so is δφ ∗ . Again this gives a conserved norm between two solutions to the equations of motion δφ and δφ ' of We note that this equation depends only the conditions at the surface of the flow. It is defined entirely in terms of the factors ( v 2 and G = ∂ ψ ( gy + 1 2 v 2 ) defined at ψ = ψ T , and is independent of the obstacles, or the flow throughout the rest of the stream except insofar as they affect the flow at the surface. This might well change if either vorticity or viscosity were introduced into the equations. This norm is crucial to the analysis of the wave equation. It is conserved (in the absence of viscosity), and in the use of such waves as models for black holes, it is this norm which determines the Bugoliubov coefficients (or the amplification factor) for waves in the vicinity of a horizon (blocking flow in the hydrodynamics sense) and determines the quantum noise (Hawking radiation) emitted by such a horizon analog. The quantum norm used in the quantization procedure is If we define a new coordinate ˆ φ = ∫ 1 v 2 dφ , the norm becomes If the surface of the flow is shallow ( dy T dx << 1) then dφ dx = v x ≈ v and ˆ φ ≈ dx v x . To relate this to the measured quantity, the vertical displacement at the surface of the waves, we must relate δφ to δy at the surface of the fluid. We have", "pages": [ 9, 10 ] }, { "title": "V. GENERAL LINEARIZED WAVES", "content": "Now, Gv x ≈ g v 2 x v 2 ≈ g (ignoring the centrifugal contribution to the effective gravity), so the norm becomes and d ˆ φ dφ v 2 ≈ dx v If we assume that the incoming wave is at a set frequency ω and take the fourier transform with respect to t, ˆ x of √ v ( ˆ ψ ) y T ( t, ˆ φ ) this becomes We can also look at the norm current. The integrand is a complete derivatives. Although this is not obvious for the terms with the tanh in them, we can use and the fact that i∂ φ tanh( iψ T ∂ φ ) can be expanded in a power series in ∂ 2 φ to show that they also a complete derivative.. Thus the integrand can be written in terms of a complete derivative of with respect to ∂ ψ and we can regard the term that is being taken the derivative of as a spatial norm current J φ so that if J t is the temporal part of the norm current, we have ∂ t J t + ∂ φ J φ = 0. If we are in a regime where δφ = Ae -iωt -kφ , (ie, a regime where the velocity v and G are both constants), then we have where v p and v g are the phase and group velocity of the wave. In a situation in which one has a wave train with some definite frequency and wave number entering a region, then the sum of all the norm currents for each k at the boundary of the region must be zero.", "pages": [ 11 ] }, { "title": "VI. BLOCKING FLOW", "content": "Let us return to the static situation. Define U = ∂ φ δφ , we have the equation As above, there is a solution if we assume that the derivatives are small, which gives For rapid variations, we have with the transition from one to the other occuring roughly when the logarithmic derivatives of the two solutions are equal (129) Defining the Froude number by F 2 = v 2 Gψ T (the square of the velocity of the fluid over the velocity of the long wavelengths in the fluid in the WKB approximation), we have Note that for a non-trivial rate of change of of the bottom, the turning point occurs well before the horizon. The ' denotes derivative with respect to φ not x. We can rewrite this approximately (assuming that v x v ≈ that 2 ln ( F ) ' < 1 F 2 ψ T and ψ T ≈ vd where d is the depth of the water at postion x . as 1 and Note that this transition occurs before Gψ T = v 2 or Froude number equals 1. The wave on the slope piles up and its frequency makes the transition to deep water wave before we hit the effective horizon. The long wavelength equation, is not that of a two dimension metric, which is always conformally flat, but can be written as a the wave equation for a three dimensional metric where all derivatives are equal to zero in the third ξ dimension for the variable δφ . The metric is where α is an arbitrary function of φ , a two dimensional conformal factor which does not affect the two dimensional wave equation This metric has surface gravity (The surface gravity is the acceleration in the metric as seen from far away. for a static time independent metric in a coordinate system which is regular across the horizon, it can be defined by κ = Γ t tt at the horizon, where Γ i jk is the Christofell symbol for the metric. Then Γ t tt = -1 2 g tφ ( ∂ φ g tt ) at the horizon.)", "pages": [ 11, 12 ] }, { "title": "VII. CONVERSION TO δy", "content": "Of course δφ is not what is actually measured in an experiment. That is the fluctuation ∆ y ( x ) which is the difference in height between the stationary flow, and the height with the wave present. We can relate this to δψ and Ψ. where y 0 is the surface for the background. Since δψ = tanh(Ψ H ∂ φ ) δφ , we have Inverting this for deep water waves, while for shallow water waves The integrand in the exponent is non-zero only in the region where the background flow is dimpled, and, since v y v x is in general very small, the exponential can be neglected in most situations. In the intermediate region, where the wave changes from shallow to deep water wave, there is no easy solution to these equations, but they can be integrated numerically.", "pages": [ 12, 13 ] }, { "title": "VIII. WAVES IN STATIONARY WATER OVER UNEVEN BOTTOM", "content": "In the limit as v 0 goes to zero, so does v with the ratio being a finite function. y obeys the equation ∂ 2 φ + ∂ 2 ψ y = 0 with the boundary conditions along the bottom that y = Y ( x ), with Y the given function of x of the bottom, and along the top, y = H , a constant. If we assume that we know Y ( φ ) (instead of Y ( x )) along the bottom, this can be solved by where and One gets rapid convergence if one starts by taking x = φ , substituting into Y ( x ( φ )) to find Y ( φ ), finding the new x ( φ ) and substituting in again. Then v y v 0 at the surface is zero, while The equation for small perturbations becomes where .If the depth is constant, the backgound ψ = y and φ = x giving the usual equation, which allows us to write For deep water waves, where the tanh is unity, this equation is exactly the same as the deep water equation for constant depth. The fact that the bottom varies makes no difference to the propagation of the waves, as one would expect. For shallow water waves, where the tanh can be approximated as the linear function in its argument, the equation becomes This allows us to determine the wave propagation over an arbitrarily defined bottom. Note that in the stationary limit, the background flow is certainly irrotational, implying that the assumptions made here should certainly be valid (of course neglecting the viscosity of the fluid). Acknowledgement This work was supported by The Canadian Institute for Advanced Research (CIfAR) and by The Natureal Science and Engineering Research Council of Canada (NSERC) and was completed while the author was a visitor at the Perimeter Institute.", "pages": [ 13, 14 ] } ]

[ { "title": "ABSTRACT", "content": "Topic of this paper is the time-evolution of intrinsic correlations of galaxy ellipticities due to peculiar motion. In our model, the galaxy ellipticities are determined from the angular momentum of their host haloes, which can be computed from the fluctuations statistics of a Gaussian random field. Subsequent peculiar motion distorts the ellipticity field and causes changes in the ellipticity correlations. Using analogies between this problem of shifted ellipticity tensors and the displacements of polarisation tensors in gravitational lensing of the cosmic microwave background we compute E -mode and B -mode spectra of the time-evolved ellipticity field, where the displacements are modelled with first and second order Lagrangian perturbation theory. For EUCLID, ellipticity correlations are decreased on large multipoles /lscript > ∼ 1000, amounting to up to 10% in the E -mode spectrum C /epsilon1 E ( /lscript ) and up to 60% in the B -mode spectrum C /epsilon1 B ( /lscript ) at /lscript /similarequal 3000 due to the dispersing e ff ect of peculiar motion. E / B -mode conversion in analogy to CMB-lensing is present but small. We conclude that distortions of the ellipticity field due to peculiar motion is not a ff ecting the prediction of ellipticity models on the scales relevant for lensing in the case of EUCLID's galaxy distribution, but should a ff ect larger scales for surveys at lower redshifts. Key words: cosmology: large-scale structure, gravitational lensing, methods: analytical", "pages": [ 1 ] }, { "title": "Aram Giahi-Saravani /star and Bjorn Malte Schafer", "content": "Astronomisches Recheninstitut, Zentrum fur Astronomie,Universitat Heidelberg, Monchhofstraße 12, 69120 Heidelberg, Germany 20 March 2018", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Weak gravitational lensing by the cosmic-large scale structure is a tool for investigating the fluctuations statistics of the cosmic density field and its dependence on the underlying cosmological model (for reviews, see Bartelmann & Schneider 2001; Bartelmann 2010). Future weak lensing surveys such as EUCLID, DES, LSST and JDEM are designed to yield sub-percent accuracy on the set of cosmological parameters in a dark energy cosmology by measuring the correlation function of the gravitationally sheared galaxy ellipticity field. A common assumption in weak cosmic lensing is that those ellipticities are intrinsically uncorrelated and that the only correlating e ff ect is weak lensing, because the light from neighbouring galaxies has to transverse the same cosmic tidal fields, leading to a correlation in change in shape. This assumption, however, is challenged on small scales by intrinsic alignment e ff ects (for a review on angular momentum models and intrinsic alignments, see Schafer 2009). Tidal shearing models for angular momentum built-up in galactic haloes predict correlated angular momenta of neighbouring galaxies. If the symmetry axis of the galactic disk is aligned with the angular momentum direction of the host halo, neighbouring galaxies are viewed under correlated angles of inclination such that their ellipticities appear correlated. This intrinsic alignment e ff ect is important on small scales because the angular momentum correlation is comparatively short ranged: it is predicted to be present on scales of about 1 Mpc / h (Crittenden et al. 2001; Natarajan et al. 2001; Schafer & Merkel 2011). Intrinsic alignments of galaxies based on angular momentum models are a relatively new topic which will undoubtedly attract much interest in the future as the weak lensing data sets provided by large-scale lensing observations will at the same time help to scrutinise intrinsic alignment models. The theory of angular momentum-induced alignments (Croft & Metzler 2000; Crittenden et al. 2001, 2002; Mackey et al. 2002) has been applied to describe contamination of weak lensing data in the convergence spectrum (Heavens et al. 2000; Heymans & Heavens 2003; Heymans et al. 2004, 2006) and bispectrum (Semboloni et al. 2008). Di ff erent schemes for removing the contamination of intrinsic alignments have been proposed, from discarding close galaxy pairs (King & Schneider 2003, 2002) to specifically designed weighting schemes for nulling out their contribution (Joachimi & Schneider 2008) or amplifying them relative to the weak lensing induced ellipticity correlations (King & Schneider 2003; Joachimi & Schneider 2010). Resulting biases on cosmological parameter estimation if intrinsic alignments remain uncorrected have been quantified (Bridle & King 2007; Joachimi & Bridle 2010; Kirk et al. 2010; Schneider & Bridle 2010). The wealth of structure in the angular momentum and ellipticity field and their alignment with large-scale tidal fields has attracted much interest from the numerical perspective (Hahn et al. 2007; Codis et al. 2012) and suggests the question if large-scale tidal fields can be reconstructed using the ellipticity field as a tracer (Lee & Pen 2000, 2001, 2007). On a more fundamental level, the investigation of tidal shearing mechanisms in di ff erent orders of perturbation theory along with the deformation of forming haloes due to tidal forces (Catelan & Theuns 1996b; Catelan 1995; Catelan et al. 2001; Catelan & Theuns 1996a; Catelan et al. 1995; Catelan & Porciani 2001; Lee et al. 2007). Numerical verification of the tidal torquing theories of angular momentum build-up has been the topic of a number of papers (Catelan & Porciani 2001; Bullock et al. 2001; Porciani et al. 2002a,b; Hahn et al. 2007) who agree that the angular momentum direction can be well described by tidal torquing whereas the amount of angular momentum might be overestimated. Because ellipticity alignments are only sensitive to the angular momentum direction, these studies provide support for using tidal torquing models with this particular application in mind. The way in which the orientation of a galactic disk is linked to the angular momentum direction of the host halo is not very clearly cut. In a small number of observations Bailin et al. (2005) found mismatches and suggest that direct linking of the symmetry axis of the disk to the host halo angular momentum would lead to overestimation of the ellipticity alignments. This e ff ect is partially covered by parameterisation, but unless the relation is better understood, angular momentum-based alignment models proved upper limits on ellipticity correlations. By now, intrinsic alignments have been measured in a number of data sets and have been found at the expected levels (Pen et al. 2000; Lee & Pen 2002; Mandelbaum et al. 2006; Hirata et al. 2007), although some studies doubt these claims (e.g. Andrae & Jahnke 2011). The point which motivated this paper is the comparatively short-ranged correlation of angular momenta and consequently of the galaxy ellipticities, which reaches out to distances of about 1 Mpc / h . If the ellipticity field is this short-ranged, and if it is distorted by the peculiar motion of galaxies, by how much do the correlations change and on what scales? We will investigate this question by employing a formalism based on lensing of the CMBpolarisation, by describing peculiar motion with Lagrangian perturbation theory, and consider the EUCLID galaxy sample as an application: After a summary of cosmology, structure formation, Lagrangian perturbation theory, angular momentum models and ellipticity correlations in Sect. 2 we describe our formalism and the results in Sect. 3. Our main findings are summarised in Sect. 4. As reference model we chose a spatially flat w CDM model with Gaussian adiabatic initial perturbations in the cold dark matter distribution. Specifically, parameters were chosen to be Ω m = 0 . 25, ns = 1, σ 8 = 0 . 8, Ω b = 0 . 04 and finally H 0 = 10 5 h m / s / Mpc, with h = 0 . 72. The dark energy equation of state is set to w = -0 . 95 and the sound speed is equal to the speed of light, cs = c , such that there is no dark energy clustering.", "pages": [ 1, 2 ] }, { "title": "2.1 Dark energy cosmologies", "content": "The dynamics of a spatially flat Friedmann-universe with dark matter and dark energy is described by the Hubble function H ( a ) = d ln a / d t , which is given by with the matter density parameter Ω m and the dark energy equation of state function w ( a ). The value w ≡ -1 corresponds to the cos- gical constant Λ . Comoving distance χ and scale factor a are related by which yields distances in unit of the Hubble distance χ H = c / H 0. For the galaxy redshift distribution n ( z )d z , we use a standard shape with parameters z 0 = 0 . 64 and β = 3 / 2, as forecasted for EUCLID (Amara & R'efr'egier 2007). The distribution can be rew ritten in terms of comoving distance using the relation p ( z )d z = p ( χ )d χ with d z / d χ = H ( χ ) / c .", "pages": [ 2 ] }, { "title": "2.2 CDMpower spectrum", "content": "The spectrum P ( k ) describes the fluctuation amplitude of the Gaussian, statistically homogeneous density field δ , 〈 δ ( k ) δ ( k ' ) 〉 = (2 π ) 3 δ D ( k + k ' ) P ( k ), and is given by the ansatz with the transfer function T ( k ). This transfer function is approximated by (see Bardeen et al. 1986). The wave vector k = q Γ is measured in units of the shape parameter Γ . Sugiyama (1995) describe corrections due to a non-zero baryon density Ω b , The spectrum P ( k ) is normalised to the variance σ 8 on the scale R = 8 Mpc / h , with a Fourier transformed spherical top hat filter function, W ( x ) = 3 j 1( x ) / x . j /lscript ( x ) is the spherical Bessel function of the first kind of order /lscript (Abramowitz & Stegun 1972).", "pages": [ 2 ] }, { "title": "2.3 Structure growth with clustering dark energy", "content": "The growth of the density field in the linear regime, δ ( x , a ) = D + ( a ) δ ( x , a = 1), is given by the growth function D + ( a ), which follows as a solution to the growth equation (Turner & White 1997; Wang & Steinhardt 1998; Linder & Jenkins 2003),", "pages": [ 2 ] }, { "title": "2.4 Lagrangian perturbation theory", "content": "The peculiar motion of galaxies can be described using Lagrangian perturbation theory (LPT) if the flow of dark matter and of the advected galaxies is irrotational and nonlinearities are weak. In this limit, galaxies follow straight lines given by the gradient of the Zel'dovich potential Φ 1 to first order (1LPT, Zel'Dovich 1970; Doroshkevich 1970; Buchert 1989; Moutarde et al. 1991; Bernardeau et al. 2002), where Φ 1 is the solution to the Poisson equation, ∆Φ 1 = ∆Φ = δ . This solution can be improved by adding second order corrections to Lagrangian perturbation theory (2LPT, Buchert 1994; Melott et al. 1995; Bouchet et al. 1995), with the second order potential Φ 2 (Buchert et al. 1994; Bouchet et al. 1995), The time dependences are given by D 1( a ) = D + ( a ) and D 2( a ) = -3 / 7 D 2 + ( a ) Ω -1 / 143 m (for a low Ω m -cosmology with a cosmological constant Λ , see Bouchet et al. 1992). The solution to the latter relation can be written down in Fourier-space, where the products of tidal fields become convolutions, PSfrag replacements where the mode coupling function Qij ( k , k ' ) becomes: with P (1) Φ ( k ) = P ( k ) / k 4 as a consequence of the Poisson equation and with P (2) Φ ( k ) which can be derived to follow by application of the Wick-theorem (for a proof, see Durrer 2008). The integration is most e ffi ciently carried out using cylindrical coordinates aligned with k such that d 3 k ' = 2 π ( k ' ) 2 d k ' dcos θ using azimuthal symmetry, with θ being the angle between k and k ' . Fig. 1 gives an impression of the spectrum P (1) Φ ( k ) and of the 2LPT-corrections P (2) Φ ( k ) relative to 1LPT. We plot k 4 P ( i ) Φ ( k ), i = 1 , 2 which is equal to the CDM spectrum P ( k ) for the 1LPT result due to the Poisson equation. The 2LPT-spectrum is smaller on almost all scales by up to an order of magnitude and is only similar in amplitude on spatial scales of about 1 Mpc / h .", "pages": [ 2, 3 ] }, { "title": "2.5 Angular momentum from tidal shearing", "content": "Angular momenta of dark matter haloes are introduced by tidal shearing, where the di ff erential motion of a protohalo gives rise to a torquing moment (Hoyle 1949; Sciama 1955; Peebles 1969; Doroshkevich 1970; White 1984): This relation reflects the interesting misalignment property of the shear and inertia eigensystems necessary for angular momentum generation (Schafer & Merkel 2011): Only the antisymmetric tensor X -βγ = ∑ δ ( I βδ Φ δγ -Φ βδ I δγ ) / 2 is relevant for the angular momentum, L α ∝ X -βγ , because the contraction of the symmetric tensor X + βγ = ∑ δ ( I βδ Φ δγ + Φ βδ I δγ ) / 2 with the antisymmetric /epsilon1αβδ vanishes. The antisymmetric tensor X -is equal to the commutator [ I βδ, Φ δγ ] which suggests that for angular momentum generation, the tidal shear and the inertia are not allowed to be simultaneously diagonalisable and may not have a common eigensystem. Angular momenta L are described as being coupled to the tidal shear by means of a Gaussian random process p ( L | Φ i j )d L involving tidal fields Φ i j shaping the covariance cov( L ) i j of the Gaussian distribution (Lee & Pen 2001), with the misalignment parameter a , which describes the average orientation of the protohalo's inertia to the tidal shear eigensystem. a has been measured in numerical simulation to be close to 0.25 which we will assume in this work. ˆ Φ is the unit normalised traceless tidal shear with the properties tr( ˆ Φ ) = 0 and tr( ˆ Φ 2 ) = 1. This description is valid on scales where the correlations between tidal shears are negligible.", "pages": [ 3 ] }, { "title": "2.6 Intrinsic ellipticity correlations", "content": "Ellipticity correlations between galaxies are traced back to correlated angular momenta of their host haloes. CDM haloes acquire their angular momentum by tidal shearing and due to the fact that neighbouring galaxies experience correlated tidal fields, their angular momenta are correlated in consequence. The angular momentum L in turn determines the angle of inclination at which the galactic disk is viewed, and ultimately the ellipticity /epsilon1 (Heavens et al. 2000; Crittenden et al. 2001, 2002; Mackey et al. 2002; Heymans & Heavens 2003): with the angular momentum direction ˆ L = L / L and the coordinate system being aligned with its z -axis being parallel to the line of sight. A rotation of the coordinate frame by ϕ causes the complex ellipticity to rotate twice as fast, /epsilon1 → exp(2i ϕ ) /epsilon1 , in accordance with the spin-2 property of the ellipticity field. α is a free parameter weakening the dependence between inclination angle and ellipticity for thick galactic disks and has been determined to be α = 0 . 75 in the APM sample (Crittenden et al. 2001). In this work we use the angular momentum-based ellipticity correlation model proposed by Crittenden et al. (2001), who trace ellipticity correlations back to tidal shear correlations using the conditional probability distribution p ( L | Φ i j )d L introduced by Lee & Pen (2001): In this model, the distribution p ( L | Φ i j )d L is assumed as being Gaussian which is then being marginalised over the magnitude of the angular momentum vector, retaining only its directional dependence. Writing down the ellipticity components as a function of the angular momentum direction and employing the covariance 〈 LiLj 〉 as a function of the squared tidal shear tensor, as advocated by Lee and Pen, it is possible to relate the tidal shear correlations to the spectrum of the density field. With this relation, one can write down a correlation function of the ellipticity field as a function of moments ζ n ( r ) (see Crittenden et al. 2001) of the tidal shear field and finally to carry out a Limber projection for obtaining the angular correlation function. For the parameter a we chose the value 0.25 supported by numerical simulations. Ellipticity correlations between two points θ 1 and θ 2 separated by the distance θ are described in terms of two correlation functions ξ ± ( θ ), which are formed from the variances of the ellipticity components /epsilon1 + and /epsilon1 × using 〈 /epsilon1 + /epsilon1 × 〉 = 0. They can be transformed to the spectra C /epsilon1 E ( /lscript ) and C /epsilon1 B ( /lscript ) of the gradient and vorticity modes of the ellipticity field, by Fourier transform (Kaiser 1992; Schneider et al. 2002; Schneider & Kilbinger 2007; Fu & Kilbinger 2010). Fig. 6 shows intrinsic ellipticity spectra C /epsilon1 E ( /lscript ) and C /epsilon1 B ( /lscript ) for the EUCLID galaxy sample with its median redshift at z med = 0 . 9. The spectra are constant and equal in amplitude up to multipoles of /lscript /similarequal 100, indicating the absence of correlations such that on each scale on measures the variance of the uncorrelated ellipticity field. Correlations become important on angular scales /lscript > ∼ 300 where the spectra level o ff and decrease from multipoles of /lscript > ∼ 3000 on very rapidly. In the peak region, the ellipticity E -modes have an amplitude larger than the B -modes by about an order of magnitude.", "pages": [ 3, 4 ] }, { "title": "3.1 Analogy between ellipticities and polarisation", "content": "The evolution of the angular ellipticity spectra due to peculiar motion of the galaxies are described in our model by drawing an analogy to lensing of the polarisation modes of the cosmic microwave background. Both the galaxy ellipticities and the Stokes-parameters of the CMB-polarisation form a tensorial spin-2 field, which means that rotations of the coordinate frame by an angle ϕ give rise to a transformation of the tensor components as /epsilon1 → exp(2i ϕ ) /epsilon1 and P → exp(2i ϕ ) P , when the ellipticity is written as a complex ellipticity /epsilon1 = /epsilon1 + + i /epsilon1 × and the polarisation tensor P is composed of the Stokes parameters Q and U according to P = U + i Q . Peculiar motion as well as gravitational lensing introduces a shift in the position by an angle α such that the ellipticity /epsilon1 is not observed at the position θ where the galaxy was formed, but rather has been displaced /epsilon1 ( θ ) → /epsilon1 ( θ + α ). The correlation properties of such a distorted field can be computed using the formalism developed for CMB lensing, which allows the computation of correlation of the lensed polarisation field, P ( θ ) → P ( θ + α ), where α refers now to the lensing deflection angle. Our formalism will be built in complete analogy and computes the shifting angle from the peculiar velocity, which in turn is derived from a velocity potential using Lagrangian perturbation theory for the description of peculiar motion.", "pages": [ 4 ] }, { "title": "3.2 Formalism for displacing the ellipticities", "content": "By drawing analogies between the peculiar motion of galaxies causing displacements in the ellipticities, /epsilon1 ( θ ) → /epsilon1 ( θ + α ) and the lensing of the polarisation of the CMB, P ( θ ) → P ( θ + α ) it becomes possible to derive spectra of the evolved ellipticity field. Peculiar motion by D + ( a ) ∇ Φ changes the position of a galaxy by a shifting angle α = D + ∇ Φ /χ if the galaxy is situated at a comoving distance χ . The angular displacement field α can be derived from a displacement potential ψ = D +Φ /χ 2 by angular derivation, such that α = ∇ θψ , because ∇ θ = χ ∇ . Generalising this argument to a galaxy population which is described by a normalised distribution n ( χ )d χ in comoving distance χ one obtains an expression for the angular displacement potential, which replaces the lensing potential in the case of gravitational lensing of the CMB. The statistical properties of ψ , which is a Gaussian random field, are described by the spectrum C ψ ( /lscript ), which results from carrying out a Limber-projection of ψ . The spectrum C α ( /lscript ) is related to C ψ ( /lscript ) by C α ( /lscript ) = l 2 C ψ ( /lscript ) as a consequence of the relation α = ∇ θψ . The angular spectrum C ψ ( /lscript ) of the displacement potential ψ resulting from the Limber-projection of P Φ ( k ) is depicted in Fig. 2 along with the spectrum C α ( /lscript ) = /lscript 2 C ψ ( /lscript ) of the displacement angle α . Clearly, the 1LPT-result dominates over the 2LPT result by more than one order of magnitude, as already suggested by Fig. 1. The similarity of the plot to the analogous quanities in CMB-lensing is striking. Correlations between the components of the shifting angle α at two positions θ 1 and θ 2 are described by (Seljak 1996) with θ = θ 2 -θ 1, and correlation functions of the displacement angle which are defined as and We introduce the abbreviation σ 2 ( θ ) = C 0(0) -C 0( θ ) in complete analogy to CMB-lensing for describing uncorrelated displacements. The characteristic function of a Gaussian displacement field α would then be: replacements replacements In the case of CMB-lensing, non-Gaussian contributions have been shown to have negligible e ff ect on the deflection angle statistic (Carbone et al. 2009; Merkel & Schafer 2011) and in the case of weak cosmic shear, analogous arguments about the sparcity of strong deflections apply equally (Hamana et al. 2005). Fig. 3 shows the quantities σ 2 ( θ ) = C 0(0) -C 0( θ ) and C 2( θ ) used in this formalism, for both 1LPT and 2LPT. Again, we would like to draw the reader's attention to the similarity between our results and the formally equivalent result in CMB-lensing and to the domination of the 1LPT results over the 2LPT spectra. The correlation properties of the shifted ellipticity field can be described using the two correlation functions ξ ± ( θ ), where the points at which the ellipticities are observed, are shifted by exactly the angle α . Substituting the correlation function for the deflection angle in the Fourier-transforms of the above expressions yields the correlation functions ξ ' ± ( θ ) of the shifted ellipticity field. They can be transformed to E -mode and B -mode spectra with the standard transformation written down in eqns. (21) and (22). In summary, the E -mode and B -mode spectra of the shifted ellipticity field can be written concisely in a matrix notation: This notation shows explicitly the mixing between scales due to the convolution integral and the conversion between C /epsilon1 E ( /lscript ) and C /epsilon1 B ( /lscript ) under the influence of W -( /lscript, /lscript ' ), which is the non-diagonal entry of the mixing matrix. The kernels W ± ( /lscript, /lscript ' are given by with the functions which describe uncorrelated shifting due to σ 2 ( θ ) and correlated displacements due to C 2( θ ). We abbreviated Js ( x ) = J 2( x ) + J 6( x ). In the limit of no shifting, C 0( θ ) = C 2( θ ) = 0 such that W + ( /lscript, /lscript ' ) = δ ( /lscript -/lscript ' ) //lscript and W -( /lscript, /lscript ' ) = 0, due to the orthogonality relations of the cylindrical Bessel functions, In this limit, the convolution is reduced to a Dirac δ D -function and the mixing matrix is the unit matrix, so that the E -mode and B -mode amplitudes are conserved. We have verified that higher-oder corrections arising in the transformation of correlation functions do have a negligible e ff ect for the evolved ellipticity correlations (Challinor & Lewis 2005; Lewis & Challinor 2006).", "pages": [ 4, 5 ] }, { "title": "3.3 E / B -mode conversion", "content": "Figs. 4 and 5 show the mode coupling kernels W + ( /lscript, /lscript ' ) and W -( /lscript, /lscript ' ) where for simplicity we focus on 1LPT because the contributions due to 2LPT are comparatively small. From Fig. 4 we see that the power of the W + ( /lscript, /lscript ' )-kernel is mainly distributed along the diagonal and increasing with multipole number, with maximum contribution from 300 < ∼ /lscript < ∼ 3000. The o ff -diagonal contribution creates a convolution (eqn. 31) between the spectra at di ff erent multipoles, mediated by W + ( /lscript, /lscript ' ). In contrast, the mode coupling kernel W -( /lscript, /lscript ' ) (Fig. 5), which is responsible for the E / B -conversion, shows a lateral pattern which is three orders of magnitude smaller in amplitude and decreasing with higher multipole numbers ( /lscript, /lscript ' ).", "pages": [ 5 ] }, { "title": "3.4 Ellipticity spectra", "content": "The final result is given in Fig. 6, which compares the initial ellipticity spectra C /epsilon1 E ( /lscript ) and C /epsilon1 B ( /lscript ) of the ellipticity field as predicted by the angular momentum model, and the evolved spectra C ' E ( /lscript ) and C ' B ( /lscript ) due to peculiar motion. For comparison with weak lensing, PSfrag replacements multipole /lscript multipole /lscript ' PSfrag replacements PSfrag replacements multipole /lscript multipole /lscript ' PSfrag replacements we plot the weak convergence spectrum C κ ( /lscript ) expected from the EUCLID galaxy sample in comparison, for a nonlinear CDM spectrum (using the parameterisation by Smith et al. 2003). The first observation is that ellipticity correlations reach amplitudes similat to those of the weak lensing convergence in the nonlinear part corresponding to amplitudes /lscript < ∼ 300, and that the intrinsic E -mode spectrum C /epsilon1 E ( /lscript ) is larger than the B -mode spectrum C /epsilon1 B ( /lscript ) by about an order of magnitude in this regime. On larger angular scales, there are no appreciable ellipticity correlations and one e ff ectively observes the variance of the ellipticity field for uncorrelated objects. Consequently, the spectra have identical amplitudes and are e ff ectively constant. In this regime, the shifting e ff ect is not able to affect the galaxies, which is a well-known result in CMB-lensing, where scale free-spectra are invariant (Lewis & Challinor 2006): The mode-conversion mechanism is une ff ective if the spectra are equal, C /epsilon1 E ( /lscript ) = C /epsilon1 B ( /lscript ), and the convolution with W + ( /lscript, /lscript ' ) is not able to redistribute amplitudes. In contrast, both spectra are a ff ected on multipoles /lscript > 1000, where in particular C ' B ( /lscript ) has decreased relative to C /epsilon1 B ( /lscript ). evolved and initial E -mode and B -mode spectra. As already indicated by Fig. 6, we see a significant decrease for l > 1000 of up to 10% for the E - and 60% for the B -modes at l /similarequal 3000. The ratios C ' B ( /lscript ) / C ' E ( /lscript ) and C /epsilon1 B ( /lscript ) / C /epsilon1 E ( /lscript ) of intrinsic and evolved spectra are similar up to multipoles of /lscript /similarequal 1000, where they separate and indicate that the newly generated B -modes are small and that the B -mode spectra are more strongly a ff ected. For EUCLID's weak lensing application, changes in the ellipticity spectra are a ff ecting scales where the shape noise starts dominating, but for shallower surveys, lower multipoles would be a ff ected by the peculiar motion e ff ect. Finally, Fig. 8 gives an impression of the mode conversion mechanism, where we plot evolved spectra C ' E ( /lscript ) and C ' B ( /lscript ), when the E -mode or the B -mode in the initial spectra was deliberately replacements set to zero, i.e. C /epsilon1 E ( /lscript ) = 0 in the first and C /epsilon1 B ( /lscript ) = 0 in the second case. Even in the absence of a particular initial mode we observe power in the corresponding evolved spectrum, as a consequence of E / B -coupling introduced by peculiar motion.", "pages": [ 5, 6, 7 ] }, { "title": "4 SUMMARY", "content": "The topic of this paper is the evolution of intrinsic ellipticity correlation between galaxies due to peculiar motion. Intrinsic ellipticity correlations are derived in the framework of angular momentum models, which explain these correlations by correlated tidal shears experienced by the protohaloes in acquiring their angular momenta. Because the symmetry axis of the galactic disk is related to the angular momentum direction of the host halo, correlated angular momenta give rise to correlated angles of inclination and hence correlated ellipticities. (i) Peculiar motion of galaxies changes the correlation properties of the ellipticity field by displacing the galaxies and distorting the ellipticity field. We describe the peculiar motion by Lagrangian perturbation theory and derive corresponding displacement angles along with their statistical properties for the EUCLID galaxy sample. The formalism for evolving the ellipticity spectra uses an analogy to the formalism describing lensing of the CMB polarisation spectra. Both quantities, the ellipticity field as well as the polarisation field, have the same symmetry properties, being of spin 2. The loci at which ellipticities and polarisations are measured are displaced by peculiar motion in the first and by gravitational lensing in the second case. Because the peculiar motion field in the quasilinear regime is a flow resulting from a velocity potential which corresponds to the lensing potential, is it possible to derive all necessary quanities in complete analogy. (ii) Peculiar motion has two e ff ects on the ellipticity spectra: There is a convolution of the spectra and a conversion between E -modes and B -modes of the ellipticity field. Both e ff ects become important on angular scales /lscript > 1000, because on smaller multipoles, the spectra are e ff ectively constant and equally large. In particular the spectrum C /epsilon1 B ( /lscript ) is strongly a ff ected and looses amplitude: For the EUCLID galaxy sample we measure decrements by about 10% for C /epsilon1 E ( /lscript ) and 60% for C /epsilon1 B ( /lscript ). The mode-conversion mechanism is comparatively weak and we tested it by deliberately setting the initial spectra C /epsilon1 E ( /lscript ) and C /epsilon1 B ( /lscript ) to zero. (iii) Second order corrections in the dynamical model were found to be negligibly small in comparison to first order Lagrangian perturbation theory. Likewise, we made sure that higher-order corrections in the transformation of the ellipticity spectra had a minor e ff ect on the evolved ellipticity spectra. We conclude that in principle the dispersing e ff ect of peculiar motion weakens intrinsic ellipticity correlations and make them less troublesome for analysing weak lensing data. For the case of EUCLID we see changes in the spectra on scales where the shape noise is already dominating. A natural extention to this investigation would comprise the shifting and distorting e ff ect of weak gravitational lensing, and ultimately the usage of analysis methods conceived for the polarisation of the CMB for investigating intrinsic ellipticity correlations.", "pages": [ 7 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "Our work was supported by the German Research Foundation (DFG) within the framework of the excellence initiative through the Heidelberg Graduate School of Fundamental Physics. In particular, we acknowledge funding from the FRONTIER-programme. We would like to thank Philipp M. Merkel for his suggestions and advice on numerical computations.", "pages": [ 7 ] }, { "title": "REFERENCES", "content": "Abramowitz M., Stegun I. A., 1972, Handbook of Mathematical Functions. Handbook of Mathematical Functions, New York: Dover, 1972 Amara A., R'efr'egier A., 2007, MNRAS, 381, 1018 Andrae R., Jahnke K., 2011, MNRAS, p. 1665 Bailin J., Kawata D., Gibson B. K., Steinmetz M., Navarro J. F., Brook C. B., Gill S. P. D., Ibata R. A., Knebe A., Lewis G. F., Okamoto T., 2005, ApJL, 627, L17 Bardeen J. M., Bond J. R., Kaiser N., Szalay A. S., 1986, ApJ, 304, 15 Bartelmann M., 2010, Classical and Quantum Gravity, 27, 233001 Bartelmann M., Schneider P., 2001, Physics Reports, 340, 291 Bernardeau F., Colombi S., Gazta˜naga E., Scoccimarro R., 2002, Physics Reports, 367, 1 Bouchet F. R., Colombi S., Hivon E., Juszkiewicz R., 1995, A&A, 296, 575 Bouchet F. R., Juszkiewicz R., Colombi S., Pellat R., 1992, ApJL, 394, L5 Bridle S., King L., 2007, New Journal of Physics, 9, 444 Buchert T., 1989, A&A, 223, 9 Buchert T., 1994, MNRAS, 267, 811 Buchert T., Melott A. L., Weiss A. G., 1994, A&A, 288, 349 Bullock J. S., Dekel A., Kolatt T. S., Kravtsov A. V., Klypin A. A., Porciani C., Primack J. R., 2001, ApJ, 555, 240 Carbone C., Baccigalupi C., Bartelmann M., Matarrese S., Springel V., 2009, MNRAS, 396, 668 Catelan P., 1995, MNRAS, 276, 115 Catelan P., Kamionkowski M., Blandford R. D., 2001, MNRAS, 320, L7 Catelan P., Lucchin F., Matarrese S., Moscardini L., 1995, MN- RAS, 276, 39", "pages": [ 7 ] }, { "title": "8 A. Giahi-Saravani and B.M. Schafer", "content": "Schneider P., Kilbinger M., 2007, A&A, 462, 841 Schneider P., van Waerbeke L., Mellier Y., 2002, A&A, 389, 729 Sciama D. W., 1955, MNRAS, 115, 2 Seljak U., 1996, ApJ, 463, 1 Semboloni E., Heymans C., van Waerbeke L., Schneider P., 2008, ArXiv 0802.3978, 802 Smith R. E., Peacock J. A., Jenkins A., White S. D. M., Frenk C. S., Pearce F. R., Thomas P. A., Efstathiou G., Couchman H. M. P., 2003, MNRAS, 341, 1311 Sugiyama N., 1995, ApJS, 100, 281 Turner M. S., White M., 1997, Phys. Rev. D, 56, 4439 Wang L., Steinhardt P. J., 1998, ApJ, 508, 483 White S. D. M., 1984, ApJ, 286, 38 Zel'Dovich Y. B., 1970, A&A, 5, 84 This paper has been typeset from a T E X / L A T E X file prepared by the author.", "pages": [ 8 ] } ]

[ { "title": "ABSTRACT", "content": "We compare the outer radius of the accretion disc in the intermediate-mass black hole candidate HLX-1 as estimated from the UV/optical continuum, with the values estimated from its outburst decline timescales. We fit the Swift 2010 outburst decline lightcurve with an exponential decay, a knee and a linear decay. We find that the disk has an outer radius 10 12 cm /lessorsimilar R out /lessorsimilar 10 13 cm, only an order of magnitude larger than typical accretion discs in the high/soft state of Galactic black holes. By contrast, the semimajor axis is ≈ a few 10 14 cm. This discrepancy can be explained with a highly eccentric orbit. We estimate the tidal truncation radius and circularization radius around the black hole at periastron, and impose that they are similar or smaller than the outer disk radius. We obtain that e /greaterorsimilar 0 . 95, that the radius of the donor star is /lessorsimilar a few solar radii, and that the donor star is not at risk of tidal disruption. If the companion star fills its Roche lobe and impulsively transfers mass only around periastron, secular evolution of the orbit is expected to increase eccentricity and semimajor axis even further. We speculate that such extremely eccentric systems may have the same origin as the S stars in the Galactic centre. Key words: accretion, accretion discs - X-rays: individual: HLX-1 - black hole physics.", "pages": [ 1 ] }, { "title": "Roberto Soria 1 /star", "content": "1 International Centre for Radio Astronomy Research, Curtin University, GPO Box U1987, Perth, WA 6845, Australia Accepted 2012 October 15; Received 2012 October 4; in original form 2012 August 15", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "The point-like X-ray source 2XMMJ011028.1 -460421 (henceforth, HLX-1 for simplicity) is the strongest intermediate-mass black hole (IMBH) candidate known to date (Farrell et al. 2009; Wiersema et al. 2010; Davis et al. 2011; Servillat et al. 2011). It is seen in the sky at a distance of ≈ 8 '' from the nucleus of the S0 galaxy ESO 243-49 (redshift z = 0 . 0224, luminosity distance ≈ 95 Mpc, distance modulus ≈ 34 . 89 mag; at this distance, 1 '' ≈ 460 pc). Its X-ray luminosity and spectral variability (Farrell et al. 2009; Godet et al. 2009; Servillat et al. 2011) and its radio flares detected in association with the X-ray outbursts (Webb et al. 2012) are consistent with the canonical state transitions and jet properties of an accreting BH. With a peak X-ray luminosity ≈ 10 42 erg s -1 , the BH mass required to be consistent with the Eddington limit is ∼ 10 4 M /circledot . A similar value is obtained from spectral modelling of the thermal X-ray component, which is consistent with emission from an accretion disc (Farrell et al. 2009; Davis et al. 2011; Servillat et al. 2011). If these BH mass estimates are correct, HLX-1 is way too massive to have been formed from any stellar evolution process. A more likely scenario is that it is the nuclear BH (perhaps still surrounded by its own nuclear star cluster) of a disrupted dwarf satellite galaxy, accreted by ESO243-49 (Mapelli et al. 2012; King & Dehnen 2005). HLX-1 has a point-like, blue optical counterpart ( B ∼ V ∼ 24 mag near the outburst peak; Farrell et al. 2012; Soria et al. 2012, 2010). The presence of H α emission at a redshift consistent with that of ESO 243-49 (Wiersema et al. 2010) is perhaps the strongest argument for a true physical association. It is still debated whether the optical continuum emission is dominated by the outer regions of the BH accretion disc, or by a young star cluster around the BH (Soria et al. 2012; Farrell et al. 2012). In the absence of phase-resolved dynamical measurements of the BH motion, we can use the Swift X-ray lightcurve properties to constrain the system parameters. The X-ray flux shows recurrent outbursts every ≈ (366 ± 4) d (seen every late August in 2009, 2010, 2011 and 2012), due either to some kind of disc instability, or to a periodic enhancement of the accretion rate. Several alternative scenarios were considered and discussed by Lasota et al. (2011), who favoured a model in which enhanced mass transfer into a quasi-permanent accretion disc is triggered by the passage at periastron of an asymptotic giant branch (AGB) star on an eccentric orbit ( e ∼ 0 . 7). Since the publication of that work, the detection of the third and fourth consecutive outbursts (see Godet et al. 2012 for the first report of this year's outburst) has clinched the interpretation of the recurrence timescale as the binary period. Furthermore, additional op- photometric results have been published, based on data from the Hubble Space Telescope (HST) (Farrell et al. 2012) and from the European Southern Observatory (ESO)'s Very Large Telescope (VLT) (Soria et al. 2012). Thus, in this paper we revisit and update Lasota et al. (2011)'s orbital models and constraints in the light of the new results.", "pages": [ 1, 2 ] }, { "title": "2.1 Predictions for a standard disc model", "content": "At a distance of 95 Mpc, the characteristic size of the region responsible for most of the soft, thermal ( kT ≈ 0 . 2 keV) Xray emission is ∼ a few 10 9 cm (inferred from fits to XMMNewton , Chandra , Swift spectra), and is consistent with being constant during the decline of individual outbursts, and over the three recorded outbursts (Farrell et al. 2009; Servillat et al. 2011; Davis et al. 2011; Soria et al. 2011, 2012; Farrell et al. 2012). This suggests that the soft X-ray emission traces the true inner radius of the disc, bounded by the innermost stable circular orbit around the BH. Instead, much less is known about the outer disc radius, from UV/optical/IR observations; it is still debated how much of the blue optical emission comes from an irradiated disc, and how much from a possible cluster of young stars around the BH. If the disc is the dominant UV/optical emitter, the HST and VLT studies of Farrell et al. (2012) and Soria et al. (2012), respectively, agree on an outer disc radius ≈ 10 13 cm ∼ 1 au 1 . If a substantial contribution comes from unresolved young stars, we can take that value as an upper limit to the true disc size. A ratio of outer/inner disc radii ∼ 10 3 is significantly smaller than observed in transient Galactic BHs with Roche-lobe-filling donors, where typical outer radii are ∼ 10 11 cm ∼ a few 10 4 times the innermost stable circular orbit (Zurita Heras et al. 2011; Hynes et al. 2002, 1998). This serves as a warning that we have to disentangle what scales with BH mass and what does not, when using scaledup Galactic BH models to interpret HLX-1. While the inner disc depends directly on the BH mass, the outer disc depends mostly on the donor star and binary separation. There is an alternative way to estimate the outer disc size, based on the X-ray outburst decline timescale. Following King & Ritter (1998) and Frank et al. (2002), we assume that the outbursting disc is approximately in a steady state with surface density where ˙ M BH is the central accretion rate and ν the kinematic viscosity. When the whole disc from R in to R out is in a hot, high-viscosity state, the total mass in the disc 1 Note for the arXiv version: after this paper was accepted in MNRAS, a re-analysis of the HST data with a different disc irradiation model, by Mapelli et al. (submitted to MNRAS), showed that they are consistent with an outer disk radius ≈ 3 . 5 × 10 13 cm, for a viewing angle i = 45 · ; the radius is smaller for a face-on view, which we consider a more likely scenario, given the narrow full-width-half-maximum of the H α emission line (work by Hau & Soria, in preparation). Even if we adopt this slightly larger upper limit for the disc radius, the argument for a high eccentricity discussed in Section 3 remains unchanged. where we have neglected other sources of mass loss from the disc apart from BH accretion. In Eq.(2), ν is interpreted as an average value of the kinematic viscosity over the whole disc; in practice, we take the value of ν near the outer edge of the disc (King & Ritter 1998). Integrating Eq.(2), we obtain the well-known exponential decline for the disc mass and consequently also for the accretion rate and the outburst luminosity L ∼ L X ∼ 0 . 1 ˙ M BH c 2 . In summary, we expect to see a luminosity where L X , 0 is the value at the outburst peak, declining on a timescale as long as the rate at which the disc mass is depleted during the outburst decline is much larger than any ongoing transfer of mass from the donor star. For the viscosity at the outer edge of the disc, we take the usual parameterization ν = αc s H (Shakura & Sunyaev 1973), where α is the viscosity coefficient in the hot state, c s is the sound speed and H the vertical scaleheight. In the simplest, order-of-magnitude approximation, we can take an outer disc temperature ≈ 10 4 K (enough to keep it in the hot state), corresponding to c s ≈ 2 × 10 6 cm s -1 , and a vertical height H ≈ 0 . 1 R . This gives, from Eq.(6): which we shall directly compare with the observations. If we adopt the Shakura-Sunyaev disc solution (Shakura & Sunyaev 1973; Frank et al. 2002), with Kramers opacity, we can write where ˙ M 16 is the accretion rate in units of 10 16 g s -1 , m 1 is the BH mass in solar units, R 10 ≡ R out / (10 10 cm), etc. Then, from Eq.(6): that is The peak luminosity L X , 0 in Eq.(2) can be left as a purely observational parameter, or it can itself be expressed as a function of outer disc radius, viscosity, density and BH mass, if we assume that the outburst is triggered via the dwarf-nova instability (Cannizzo 1993). (More precisely, if we assume that the outburst starts when the enhanced mass transfer due to periastron passage pushes the disk from the cold to the hot state.) In that case, the surface density at any radius immediately before the start of the outburst approaches the maximum value allowed by the S-curve in the surface-temperature phase space (King & Ritter 1998; Frank et al. 2002): where α c ∼ 0 . 01 is the viscosity parameter in the cold disc state. Taking for simplicity H ≈ bR , where b is a constant ∼ 0 . 1, we can then express the maximum volume density at the start of the outburst as ρ max = Σ max /H ≈ Σ max / ( bR ), which is essentially independent of R , given the expression for Σ max in Eq.(11). It is then easy to integrate the total disc mass at the start of the outburst: the disc mass at later times the accretion rate (cf. Eq.(4)) and the peak luminosity where Σ max comes from Eq.(11). So far, we have assumed that the whole disc is in the hot state; this is usually the case in the early part of an outburst, especially when the outer edge of the disc is kept in the hot state by X-ray irradiation. The exponential decay continues until the outer disc annuli can no longer be kept in the hot state, so that hydrogen recombines and viscosity drops. From that moment, the outer edge of the hot disc R h < R out . The contribution to the accretion rate and to the continuum X-ray/UV/optical emission from the outer (cold, low-viscosity) annuli at R h < R < R out becomes negligible. It was shown by King & Ritter (1998) that the central accretion rate in this second phase of the decline is where t 1 is the time after which the outer disc is no longer in the hot state, and C parameterizes the fraction of X-ray luminosity intercepted and thermalized in the outer disc. As C can be taken as a constant, Eq.(16) shows that the accretion rate and luminosity decline in the late part of the outburst is linear. Most importantly for our current purpose, the slope of the linear decline is such that t (King & Ritter 1998), where t end is the (extrapolated) time in which the accretion rate and luminosity go to zero. Finally, we need to consider the case when there is ongoing mass transfer ˙ M 2 from the donor star during the outburst. In that case, the asymptotic value of the luminosity in the exponential decline is not zero but L 2 ≈ 0 . 1( -˙ M 2 ) c 2 (assuming a standard radiative efficiency ≈ 0 . 1). Recalling that t 1 is the time when the lightcurve switches from an exponential to a linear decline, and defining L 1 ≡ L ( t 1 ), Eq.(5) is modified as (Powell et al. 2007) After the transition to a linear regime, the luminosity is Note that if -˙ M 2 > 0, the first derivative of the luminosity is discontinuous at t = t 1 (Powell et al. 2007), because the gradient of the exponential decay is that is flatter than the gradient of the linear decay Therefore, the exponential-to-linear transition is often referred to as the 'knee' in the lightcurve of transient X-ray binaries.", "pages": [ 2, 3 ] }, { "title": "2.2 Comparison with the observations", "content": "We shall now fit the X-ray lightcurve to obtain two independent estimates of the viscous timescale τ e , from the exponential and the linear regime, and use them to constrain R out from Eq.(10), or, using a simpler approximation for the scale-height, from Eq.(7). We shall then derive an independent estimate of R out from the expression for the peak luminosity in Eq.(15). We studied the publicly available 2 Swift X-Ray Telescope (XRT; Burrows et al. 2005) data for the 2010 outburst, because it is the same outburst for which we obtained constraints on R out from the optical continuum (Soria et al. 2012; Farrell et al. 2012). We used the online Swift /XRT data product generator 3 (Evans et al. 2007, 2009) to extract a lightcurve in the 0 . 3-10 keV band. We fitted the lightcurve with an initial exponential decay, a knee and a linear decay (Fig. 1). The shape of the X-ray outburst lightcurve of HLX-1 is remarkably similar to those of several transient Galactic X-ray binaries (BHs and neutron stars), modelled by Powell et al. (2007), which were successfully used to constrain the size of their accretion discs. For the exponential part, we obtain a best-fitting timescale τ e = R 2 out / (3 ν ) = 3 . 7 +5 . 0 -1 . 5 × 10 6 s (90% confidence limit). For the linear part, we have τ e = 3 . 5 +1 . 0 -0 . 8 × 10 6 s. Assuming a peak luminosity ≈ 10 42 erg s -1 in the 0 . 3-10 keV band, and a bolometric luminosity a factor of 2 higher, implies an accretion rate ˙ M ≈ 2 × 10 22 g s -1 at standard efficiency. The viscosity parameter α /lessorsimilar 1, and more likely α ∼ 0 . 3 (Frank et al. 2002). From Eq.(10), this implies R out ≈ 10 12 cm, only very weakly dependent on BH mass and accretion rate. Using the approximation in Eq.(7), we also obtain R out ∼ 10 12 cm. From the peak luminosity (Eq.(15)), for α ∼ 0 . 3 we obtain R out ∼ 4 × 10 12 cm. We also analyzed the lightcurves for the 2009 and 2011 outbursts (the 2012 outburst was still ongoing as this paper went to press). They are more noisy, less easy to interpret in terms of exponential and linear branches. However, for both of them it is possible to estimate an e-folding decline timescale, roughly corresponding to the exponential timescale determined for the 2010 outburst. The timescales are ≈ 5 × 10 6 s and ≈ 3 × 10 6 s for 2009 and 2011, respectively, and the peak luminosities are approximately the same in all three outbursts. Thus, we also estimate an outer radius ∼ 10 12 cm in the 2009 and 2011 outbursts, in the standard disc approximation. Those values are almost one order of magnitude smaller than what we estimated by assuming that most of the continuum emission comes from the hot disc; and the latter was already a surprisingly small radius compared with the binary system parameters (see Section 3). We note that both the HST and VLT observations (Soria et al. 2012; Farrell et al. 2012) were taken during the exponential part of the decline, that is when the whole disc was in a hot state. Therefore, the estimates of R out from the optical/UV continuum should be comparable to those from the X-ray lightcurve. New optical observations early in the next outburst will hopefully allow us to measure the true outer disc size and luminosity.", "pages": [ 3, 4 ] }, { "title": "3 ORBITAL PARAMETERS", "content": "We can now compare the size of the disc estimated from optical and X-ray flux measurements (10 12 cm /lessorsimilar R out /lessorsimilar 10 13 cm) with the characteristic size of the binary system. Because of the sharpness of the outbursts rise and decline, reminiscent of Galactic X-ray binaries, we assume that the BH is accreting from a single donor star rather than AGNlike gas inflows. We also assume that the outburst recurrence timescale ∼ 370 d corresponds to the binary period. Then, the semimajor axis a of the binary is (Newton 1687), where q = M 2 /M BH and m ≡ ( M BH + M 2 ) /M /circledot ≡ M/M /circledot . Typical values for HLX-1 in the intermediate-mass BH scenario are q ∼ 10 -3 and m 1 / 3 ∼ 10-20. Therefore, in the most accepted scenario, the semimajor axis is at least 10, and possibly up to 100 times larger than the disc radius. This mismatch clearly suggests an eccentric orbit (Lasota et al. 2011), in which the characteristic disc size is determined by the periastron separation R per = (1 -e ) a , with eccentricity e /greaterorsimilar 0 . 9. The amount of mass transferred to the BH in each outburst suggests that the donor star overflows its instantaneous Roche lobe every time it passes at periastron. In Roche-lobe mass transfer systems, the outer edge of the accretion disc is generally identified with the largest stable non-intersecting orbit (tidal truncation radius R T ). For mass ratios M 2 /M 1 /lessmuch 1, R T ≈ 0 . 48 a (Paczynski 1977; Papaloizou & Pringle 1977; Whitehurst 1988; Warner 1995). If unstable orbits are also allowed, the disc may expand up to R T ≈ 0 . 60 a/ (1+ q ) (Warner 1995, and references therein). If we take the periastron distance as the instantaneous binary separation during the phase of Roche-lobe overflow, this corresponds to an expected disc size R out ≈ 0 . 6(1 -e ) a . For an observed disc size R out = 10 13 cm, the tidal radius condition would require an eccentricity e ≈ 0 . 89 for a BH mass = 1000 M /circledot , and e ≈ 0 . 95 for a BH mass = 10 4 M /circledot . If we take the lower bound to our observed disc size R out = 10 12 cm, we need e ≈ 0 . 989 or e ≈ 0 . 995, respectively. We can argue that the tidal truncation constraint is not relevant to the case of HLX-1, where mass transfer may occur impulsively near periastron, and the timescale for the disc to expand to its tidal truncation radius is similar to the timescale for the disc matter to be accreted and for the binary orbit to expand after periastron. In other words, in HLX-1 the disc may look small because it did not have time to grow to its tidal truncation radius. Instead, the circularization radius provides a stronger lower limit to the predicted disc size, and is applicable to any system where mass transfer occurs through the Lagrangian point L 1 . The circularization radius R cir is defined via the conservation of angular momentum equation where v φ is the orbital velocity of the accretion stream around the BH, X L1 R per the distance between the BH and the Lagrangian point L 1 , and Ω( R per ) the angular velocity of the donor star at periastron. Here, we must be careful not to use the well-known fitting formula for X L1 ≈ 0 . 500 -0 . 227 log q (Frank et al. 2002) because it applies only for q > 0 . 1 and for circular orbits. Instead, we need to compute X L1 from Eq.(A13) of Sepinsky et al. (2007), valid for eccentric orbits: Here, f is the ratio of the rotational angular velocity of the donor star to its orbital angular velocity, at periastron. It parameterizes the degree of tidal locking; a star with f = 1 is rotating synchronously with the orbit, at periastron. A list of X L1 solutions for characteristic values of q and e is given in Table 1. The orbital velocity of the accretion stream around the BH and the orbital angular velocity at periastron For the sake of our numerical estimate, we take X L1 = 0 . 95, a typical value in the range of parameters thought to be relevant for HLX-1 (Table 1). Then, substituting into Eq.(23), we have: Assuming that the BH mass is in the range ∼ 10 3 -10 4 M /circledot , Eq.(27) gives us a strong constraint on e , by imposing that the circularization radius is smaller than the observed outer disc size. For example, assuming M BH = 5 × 10 3 M /circledot , a circularization radius R cir = 10 13 cm (at the upper end of our disc size estimates) requires e ≈ 0 . 97; for R cir = 10 12 cm (at the lower end of our disc size estimates), e ≈ 0 . 997. The corresponding periastron distances 4 R per = (1 -e ) a ≈ 6 × 10 12 cm for e = 0 . 97, and R per ≈ 6 × 10 11 cm for e = 0 . 997. The distance between L 1 and the centre of the donor star places an upper limit on its radius; characteristic values of its instantaneous volume-averaged Roche lobe radius at periastron can be obtained from Eggleton (1983). For typical values q ∼ 10 -4 -10 -3 , the secondary gets squeezed to a radius ≈ (0 . 02-0 . 05) R per . Since we have assumed that the secondary fills the Roche lobe and dumps mass into the BH only near periastron, this radius must also be similar to the size of the donor star. For example, if q = 2 × 10 -4 and e = 0 . 97, the donor star must have a radius ≈ 4 R /circledot , consistent with main sequence and subgiant stars. 4 Note that for e /greaterorsimilar 0 . 2, the periastrion distance between the two stars is always smaller than the circularization radius around the accreting primary. This is because the angular momentum of the secondary at periastron, and of the matter transiting through the L 1 point, is larger than the angular momentum of a circular orbit with the same binary separation. It is not a problem, because by the time the accretion stream has completed a full orbit around the BH and formed a ring, the secondary has moved away from periastron and the primary's Roche lobe has widened. The values of e estimated here from disc size arguments are much more extreme than what was suggested in Lasota et al. (2011). They may seem implausible, knowing that tidal forces tend to circularize orbits in X-ray binaries. However, Sepinsky et al. (2007, 2009) showed that in the case of a donor star that transfers mass impulsively only at periastron, with q /lessorsimilar 1 -0 . 4 e +0 . 18 e 2 , the secular evolution of the orbit leads to an increase of both eccentricity and semimajor axis, even when the opposite effect of tidal forces is taken into account. If tidal circularization is neglected, the eccentricity increases as and the semimajor axis (and hence the binary period) as It is then easy to show from Eq.(28,29) that the periastron distance R per = (1 -e ) a , and therefore also the size of the secondary's Roche lobe at periastron, remains unchanged. Finally, we need to assess what donor stars can survive on such eccentric orbits with small periastron distance, avoiding tidal disruption. The condition for survival is that the periastron distance R per is larger than the tidal disruption radius (Rees 1988) By substituting R 2 /lessorsimilar 0 . 05 R per into Eq.(31), we can re-cast the tidal survival condition as easily satisfied in the likely mass range of HLX-1.", "pages": [ 4, 5 ] }, { "title": "4 CONCLUSIONS", "content": "We compared the estimates of the disc size from the optical continuum flux ( R out /lessorsimilar 10 13 cm) with those obtained by fitting the X-ray luminosity decline after an outburst. We found that, at least for the 2010 outburst, the decline displays the standard sequence of exponential phase, knee, linear phase often seen in Galactic X-ray binaries; this strengthens the interpretation that the thermal X-ray emission in HLX-1 comes from a disc, and the decline timescale corresponds to its viscous timescale. For all three outbursts observed to-date, the timescale is very short, ∼ 4-8 weeks. This is similar or only slightly longer than what is typically observed in Galactic X-ray transients, despite that fact that both the orbital period and the BH mass (and, hence, the semimajor axis) of HLX-1 are /greaterorsimilar 100 times larger. The outer disc radius estimated from the viscous timescale is R out ∼ 10 12 cm, if the viscosity parameter is similar to the values usually estimated for Galactic BH transients in a high state. We cannot rule out that the fast accretion of the disc matter in HLX-1 may be partly due to an effective viscosity α eff /greaterorsimilar 1, higher than in the Shakura-Sunyaev prescription. But we argue that even if we assume the upper disc size estimate R out ∼ 10 13 cm, a highly eccentric orbit is required to explain the small disc size. To quantify the eccentricity, we calculated the characteristic length-scales of the binary system, as a function of BH mass and eccentricity. If the disc extends at least as far as the circularization radius (as is usually the case in X-ray binaries with Roche-lobe mass transfer), we obtain that R cir ∼ (1 -e 2 ) a , and therefore e /greaterorsimilar 0 . 95 for a BH mass /greaterorsimilar 10 3 M /circledot . We argued that X-ray binaries with such extreme values of e are the most likely evolutionary endpoint of systems with q /lessmuch 1 and a moderately eccentric initial orbit, such that Roche-lobe-overflow mass transfer occurs only impulsively near periastron. Secular evolution will tend to make the orbit more and more eccentric, by increasing the semimajor axis and the binary period, at constant periastron distance. The small periastron distance required to explain the HLX-1 observations sets an upper limit to the current radius of the donor star R /lessorsimilar a few R /circledot , ruling out supergiants, red giants and AGB stars. Possible donors are main sequence (B type or later) or subgiants. The compactness of the donor star, and the fact that secular orbital evolution due to mass transfer will not change the periastron distance, imply that the companion star in HLX-1 is not at immediate risk of tidal disruption, and will not be in the near future. In other words, we are not observing HLX-1 in a peculiar moment of its evolution, immediately prior to tidal break-up of the donor star. HLX-1 appears to be a stable system, with a lifetime for X-ray outbursts determined primarily by the mass transfer timescale from the donor; at a rate ∼ 10 -5 M /circledot yr -1 (averaged over the binary period), it may last for another ∼ 10 5 -10 6 yr, during which its semimajor axis and binary period (and hence, interval between outbursts) will continue to increase. Eccentricities /greaterorsimilar 0 . 95 may seem implausibly extreme, but there is at least one class of stellar objects where they are the norm: S stars observed on highly eccentric orbits within 0.01 pc of the Galactic nuclear BH (Alexander 2005; Gillessen et al. 2009). A possible scenario for the origin of Galactic S stars is the tidal disruption of a stellar binary system near the BH, which produces an escaping, hypervelocity star, and a more tightly bound star on a very eccentric orbit, theoretically as high as e ≈ 0 . 99 (Lockmann et al. 2008). Observationally, the most eccentric, bound S star for which orbital parameters have been reliably determined has e ≈ 0 . 96 (Gillessen et al. 2009). We speculate that intermediate-mass BHs in star clusters may also capture stellar companions on very eccentric orbits through a similar process.", "pages": [ 5, 6 ] }, { "title": "ACKNOWLEDGMENTS", "content": "I thank Guillaume Dubus, Sean Farrell, Jeanette Gladstone, Pasi Hakala, Michela Mapelli, George Hau, Albert Kong, Tom Russell, Luca Zampieri for insightful discussions on the nature of HLX-1. This work was improved also thanks to the feedback received from several colleagues after my presentation at the IAU Symposium 290 in Beijing.", "pages": [ 6 ] }, { "title": "REFERENCES", "content": "Alexander T., 2005, Phys. Rep., 419, 65 Burrows D.N., 2005, SSRv, 120, 165", "pages": [ 6 ] } ]

[ { "title": "ABSTRACT", "content": "We examine the changes in the properties of galactic bulges and discs with environment for a volume-limited sample of 12500 nearby galaxies from SDSS. We focus on galaxies with classical bulges. Classical bulges seem to have the same formation history as ellipticals of the same mass, and we test if environment determines whether or not a classical bulge possesses a disc. Using the projected fifth nearest neighbour density as a measure of local environment, we look for correlations with environment at fixed bulge stellar mass. In groups with fewer than 20 members, we find no evidence for changes in disc morphology with local density. At fixed bulge mass, disc mass and disc scale length are independent of local density. However, disc colour does increase (∆( g -r ) ∼ 0 . 05 mag) as a function of local density in relatively poor groups. Therefore, the colour-density relation for classical bulge+disc galaxies in the field and in poor groups is due solely to changes in disc colour with density. In contrast, we find no correlations between disc colour and local density for classical bulge+disc galaxies in large, relaxed groups and clusters. However, there is a weak correlation between disc mass and group crossing time, suggesting morphological transformation takes places in rich groups. Our results add to the evidence that star formation is quenched in group environments, instead of clusters, and that star formation quenching and morphological transformation are separate processes. Overall, we show that environment has two effects on galactic discs: relatively low density environments can quench star formation in discs, while processes occurring in higher density environments contribute to the morphological transformation from disc-dominated systems to bulge-dominated systems. Key words: galaxies: structure - galaxies: bulges - galaxies: formation - galaxies: photometry", "pages": [ 1 ] }, { "title": "C. N. Lackner 1 , 2 /star and J. E. Gunn 1", "content": "1 Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544 2 Kavli Institute for the Physics and Mathematics of the Universe, Todai Institutes for Advanced Study, the University of Tokyo, Kashiwa, Japan 277-85823 (Kavli IPMU, WPI) 30 August 2021", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Galaxy morphology, stellar mass, star formation rate, and projected number density are all known to correlate. Generally, massive galaxies are bulge-dominated (de Vaucouleurs 1961; Blanton et al. 2003), not forming stars (red) (Strateva et al. 2001; Kauffmann et al. 2004; Baldry et al. 2006), and reside in high density regions (Oemler 1974; Dressler 1980; Postman & Geller 1984; Goto et al. 2003; Lewis et al. 2002; G'omez et al. 2003; Yang et al. 2007; Bamford et al. 2009). Low mass galaxies are disc-dominated, star-forming (blue), and reside in low density regions. There are, of course, exceptions: passive discs make up a significant fraction of the red sequence (e.g. Bernardi et al. 2003a; Maller et al. 2009), blue ellipticals are still forming stars (Schawinski et al. 2009), and there are many early-type, passively evolving galaxies in relatively low density regions (e.g. Mulchaey & Zabludoff 1999). For galaxies which follow the general pattern, it is unclear which of the correlations mentioned above are the result of physical processes, and which, if any, are simply consequences of other correlations. Morphology, star formation, and density are all strongly correlated with stellar mass (e.g. Hamilton 1988; Brinchmann & Ellis 2000; Blanton et al. 2003; Kauffmann et al. 2004; Blanton et al. 2005; Thomas et al. 2005), and correlations between these properties are partially due to their correlations with stellar mass. Yet, at fixed stellar mass, studies have found correlations between density and morphology (Bamford et al. 2009), density and stellar age (Thomas et al. 2005; Cooper et al. 2010), density and colour (Balogh et al. 2004; Skibba et al. 2009; Cibinel et al. 2012), and density and star formation rate (Kauffmann et al. 2004; Christlein & Zabludoff 2005). Furthermore, at fixed lumi- nosity, neither blue nor red galaxy colours seem to depend on density, i.e. blue galaxies do not get redder as a function of density, only their number fraction decreases (Balogh et al. 2004; Hogg et al. 2003). Furthermore, correlations with morphology and density seem to disappear for high mass galaxies (e.g. Tasca et al. 2009; Grutzbauch et al. 2011). The local correlations between galaxy properties and density extend to higher redshift; the morphology-density and colour-density relations are in place by z ≈ 1 (Dressler et al. 1997; Postman et al. 2005; Treu et al. 2003; Smith et al. 2005), but the relations do evolve with redshift. The fraction of blue galaxies in high density regions increases with redshift (e.g. Butcher & Oemler 1978), while the fraction of S0s and red discs galaxies decreases with increasing redshift (Smith et al. 2005; Moran et al. 2007; Bundy et al. 2010; Bruce et al. 2012). In addition, the hierarchical growth of structure implies that galaxies generally move from low density regions to high density regions as a function of time. Therefore, any environmental effects and trends will be more pronounced today than in the past (Tasca et al. 2009). From these observations, a general outline of galaxy evolution has been developed. At early times, galaxies are blue, intensely star-forming, disc systems. As galaxies become more massive, star formation is quenched (by internal feedback mechanisms), creating a population of red, massive galaxies. This mass quenching (Peng et al. 2010) is compounded by environmental effects. At fixed mass, galaxies in higher density regions become red and bulge-dominated earlier, creating the colour-density and morphology-density relations. Star formation quenching is thought to occur before morphological transformation, which leads to an increase in the fraction of S0s at intermediate densities (e.g. Dressler 1980; McIntosh et al. 2004; Cooper et al. 2006; Moran et al. 2007; Bundy et al. 2006). In order to separate the effects of environment from the effects of stellar mass, we examine correlations between galaxy morphologies, colours, and local density at fixed stellar mass. The physical processes responsible for the environmentdriven transformations are unknown, although there are many candidates. Processes can be divided into those which truncate star formation, and those which also cause morphological transformations (see Boselli & Gavazzi 2006, for a review of these processes). Ram-pressure stripping of ISM from galaxies entering clusters (Gunn & Gott 1972) and the removal of hot halo gas (strangulation) (Larson et al. 1980; Balogh & Morris 2000) both act to truncate star formation in discs and can transform spiral discs into S0s, but do not drastically alter a galaxy's stellar disc. Tidal stripping by the cluster potential (Merritt 1984) and high speed encounters with other cluster galaxies (harassment) (Moore et al. 1996, 1998, 1999) both act to transform disc-dominated galaxies into bulge-dominated galaxies. All these processes act on a variety of timescales and require different minimum local densities in order to be effective. It is likely that more than one process is responsible for the morphology-density relation. In this work, we examine low redshift galaxies with both a bulge and disc, and study the correlations of the separate bulge and disc properties with local density. These galaxies, which include S0s, may represent a transition from discdominated to bulge-dominated, and the environmental processes enumerated above should have observable effects on the discs and possibly the bulges of these transitional galaxies. For the bulge and disc properties, we use the bulge+disc decompositions from our earlier work (Lackner & Gunn 2012, hereafter L12). L12 presents bulge+disc decompositions for nearly 72 , 000 low redshift (0 . 002 < z < 0 . 05) galaxies from the Sloan Digital Sky Survey (SDSS). The galaxies we use for in this work are a luminous subsample from L12. We focus on galaxies which host classical bulges. These bulges have properties and, presumably, formation histories identical to elliptical galaxies of the same mass . Classical bulges are concentrated, pressure-supported systems (Falc'on-Barroso et al. 2002; MacArthur et al. 2008), with old stellar populations (Peletier et al. 1999; Moorthy & Holtzman 2006; MacArthur et al. 2010, but see Gadotti 2009). In L12, we model the light profiles of both classical bulges and ellipticals using a de Vaucouleurs profile (but see Caon et al. 1993). We show in L12 that classical bulges and ellipticals follow the same size-density relation (Kormendy 1977). Taken together, these studies support the assertion that classical bulges and ellipticals of the same stellar mass are indistinguishable. Therefore, by exploring how local density affects the discs around classical bulges, we can determine if changes in environment correspond to a transition from classical bulges with discs to disc-less elliptical galaxies. To date, there have been a handful of studies which explore the effects of environment on bulges and discs separately. Both McIntosh et al. (2004) and Hudson et al. (2010) show that disc colour is a function of cluster radius. McIntosh et al. (2004) also show that the amount of substructure in discs declines with increasing density, further demonstrating that star formation is quenched in dense environments. The sample we use in this work is considerably larger than the samples in previous studies, and it covers the entire spectrum of local densities, not just rich clusters and the field. Furthermore, our large sample can easily be divided into subsamples of constant bulge mass, eliminating trends with stellar mass and environment, and still yield statistically significant results. Below, we use our sample of classical bulge+disc galaxies to determine whether the colour-density relation for these galaxies is due to changes in bulge or disc colour, indicating star formation truncation, or changes in bulgeto-total ratio, indicating morphological transformation. We find that the colour-density relation for these galaxies at fixed bulge mass is due entirely to changes in disc colour, not changes in disc mass or size. Next, we divide the sample into rich and poor groups to determine if the trends in disc properties with environment depend on group size or halo mass. We find that while disc colour is a function of local density in relatively poor groups, disc colour is independent of local density in larger, relaxed groups and clusters. However, Disc mass decreases slightly with increasing local density in large groups and clusters, while disc mass is independent of local density for galaxies in the field and in poor groups. From these results, we conclude that environment-driven star formation quenching occurs in relatively low density environments, while structural changes to discs only occur in higher density environments. In order to perform the studies detailed above, we require a robust sample of classical bulge+disc galaxies. It is especially important to distinguish classical bulge+disc galaxies from disc-less ellipticals. We present a probabilistic method for separating classical bulge hosts and ellipticals using bulge+disc decompositions in Section 2.1.1. The remainder of Section 2 details the environment metrics employed. In order to determine group membership and local density, we use an updated group catalogue from A. Berlind, (priv. comm.), which is based on the group catalogues presented in Berlind et al. (2006). Section 3 presents the morphologydensity relation and colour-density relation for our sample. The results of this section help to confirm the assignment of galaxy morphologies in Section 2.1.1. Sections 4 and 5 are devoted to the correlations between environment and bulge and disc properties. Section 4 focuses on correlations with projected local density, while Section 5 discusses classical bulge+disc galaxies in relatively rich groups. Throughout this paper we use the ΛCDM cosmology: Ω m = 0 . 3, H 0 = 70 km s -1 Mpc -1 , and Ω λ = 0 . 7.", "pages": [ 1, 2, 3 ] }, { "title": "2 SAMPLE", "content": "The sample used in the work consists of a luminous subsample of galaxies from our earlier bulge+disc decompositions matched to an updated group catalogue from A. Berlind (priv. comm.). This group catalogue employs the method presented in Berlind et al. (2006), but uses data from SDSS data release 7. The sample contains 29781 galaxies and is complete to an absolute 0 . 1 r -band magnitude of -19 . 77 and covers the redshift range 0 . 02 /lessorequalslant z < 0 . 05. Below, we describe the bulge+disc decompositions and the environmental information extracted from the group catalogue.", "pages": [ 3 ] }, { "title": "2.1 Bulge-Disc Decompositions", "content": "For the bulge and disc properties, we use the results of our earlier work (L12). L12 presents bulge+disc decompositions for 72 , 000 galaxies from SDSS data release 8 (the data is the same as that in data release 7, but the reductions have been improved). The galaxies have redshifts between 0 . 003 and 0 . 05. All of the galaxies are in the SDSS spectroscopic sample, which implies a limiting magnitude of m r < 17 . 77. Two dimensional bulge-disc decompositions are performed for the r -band images. The results are then linearly scaled to fit the galaxy images in the u , g , i , and z bands, yielding colours for the bulge and disc components. Because we only linearly scale the fits in each band, our bulge+disc models do not take into account colour gradients within each component. Each galaxy is fit with 5 different models: a de Vaucouleurs bulge and an exponential disc ( n b = 4 B+D), an exponential bulge and exponential disc ( n b = 1 B+D), a single de Vaucouleurs profile, a single exponential profile, and a single S'ersic profile. The two bulge+disc models allow us to fit both elliptical-like, pressure-supported classical bulges (de Vaucouleurs profile) and disc-like, rotationally-supported pseudo-bulges (exponential profile) (Kormendy 1977, 1993; Fisher & Drory 2008). Pseudo-bulges are thought to arise from secular processes within discs, such as bar-driven instabilities (e.g. Kormendy & Kennicutt 2004; Athanassoula 2005; Weinzirl et al. 2009), and, as such, have very different formation histories than classical bulges (but see Elmegreen et al. 2009). Often, pseudo-bulges are still forming stars today (Kormendy & Kennicutt 2004; Fisher 2006). Since pseudo-bulges are a disc phenomenon, we do not include them in our sample of bulge+disc galaxies. For each galaxy, we tabulate the bulge and disc magnitudes and colours in all 5 SDSS bands. These values are Galactic extinction corrected (Schlegel et al. 1998) and kcorrected to z = 0 using the IDL package kcorrect v4_2 (Blanton & Roweis 2007). In addition, we correct the colours and magnitudes of galaxies with discs for intrinsic extinction using corrections from Maller et al. (2009) and L12. These corrections remove trends in colours with disc inclination, but they do not correct for extinction due to dust in face-on discs. Finally, we calculate the stellar masses for the bulge and disc using the relation from Bell et al. (2003): where 0 . 15 accounts for the difference between the diet Salpeter initial mass function (IMF) used by Bell et al. (2003) and the Kroupa IMF (Kroupa 2002) we employ. The colours and magnitudes used for the stellar mass are not corrected for intrinsic extinction, in keeping with the derivation of the relation in Bell et al. (2003). Because the mass-tolight ratio is a convex function of galaxy colour, the sum of the masses of the bulge and disc is always slightly larger than the mass measured using the total galaxy colour and magnitude. For most of the galaxies, this difference is small; the median M total / ( M disc + M bulge ) is 0 . 9996, and for 95 per cent of the galaxies, this ratio is between 0 . 85 and 1 . 0.", "pages": [ 3 ] }, { "title": "2.1.1 Classifying Galaxies", "content": "Since we fit each galaxy with five different models, we require a method for selecting the best-fitting model. In L12, we show that the χ 2 values of the various model fits are indistinguishable at the resolution of SDSS (but see Simard et al. 2011). Instead, we develop an algorithm that relies on the sizes, shapes, and colours of the bulges and discs in order to select the best-fitting, physically-sensible model for each galaxy. In this work, we present a simplified classification algorithm that attempts to classify most of the galaxies in our sample and emphasises the distinction between classical bulge galaxies and ellipticals. Additionally, instead of assigning each galaxy a best-fitting model, we assign each galaxy a probability of being fit by each model (often, the probability is unity for one of the models). Although this does not allow us to accurately classify a given galaxy, it does allow us to study the properties of a large sample of galaxies. When we examine properties of bulges and discs, we weight each galaxy by its probability of having a bulge and a disc. We separate galaxies into five different categories: bulge-less disc galaxies, disc-less ellipticals, classical bulge+disc galaxies, pseudo-bulge+disc galaxies, and unclassifiable galaxies. Our goal is to assemble a sample of galaxies which are accurately modelled by a classical bulge plus a disc. A brief outline of the classification is given below and summarised in Table 1. Details can be found in Appendix A. First, we identify bulge-less and disc-less galaxies. Bulge-less disc galaxies are defined to have B/T < 10 per cent. Disc-less galaxies (ellipticals) are more difficult to identify. The few galaxies with B/T > 90 per cent are considered ellipticals. These make up only 6 per cent of the our sample. As shown in L12 (see also Allen et al. 2006), elliptical galaxies are often best fit by a de Vaucouleurs component along with a low surface brightness exponential component. This 'disc' is not a physical disc and has several possible origins, i.e. the outer halo of ellipticals, a S'ersic index larger than four, and/or inadequate sky subtraction around bright galaxies in SDSS. These model 'discs' make ellipticals indistinguishable from face-on bulge+disc galaxies based on the 2-dimensional bulge+disc decomposition alone. However, the distributions of inclination angles for real and spurious discs will be different; the former will be randomly oriented, while the latter will be preferentially faceon. We use this fact to statistically separate ellipticals from face-on classical bulge host galaxies. Galaxies with a small measured disc axis ratio have a high probability of being a bulge+disc galaxy, while galaxies with a large disc axis ratio (face-on) might be either a bulge+disc galaxy or an elliptical with an exponential halo. For this statistical separation, we examine galaxies with 0 . 1 < B/T < 0 . 9 and ( u -r ) > 2 . 22 for both the bulge and disc component. This means all the ellipticals we find will be red. Restricting ourselves to red galaxies will enhance the fraction of ellipticals relative to bulge+disc galaxies, making the two inclination angle distributions easier to fit. Blue ellipticals are relatively rare (Schawinski et al. 2009), and, therefore, are a small contamination in our sample of classical bulge host galaxies. After setting aside ellipticals, we distinguish between galaxies with quiescent bulges and those with young, star-forming bulges based on the 4000 ˚ A break strength ( D n (4000)) measured by SDSS 1 . Based on results from morphology studies at higher resolution (e.g. Fisher 2006), we associate star-forming bulges with pseudo-bulges and quies- ent bulges with classical bulges. As above, we assign each galaxy a probability of having a classical bulge or pseudobulge based on its D n (4000) (see Fig. A3). Since pseudobulges are a disc phenomenon, we consider galaxies with pseudo-bulges to be bulge-less and exclude them from our analysis. Because 90 per cent of the star-forming bulges are less massive than the median classical, quiescent bulge in our sample, excluding pseudo-bulge hosts does not significantly affect correlations in galaxy properties with local density at fixed bulge mass . However, the separation of quiescent and star-forming bulges will eliminate any star-forming classical bulges from our sample. Finally, we exclude galaxies which are not well fit by any of the above models. These galaxies are modelled by a single S'ersic profile. Unclassifiable galaxies make up less than 10% of our sample. Seventy-five percent of unclassifiable galaxies have a S'ersic index less than 2 . 3 and the same fraction lie in the blue cloud ( u -r < 2 . 22). These galaxies are probably disc-like irregulars, which are unlikely to have a well-defined bulge and disc. The remaining 25 per cent of unclassifiable galaxies are mostly merger remnants, starbursts, and other complicated morphologies. None the less, because the majority of unclassifiable galaxies exhibit disc-like properties, we group them with other bulge-less galaxies. Below, we use a bright ( M 0 . 1 r < -19 . 77) subsample of 29781 galaxies from the L12 sample. This sample contains 12523 classical bulge+disc galaxies, 3681 ellipticals, 2214 pseudo-bulge galaxies, 8505 bulge-less galaxies, and 2857 unclassifiable galaxies. These numbers are inexact, since we only assign each galaxy a probability of being a certain type.", "pages": [ 3, 4 ] }, { "title": "2.2 Group Catalogues", "content": "We study the environmental properties of bulges and discs using the group catalogue from A. Berlind (priv. comm.). This catalogue is built using the methods in Berlind et al. (2006), but is based on SDSS data release 7 instead of data release 4. The group catalogue allows us to relate galaxy properties to their host group (and dark matter halo) properties as well as to study the properties of galaxies as a function of intra-group environment. We select this group catalogue from the many group catalogues available for SDSS data because it extends to low redshift, and overlaps significantly with our sample of bulge+disc galaxies. The group catalogue is volume-limited and includes all galaxies with absolute magnitudes M 0 . 1 r /lessorequalslant -19 . 77 in the redshift range 0 . 02 < z < 0 . 067. We include isolated galaxies in the group catalogue (groups of richness one). The catalogue is created using a friends-of-friends algorithm to determine group membership (see Berlind et al. 2006, for details). Two galaxies are linked if the projected and transverse distances between them are smaller than b ⊥ ¯ n -1 / 3 g and b ‖ ¯ n -1 / 3 g , respectively, where b ‖ , ⊥ are the linking lengths and ¯ n g is the average galaxy number density in the sample. Here, ¯ n g = 5 . 275 × 10 -3 Mpc -3 , b ‖ = 0 . 14, and b ⊥ = 0 . 75, which corresponds to physical linking lengths of 0 . 8 Mpc in the transverse direction and 300 km s -1 along the line of sight. In the group catalogue, there are 90893 galaxies, of which 29781 have a bulge+disc model fit from L12. This subsample includes 83 per cent of the galaxies below z = 0 . 05 with spectroscopic redshifts in the group catalogues. Galax- ies without spectroscopic redshifts (due to fibre collisions in the SDSS spectrograph) make up 4 per cent of the sample below z = 0 . 05, and are therefore a small omission. The remaining 6124 galaxies are missing from the L12 bulge+disc sample because cuts in the galaxy axis ratio (1289), model fits with surface brightness consistent with zero (2830), and galaxies which did not make our quality cuts due to problems with deblending and cosmic ray removal. Despite the missing galaxies, the bulge+disc matched sample is a representative subsample of the group catalogue for z /lessorequalslant 0 . 05. This is demonstrated by Fig. 1, which shows the distributions of group sizes and projected fifth nearest neighbour densities (Σ 5 , see § 2.3 for explanation) for the full group catalogue and for the subsample which overlaps with the bulge+disc decomposition sample. The bulge+disc sample follows essentially the same distributions in Σ 5 and N gal as the full group catalogue. A Kolmogorov-Smirnov (KS) test shows that the distributions of Σ 5 (lower panel) are indistinguishable. The distributions of group sizes are not identical; a KS test yields a probability of 3 × 10 -5 that the distributions are the same. The bulge+disc matched sample is missing galaxies in mid-sized groups, but the number of missing galaxies is small. Furthermore, for the larger groups, the missing galaxies are not a function of position in the group or of local density. This is not surprising, since the number of galaxies missing a spectrum due to fibre collisions is a small fraction (4 to 6 per cent) of the sample. If fibre collisions were more prevalent, we would expect to be missing more galaxies in high density regions than in low density regions. We also check that matching the L12 sample to the group catalogue does not change the distributions of galaxy morphology as a function of magnitude. The top panel of Fig. 2 shows the absolute magnitude distribution of the total r L12 sample, down to a magnitude of M r = -19 . 77. The lower panel shows the same for the galaxies from the group catalogue. Although there are small differences at the faint end, the distributions of galaxy morphology as a function of magnitude are essentially the same for the both samples. The fraction of galaxies of each type in the two samples is the same to within 450 galaxies, or 1 . 5 per cent of the grouped sample. In this work, we address the effects of environment on galaxy properties at fixed bulge mass. Therefore, although it is important that certain types of galaxies are not systematically excluded as a function of environment, we are not concerned with the overall completeness of the sample. We are only concerned that the galaxies in the different environments are a representative sample, which is demonstrated by Figs. 1 and 2.", "pages": [ 4, 5 ] }, { "title": "2.3 Measuring environment", "content": "In order to examine the correlations between galaxy properties and environment, we use several measures of environment, including both group halo properties and more local measures of environment. Directly from the group catalogue, we obtain the group richness, N gal , the group line-of-sight velocity dispersion, and the total stellar mass in galaxies brighter than M 0 . 1 r = -19 . 77. Using the relation between total group stellar mass and halo mass from Leauthaud et al. (2012), we calculate the group dark matter mass, M 200 , defined as the mass enclosed in a region 200 times denser than the critical density 2 . We do not take into account the difference in stellar mass completeness between our sample and that used in Leauthaud et al. (2012), nor do we correct the stellar masses for contamination from non-group galaxies. However, we expect the corrections to the stellar masses to be small (see Leauthaud et al. 2012). We can compute the line-of-sight velocity dispersion ( σ ) for a halo of a given mass using the M halo -σ relation from Yang et al. (2007). For groups with more than 10 galaxies, the σ obtained from the dark matter halo mass is a factor of 1 . 4 larger than the σ measured directly from the galaxies. Half of this discrepancy is due to the small value for the line-of-sight linking length used to build the group catalogue. The small b ‖ biases the measured velocity dispersion down by ∼ 20 per cent (Berlind et al. 2006). In the following analysis, we do not make extensive use of the dark matter halo mass. We do use the group velocity dispersion, but since we are only interested in making comparisons between different environments, the absolute values of σ are not relevant. The group catalogue also contains the the projected distance, R p , from each galaxy to its host group centre (the number-weighted mean angular position). We use this distance, along with the velocity dispersion of the group to define the crossing time t cross = R p /σ , where σ is the the group velocity dispersion, measured directly from the galaxy redshifts (see Berlind et al. 2006). Clearly, t cross is only a sensible measure of environment for relatively large groups and clusters, which have a definite centre. While the numberweighted (or mass-weighted) centre of a group always is welldefined, the FoF grouping algorithm does not guarantee it is physically meaningful. This is especially true for non-relaxed groups. We will address this problem in our investigation of intra-group trends by limiting our sample to galaxies from relaxed groups ( § 5.1). Finally, in addition to the halo properties, we measure the local surface density around each galaxy. We use the projected fifth nearest neighbour density (Σ 5 [Mpc -2 ]). Neighbours are selected from the volume-limited group catalogue in a redshift slice which has a width equal to the velocity dispersion of a galaxy's host group. The minimum width is 300 km s -1 , the line-of-sight linking length used in the group catalogue. For galaxies in large clusters, Σ 5 is a measure of their immediate vicinity, not the underlying large-scale dark matter density field (Muldrew et al. 2012). In the sections below, we rely mostly on Σ 5 as a metric for environment. We will demonstrate that our results are essentially unchanged if group richness, N gal , is substituted for Σ 5 . In § 5.1 we explore trends in galaxy properties within massive, relaxed groups and use the crossing time, t cross , as a measure of galaxy environment.", "pages": [ 5, 6 ] }, { "title": "3 WHOLE GALAXY PROPERTIES WITH Σ 5", "content": "Before examining trends in bulge and disc properties with density, we confirm trends in whole galaxy properties with density. The low redshift morphology-density and colour-density relations have been measured numerous times using various data sets (e.g. Dressler 1980; Goto et al. 2003; Balogh et al. 2004; De Propris et al. 2004; Kauffmann et al. 2004; Tanaka et al. 2004; Weinmann et al. 2006; Blanton et al. 2005; Hansen et al. 2009; Skibba et al. 2009; Bamford et al. 2009). First, we confirm that the morphological classifications described in § 2.1.1 follow the expected morphology-density relation. Fig. 3 shows the distribution of the five morphological types as a function of Σ 5 . The trends with density are in agreement with the morphological trends from other studies (e.g. Dressler 1980; Goto et al. 2003; Bamford et al. 2009). Our results are in close agreement with those from Goto et al. (2003) who find that early and intermediate disc galaxies (Sa and S0) dominate in almost all environments, while late-type discs drop off quickly at high densities. In our sample, classical bulge galaxies dominate at all but the lowest densities. The fraction of classical bulge host galaxies rises quickly above Σ 5 ≈ 1 Mpc -2 . Figure 3 is missing roughly 1300 galaxies which have axis ratios below 0 . 2. These galaxies are excluded from the bulge+disc catalogue in L12 because our exponential disc model is unsuitable for edge-on discs. Including these galaxies slightly decreases the fraction of elliptical galaxies, but the changes are small and the overall trends remain the same. Note that although both Goto et al. (2003) and this work use Σ 5 to measure local environment, the numerical values cannot be directly compared since Σ 5 depends on the lower flux limit of the galaxy catalogue used and the width of the redshift interval used. Goto et al. (2003) measure the distance to the fifth nearest neighbour within a redshift slice of ± 1000 km s -1 , while the redshift slice we use depends on the galaxy's host group velocity dispersion and is typically smaller. Fig. 4 shows the morphology trends for galaxies divided into three stellar mass bins. The stellar mass is computed using the most likely model for each galaxy (although the difference in total stellar mass between two reasonable model fits is always small). The trends with environment are the same for all three mass bins, but the fraction of early/late type galaxies is a strong function of mass in all environments, in agreement with previous results (e.g. Kauffmann et al. 2004; Blanton & Moustakas 2009; Bamford et al. 2009). Classical bulges dominate over all other types of galaxies in the two highest mass bins. Since colour and morphology are generally correlated, the morphology-density relation implies a relation between local density and galaxy colour such that red galaxies dominate in high density regions while blue galaxies are dominant in the field (G'omez et al. 2003; Hansen et al. 2009; Balogh et al. 2004; Baldry et al. 2006; Skibba et al. 2009). There is evidence that the correlation between colour and density is more fundamental than the correlation between morphology and density (Kauffmann et al. 2004; Christlein & Zabludoff 2005; Skibba et al. 2009). We show the fraction of red ( u -r > 2 . 22) galaxies as a function of environment and mass in Fig. 5. In agreement with previous work, we find the fraction of red galaxies is an increasing function of galaxy mass and environment. For all masses, the fraction of red galaxies increases sharply near Σ 5 ≈ 1 Mpc -2 , in agreement with results from G'omez et al. (2003), who find a break in the star formation rate-density relation at approximately the same local density. Figs. 3 and 5 show that both the colour-density and morphology-density relations hold for galaxies of constant stellar mass (Bamford et al. 2009; Skibba et al. 2009). Additionally, comparing the two figures demonstrates that the red fraction of galaxies increases sharply at the same local density as the classical bulge host and elliptical galaxy fractions. We note ellipticals are not the majority of red galaxies, but only contribute 25 per cent by number; the remainder of red galaxies are classical bulge hosts, with a small contribution (5 per cent) from unclassifiable galaxies. In the following section, we explore the colour-density and morphology-density relations for classical bulge host galaxies. The colours of these galaxies follow the same trends as the colours of the galaxy population as a whole. Using bulge+disc decompositions, we can study the colourdensity relations for bulges and discs separately, and determine whether changes in galaxy morphology (e.g. B/T ratio) or changes in the component colours drive the increase in the fraction of galaxies with red integrated colour as a function of Σ 5 .", "pages": [ 6, 7 ] }, { "title": "4 BULGE AND DISC PROPERTIES WITH Σ 5", "content": "Many studies show that a galaxy's properties are largely determined by a its stellar mass (e.g. Brinchmann & Ellis 2000; Kauffmann et al. 2004) and dark matter host halo mass (e.g. Blanton et al. 2006; Weinmann et al. 2006). Since the most massive haloes are strongly clustered, galaxy mass and environment are strongly correlated. However, there are correlations with environment at fixed stellar mass (e.g. Balogh et al. 2004; Bamford et al. 2009; Peng et al. 2011; Grutzbauch et al. 2011; Cooper et al. 2010; Cibinel et al. 2012), and these are the trends we explore. Below, we show trends with environment at fixed bulge mass; however, our results are essentially unchanged if we choose to hold total galaxy stellar mass fixed. In this section, we focus on our sample of 12500 galaxies which have both a classical bulge and a disc. When calculating medians and Spearman rank correlation coefficients, we weight each galaxy by its probability of having a classical bulge and a disc, as explained in § 2.1.1. The left panel in Fig. 6 shows the colours of galaxies with classical bulges and discs as a function of Σ 5 . We divide the galaxies into quartiles based on bulge mass. We remove galaxies with the most and least massive 0 . 5 per cent of bulges in order to eliminate galaxies with large errors in modelled colours and masses. The mass bin divisions are at log M bulge /M /circledot = [9 . 3 , 10 . 1 , 10 . 3 , 10 . 6 , 11 . 2]. These mass bins are then divided into six bins of equal galaxy number at different Σ 5 . Thus, each point in Fig. 6 represents 1 / 24 of the sample ( ∼ 520 galaxies). These points are the weighted median colour in each bin. We define the weighted median as the value for which the sum of weights for values smaller than the median is equal to the sum of weights for values larger than the median. The thin lines in Fig. 6 denote the similarly-defined weighted inter-quartile ranges. Although there is significant scatter in the colour, the trends with environment are statistically significant 3 . The trend in total galaxy colour is simply the colour-density relation for classical bulge+disc galaxies. The weighted Spearman rank coefficients are given in Table 2. We also calculate the bestfitting linear slopes of the relation (see Table 2). In order to reduce the errors on the slope, we remove points offset from the fitted relation by more than 3 standard deviations and then re-fit the relation using the slightly smaller sample. Despite being statistically significant, the change in median integrated colour is small, ∼ 0 . 01 -0 . 03 per dex in Σ 5 . The change is largest for galaxies which host low mass bulges. This is in agreement with the conclusions of Bamford et al. (2009), who show that the colour-density relation is most significant for lower mass galaxies. The middle and right panels of Fig. 6 show the colour density relation for the bulge and disc components of these galaxies separately. It is immediately evident that the change in disc colour with Σ 5 must contribute significantly to the change in total galaxy colour. In fact, we demonstrate below that the change in disc colour is the only contribution to the change in total colour. In the following subsections, we will discuss the changes in bulge properties ( § 4.1), and the changes in disc properties ( § 4.2) as functions of Σ 5 .", "pages": [ 7, 8 ] }, { "title": "4.1 Bulges", "content": "The classical bulges in Fig. 6 were selected to have large 4000 ˚ A breaks, similar to ellipticals. Elliptical galaxy colours depend on galaxy mass (e.g. de Vaucouleurs 1961; Brinchmann & Ellis 2000), but only weakly, if at all, on environment (e.g. Dressler et al. 1987; Bernardi et al. 2003b; Balogh et al. 2004; Trager et al. 2008; Hansen et al. 2009). The central panel in Fig. 6 shows that the median bulge colours lie on the red sequence ( g -r ∼ 0 . 8), and that there is a small increase (∆( g -r ) ≈ 0 . 03) in median colour from the lowest mass bulges to the highest mass bulges. The weakness of this trend may be due in part to large scatter in the model bulge colours compared to the scatter in model elliptical colours (see fig. 32 in L12). In addition to trends in colour with bulge mass, there is a weak but statistically significant anti-correlation between environment and bulge colour; bulges in low density regions are redder than those in higher density regions. The Spearman rank correlation coefficients for this trend are listed in Table 2. This result is in contrast to the results of Hudson et al. (2010) who find no variation in median bulge colour with increasing local density. This trend is strongest for the second lowest bulge mass bin (red circles). Since pseudo-bulges are typically blue and are not as strongly clustered as classical bulge galaxies (see Figs. 3 and 4), contamination from pseudo-bulges will tend to flatten the trend in bulge colour for the lowest mass bulges, possibly explaining the weaker trend for the lowest mass bulges. Although the bulge colours decrease as a function of environment, there is no significant correlation between Σ 5 and SDSS fibre colours, and the correlation between Σ 5 and D n (4000) is in the opposite sense (see Fig. 8). However, if we replace the L12 bulge+disc decompositions with those from Simard et al. (2011), the same trend in bulge colour with density is recovered. We propose that the decrease in bulge colour with environment is an artefact of 2-dimensional bulge+disc decompositions. Specifically, the colours of small bulges in blue discs are biased redwards by the bulge+disc decomposition. The disc model used in L12 (and Simard et al. (2011)) is an exponential profile, which is centrally peaked. Therefore, the disc can contribute significantly to the central flux of the galaxy. Moreover, since the disc is usually bluer than the bulge (especially in low density regions), the bulge+disc decomposition will attribute blue light from the central region to the disc, thus making the bulge appear redder. This effect will be largest for the smallest bulges in the bluest discs. Since the bulge-to-total flux ratio and disc colour are both increasing functions of environment, the bias in bulge colour will be largest for small bulges in low density environments. We test this explanation by refitting classical bulge galaxies with models that suppress the disc flux in the central region. In the new models, the disc flux goes rapidly to zero within one bulge scale radius, I disc ∝ ( r/r bulge ) / [1 + ( r/r bulge ) 4 ] 0 . 25 . Keeping all the other model parameters fixed, we linearly scale the bulge and disc flux of this new model to fit each galaxy. By design, the new fits have larger bulge-to-total flux ratios, and the differences are largest for galaxies with small bulges. The resulting trends in bulge colour with environment for these disc-suppressed models are shown in Fig. 7. There is only a statistically significant trend with mass for galaxies with 10 . 2 < log M ∗ /lessorequalslant 10 . 5 (note that because the bulge and disc masses have been recomputed, the mass quartiles in Fig. 7 are different from those in Fig. 6). The choice of parametrisation for the central region of the disc plays an important role in the bulge colours. In general, it is unknown whether the stellar disc continues unchanged through the bulge or if the disc only exists outside the central bulge. In the Sombrero galaxy, photometry and spectroscopy show an inner cutoff for the stellar disc, suggesting the disc-suppressed models may be a better choice for early-type disc galaxies (Emsellem et al. 1996). In this work, we will continue to use the centrally-peaked model for the disc, knowing that the colours of small bulges will be biased redwards. This bias will also occur in the opposite case, when a small disc surrounds a large, red bulge. However, we do not expect the bias in disc colours of galaxies with large bulges to be as severe. Since the majority of the bulge flux comes from regions above and below the disc, there is no physical basis for the bias in disc colour, as in the case for small bulges.", "pages": [ 8, 9 ] }, { "title": "4.1.1 Spectroscopic Properties", "content": "Recent work has shown statistically significant positive correlations between stellar age and local density (e.g. Thomas et al. 2005; Bernardi et al. 2006; Clemens et al. 2006; Smith et al. 2006; Cooper et al. 2010) and between metallicity and local density (Cooper et al. 2010) for early type, or red sequence, galaxies. Our sample of classical bulge+disc galaxies contains both early type, passively evolving, galaxies and galaxies with ongoing star formation in their discs. However, we expect all classical bulges and ellipticals to follow the same relations with stellar mass and environment. In order to examine the stellar population properties of classical bulges more closely, we use the SDSS fibre spectra and the line indices and metallicities reported in the MPA/JHU SDSS spectroscopic catalogue. At low redshift, the fibre spectra are dominated by bulge stellar light, but the mixture of bulge and disc light in the fibre is a function of redshift. These aperture effects may influence correlations between measured line indices and Σ 5 . For classical bulge+disc galaxies in our sample, the median ( B/T ) r in the inner 3 arcsec is 0 . 74, and eighty per cent of the galaxies have ( B/T ) r within 3 arcsec larger than 0 . 5. For galaxies at z > 0 . 04, the median (3 arcsec) ( B/T ) r is only slightly smaller at 0 . 72. Therefore, the 3 arcsec SDSS spectroscopic fibres are dominated by bulge stellar light, even at the highest redshift ( z = 0 . 05) in our sample. Since our sample is in the regime where angular size scales linearly with distance, the spectroscopic measure- ments will include more disc flux at higher redshift. This effect is partially countered by a slight bias toward larger bulges at higher redshifts; in this sample, the average physical bulge size increases by 8 per cent from z = 0 . 02 to z = 0 . 05, due to the fact that small bulges at high redshift are more difficult to accurately fit. However, there is no statistically significant trend in Σ 5 with redshift. Therefore, even though the the spectroscopic properties do change with redshift, the correlations between spectroscopic properties and Σ 5 will be unaffected, since Σ 5 and z are uncorrelated. The top panel of Fig. 8 shows the 4000 ˚ A break (D n (4000)) (Balogh et al. 1999) as a function of Σ 5 for four different bulge masses. In this figure, we include both classical bulges and ellipticals, since we expect the stellar populations of classical bulges and ellipticals to be the same at a given mass (MacArthur et al. 2010). Indeed, excluding ellipticals from Fig. 8 does not noticeably alter the results. The four mass bins used are the same as in Fig. 6. The increase in D n (4000) as a function of density is statistically significant for all bulge masses, but the trend is strongest for the lowest mass bulges. These results imply that bulges in high density environments have had less recent star formation than those in lower density environments. The same results are obtained if we plot the line index H δ A as a proxy for stellar age (the equivalent width of H δ is anti-correlated with environment) (Kauffmann et al. 2003). The increase in stellar age as a function of local density at fixed bulge mass is in agreement with previous results (Trager et al. 2000; Kauffmann et al. 2004; Thomas et al. 2005; Clemens et al. 2006; Bernardi et al. 2006; Smith et al. 2006; Cooper et al. 2010, but see Thomas et al. 2007). Cooper et al. (2010) note that the trend in age of red sequence galaxies with density is evidence for galaxy assembly bias (Croton et al. 2007); namely, older galaxies are more strongly clustered than younger ones at fixed stellar mass. The lower panel of Fig. 8 shows stellar metallicity as a function of Σ 5 . The values for log Z/Z /circledot are taken from Gallazzi et al. (2005, 2006) 4 . The metallicities are only computed for galaxies from SDSS data release 4, which includes approximately half of the sample used in this paper. Gallazzi et al. (2005) compute the stellar ages and metallicities by fitting model spectra from Bruzual & Charlot (2003) to a combination of iron and magnesium line indices, the 4000 ˚ A break, and three Balmer lines. The trend in stellar metallicity at fixed bulge mass is statistically significant for all but the second lowest mass bin (red circles). The trends are strongest for the highest mass bin, where the metallicity changes by 0 . 01 dex per decade in Σ 5 . Despite being statistically significant, the change in metallicity is quite small. The increasing trend in stellar ages and metallicities is orthogonal to the age-metallicity degeneracy (Worthey 1994; Gallazzi et al. 2005), which makes the results more robust. The small increase in metallicity is in agreement with results from Cooper et al. (2010) (but see Thomas et al. 2005; Smith et al. 2006), as well as studies of the gas-phase metallicity which show an increase in metallicity in star-forming galaxies as a function of local density (Cooper et al. 2008). These results demonstrate that bulges in high density re- gions formed earlier and with higher star formation rates (see Cooper et al. 2010). The increases in stellar age and metallicity with increasing density suggest that classical bulges should be redder in higher density regions. This disagrees with the results from § 4.1, where we find that bulge colours may even be slightly bluer in high density regions. This disagreement is likely due to biases in our model bulge colours.", "pages": [ 10, 11 ] }, { "title": "4.2 Discs", "content": "In the previous section, we show that the change in bulge colours with local density is small, and any statistically significant change is probably due to our choice of disc profile for galaxies with small bulges. On the other hand, the trends in disc colour with density are statistically significant. These are shown in the right panel of Fig. 6. The Spearman rank coefficients and slopes of the linear relation between log Σ 5 and ( g -r ) disc are given in Table 2. Although the correlations are statistically significant, the changes in disc colour are small; the colour only increases by ∼ 0 . 015 mag per dex in Σ 5 . However, this change is 2 -3 times larger than the change in colour measured for bulges. For all four bulge mass bins, the slope of the relation between log Σ 5 and disc colour is comparable or larger than the slope in total galaxy colour, which suggests the change in integrated galaxy colour can be fully explained by the change in disc colour. We will return to this conclusion in § 4.2.1. As with the bulge colours, there is significant scatter in disc colour at fixed bulge mass; the inter-quartile range of ( g -r ) disc colours is typically 0 . 1 -0 . 2 mag. This scatter is due to the scatter in disc mass at fixed bulge mass and fixed Σ 5 . Disc mass and disc colour are strongly correlated (e.g. de Jong 1994), so a range of disc masses will yield a range of disc colours. We show below that the correlation with disc mass and environment does not contribute to the trend in disc colour with environment. In Fig. 6, we plot the relation between Σ 5 and disc colour for bulge-less discs, including pseudo-bulges and unclassifiable galaxies (grey crosses and dashed lines). This relation is offset to lower values of Σ 5 since bulge-less discs are typically found in lower density environments. None the less, discs in galaxies without prominent bulges follow the same colour-density relation as discs with large classical bulges. Like the change in total galaxy colour, the change in disc colour is largest for discs around the smallest bulges. This does not extend to bulge-less disc galaxies, for which the relation between log Σ 5 and ( g -r ) disc is not as steep as the relation for galaxies with low, but non-zero, mass bulges (see Fig. 6 and Table 2). However, since each bulge mass bin includes a large range of disc masses, this does not contradict the observation that the colour-density relation is strongest for low- (total) mass galaxies (Bamford et al. 2009; Tasca et al. 2009). The trend of increasing disc colour with environment is a signature of star formation being halted in denser environments. Our results are in agreement with earlier studies of discs becoming redder in cluster environments (e.g. Hashimoto et al. 1998; McIntosh et al. 2004; Hudson et al. 2010) and the rise of red disc galaxies (anaemic spirals) with increasing local density (e.g. Dressler 1980; Goto et al. 2003; G'omez et al. 2003; Bamford et al. 2009). Two mod- ls for removing gas from galaxies in dense environments include ram-pressure stripping by the intra-group medium (Gunn & Gott 1972), and the removal of the hot halo gas supply around disc galaxies (strangulation) (Larson et al. 1980). These two mechanisms have different timescales for shutting off star formation; ram-pressure stripping almost immediately ends star formation, while galaxies losing their halo gas reservoirs undergo an exponential decay in star formation rate (Balogh et al. 2000; van den Bosch et al. 2008). From the disc fading shown here, these two processes are indistinguishable. We return to the differences between rampressure stripping and strangulation in § 5 when discussing the correlation between disc colour and Σ 5 for rich and poor groups, separately. Fig. 6 shows that the most massive bulges in our sample also have the bluest median disc colours (black crosses in Fig. 6). This seemingly contradicts observations that find redder, and presumably more massive, bulges have redder discs (de Jong 1994; Peletier & Balcells 1996; Wyse et al. 1997; Cameron et al. 2009). However, Fig. 6 and 7, show little change in bulge colour as a function of bulge mass. Furthermore, while we separate the sample based on bulge mass, the disc mass does not monotonically increase with bulge mass. The trend of decreasing disc colour with increasing bulge mass exists even if the inclination correction is removed (see Fig. 12). This trend may be partially explained by the same modelling artefact which affects bulge colours ( § 4.1). In this case, the subtraction of a large, red bulge from an image will leave behind an abnormally blue disc. This modelling effect will be strongest for galaxies with large red bulges, and relatively small discs.", "pages": [ 11 ] }, { "title": "4.2.1 Disc mass and size", "content": "In § 4.2, we show that disc colour and environment are significantly correlated, thus disc fading is a significant contribution to the colour-density relation. However, morphological transformation could also play a role. If discs are being stripped at the same time they are fading, we expect that the mass contribution from the disc to be a decreasing function of Σ 5 . Since bulges are typically redder than discs, decreases in disc mass will lead to increases in total galaxy colour. In Fig. 9 we show the disc-to-total stellar mass ratio ( D/T ) for galaxies in four bulge mass bins. The mass bins are the same as in Fig. 6. We find no statistically significant trend in D/T with Σ 5 . For the highest bulge mass bin (black × s), the median D/T is above 10 per cent, the cutoff D/T below which we classify galaxies as ellipticals. Therefore, it is unlikely that the constant D/T at high bulge mass is due to the minimum detectable D/T . Although we have separated the galaxies by bulge mass, changes in median bulge mass within each bin could affect the relation between D/T and Σ 5 . We confirm there is no statistically significant change in median bulge mass in each of the four mass bins as a function of Σ 5 . Therefore, at fixed bulge mass, the disc mass is independent of local density, and the changes in total galaxy colour as a function of Σ 5 are due solely to the changes in disc colour. Fig. 10 strengthens the argument against morphological transformations by demonstrating there are no statistically significant correlations between disc half-light radius (measured in the r band) and Σ 5 at fixed bulge mass. Neither the disc size nor mass changes significantly with increased density, as would be expected if galaxy harassment or mergers play a major role in galaxy evolution in high density environments. These results are in agreement with results from McIntosh et al. (2004). They find large differences in star formation rates between cluster galaxies and field galaxies, as evidenced by differences in disc colours and disc structures (e.g. spiral arms) between cluster and field galaxies, but they do not find changes in the bulge-total-ratio distributions for field and cluster galaxies at fixed galaxy colour. It is important to keep in mind that Figs. 9 and 10 are based entirely on galaxies with classical bulges and discs. If we include ellipticals ( D/T = 0) in the sample, the median D/T is a very weakly decreasing function of Σ 5 , but only for the highest mass bulges. This is expected since ellipticals are more strongly clustered than typical bulge+disc galaxies (see Fig. 4). Similarly, if we include bulge-less galaxies in Fig. 9 (and separate galaxies into bins of constant total stellar mass instead of constant bulge mass), the median D/T is a decreasing function of Σ 5 . By focusing on galaxies with a bulge and a disc, we are investigating whether there are morphological transformations within this population that help explain the transition from disc-dominated galaxies in the field to bulge-dominated galaxies in large groups and clusters. The lack of evolution in D/T with increasing Σ 5 indicates that either higher density environments do not lead to disc destruction or the transition from a disc-dominated galaxy to a bulge-dominated galaxy occurs on very short timescales, and we do not observe any galaxies in this transitional phase.", "pages": [ 11, 12 ] }, { "title": "4.2.2 Inclination Angle", "content": "The previous section demonstrates that the changes in total galaxy colour for bulge+disc galaxies are due to changes in disc colour. However, these results could be biased by the inclusion of ellipticals in high density regions. In this section, we address this concern by dividing the sample into three bins of disc inclination angle. If we mistakenly model the elliptical outskirts and halo as a disc, we would expect no colour change with environment for face-on discs, where contamination from ellipticals is most significant. Conversely, if contamination from ellipticals is zero and our sample does not suffer from any biases based on disc inclination, we expect the same trends in bulge and disc properties with environment, irrespective of disc inclination. Figures 11 and 12 show the disc colour and disc-to-total stellar mass ratio as a function of Σ 5 for three bins in disc inclination angle. The galaxies plotted are the same as in previous figures (classical bulge+disc galaxies). The bins in inclination are chosen such that each panel shows the same number of galaxies. Assuming a flat disc, the inclination angle limits for each panel are (76 · , 63 · , 46 · , 20 · ), where 0 · is face-on. From Fig. A2, it is clear that galaxies in the leftmost panels of Figs. 11 and 12, with q d < 0 . 46, have a very low probability of being ellipticals, while the galaxies in the rightmost panel are the most likely to include ellipticals. The galaxies are divided into four bulge mass bins, using the same divisions as in Fig. 6. Note that the bulge mass bins do not all have the same number of galaxies, as they did above; at low q d (highly inclined galaxies), there are more massive bulges, while at high q d (face-on galaxies), there are more low mass bulges. However, the number of galaxies in each bin differs by at most ∼ 200, and each bin has 800 galaxies on average. Therefore, any bias introduced by changes in average bulge mass with disc inclination will be small. Figure 11 shows no statistically significant trend in D/T as a function of density. Furthermore, D/T is independent of disc inclination angle. This supports our previous conclusions. Namely, at constant bulge mass, disc mass is not a function of environment. The dependence of disc colour on environment and disc inclination is less straightforward. Although the rank correlation coefficients are always positive, the trends are only statistically significant for the low mass bulges or highly inclined discs (see Table 3). This hints at contamination from elliptical galaxies in the highest q d and highest bulge mass bins. For the most inclined galaxies, where contamination from ellipticals is minimal, the slopes in disc colour reported in Table 3 agree with those reported for disc colours in Table 2. Additionally, for the lowest mass bulges (blue squares), the slope in ( g -r ) disc with Σ 5 is essentially independent of q d . Although the trends for the higher mass bulges are not always statistically significant, the slopes measured are also in reasonable agreement with those for the whole sample. Together with the lack of correlation between D/T and Σ 5 at fixed q d , the trends in disc colour with Σ 5 at fixed q d reinforce our conclusion that the changes in bulge+disc galaxy colour, while small, are entirely due to changes in disc colour with Σ 5 . The colours plotted in Fig. 12 are not corrected for disc inclination, which accounts for the differences in the median disc colour across the three panels. Without the intrinsic inclination correction, the redder colours for inclined discs are expected. However, using the uncorrected colours will allow us to verify the disc inclination correction used in the previous sections. Fig. 12 shows that the change in median colour as a function of q d is largest for the the two middle mass bins (red circles and green triangles). In this mass range, the median disc colour decreases by ∼ 0 . 04 mag from the highest inclination to the lowest inclination galaxies. For the high and low mass bins, the median disc colours only decrease 0 . 02 mag. This demonstrates that the intrinsic inclination correction should be a function of bulge mass; galaxies with very low mass bulges and very high mass bulges seem to suffer less intrinsic extinction in their discs. Following Maller et al. (2009), our extinction correction addresses the low bulge mass effect. The correction includes a term that depends on K -band magnitude such that higher mass galaxies have a larger extinction correction. Thus, our extinction correction is typically too large for galaxies with massive bulges. Finally, the lack of extinction due to 'discs' around high mass bulges may be due to contamination from ellipticals, where we expect little extinction. Reddening of discs is both a function of disc inclination and a function of the amount of dust in the disc. Above, we argue that discs in galaxies with different bulge masses suffer from different amounts of extinction. We can also use the differences in disc colour across the three panels in Fig. 12 to explore changes in the amount of extinction as a function of local density. In principle, the change in the correlation between the uncorrected disc colour and Σ 5 as a function of q d could be used to measure the change in extinction as a function of local density. However, this requires a detailed understanding of how extinction depends on both the amount of dust and the disc inclination. Reddening is a complicated, possibly non-monotonic, function of these variables (see Tuffs et al. 2004, for one parametrisation). Therefore, we limit ourselves to a simple test; if the extinction in high density environments is negligible, then the disc colour in the highest density environments should be independent of disc inclination. Fig. 12 shows this is not true, with the possible exception of the highest bulge mass galaxies (black crosses). As explained above, this mass bin is likely to contain misrepresented ellipticals, which would counter the change in disc colour with disc inclination. Therefore, any contamination from ellipticals strengthens the argument that discs in high density regions are not free from extinction.", "pages": [ 12, 13, 14 ] }, { "title": "5 OTHER ENVIRONMENT MEASURES", "content": "As discussed in § 2.3, we compute environment metrics in addition to the local density measure, Σ 5 . Below, we use the group richness, N gal , as our environment measure. We define richness as the number of galaxies in a group above the group catalogue magnitude limit, M 0 . 1 r = -19 . 77. We do not correct N gal for contamination, nor do we attempt to include lower luminosity galaxies in N gal . For rich groups, N gal is closely related to a galaxy's host dark matter halo mass, and is notably different from Σ 5 , which measures the local (intra-group) density around a galaxy. Since many of the effects of environment depend, at least indirectly, on the group potential, correlations between bulge and disc properties and N gal may help determine which environmental effects are most relevant. In Section 5.2, we use crossing time, t cross as a proxy for environment within relaxed, relatively rich groups. Unlike N gal , t cross is different for each galaxy in a group. It measures how long a galaxy has been affected by its host group. In Fig. 13 we plot the trends in total, bulge, and disc colour as a function of group richness, N gal , instead of Σ 5 . Unsurprisingly, the trends in galaxy colours with host group richness are similar to the trends with Σ 5 ; total galaxy colour and disc colour are increasing functions of richness, while the bulge colour decreases slightly with increasing N gal . The latter effect is explained by the modelling bias we discuss in § 4.1. Additionally, we find no significant correlation between D/T and N gal , in agreement with the lack of correlation in Fig. 9. However, the plots in Fig. 13 do indicate that galaxy colours (and disc colours) do not redden above N gal ∼ 20. This is in agreement with results from previous studies (e.g. Balogh et al. 2004; van den Bosch et al. 2008). Balogh et al. (2004) show that the fraction of red galaxies is independent of cluster velocity dispersion for σ > 250 km s -1 , which corresponds to N gal ≈ 15 -25. Thus, the majority of the colour evolution of bulge+disc galaxies occurs in smaller groups. We can test this hypothesis by examining the colourdensity relation in relatively rich groups and poor groups separately. This test is shown in Fig. 14; the left panel shows the relation between disc colour and Σ 5 for groups with at most 20 members, while the right panel shows the same for larger groups. There is no statistically significant correlation between disc colour and Σ 5 in the large groups, but the trends are statistically significant for galaxies in the field 0.8 and in smaller groups. Therefore, the transformation of disc colour does not require rich clusters, and star formation in discs is effectively halted by group density environments. This is in agreement with other studies of colour and environment (e.g. Zabludoff & Mulchaey 1998; G'omez et al. 2003; Cooper et al. 2006; Blanton & Berlind 2007). However, Fig. 14 does show a significant offset ( /greaterorsimilar 0 . 01) in disc colour between galaxies in poor groups just below Σ 5 ≈ 1 Mpc -2 and galaxies in rich groups at Σ 5 /greaterorsimilar 1 Mpc -2 . This density threshold is close to the density of the inflection point in the red fraction seen in Fig. 5. ⊙ In addition to examining the changes in colour as a function of environment, we show the changes in morphological types as a function of N gal and dark matter halo mass in Figs. 15 and 16. In general, the trends in morphological type fraction are in agreement with those in Fig. 3. For the largest groups, there is little change in the elliptical galaxy fraction as a function of halo mass, in agreement with Hoyle et al. (2012), who find that the fraction of early type galaxies is ≈ 0 . 2 and independent of halo mass for masses above ∼ 10 13 M /circledot . Hoyle et al. (2012) use morphologies from the Galaxy Zoo project (Lintott et al. 2011), which are qualitative classifications, and cannot distinguish between face-on S0s and elliptical galaxies. As such, their definition of early type galaxies only corresponds approximately to our definition of ellipticals. There are certainly some early type galaxies in our classical bulge+disc category. In Fig. 15, the downturn in the elliptical fraction in rich groups is partially due to the inclusion of large, but not necessarily bound structures, in the FoF group catalogue. We work to eliminate these non-virialised groupings the next section.", "pages": [ 14, 15, 16 ] }, { "title": "5.1 Round groups", "content": "Although we show that the change in disc colour occurs in relatively small groups, the original morphologydensity relation was constructed using cluster-centric distance as a proxy for environment (Oemler 1974). Indeed, many studies compare galaxies on the outskirts of massive clusters to those at the centres (e.g. Dressler 1980; Postman & Geller 1984; Whitmore & Gilmore 1991; G'omez et al. 2003; Treu et al. 2003; Smith et al. 2006; Weinmann et al. 2006; Trager et al. 2008; Hudson et al. 2010). Although our sample does not extend to significant redshifts, there are several massive groups included in the group catalogue which we use to study trends in bulge and disc properties with local density within groups. These trends may help reduce the scatter in the relations with Σ 5 and N gal shown in Figs. 6 and 13. In order to study trends within clusters, we first limit our sample to groups with at least 20 members. However, as discussed in § 2.3, the FoF grouping algorithm does not necessarily identify relaxed or even bound systems; filaments and merging galaxy groups are often identified by the FoF algorithm as galaxy groups. Furthermore, many large galaxy groups do not have well-defined centres (e.g. Zabludoff & Mulchaey 1998; Skibba et al. 2011), making trends in galaxy properties with distance from the group centre almost meaningless. In order to use environment measures such as the crossing time or distance from the group centre, we select a sample of groups that have a symmetric projected distribution of galaxies. This eliminates groups in the process of forming or merging, for which the distance from the cluster centre has little physical meaning. If a group is relaxed and approximately spherical, we expect that galaxies should have a rotationally symmetric distribution on the sky. We generalise the requirement of rotational symmetry by allowing galaxies to be evenly distributed along elliptical contours, instead of just circular ones. In practise, we accomplish this by compressing the coordinates along the minor axis of the group, making an elliptical group appear spherical. Allowing groups to be elliptical in projection does not accurately represent oblate, prolate, or triaxial groups in projection (expect from special viewing angles), but it does account for some of the asymmetries real groups. We test whether a group is symmetric by comparing the distribution of galaxies observed for a given group, to a distribution drawn from a rotationally symmetric radial profile. The functional form of the radial profile, however, is uncertain (e.g. Adami et al. 2001). Instead of relying on an analytic function for the profile, we create separate comparison radial profiles for each galaxy group by fitting a smooth curve to the binned radial distribution of galaxies in each group. To limit the Poisson noise, this fit is smoothed using a Gaussian filter with a width ( σ ) equal to half the rootmean-squared (rms) radius, R rms , of the group. We then compare the actual distribution of galaxies to the smoothed radial distribution using a χ 2 test: χ 2 = Σ i,j [( N obs .,i,j -ρ i,j N gal ) 2 / ( ρ i,j N gal )], where N obs .,i,j is the number of galaxies observed in a region ( i, j ) and ρ i,j N gal is the expected number of galaxies from the rotationally symmetric distribution. The denominator is the Poisson uncertainty in the number of galaxies. The size of the ( i, j ) region used in the sum will affect the χ 2 value; if the region is too large, the two distributions will trivially agree. We use a square region which is 0 . 5 R rms on a side. This is large enough to ensure most regions in the sum have at least one galaxy, but small enough to distinguish between rotationally symmetric and asymmetric groups. If the galaxy distribution is rotationally symmetric, it will be statistically indistinguishable from the smoothed radial distribution. On the other hand, if the galaxy distribution is markedly non-symmetric (e.g. bimodal), the χ 2 value will be large. We consider a group to be round if the χ 2 value from this test has a probability between 0 . 1 and 0 . 9. This selects 95 of the 197 groups with at least 20 members in the group catalogue. Below, we designate these groups round groups. Of these groups, 46 are represented in the bulge+disc catalogue; the remainder are at redshifts greater than 0 . 05. There are ∼ 720 classical bulge+disc galaxies in round groups, compared to 1470 classical bulge+disc galaxies in groups with N gal > 20.", "pages": [ 16 ] }, { "title": "5.2 Intra-group trends", "content": "In order to examine the effect of intra-group environment on bulges and discs, we define the crossing time ( t cross ) for each galaxy as the distance from the galaxy to the group centre divided by the line-of-sight velocity dispersion of the group. For a single galaxy, t cross is unphysical, since the galaxy is typically not moving radially with the group velocity dispersion. However, for a large sample of galaxies, t cross is a measure of how long galaxies have been affected by the group environment. Crossing time is anti-correlated with local density Σ 5 , although there is significant scatter. We have not divided our sample into central galaxies and satellite galaxies as previous studies have done (e.g. van den Bosch et al. 2008; Skibba & Sheth 2009; Skibba 2009; Peng et al. 2011). However, the number of groups is much smaller than the number of galaxies in this sample, so the contribution from central galaxies is small. Fig. 17 shows the relation between galaxy colours and t cross for classical bulge+disc galaxies in round groups with N gal > 20. Since these constraints greatly reduce the sample size ( ∼ 720 classical bulge+disc galaxies), we only divide the galaxies into three mass bins and only show three median points in t cross . The largest crossing times in our sample are less than 40 per cent of a Hubble time, in agreement with those found at higher redshift (Grutzbauch et al. 2011) and with the timescale for relaxation (Gunn & Gott 1972; Ferguson & Sandage 1990). Crossing times shorter than the Hubble time indicate that the galaxies are not falling into a group for the first time, and have probably orbited the group centre at least once. The crossing time gives an indication of how long a galaxy has been affected by the group environment; galax- ies with the shortest crossing times have presumably passed through the centre of the group most often and have experienced the highest average local density. In addition, a small t cross implies a high mass density within the group ( t cross ∝ ρ -1 / 2 ); galaxies in compact groups will have smaller crossing times. Therefore, any trends in galaxy properties with t cross will be sensitive to how the environmental processes causing the trends depend on local density and time, although the exact dependencies are not straightforward. In addition, the median distance to the group centre in our sample of round groups is two thirds of the group virial radius. The galaxies in our sample are all much closer to their host group centre than 3 -4 virial radii, the distance at which G'omez et al. (2003) find a large change in the star formation rate. Samples that extend to much larger radii are probably needed in order to see large trends in galaxy colours with group-centric distance. The left panel in Fig. 17 shows that the total galaxy colour is anti-correlated with t cross ; galaxies with short group crossing times typically have redder colours. The correlation is only statistically significant for the highest mass bulges. If we do not limit the sample to galaxies from round groups, the statistically significance of all the trends decreases. The middle panel in Fig. 17 shows no statistically significant trend in bulge colour with crossing time. For the smaller bulges, there is a weak correlation with t cross , but, as shown in § 4.1, these trends are likely due to uncertainties in the disc model. The final panel in Fig. 17 shows the disc colours as a function of t cross . In this case, none of the correlations are statistically significant (for the highest mass bulges, t cross and disc colour are anti-correlated with 2 σ significance). These weak trends in disc colour are in agreement with the lack of correlation between ( g -r ) disc and Σ 5 for all groups with more than 20 members (see Fig. 14). There are significant colour differences between discs in large groups and discs in the field. Isolated classical bulge+disc galaxy colours are denoted by solid points in Fig. 17, and, at least for low mass bulges, discs in the field are significantly bluer than discs in groups. This agrees with our earlier conclusion that most disc colour evolution occurs in smaller groups (see also van den Bosch et al. 2008; Cooper et al. 2006). The lack of correlation between disc colour and t cross is in contrast with the results of Hudson et al. (2010), who find a statistically significant correlation between disc colour and a galaxy's distance to the group centre. This is probably due to the selection of the samples; Hudson et al. (2010) do not divide their sample into galaxies with bulges and discs only. If we include bulge-less galaxies in our sample, there is a statistically significant correlation with disc colour and crossing time. Fig. 18 shows the change in disc-to-total mass ratio and disc scale length as a function of t cross . Unlike the case of the relations with Σ 5 , there are statistically significant cor- elations between disc mass and t cross and between disc size and t cross . The positive correlation between D/T and t cross is statistically significant for the highest and lowest mass bins, and significant at the 2 σ level for the middle mass bin. For the highest mass bulges, the disc-to-total mass ratio increases by 6 per cent from t cross = 0 to t cross = 0 . 3 /H 0 . There is no change in average bulge mass as a function of t cross , so the change in D/T is due entirely to changes in the disc mass. In addition, the lower panel of Fig. 18 shows a correlation between t cross and the disc R eff for the most massive bulges, which is significant at the 2 . 3 σ level. Although these trends with t cross are weak, they are also present if we substitute Σ 5 for t cross (correlations become anti-correlations). These trends suggest that processes active in the highest density regions in groups either suppress disc formation or destroy discs around infalling galaxies. We can expand our sample to include ellipticals ( D/T = 0), to determine if the trends in D/T and t cross continue until the disc is negligible. In this case, the correlation between D/T and t cross for the highest bulge mass bin becomes insignificant. Thus, at least for high mass bulges, the trend in D/T with t cross does not extend to disc-less ellipticals. Together, Figs. 17 and 18 tentatively suggest that morphological changes play some role in the colour-density relation for galaxies in rich groups. This seems to conflict with the conclusions in § 4.2, which show no significant trends in disc mass with density. Taken together, these results suggest that star formation quenching and morphological transformation are separate physical processes and that these transitions take place in different environments. This agrees with previous studies that find an increase in the number of passive, red discs as a function of time (Dressler et al. 1997; Moran et al. 2007; Bundy et al. 2010) and environment (Goto et al. 2003; McIntosh et al. 2004; Bundy et al. 2006; Bamford et al. 2009). The existence of these nonstar forming disc galaxies demonstrates that star formation quenching occurs before morphological transformation. In addition, Skibba et al. (2009) note that there is a weak correlation between red galaxy morphology and density at small scales, in agreement with our findings. Namely, we demonstrate that morphological transformation may be taking place in rich groups and clusters, and that these transformations are not associated with star formation quenching since disc colours are not correlated with t cross . Galaxy harassment, i.e. high-speed encounters between galaxies, is a plausible explanation for the decrease in disc mass we observe. The number of encounters a galaxy experiences will increase with galaxy number density, and galaxies with short crossing times are exposed to the highest average densities. Thus galaxy harassment would yield correlations between t cross and disc mass and disc size similar to those in Fig. 18.", "pages": [ 16, 17, 18 ] }, { "title": "6 SUMMARY", "content": "In this work, we examine the changes in bulge and disc properties for a sample of 12500 galaxies as functions of local projected density. Since galaxy mass and environment are strongly correlated, we divide the sample into bins of equal bulge mass in order to study any residual trends in bulge and disc properties with local density. Using 2-dimensional bulge+disc decompositions, we are able to study the colour- density and morphology-density relations for bulges and discs, separately. Our sample consists of galaxies with a classical, elliptical-like bulge surrounded by a disc. Classical bulges are observed to have the same characteristics as ellipticals of the same mass; the only difference is the encompassing disc. By studying the properties of classical bulge hosts in different environments, we can deduce if local density is the determining factor in whether or not a classical bulge acquires and retains a disc. In addition, we study whether the population of classical bulge+disc galaxies is undergoing a transition from disc-dominated and star-forming in low-density regions to bulge-dominated and passive in highdensity regions, as is expected from the morphology-density and colour-density relations. Based on our results, we can draw two conclusions about the effects of environment on discs around classical bulges. First, both the colour and the mass of these discs changes with increasing density, but these changes occur at different densities and, presumably, on different time-scales. This suggests that star formation quenching and morphological transformation are caused by different physical processes, in agreement with many previous studies of galactic environment (e.g. Goto et al. 2003; G'omez et al. 2003; McIntosh et al. 2004; Kauffmann et al. 2004; Christlein & Zabludoff 2005; Cooper et al. 2006; Bundy et al. 2006; van den Bosch et al. 2008; Bamford et al. 2009; Skibba et al. 2009; Hudson et al. 2010). Second, although disc properties are a function of environment, the changes in disc colour and disc mass are insufficient to explain why classical bulges have discs and ellipticals do not have discs. While environment clearly affects disc formation and evolution around classical bulges, there must be other parameters that determine whether or not a classical bulge is surrounded by a disc. The first piece of evidence for our conclusions comes from separating the colour-density relation for bulge+disc galaxies into relations for bulges and discs. We find that the correlation between total galaxy g -r and local density is due entirely to changes in disc colour with density. The small change in bulge colour with increasing density depends on our choice of disc model, but this choice does not noticeably affect disc colours. At fixed bulge mass, disc mass and disc scale length are independent of local density. Thus, there is no evidence for morphological transformation as a function of local density, but there is evidence for star formation quenching in discs with increased local density. Morphological transformation and star formation quenching must occur separately, and any process responsible for star formation quenching cannot dramatically alter stellar discs. Even for low bulge masses, where the colour-density relation is strongest, the change in disc colour with increasing density is small; ( g -r ) disc increases by ∼ 0 . 05 mag over two decades in projected galaxy number density. This colour change is not enough to explain the colour-density relation for the entire galaxy population, which is twice as steep. Therefore, the correlations between stellar mass and density and between stellar mass and galaxy colour contribute significantly to the colour-density relation for classical bulge+disc galaxies. We expect processes internal to galaxies, which may depend on stellar mass, are at least as important in moderating star formation as external, environment-related, processes. The colour-density relation for discs is not linear. By dividing the sample into groups with more or fewer than 20 members, we show that the relation between density and disc colour disappears for galaxies in rich groups, regardless of bulge mass. This is in agreement with results that star formation is quenched well before galaxies enter massive clusters (G'omez et al. 2003; Cooper et al. 2006; McIntosh et al. 2004). It also puts limits on the physical processes responsible for stopping star formation. For example, ram-pressure stripping requires cluster-scale velocities and is unlikely to account for star formation cessation in smaller groups. Strangulation, on the other hand, is effective at relatively low densities and is, therefore, a possible cause of star formation quenching in discs. Although our analysis is not sensitive to colour gradients in discs, changes in these gradients as a function of environment may put additional constraints on the physical processes responsible for disc fading as a function of local density (see Roediger et al. 2011). After determining that star formation quenching occurs outside rich groups, we examine trends in bulge and disc properties within rich groups. Instead of local projected density, we use the group crossing time, t cross , as a measure of environment. Since t cross depends on distance to the group centre, we restrict our sample to 'round' groups, which have a smooth angular distribution of galaxies. Even for these groups, the luminosity-weighted group centre may not be physically relevant; it is not unusual for the most massive galaxy to be offset from the group centre (Skibba et al. 2011; George et al. 2012). The average crossing time for galaxies in this restricted sample is roughly one third of a Hubble time, suggesting most of these galaxies have already orbited their host group at least once. We find no statistically significant correlations with disc colour and t cross , but we do find moderately significant (2 -3 σ ) correlations between disc-to-total mass ratio ( D/T ) and t cross , and between disc scale length and t cross , at fixed bulge mass. The trends in D/T are not a result of correlations between group richness and t cross , or between richness and D/T . In addition, the correlations are also present at the same significance if we replace t cross with local projected density. Like the changes in disc colour as a function of projected density, the changes in disc mass as a function of t cross are small; the median D/T decreases by less than 10 per cent from the largest to the smallest crossing times. None the less, we infer that bulge+disc galaxies do undergo morphological transformations in large groups, but they do not undergo star formation quenching at the same time. This is direct evidence for the separation of morphological transformation and star formation quenching. The above results demonstrate that environment has two distinct effects on the discs surrounding classical bulges. First, disc star formation is truncated in (relatively) poor groups, leading to the colour-density relation for discs (Fig. 6). Although gas-stripping requires relatively high velocities, tidal interactions and heating by the intra-group median may remove a disc's outer halo gas supply (strangulation), thus quenching star formation over several gigayears (Larson et al. 1980). This preprocessing of galaxies in small groups has been suggested before as the origin of S0 galaxies (e.g. Dressler et al. 1997). The second effect is a morphological transformation. As these quenched galaxies enter higher density environments in larger groups, the stellar disc is disrupted over several orbits, leading to the observed D/T -t cross correlation. This disruption may be caused by galaxy harassment (Moore et al. 1996), which is most effective for galaxies in high density regions with short crossing times. Although environment does affect discs around classical bulges, these effects are small. Therefore, while statistically significant, the changes in disc colour with density and disc mass with crossing time are insufficient to explain the full range of the colour-density relation and morphologydensity relation. Furthermore, since the changes are small, present-day local density cannot be the determining factor in whether or not a classical bulge has a disc. There must be other processes, unrelated to present-day environment, which regulate disc formation around classical bulges. None the less, because classical bulges and ellipticals of the same mass seem to have the same formation history, the processes that regulate disc formation are probably external to the galaxy. By only examining trends with density at fixed bulge stellar mass, we cannot explore trends in bulge mass with density. Environmental processes, such as ram pressure stripping, may drive bulge growth (e.g. Tonnesen & Bryan 2009). We plan to explore the effects of environment on bulges at fixed total stellar mass in a later paper. One way to further explore disc formation around classical bulges would be to look for evolution in discs around classical bulges as a function of redshift. In this paper, we present correlations between disc properties with environment in the local universe. In drawing conclusions, we have assumed these correlations are signatures of the evolution of bulges and discs as galaxies move from low density environments to high density environments. However, these conclusions need to be supplemented with observations of bulges and discs at higher redshifts. The comparison data needed for such studies is easily available from space-based optical and near-infrared surveys. For example, using data from the CANDELS survey, Bruce et al. (2012) presents bulge+disc decompositions for massive galaxies beyond z = 1. Careful comparisons of bulges and discs at different redshifts and in different environments will better constrain what effect environment has on the evolution of galaxies and their component bulges and discs.", "pages": [ 18, 19 ] }, { "title": "ACKNOWLEDGMENTS", "content": "CNL is supported by NSF grant AST0908368. This work makes extensive use of data from SDSSIII. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/. SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSSIII Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, University of Cambridge, University of Florida, the French Participation Group, the German Participation Group, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Labora- tory, Max Planck Institute for Astrophysics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.", "pages": [ 19, 20 ] }, { "title": "REFERENCES", "content": "Brinchmann J., Ellis R. S., 2000, ApJ, 536, L77 Bruce V. A., Dunlop J. S., Cirasuolo M., McLure R. J., Targett T. A., Bell E. F., Croton D. J., Dekel A., Faber S. M., Ferguson H. C., Grogin N. A., Kocevski D. D., Koekemoer A. M., Koo D. C., Lai K., Lotz J. M., McGrath E. J., Newman J. A., van der Wel A., 2012, preprint, (arXiv:1206.4322) Butcher H., Oemler Jr. A., 1978, ApJ, 219, 18 Cooper M. C., Newman J. A., Croton D. J., Weiner B. J., Willmer C. N. A., Gerke B. F., Madgwick D. S., Faber S. M., Davis M., Coil A. L., Finkbeiner D. P., Guhathakurta P., Koo D. C., 2006, MNRAS, 370, 198 Cooper M. C., Tremonti C. A., Newman J. A., Zabludoff A. I., 2008, MNRAS, 390, 245 De Propris R., Colless M., Peacock J. A., Couch W. J., Driver S. P., Balogh M. L., Baldry I. K., Baugh C. M., Bland-Hawthorn J., Bridges T., Cannon R., Cole S., Collins C., Cross N., Dalton G., Efstathiou G., Ellis R. S., et al., 2004, MNRAS, 351, 125 Dressler A., 1980, ApJ, 236, 351 Dressler A., Lynden-Bell D., Burstein D., Davies R. L., Faber S. M., Terlevich R., Wegner G., 1987, ApJ, 313, 42 Dressler A., Oemler Jr. A., Couch W. J., Smail I., Ellis R. S., Barger A., Butcher H., Poggianti B. M., Sharples R. M., 1997, ApJ, 490, 577 Elmegreen B. G., Elmegreen D. M., Fernandez M. X., Lemonias J. J., 2009, ApJ, 692, 12 ner G. A., 2006, MNRAS, 369, 1419 Strateva I., Ivezi'c ˇ Z., Knapp G. R., Narayanan V. K., Strauss M. A., Gunn J. E., Lupton R. H., Schlegel D., Bahcall N. A., Brinkmann J., Brunner R. J., Budav'ari T., et al., 2001, AJ, 122, 1861 Tanaka M., Goto T., Okamura S., Shimasaku K., Brinkmann J., 2004, AJ, 128, 2677 Tasca L. A. M., Kneib J.-P., Iovino A., Le F'evre O., Kovaˇc K., Bolzonella M., Lilly S. J., Abraham R. G., Cassata P., Cucciati O., Guzzo L., Tresse L., Zamorani G., Capak P., Garilli B., et al., 2009, A&A, 503, 379 Thomas D., Maraston C., Bender R., Mendes de Oliveira C., 2005, ApJ, 621, 673 Thomas D., Maraston C., Schawinski K., Sarzi M., Joo S.J., Kaviraj S., Yi S. K., 2007, in Vazdekis A., Peletier R. F., eds, IAU Symposium Vol. 241 of IAU Symposium, Environment and the epochs of galaxy formation in the SDSS era. pp 546-550 Tonnesen S., Bryan G. L., 2009, ApJ, 694, 789 Trager S. C., Faber S. M., Dressler A., 2008, MNRAS, 386, 715 Trager S. C., Faber S. M., Worthey G., Gonz'alez J. J., 2000, AJ, 120, 165 Tremonti C. A., Heckman T. M., Kauffmann G., Brinchmann J., Charlot S., White S. D. M., Seibert M., Peng E. W., Schlegel D. J., Uomoto A., Fukugita M., Brinkmann J., 2004, ApJ, 613, 898 Treu T., Ellis R. S., Kneib J.-P., Dressler A., Smail I., Czoske O., Oemler A., Natarajan P., 2003, ApJ, 591, 53 Tuffs R. J., Popescu C. C., Volk H. J., Kylafis N. D., Dopita M. A., 2004, A&A, 419, 821 van den Bosch F. C., Aquino D., Yang X., Mo H. J., Pasquali A., McIntosh D. H., Weinmann S. M., Kang X., 2008, MNRAS, 387, 79 Weinmann S. M., van den Bosch F. C., Yang X., Mo H. J., 2006, MNRAS, 366, 2 Weinzirl T., Jogee S., Khochfar S., Burkert A., Kormendy J., 2009, ApJ, 696, 411 Whitmore B. C., Gilmore D. M., 1991, ApJ, 367, 64 Worthey G., 1994, ApJS, 95, 107 Wyse R. F. G., Gilmore G., Franx M., 1997, ARA&A, 35, 637 Yang X., Mo H. J., van den Bosch F. C., Pasquali A., Li C., Barden M., 2007, ApJ, 671, 153 Zabludoff A. I., Mulchaey J. S., 1998, ApJ, 496, 39", "pages": [ 20, 22 ] }, { "title": "APPENDIX A: CLASSIFICATION OF GALAXIES", "content": "Below, we describe the galaxy classification scheme briefly outlined in § 2.1.1. We explain the classification of each type of galaxy, paying special attention to the probabilistic separation of ellipticals, classical bulge+disc galaxies, and pseudo-bulge+disc galaxies. r bulge component. These galaxies account for 37 per cent of the L12 sample (26 , 882 galaxies). These galaxies are not strictly bulge-less, but include galaxies with bulges too small to detect using bulge+disc decompositions. Fig. A1 shows that these galaxies are predominately intrinsically (and apparently) faint, as expected for disc galaxies with, at most, very small bulges. Because we use a bright ( M r < -19 . 77) subset of galaxies from L12, bulge-less galaxies are a smaller fraction (28 per cent) of our sample than of the whole L12 sample. (ii) Ellipticals and red disc galaxies: Next, we select a sample of elliptical galaxies. As above, galaxies with ( B/T ) r > 0 . 9 are assumed to be disc-less elliptical galaxies. As discussed in L12 (see also Allen et al. 2006), 2dimensional bulge+disc models of elliptical galaxies usually consist of a de Vaucouleurs profile and a low surface brightness 'disc'. This model 'disc' component has several possible origins: the outer halo of ellipticals, a S'ersic index larger than 4, and/or inadequate sky subtraction around bright galaxies in SDSS. This model 'disc' component makes elliptical galaxies indistinguishable from dustless, face-on, red classical bulge+disc galaxies based on photometry alone. Therefore, we must use other information to separate red ellipticals from face-on red classical bulge+disc galaxies. Below, we describe a method for separating face-on bulge+disc galaxies from ellipticals. We only apply this separation to galaxies with red colours for both the model de Vaucouleurs bulge and model disc: u -r > 2 . 22 (Strateva et al. 2001). This colour cut ensures that the outer exponential halo around elliptical galaxies will not be star-forming. However, we cannot identify blue ellipticals in our sample using this colour cut. Despite their similarities, we can separate red bulge+disc galaxies and ellipticals based on the statistics of the modelled disc axis ratios, q d . If the discs are randomly oriented, then q d should be uniformly distributed for values larger than the disc scale height to scale length ratio, q z . The probability density function is given by f ( q d ) = q d / √ ( q 2 d -q 2 z )(1 -q 2 z ). We use q z = 0 . 1, which is smaller than the average value q z = 0 . 14 ± 0 . 04 (Kregel et al. 2002). However, any q z less than 0.2 has a negligible effect on the results. If the bulge+disc models are applied to ellipticals, then the distribution of q d will be skewed to higher values, mimicking face-on discs. Although it is impossible to distinguish a dustless, face-on classical bulge+disc galaxy from an elliptical, we can assign each galaxy a probability of being an elliptical or bulge+disc galaxy based on its model q d . Fig. A2 shows the statistical separation of red bulge+disc and ellipticals for galaxies divided into three stellar mass bins. The upper panels show the distribution of q d for galaxies with red ( u -r > 2 . 22) model de Vaucouleurs bulge and exponential disc colours, and 0 . 1 < ( B/T ) r < 0 . 9. Each distribution of axis ratios has two contributions, one from disc galaxies, which follows the distribution for randomly oriented discs, and one from ellipticals, which we model as a Gaussian distribution in q d with an arbitrary centroid and width. We separately fit each distribution of q d in Fig. A2 with a linear combination of these two probability density functions. Since small measured axis ratios are often due to poor fits, we only use galaxies with q d > 0 . 25 for the fitting. The fractional contribution from each function gives the probability that a galaxy with a given q d is an elliptical or a red classical bulge+red disc galaxy (lower panels Fig. A2). For low mass galaxies, the distribution of axis ratios does not show any contribution from ellipticals. At any stellar mass, a small value for q d implies a galaxy has a real disc. For 0 . 6 /lessorsimilar q d /lessorsimilar 0 . 8, the probability that a galaxy is an elliptical rises to ∼ 50 per cent for the highest mass galaxies. This method of distinguishing between bulge+disc galaxies and ellipticals does not allow us to definitively assign a galaxy to either category, but we can still examine the properties of the whole sample, simply by weighting each galaxy by its likelihood of being an red bulge+red disc galaxy or an elliptical. Counting all galaxies with B/T > 0 . 9 as ellipticals, we find that 9 . 4 per cent (6800 galaxies) of the L12 sample are red bulge+red disc galaxies and 7 . 3 per cent (5300 galaxies) are ellipticals. When selecting galaxies with red bulge and disc components for Fig. A2, we first correct the bulge and disc colours for inclination in order to eliminate edge-on, dusty galaxies. However, if the inclination corrections are too large, there will be fewer galaxies with small q d , and the fraction of ellipticals will be enhanced. Therefore, we iteratively adjust the inclination corrections in u and r such that, after identifying elliptical galaxies, there is no residual trend in corrected disc colour with disc inclination for red bulge+red disc galaxies. The final adjustments make the inclination corrections in u and r smaller than those in L12, but the adjustments are minimal. They change the number of ellipticals in sample by less than 1 per cent of the total sample (500 galaxies). If we used no inclination correction, the number of ellipticals in the sample would decrease by 1000 galaxies. (iii) Classical and pseudo-bulges: After identifying ellipticals and red classical bulge+disc galaxies, we classify the remaining galaxies as either classical bulge hosts or pseudobulge hosts. We separate classical bulges and pseudo-bulges based on the age of the central stellar populations. Fisher (2006) show that pseudo-bulges selected by morphology have higher central star formation rates than classical bulges, in agreement with the notion that pseudo-bulges form via secular processes in discs which drive gas inwards and enhance the central star formation (Kormendy & Kennicutt 2004). Fig. A3 shows the distribution of the 4000 ˚ A break n strength for the classical and pseudo-bulge galaxies. These values are taken from the MPA/JHU catalogue of measured spectroscopic quantities from SDSS (Tremonti et al. 2004; Aihara et al. 2011) 5 . The value plotted is the narrow definition of the 4000 ˚ A break, D n (4000) (Balogh et al. 1999). The SDSS spectrograph uses 3 arcsec fibres; two-thirds of the galaxies shown have a bulge-to-total flux ratio within 3 arcsec larger than 0 . 5. Therefore, the fibre spectroscopic quantities are typically dominated by the stellar light from the bulge. The right panel of Fig. A3 shows the distribution of disc axis ratios for classical and pseudo-bulge host discs. This is one test of our separation of pseudo-bulges and classical bulges, as we expect the distributions to match the flat distribution for randomly oriented discs. Although the distribution of q d for classical bulge host discs together with pseudo-bulge host discs is flat, the separate distributions for classical and pseudo-bulge hosts are not. The pseudo-bulge host discs have axis ratios which are too large. This is expected, since pseudo-bulges are typically flattened, making them difficult to detect in edge-on discs. Edge-on pseudo bulge hosts are more likely to be considered bulge-less galaxies. The slight excess of low axis ratio discs around classical bulges is due to the inclusion of inclined, but dust-poor, red bulge+red disc galaxies. When selecting red bulge+disc galaxies and ellipticals above, we included an inclination correction, although we expect that some bulge+disc (and most ellipticals) will be essentially dust-free. Correcting these galaxies for inclination shifts their colours bluewards, removing them from the red sample. This is especially true for highly-inclined (low q d ) galaxies, where the colour corrections are largest. Therefore, the excess of small q d classical bulges galaxies consists mainly of red classical bulge+red disc galaxies. Since all these galaxies host classical bulges, it is not important to accurately separate them based on colour. The physical origins of classical and pseudo-bulges are very different. Classical bulges are thought to have formed by the same mechanisms as ellipticals, while pseudo-bulges arise from secular processes in discs (disc instabilities, bars, etc) (e.g. Kormendy & Kennicutt 2004). Furthermore, the absolute magnitude distribution of pseudo-bulge galaxies in Fig. A1 is more similar to that of bulge-less and unclassifiable galaxies than it is to classical bulge hosts. In this work, we will consider galaxies with pseudo-bulges as a subset of bulge-less disc galaxies. Because pseudo-bulges galaxies are relatively faint, they are a small fraction (8 per cent) of the bright ( M r /lessorsimilar -19 . 77) subsample from L12 used below. Therefore, the exclusion of pseudo-bulge hosts from bulge+disc galaxies does significantly affect our results. (iv) Unclassifiable galaxies: The unclassifiable galaxies in our sample are modelled by a single S'ersic profile. These are galaxies which are not well fit by any of the other models, i.e. they have model bulges which are larger than their discs, or they have fluxes in g , r , or i consistent with zero for the bulge or disc (or both). They are given a probability of being fit by a S'ersic profile equal to unity. Of the 72658 galaxies in L12, 12459 (17 per cent) are deemed unclassifiable. The average unclassifiable galaxy is 0 . 4 magnitudes fainter than the average galaxy in the sample (see Fig. A1). Seventy-five percent of unclassifiable galaxies have a S'ersic index less than 2 . 3 and the same fraction lie in the blue cloud ( u -r < 2 . 22). These galaxies are probably disc-like irregulars, which are unlikely to have a well-defined bulge and disc. The remaining 25 per cent of unclassifiable galaxies are mostly merger remnants, starbursts, and other complicated morphologies. None the less, because the majority of unclassifiable galaxies exhibit disc-like properties, we group them with other bulge-less galaxies described above. Since the sample used here is a bright subsample of the L12 sample, the unclassifiable galaxies make up a smaller fraction (less than 10 per cent) of the bright subsample than of the whole L12 sample.", "pages": [ 22, 23, 24 ] } ]

[ { "title": "ABSTRACT", "content": "To evaluate the capability of numerical cloud forecast as a meteorological reference for astronomical observation, we compare the cloud forecast from NCEP Global Forecast System (GFS) model for total, layer and convective cloud with normalized satellite observation from the International Satellite Cloud Climatology Project (ISCCP), for the period of July 2005 to June 2008. In general, the model forecast is consistent with the ISCCP observation. For total cloud cover, our result shows the goodness of the GFS model forecast with a mean error within ± 15% in most areas. The global mean probability of < 30% forecast error (polar regions excluded) declines from 73% to 58% throughout the 180 h forecast period, and is more skilled than ISCCP-based climatology forecast up to τ ∼ 120 h . The comparison on layer clouds reveals a distinct negative regional tendency for low cloud forecast and a questionable positive global tendency for high cloud forecast. Fractional and binary comparisons are performed on convective cloud forecast and revealed the GFS model can only identify less than half of such cloud. In short, our result suggests that the GFS model can provide satisfactory worldwide total cloud forecast up to a week ahead for observation scheduling purpose, but layer and convective cloud forecast is less reliable than the total cloud forecast. Key words: atmospheric effects, site testing.", "pages": [ 1 ] }, { "title": "Q.-Z. Ye /star , 1 , 2 and S.-S. Chen 1 , 3", "content": "Accepted 1970 January 1. Received 1970 January 1; in original form 1970 January 1", "pages": [ 1 ] }, { "title": "1 MOTIVATION", "content": "The most important prerequisite of most successful astronomical observation, mainly optical observation, is no doubt a cloudless sky (see the story of Guillaume Le Gentil in the 1761/69 Transit of Venus, cf. Sawyer Hogg 1951, for a rather unlucky example). However, cloud cover forecast had been difficult for a long time as limited by theoretical understanding of mesoscale circulation (i.e. modeling) and computation ability. Numerical simultaneous cloud cover forecast for astronomical observation had not been practically useful until the very end of 20th century. In the meteorological context, cloud plays a key role in radiation balance of the Earth, and is widely accepted to be the main source of uncertainty of global weather predictions (cf. Stubenrauch et al. 1999; Stephens 2005). It gathered much attention from atmospheric sciences community, both from modeling group and from observational group. The attempt to 'parameterize' cloud activity started in the 1980s (cf. Sundqvist et al. 1989, and references therein). Several cloud schemes have been proposed since then, followed by ground- and space-based evaluation in aim for their refinement (e.g. Hinkelman et al. 1999; Luo et al. 2005; Yang et al. 2006). Although many amateur and professional observatories have been taken advantage of open access to the outputs of major numerical weather models for a decade, the reliability of such forecast is poorly understood. The meteorology community is mostly interested in particular mesoscale events and/or particular regions, and pays relatively little attention to the model performance over broader environment; while there are only two studies from the astronomy community that have investigated this topic: Erasmus & Sarazin (2001), which suggested that only 15-25% of cloudy nights at the European Southern Observatory sites could be identified by the European Centre for Medium-Range Weather Forecasting (ECMWF) model; and our earlier study (Ye 2011), which suggested a high detection rate accompanied with a moderate false alarm rate from the NCEP Global Forecast System (GFS) model, based on the cloud observations from several astronomical observatories. Even so, as the two studies are both limited at spatial densities and scales, investigations of numerical cloud forecasts over global scale are still lacking. This study is therefore carried out in aim to assess the cloud forecast ability of a global numerical model to provide insight on its reliability as a reference for astronomical observation. In order to do this, we need: i) output from global numerical model as forecast data, and ii) observational data with appreciable temporal and spatial coverage. The selection and reduction of such data will be discussed in the following section.", "pages": [ 2, 3 ] }, { "title": "2.1 Modeling Data", "content": "The ECMWF and GFS models mentioned in § 1 are among the major global numerical weather models used for daily weather forecast. In this study, we follow our earlier study and use the GFS model as well. Firstly, we retrieve the GFS data in grid 003 (1 · × 1 · grid) for the period of July 2005 to June 2008 from the National Operational Model Archive & Distribution System (NOMADS; see Rutledge et al. 2006). The data is in 3-hourly interval for forecast lead time ( τ ) up to 180h and is produced four times per day at 00Z, 06Z, 12Z and 18Z (we refer each of them as one 'initialization' hereafter). The data is then weightaveraged into the spatial resolution of 2 . 5 · to match the definition of the observational data that to be described in § 2.2. The forecast cloud fraction in decimals, C , is computed using the Xu-Randall cloud scheme (Xu & Randall 1996): where RH is the environmental relative humidity, q l is the liquid water mixing ratio, q s is the saturation specific humidity, and k 1 = 0 . 25, k 2 = 100, and k 3 = 0 . 49 are empirical coefficients. q s is calculated with respect to water phase or ice phase and environmental temperature. We will deal with five types of cloud cover in this study: total cloud cover, which accounts for cloud cover over the entire atmospheric column and is most closely related to observational astronomy; layer cloud cover (low, mid, and high level), that divides according to the cloud-top pressure (680hPa, 440hPa, and < 440hPa), which is particularly useful for observatories that mostly affected by a certain cloud type (e.g. high cloud for high altitude observatories); and convective cloud, which is mostly associated with convective weather that can be an unwarned threat for astronomical observation. The GFS model divides the whole atmospheric column into 64 sub-layers for simulation, and cloud cover is derived under the assumption that clouds in all sub-layers for the cor- c © 1970 RAS, MNRAS 000 , 1-16", "pages": [ 3 ] }, { "title": "4 Q.-Z. Ye & S.-S. Chen", "content": "responding layer are maximally randomly overlapped (Yang et al. 2006). The exception is convective cloud, which is derived based on the method proposed by Pan & Wu (1995).", "pages": [ 4 ] }, { "title": "2.2 Observational Data", "content": "Rather than using the 'traditional' surface observation, we decide to use calibrated satellite observation in this study. The reason is that standard cloud observation is not practiced by a number of surface meteorological stations, as such observation needs to be carried out by expertise observers, therefore qualified observations are mostly limited to inhabited area with dense population, of which is avoid by most astronomical observatories. Satellite observation, on the other hand, has a better temporal and spatial coverage. Since it is carried out by robotic observers and is reduced following identical algorithms, it is also easier to determine the scale of uncertainty. A good source of such data is the International Satellite Cloud Climatology Project (ISCCP; cf. Schiffer & Rossow 1983; Rossow & Schiffer 1999), which we will use for our study. We retrieve the 3-hourly ISCCP D1 data from the ISCCP database for the same time period as the GFS data. The D1 dataset is in a spatial resolution of 280km and includes cloud cover data for total, low, mid and high cloud as well as convective cloud. The data is determined from raw satellite observation of cloud top pressure and optical thickness, and is in the same definition of the GFS data that we used. We then transform the ISCCP data from equal-area grid into equal-angle grid following the method by Rossow et al. (1996), to match the projection setup of the GFS data. The uncertainty of ISCCP data is estimated to be ∼ 0 . 15 for individual cases and less than ∼ 0 . 05 for 30-day means (Schiffer & Rossow 1983; Rossow & Schiffer 1999). However, additional studies did reveal some observational tendencies for each cloud type: CCP total clouds for polar regions are 0.15-0.25 too low in summer and 0.05-0.10 too high in winter; and As noted by Curry & Ebert (1992); Rossow et al. (1993), the ISCCP observations over the polar regions suffered from strong seasonal tendencies due to low visual and thermal contrast between surface and clouds; therefore, we will focus at the region between 60 · S and 60 · N in our study, despite we will still include results from polar regions in figures.", "pages": [ 4, 5 ] }, { "title": "3.1 Evaluation Methodology", "content": "We generate over 1 million GFS-ISCCP data pairs of every initialization, forecast time point and cloud type for the entire time period of interest. To get a more objective evaluation of the model forecast skill for total cloud cover, we compose three additional models that are to be compared with the ISCCP data: Limiting by computational resource, we randomly choose July 2006 for such comparisons. We will show that monthly variation of forecast error is not significant in the period of study, so the July 2006 result is representative. We use a different evaluation scheme for convective cloud. Although we are dealing with the term 'convective cloud cover' or 'fractional convective cloud', there is virtually no scientific/observational meaning in this term. It is due to the small scales of most convective clouds comparing with the spatial resolution of global model or satellite camera (which are c © 1970 RAS, MNRAS 000 , 1-16 120 mostly at tens of kilometers), so such cloud can only be represented in fractional numbers in model outputs or observations. Therefore, in addition to fractional comparisons, we also binary degenerate the modeling and observational data, so binary statistical indicators can be used to assess the forecast skill (see § 3.4.2).", "pages": [ 5, 6 ] }, { "title": "3.2.1 General Forecast Accuracy", "content": "In Table 1 and Table 2, our analysis shows that the global mean fc -obs (excluding polar regions) varies from -6.43% to -8.93% depending on initializations. Negative values are mostly contributed by low fc -obs values over ocean, especially western coastal regions at mid latitude. The forecast over land matches the observation well (between -0 . 5% and +1% throughout the forecast period), but it may be positively biased according to the suggested underestimation of ISCCP data over land by Rossow et al. (1993). Meanwhile, the difference between each initialization is not significant comparing to the fc -obs tendency (only ∼ 3% or less). We also find the negative tendency over the southern hemisphere is stronger than that of northern hemisphere. Figure 2 shows the probability of < 30% forecast error gradually decays from ∼ 73% at τ = 3 h to ∼ 58% at τ = 180 h . One may argue that ± 30% has covered 3/5 of the possible range (0% -100%) which may lift the probability; however, cloud cover is not uniformly distributed: in a significant number of occasions, the cloud cover is either close to 0% or 100%, so ± 30% is a fairly reasonable constraint. We will also show that the GFS model is solidly better than the random-guess model in later section.", "pages": [ 6, 7 ] }, { "title": "3.2.2 Daily and Seasonal Variation", "content": "Since most astronomical optical observations are conducted in night hours, we are interested in examining the daily variation of forecast accuracy. We divide the entire globe into 24 time", "pages": [ 7 ] }, { "title": "8 Q.-Z. Ye & S.-S. Chen", "content": "Month zones with equally longitudinal spacing, and average the fc -obs series with respect to local hour in each time zone. As illustrated in Figure 3, the daily fc -obs value varies between -15% to -5% for the entire globe (excluding polar regions) and ocean, but for land it varies from -13% in the morning to +5% to +10% in night hours. However, considering a ∼ 7% underestimation of ISCCP data over land in day and ∼ 12% underestimation in night (Rossow et al. 1993), the actual fc -obs could be as low as -25% in day but near 0% in night. ISCCP tendency can be as large as 9%, with a reverse maximum-minimum setup ( -6% in summer and -15% in winter) than Figure 4. In such a case, it is possible that the GFS model forecast is closer to actual situation in summer, rather than in winter as suggested in the figure. We also attempt to identify any annual variation (Figure 5), but no significant feature can be noted, possibly due to insufficient temporal coverage.", "pages": [ 8, 9 ] }, { "title": "3.2.3 Comparison with Climatology, Randomize and Persistence Models", "content": "As described before, we generated three additional models and will use them to compare with the ISCCP data for July 2006. The result is shown in Figure 6 and, in our opinion, is comparable with Figure 2 despite different temporal coverage, as we have shown the seasonal variation to be insignificant comparing to overall fc -obs tendency. We can see the GFS model is superior than all three other models in most time points. Statistically, the persistence model is best of all for τ < 6 h , but this is not meaningful as the GFS model data is not available after approximately 4-5h of the respective initialization time due to computational layover. We note that the ISCCP climatology model becomes better than the GFS model after τ ∼ 120 h , and that both models are more skilled than random guess at all times. From this result, we can conclude that the GFS model is skilled, and performs better than the ISCCP climatology model until τ ∼ 120 h .", "pages": [ 9 ] }, { "title": "3.3 Layer Cloud", "content": "The evaluation result (Table 3, Figure 7, Figure 8 and Figure 9) has revealed the global overestimation of high cloud from the GFS model, but this result is affected by underestimation of similar scale in ISCCP data (Rossow & Schiffer 1999) and is questionable. We can also identify underestimation of low cloud off the west coast of major continents at mid latitude, which seems to be the major contribution to the underestimation of total cloud in these regions. At high latitude, the model tends to overestimate the low cloud. The probability of < 30% error forecast for low cloud is significantly lower than other two clouds (Figure 8), but does not appear to affect the total cloud at a small τ . This behavior, together with the unexpected negative fc -obs for total cloud (while fc -obs for layer clouds are mostly positive), could indicate that the assumption of maximally randomly overlaps (see § 2.1) between each cloud layers may not apply at all times.", "pages": [ 10 ] }, { "title": "3.4.1 Fractional Evaluation", "content": "Direct fractional comparison (Table 4 and Figure 10) shows that model overestimation only occurs in tropical area (within 15 · latitude in both hemispheres), while moderate to strong underestimation (mostly 10 -20%) occurs in mid-to-high latitude area. Considering the fact that convective weather is frequent in tropic area throughout the year but is relatively rare in mid-to-high latitude area, we may conclude that the current model misses and/or underestimates a significant fraction of convective cloud in mid-to-high latitude area, while it tends to overestimate the number and/or intensity of convective cloud in tropical area.", "pages": [ 10 ] }, { "title": "12 Q.-Z. Ye & S.-S. Chen", "content": "90 60 30", "pages": [ 12 ] }, { "title": "3.4.2 Binary Evaluation", "content": "An alternative way to evaluate the GFS convective cloud forecast is to degenerate the forecast and observation into binary value, so we can focus on the occurrence of such cloud rather than a mixture of occurrence and intensity. As discussed in earlier section, we set a cut-off limit at 1% for convective cloud fraction to put both the modeling and observational data into the categories of 'convective cloud' and 'no convective cloud'. We use the following statistical indicators for evaluation: Proportion of Perfect Forecasts (PPF), Probability of Detection (POD), False Alarm Rate (FAR) and Frequency Bias Index (FBI). In the following expressions, H indicates 'hits' (forecasted and observed), F indicates 'false alarms' (forecasted but not observed), M indicates 'missed' (not forecasted but observed), and Z indicates to not-forecasted and not-observed events. The result is shown in Figure 11. As we have shown that tropical overestimation only composed a relatively small fraction in the sample, the setup of Figure 11 more or less represents the situation of mid-to-high latitude area. This is affirmed by the FBI value which lies way below 1, suggesting a global underestimation of convective cloud. The PPF varies around 0.6 and creates a decent picture of the forecast ability, but the rare occurrence of convective cloud in most areas will lead to a significant fraction of PPF being contributed from 'Z' events (not-forecasted and not-observed events), so we must take POD and FAR into account for a unbiased view. From POD we notice that only slightly less than half of the convective cloud can be detected by the model. Since convective cloud is © 1970 RAS, MNRAS 000 , 1-16 c", "pages": [ 12, 13 ] }, { "title": "14 Q.-Z. Ye & S.-S. Chen", "content": "most commonly seen in tropical area, the POD for mid-to-high latitude area must be lower. However, globally speaking, the model is still well skilled as the FAR is about 2 times lower than the POD.", "pages": [ 14 ] }, { "title": "4 CONCLUDING REMARKS", "content": "To study the reliability of cloud forecast from the GFS model as a reference of astronomical observations, we analyzed a 3-year sample composed with the GFS modeling data and satellite observation data from ISCCP. The results are summarized as follow: In all, we observed a good overall consistency between the GFS model forecast and ISCCP observation throughout the time period of interest. For total cloud cover, our result suggested a satisfactory performance of the GFS model for the need of observation scheduling up to a week ahead. However, for layer and convective cloud, which can be considerably important for observatories located in certain environment (for example, high cloud for high altitude observatories), the success rate of the GFS model is relatively less satisfying than that of total cloud forecast.", "pages": [ 14, 15 ] }, { "title": "ACKNOWLEDGMENT", "content": "We would like to thank several anonymous reviewers as well as Fangling Yang and William Rossow for their constructive comments and discussions that help us to make significant improvements of this work. We also would like to thank Chen Junwen from Atmospheric Exploration Laboratory (EESAEL) at Sun Yat-sen University, Cui Chenzhou (National Astronomical Observatory), Lin Qing and Tang Haiming (Shanghai Astronomical Observatory), for providing assistance on computational resources. We would like to especially thank the 'meteo cat' who had visited the EESAEL a few times and brought some joys to us. The data analysis of this work was accomplished with the computer resource at EESAEL. The GFS data were obtained from NOMADS (National Operational Model Archive and Distribution System) at NOAA/NESDIS/NCDC. The ISCCP D1 data were obtained from the International Satellite Cloud Climatology Project data archives at NOAA/NESDIS/NCDC Satellite Services Group, [email protected], on July to October, 2010.", "pages": [ 15 ] }, { "title": "REFERENCES", "content": "Curry J. A., Ebert E. E., 1992, Journal of Climate, 5, 1267 Erasmus D. A., Sarazin M. S., 2001, in Russell J. E., Schaefer K., Lado-Bordowsky O., eds, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol. 4168 of Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Forecasting precipitable water vapor and cirrus cloud cover for astronomical observatories: satellite image processing guided by synoptic model dissemination data. pp 317-328 Hahn C. J., Warren S. G., London J., 1995, Journal of Climate, 8, 1429 Hinkelman L. M., Ackerman T. P., Marchand R. T., 1999, J. Geophys. Res., 104, 19535 Luo Y., Krueger S. K., Moorthi S., 2005, Journal of the Atmospheric Sciences, 62, 1428 Pan H.-L., Wu W.-S., 1995, NMC Office Note, 409, 40 Rossow W. B., Schiffer R. A., 1999, Bulletin of the American Meteorological Society, 80, 2261 Rossow W. B., Walker A. W., Beuschel D. E., Roiter M. D., 1996, WMO/TD, 737, 115 c © 1970 RAS, MNRAS 000 , 1-16 16 Q.-Z. Ye & S.-S. Chen Rossow W. B., Walker A. W., Garder L. C., 1993, Journal of Climate, 6, 2394 Rutledge G. K., Alpert J., Ebisuzaki W., 2006, Bulletin of the American Meteorological Society, 87, 327 Sawyer Hogg H., 1951, JRASC, 45, 173 Schiffer R. A., Rossow W. B., 1983, Bulletin of the American Meteorological Society, 64, 779 Stephens G. L., 2005, Journal of Climate, 18, 237 Stubenrauch C. J., Rossow W. B., Cheruy F., Chedin A., Scott N. A., 1999, Journal of Climate, 12, 2214 Sundqvist H., Berge E., Kristj'ansson J. E., 1989, Monthly Weather Review, 117, 1641 Wielicki B. A., Parker L., 1992, J. Geophys. Res., 97, 12799 Xu K. M., Randall D. A., 1996, Journal of the Atmospheric Sciences, 53, 3084 Yang F., Pan H.-L., Krueger S. K., Moorthi S., Lord S. J., 2006, Monthly Weather Review, 134, 3668 Ye Q.-Z., 2011, PASP, 123, 113", "pages": [ 15, 16 ] } ]

[ { "title": "ABSTRACT", "content": "An increasing number of OB stars have been shown to possess magnetic fields. Although the sample remains small, it is surprising that the magnetic and X-ray properties of these stars appear to be far less correlated than expected. This contradicts model predictions, which generally indicate that the X-rays from magnetic stars to be harder and more luminous than their non-magnetic counterparts. Instead, the X-ray properties of magnetic OB stars are quite diverse. τ Sco is one example where the expectations are better met. This bright main sequence, early B star has been studied extensively in a variety of wavebands. It has a surface magnetic field of around 500 G, and Zeeman Doppler tomography has revealed an unusual field configuration. Furthermore, τ Sco displays an unusually hard X-ray spectrum, much harder than similar, non-magnetic OB stars. In addition, the profiles of its UV P Cygni wind lines have long been known to possess a peculiar morphology. Recently, two stars, HD 66665 and HD 63425, whose spectral types and UV wind line profiles are similar to those of τ Sco, have also been determined to be magnetic. In the hope of establishing a magnetic field - X-ray connection for at least a sub-set of the magnetic stars, we obtained XMM-Newton EPIC spectra of these two objects. Our results for HD 66665 are somewhat inconclusive. No especially strong hard component is detected; however, the number of source counts is insufficient to rule out hard emission. longer exposure is needed to assess the nature of the X-rays from this star. On the other hand, we do find that HD 63425 has a substantial hard X-ray component, thereby bolstering its close similarity to τ Sco. Key words: stars: early-type; stars: individual: HD63425, HD66665; stars: magnetic field; X-rays: stars", "pages": [ 1 ] }, { "title": "R. Ignace 1 /star , L. M. Oskinova 2 , and D. Massa 3", "content": "1 Physics & Astronomy, East Tennessee State University, Johnson City, TN, USA 2 Institute for Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany 3 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA 18 August 2018", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "In spite of their apparent simplicity, near main sequence B stars exhibit a range of properties that are not well understood. Among the challenges include: surprisingly low wind mass-loss rates and wind terminal speeds (Petit et al. 2011; Oskinova et al. 2011), a full understanding of the causes and evolution of the Be phenomenon (e.g., Porter 1999; Brown, Cassinelli, & Maheswaran 2008; Townsend, Owocki, & Howarth 2004; Carciofi et al. 2009; Wisniewski et al. 2010), and the nature of magnetism being detected among some B stars (Hubrig et al. 2006; Rivinius et al. 2010; Petit et al. 2011; Oksala et al. 2012; Grunhut et al. 2012). In regard to their mass loss, a number of main sequence B stars display mass-loss rates ˙ M that are an order of magnitude lower than theoretical expectations. Examples include detailed spectral analyses of five B stars described in Oskinova et al. (2011): τ Sco, β Cep, ξ 1 CMa, V2052 Oph, and ζ Cas. Although β Cep is a giant, the other four are B0-2 main sequence stars. In addition, the early BV stars HD 66665 and HD 63425, which are the subject of this paper, were analyzed by Petit et al. (2011) and found to have low ˙ M values, an order magnitude lower than predicted by Vink, de Koter, & Lamers (2000). Notably, the presence of X-rays in their winds played an important, if not central, role in achieving satisfactory fits to observed UV and optical spectra. It seems then that at least some B stars exhibit the same 'weak wind' problem seen in the less luminous O stars (e.g., Martins et al. 2005; Marcolino et al. 2009; Muijres et al. 2012; Lucy 2012; Huenemoerder et al. 2012). Then there is the occurrence, properties, and evolutionary influence of magnetic fields among B stars. Our contribution to this issue has been to study the X-ray emissions from magnetic B stars in order to identify relationships (or the absence thereof) between known magnetic properties and measured X-ray characteristics (Ignace et al. 2010; Oskinova et al. 2011). Our attempt to draw connections between magnetism and Xrays among the B stars was inspired by several by successes in relating X-ray properties to magnetospheric models for early-type stars with strong magnetic fields. Magneto-hydrodynamical simulations of the early O star θ Ori 1 appear capable of matching the observed X-ray variations, both in broad band terms as well as in emission lines (Gagne et al. 2005; ud-Doula 2012). The Bp star σ Ori E has a very strong surface magnetic field of ≈ 20 , 000 G that motivated a semi-analytic approach called the Rigidly Rotating Magnetosphere (RRM) model (Townsend & Owocki 2005). The RRM has been broadly successful in explaining observed H α variations, the polarization light curve, and with extension to time-dependent hydrodynamics, the star's broad X-ray properties (Townsend, Owocki, & ud-Doula 2007). Indeed, the approach has even been able to explain the measured rotational spin-down rate of the star (Townsend et al. 2010). Although MHD simulations still face challenges with producing detailed quantitative matches to observed X-rays of magnetic massive stars (e.g., Naz'e et al. 2010), the models are a work in progress that offer a promising framework in which to interpret the observations. With this framework, it was thought that deeper insights into the relationship between magnetic and X-ray characteristics could be gained through a study of B stars with weaker yet moderately strong surface magnetic fields and lower wind mass-loss rates. To this end, the efforts of the MiMeS (e.g., Grunhut & Wade 2012) and Magori (Scholler et al. 2011) collaborations to detect, characterize, and catalog the magnetic properties of early-type stars have been indispensable. Unfortunately, a clear relation between stellar magnetism and X-ray fluxes has not emerged. Indeed, the apparent absence of expected relationships between magnetic and X-ray properties has been a surprise (Favata et al. 2009; Ignace et al. 2010; Oskinova et al. 2011). Despite the lack of an overall connection, it may be that certain types of magnetic stars do exhibit one. For example, τ Sco is a magnetic star with unusual UV wind lines, and is also notable for having an unusually hard component to its X-ray emission for a massive star, especially for an early B type star that is believed to be single (e.g., Cassinelli et al. 1994; Cohen et al. 2003; Mewe et al. 2003; Ignace et al. 2010). Recently, two other stars, HD 66665 and HD 63425, were identified as having UV wind lines with the same peculiar morphology seen in τ Sco. This motivated Petit et al. (2011) to observe both stars, and both yielded significant positive detections of surface magnetic fields. We refer to Petit et al. (2011) for a discussion and spectral analysis of HD 66665 and HD 63425, and to Oskinova et al. (2011) for a spectral analysis of τ Sco. The question that naturally arises is whether or not HD 66665 and HD 63425 are also hard sources of X-rays like τ Sco. If so, the discovery would produce a rare example among massive stars of a relationship involving stellar magnetism, X-ray emissions, and UV line profile morphology. We report here on data obtained with the XMM-Newton in an effort to characterize the X-ray luminosities and hot plasma temperatures for HD 66665 and HD 63425. Section 2 details the acquisition and reduction of data obtained with the EPIC detectors. Section 3 presents an analysis of the X-ray spectra. An assessment of whether HD 66665 and HD 63425 are indeed hard sources is given in Section 4, followed by concluding remarks in section 5.", "pages": [ 1, 2 ] }, { "title": "2 OBSERVATIONS AND DATA REDUCTION", "content": "We obtained dedicated XMM-Newton observations of HD 63425 and HD66665. Stellar and wind properties of our target stars are given in Table 2. All three (MOS1, MOS2, and PN) European Photon Imaging Cameras (EPICs) were operated in the standard, full- a In the 0.3-7.0 keV band; background subtracted. frame mode and a medium UV filter. A log of observations is shown in Table 2. The data were analyzed using the software SAS 10.0. The time periods when the particle background was high were excluded from the analysis. Both stars were detected by the standard source detection software. The exposure times and EPIC PN count rates for our program stars are given in Table 2. A bright patch of diffuse X-ray emission with diameter of ≈ 4 arcmin is present in the EPIC images of HD 63425. The spectrum of the diffuse emission was found to be well fitted with a two temperature plasma having components kT 1 ≈ 0 . 7 keV and kT 2 ≈ 5 . 4 keV. The X-ray temperature, flux, brightness distribution, and comparison with optical and IR images indicate that this diffuse emission is most likely due to a massive galaxy cluster at z > 0 . 3 (A. Finoguenov, private comm.). The spectrum of HD63425 was extracted from a region with a diameter of ≈ 15 '' . The X-ray background was chosen from a nearby area in the diffuse X-ray source. Thus, it is possible that the hard stellar X-ray emission for HD 63425 is over-subtracted because of the hard background diffuse radiation. Therefore, the X-ray spectrum of HD63425 presented here provides only a conservative estimate of the hottest temperature plasma component. The X-ray point source with the coordinates of HD 66665 is well isolated, and there was no difficulty in obtaining its spectrum using the standard procedure and determining the X-ray background from a nearby region free of X-ray sources.", "pages": [ 2 ] }, { "title": "3 RESULTS", "content": "To analyze the spectra we used the standard spectral fitting software XSPEC (Arnaud 1996). The number of counts per bin in the spectra of HD63425 and HD 66665 is small; therefore, we used the Cashstatistic (Cash 1979) for spectral fitting. Using the neutral hydrogen column density as a fitting parameter does not yield a sensible con- straint on its value; therefore, N H was fixed at its interstellar value (see Tab. 1). Our targets are known magnetic stars, and peculiar abundances are often found in such stars, typically explained as arising from diffusion processes which allow heavier elements to sink in the atmosphere under the influence of gravity, while lighter elements are elevated to the surface by radiation pressure (e.g., Morel et al. 2008). It is usual for a magnetic star to show an overabundance of nitrogen, and sometimes helium. For example, Morel (2011) find that the abundance ratio [N/C] is higher than solar in HD66665, while [N/O] is nearly solar. We are not aware of any abundance studies for HD 63425. The quality of the X-ray spectra of our program stars are not sufficient to constrain abundances. We carried out tests that showed that the overabundance of N by a factor of a few does not significantly change the results of our spectral fits. Therefore, abundances for our two target stars were set to solar values based on Asplund (2009).", "pages": [ 2, 3 ] }, { "title": "3.1 HD63425", "content": "Our XMM-Newton observation detected the X-ray emission from HD63425 for the first time. The 90% confidence range for the unabsorbed X-ray flux, meaning the intrinsic flux of the star after correcting for interstellar absorption, is 1 . 31 -1 . 71 × 10 -13 erg s -1 cm -2 . Assuming a distance of d = 1 . 136 kpc, the X-ray luminosity of HD 63425 is L X ≈ 2 × 10 31 erg s -1 with an error of about 15%; this X-ray luminosity is comparable to the value for τ Sco. The observed EPIC spectra of HD 63425 and the fitted model are shown in Figure 1. A two temperature plasma model can reproduce the observed spectrum quite well (see Tab. 3).", "pages": [ 3 ] }, { "title": "3.2 HD66665", "content": "Our XMM-Newton observation detected the X-ray emission from HD66665 also for the first time. The source has only a modest count rate (see Tab. 2). The unabsorbed X-ray flux is 2 . 0 -4 . 8 × 10 -14 erg s -1 cm -2 . Assuming a distance of d = 1 -2 kpc, the X-ray luminosity of HD 66665 falls in the conservative range of L X ≈ 2 -22 × 10 30 erg s -1 . The EPIC spectra of HD 66665 can be well described using a two temperature plasma model (see Tab. 3). The observed spectra and a model fit are shown in Figure 2. It is interesting to note that the emission measures of hotter and cooler plasma components are quite similar. This is in contrast to other magnetic B-type stars, where the softer component usually has much larger emission measure (c.f., Oskinova et al. 2011); however, τ Sco is one notable exception to this rule. Although the two-temmperature fit is statistically acceptable, it seems that the model doesn't reproduce well the spectral shape at energies above 2 keV. We attempted to find a three-temperature model fit or a power-law fit, but these additional model components were essentially unconstrained. Thus, while it appears that there are indications of a harder component being present in the spectrum of HD66665, it must be confirmed by better quality data.", "pages": [ 3 ] }, { "title": "4.1 Comparison of Spectra", "content": "Figure 3 displays a comparison of the X-ray spectra of the two τ Sco analogues against τ Sco itself, as well as two other reference objects, ξ 1 CMa and β CMa. Each source spectrum has been normalized to unit area for the sake of comparison. The spectrum of τ Sco is shown as the hatched region in each panel of this figure. The two analogue objects are shown at bottom; the other two reference objects at top. The B star ξ 1 CMa is a magnetic star with a surface field of about 1,450 G (Hubrig et al. 2006; Fortune-Ravard et al. 2011); its X-ray properties have been reported in Oskinova et al. (2011). The source β CMa is a giant B star that does not, so far, have a detectable magnetic field (Hubrig 2006). The star has been observed with the XMM-EPIC (PI: W. Waldron), but a detailed analysis has not been reported in the literature. Here we present only a preliminary spectrum of β CMa for the purpose of having a high signal-to-noise X-ray spectrum with (a) the same instrument as our analogues sources and (b) which is known not to have a significant surface magnetic field. Normalization of the spectrum accentuates differences in the spectral energy distributions between the repective sources and τ Sco. (Note: With an EPIC/PN spectrum of over 100,000 X-ray counts, the S/N of τ Sco's spectrum is so much higher than the other stars that we do not show error bars.) The spectra of both HD 63425 and ξ 1 CMa closely hug the shape of τ Sco's spectrum. By contrast both HD 66665 and β CMa show peak values that are shifted to softer energies and a relative deficit of quite hot gas as compared to τ Sco. There are two main comments to be made at this point. First, β CMa is at an extremely low interstellar hydrogen column density, approximately two orders of magnitude lower than the other four stars (see Tab. 3). In fact, because of its low column density, β CMa was one of only two massive stars observed with the EUVE (Cassinelli et al. 1996). For the present analysis, the low column results in minimal attenuation of the softer X-ray emissions from this star, which naturally shifts the X-ray spectral peak of β CMa to lower energies. Still, as will be discussed, β CMa lacks a substantial hard component to its X-ray spectrum. The second point is that the overall counts for HD 66665 are low, lowest of all five sources in this report. The lower-thanexpected count rate of HD 66665 suggests that hard emission could be present but not detected. In effect, a low level of hard emission above 1.5 keV could be present intrinsically, but lost in the background noise owing to insufficient counts. As a result, only the dominant softer component survives in the data reduction. Our main conclusion for HD 66665 is that its spectrum does not provide evidence of hard emission, but that a longer exposure is needed to determine confidently whether or not hard emission is produced by the system.", "pages": [ 3, 4 ] }, { "title": "4.2 Statistical Analysis", "content": "With the exception of τ Sco, we have made two-temperature fits to our sources. For τ Sco, the quality of the spectrum is so high, at over 100,000 counts detected, that a two-temperature fit produces a poor match to the spectrum. In this case we use the fourtemperature fit of Mewe et al. (2003) in the following discussion of source X-ray properties. X-ray spectral characteristics are given for the five stars under discussion in Table 3. The table lists the hydrogen column density, X-ray count rate in EPIC/PN, the X-ray luminosity from EPIC/PN, and the temperatures (as kT in keV) and relative emission measures of the two temperature fits. Also listed is an emission-measureweighted average temperature, defined by (2) where EM T is the total emission measure. In the case of τ Sco, the star has one measure of temperature of relatively low value, typical of other OB stars, and three higher temperature components. Those three higher ones have been emission-measure averaged according to the values quoted by Mewe et al. (see their Tab. 2), and given in the table simply as kT 2 . As mentioned in the previous section, there is some concern for HD 66665 that its apparent lack of quite hard emission is an artifact of its low quality spectrum. To illustrate Figure 4 plots the two kT values for each source against the total number of detected X-ray counts. It seems clear that the failure to detect a hot component in β CMa, the lone non-magnetic star in this sample, is not a question of sufficient counts. Both HD 63425 and ξ 1 CMa have lower total counts in EPIC/PN yet substantially hotter kT 2 values. The kT 2 value for HD 66665 is lowest among the magnetic stars, but suspiciously also has the lowest total counts. The fact that the spectrum of HD 66665 is fit by roughly equal amounts of soft and hard emissions is anomalous among single OB stars, and tantalizingly suggestive that hotter gas may be present in HD 66665 but was simply not detected. We feel strongly that a longer exposure spectrum is needed to determine whether or not HD 66665 has a hot component, similar to τ Sco. Clearly, HD 63425 does have a hot component similar to τ Sco. To place these claims on a more quantitative level, consider the following analysis of the emission-measure-weighted kT values, 〈 kT 〉 , for our sources as compared to our reference non-magnetic B star, β CMa. For this purpose we introduce a difference parameter ∆ kT as The error in the difference ∆ kT is given by σ ∆ , with Then the significance of the difference in weighted kT values can be evaluated from the ratio ∆ kT/σ ∆ , as provided in Table 3. The overall harder emission observed in τ Sco and HD 63425 as distinct from β CMa is found to be significant at confidence levels of roughly 7 σ ∆ and 3 σ ∆ , respectively. For ξ 1 CMa the significance is actually less at just under 2 σ ∆ . For HD 66665 the distinction is formally not significant at all because of the larger error in the determination of 〈 kT 〉 for this star.", "pages": [ 4, 5 ] }, { "title": "5 CONCLUSIONS", "content": "The two B stars HD 63425 and HD 66665 have been identified as analogues to the B star, τ Sco, initially owing to similarity between UV P Cygni spectral lines. Given the successful detection of magnetism in τ Sco (Donati et al. 2006), Petit et al. (2011) reported the search and detection of magnetism in HD 63425 and HD 66665 that strengthens the physical connection that these three stars appear to share. The magnetic detections prompted an investigation of the Xray properties of HD 63425 and HD 66665. The detection of hard emission with a substantial relative emission measure for HD63425 appears to solidify its observational status as a bona fide τ Sco analogue by virtue of having (a) peculiar UV wind lines, (b) a substantial surface magnetic field, and (c) hard X-ray emission well in excess of values typically seen in single OB stars. For HD 66665 our analysis does not provide evidence of a substantial hard component. However, our observation led to fewer counts than expected, and we view the experiment as inconclusive. A longer exposure is needed to verify whether or not the star possesses significant plasma of abnormally high temperature as in the case of τ Sco. Given the success in verifying the analogue nature of HD 63425, it now seems ripe, from an empirical point of view, to suggest that τ Sco may be a prototype for a new class of magnetic B stars. Exactly how the wind and magnetic field interacts to produce the observed hard X-ray emissions and pecular UV line morphologies remain important open questions. Zeeman Doppler imaging of τ Sco has revealed a quite complex magnetic field topology (Donati et al. 2006). The star's surface distribution is more complicated than a simple dipole field like those used in current models of magnetized stellar winds ud-Doula & Owocki 2002; Townsend, Owocki, & ud-Doula 2007). What can be concluded is that there is a subset of magnetic B stars taht share the properties of τ Sco; and that adopting β CMa as a reference non-magnetic B star, it appears that the magnetic stars τ Sco, HD 63425, and ξ 1 CMa are all comparatively hard and X-ray luminous (in terms of L X /L ∗ ), as one might generally expect from models of magnetically channeled wind flow. In the future more intensive monitoring of HD 63425 and HD 66665 is needed to discern the detailed mag- eld geometries of these stars, and more data are needed to confirm the X-ray nature of HD 66665.", "pages": [ 5, 6 ] }, { "title": "ACKNOWLEDGMENTS", "content": "Based on observations obtained with XMM-Newton , an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA. This research has made use of NASA's Astrophysics Data System Service and the SIMBAD database, operated at CDS, Strasbourg, France. We are greatful to Alexis Finoguenov for his expert opinion on galaxy cluster Xray emission. Funding for this research has been provided by DLR grant 50 OR 1101 (LMO).", "pages": [ 6 ] }, { "title": "REFERENCES", "content": "Arnaud K. A., 1996, in Astronomical Society of the Pacific Conference Series, Vol. 101, Astronomical Data Analysis Software and Systems V, G. H. Jacoby & J. Barnes, ed., pp. 17 Diplas, A., Savage, B. D., 1994, ApJS, 93, 211 Petit, V., Massa, D. L., Marcolino, W. L. F., Wade, G. A., Ignace, R., 2011, MNRAS, 412, L45 Porter, J. M., 1999, A&A, 348, 512 Porter, J. M., Rivinius, Th., 2003, PASP, 115, 1153 Townsend, R. H. D., Oksala, M. E., Cohen, D. H., Owocki, S. P., ud-Doula, A., 2010, ApJ, 714, L318 Townsend, R. H. D., Owocki, S. P., Howarth, I. D., 2004, MNRAS, 350, 189 Townsend, R. H. D., Owocki, S. P., 2005, MNRAS, 357, 251 Townsend, R. H. D., Owocki, S. P., ud-Doula, A., 2007, MNRAS, 382, 139 ud-Doula, A., 2012, in Stellar Polarimetry: From Birth to Death, (eds) J. Hoffman, B. Whitney, J. Bjorkman, AIP Conf. Proc. #1429, 282 ud-Doula, A., Owocki, S. P., 2002, ApJ, 576, 413 Vink, J. S., de Koter, A., Lamers, H. J. G. L. M., 2000, A&A, 362, 295 Wisniewski, J. P., Draper, Z. H., Bjorkman, K. S., Meade, M. R., Bjorkman, J. E., Kowalski, A. F., 2010, ApJ, 709, 1306", "pages": [ 6, 7 ] } ]

[ { "title": "ABSTRACT", "content": "Cosmic rays that escape their acceleration site interact with the ambient medium and produce gamma rays as the result of inelastic proton-proton collisions. The detection of such diffuse emission may reveal the presence of an accelerator of cosmic rays, and also constrain the cosmic ray diffusion coefficient in its vicinity. Preliminary results in this direction have been obtained in the last years from studies of the gamma-ray emission from molecular clouds located in the vicinity of supernova remnants, which are the prime candidate for cosmic ray production. Hints have been found for a significant suppression of the diffusion coefficient with respect to the average one in the Galaxy. However, most of these studies rely on the assumption of isotropic diffusion, which may not be very well justified. Here, we extend this study to the case in which cosmic rays that escape an accelerator diffuse preferentially along the magnetic field lines. As a first approximation, we further assume that particles are strongly magnetized and that their transport perpendicular to the magnetic field is mainly due to the wandering of the field lines. The resulting spatial distribution of runaway cosmic rays around the accelerator is, in this case, strongly anisotropic. An application of the model to the case of the supernova remnant W28 demonstrates how the estimates of the diffusion coefficient from gamma-ray observations strongly depend on the assumptions made on the isotropy (or anisotropy) of diffusion. For higher levels of anisotropy of the diffusion, larger values of the diffusion coefficient are found to provide a good fit to data. Thus, detailed models for the propagation of cosmic rays are needed in order to interpret in a correct way the gamma-ray observations. Key words: cosmic rays - gamma rays - ISM: supernova remnants.", "pages": [ 1 ] }, { "title": "L. Nava 1 /star and S. Gabici 1", "content": "1 APC, AstroParticule et Cosmologie, Universit'e Paris Diderot, CNRS/IN2P3, CEA/Irfu, Observatoire de Paris, Sorbonne Paris Cit'e, 10, rue Alice Domon et L'eonie Duquet, 75205 Paris Cedex 13, France", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Galactic Cosmic Rays (CRs) are mainly constituted by relativistic protons and are believed to be accelerated at SuperNova Remnant (SNR) shocks via first order Fermi mechanism (Hillas 2005). Though very popular, this scenario still needs to be conclusively proven by observations. If CRs are indeed accelerated at SNRs, these objects must be gamma-ray sources. This is because the CRs accelerated at the shock undergo inelastic proton-proton interactions with the ambient medium and produce neutral pions which in turn decay into gamma rays (Drury et al. 1994; Naito & Takahara 1994). Several SNRs have been detected in gamma rays at both TeV (e.g. Hinton & Hofmann 2009) and GeV (e.g. Giordano 2011) energies, in agreement with such expectations. However, it is often difficult to determine whether the origin of the gamma-ray emission is hadronic, and thus related to the acceleration of CRs, or due to leptonic mechanisms such as inverse Compton scattering. For this reason, multi-wavelength studies of SNRs have been extensively carried out in an attempt to solve this degeneracy. Though for some individual SNRs it has been possible to ascribe the gamma-ray emission exclusively and quite confidently to hadronic (e.g. Acciari et al. 2011; Morlino & Caprioli 2012) or leptonic (e.g. Abdo et al 2011; Ellison et al 2010) processes, in other cases this ambiguity remains a problem. An alternative way to reveal the presence of a CR source is by searching for the radiation produced by CRs that escape the acceleration site (Aharonian & Atoyan 1996; Gabici & Aharonian 2007; Rodriguez Marrero et al. 2008; Gabici et al. 2009). At some stage of the dynamical evolution of the SNR, CRs are expected to leave the shock region and escape into the interstellar medium. The details of the escape mechanism are still not very well understood (see e.g. Gabici 2011, and references therein), but it is generally believed that the ability of a SNR in confining particles decreases gradually with the shock speed, with higher energy particles leaving the shock earlier than low energy ones. Once escaped, CRs diffuse away from the SNR and produce gamma-rays in interactions with the ambient gas. To date, some possible evidence for particle escape from SNRs has been pushed forward by the observation of diffuse gamma-ray emission from the vicinity of the shell of the SNRs W28 (Aharonian et al. 2008; Giuliani et al. 2010; Abdo et al. 2010) and W44 (Uchiyama et al. 2012). In both cases, the emission is clearly located outside of the shell and it is spatially coincident with the location of massive Molecular Clouds (MCs). This would favor a scenario in which the MCs are illuminated by the runaway CRs and, being very massive, become prominent gamma-ray sources (for the case of W28 see e.g. Gabici et al. 2010, and discussion in Section 4). Besides revealing the presence of a CR source, the gamma-ray emission from runaway CRs can also be used to constrain the particles' diffusion coefficient in the region surrounding the accelerator. This is very important for several reasons: first of all, a theoretical determination of the diffusion coefficient is a very complex task (see e.g. Yan & Lazarian 2004, 2008) and observational constraints are needed in order to guide and constrain models. In addition to that, the diffusion of CRs is believed to be a non-linear process in which CRs themselves generate via streaming instability the turbulence they scatter off (e.g. Kulsrud & Pearce 1969). This is particularly relevant close to CR sources, where the CR density is expected to be very high, and possibly sufficient to suppress significantly the diffusion coefficient through streaming instability (Ptuskin et al. 2008; Malkov et al. 2012). Thus, an empirical determination of the diffusion coefficient can reveal precious information on the ways in which particles and waves interact in astrophysical plasmas. Most of the studies aimed at the determination of the CR diffusion coefficient from gamma-ray observations rely on the assumption of isotropic diffusion (Torres et al. 2008; Fujita et al. 2009; Li & Chen 2010; Gabici et al. 2010; Ohira et al. 2011; Yan et al. 2012). The common rationale of these approaches can be summarized as follows: if SNRs are the sources of CRs, they have to convert a fraction η ≈ 10% of their explosion energy E SN = 10 51 E 51 erg into accelerated particles. If the diffusion of CRs proceeds isotropically, after a time t from escape CRs of a given energy E are distributed roughly homogeneously within a distance R d ( E ) ≈ √ 6 D ( E ) × t from the SNR. Here, D ( E ) is the energy dependent diffusion coefficient of CRs. Gamma-ray observations of MCs located in the vicinity of W28 or W44 tells us which is the CR density n CR ( E ) needed to explain the observed emission. According to what said above, such density has to be of the order of n CR ( E ) ≈ f sp ( E ) ηE SN /R 3 d , where the factor f sp ( E ) contains the information on the shape of the spectral energy distribution of escaping CRs. For aged SNRs such as W28 and W44, the time t after the escape can be identified with the SNR age t age , and thus an expression for the diffusion coefficient can be obtained, and reads: D ≈ ( f sp ηE SN /n CR ) 2 / 3 t -1 age . Since the values of all the physical quantities present on the right side of the equation can be inferred from observations, the expression provides a direct estimate of the diffusion coefficient. As an example, we summarize here the results obtained by Gabici et al. (2010) in interpreting the gamma-ray emis- sion observed from the MCs located close to the SNR W28. They obtained a good fit to the gamma-ray spetrum measured by H.E.S.S. at photon energies /greaterorsimilar 300 GeV by assuming a diffusion coefficient for ≈ 3 TeV CRs of the order of D (3 TeV) ≈ 5 × 10 27 ( η/ 0 . 1) 2 / 3 cm 2 /s. The corresponding diffusion length R d of these particles is of the order of 100 pc. Also the broad band gamma-ray spectrum from GeV to TeV energies can be fitted by adjusting the energy dependence of the diffusion coefficient and the distances between the SNR and the clouds. The important point here is the fact that the estimated diffusion coefficient is more than one order of magnitude smaller than the one normally adopted to describe the propagation of CRs of energy /greaterorsimilar TeV in the galactic disk, which is ≈ 10 29 cm 2 /s (Strong et al. 2007; Castellina & Donato 2012). These results are very similar to the ones obtained by other authors by means of similar modeling (Fujita et al. 2009; Li & Chen 2010; Ohira et al. 2011; Yan et al. 2012) and seem to point toward a drop of the diffusion close to the SNR W28. However, the validity of the assumption of isotropic diffusion of CRs needs to be discussed. In fact, if the intensity of the turbulent field δB on scales resonant with the Larmor radius of particles is significantly smaller than the mean large scale field B 0 (i.e. if δB/B 0 /lessmuch 1), then CR diffusion becomes anisotropic , with particles diffusing preferentially along the magnetic field lines (e.g. Casse et al. 2002). In the limiting (but still reasonable) case in which the perpendicular diffusion coefficient can be set equal to zero, the transport of CRs across the mean field is mainly due to the wandering of magnetic field lines (Jokipii & Parker 1969). This is the situation that we investigate in this paper. To give a qualitative idea of the role that anisotropic diffusion can play in these study, let us consider an idealized case in which particles that escape a SNR diffuse along a magnetic flux tube characterized by a very long coherence length (i.e. the magnetic flux tube is preserved for a long distance). In this case, after a time t particle will diffuse up to a distance R d ≈ √ 2 D ‖ × t along the tube (here D ‖ is the parallel diffusion coefficient), while their transverse distribution will be equal to the radius of the SNR shock at the time of their escape, R sh , which is of the order of ≈ 110 pc. Thus, the enhanced CR density in the flux tube will be proportional to n CR ∝ ( R d R 2 sh ) -1 instead of ∝ R -3 d as in the isotropic case. It is easy to see that the estimates of the diffusion coefficient based on the two opposite assumptions of isotropic and one-dimensional diffusion will differ by a factor of ≈ ( R d /R sh ) 4 / 3 , which can be much larger than an order of magnitude! Thus, it is of paramount importance to investigate how the interpretation of gamma-ray observations depends on the assumptions made concerning CR diffusion. In Section 2 we develop a model for CR propagation in which CRs are strongly magnetized and diffuse uniquely along the magnetic field lines. The wandering of the field lines is also taken into account, and a diffusion coefficient D m for the magnetic field lines that depends on the properties of the turbulent field is defined (see e.g. Duffy et al. 1995). In Section 3 the model is used to predict the spatial distribution of runaway CRs and their spectrum. Finally, we apply the model to the case of the SNR W28 in Section 4. A good fit to data is obtained, and the estimate of the parallel diffusion coefficient is found to depend on the level of anisotropy of the diffusion. For higher level of anisotropy, i.e. smaller values of D m , larger values of the particles' diffusion coefficient are needed in order to fit data. A discussion of the results and of future perspectives in this line of research can be found in Section 5.", "pages": [ 1, 2, 3 ] }, { "title": "2 COSMIC-RAY TRANSPORT IN THE PRESENCE OF MAGNETIC FIELD LINE WANDERING", "content": "Consider a magnetic flux tube whose mean magnetic field B 0 is assumed to lie along the z -axis, perpendicular to the ( x, y ) plane. The wandering of magnetic field lines is due to long wavelength perturbations, i.e. perturbations on scales much larger than the particles' gyroradii, with root mean square amplitude δB . The condition for the validity of quasi-linear theory is δB/B 0 /lessmuch L ⊥ /L ‖ , where L ⊥ and L ‖ are the field coherence lengths perpendicular and parallel to B 0 , respectively (Kadomtsev & Pogutse 1979). According to quasi-linear theory, the field lines passing in the vicinity of ( x 0 , y 0 ) at z = 0 are spread over a larger region as they reach a given z . The probability distribution describing this spreading of field lines is gaussian and characterized by 〈 ( x -x 0 ) 2 〉 = 〈 ( y -y 0 ) 2 〉 = 2 D m z , where brackets indicate an ensemble average and D m is a diffusion coefficient for field lines (Jokipii & Parker 1969). For a broad band Fourier spectrum of the perturbation, the coherence lengths can be expressed as L ⊥ , ‖ = 2 π/ ∆ k ⊥ , ‖ ≈ 2 π/k ⊥ , ‖ , with k ⊥ and k ‖ the characteristic wave-vectors of the perturbation (Achterberg & Ball 1994). Under these circumstances, the diffusion coefficient is D m = ( δB/B 0 ) 2 L ‖ / 4 (Kadomtsev & Pogutse 1979; Duffy et al. 1995). It is also possible to define the Lyapunov length λ L = L 2 ⊥ ( δB/B 0 ) -2 /L ‖ which describes the exponential separation of field lines whose initial separation is smaller than L ⊥ (Isichenko 1991). This can be interpreted as the length above which the flux tube is disrupted by field line divergence. For fiducial values of the parameters the length of the flux tube is of the order of a few hundred parsecs (see e.g. Ptuskin et al. 2008). We are interested here in studying the propagation of CRs in the presence of magnetic field line wandering. Energetic particles diffuse along and across the magnetic field line as the result of resonant interactions with magnetic field perturbations. Such perturbations are characterized by length scales of the order of the particles' Larmor radii. Such scales are much smaller than the ones responsible for field line wandering. According to quasi-linear theory, the ratio between the parallel to perpendicular diffusion coefficient is D ‖ /D ⊥ = 1 + ( λ ‖ /r g ) 2 , where λ ‖ is the particle's mean free path along the field line and r g is its gyroradius. In the interstellar medium it is believed that λ ‖ /greatermuch r g which implies D ⊥ /lessmuch D ‖ (e.g. Casse et al. 2002). Thus, in the following we will neglect the diffusion of particles perpendicular to the field lines. In other words, a given particle remains attached to the same field line. Under this conditions, in a time interval ∆ t a particle diffuses along a given field line a distance 〈 (∆ z ) 2 〉 = 2 D ‖ ∆ t , but over such a distance ∆ z along the z -axis the field line is displaced by an amount 〈 (∆ x ) 2 〉 = 2 D m ∆ z . This leads to (Getmantsev 1963; Rechester & Rosenbluth 1978; Chuvilgin & Ptuskin 1993): which describes a sub-diffusive transport of particles perpendicular to the mean magnetic field B 0 . This behavior of energetic particles due to the combination of the particle diffusion along the field lines and the random walk of the field lines themselves has been often referred to as compound diffusion , or anomalous diffusion . Models of compound diffusion have been developed and used in a great variety of contexts, to study phenomena like the heat transport in Tokamak (Rechester & Rosenbluth 1978; Isichenko 1991), the propagation of energetic particles in the solar wind (Jokipii & Parker 1969; Zimbardo et al. 2006), the confinement of CRs in the Galaxy (Getmantsev 1963; Lingenfelter et al. 1971; Chuvilgin & Ptuskin 1993) and their acceleration at astrophysical shocks (Achterberg & Ball 1994; Duffy et al. 1995; Kirk et al. 1996). In this paper, we apply the formalism of compound diffusion to another context, which is the propagation of CRs in the vicinity of their sources, i.e. SNRs, after they escape the acceleration region. We will show that in this situation an accurate modeling is needed in order to interpret in a correct way the gamma-ray observations of MCs located in the vicinity of SNRs. In order to describe the compound diffusion of CRs we adopt the mathematical formalism developed by Webb et al. (2006) and we define P FRW ( x | z ) as the probability to find a field line displaced by an amount ∆ x after a step of length z along the direction of the umperturbed field B 0 . From what said above, it follows that: which corresponds to a diffusive behavior of field lines (FLW stands for Field line Random Walk). A similar equation holds for the displacement ∆ y . This has to be combined with the probability P ‖ ( z | ∆ t ) that a particle moves a distance z along the field line in a time ∆ t = t -t 0 . For diffusive transport of particles along the field we have: The probability for a particle to reach the position ( x, y, z ) at the time t , when its position at the time t 0 was ( x 0 , y 0 , z 0 = 0), is then the product of P FRW with P ‖ : In order to model the escape of CRs from a SNRs we assume, following Ptuskin et al. (2008), that particles are injected in the flux tube in the xy -plane at z = 0, within a circular region whose radius is equal to the SNR shock radius R sh ( E ). Since CRs of different energy are expected to escape the remnant at different times, the radius of the injection region is an energy dependent quantity. Following Gabici et al. (2009) we assume a power law scaling to connect the particle energy of the runaway CRs with the time after the supernova explosion: where the implicit assumption has been made that the maximum energy of CRs accelerated in a SNR E MAX is reached at the time t Sed which marks the transition between the free expansion and the Sedov phases of the SNR evolution, and that CRs are gradually released in the interstellar medium from that time on. The Sedov phase is characterized by a scaling R sh ∝ t 2 / 5 which gives: which is what is assumed in the following. Other parameterizations of the escape time of CRs can be easily implemented. The spatial distribution of CRs can now be obtained by integrating the probability function given by Eq. 4 within the range R 0 = √ ( x 2 0 + y 2 0 ) /lessorequalslant R sh ( t esc ( E )). To do so, it is convenient to adopt a cylindrical coordinate system and express the Field line Random Walk part of Eq. 4 as a function of the quantities R = √ x 2 + y 2 , R 0 = √ ( x 2 0 + y 2 0 ) and cos (∆ ϕ ) = ( xx 0 + yy 0 ) / ( RR 0 ) which leads to: where it has been assumed that the total spectrum of CRs released in the interstellar medium during the whole life of the SNR is a power law AE -Γ with normalization: /negationslash Here, E SN it the supernova explosion energy, η is the fraction of this energy converted into CRs which are released in the interstellar medium, and E MIN and E MAX represent the extension in energy of the CR spectrum. To describe the diffusion of CRs along field lines we adopt a diffusion coefficient which is a power-law in energy: where ˜ D ‖ and s are considered here as free parameters. Before proceeding in computing the spatial distribution of CRs around a SNR we notice that, for z /lessmuch L ‖ , Eq. 2 does not provide a good description of the field line wandering. The reason is that in this regime the lateral displacement of a field line after a step z along B 0 is of the order of ≈ ( δB/B 0 ) z = bz , since b = δB/B 0 represents the angle between the unperturbed ( B 0 ) and total ( B 0 + δB ) magnetic field (Isichenko 1991). An accurate and quantitative analysis of the behavior of a magnetic flux tube in this regime goes beyond the scope of this paper (see Isichenko 1991, for a more detailed discussion). However, in order to describe this regime in a qualitative way, for z /lessmuch L ‖ we substitute Eq. 2 with: where ϑ [ s ] is the Heaviside function, equal to 1 for s > 0 and 0 for s < 0. Eq. 9 roughly mimics the behavior of a flux tube characterized by a opening angle b . In the intermediate region z ≈ L ‖ we use an interpolating function to bridge the behaviors described by Eqns. 2 and 9.", "pages": [ 3, 4 ] }, { "title": "3 RESULTS", "content": "In this Section we compute the spatial distribution of CRs expected in the vicinity of a SNR at a given time after the explosion. We consider a typical supernova, characterized by the following fiducial values of parameters: an explosion energy of E SN = 10 51 erg, a mass of the ejecta equal to M ej = 1 . 4 M /circledot , and a density of the circumstellar medium n 0 = 1cm -3 . We further assume that a fraction η = 0 . 1 of the supernova explosion energy is converted into CRs, which are injected in the interstellar medium with a power law differential energy spectrum dN/dE ∝ E -α which extends from E MIN = 1GeV to E MAX = 5PeV (approximately the position of the knee in the CR spectrum). It is known from CR data that α should be in the range ≈ 2 . 1 -2 . 4 (Castellina & Donato 2012; Strong et al. 2007). We adopt here α = 2 . 2 as a representative value. As described in Sec. 2, CRs are gradually released from the SNR during the Sedov phase that goes from t ≈ 280 yr to t ≈ 3 . 6 × 10 4 yr (Cioffi et al. 1988). For the parallel diffusion coefficient of CRs (Eq. 8) we assume D ‖ = 10 28 cm 2 /s and s = 0 . 5. ˜ As a first step, we compare in Fig. 1 the results that are obtained if an isotropic diffusion coefficient is assumed (as, e.g., in Aharonian & Atoyan 1996; Gabici et al. 2009), with the ones obtained for the anisotropic diffusion model that we consider in this paper. In both panels of Fig. 1, the SNR is located at the centre of the field and the color code refers to the excess of CRs with respect to the average density of CRs in the Galaxy, which is (e.g. Particle Data Group 2008): Over-densities are plotted for a particle energy of 1 TeV and for a time t = 10kyr after the supernova explosion. Here the diffusion coefficient of the magnetic field lines is set equal to D m = 1pc, with b 2 = ( δB/B 0 ) 2 = 0 . 2 (different values of D m will be explored in the following). This corresponds to a parallel coherence length of the perturbation of L ‖ = 20pc. The spatial distribution of CRs is strikingly different in the two scenarios: spherically symmetric in the left panel, and strongly elongated in the direction of the magnetic field flux tube in the right panel. A filamentary diffusion of CRs was also found in the numerical simulations by Giacinti et al. (2012). The same parameters have been used to compute the over-densities in the two scenarios, with the exception of the CR diffusion coefficient, which in the left panel has been assumed to be isotropic and equal to D = ˜ D ( E/ 10 GeV) 0 . 5 cm 2 /s with ˜ D = 5 × 10 26 cm 2 /s. The choice of two significantly different values for ˜ D and ˜ D ‖ , with D /lessmuch D ‖ has been made in order to obtain the same level of CR over-density in the vicinity of the SNR. As an example, the black cross in Fig. 1 shows a position, located 30 pc away from the centre of the explosion, where the CR over-density is identical in the two panels. To get comparable values for the CR over-density, a much smaller (isotropic) diffusion coefficient ˜ D is needed in order to compensate for the larger solid angle over which CRs can propagate. As already stressed in the introduction, this fact must be taken into account when interpreting the gamma-ray observations of molecular clouds illuminated by CRs escaping from SNRs. This will be discussed in Sec. 4, when the model developed here will be applied to fit the gamma-ray observations of the SNR W28. In Fig. 2 we show the CR over-density around the SNR for different values of the particle energy and of the time after the supernova explosion. The upper, middle, and lower panels refer to a time of 6, 19, and 60 kyr after the explosion, respectively. Plots on the first, second, and third column refer to particle energies of 30 GeV, 3 TeV, and 300 TeV, respectively. The escape of CRs is described by Eq. 5, which states that higher energy CRs are released first, and lower energy CRs escape at later times. This is the reason why there is no CR excess in the top-left panel of Fig. 2: for the choice of parameters made here, for a SNR age of 6 kyr particles with an energy of 30 GeV are still confined within the SNR shock. As the age of the SNR increases, CRs diffuse further away along the flux tube and fill a broader and broader region. As a consequence of that, the CR overdensity decreases accordingly. It is evident from these maps that a molecular cloud located in the vicinity of the SNR can be illuminated by the escaping CRs and become a bright gamma-ray source only if it is located within the flux tube. Anearby cloud which is not magnetically connected with the SNR will not be illuminated by CRs, despite its proximity to the SNR. All the plots in Fig. 2 refer to a region of size ≈ 200 pc around the SNR. As said in Sec. 2, this roughly represents the expected length of a magnetic flux tube in the Galaxy (e.g. Ptuskin et al. 2008). For distances larger than a few hundred parsecs from the SNR, the flux tube loses its identity and it is disrupted due to the exponential divergence of field lines. Thus, the results presented in this paper are accurate and reliable for distances up to a few hundred parsecs and less. Since we assumed here that CRs are strongly magnetized, i.e. their remain attached to field lines, their transport across the mean magnetic field is solely governed by the diffusion of field lines, which is described by the diffusion coefficient D m . This quantity determines how quickly the field lines diverge as a function of the displacement z along the mean field. This is demonstrated in Fig. 3, where the CR over-density for particles of energy 3 TeV is shown. The age of the SNR is 19 kyr. Three different values for D m are considered: 0.5, 1, and 2 pc for the left, middle, and right panel, respectively. Unfortunately, our knowledge of the properties of the interstellar magnetic field is not good enough to allow a determination or an estimate of this parameter. For a SNR located in a diffuse interstellar medium (i.e. with no massive molecular clouds) the morphology of the resulting gammaray emission due to CR proton-proton interaction with the ambient gas would closely follow the spatial distribution of CRs. Thus, observing the diffuse gamma-ray emission generated by runaway CRs around SNRs might serve as a tool to explore the structure of the interstellar magnetic field. The detection of such diffuse emission is within the capabilities of future gamma-ray instruments such as the Cherenkov Telescope Array (Acero et al. 2012; Casanova et al. 2010). Finally, we show in Fig. 4 the spectra of escaping CRs at different distances from the SNR and at different times after the supernova explosion. Each panel refers to a different epoch: 6, 19, and 60 kyr for the top, middle, and bottom panel, respectively. Solid curves show the spectra at three different positions on the z -axis: 40 pc, 100 pc, and 200 pc. For each of these positions we also show the spectra at differ- ent distances from the z -axis: 25 pc (dotted lines) and 50 pc (dashed lines). At high energies, in almost all cases the energy spectra are power laws with slope ≈ α + s/ 2, where α is the slope of the injection spectrum of runaway CRs and s is the slope of the energy dependent diffusion coefficient (Eq. 8). Such a behavior can be inferred from Eq. 3. If one moves to larger values of z , the on-axis (i.e. R = 0) high energy spectrum preserves the same slope, but its normalization decreases as ≈ 1 /z . This is due to the fact that the transverse section of the magnetic flux tube is increasing proportionally to z , while the CR intensity along a field line is independent of z for z 2 /lessmuch 4 D ‖ t (see Eqns. 2 and 3). A feature common to all the spectra plotted in Fig. 4 is the presence of a low energy cutoff. The cutoff is due to the fact that at a given time, only particles of sufficiently large energy had enough time to propagate over the distance z . The cutoff is moving towards lower energies if a longer time is considered, because particles with lower energies have then the time to reach a given position z along the axis. Finally, some curves (for example the ones with z = 40 pc and R = 25pc) exhibit a quite sharp high energy cutoff. This cutoff is due to the fact that particles of different energy are injected within different transverse sections of the flux tube. Higher energy particles are released earlier from the SNR, when the shock radius is smaller, lower energy ones are injected later, when the shock radius is larger. While diffusing along the field lines, CRs are displaced in the transverse direction due to field line wandering. Higher energy particles, which have been injected in a smaller region around z = 0, need on average a larger transverse displacement in order to reach a given distance ˆ R from the z -axis. Thus, for small enough z , the opening of the magnetic flux tube might not be enough to allow high energy particles to reach ˆ R , and this explains the presence of the cutoff. In the next Section we apply the model developed above to a specific object, i.e. the SNR W28 and the molecular clouds located in its proximity. Such clouds have been detected in gamma rays and this has been interpreted by many authors as the result of their being illuminated by CRs that escaped the SNR. We will demonstrate that a good agreement can be reached between the predictions of the model and observations and we will discuss the impact of this on the attempts to derive the particle diffusion coefficient close to SNRs by means of gamma-ray observations.", "pages": [ 4, 5, 6, 7 ] }, { "title": "4 APPLICATION TO THE SUPERNOVA REMNANT W28", "content": "W28 is an old SNR in its radiative phase of evolution, located in a region rich of dense molecular gas with average density /greaterorsimilar 5 cm -3 . At an estimated distance of ∼ 2 kpc the SNR shock radius is ∼ 12 pc and its velocity ∼ 80 km / s (e.g. Rho & Borkowski 2002). By using the dynamical model by Cioffi et al. (1988) and assuming that the mass of the supernova ejecta is ∼ 1 . 4 M /circledot , it is possible to infer the supernova explosion energy ( E SN ∼ 0 . 4 × 10 51 erg), initial velocity ( ∼ 5500 km / s), and age ( t age ∼ 4 . 4 × 10 4 yr). Gamma ray emission has been detected from the region surrounding W28 both at TeV (Aharonian et al. 2008) and GeV energies (Abdo et al. 2010; Giuliani et al. 2010), by HESS, FERMI, and AGILE, respectively. The TeV emission correlates quite well with the position of three massive molecular clouds, one of which is interacting with the north-eastern part of the shell (and corresponds to the TeV source HESS J1801-233), and the other two being located to the south of the SNR (TeV sources HESS J1800-240 A and B). The masses of these clouds can be estimated from CO measurements and result in ≈ 5, 6, and 4 × 10 4 M /circledot , respectively, and their projected distances from the centre of the SNR are ≈ 12, 20, and 20 pc, respectively (Aharonian et al. 2008). The GeV emission roughly mimics the TeV one, except for the fact that no significant emission is detected at the position of HESS J1800-240 A. The gamma-ray emission from the clouds in the W28 region has been interpreted by many authors as the result of the interaction of CRs that escaped W28 with the dense gas in the cloud (Fujita et al. 2009; Li & Chen 2010; Gabici et al. 2010; Ohira et al. 2011; Yan et al. 2012). All these approaches started from the assumption of isotropic diffusion of CRs, and a general consensus was found on the fact that, in order to fit observations, the diffusion coefficient had to be suppressed by a factor of ≈ 10 ... 100 with respect to the average value in the Galaxy, which is D gal ≈ D 0 ( E/ 10 GeV) δ with D 0 ≈ 10 28 ... 10 29 cm 2 /s and s ≈ 0 . 3 ... 0 . 7 (Castellina & Donato 2012). In this section we take a different approach and we ap- ply the model developed in Sec. 2 to estimate the spectrum of CRs and derive the γ -ray emission expected from the clouds in the W28 region. This approach is radically different from the ones mentioned above because it relies on the more physical assumption that the diffusion of CRs is not isotropic, but proceeds mainly along the magnetic field lines. The results of our modeling are shown in Fig. 5, where the gamma-ray data from the three molecular clouds are plotted as blue open symbols (data from FERMI) and black filled dots (data from HESS). The emission from the sources HESS J1801-233, and HESS J1800-240 A and B is plotted in the top, middle, and bottom panel, respectively. The black dashed lines represent the contribution to the gamma-ray emission from the proton-proton interactions of the CRs in the galactic background with the inter-cloud gas. The blue long-dashed lines represents the contribution to the emission from the runaway CRs that escaped from W28. The solid red line is the total emission. The gamma-ray fluxes have been computed following Kamae et al. (2006) with an additional multiplicative factor 1.5 to take into account elements heavier than hydrogen in both cosmic rays and ambient gas (Mori 1997). A good agreement with observations is obtained is a parallel diffusion coefficient ˜ D ‖ = 10 28 cm 2 /s with s = 0 . 5 is adopted, together with a diffusion coefficient for field lines D m = 1pc with b 2 = ( δB/B 0 ) 2 = 0 . 2. Moreover, we assumed that ≈ 20% of the total explosion energy has been converted into CRs with a spectrum proportional to E -2 . 2 and extending from 1 GeV to 5 PeV. In order to be illuminated by the escaping CRs, the three molecular clouds have to be located in the proximity of the axis of the magnetic flux tube (i.e. the direction of the local mean field). The spectra reported in the figure refers to the positions z = 10, 165, and 35 pc and R = 6 . 5, 0, and 14 pc (top to bottom panel, respectively). It has to be noted that, due to the number of parameters involved in the model, other sets of parameter values might be found that provide an equally satisfactory fit to data. This is not surprising, given that several previous modelings of this source provided an equally good fit to data by using a radically different picture (i.e. isotropic diffusion of CRs) for the transport of particles. Moreover, while a quite small normalization of the (isotropic) diffusion coefficient, roughly of the order of ˜ D ≈ 5 × 10 26 cm 2 /s had to be adopted in order to fit data satisfactorily, in the anisotropic case we obtain a good agreement with data for a significantly larger value of the (parallel) diffusion coefficient of ˜ D ‖ ≈ 10 28 cm 2 /s. It might be noticed that this number is close to the standard values inferred for the diffusion of CRs in the Galaxy. Thus, any attempt to constrain the CR diffusion coefficient from the observations of gamma-rays from the vicinity of SNRs needs to take into account that an intrinsic uncertainty exists, and it is related to the unknown nature of the CR transport in the interstellar medium, and in particular to the unknown relative relevance of the transport parallel and perpendicular to the magnetic field lines.", "pages": [ 7, 8 ] }, { "title": "5 CONCLUSIONS", "content": "The details of the transport of CRs in the Galaxy are still little understood. Studies of the composition of CRs provide us with an estimate of the average confinement time of CRs within the Galaxy, which can be translated into a spatially averaged diffusion coefficient for CRs (e.g. Strong et al. 2007; Castellina & Donato 2012). Whether the CR diffusion coefficient has large spatial variations or it is rather uniform throughout the Galaxy is not known, thought a suppression of diffusion close to CR sources might be expected due to CR streaming instability (Ptuskin et al. 2008; Malkov et al. 2012). To this purpose, the detection of gamma-ray emission from the vicinity of CR accelerators might be used to constrain the CR diffusion coefficient, and thus assess the importance of such suppression (e.g. Aharonian & Atoyan 1996; Gabici et al. 2009). This is because CRs escaping the accelerators would produce gamma rays via proton-proton interactions with the ambient medium. Both the morphology of the resulting emission and its spectrum would depend on the functional form (i.e. energy dependence, level of anisotropy) of the diffusion coefficient. An object that has been extensively investigated in this context is the SNR W28. Three massive molecular clouds, with total mass in the ≈ 10 5 M /circledot range, are located in the vicinity of the SNR shell and emit gamma rays. This has been interpreted as the result of the illumination of the clouds by the CRs that escaped the SNR. Several models have been proposed to fit these observations, and all of them are based on the assumption that the diffusion of CRs proceeds isotropically (e.g. Giuliani et al. 2010; Gabici et al. 2010, and see Sec. 4 for a complete list of references). There is a general consensus on the fact that the (isotropic) diffusion coefficient has to be suppressed by a factor of ≈ 10 ... 100 with respect to the average Galactic one in order to explain the observations. This implies coefficients in the range D ≈ 10 26 ... 10 27 cm 2 /s. ˜ In this paper, the assumption of isotropy of diffusion has been relaxed, and a more physically motivated situation have been investigated, in which CRs propagate mainly along the magnetic field lines. We considered here the limiting scenario in which the diffusion of CRs across field lines is very small and thus can be neglected. In such a situation, the transverse displacement of CRs is uniquely due to the wandering of the field lines (Jokipii & Parker 1969). Spectra and morphology of the spatial distribution of CRs around SNRs have been computed and described. The main feature is the elongated, filamentary distribution of CRs, as opposed to the spherical distribution found in the case of isotropic diffusion. In order to fit the gamma-ray data from the W28 region within this scenario, one has to assume that the molecular clouds in its vicinity are magnetically connected to the SNR through a magnetic field flux tube. If this is the case, an accurate fit to data can be obtained. Under this assumption, particles are bound to the flux tube and thus forced to propagate along a specific direction. For plausible values of the diffusion coefficient of magnetic field lines, in order to obtain the correct CR over-density at the location of the molecular clouds a large (parallel) diffusion coefficient of the order of D ‖ ≈ 10 28 cm 2 /s has to be adopted. ˜ The fact that a very good agreement has been found with data in the two radically different scenarios characterized by isotropic and anisotropic diffusion tells us that more data needs to be collected from more SNRs in order to infer with reasonable confidence the properties of the diffusion of particles escaping their accelerators. The diffuse emission that these runaway particles would produce in their interaction with the ambient gas is, even in the absence of very massive clouds, within the capabilities of the Cherenkov Telescope Array (Acero et al. 2012; Casanova et al. 2010). These observations will provide us with precious informations about the properties of the transport of CRs in the Galaxy, but also with a direct evidence for the fact that SNRs are indeed the accelerators of galactic CRs.", "pages": [ 8, 9 ] }, { "title": "ACKNOWLEDGMENTS", "content": "We thank F. Casse, A. Marcowith, F. Piazza, V. Ptuskin, R. Schlickeiser, and L. Sironi for helpful discussions. The work of LN and SG has been supported by ANR under the JCJC Programme.", "pages": [ 9 ] } ]

[ { "title": "ABSTRACT", "content": "We present a complete spectral analysis of an XMM-Newton and Chandra campaign of the obscured AGN in NGC 4507, consisting of six observations spanning a period of six months, ranging from June 2010 to December 2010. We detect strong absorption variability on time scales between 1.5 and 4 months, suggesting that the obscuring material consists of gas clouds at parsec-scale distance. The lack of significant variability on shorter time scales suggests that this event is not due to absorption by broad line region clouds, which was instead found in other studies of similar sources. This shows that a single, universal structure of the absorber (either BLR clouds, or the parsec-scale torus) is not enough to reproduce the observed complexity of the X-ray absorption features of this AGN. Key words: Galaxies: active - Galaxies: Seyfert - Galaxies: accretion", "pages": [ 1 ] }, { "title": "X-ray absorption variability in NGC 4507", "content": "Andrea Marinucci 1 , 2 /star , Guido Risaliti 2 , 3 , Junfeng Wang 2 , Stefano Bianchi 1 , 2 1 2 3 Martin Elvis , Giorgio Matt , Emanuele Nardini , Valentina Braito 1 Dipartimento di Fisica, Universit'a degli Studi Roma Tre, via della Vasca Navale 84, 00146 Roma, Italy 2 Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge MA 02138, USA 3 INAF - Osservatorio Astrofisico di Arcetri, L.go E. Fermi 5, Firenze, Italy 4 INAF - Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807, Merate, Italy 5 July 2018", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "The standard Unification Model for active galactic nuclei (AGN) assumes the same internal structure for Seyfert 2 and Seyfert 1 galaxies (Antonucci 1993), with all the type 1/2 observational differences ascribed to an axisymmetric absorber/reflector, located between the broad line region and the narrow line region, in order to obscure the former, but not the latter. An early developed, natural geometrical and physical scenario is that of a homogeneous torus on a parsec scale (Krolik & Begelman 1988); however recent studies on X-ray absorbing column density changes performed with Chandra, XMM-Newton and Suzaku satellites ruled out a universal geometrical structure of the circumnuclear absorber. Absorption variability is common (almost ubiquitous) when we compare observations months to years apart (Risaliti et al. 2002), and, most notably, has been found on time scales of hours to days in several sources, such as NGC1365 (Risaliti et al. 2005, 2007, 2009), NGC 4388 (Elvis et al. 2004), NGC 4151 (Puccetti et al. 2007) and NGC 7582 (Bianchi et al. 2009). NGC 4507 is a nearby (z=0.0118) barred spiral galaxy and one of the X-ray brightest Compton-thin Seyfert 2s, despite the heavy obscuration ( N H ∼ 4 -9 × 10 23 cm -2 ). It was first observed in the X-rays by Einstein (L X = 2 . 8 × 10 41 erg s -1 , Kriss et al. 1980) and then in 1990 with Ginga (Awaki et al. 1991), showing a strongly absorbed ( N H ∼ 5 × 10 23 cm -2 ) power law continuum and a prominent iron K α line. In 1994 ASCA also revealed a strong X-ray excess and an intense emission line (identified as Ne ix ) at ∼ 0 . 9 keV (Comastri et al. 1998). Risaliti (2002) reported then three BeppoSAX observations of the source, confirming the obscuring column densities observed in the previous two observations. In 2001 NGC 4507 has been observed for the first time with the most sensitive X-ray satellites, XMMNewton and Chandra/ACIS-S HETG (Matt et al. 2004) confirming once again the clear Compton-thin state of the spectrum, with a Γ = 1 . 8 +0 . 1 -0 . 2 power law absorbed by an N H = 4 . 4 +0 . 5 -0 . 6 × 10 23 cm -2 . In 2007 a ∼ 3 days observation with Suzaku has been performed and it revealed a much larger absorbing column density ( N H ∼ 9 × 10 23 cm -2 ) with respect to the earlier observations, but no changes within the 3 days of monitoring (Braito et al. 2012, hereafter B12). Summarizing, NGC 4507 showed strong variations in time scales of years (from 1990 until 2007) but none in shorter time scales (3 days observation with Suzaku). This excluded a possible sub-parsec scale absorbing structure, but leaves the actual size and distance of the absorber largely unconstrained. In this paper we report the study of five XMMNewton observations spanning a period of 6 weeks, between June and August 2010, and a Chandra observation performed 4 months later, in December 2010. This observational campaign has been designed to fill the time gap between days and years in the previous observations, so constraining the location of the absorber.", "pages": [ 1, 2 ] }, { "title": "2 OBSERVATIONS AND DATA REDUCTION", "content": "The 5 XMMNewton observations analysed in this paper were performed on 2010 June 24 (obsid 0653870201), July 3 (obsid 0653870301), July 13 (obsid 0653870401), July 23 (obsid 0653870501), August 3 (obsid 0653870601) with the EPIC CCD cameras, the PN (Struder et al. 2001) and the two MOS (Turner et al. 2001), operated in large window and medium filter mode. The extraction radii and the optimal time cuts for flaring particle background were computed with SAS 11 (Gabriel et al. 2004) via an iterative process which leads to a maximization of the Signal-to-Noise Ratio (SNR), similarly to that described in Piconcelli et al. (2004). After this process, the net exposure times for the 5 different observations were 16 ks, 13 ks, 13 ks, 13 ks and 17 ks for the PN, respectively. The resulting optimal extraction radii are 40 arcsec for the first three observations, 30 arcsec for the fourth one, 26 arcsec for the last one. The background spectra were extracted from source-free circular regions with a radius of about 50 arcsec for all the 5 observations. We also re-extracted the data from a previous XMMNewton observation (obsid 0006220201), with a net exposure of about 36 ks, adopting an extraction radius of 40 arcsec for the source and 42 arcsec for the background. The analysis of the last set of data is discussed in Matt et al. (2004). Chandra observed the source on December 2, 2010 for 44 ks, with the Advanced CCD Imaging Spectrometer (ACIS: Garmire et al. 2003). Data were reduced with the Chandra Interactive Analysis of Observations (CIAO: Fruscione et al. 2006) 4.4 and the Chandra Calibration Data Base (CALDB) 4.4.6 database, adopting standard procedures, using a 2 arcsec and 10 arcsec extraction radii for the source and background, respectively. Spectra were binned in order to over-sample the instrumental resolution by at least a factor of 3 and to have no less than 30 counts in each background-subtracted spectral channel. This allows the applicability of the χ 2 statistics.", "pages": [ 2 ] }, { "title": "3 DATA ANALYSIS", "content": "The adopted cosmological parameters are H 0 = 70 km s -1 Mpc -1 , Ω Λ = 0 . 73 and Ω m = 0 . 27, i.e. the default ones in xspec 12.7.0 (Arnaud 1996). Errors correspond to the 90% confidence level for one interesting parameter (∆ χ 2 = 2 . 7), if not otherwise stated.", "pages": [ 2 ] }, { "title": "3.1 EPIC PN/MOS spectral analysis", "content": "The soft 0.5-3.0 keV spectrum presents a strong 'soft excess', which appears dominated by emission lines from an highly ionized gas, as observed in most X-ray obscured AGN (Turner et al. 1997; Guainazzi & Bianchi 2007). Emission lines from H-like and He-like C, N, O, and Ne, as well as from the Fe L-shell, have been detected and reported in previous observations (Matt et al. 2004). Following Matt et al. (2004) the baseline model we used to fit the 0.5-10 keV spectra can be roughly expressed by the following general formula: where σ ( E ) is the photoelectric cross-section (abundances as in Anders & Grevesse 1989), N G H is the Galactic absorbing column density along the line of sight to the source (Dickey & Lockman 1990); Ph C is the emission from a photoionised gas reproduced with self-consistent cloudy models as described in Bianchi et al. (2010) and Marinucci et al. (2011) while C is the emission from a collisionally-ionised diffuse gas ( apec model, Smith et al. 2001); N H is the neutral absorbing column density at the redshift of the source; B is the normalization of the primary powerlaw with slope Γ; R (Γ) is the Compton scattering from the inner layer of the circumnuclear torus, modelled in Xspec with pexrav (Magdziarz & Zdziarski 1995); G i ( E ) are Gaussian profiles, corresponding to required emission lines of high-Z elements such as the Fe K α at 6.4 keV, Fe K β at 7.058 keV, Fe xxvi at 6.966 keV and the forbidden line of the Fe xxv K α triplet (see table 1). A further Gaussian emission line has been used to reproduce the Compton Shoulder (CS) redwards of the Iron line core, as expected on theoretical grounds, with energy fixed at 6.3 keV and σ =40 eV (Matt 2002). The reflected, Compton scattered emission has been modeled using the pexrav model with Γ=1.8 and normalization fixed to the values measured with the broadband Suzaku observation (B12). The previous model has been used to fit, in first place, the 5 separate EPIC PN/MOS observations. The 5 spectral fits from our campaign in Summer 2010 (labeled as Obs. 1-5 in Table 1) have good χ 2 / d.o.f. and do not present any strong evidence of variations in either the absorbing column density or Γ. The Obs. 2 data set presents strong residuals mainly between 1.5 and 3.0 keV, more evident in the EPIC-MOS spectra. Since they are located in very narrow bins at ∼ 2 . 8, ∼ 1 . 6 and ∼ 1 . 9 keV and they cannot be ascribed to known spectral features we believe they are due to background/calibration issues. Two different photoionised phases and a collisional one are needed to model the 0.5-3.0 keV spectra of NGC 4507. Table 1 clearly shows that the soft X-ray emitting gas has not varied during the monitoring. This suggests that the interveining absorbing material, responsible for the column density variation between June and December 2010, might be much closer to the X-ray source than the circumnuclear matter responsible for the soft emission. As already discussed in Bianchi et al. (2006) and Bianchi & Guainazzi (2007) this gas is likely coincident with the NLR. Further studies on the phoionisation mechanisms and extended emission in NGC 4507 will be discussed in detail in the future (Wang et al., in preparation). We then analysed the 10 spectra (5 EPIC PN and 5 MOS1+2, labeled as Set 1 in Table 1) simultaneously, using the model described above and linking all the parameters, except for the flux normalizations. The baseline model reproduces the 5 sets of data and some residuals are present around 1.8 keV. Indeed, the addition of a line at 1.77 ± 0 . 03 keV is required (∆ χ 2 = 44, with a significance greater than 99.99%, according to F -test 1 ); it can be identified as Si K α , with a corresponding flux of 2 . 0 ± 0 . 4 × 10 -6 ph cm -2 s -1 . The primary powerlaw (Γ = 1 . 8 +0 . 2 -0 . 2 ) is absorbed by a column density of 9 . 0 ± 0 . 5 × 10 23 cm -2 . The photon index is in agreement with previous studies on this source, on the contrary a clear variation in the N H can be noticed with respect to the old 2004 XMM-Newton observation, but it is fully compatible with the Suzaku observation in 2007 (Fig. 4). The addition of a CS redwards of the Iron K α line core is required by the fit (∆ χ 2 = 20 with a significance greater than 99.99%) and its flux, being 24 ± 7% of the flux of the narrow core of the Fe K α , is consistent with expectations. Both Iron K α and CS fluxes are in agreement with the ones found in the previous XMM observation (Matt et al. 2004). The equivalent widths of the high energy emission lines are in full agreement with the ones we find when we analyse the 5 sets of data separately. We only find upper limits on fluxes ( < 0 . 5 × 10 -6 ph cm -2 s -1 ) and EWs ( < 50 eV) of the Fe xxvi emission lines at 6.966 keV. As a last cross-check, we let the 5 different absorbing column density parameters free to vary in our fit, to check whether a possible variation may have been occurred in the 1.5 months monitoring. The best fit values of the 5 different N H do not show any significant variation with respect to the best fit value for the whole data set (9 . 0 ± 0 . 5 × 10 23 cm -2 ) with a non significant improvement of the fit (∆ χ 2 = 5 with four more parameters and a significance lower than 8%). This result brings further evidence to the argument that absorption variability on short time scales (hours-days) can be ruled out in our analysis of NGC 4507. If the intervening absorbing material had varied on such short time scales we would not have measured a constant column density in a 1.5 months monitoring. The measured values would have been completely scattered over the range of values observed in the past (4 -9 × 10 23 cm -2 ). 1 The F-test is not a reliable test for the significance of emission lines unless their normalizations are allowed to be negative (Protassov et al. 2002).", "pages": [ 2, 3 ] }, { "title": "3.2 Chandra ACIS spectral analysis", "content": "The baseline model is the same we used to fit the XMMNewton data. The overall fit is very good ( χ 2 =327/329) and the addition of a further emission line at 1.81 ± 0 . 02 keV is required (∆ χ 2 = 14 with a significance greater than 99.99%), with a flux of 4 +1 -2 × 10 -6 phcm -2 s -1 , marginally consistent with the emission from Si K α already found in XMMNewton best fits. The soft X-ray spectrum is produced by two photoionised phases, while the contribution by a collisional gas is not required by the fit. Equivalent widths, fluxes and energy centroids of the emission lines found in the 5 7.5 keV energy range are fully consistent with the results reported above. Best fit values are shown in Table 1 and the column is labeled as Obs. 6. The reflected primary continuum has been fitted as described before (see Sect. 3.1): the best fit value of the absorbing column density is 6 . 5 ± 0 . 7 × 10 23 cm -2 , leading to a 2 . 5 × 10 23 cm -2 variation at a 3 σ confidence level in a time scale ranging between 1.5 (time interval between the first and the fifth XMM-Newton observation) and 4 months (time interval between the last XMM-Newton observation and the one with Chandra). In Fig. 2 we show the influence of the change in N H on the spectral shape between 3 and 10 keV. Considering the partial degeneracy between the column density and the spectral slope, the significance of the variation is even stronger, as illustrated by the Γ-N H contour plots shown in Fig. 3.", "pages": [ 3 ] }, { "title": "4 PHYSICAL DISCUSSION", "content": "From the X-ray data analysis presented above a column density variation in a time scale of months is evident. We are going to describe, in the following, the physical implications of this result. Changes in absorbing column density are due to two different physical processes: a variation of the ionizing primary radiation, which causes the variation in the ionization state of the absorber or variations in the amount of absorbing gas along the line of sight. In the case of NGC 4507 the first physical scenario can be clearly ruled out because the difference in the 2-10 keV fluxes between the observations is not significant enough to justify a variation in the primary ionizing radiation. The black hole mass of NGC 4507 is estimated by means of stellar velocity dispersion to be 4 . 5 × 10 7 M /circledot (Marinucci et al. 2012). We assume the dimensions of the X-ray emitting source D S to be 10 R G . This is in agreement with continuum variability studies and disk-corona emission models, all suggesting a compact central X-ray source, confined within a few R G from the central black hole. We also assume that the size of the obscuring cloud D C ∼ D S . A schematic view of the geometrical structure is shown in Fig. 5. The transverse velocity v K for one obscuring cloud is then simply given by the linear dimension of the X-ray source, D S , divided by the crossing time T cr : where we introduced the adimensional parameters M 7 . 65 = M bh / 10 7 . 65 M /circledot and T 4 /similarequal 1 × 10 7 s = 4 months. If we then consider the absorbing material located at a distance R from the central X-ray source, moving with Keplerian velocity, we can calculate R with the simple formula: where R 10 = D S / 10 R G . In the case of NGC 4507 the lower limit on the crossing time is 1.5 months, the time interval between the first and last observation of our XMM-Newton campaign, during which the column density is maximal and nearly constant. The upper limit to the uncovering time ( ∼ D S /v K ) is 4 months, time interval between the fifth XMM-Newton observation (August 2011) and the Chandra one (December 2011). Using these limits in the relations above we get a lower limit of R = 7 pc M 7 . 65 R -2 10 and an upper limit of R = 40 pc M 7 . 65 R -2 10 . These distances imply that the obscuring material is located well outside the BLR. To be consistent with the BLR location the obscuring cloud should have a linear size D C > 20 D S . If we assume a typical BLR density n e ∼ 10 9 cm -3 (Osterbrock 1989) and a N H = 2 . 5 × 10 23 cm -2 (difference between XMM-Newton and Chandra spectra) we get a linear size for the obscuring cloud D C < 3 . 5 D S : these two values are clearly inconsistent. The Suzaku observation is the only one that really probes the typical BLR timescales in NGC 4507, but with an average column density of ∼ 10 24 cm -2 it is not possible to disentangle any inner component with N H ∼ 10 22 -10 23 cm -2 . We reduced and analyzed two 10ks long Swift-XRT observations on the 24 th and 30 th of December 2005 ( obsid 00035465001 and 00035465003 respectively), to check for column density variations on timescales of days. The baseline model we used to reproduce the data is a simple absorbed power law, a Gaussian emission line and a further soft power law for spectral features below ∼ 3 keV. Data quality does not allow for a clear investigation of the Γ-N H parameter space, so we adopted a fixed value of Γ = 1 . 8. Only a marginal column density variation (at 1 σ confidence level) is found between the two Swift observations and the two measurements of N H are consistent at the 90% confidence level (Fig. 4). The lack of significant variability on short time scales does not imply that a BLR component does not exist at all, it suggests indeed a much more complex environment of absorbing structures, as already discussed in Nardini & Risaliti (2011) for the dwarf Seyfert galaxy NGC 4395. The obscuring clouds' velocities we measured (70-170 km/s M 7 . 65 ) are at least one order of magnitude smaller than the typical BLR clouds' velocities. The absorption variability is due to circumnuclear material which is located at distances consistent with the putative torus (Antonucci 1993). Such material cannot be located at much larger scales (i.e. dust lanes) since the NLR is not significantly affected by reddening, as inferred by the observed H α /H β ratio (Kewley et al. 2001). Since the presence of a Compton-thick material around the nucleus is invariably accompanied by a neutral iron narrow K α -emission line and a cold reflection emission it is interesting to point out the fact that the spatial scale of the obscuring material in NGC 4507 is consistent with the distance of the reflector observed with Chandra in the nearby Compton-thick Sy2 NGC 4945 (Marinucci et al. 2012). The existence of a more complex structure surrounding the central engine of AGN rather than the one predicted by the unification model has been proposed and widely discussed in the past few years (Maiolino & Rieke 1995; Elvis 2000; Matt 2000) and recently in Bianchi et al. (2007); Risaliti & Elvis (2010); Bianchi et al. (2012); Elvis (2012). Accordingly to these models, only absorbing matter on very different scales can be responsible for the wide phenomenol- ogy of column density variations, leading to an overall scenario where different absorbers/reflectors are responsible for the spectral changes in the Seyfert 2 galaxies observed so far. The column density variation we measured in NGC 4507 is a further piece that can be added to the puzzle. In our analysis the lack of any variation on short time scales (hours, days) excludes an absorber located in the BLR while the change on time scales of months leads to an absorber much farther from the X-ray source, differently with respect to other variable objects (e.g. NGC 1365, NGC 4151, UGC 4203: Risaliti et al. 2005, 2007, 2009; Puccetti et al. 2007; Risaliti et al. 2010) on long time scales (months, years) and rapidly changing on short ones observed so far. Our analysis provides further evidence that a universal circumnuclear structure of absorbing matter is therefore not suited for taking into account all the observed phenomenology on absorption variability in AGN. While it is true that absorption must occur on different scales, not all the objects present evidence of absorption from all the possible scales.", "pages": [ 3, 4, 5, 6 ] }, { "title": "5 CONCLUSIONS", "content": "We reported in this paper the analysis of 5 XMM-Newton observations spanning a period of 6 weeks and a Chandra observation performed 4 months afterwards of the obscured AGN in NGC 4507. This source had shown strong column density variations in time scales of years (from 1990 until 2007) and none in shorter time scales during the 3 days of Suzaku monitoring. We therefore investigated time scales ranging from 1.5 up to 4 months, looking for absorbing structures located farther from the innermost X-ray source. Our results can be summarized as follows: is located well outside the BLR, and suggest a much more complex environment of absorbing structures; In the next future, following the results presented in Risaliti (2002), a monitoring of all the sources that have shown changes on time scales of years but none on shorter (hoursdays) can be performed. A broad band, time-resolved study of AGN is fundamental for a better understanding of the complex circumnuclear material and its interaction and response to the primary radiation.", "pages": [ 6 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "The authors thank both the referees for their comments, that greatly improved this paper. This work was partially supported by NASA grants NNX11AC85G and GO213124X.", "pages": [ 6 ] }, { "title": "REFERENCES", "content": "Anders E., Grevesse N., 1989, Geochim. Cosmochim. Acta, 53, 197 Antonucci R., 1993, ARA&A, 31, 473 Awaki H., Kunieda H., Tawara Y., Koyama K., 1991, PASJ, 43, L37 Bianchi S., Chiaberge M., Evans D. A., Guainazzi M., Baldi R. D., Matt G., Piconcelli E., 2010, MNRAS, pp 418-+ Bianchi S., Chiaberge M., Piconcelli E., Guainazzi M., 2007, MNRAS, 374, 697 Bianchi S., Guainazzi M., 2007, in di Salvo T., Israel G. L., Piersant L., Burderi L., Matt G., Tornambe A., Menna M. T., eds, The Multicolored Landscape of Compact Objects and Their Explosive Origins Vol. 924 of American Institute of Physics Conference Series, The nature of the soft X-ray emission in obscured AGN. pp 822-829 Bianchi S., Guainazzi M., Chiaberge M., 2006, A&A, 448, 499 Dickey J. M., Lockman F. J., 1990, ARA&A, 28, 215 Elvis M., 2000, ApJ, 545, 63 Elvis M., 2012, in Chartas G., Hamann F., Leighly K. M., eds, AGN Winds in Charleston Vol. 460 of Astronomical Society of the Pacific Conference Series, Quasar Structure Emerges from the Three Forms of Radiation Pressure. p. 186 Elvis M., Risaliti G., Nicastro F., Miller J. M., Fiore F., Puccetti S., 2004, ApJ, 615, L25 Fruscione A., McDowell J. C., Allen G. E., Brickhouse N. S., Burke D. J., Davis J. E., Durham N., Elvis M., Galle E. C., Harris D. E., Huenemoerder D. P., Houck J. C., Ishibashi B., Karovska M., Nicastro F., Noble M. S., Nowak M. A., Primini F. A., 2006, in Observatory Operations: Strategies, Processes, and Systems. Edited by Silva, David R.; Doxsey, Rodger E.. Proceedings of the SPIE, Volume 6270, pp. 62701V (2006). Vol. 6270 of Presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference, CIAO: Chandra's data analysis system Gabriel C., Denby M., Fyfe D. J., Hoar J., Ibarra A., Ojero E., Osborne J., Saxton R. D., Lammers U., Vacanti G., 2004, in F. Ochsenbein, M. G. Allen, & D. Egret ed., Astronomical Data Analysis Software and Systems (ADASS) XIII Vol. 314 of Astronomical Society of the Pacific Conference Series, The XMM-Newton SAS - Distributed Development and Maintenance of a Large Science Analysis System: A Critical Analysis. pp 759-+ Garmire G. P., Bautz M. W., Ford P. G., Nousek J. A., Ricker G. R., 2003, in X-Ray and Gamma-Ray Telescopes and Instruments for Astronomy. Edited by Joachim E. Truemper, Harvey D. Tananbaum. Proceedings of the SPIE, Volume 4851, p. 28-44 Advanced CCD imaging spectrometer (ACIS) instrument on the Chandra X-ray Observatory Guainazzi M., Bianchi S., 2007, MNRAS, 374, 1290 Kewley L. J., Heisler C. A., Dopita M. A., Lumsden S., 2001, ApJS, 132, 37 Kriss G. A., Canizares C. R., Ricker G. R., 1980, ApJ, 242, 492 Krolik J. H., Begelman M. C., 1988, ApJ, 329, 702 Magdziarz P., Zdziarski A. A., 1995, MNRAS, 273, 837 Maiolino R., Rieke G. H., 1995, ApJ, 454, 95 Marinucci A., Bianchi S., Matt G., Fabian A. C., Iwasawa K., Miniutti G., Piconcelli E., 2011, A&A, 526, A36+ Marinucci A., Bianchi S., Nicastro F., Matt G., Goulding A. D., 2012, ApJ, 748, 130 Marinucci A., Risaliti G., Wang J., Nardini E., Elvis M., Fabbiano G., Bianchi S., Matt G., 2012, MNRAS, 423, L6 Matt G., 2000, A&A, 355, L31 Matt G., 2002, MNRAS, 337, 147 Matt G., Bianchi S., D'Ammando F., Martocchia A., 2004, A&A, 421, 473 Nardini E., Risaliti G., 2011, MNRAS, 417, 2571 Osterbrock D. E., 1989, Astrophysics of gaseous nebulae and active galactic nuclei Piconcelli E., Jimenez-Bail'on E., Guainazzi M., Schartel N., Rodr'ıguez-Pascual P. M., Santos-Lle'o M., 2004, MNRAS, 351, 161 Protassov R., van Dyk D. A., Connors A., Kashyap V. L., Siemiginowska A., 2002, ApJ, 571, 545 Puccetti S., Fiore F., Risaliti G., Capalbi M., Elvis M., Nicastro F., 2007, MNRAS, 377, 607 Risaliti G., 2002, A&A, 386, 379 Risaliti G., Elvis M., 2010, A&A, 516, A89 Risaliti G., Elvis M., Bianchi S., Matt G., 2010, MNRAS, 406, L20 Risaliti G., Elvis M., Fabbiano G., Baldi A., Zezas A., 2005, ApJ, 623, L93 Risaliti G., Elvis M., Fabbiano G., Baldi A., Zezas A., Sal- vati M., 2007, ApJ, 659, L111 Risaliti G., Elvis M., Nicastro F., 2002, ApJ, 571, 234 Risaliti G., Miniutti G., Elvis M., Fabbiano G., Salvati M., Baldi A., Braito V., Bianchi S., Matt G., Reeves J., Soria R., Zezas A., 2009, ApJ, 696, 160 Smith R. K., Brickhouse N. S., Liedahl D. A., Raymond J. C., 2001, ApJ, 556, L91 Struder L., Briel U., Dennerl K., Hartmann R., Kendziorra E., Meidinger N., Pfeffermann E., Reppin C., Aschenbach B., Bornemann W., Brauninger H., Burkert W., Elender M., 2001, A&A, 365, L18 Turner M. J. L., Abbey A., Arnaud M., Balasini M., Barbera M., Belsole E., Bennie P. J., Bernard J. P., Bignami G. F., Boer M., Briel U., Butler I., Cara C., Chabaud C., Cole R., Collura A., Conte M., Cros A., Denby 2001, A&A, 365, L27 Turner T. J., George I. M., Nandra K., Mushotzky R. F., 1997, ApJS, 113, 23", "pages": [ 6, 7 ] } ]

[ { "title": "ABSTRACT", "content": "Tentative line emission at 111 and 129 GeV from 16 unassociated Fermi -LAT point sources has been reported recently by Su & Finkbeiner (2012c). Together with similar features seen by Fermi in a region near the Galactic Centre, the evidence has been interpreted as the spectral signature of dark matter annihilation or internal bremsstrahlung. Through a combination of supervised machine-learning algorithms and archival multiwavelength observations we find that 14 out of the 16 unassociated sources showing the line emission in the Su & Finkbeiner sample are most likely active galactic nuclei (AGN). Based on this new evidence, one must widen the range of possible solutions for the 100-140 GeV excess to include a very distinct astrophysical explanation. While we cannot rule out a dark matter origin for the line emission in the Galactic Centre, we posit that if the detection in the Su & Finkbeiner sample is indeed real it might be related to accretion, bubble, or jet activity in nearby ( z < 0 . 2) AGN. Alternatively, given the right conditions, the similarity could be due to a chance occurrence caused by extragalactic background light (EBL) absorption. Or else one must concede that the features are an artefact of instrumental or calibration issues. Key words: (cosmology:) dark matter - gamma-rays: observations - galaxies: active", "pages": [ 1 ] }, { "title": "N. Mirabal 1 , 2 /star", "content": "1 Ram'on y Cajal Fellow 2 Dpto. de F'ısica At'omica, Molecular y Nuclear, Universidad Complutense de Madrid, Spain", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Frantic activity has ensued over the past few months following the report of an excess of Fermi gammaray events clustered around 100 and 140 GeV in a region near the Galactic Centre (Weniger 2012; Tempel et al. 2012; Su & Finkbeiner 2012b), as well as in galaxy clusters (Hektor, Raidal & Tempel 2012a). Dark matter annihilation and internal bremsstrahlung have rapidly emerged as possible explanations (Bringmann et al. 2012; Weniger 2012). Alternative interpretations have also been advanced (Profumo & Linden 2012; Boyarsky, Malyshev & Ruchayskiy 2012; Aharonian, Khangulyan & Malyshev 2012). More recently things have heated up even further with a tantalising claim of similar line emission at 111 and 129 GeV in 16 unassociated sources detected in the Second Fermi -LAT catalogue (2FGL). The detection could provide independent support for a dark matter origin for the line emission seen near the Galactic Centre region (Su & Finkbeiner 2012c). Certainly, such coincidence might not only help us unlock the mysteries of dark matter, but it would also prove the existence of dark matter subhaloes (Klypin et al. 1999; Moore et al. 1999; Springel et al. 2008). In the absence of obvious flaws in the analysis, the collected evidence has risen as a sort of dark knight - albeit an indirect one - that might finally grant us non-gravitational access to dark matter. Intrigued by this possibility, we explore the nature of the 16 Fermi unassociated sources listed by Su & Finkbeiner (2012c). Based on machine-learning classification algorithms and multiwavelength examination, we show that 14 out of the 16 unassociated Fermi sources displaying the lines are likely gamma-ray AGN. Therefore, rather than strengthening the argument, the detection of an identical signal in the Su & Finkbeiner sample appears to disprove a dark matter origin for the Fermi features unless a set of very unique astrophysical conditions are met. The paper is structured as follows. In Section 2 we explain the machine-learning classifier Sibyl . In Section 3 we compile class prediction for the 16 unassociated Fermi sources listed in Su & Finkbeiner (2012c). Section 4 details multiwavelength searches for the potential counterparts of these 16 objects. Finally, we provide some interpretation in Section 5.", "pages": [ 1 ] }, { "title": "2 SIBYL", "content": "Confirmation of a truly unique type of gamma-ray source would hint that we may have finally found the much soughtafter dark matter subhaloes predicted by numerical sim- ulations. As our base for such comparison, we use Sibyl , a Random Forests classifier that generates predictions on class memberships for unassociated Fermi -LAT sources using gamma-ray spectral features extracted from the 2FGL (Mirabal et al. 2012). Only a brief description of Sibyl is presented here since it has been thoroughly covered in the literature (Mirabal et al. 2012; Hassan et al. 2012). Random Forests (RF) create an ensemble of classifiers with a tree structure, where the splits captures the complexity of the feature space among the set of training objects (Breiman 2001). To tag a new object, RF lets each classifier vote and then outputs a prediction based on the majority of votes (P > 0 . 5). RF also computes proximities between pairs of objects and quantifies which variables are instrumental to individual classification. Previously we performed a similar analysis for unassociated 2FGL sources at | b | /greaterorequalslant 10 · (Mirabal et al. 2012). Here, we extend the coverage down to | b | /greaterorequalslant 5 · in order to encompass the entire Su & Finkbeiner sample. The classifier presented has been implemented with the R randomForest package (Liaw & Wiener 2002). As in Mirabal et al. (2012), we trained Sibyl using 800 labelled AGNs (BL Lacs and flat-spectrum radio quasars only) and 108 pulsars from the the complete Fermi LAT 2FGL catalogue (Abdo et al. 2010; Ackermann et al. 2011; Nolan et al. 2012). There are additional gamma-ray classes in the 2FGL, but since the 16 unassociated sources in Su & Finkbeiner (2012c) lie at | b | /greaterorequalslant 5 · we do not expect noticeable contamination from Galactic plane sources. The main task thus is to find out whether the unassociated sample from Su & Finkbeiner (2012c) falls into these two categories or it is clearly different from these types of objects. During training and testing with the 908 labelled sources, we used a total of 7 spectral features: Index, Curve, Variability, and Flux Ratios ( FR 12 , FR 23 , FR 34 , and FR 45 ) (Mirabal et al. 2012). Assuming class bimodality, Sibyl achieves an accuracy rate of 97.1% based on majority voting (97.7% for AGNs and 96.5% for pulsars). Inspection of the results shows that misidentifications tend to occur when less than 70% of the individual classifiers (P < 0 . 7) agree on a particular prediction. Therefore we set this threshold as our internal limit for a valid prediction.", "pages": [ 1, 2 ] }, { "title": "3 APPLICATION TO THE SU & FINKBEINER SAMPLE", "content": "Initially, we want to examine whether the set of unassociated sources showing line emission at 111 and 129 GeV (Su & Finkbeiner 2012c) is distinct in any way when compared to the bulk of associated Fermi sources. For each of the 16 unassociated Fermi sources listed in Su & Finkbeiner (2012c), Sibyl provides a prediction that the object is an AGN(P AGN ) or a pulsar (P Pulsar ) based on individual votes tallied from the classifiers. We adopt a threshold P > 0 . 7 to accept a prediction i.e., at least 70% of the trees agree on the final decision. Conservatively, sources failing to reach the threshold remain formally unassociated and most likely constitute interesting gamma-ray sources. In total, Sibyl predicts 14 objects in the Su & Finkbeiner sample to be AGN. The resulting predictions and percentages of voting agreements are listed in Table 1. Only two objects 2FGL J1716.60526c and 2FGL J1721.5-0718c remain without a firm pre- diction. We note that both sources are also the only ones fitted with LogParabola functions in the Su & Finkbeiner sample. But such a pair is not uncommon as there are at least 163 associated Fermi sources with LogParabola best fittings in the 2FGL including numerous AGN, pulsars, and supernova remnants (Nolan et al. 2012). Furthermore, both sources have attached caution flags to indicate possible problems with the diffuse model that might lead to odd spectral behaviour (Nolan et al. 2012). Notably, we find that there are no outliers relative to the predicted classes among the 16 sources under scrutiny (Mirabal et al. 2012). Outliers correspond to cases far removed from the rest of the objects. To locate such cases, RF computes the outlier measure as the inverse of the average squared proximity between an individual object and all other objects (Breiman 2001; Liaw & Wiener 2002). Typically, outliers can be found with outlier measures greater than 10. In the 16 sources chosen, the largest outlier measure corresponds to 1.8. There are also no signs of potential dark matter subhalo candidates in any of the previous Fermi searches conducted to date (Buckley & Hooper 2010; Mirabal, Nieto & Pardo 2010; Nieto et al. 2011; Belikov, Hooper & Buckley 2011; Zechlin et al. 2011; Ackermann et al. 2012; Mirabal et al. 2012). Lastly, the photon flux distribution for the 16 unassociated sources is compared to the overall distribution for the full Fermi -LAT unassociated sample (573 sources) in Figure 1. Application of the Wilcoxon rank sum test returned a p-value of p = 0.07946, which indicates that the distributions are not statistically significantly different. Except for a slight mismatch at the very bright end, we find no obvious selection effects that could produce line emission in this particular set.", "pages": [ 2 ] }, { "title": "4 MULTIWAVELENGTH EXAMINATION", "content": "To the untrained eye, it might seem overly simplistic to rely on machine-learning algorithms to make a class prediction for a particular source. We simply refer the reader to the vast amount of research and applications of classification methods that have managed to reach tremendous accuracies in a variety of astrophysical subfields (Bloom & Richards 2011; Richards et al. 2012). However, one must never forget that a smart computer generated guess is no substitute for observation 1 . We take this recommendation at heart and move the association process even further by searching for the actual counterparts in archival multiwavelength observations that have partially or fully covered the Fermi 95% confidence error ellipses of the 16 unassociated sources. For this purpose, we employ a set of well-validated strategies (Mirabal et al. 2000; Reimer et al. 2001; Mukherjee et al. 2002). The wide distribution of sources over the sky results on a hodgepodge of radio and X-ray observations from various existing catalogues. For radio counterparts, we relied on measurements from the Green Bank (GB6) catalogue at 4.85 GHz (Gregory et al. 1996), the 4.85 GHZ Parkes-MITNRAO (PMN) survey catalogue (Griffith & Wright 1993), the 1.4 GHz NVSS catalogue (Condon et al. 1998), and the 843 MHz SUMSS catalogue (Mauch et al. 2003). In total, we find that 13 out of the the 16 unassociated sources have prominent potential radio counterparts within their Fermi 95% confidence error ellipses. We complement the radio matches with observations from the most ambitious X-ray program for counterpart identification currently underway (Falcone et al. 2011), which aims to image the totality of unassociated Fermi sources with the Swift X-ray telescope. To date, nine sources have been imaged by Swift with times ranging from 1.1 to 19.1 ks of useful exposure. Source extraction to identify all significant X-ray sources within the Fermi error ellipses was performed with wavdetect . Source positions and positional errors were derived using xrtcentroid . X-ray counts (0.1-2.4 keV) were extracted from a circular region with a 20 pixel radius (47 '' ). The background was extracted from an annulus with a 20 pixel (inner radius) to 30 pixel (outer radius) around the source. Throughout, we used XSELECT to include counts with grades 0-12. Six out of the nine Swift fields have single potential counterparts within their Fermi 95% confidence error ellipses. The ROSAT All-Sky Survey Faint Source Catalogue (Voges et al. 2000) adds four more potential single X-ray counterparts to the final count. Table 2 summarises the counterpart candidates in each case. Of the 16 sources, seven have both radio and X-ray tentative counterparts. Figure 2 shows X-ray flux versus radio flux density of associated Fermi AGN . Superposed are the potential seven with simultaneous radio and X-ray counterparts. The results are in line with radio and X-ray counterpart flux levels expected for typical Fermi AGN. But we must emphasise that without dedicated spectral classification in the optical these must be considered solid but tentative counterparts at this stage.", "pages": [ 3 ] }, { "title": "5 INTERPRETATION AND CONCLUSIONS", "content": "We have presented class predictions of the Random Forest classifier Sibyl for 16 unassociated Fermi sources showing line emission at 111 and 129 GeV. We find that 14 out of 16 unassociated sources in the Su & Finkbeiner sample are AGNcandidates with prediction accuracy rates greater than 97.1%. In addition, we have detected 10 X-ray and 13 radio potential counterparts distributed over the 16 unassociated Fermi 95% confidence error ellipses that would be consistent with the AGN predictions. We emphasise the word potential here as a more exhaustive detective work must be completed to confirm the appropriate counterpart for each unassociated source. It was postulated that the gamma-ray lines among the unassociated were perhaps connected to dark matter subhaloes dragged into the Galactic disc (Su & Finkbeiner 2012c). However, assuming an isotropic distribution, at least 160 Fermi AGN are expected at | b | /lessorequalslant 10 · . To date only about 100 are accounted for in the 2FGL (Ackermann et al. 2011). Thus, it makes astrophysical sense that AGN are making up an important fraction of the Su & Finkbeiner sample even at relatively low Galactic latitudes. In light of these results, the dark matter origin for the narrow gamma-ray features observed by Fermi is in question. Were these dark matter subhaloes (Baltz, Taylor & Wai 2007; Diemand et al. 2008; Kuhlen, Madau & Silk 2009), coincidence between the Galactic Centre and the Su & Finkbeiner sample would certainly confirm a dark matter particle origin (Hooper & Linden 2012). However, the interpretation changes dramatically if the unassociated sources showing an identical line signature are AGN, as implied by both machine-learning classifiers and the multiwavelength argu- ments just presented. Dark matter could be fed into AGN jets and the Galactic Centre, but such an explanation feels contrived given the hadronic and leptonic dominance in the gamma-ray photon field (Hinton & Hofmann 2009). Instead, a distinct astrophysical mechanism unrelated to dark matter annihilation and linked to nearby AGN ( z < 0 . 2 to avoid redshifted lines) such as accretion, bubble (Su et al. 2010; Profumo & Linden 2012), or jet (Su & Finkbeiner 2012a) phenomenology would appear to be more logical. However, we note that although many Fermi AGNdisplay photons above ∼ 10 GeV, only a handful of soft AGN (Γ > 2) exhibit maximum photon energies greater than 100 GeV at z > 0 . 5 (Ackermann et al. 2011). Consequently, Su & Finkbeiner (2012c) might be detecting a fiendish cluster of events imprinted by EBL absorption in the same energy band, but completely unrelated in origin to the emission observed near the Galactic Centre region. Oddly enough, the lines reported by Su & Finkbeiner (2012c) appear to be only present collectively in unassociated sources and do not appear as pronounced among associated sources, including well-known gamma-ray AGN (Su & Finkbeiner 2012c). Therefore, we must also admit the possibility that the spectral signatures detected by Fermi originate from confounding instrumental or calibration problems (Hooper & Linden 2012; Hektor, Raidal & Tempel 2012b,c; Finkbeiner, Su & Weniger 2012). The Fermi calibration team will have the final word on the matter very soon, but independent efforts must be made to scan the public Fermi archive for gamma-ray lines among individual AGN at z < 0 . 2, as well as in diffuse emission outside the Galactic plane. We shall hear more about this energy region by the end of the year with the recently unveiled H.E.S.S. II (Becherini et al. 2012; Bergstrom et al. 2012), and even more sensitive observations will be available later on after completion of the Cherenkov Telescope Array (CTA Consortium 2011). In the future, a dark knight might rise again. Until then, we eagerly await for the final chapter of this intriguing saga.", "pages": [ 3, 4 ] }, { "title": "ACKNOWLEDGMENTS", "content": "N.M. acknowledges support from the Spanish government through a Ram'on y Cajal fellowship and the ConsoliderIngenio 2010 Programme under grant MultiDark CSD200900064. We thank Doug Finkbeiner for helpful email exchanges. We acknowledge the use of public data from the Swift data archive. This research has made use of data obtained from the High Energy Astrophysics Science Archive Research Centre (HEASARC), provided by NASA's Goddard Space Flight Centre. We also thank the referee for useful suggestions and comments on the manuscript.", "pages": [ 5 ] } ]

[ { "title": "ABSTRACT", "content": "One might expect light to be scattered when it passes through a gravitational wave, and might hope that in favourable circumstances these scatterings could be observed on Earth even if the interaction occurs far away. Damour and Esposito-Farèse, and Kopeikin, Schäfer, Gwinn and Eubanks, found that there were cancellations making such effects disappointingly small. Here I show that those cancellations depend on the emission of the light occurring far behind the gravity-wave source; for light-emissions near that source, larger effects are possible. I first develop a covariant treatment of the problem in exact general relativity (the propagation of light being modelled by geometric optics), and then specialise to linearised gravity. The most promising candidates identified here for detection in the not-too-distant future would involve sufficiently tight binaries as sources of gravitational radiation, and nearby pulsars as lightsources. In some favourable but not extreme cases, I find offsets in the pulses' times of arrival at Earth by ∼ 10 -10 -10 -9 s , with periods half the binaries' periods. Key words: gravitational waves - relativity - pulsars: general", "pages": [ 1 ] }, { "title": "Adam D. Helfer glyph[star]", "content": "Department of Mathematics, University of Missouri, Columbia, MO 65211 U.S.A. Accepted xxxx. Received xxxx; in original form xxxx", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "The detection of gravitational waves is a critical test of Einstein's theory of general relativity. We have at present good indirect evidence for the existence of such waves in the orbital decays of binary pulsars (Taylor & Weisberg 1989), but no direct evidence. Unfortunately, we expect most astrophysical sources to be so distant that the waves should be very weak by the time they reach the solar system, making terrestrial detection a challenge. It is natural to ask whether gravitational waves might have effects on matter or light closer to the sources, where the waves are stronger, but with the effects being observable on the Earth. In particular, one might search for modulations in light caused by its passage through gravitational waves, for instance light from another electromagnetic source passing nearby a neutron star or binary system. (Here and throughout, 'light' means electromagnetic signals of any frequency which can be treated by geometric optics.) These modulations could include variations in intensity, in apparent position, in polarisation, in phase, or in time-dilation (often referred to in the literature as time-delay). Of course, such variations could also be produced by other means; one would need further information to conclude they were caused by gravitational waves.", "pages": [ 1 ] }, { "title": "1.1 Previous work and its implications", "content": "Such possibilities were contemplated by a number of authors, notably Sazhin (1978), Labeyrie (1993), Durrer (1994) and Fakir (1994); it seemed from these papers that some of the effects might be detectable in the near future. Concerns, though, were raised about the adequacy of those treatments, and Damour & EspositoFarèse (1998) and Kopeikin, Schäfer, Gwinn & Eubanks (1999) (see also Kaiser & Jaffe 1997), after more considered analyses, found very much smaller effects than then the optimistic ones in the earlier papers. 1 The effects were so small that they seemed to preclude the detection of gravitational waves by these means, at least for known classes of systems in the near future. This conclusion has been accepted by much of the community (Schutz 2010; Book & Flanagan 2011). But the situation is not so clear-cut. For one thing, the later authors' investigations, while overlapping the earlier ones, do not cover all of the physical possibilities they considered. Also the results as they stand strongly suggest there are essential points which have not yet been understood. We can appreciate this if we ask how such gross discrepancies between the earlier and later work could have occurred. While it is true that much of the earlier work involved rough approximations, one would think that in order to explain such a large and systematic change there must have been a shift in what was taken to be the essential physics. The earlier papers typically and very plausibly found that the light rays responded to the gravitational radiation fields of the sources, but later authors found that there were cancellations which resulted in those terms not entering. 2 Indeed, Damour and Esposito-Farèse pointed out that pure, vacuum, gravitational waves would give no overall scattering in linearized theory. More specifically, both Damour and Esposito-Farèse, and Kopeikin et al., reported that what scattering of a light-ray by an isolated source (on a straight world-line in linearised theory) does occur depends only on the source's configuration at the light-ray's point of closest approach. This is something which cannot be locally causally determined, and therefore must depend on a non-local cancellation of effects over the light-ray's trajectory. These observations suggest that the cancellations are tied to the infinite extent of the light-ray's trajectory, and may not occur for propagation along finite, or only half-infinite, segments. 3 We shall see below that this is indeed the case. In other words, the cancellations are present when the light effectively comes in from infinity, but larger effects are possible when the emission is from a finite point, in particular when it might be close to the gravitational-wave source. Such situations are very much of interest, because there are many astrophysical configurations in which there is a source of light near an expected source of gravitational radiation. This fits well with other points in the literature. One can see that many of the earlier works did, either explicitly or in effect, use scattering estimates which depended on integration over halfinfinite or finite segments rather than doubly-infinite lines, and in some cases (depending on the situation considered) the authors were correct in wanting to do so. And Damour and Esposito-Farèse (see also Faraoni 2008) correctly noted that overcoming the cancellations would depend on 'edge effects'. The paper of Kopeikin et al. is sometimes considered to show that no significantly larger effects arise from finite or half-infinite segments. This, however, is not really correct. While these authors did indeed work out formulas for the null geodesics around certain sources in linearised gravity, those formulas are very lengthy and were examined in detail only in certain limiting cases. None of those cases corresponds to the configuration of ultimate interest here, a light-emission reasonably close to the gravitational-wave source (precisely, within of the order of the impact parameter to the light-ray's point of closest approach to the gravity-wave source) received by a distant observer. In fact, the computations of Kopeikin et al. are consistent with the possibility of larger effects from restricted segments. There is, additionally, a key issue which must be properly addressed if we are to take into account the emission (or reception) of light-rays at finite events, especially events near gravitationally radiating sources. We do not expect to learn anything about gravitational radiation from the reception of one light-ray; rather, it is in the changes in the light received over time that the we hope the information is encoded. In other words, it is differential scattering effects, of successive light-rays relative to each other, which are of interest. But to treat these properly, we must know we are modelling the emissions of the successive rays accurately. This is potentially problematic when the light-sources themselves are in the region affected by the gravitational waves, because the effects of the waves on the light through their perturbations of the motion of the light-sources compete with the other contributions. Another way of saying this is that the received signal depends, not only on the gravitational field through which the light has passed, but also on the apparent acceleration of the light-emitter, as inferred from electromagnetic measurements at the receiver. Even if the emitter suffers no local acceleration, its trajectory as reconstructed from light-signals coming into the detector will appear to involve accelerations, because of the gravitational radiation the light has passed through. (And so one might want to say that the variations in the received signal are not purely due to scattering, but also to this non-local acceleration of the source. However, from a general-relativistic point of view these are two facets of the same thing.) This is a point which has only been partially attended to in the literature on scattering. 4 In certain restricted cases, where the gravitational waves have a simple form and the light-rays considered are aligned to take advantage of it, the issue has been considered (e.g. Sazhin 1978), but the pattern has been to compute the null geodesics but not examine all the details of the time-dependence of light-emission and reception. For instance, Kopeikin et al. (1999) worked out the null geodesics around certain gravitational sources in linearised theory, in terms of the geodesics' initial positions and tangents. They derived formulas for the coordinates of finite points along the geodesics, but they did not fully resolve the question of how to choose the data for a one-parameter family of null geodesics emitted at one world-line and received at another. (They addressed some aspects of the issue, with their discussion of the appropriateness of their gauge choice. That argument shows that some lightemitters may be modelled by taking the spatial coordinates, in the gauge used, fixed. For such an emitter, the initial positions of the null geodesics would simply be the points on the emitter's worldline. However, this leaves open the question of how to choose the geodesics' tangents, so as to ensure the geodesics go from the emitter to the receiver; it also leaves open the question of how to treat generic emitters, which need not have their space coordinates constant.) Developments of this approach can of course take into such variations; see Kopeikin et al. (2011). We wish then to extend the existing treatment of lightscattering by gravitational fields to propagation over finite or halfinfinite segments, accounting for the gravitational waves' effects on the light-sources in those cases. It is worthwhile considering what the best approach is.", "pages": [ 1, 2 ] }, { "title": "1.2 A more general setting", "content": "Virtually all work that has been done in this area has been done in the linearised-gravity (or at least linearised gravitational-wave) setting, and has used detailed coordinate calculations in specific gauges. There are reasons for wanting a more general, and invariant, approach. Most importantly, the core issue - how light emitted from successive events along one world-line is received along another, after passage through a region of space-time - is evidently a basic problem in relativity: it is worth treating in generality. It is a tenet of relativity that the physics is invariant, so invariant treatments focus most directly on the physics; a central goal of this paper is to give one. As emphasized earlier, the key physical observables are associated with differential scattering, of successive light-rays relative to each other. This is governed by the geodesic deviation equation, a linear ordinary differential equation involving the curvature, and it is the solution of this equation which is the central mathematical chore in an invariant approach. By contrast, gauge-based approaches focus on the metric and on integrating the null geodesics, a considerable task. It is only once these are known that one can proceed to extract the measurable differential effects (and, as noted above, this further step has not generally been completely worked out in these approaches). One sees that the invariant approach goes more directly to the observables. Also, by limiting oneself to differential effects, one expands considerably the cases which can be treated, since one need not confront the problem of finding the null geodesics explicitly. The invariant formulas derived here would, for instance, be applicable in a cosmological context, or to treat lightrays in strong gravitational fields. With the invariant formulas in hand, one can always specialise to gauge choices which may be convenient for particular problems. But not having to make such a choice too early saves much work, and does away with many conceptual issues associated with gauges. In some of the literature the motivation for the choice of gauge is to be confident that one is indeed modelling gravitational waves, as opposed to other gravitational disturbances. In this paper, that is a distinction which is best made, not at the beginning of the analysis, but rather once we have the general formulas. This is because there is no very simple invariant characterization of what counts as gravitational radiation and what does not. We first derive general, invariant, formulas; we may then specialise these to systems which are gravitationally radiating.", "pages": [ 2, 3 ] }, { "title": "1.3 Goals and results", "content": "The aims and results of this paper are: (a) To provide an invariant treatment of the effects of gravity on light, in the approximation where the electromagnetic field can be treated by geometric optics, in exact general relativity, and a specialisation of that treatment to linearised gravity. In particular, while much of the motivation for this work comes from wanting to understand gravitational waves, the approach here allows us to treat any gravitational fields in general relativity. I emphasized above that the effects of interest were all differential ones, where one light-ray is compared with a nearby one. Thus the main tool in the analysis is the geodesic deviation equation. Essentially, the task is to solve this equation with initial and final conditions corresponding to the emission of light at one worldline and its absorption at another. This approach allows us to disregard many of the coordinate issues which occur elsewhere. For instance, previous papers have tended to discuss 'time-delay' effects in terms of local coordinate times, whereas it is really the clock times of the emitter and the receiver of light which are relevant; the approach here gives an integro-differential equation directly relating the clock times. There are similar equations for the changes observed at the receiver in the wave-vector (this is equivalent to equations for the change in phase, since the wave-vector is the gradient of the phase) and the electromagnetic field (and thus the amplitudes and polarisations). In all of these cases, the changes are determined partly by the geometry of the space-time the light propagates through, partly by the world-lines of the emitter and receiver, and partly by any intrinsic changes in the field at the emitter. All of these contributions are accounted for directly. These results are closely related to those of Damour and Esposito-Farèse; the chief difference is that they worked with the field's Fourier transform and gave global results, whereas our analysis is directly in terms of the radiative data and considers not only the limiting case of a doubly-infinite geodesic, but how that limit arises. Similarly, the results here are also compatible with the computations of Kopeikin et al. (b) To give rough estimates of the effects in question in the case of quadrupolar radiation in linearised gravity. Of the various effects on the propagation of light by its passage through, or emission in, a gravitational wave field - changes in its position on the celestial sphere of the receiver, in its phase, polarisation, amplitude, or time-dilation - the last is, at present, the least difficult to imagine measuring. We shall find that changes in the position on the celestial sphere and phase are suppressed by a geometric factor (related to the 'lighthouse effect'). While changes in polarisation or (fractional) amplitude might be, in numerical terms, of the same magnitudes as those due to time-dilation, it is hard to foresee polarisation or intensity measurements accurate enough to reveal gravitational-wave effects. The time-dilation effects (often referred to as 'time-delay effects') arise from the distortion τ 1 ( τ ) of the light-emitter's proper time τ 1 as measured by light-signals arriving at the receiver's proper time τ . This function τ 1 ( τ ) depends on gravitational radiation the light encounters; in particular, we shall see that an integral of a curvature component over the geodesic contributes to d 2 τ 1 /dτ 2 . One result of this is a red-shift; also, if the emitter can be expected to give off signals at known proper times (for instance, if it is a pulsar), then the light's passage through a gravitational wave may cause the times of arrival of those signals to wander slightly from what they would have been, had no radiation been present. The detection of these effects would require training a telescope, whether optical or radio, on a light-emitter in the vicinity of a gravitational-wave source. For this reason we are interested primarily in sources which have at least a reasonable likelihood of radiating in a reasonable observing period. These will be continuous, localised, sources of gravitational waves, which unfortunately are thought to be weak. The best sort of candidate found here would involve a tight binary as the source of gravitational waves, with a nearby pulsar (perhaps as a distant tertiary) as the light-emitter. For two 5 M glyph[circledot] stars (or black holes) in a . 1 -day circular orbit, I find possible displacements ∆ τ max in pulse arrival times ∼ 2 × 10 -10 s . For (as might possibly be found in a globular cluster) a one-solar-mass star in a one-day circular orbit around a 10 4 M glyph[circledot] black hole, one has ∆ τ max ∼ 3 × 10 -9 s . For (as might happen near the galactic centre) a star of mass m in a ten-year circular orbit around a 4 × 10 6 M glyph[circledot] black-hole, one has ∆ τ max ∼ 7 × 10 -10 ( m/M glyph[circledot] ) s . These displacements would vary with a period half the orbital period.", "pages": [ 3 ] }, { "title": "1.4 Organization", "content": "The next section develops a general formalism for the analysis, valid in full general relativity, based on geodesic deviation. In Section 3, this is specialised to the case of linearised gravity; there the geodesic deviation equation can be solved in terms of curvature integrals. Section 4 considers the effects of quadrupole sources (allowing both 'electric' and 'magnetic' terms) of gravitational radiation on light-rays in the sources' wave zones. In Section 5, those results are applied to estimate the magnitudes of the effects for some expected sources of gravitational radiation. Section 6 contains a summary and discussion. The reader wishing to skip the derivations will find the formula (77) for the comparison of emitter and receiver clock-times at the end of section 4; the meanings of the symbols in this formula are recapitulated in the paragraph containing it. The rough estimate of the magnitude of the effect is (79). Sections 5 and 6, dealing with the estimates, depend (except for a few technical comments which can be skipped) only on this rough form and can be read independently of the earlier sections. Notation and conventions. Except where otherwise specified, the notation and conventions here are those of Penrose & Rindler (1984, 1986), which will also serve as a reference for all material not otherwise explained. The metric signature is + - --, and the curvature satisfies [ ∇ a , ∇ b ] v d = R abc d v c . Factors of c , the speed of light, are omitted until the end of Section 4. In keeping with this paper's main aim of an invariant treatment, the basic formalism is, in principle, the abstract-index one of Penrose and Rindler. However, in fact, in almost all cases the tensorial indices in this paper may be interpreted equally well as abstract indices or as component indices with respect to a chart. Where there is any difference, it is noted.", "pages": [ 3, 4 ] }, { "title": "2 GENERAL FORMALISM", "content": "Let us suppose we have two world-lines γ j ( τ j ) (with j = 1 , 2 ) in space-time, with τ j proper time on each. These will be the worldlines of the source and the detector of the light-rays. 5 They need not be geodesic, and they need not be in any asymptotic or weak-field region. Let p ( s, τ ) be a smooth family of light rays from γ 1 to γ 2 : for each fixed τ , the light ray runs from γ 1 ( τ 1 ( τ )) = p ( s 1 ( τ ) , τ ) to γ 2 ( τ 2 ( τ )) = p ( s 2 ( τ ) , τ ) , with s an affine parameter along the ray. Then l a = ∂ s p will be the tangent null vector, and w a = ∂ τ p will be the connecting, Jacobi, field. 6 Note that we have on γ 1 , γ 2 . We shall take τ = τ 2 , so as to index the light-rays by the receiver's proper time. There is some normalisation freedom in this: for each τ , the vector ( s 2 ( τ ) -s 1 ( τ )) l a is determined, but the individual values s 2 ( τ ) , s 1 ( τ ) , l a are not. It will be simplest to take so 5 We should write γ a j ( τ j ) for the coordinates of a curve in a chart, with a a coordinate index; similarly we would have p a ( s, τ ) for the coordinates of a family of light-rays below. (Since the coordinates themselves are not tensorial quantities, the quantities γ j ( τ j ) and p ( s, τ ) do not carry abstract indices. We shall be interested in how the light-ray varies with τ , and hence in the connecting field w a . Contracting the equation l · ∇ w a = w · ∇ l a with l a , we find l · ∇ ( w · l ) = 0 , which, with eq. (4), gives us It is dτ 1 /dτ which gives rise to what are often called 'time-delay' effects, and eq. (5) will allow us to solve for these. We shall want the connecting vector w a in terms of the curvature along the light-rays. For this, we must use the Jacobi equation in some detail. Let U a b ( s, τ ) , V a b ( s, τ ) be solutions, so ( l · ∇ ) 2 U a b = l p l q R pcq a U c b , ( l · ∇ ) 2 V a b = l p l q R pcq a V c b , with and Then the connecting field is for some α a , β a (elements of the tangent space at γ 1 ( τ 1 ) ). We can find α a , β a by using eq. (4); this gives Note that l · ∇ ( l · w ) = 0 , applied to eq. (10) and evaluated at s = s 1 , implies l · β = 0 . Of course, at a conjugate point V a b will not be invertible; conjugate points will be discussed elsewhere.", "pages": [ 4 ] }, { "title": "2.1 Time dilation", "content": "Much of the literature is phrased in terms of 'time delays', the delay being taken to be the difference in the coordinate times of emission and reception. This is not an invariant concept, and so it is then corrected (or corrections are at least as a matter of principle considered) to take into account differences between the clock times and the coordinate times. 7 We will work with the clock times directly, the basic quantity of physical interest being τ 1 ( τ ) , the time along γ 1 at which a signal was sent to arrive at time τ on γ 2 . Then dτ 1 /dτ , the time-dilation or red-shift factor. 8 It is changes in this quantity which may carry the imprint of gravitational radiation. We have Expressing this in terms of U and V , we have The last terms of course vanish when the source and the emitter are freely falling. (In eqs. (13), (14), and subsequently, the accelerations ¨ γ a j = ˙ γ j · ∇ ˙ γ a j are taken with respect to the proper time τ j along the corresponding world-line.)", "pages": [ 4, 5 ] }, { "title": "2.2 Change in wave-vector", "content": "In the geometric-optics approximation, the wave-vector is where ω 1 is the angular frequency with respect to the frame of the emitter. We are interested in the time-dependence of k a at the detector's world-line, γ 2 ( τ ) . For simplicity, we will assume here that ω 1 is constant. (Otherwise, in what follows, one simply gets an extra term, from the product rule.) Then applying w · ∇ to eq. (15), we have The last term in the brackets is proportional to the acceleration of γ 1 , and vanishes if that world-line is freely falling. Also, the equation (14) for the time-dilation can be regarded as a consequence of eq. (16), since d 2 τ 1 /dτ 2 = ˙ γ 2 · ∇ ( ˙ γ 2 · k/ω 1 ) . Finally, a physicist receiving signals must decide how to compare successive measurements of k a along γ 2 . If γ 2 is a geodesic, there is a natural choice: parallel-transport, which leads to the differential formula (16). If, however, γ 2 is not a geodesic, one might prefer to use Fermi-Walker transport. It is easy enough to interconvert the two, the Fermi-Walker derivative of k a along γ 2 being so one would supplement (16) by an additional term. The choice of which quantity to use is really a question of which measures of the change one is most interested in reporting. The covariant derivative gives us ones less sensitive to the geometry of the worldline γ 2 ; the Fermi-Walker derivative is more frame-dependent but measures more directly the changes in frequency and spatial wavevector relative to the observer. The same principle will apply to changes in the field.", "pages": [ 5 ] }, { "title": "2.3 Change in the field", "content": "We may also analyse the effects on the received electromagnetic field of its propagation through the gravitational field; this includes changes in amplitude and polarisation. As is well-known, in geometric optics the field F ab is transverse to the direction l a of propagation. That is, however, not a relativistic formulation of the condition; relativistically the statement is that l a must be a repeated principle null direction of F ab (Pirani 1965; Penrose & Rindler 1986). We may express this conveniently by choosing a complex null vector m a (covariantly constant along l a and normalised to l · m = 0 , m · m = -1 ); then the field must have the form for a scalar field φ . One can fix the freedom in the choice of m a to have φ glyph[greaterorequalslant] 0 ; then the two-form l a m b -m a l b carries the polarisation information. One has where ρ = -(1 / 2) ∇· l is the convergence of l a . It is convenient to introduce a luminosity distance r such that l ·∇ r = -ρ r . Then l ·∇ ( r F ab ) = 0 and r F ab is parallel-transported along the null geodesic. This means that the field F ab diverges as 1 / r at γ 1 , but that is simply a mathematical artefact of modelling the emitter as a point source. Really, one should imagine a finite surface of emission, and r F ab taking direction-dependent limits as one approaches this surface. The natural normalisation for r is then with respect to the world-line of the electromagnetic source, that is ( ˙ γ 1 · l ) -1 l ·∇ r ∣ ∣ ∣ s 1 = 1 . The direction-dependent limit of r F ab on γ 1 is a measure of the intrinsic strength of the field at the source, which will vary along the world-line. We can express r in terms of the quantities already given. Since ρ measures -1 / 2 the logarithmic rate of increase of the surface area element along the rays abreast l a , the luminosity distance is the square root of the area element. Taking into account the normalisation, we have We have then Integrating this, we have where we understand that parallel propagation along the null geodesic has been used to identify w · ∇ ( r F ab ) ∣ ∣ ∣ s 1 with a tensor at γ 2 , as well as to define the integrals. 9 Then The last term is a pleasingly clean Lorentz transformation derived from a curvature integral; the middle term corresponds to changes in the received field due to intrinsic changes in the source. The first term, due to changes in luminosity distance, can be expressed (somewhat lengthily) in terms of the data we have, as follows. We have, from eq. (20), Here Differentiating the normalisation conditions for m a , we find where the terms proportional to l a or l b will not contribute to eq. (24), because those are eigenvectors of V p q . Finally, we must compute ˙ γ 2 ·∇ V a b = w ·∇ V a b ∣ ∣ ∣ γ 2 . Differentiating the Jacobi equation (which V a b satisfies) and working out some commutators, we find 9 Thus an integral written as ∫ s 2 s 1 Q ab (' s ) d ' s is really ∫ s 2 s 1 P a c (' s ) P b d (' s ) Q cd (' s ) d ' s , where P a c (' s ) λ c is the result of parallelpropagating λ c to γ 2 along the null geodesic p (' s, τ ) ; similarly we should have P a c ( s 1 ) P b d ( s 1 ) w · ∇ ( r F cd ) ∣ ∣ ∣ s 1 for w · ∇ ( r F ab ) ∣ ∣ ∣ s 1 . where We can regard eq. (27) as a sort of inhomogeneous Jacobi equation with source S a b , and solve it by variation of parameters. The result is where the indices have been omitted (and matrix operations are to be understood throughout) in the interest of clarity. Combining eqs. (25), (26), (29) (evaluated at s = s 2 ) and (28) gives the first term on the right in eq. (23), and this completes the formula for the change in the field at the receiver.", "pages": [ 5, 6 ] }, { "title": "3 PASSAGE TO LINEARISED GRAVITY", "content": "We will now specialise to linearised gravity. Thus we regard the metric as a first-order perturbation of the Minkowskian one. We shall nevertheless avoid an explicit choice of gauge, and continue to present the results in an invariant form, in order to keep the geometry and physics as clear as possible. There is an overall issue to keep in mind in such schemes: in general, a quantity of interest will have both zeroth- and firstorder terms. If it is not a scalar, and its zeroth-order term is nonvanishing, then a first-order gauge change will in general add in a portion of the zeroth-order term to the first-order term. This means that the decomposition into zeroth- and first-order terms is not invariant unless the zeroth-order term vanishes. In what follows, many of the contributions we compute will be purely first-order, and thus will have invariant interpretations. However, for each of the effects there will also be zeroth-order terms. (For instance, if γ 1 and γ 2 are skew time-like geodesics in Minkowski space, then τ 1 ( τ ) will incorporate a time-dependent Doppler effect.) Thus there will be certain contributions which have no invariant decomposition into zeroth- and first-order effects; for these, any attempt to specify them in terms of a background Minkowski geometry will require choosing a gauge. The place this mixing of zeroth- and first-order terms will show up is when we use parallel transport along the null geodesic to identify the tangent ˙ γ a 1 to γ 1 with a vector at γ 2 . In fact, the quantity which will enter is where P a b is the parallel-propagator from γ 1 ( τ 1 ) to γ 2 ( τ ) along p ( s, τ ) . In the previous section, I did not write P a b explicitly, but here it is best to do so, to guard against the temptation to use a background Minkowski structure to subtract ( dτ 1 /dτ ) ˙ γ a 1 from ˙ γ a 2 (and thus neglect a potential first-order part). In general, I will not write the parallel-propagator factors in terms which are already first-order (since the omitted corrections would be of higher order), but I will keep them in zeroth-order terms. For later reference, note that (In keeping with the remarks above, since the curvature is firstorder, I have omitted the parallel-propagator terms which should properly appear in the integrand.) From this, we have In these equations, and in others that follow, the field w a appears within terms which are already first-order. In such terms, we need only the zeroth-order expression for w a ; this is which linearly interpolates between the values of w a at the ends. (Because this will be used only when multiplied by first-order factors, we do not write the parallel propagators which transport the vectors at the ends of the null geodesic to p (' s, τ ) .) The remaining quantities we shall need are the solutions U a b , V a b to the Jacobi equation, the vector β a which is one of the initial data (the vector α a = ( dτ 1 /dτ ) ˙ γ a 1 is already known), and the luminosity distance r . To first order in the metric perturbation, we have where now P c b = P c b ( s ) is the parallel-propagator along p ( s, τ ) from γ 1 ( τ 1 ( τ )) = p ( s 1 , τ ) to p ( s, τ ) and with H the Heaviside step-function, and the curvature is evaluated at p (' s, τ ) in the integrands. We find, in the linear approximation, that where on the right the first term contains the zeroth-order contribution, and u a b , v a b are evaluated at s = s 2 . Finally, from eqs. (37) and (20), we find where T ab is the stress-energy and we have used Einstein's equation.", "pages": [ 6, 7 ] }, { "title": "3.1 Basic results and discussion", "content": "With the results of the beginning of this section and a bit of work using eqs. (11), (12), (14), we have A similar computation, using eq. (16), gives us the rate of change of the wave-vector k a = ω 1 l a . We have And for the change in field, we find Equations (40), (41) and (42) describe basic observables, in the limit of linearised gravity. There are several points to make about these results: (a) The possibility of ascribing an observation of one the lefthand quantities to gravitational radiation relies on having some sort of extra information allowing one to discriminate between the various terms on the right. In most cases, we must assume that timedependence of the gravitational waves is enough different from those of the other physical processes that this can be used. (b) The equations contain terms proportional to δ a or δ · δ divided by l · ˙ γ 1 ( s 2 -s 1 ) . These terms typically are suppressed when the observer is at very great distances from the emitter. (In fact, in view of the formulas (31), (32), (39), for monochromatic waves of angular frequency ω , the first-order parts of these terms are typically suppressed by factors of ( ω r ) -1 .) In such cases, the zerothorder contribution to d 2 τ 1 /dτ 2 may become effectively negligible, leaving a geometrically pure first-order curvature-integral term. (c) In vacuum, the second term on the right-hand side of eq. (42) will vanish. Note that, apart from this term, all of the curvatureintegral terms are sums of for n = 0 , 1 , 2 , where we have used the formula (33) for v a . (Note that the form (18) of F ab implies this for (42).) (d) Many traditional approaches to these problems aim to work out what we might call the long-range scattering, corresponding to the receiver and emitter receding to great distances along the same null geodesic. 10 The formulas here show, however, that only in restricted circumstances will this limit exist. For the long-range scattering, we want to examine what happens as s 1 → -∞ , s 2 → + ∞ . Of course, we must assume that the contributions from the accelerations ¨ γ a 1 , ¨ γ a 2 are negligible (or at least are stable under the limit), and that the δ a -dependent terms drop out. But even then the remaining curvature integrals will not in general stabilise. This is because they depend on the quantity v a (' s ) (eq. (33)) which interpolates from ( dτ 1 /dτ ) ˙ γ a 1 at γ 1 to ˙ γ a 2 at γ 2 . This quantity has no well-defined limit point-wise in ' s as s 1 → -∞ , s 2 → + ∞ independently. In other words, the contributions of the curvature integrals are in principle sensitive to the choices of of s 1 and s 2 in the asymptotic regime. On the other hand, in many cases of interest ˙ γ a 1 and ˙ γ a 2 will differ by only sub-relativistic effects, and then v a (' s ) will be nearly constant along the null geodesic. Then the curvature integrals contributing to the time-dilation(40) and field change (42) will (assuming the curvature falls off suitably) stabilise as s 1 → -∞ , s 2 → + ∞ . Below, we shall mostly be interested in the case where the receiver is removed to arbitrarily great distances, but the emitter is held fixed. In this case we will have v a → ( dτ 1 /dτ ) ˙ γ a 1 , and only the integrals (43) for n = 0 will contribute. (e) Besides the implicit dependence of v a (' s ) on s 1 , s 2 , the curvature integrals in the expression (41) for the change in wavevector involve explicit factors (' s -s 1 ) / ( s 2 -s 1 ) , ( s 2 -' s ) / ( s 2 -s 1 ) . These factors are of geometric origin, and express the fact that change in angle perceived by a distant observer will be of order half the full scattering angle multiplied by the ratio ( distance of source to scatterer ) / ( distance of source to receiver ) . In practice this means that if the source of the light is much closer to the source of the gravitational waves than it is to the Earth, the angular change due to the scattering is correspondingly reduced. Thus attempts to measure angular deflections due to gravitational waves are at a geometric disadvantage relative to measurements of changes in time-dilation or field. (This point can also be deduced from the formulas in Kopeikin et al. 1999.) (f) Below, we shall be interested in the case where the receiver is very distant but the emitter is not, so s 2 → + ∞ but s 1 is finite. In this case we will have v a → ( dτ 1 /dτ ) ˙ γ a 1 , and the change in wave-vector will be suppressed.", "pages": [ 7, 8 ] }, { "title": "4 QUADRUPOLE SOURCES", "content": "With the formulas derived above, the analysis of the effects at the linearised level in any given space-time reduces to the computation of certain moments of the curvature over the relevant segment of the light-ray's trajectory. The curvature can itself be expressed as a retarded field due to sources, plus a possible pure radiation term. The results of this can be quite complicated, even in simple cases, because of the time-dependence of the curvature and the different components which enter, and the fact that we wish to take the point of emission of the light to be finite. However, for the remainder of 10 It might be tempting to call this the total scattering, but that would be misleading in this context, because we still track here only the differential effects, as the light-rays vary, of the scattering. this paper, the aim will not be detailed modelling but simply rough estimates of the scales of the effects. I shall here work out the leading contributions in a simple but important case: a pure quadrupole field, from an isolated source, with the null geodesics in the radiation zone in the sense that ωb glyph[greatermuch] 1 , where ω is the angular frequency of any component of the gravitational wave and b is the null geodesic's impact parameter relative to the quadrupole source. The point of reception will be taken to be very far away from the source; this corresponds to the limit s 2 → + ∞ discussed earlier, but s 1 will be held finite (recall s 1 , s 2 are the affine parameters specifying the null geodesic segment from emission to reception). In this case, the only curvature integral (43) we have to compute is I (0) bd (because within the integrands we have v a → ( dτ 1 /dτ ) ˙ γ a 1 , (( s 2 -' s ) / ( s 2 -s 1 )) → 1 , ((' s -s 1 ) / ( s 2 -s 1 )) → 0 ). Of course, real sources have monopole and perhaps dipole as well as quadrupole components; however, these contribute only stationary terms to the field, and in any event in the linear approximation those can simply be added to the quadrupole effects.", "pages": [ 8 ] }, { "title": "4.1 Quadrupole fields", "content": "By a quadrupole field I mean a linearised gravitational field in an appropriate j = 2 representation of the rotation group; both 'electric' and 'magnetic' quadrupoles (often called mass quadrupole and current quadrupole terms) are allowed. The treatment here is chosen to fit with rest of this paper's formalism; other forms are given in Regge & Wheeler (1957); Pirani (1965); Thorne (1980). Let us first consider the 'electric' part, which we idealise as a pure quadrupole at the spatial origin. (Since we are only interested in the field outside the source, this is adequate.) Let the quadrupole moment be Q el ab ( t ) , with arbitrary time-dependence. (Here t is the the coordinate time at the spatial origin, and Q el ab is symmetric, trace-free and orthogonal to t a .) It makes a contribution to the stress-energy, where δ (3) is the spatial delta-function. One easily verifies that ∫ ( t p t q T el pq )( x a -tt a )( x b -tt b ) d 3 x = Q el ab -the mass quadrupole is indeed Q el ab . The full curvature tensor can easily be worked out by standard means. 11 It is convenient to introduce a null tetrad l a , m a , m a , n a , with l a = t a +ˆ r a , for ˆ r a a unit space-like radially outward vector, n a = t a -ˆ r a , and m a = 2 -1 / 2 ( ∂ θ -i csc θ∂ φ ) . Then the radiative (order r -1 ) term is where the superscript (4) indicates the order of differentiation with respect to u . (Note that the polarisation factors l [ a m b ] are the same as in the electromagnetic case.) One does not actually need the detailed form of the m a vectors in computations; the combination m p m b entering here may be written as Weyl curvatures of 'magnetic' type can be obtained, in linearised theory in the vacuum, by dualizing the electric ones. Thus glyph[negationslash] 11 One point to be careful of in these calculations is that f ( u ) ∇ a δ (3) = f ( t ) ∇ a δ (3) in general, as becomes clear by multiplying by a test function and integrating by parts. a 'magnetic' quadrupole field will, in the vacuum region, be given by where Q mag pq is referred to as the magnetic part of the quadrupole moment. 12 The form of the source for this term is different from (44); one can check that the corresponding contribution to the stress-energy is It turns out that Q mag ab is essentially a first spatial moment of the angular momentum density. To see this, note that L a = -glyph[epsilon1] pqr a ( t p )( x q )( t c T c r ) can be interpreted as the angular momentum density with respect to the spatial origin (the minus sign giving the usual convention for the angular momentum as a spatial vector). Then a short calculation shows One may take a complex quadrupole moment Q ab = Q el ab + iQ mag ab ; then the curvature in the radiation zone is", "pages": [ 8, 9 ] }, { "title": "4.2 The light-rays", "content": "We wish to study the differential scattering of light-rays which pass through the gravitational wave source's radiation zone and are received at some great distance. In this subsection, we work out the appropriate parametrization of those rays. Since we are to compute an integral of the curvature, which is a first-order quantity, it is enough to know the geometry of the ray and of the receiver to zeroth order. Let the Bondi coordinates, centred at the world-line of the gravitational-wave source, be ( u, r, θ, φ ) . Actually, we will not need to write ( θ, φ ) explicitly; we may represent them by their corresponding null vector ' l a = ' l a ( θ, φ ) , normalised by ' l · t = 1 , with t a the unit future-directed time-like vector characterising the Bondi frame. A point in Minkowski space is thus specified as ut a + r ' l a . We will suppose the light-ray is received at an event where r 2 is very large. In general, the equation of a light-ray may be expressed conveniently as where u is the retarded time the light-ray tends to as s → + ∞ , the null future-directed vectors l a 0 , l a 1 are normalised by l 0 · l 1 = l 0 · t = l 1 · t = 1 (so the spatial parts of l a 0 , l a 1 are orthogonal); then b is the ray's impact parameter, its point of closest approach to the spatial origin occurs at s = 0 , and l a 0 , l a 1 code the direction of closest approach and the direction of the ray. We have r = √ b 2 + s 2 and ' l a = ( b/r ) l a 0 +( s/r ) l a 1 +(1 -b/r -s/r ) t a . 12 The sign here is fixed by the convention that it is the real and imaginary parts of the Bondi shear which determine the electric and magnetic parts of the curvature. Requiring the ray to be received at p a ( s 2 ) = γ a 2 ( τ ) , we find where 'mod t a ' means up to terms proportional to t a . Solving these equations perturbatively in b/r 2 , we have To this order, the vector l a 0 is (apart from the normalisations specified above) unrestricted; the quantity ( b/r 2 ) l a 0 specifies the apparent direction of the light-ray, relative to the direction of the source, at the receiver. The light-ray's trajectory is thus meeting the receiver at s = s 2 = r 2 . The point of emission will be at a parameter value s 1 , the point of closest approach to the spatial origin, as noted above, would be s = 0 . Expressing p a ( s ) in Bondi coordinates, we have where In the computations to follow, it will be convenient to put s = b sinh ξ . (64) Then r = b cosh ξ and bs -√ b 2 + s 2 = -be -ξ and we define note that S is an increasing function of s , and that", "pages": [ 9, 10 ] }, { "title": "4.3 The curvature integral", "content": "As noted above, the only curvature integral we require is where l a = l a 1 . The integrand, in the radiation zone, is where the accents indicate the vectors evaluated at ' s along the null geodesic. Because we wish to evaluate I (0) bd in the case ωb glyph[greatermuch] 1 , where ω is the angular frequency of any contributing component to the gravitational radiation field, we shall for the moment just work with one Fourier component, putting e iωu K pq (where K pq is constant, symmetric, trace-free, and orthogonal to t a ) in place of Q pq ; after using the condition ωb glyph[greatermuch] 1 we will restore Q pq . Then making use of eqs. (61) and (65), we find In this form, the integral is proportional to e -iωbS , and we may regard it as effecting a Fourier transform. We are interested in the behaviour of this for large ωb , which is to say the high-frequency regime. The function to be transformed is smooth except for being cut off at the end-points S = S 1 , S = 0 ; it is the non-smooth behaviour at these end-points which will give the leading contribution. In fact, it is the lower end-point S = S 1 which makes the dominant contribution, for a little algebra shows the integrand tends continuously to zero as S ↑ 0 . We have then the Fourier transform of a function with a jump discontinuity at S = S 1 ; this is where the tilde denotes asymptotic expansion for ωb glyph[greatermuch] 1 and we have restored Q pq . We will be most interested in the case of time-delays, for which the curvature-integral contribution is Then the combination of vectors entering into eq. (70) becomes where terms proportional to t p have been dropped because they will not contribute when contracted with Q pq and is a purely spatial vector equal to l 0 × l 1 , where l 0 , l 1 are the spatial parts of l a 0 , l a 1 . Writing now r 1 = ' r ∣ ∣ ∣ ' s = s 1 = √ b 2 +( s 1 ) 2 , we have or Here Q (3) pq is evaluated at u = u 2 + s 1 -r 1 . We may recast the foregoing in terms of the angle θ of the light-emission from the point of closest approach relative to the source (see Fig. 1), so (where b is the impact parameter); then we have Formula (77) is the main result, relating the clock times τ 1 of the emitter (at radius r 1 ) and τ of the receiver, as influenced by gravitational mass ('electric') Q el ab and current ('magnetic') Q mag ab quadrupole sources (evaluated at retarded time u = u 2 + s 1 -r 1 = u 2 -(1 -sin θ ) r 1 , where u 2 is the observer's retarded time). Recall that here l a 1 is the null geodesic's tangent and l a 0 is the null vector whose spatial part l 0 is a unit vector from the source to the geodesic's point of closest approach, and C a is a spatial vector l 0 × l 1 normal to the plane containing the source and the light-ray. Perhaps the most striking feature of this result is the 'forwardbackward' asymmetry represented by the overall factor (1+sin θ ) , which enhances effects from light-emitters on the portion of the light-ray outgoing from the gravitational-wave source ( 0 < θ < π/ 2 ) relative to those from the incoming portion ( -π/ 2 < θ < 0 ). This is a relativistic effect arising from the use of light-signals to probe the space-time curvature. For gravitational waves of a given frequency with respect to t a (the gravitational source's frame), the frequency with respect to an affine parameter along the lightray will be larger along the ingoing portion than along the outgoing one, and the effects due to those higher-frequency terms more nearly average out. For the scaling of eq. (77) with distance, for light-emitters very distant from the light-ray's point of closest approach to the gravitational-wave source, that is r 1 glyph[greatermuch] b , we have 1 + sin θ = 1 ± √ 1 -( b/r 1 ) 2 and Thus the rough magnitude of the effect will be where the speed of light has been given explicitly. (For rough estimates, the precise choice of norm for the tensor is not very important; any L p norm in terms of a standard Euclidean basis will do.) While the upper line corresponds to the scaling found by Damour and Esposito-Farèse, we see that the fall-off for light-sources along the outgoing portion of the ray is much softer, having the ∼ r -1 behaviour characteristic of radiative effects. While the appearance of this radiative scaling is certainly of interest, we shall see below that even in favourable circumstances the effects are small; we shall therefore concentrate, in the following sections, with the case of moderate θ , corresponding to lightemitters with r 1 ∼ b , the middle line of eq. (79). Finally, two remarks about the angular dependence of the effects through the factor in curly braces in eq. (77). First, the variations of this term for moderate θ means that different lightemitters in this regime will probe the different components of the quadrupole tensors. Second, one might have thought that, for lightsources further away, the trigonometric factors would tend to suppress the dependences on l a 0 , the vector from the origin to the point of closest approach, and lead to a dependence of the effects primarily on l a 1 , the tangent to the light-ray. The opposite is true of the factor in curly braces in eq. (77), however. This is a direct consequence of the transversality of the waves; the components of the curvature that enter are orthogonal to the position-vector relative to the origin.", "pages": [ 10, 11 ] }, { "title": "5 ESTIMATES", "content": "Of the various possible modulations of the light by its passage through, and emission within, the gravitational radiation - changes in the received light's amplitude, polarisation, phase, location on the receiver's sky, and time-dilation - it seems that in most cases time-dilation will be the most promising for detection (but still, as we shall see, quite challenging). As pointed out earlier, changes in the location and phase will be suppressed by a geometric factor. Changes in amplitude and polarisation would be so small they would probably be too hard to detect. In this section I shall estimate two time-dilation effects, redshift and pulse time-of-arrival offsets, in some cases of interest. First some rough general formulas for these will be derived; we will see that they are expressed naturally in terms of a dimensionless intrinsic measure of gravitational-wave strength, the Bondi news (essentially GQ (3) ab /c 5 in our case). While this quantity is central in much of gravitational radiation theory, few numerical values for it have appeared in the astrophysical literature, so samples of these are given. I then discuss the pulsar timing resolutions appropriate to the detection of time-of-arrival wanderings (unfortunately, the extraordinary long-term stability of pulsars does not help directly with this), and finally estimate those wanderings.", "pages": [ 11 ] }, { "title": "5.1 Formulas for the red-shift and time-offsets", "content": "The entire light-signal from the emitter is subject to a time-dilation, which will itself be a function of time. If the emitted signal is a ( τ 1 ) in the frame of the emitter, then, the received signal will be (neglecting other effects) a ( τ 1 ( τ )) ; that is, there will be a distortion due to the relative differences in the flows of time. One effect this would give rise to would be a time-dependent red-shift. Or if one knew the light-source was, in its own frame, emitting regular signals (for instance, if it were a pulsar), the effect of the gravitational radiation would be to make the times of receipt of these wander slightly from complete regularity. Where linearised gravity is adequate, the equation governing this was (40), and it contained three sorts of contributions: terms due to possible covariant accelerations of the emitter or the receiver; a sort of kinematic term due to the possible boost of the receiver relative to the emitter; and the curvature integral estimated in the previous section. It is the last which will be important, as will now be explained. Recall that our main hope for detecting gravitational waves by this effect comes not from relative magnitudes of these terms (the gravitational acceleration at the surface of the Earth is far larger than the expected gravitational-radiation effects), but from the different time-dependences of the terms. We must assume that we can account for any source and receiver accelerations well enough to distinguish the effects from those of possible gravitational waves. The kinematic term δ · δ/ ( l · ˙ γ 1 ( s 2 -s 1 )) requires a bit more discussion, though. Here δ a is the difference between ( dτ 1 /dτ ) ˙ γ a 1 and ˙ γ a 2 , parallel-propagated along the null geodesic. While in the situations we shall consider the zeroth-order contribution to this will have a different time-dependence than the gravitational-wave effects, the first-order contribution will be affected by the gravitational radiation. However the corresponding effects are suppressed by a factor of ( ωr/c ) -1 (essentially because they involve velocities rather than accelerations), as inspection of eqs. (32), (41) shows; see the discussion under point (b) in 3.1. We thus consider only the curvature-integral contributions to the time-dilation. Assuming that the light-source is not moving ultra-relativistically with respect to the Earth, we see from eq. (79) that the scale of the effects is, in the case of light-sources ∼ b from the gravitational-wave source, where b is the light-ray's impact parameter, where the first factor on the right has units inverse time and the second is dimensionless (recall Q (3) ab is the third time-derivative of the quadrupole moment). 13 The second factor on the right in eq. (80) is a dimensionless measure of the intrinsic strength of the gravitational radiation; it is essentially an average value of the magnitude of the Bondi news N , a key quantity in gravitational radiation theory which will be discussed a bit further below. Because this is such central concept, we now switch to expressing the quantities of interest in terms of this average news Then the rough magnitudes of the red-shifts will be and the magnitude the time-of-arrival wandering will be where ω is the angular frequency of the wave and b is impact parameter of the null geodesic from the gravitational wave source. (Again we assume the emission occurs near the point of closest approach - more generally one would replace the factors of 1 /b with (1 + sin θ ) /r 1 , in accord with (77) -, and that the motion of the receiver relative to the emitter is sub-relativistic.) We recall that the analysis of the previous section assumed that the light emission occurred in the gravitational wave zone. This means that we must have ωb/c glyph[greaterorsimilar] 1 . (Emissions from points closer to the gravitational source could be studied by the general formulas given earlier, but they would correspond to near- or intermediate-zone gravitational disturbances, which had not propagated far enough for their wave character to be fully developed.) Thus we have and for rough estimates of the largest possible red-shift and time-offset due to light emissions near the geodesic's point of closest approach, in the gravitational wave zone, for a given gravitationalwave source.", "pages": [ 11, 12 ] }, { "title": "5.2 Source types and intrinsic wave strengths", "content": "In much of the literature, gravitational radiation is estimated by a combination h of the linearised metric components at the Earth. While useful for discussions involving terrestrial detectors, this measure is neither invariant nor intrinsic; it is not well-suited for the present considerations. The measure which is appropriate is the Bondi news N , which we have already mentioned. This is a function of retarded time and angle; it has a suitable invariance (Bondi-Metzner-Sachs covariance), is intrinsic and dimensionless (Penrose & Rindler 1986). The gravitational luminosity is ( c 5 / 4 πG ) ∮ | N | 2 (over the sphere of directions). In the quadrupole approximation one has N = -( G/c 5 ) Q (3) ab m a m b (where m a is a complex null vector coding the direction), and so the quantity | N | av given in eq. (81) is an average of | N | over directions. There are, however, few numerical values of the news in the literature. Table 1 gives some rough estimates for the news, using tional rate of change of the electromagnetic field components, induced by the light's passage through, and emission in, the gravitational radiation. | N | av = ( GL/c 5 ) 1 / 2 where L is the gravitational luminosity (and, for the last four lines, the quadrupole approximation). Note that the first three cases correspond to burst-type sources. It is clear from Table 1 that if we should be lucky enough to observe light-signals from the vicinity of an extreme gravitationalwave event, such as an asymmetric supernova or colliding black holes of comparable mass, a great deal of information could be gained. However, because the gravitational waves decay very quickly in such cases (typically on a time-scale of order the lightcrossing time associated with the mass, that is GM/c 3 ), one would have to already have the telescope trained on the object, and in general this would require extraordinary serendipity. An exception would be if we could detect an inspiralling system and thus be prepared to monitor light-sources in its vicinity when strong gravitational waves were produced. For the rest of this paper, we leave aside the possibility of gravitational-wave sources with news any significant fraction of unity. The best other candidates for detection appear to have binaries as sources of gravitational waves, corresponding to the last two lines in Table 1. For these, we have for the case of equal masses in a circular orbit and for highly unequal masses M glyph[greatermuch] m in a circular orbit; in each case P is the period.", "pages": [ 12 ] }, { "title": "5.3 Red-shifts", "content": "For neutron-star glitches, the factor 8 × 10 -5 in the third line of Table 1 corresponds to a spectral resolving power of 20 km s -1 . The red-shift will be smaller by a factor ( c/ωb ) , and for a lightsource ten light-seconds away from the glitching neutron star (say, a companion) this is likely ∼ 10 -4 (taking 1 /ω ∼ 1 ms), giving | z | ∼ 2 m s -1 /c . Radial components of stellar velocities are measured, interferometrically, to 1 -3 m s -1 (Butler et al. 1996; Rupprecht et al. 2004); however, such precisions typically require integration times of a minute or more, whereas the glitches are thought to decay over milliseconds. For the other sources we considered, the red-shifts would be smaller, and hence we now move on to the possibilities of detecting the wanderings of times-of-arrival of signals from pulsars.", "pages": [ 12 ] }, { "title": "5.4 Pulse time-of-arrival offsets", "content": "We now come to the possibility of measuring the offsets caused in pulsar signals times-of-arrival caused by their passage through gravitational waves. We saw above (eq. (85)) that the maximum offset was roughly ω -1 | N | av . The effect is therefore determined by a ratio of the intrinsic gravitational wave strength | N | av to the gravitational wave period ω , and we must consider this for different classes of sources. First, however, let us consider what temporal resolution we might get from the pulsar signals.", "pages": [ 12 ] }, { "title": "5.4.1 Millisecond pulsars as clocks", "content": "As account of pulsar and timing is given by Lorimer & Kramer (2005). While millisecond pulsars can have extraordinary long- term stability, that stability is essentially a secular effect, based on measuring the times and numbers of pulses between initial and final pulses years apart, and having only to reckon with the uncertainties in locating the times of arrival of those initial and final signals. Except in searching for very low frequency gravitational waves (or 'memory' effects, which are essentially zero-frequency phenomena), that stability is not of direct help. We must consider timing over shorter scales. There are two key issues: that pulsars are weak radio sources, and that their signals over a period appear to be subject to intrinsic variations, jitter and drifting sub-pulses. Because of these, one cannot extract very accurate times of arrival from individual pulses; instead one must fold them, that is, integrate (typically over a few minutes) with a mean periodicity removed in order to get a good pulse profile. One may then have a resolution around ∆ τ pulsar ∼ 10 -7 -10 -6 s . If we search for effects corresponding to shifts in the times of arrival of order ∆ τ glyph[lessmuch] ∆ τ pulsar , we need of order (∆ τ pulsar / ∆ τ ) 2 samplings, each of some minutes' length. There are about 5 × 10 5 minutes in a year, and so it would be very hard to track changes smaller than ∼ 10 -10 s by this crude statistical means with current technology. However, were we to learn more about the jitter and sub-pulse drift, we might be able to do better. Most of the effects considered here will have fairly well-defined periods, and the question is whether those Fourier components could be distinguished from the jitter and drift. These displacements of the times-of-arrival would show up in the Fourier domain as splittings of the angular frequencies of the components of the signal by ± ω , or as peaks in the Fourier-transformed residuals, at angular frequencies ± ω .", "pages": [ 12, 13 ] }, { "title": "5.4.2 Long-duration waves from neutron stars", "content": "The case where the pulsar is one member of a binary, the other member being a neutron star gravitationally radiating due to an asymmetry or instability (fourth line of Table 1), turns out not to be very promising. (Note that here the gravitational radiation in question is not due to the binary system.) Taking the case of a non- axisymmetric neutron star considered by Prix (2009), we have where P is the period of rotation. Using Table 1, we see that in optimistic circumstances for persistent waves, we might have effects as large as ∼ 4 × 10 -16 s . This is very small.", "pages": [ 13 ] }, { "title": "5.4.3 Pulsar detection of binary emission", "content": "It is more promising to consider as a source of gravitational waves a tight binary, and assume that there is a nearby pulsar. This pulsar could be a (relatively distant) tertiary, or not be directly gravitation- ally bound but still close by (as might happen if, for example, both the binary and the pulsar were in a globular cluster). Recalling that ωb/c glyph[greaterorsimilar] 1 for the wave character of the binary-induced changes in curvature to be developed at the pulsar, and using eqs. (86), (85), we see that for a binary of equal masses, each M , in a circular orbit, the maximum offset in pulse time-of-arrival will be Figure 2 shows contours for ∆ τ max for (89). For example, if we had the component masses M = 5 M glyph[circledot] and period P = . 1 d , we would have ∆ τ max ∼ 2 × 10 -10 s . This (rough) estimate of the maximum time-displacement corresponds to the pulsar being at the inner part of the gravitational radiation zone, that is ∼ cP/ (2 π ) ∼ 30 l-min away from the binary. If the pulsar were as far away as 10 -2 pc (representative of interstellar distances in a globular cluster), one would have ∆ τ ∼ 3 × 10 -13 s . In the case M glyph[greatermuch] m of one mass dominating the other we have (from eqs. (87), (85)) Contours of ∆ τ max for this case, for the secondary of mass m = M glyph[circledot] , are given in Fig. 3. For instance, if one could find a black hole of mass M = 10 4 M glyph[circledot] (in, say, a globular cluster), with a star with mass m = M glyph[circledot] orbiting in a one-day period, one would have ∆ τ max ∼ 3 × 10 -9 s ; again, this limit is set when the pulsar is at the inner part of the gravitational wave zone, in this case a few light-days away from the radiating binary. For a solar-mass star in a ten-year circular orbit about the M = 4 × 10 6 M glyph[circledot] presumed black hole at the galactic centre, we should have ∆ τ max ∼ 7 × 10 -10 s , and this would apply to pulsars around cP/ (2 π ) ∼ 1 . 6 l-yr from that system. It should be emphasized that these estimates are very rough.", "pages": [ 13, 14 ] }, { "title": "6 SUMMARY AND DISCUSSION", "content": "The first main goal of this paper was to present an invariant framework for treating the differential scattering of light-rays in exact general relativity. This allows one to keep track of the different physical contributions to the various scattering effects, and to focus on quantities of direct physical interest. The result of greatest near-term possible observational consequence is that light emitted from the vicinities of gravitational-wave sources may be scattered by much larger amounts than those discussed by Damour and Esposito-Farèse, and by Kopeikin, Schäfer, Gwinn and Eubanks. The results here are however compatible with those authors'; one can understand them as due to 'edge effects', whose possibility was explicitly noted by Damour and EspositoFar'see, and which could be treated within the general formalism of Kopeikin et al. An equivalent statement is that for light emissions at finite distances from the gravitational-wave sources, the scattering depends on the gravitational radiation field, in contrast to the cancellations found by Damour, Kopeikin et al. which occur in the limit of infinite distances. The effects do come out to be, roughly, of the scale of those predicted in some of the still earlier papers (Sazhin 1978; Fakir 1994), and in a very rough sense one may say that this is because those papers did have the scattering respond to the gravitational radiation fields; however (again in accord with Damour, Kopeikin et al.) the details of the arguments of those papers are not really compatible with those here. The main physical issue which was not addressed systematically in earlier work was the effect of the gravitational waves on the motion of the light-sources. (See however Kopeikin et al. 2011 for an exception.) This showed up, mathematically, in leaving the results in terms of the coordinate times rather than clock times of the light-emitter and receiver. Such formulas are of direct physical significance only if the coordinates are adapted to the motions of both the light-emitter and receiver. This issue is particularly problematic when the light-emitter is close enough to the gravitational-wave source that that gravitational effect on the light-emitter's motion must be accounted for. The present approach works with the clock times and can account for any motions of the light-emitter and receiver. Some candidates for the possible observation of these scattering effects were considered, in a quadrupole approximation for the gravitational waves. It was found that the best effects to search for were based on the relative distortion of the emitter's proper time τ 1 as measured by light-signals incoming to the receiver at its proper time τ . Because these observations would require training a telescope on the emitter, one is driven to look for specific likely sources of gravitational radiation, and these are in general weak ones. The most promising sort identified here would be a tight binary, the light-emitter being a pulsar orbiting the binary as a distant tertiary, or not gravitationally bound but close by. In an example with favourable but not extreme parameters (a binary of two 5 M glyph[circledot] stars with a . 1 d period, and a pulsar about 30 l-min away), we found offsets in the times of arrival of the pulsar pulses of the order of ∼ 10 -10 s , with the offsets varying with a period around an hour. If we a were able to find a solar-mass star orbiting a 10 4 M glyph[circledot] black hole with a one-day period, for example in a globular cluster, pulsars a few light-days away might have their pulse times-of-arrival offset by ∼ 10 -9 s .", "pages": [ 14 ] }, { "title": "ACKNOWLEDGMENTS", "content": "I thank Sergei Kopeikin and Bahram Mashhoon for drawing my attention to earlier work on this and for a helpful discussion. I thank Duncan Lorimer for a helpful electronic exchange about pulsar timing measurements, and an anonymous referee for constructive suggestions about the exposition. The graphs were prepared with VEUSZ. This work was supported in part by the University of Missouri Research Board.", "pages": [ 15 ] }, { "title": "REFERENCES", "content": "Andersson N., Ferrari V., Jones D. I., Kokkotas K. D., Krishnan B., Read J. S., Rezzolla L., Zink B., 2011, General Relativity and Gravitation, 43, 409 Book L. G., Flanagan É. É., 2011, Phys. Rev. D, 83, 024024 Damour T., Esposito-Farèse G., 1998, Phys. Rev. D, 58, 044003 Durrer R., 1994, Phys. Rev. Lett., 72, 3301 Fakir R., 1994, Phys. Rev. D, 50, 3795 Faraoni V., 2008, New Astronomy, 13, 178 Hellings R. W., Downs G. S., 1983, ApJ, 265, L39 Kaiser N., Jaffe A. H., 1997, ApJ, 484, 545 Kokkotas K. D., Apostolatos T. A., Andersson N., 2001, MNRAS, 320, 307 Kopeikin S., Efroimsky M., Kaplan G., 2011, Relativistic Celestial Mechanics of the Solar System. Wiley. Kopeikin S., Mashhoon B., 2002, Phys. Rev. D, 65, 064025 Kopeikin S. M., Schäfer G., Gwinn C. R., Eubanks T. M., 1999, Phys. Rev. D, 59, 084023 Labeyrie A., 1993, Astronomy and Astrophysics, 268, 823 Lesovik G. B., Lebedev A. V., Mounutcharyan V., Martin T., 2005, Phys. Rev. D, 71, 122001 Lorimer D. R., Kramer M., 2005, Handbook of Pulsar Astronomy. Cambridge University Press Misner C. W., Thorne K. S., Wheeler J. A., 1973, Gravitation. W.H. Freeman and Co. Penrose R., Rindler W., 1984, Spinors and space-time, vol. 1: Two-spinor calculus and relativistic fields. Cambridge University Press Penrose R., Rindler W., 1986, Spinors and space-time, vol. 2: spinor and twistor methods in space-time geometry. Cambridge University Press Pirani F. A. E., 1965, Introduction to Gravitational Radiation Theory (Notes by J. J. Marek and the Lecturer). Prentice-Hall, p. 249 Prasanna A. R., Mohanty S., 2002, EPL (Europhysics Letters), 60, 651 Prix R., 2004, in W. Beecker ed., Astrophysics and Space Science Library, Vol. 357, Gravitational Waves from Spinning Neutron Stars. p. 651 Regge T., Wheeler J. A., 1957, Phys. Rev., 108, 1063 Rupprecht G., Pepe F., Mayor M., Queloz D., Bouchy F., Avila G., Benz W., Bertaux J.-L., Bonfils X., Dall T., Delabre B., Dekker H., Eckert W., Fleury M., Gilliotte A., Gojak D., Guzman J. C., Kohler D., Lizon J.-L., Lo Curto G., Longinotti A., Lovis C., Megevand D., Pasquini L., Reyes J., Sivan J.-P., Sosnowska D., Soto R., Udry S., Van Kesteren A., Weber L., Weilenmann U., 2004, in A. F. M. Moorwood & M. Iye ed., Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol. 5492, The exoplanet hunter HARPS: performance and first results. pp 148-159 Sazhin M. V., 1978, SvA, 22, 36 Schutz B. F., 2010, in S. A. Klioner, P. K. Seidelmann, & M. H. Soffel ed., IAU Symposium Vol. 261 of IAU Symposium, Astrometric and timing effects of gravitational waves. pp 234239 Shapiro I. I., 1964, Physical Review Letters, 13, 789 Taylor J. H., Weisberg J. M., 1989, ApJ, 345, 434 Thorne K. S., 1980, Reviews of Modern Physics, 52, 299", "pages": [ 15 ] } ]

[ { "title": "ABSTRACT", "content": "In previous works we proposed the Reverse Zeldovich Approximation (RZA) method, which can be used to estimate the cosmological initial conditions underlying the galaxy distribution in the Local Universe using peculiar velocity data. In this paper, we apply the technique to run constrained cosmological simulations from the RZA-reconstructed initial conditions, designed to reproduce the large-scale structure of the Local Universe. We test the method with mock peculiar velocity catalogues extracted from a reference simulation. We first reconstruct the initial conditions of this reference simulation using the mock data, and then run the reconstructed initial conditions forward in time until z = 0. We compare the resulting constrained simulations with the original simulation at z = 0 to test the accuracy of this method. We also compare them with constrained simulations run from the mock data without the addition of RZA, i.e. using only the currently established constrained realizations (CR) method. Our resimulations are able to correctly recover the evolution of the large-scale structure underlying the data. The results show that the addition of RZA to the CR method significantly improves both the reconstruction of the initial conditions and the accuracy of the obtained constrained resimulations. Haloes from the original simulation are recovered in the re-simulations with an average accuracy of ≈ 2 Mpc / h on their position and a factor of 2 in mass, down to haloes with a mass of ≈ 10 14 M /circledot / h . In comparison, without RZA the re-simulations recover only the most massive haloes with masses of ≈ 5 · 10 14 M /circledot / h and higher, and with a systematic shift on their position of about ≈ 10 Mpc / h due to the cosmic displacement field. We show that with the additional Lagrangian reconstruction step introduced by the RZA, this shift can be removed. Key words: cosmology: theory - dark matter - large-scale structure of Universe - galaxies: haloes - methods: numerical", "pages": [ 1 ] }, { "title": "Reconstructing cosmological initial conditions from galaxy peculiar velocities. III. Constrained simulations", "content": "Timur Doumler 1 , 2 , Stefan Gottlober 2 , Yehuda Ho ff man 3 , and H'el'ene Courtois 1 1 Universit'e Lyon 1, CNRS / IN2P3, Institut de Physique Nucl'eaire, 69622 Villeurbanne, Lyon, France 2 Leibniz-Institut fur Astrophysik Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany 3 Racah Institute of Physics, Hebrew University, Jerusalem 91904, Israel", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "While numerical simulations have developed into a cornerstone of studying the large-scale structure (LSS) of the Universe, there is still a long way to go towards reconciling the predictions drawn from such cosmological simulations with observational data. On the one hand, the general properties of large-scale matter clustering and interacting are now very well understood. This process is typically simulated by generating a random realization of the primordial density fluctuations of the Universe, and then integrating it forward in time using N -body techniques and the physics defined by the concordance Λ CDM model. These simulations produce model universes which are statistically in good agreement with the observed LSS. On the other hand, the region best studied observationally - the Local Universe - shows many properties that cannot be directly modelled with such random realizations. In order to explain the properties and dynamics of the Local Group, i.e. the Milky Way and Andromeda galaxies and their satellites, one has to study its formation history, which seems to be tightly connected to the peculiar alignment of the large-scale structure in the Local Universe, such as the Local Void, the Local Supercluster (LSC), and farther away structures like the Great Attractor (GA) and Perseus-Pisces cluster. This also includes studies of the Local Universe's velocity field, since it directly traces the gravitational potential and therefore the total matter distribution. The ansatz of the CLUES project 1 , which provides the framework for the study presented here, is to conduct constrained simulations that are able to reproduce the LSS of the observed Local Universe (Klypin et al. 2003; Gottlober et al. 2010). Such simulations serve as an ideal laboratory for studying structure formation in our cosmological neighbourhood (Libeskind et al. 2010; Forero-Romero et al. 2011; Knebe et al. 2011a,b). They are constructed by constraining the initial conditions using observational data (Ho ff man & Ribak 1991; Bistolas & Ho ff man 1998; Zaroubi et al. 1995, 1999), in particular we will use the measured radial peculiar velocities of galaxies in the Local Universe (Tully et al. 2008, 2009; Courtois et al. 2012) as the input data. To construct initial conditions for these simulations, the constrained realizations (CR) algorithm (Ho ff man & Ribak 1991) is used, which combines a Bayesian reconstruction from the data with a random component created with a conventional initial conditions generator. From previous CLUES simulations (Klypin et al. 2003; Gottlober et al. 2010) we know that the technique of using peculiar velocities as constraints has its limits. These simulations used radial velocities from the now outdated MARK III (Willick et al. 1997), SBF (Tonry et al. 2001), and Karachentsev et al. (2004) catalogues. To obtain reasonable reproductions of the Local Universe's structure, these velocity constraints needed to be complemented by additional density constraints, which were drawn from X-ray selected cluster data (Reiprich & Bohringer 2002). But even in this case only the few most massive clusters of the Local Universe, namely the LSC, the GA, and for larger boxsizes the Coma and Perseus-Pisces superclusters, were robust features appearing in the constrained realizations, with smaller scales essentially unconstrained and dominated by the random component. It is important to understand whether these limitations stem from insufficient accuracy of the observational data used as constraints, the method employed to construct the constrained initial conditions, or some fundamental physical limitation. Through a collaboration of CLUES with the observational Cosmicflows program (Courtois 2011a,b; Courtois & Tully 2012; Tully & Courtois 2012), it will become possible to use significantly higher-quality data with many more peculiar velocity datapoints out to higher distances, with better sky coverage and much smaller observational errors, for generating constrained simulations. Here, we want to investigate how much increase in accuracy we can expect from constrained simulations set up with this new data compared to what we have now, and how the method itself of setting up constrained initial conditions can be improved in order to optimally utilize the additional information contained in the data. With these studies, we hope to pave the way for a new generation of accurate constrained simulations that will be produced by our collaboration during the next years, to provide a powerful framework in which the Local Universe can be studied in detail. This work is the third in a series of papers on this subject. In the first paper (Doumler et al. 2012a, from here on Paper I) we presented the Reverse Zeldovich Approximation (RZA) method, which significantly increases the quality of reconstructed initial conditions obtained from peculiar velocity data. This is accomplished by a Lagrangian reconstruction of the primordial density distribution underlying the observed velocity field, essentially shifting the data back in time and thus provide better constraints for initial conditions than the original dataset observed at z = 0. In the second paper (Doumler et al. 2012b, from here on Paper II) we then investigated the impact of observational errors on the RZA method. Here, we study how well constrained simulations can reproduce the underlying universe, if their initial conditions were constructed by employing the RZA method. For this we set up a detailed test by using realistic mock peculiar velocity data drawn from a test simulation snapshot at z = 0. We use the data to generate constrained initial conditions at some early initial redshift z init and run them forward again with an N -body code. This evolved re-simulation is then compared to the original simulation at z = 0. We do this for both the previous CLUES method and our new RZA method to compare by how much the simulation accuracy improves by performing Lagrangian reconstruction on the data. We also compare two mocks of di ff erent quality, to estimate how much is gained by using more accurate data. The outline of this paper is as follows. In Section 2, we briefly review our method of generating constrained initial conditions from the data, describe the setup of this test, and present the set of resimulations we conducted. In Section 3, we study the accuracy of these re-simulations compared to the original reference simulation and present our findings. We summarize and discuss our results in Section 4.", "pages": [ 1, 2 ] }, { "title": "2.1 Initial conditions with RZA", "content": "Our method of RZA reconstruction and subsequent generation of constrained initial conditions is described in detail in Paper I; we give only a brief summary here. We start with a set of datapoints, i.e. radial peculiar velocities v r at discrete positions r at z = 0. We first apply a grouping procedure to the data in order to 'linearize' it, i.e. to remove virial motions and other small-scale interactions inside galaxy groups and clusters and to produce a data set that traces the coherent large-scale velocity field. We then reconstruct the three-dimensional peculiar velocities u ( r ) at positions r by using the Wiener Filter (WF). The WF produces an estimate u WF ( r ) based on the correlation function given by the assumed prior model, which is defined through the cosmological parameters and power spectrum P ( k ). It also filters out noise due to observational errors from the data. In order to construct cosmological initial conditions at some early redshift z init, we need to obtain a suitable set of constraints. We do this with the RZA method: we estimate the initial position x init at z init of our peculiar velocity field tracers, which are located at r at z = 0, with The reconstructed initial positions x init are not to be understood as actual positions of the observed galaxies at early times - at typical initial redshifts z init, no galaxies have formed yet. They are rather interpreted as a way to trace the estimated initial peculiar velocity field at z init. In order to construct the full set of constrained initial conditions, we then shift the original datapoints v r 'back in time' to x init and use them as constraints to construct a constrained realization with the Ho ff man & Ribak (1991) method. We only change the position of the constraints from r to x init, but preserve the amplitude of the velocity and the direction of the component that is constrained (note that it is not the radial direction anymore w.r.t. the observer because the position has changed). The constrained realization is then obtained by evaluating where δ RR is an independently generated random realization, ci are the RZA-shifted datapoints, ˜ ci are the corresponding values of the same quantities in the random realization, and the angled brackets denote the values of the correlation functions of the di ff erent quantities, defined by P ( k ). In order to construct an actual set of initial conditions for an N -body simulation, we scale δ RR to z init and solve for the displacement field, ψ ( r ) = -∇ -1 δ ( r ). We then place particles on a grid with displacements ψ according to the established Zeldovich-approximation method (Efstathiou et al. 1985).", "pages": [ 2, 3 ] }, { "title": "2.2 Mock data", "content": "As we already did in Papers I and II, we use the BOX160 simulation conducted by the CLUES project as the source of our mock galaxy peculiar velocity data. Again, we refer the reader to these papers for details. The BOX160 is a constrained simulation of the Local Universe with a boxsize of 160 Mpc / h , set up with the WMAP3 cosmological parameters. The simulation contains a large-scale structure resembling the observed Local Universe (see Figure 1 and Cuesta et al. (2011)). The simulation contains a configuration of objects corresponding to the LSC, GA, Coma, and Perseus-Pisces clusters (labelled in Figure 1), and a Local Group (LG) candidate. We select the position of this LG object (marked as a white cross in Figure 1) as the mock observer and generate from there realistic mock observational catalogues of galaxy peculiar velocities. A detailed description of how the mocks are built can be found in Paper I. Here we want to mention again that the mocks realistically reproduce features of real observational catalogues, such as a limited distance, knowledge of only the radial component v r of u , sparse sampling, and observational errors due to inaccurate galaxy distance measurements. In this paper, we concentrate on the particular mocks C30 10 and E60 10, which are designed to mimic the current Cosmicflows-1 catalogue and the upcoming Cosmicflows2 catalogue, respectively. After grouping, the catalogue C30 10 contains 588 radial peculiar velocity datapoints within a relatively small radius of 30 Mpc / h from the mock observer. This is a similar quality like the datasets that were previously used to construct constrained simulations. The E60 10, on the other hand, contains 7632 datapoints within 60 Mpc / h , and models the quality of the upcoming new Cosmicflows-2 dataset, which we plan to use for constrained simulations of the Local Universe in future work. The data zone radii of 30 and 60 Mpc / h , respectively, are marked in Figure 1 with dashed white circles. Since we want to study specifically how well realizations of cosmological initial conditions can be constrained with peculiar velocity data, and how the RZA method performs in this context, in this work we do not use any other types of constraints such as cluster density constraints for our re-simulations.", "pages": [ 3 ] }, { "title": "2.3 The set of constrained realizations", "content": "For each of the two mocks, C30 10 and E60 10, we construct several constrained realizations of cosmological initial conditions. We use the method outlined in Section 2.1, which combines the CR method with RZA reconstruction. In the following, we refer to this procedure as 'Method II'. Furthermore, we also generate initial conditions with the method previously used for CLUES simulations (Klypin et al. 2003; Gottlober et al. 2010), which we call here 'Method I'. It consists of using the peculiar velocity datapoints at z = 0 directly as constraints for Eq. (2), omitting the RZA shift. The main drawback of Method I is that it treats the peculiar velocities as though linear theory would be valid at all scales at z = 0, neglecting all higher-order e ff ects such as the cosmological displacement field ψ . This leads not only to a poorer reconstruction quality, but also to a systematic position error of the clusters recovered in such simulations (compared to their observed counterparts). Typically, the object's positions will be o ff by the amplitude of ψ , which is about 10 Mpc / h on average at z = 0. These shifts were observed in all previous CLUES simulations. We showed in Paper I that RZA can compensate for the shifts; here, we want to demonstrate how this improved method a ff ects the outcome of evolved constrained simulations at z = 0. Having two di ff erent mock catalogues and two di ff erent methods to generate ICs, we also want to study the impact of the random component δ RR in Eq. (2). The peculiar velocity constraints ci affect only large scales from ≈ 5 Mpc / h upwards, and only in regions of the box well covered by the data; all other structures emerging in the constrained simulation will have their origin in the particular realization of the random component. We therefore expect that the random seed has a large impact on the outcome of the simulation. Varying the seed while keeping all other parameters such as the constraints constant allows us to estimate how robustly structures that are constrained by the data are actually recovered in the constrained simulations. For each mock-method combination, we created six di ff erent realizations with di ff erent seeds for the random component. We have therefore a set of 24 di ff erent realizations of initial conditions. Weuse this set to test our method on scales smaller than the box. In all cosmological simulations - constrained as well as unconstrained - due to periodic boundary conditions the dynamics on the scale of the box is incorrect. Therefore, one must expect that also in future constrained simulations based on observational data the bulk flow on scales of the order of the simulation box will be incorrect. We construct the initial conditions on a regular cubic grid with a resolution of N = 256 3 and a boxsize of L = 160 Mpc / h , matching the boxsize of the 'source simulation' BOX160 2 . All realiza- tions were constructed with our newly developed numerical code IC e C o R e (see Paper I for details). For the 24 di ff erent realizations, we assumed the following naming convention. We abbreviate the six di ff erent seeds as A through F. We then add the method and seed numbers to the end of the mock name, so that for example C30 10 II A refers to the first out of six realizations that were constructed with constraints from the C30 10 mock using Method II. Wethen ran each of the generated constrained realizations of initial conditions forward until z = 0 with the simulation code Gadget-2 (Springel 2005), using collisionless particles only (no SPH particles) with a resolution of N = 256 3 particles. We also used the same cosmological parameters and initial power spectrum P ( k ) that was used for the BOX160 simulation, to be fully consistent on the assumed cosmological model. In fact, assuming a di ff erent cosmology would change the result. For example, increasing substantially the normalization of the power spectrum without changing the constraints leads to a faster evolution. Instead of a local group like object one would find at the same place a massive large halo of about the total mass of the group.", "pages": [ 3, 4 ] }, { "title": "3.1 Scatter and mass function", "content": "Figure 2 shows a cell-to-cell comparison between the evolved constrained resimulations obtained from the C30 10 mock and the original field of BOX160 in the constrained volume (out to 30 Mpc / h distance from the mock observer), both for Method I (without RZA) and Method II (with RZA). Figure 3 displays the same for the resimulations from the E60 10 mock. It can be seen that all mocks and methods produce density and velocity fields that are well-correlated with the original BOX160 universe at z = 0. However, especially for the more accurate E60 10 mock, the added RZAreconstruction that was used to obtain the initial conditions in Method II significantly improves the correlation over the one obtained with Method I (without RZA). For the E60 10 mock, the peculiar velocity field is reproduced accurately down to an accuracy of about 1 / 4 of its total standard deviation with Method II that uses RZA. This is an impressive improvement of the reconstruction quality. From the correlation of the density fields, it can be seen that Method I, which does not use RZA, significantly underestimates the total density of the constrained region, which is overdense compared to the cosmic mean. The RZA method, on the other hand, accurately reproduces this total overdensity, for both the C30 10 and E60 10 resimulations. The total overdensity as well a ff ects the abundance of dark matter haloes. Figure 4 shows the binned dark matter halo mass functions of all realizations obtained from the C30 10 mock catalogue, both with and without RZA. The mass functions of the individual realizations are shown with coloured lines and points, their average is represented by the thick dashed black line, and the actual mass function of the BOX160 inside the 30 Mpc / h region is shown with the solid dashed black line. This is an overdense region in the BOX160 simulation: the mass function of this region lies significantly above the cosmic mean (grey line). The volume contains about 2.5 times more haloes with masses > 10 13 M /circledot / h than a region of mean cosmic density would is impractical to run a large number of re-simulations with such a resolution, we limited ourselves to N = 256 3 and also re-ran the BOX160 with Gadget-2 and this lower resolution to serve as the reference simulation. This way we can stay fully consistent on the parameters of all simulations. with the given cosmological model and boxsize. The constrained resimulations without RZA only partially follow that behaviour. Their average mass function is also above the cosmic mean, but still systematically underestimates the actual halo abundance in the corresponding region of the original BOX160 simulation. On the other hand, the resimulations where RZA was used for constructing the initial conditions follow the mass function of BOX160 much more closely.", "pages": [ 4 ] }, { "title": "3.2 The BOX160 universe", "content": "The large-scale structure of the 'local universe' for our test case, i.e. the region around the mock observer in the BOX160 simulation, is shaped by several dominant structures that to some degree resemble the observed Local Universe. The 'Milky Way' analogue halo that was chosen as the position of the mock observer ( X = 75; Y = 64; Z = 80) lies in the vicinity of a 'Virgo' cluster with a virial mass of 3 . 25 × 10 14 M /circledot , which is embedded in a thick filament parallel to the X -axis. The local flow of the mock observer is directed towards this structure, resembling the observed Virgocentric infall (Tully et al. 2008). On a larger scale, the whole region is dominated by a flow towards the 'Great attractor' (GA), a massive structure at about X = 30, Y = 60 that lies outside of the 30 Mpc / h data zone of the C30 10 mock. This configuration can be seen in the top left map of Figure 5, which is a zoom-in from Figure 1. The BOX160 Virgo cluster is however not the most massive structure within 30 Mpc / h , as there is an even more massive region at a distance of slightly less than 30 Mpc / h , lying in a direction in between Virgo and the GA, that contains two clusters with masses of 6 . 06 × 10 14 M /circledot / h and 5 . 20 × 10 14 M /circledot / h , respectively sun (their Z positions are just outside of the slice plotted in Figure 5). We associate them with the Centaurus and Hydra clusters. They in turn cause a significant infall flow towards them on the surrounding structure. In the other direction (towards negative Y ), there is also a large void that contributes to the shape of the large-scale velocity field by creating a push outwards of it, although this particular feature in the BOX160 may not be as dominant as the observed Local Void (Tully et al. 2008). Constrained resimulations of this test universe should be able to recover all these characteristic structures. BOX160 contains this specific configuration because as already mentioned it is itself a constrained simulation of the Local Universe. It is not entirely accurate, for example the virial mass of the BOX160 Virgo cluster is 2 - 3 times less than the estimated virial mass of its observed counterpart (Fouqu'e et al. 2001), but the described main characteristics of the large-scale structure are present. Ignoring the origin of BOX160 for this test and taking its large-scale matter distribution as 'given', we try to reproduce it in the constrained resimulations, which should retain the main characteristics that BOX160 shares with the observed Local Universe in this 'second pass' of the reconstruction-resimulation cycle.", "pages": [ 4, 5, 6 ] }, { "title": "3.3 The re-simulated Virgo cluster", "content": "The other panels of Figure 5 show three constrained realizations obtained from the C30 10 mock catalogue without RZA (top right), with RZA (bottom left), and from the larger-volume E60 10 mock from ICs with RZA (Method II) 1 catalogue with RZA, in the 10 Mpc / h thick slice that should contain the BOX160 Virgo cluster. One can immediately notice that, while all three simulations faithfully reproduce the large-scale flow towards the GA, the resimulation that was constructed without RZA fails to create a Virgo cluster. Apparently, the mass of the BOX160 Virgo cluster of 3 . 25 × 10 14 M /circledot / h lies below the scale that could be reproduced with the non-RZA Method I of generating constrained initial conditions. Why then does the BOX160, which was created with the same method, have a Virgo cluster in the first place? There are three reasons: First, the observed Virgo cluster has a mass that is 2 - 3 times higher, which places it in the recoverable mass range. Second, we observe a scatter in the recovered masses and BOX160 is a realisation with high Virgo mass. Third, in the BOX160, the Virgo cluster was additionally constrained by density constraints placed on its observed position. In our setup, we do not use such additional constraints. The non-RZA resimulations show some overdensity in that region, and there is also the tendency of a flow in that direction, but there is not enough overdensity to form a cluster of comparable mass. On the other hand, both resimulations in Figure 5 that include RZA reconstruction have a cluster at that position, which is similarly embedded in a thick filament, with a strong flow towards it from the position of the mock observer. In order to understand more clearly how robustly the Virgo cluster is recovered in the constrained resimulations we searched for it in all resimulations by searching for haloes that would be within 10 Mpc / h of the original BOX160 Virgo's position and would have a mass of at least a 10 14 M /circledot / h . The result was that in all realizations using RZA, both from the C30 10 mock and the E60 10 mock, we could find a cluster corresponding to Virgo. These objects are listed in Table 1. On the other hand, we could not find such an object in any of the realizations created without RZA, which clearly displays that the RZA improves recovering the original structures in constrained simulations. The resimulated Virgo is not exactly at its BOX160 position in the resimulations; the error lies between 1.7 and 6.3 Mpc / h and varies with the random seed. The average position of all resimulated Virgos is only about 2 Mpc / h away from its original position in BOX160, so this fluctuation is probably due to the influence of the random modes rather than a systematic shift. It is interesting to note that in the RZA resimulations, all the Virgo reproductions have a lower mass than the original BOX160 Virgo, just as the latter has a lower mass than the observed Virgo cluster. This may be connected to the findings of Courtois et al. (2012), who analysed the Cosmicflows-1 distance catalogue with the Wiener filter and found that the Virgo cluster is not dominating the peculiar velocity field as much as expected. It may therefore be harder to recover accurately in a constrained simulation. Another possibility to estimate the robustness of the reconstruction is to compute an average of the density and velocity fields over the di ff erent evolved realizations. This way, structures coming from the random component will tend to average out and be suppressed, while structures appearing consistently in every realization will be enhanced. Figure 6 shows the same slice as in Figure 5, but averaged over all six realizations A - F for the C30 10 I (without RZA), C30 10 II (with RZA), and E60 10 II (with RZA, larger mock). The position of the original BOX160 Virgo cluster is marked with a blue cross. The resimulations done without RZA (top right) show some overdensity in this region, and a tendency of a flow towards it, but the overdensity is not high enough to create a cluster of mass comparable to Virgo in any of the nonRZA realizations. The density peak is much more pronounced in the simulations with RZA (bottom panels), which all have a massive object near to that location. We also see that other structures with less mass, such as the overdense region below Virgo around X = 90 Y = 40, are not present in the averaged fields, which means that they lie in a mass range that is too low to be recovered by the reconstruction of initial conditions for any of the methods / mock catalogues.", "pages": [ 6, 7 ] }, { "title": "3.4 Positions and masses of other clusters", "content": "To obtain a more precise estimate about the mass scale on which objects can be consistently recovered in the constrained resimulations using radial peculiar velocity constraints, we search for other massive clusters in the data zone. The largest overdense region within the data zone is dominated by the Centaurus and Hydra clusters, with masses of 6 . 06 × 10 14 M /circledot / h and 5 . 20 × 10 14 M /circledot / h , respectively. These two clusters are separated by a distance of 10 Mpc / h . This overdense region is robustly recovered in all simulations including the ones from Method I not using RZA. It therefore lies above the minimum mass scale that can be recovered without RZA. Figure 7 shows the corresponding slice of the density and velocity fields that contains the Hydra and Centaurus clusters, averaged over the di ff erent realizations 3 . All resimulations show a 3 Note that in the observational data the Hydra / Centaurus clusters and the Virgo cluster all lie within the supergalactic plane around SGZ = 0 (see e.g. Courtois et al. 2012). On the other hand, while BOX160 uses the same coordinate system, there the Hydra / Centaurus clusters and the Virgo cluster are not located at the same Z ( = in the same X / Y plane), but actually in planes about 10 - 15 Mpc / h apart from each other in the Z dimension. This is another example of the systematic shifts that occur in constrained simulations from peculiar velocity data if one does not use Lagrangian reconstruction such as RZA. massive overdensity in this region. However in the Method I resimulations there is a clear systematic shift in position. Further, in the averaged map one can see only one smeared out peak instead of two, which is additionally shifted towards positive X and negative Y . In two of the six C30 10 I realizations (B and E), we can find only one massive cluster around the region where the HydraCentaurus pair should be. In these realizations, either the two overdense regions merged during their evolution due to shifts in their position and / or displacement, or already in the reconstructed initial conditions the accuracy was not su ffi cient to robustly resolve them into two separate peaks. On the other hand, in the RZA resimulations, both clusters are resolved robustly and show up as separate peaks that are very close to their intended position; in every realization using Method II, we can find appropriate objects for Hydra and Centaurus at z = 0 within a few Mpc / h of their original positions, just like for the Virgo cluster. Like in the case of Virgo, the masses of the resimulated Hydra / Centaurus clusters at z = 0 show both a scatter and a systematic deviation from the original masses of the original BOX160 clusters of 6 . 06 × 10 14 M /circledot / h and 5 . 20 × 10 14 M /circledot / h , respectively. The average mass of the resimulated Centaurus is at 90% of the original mass in BOX160, with a scatter within a factor of two. Interestingly, the resimulated Hydra cluster is more massive than it should be, at 173% of the original mass on average. In three out of the six C30 10 II realizations it is the more massive of the pair, although it should be the less massive, and in four out of six it even has a mass slightly above 10 15 M /circledot / h . As it is in the case of the Virgo cluster, this systematic error in the cluster masses does not improve if one goes from the C30 10 II realizations to the E60 10 II realizations. If we add more constraints and increase the data volume, the mass scale that can be reproduced with the RZA method increases noticeably. For the C30 10 II simulations, we can always find a Virgo cluster, but already the next massive cluster (the fourth-massive halo within 30 Mpc / h ) with 2 . 41 × 10 14 M /circledot / h cannot be unambiguously identified in one of the six realizations, the C30 10 II E. Namely, within a distance of 10 Mpc / h and a mass within a factor of 3 there is no matching object in this realization. The next-massive clusters (ranked 5 to 7 by mass within 30 Mpc / h ) cannot be reliably found anymore. On the other hand, if we go to the E60 10 II simulations, we find resimulated counterparts for all seven most massive clusters (the seventh having a mass of 0 . 96 × 10 14 M /circledot / h ) in all six realizations, all within a factor of 3 in mass and within 7 Mpc / h in distance. The seventh most massive halo with 0 . 96 × 10 14 M /circledot / h appears as a clear peak in the averaged map of E60 10 II realizations (Figure 7, bottom right, labelled 'cZ') close to its original position. For even lighter objects, it is not possible anymore to find unambiguous counterparts in the E60 10 II realizations. This is aggravated by the fact that below a certain mass there is a quickly increasing probability to find a seemingly matching, but actually randomly created object to appear around the right position. The origin of the systematic errors on the reproduced objects' masses can be understood from the non-linearity of the structure formation process. The typical mass scatter within a factor of two that we observe here is consistent with the findings of Ludlow & Porciani (2011), who studied the relation between virialised haloes and their protohalo peaks in the initial conditions. They found that a protohalo peak on some fixed scale determines the mass of the resulting halo only within a factor of two. This explains the cluster mass discrepancies in our constrained resimulations. The exact virial mass of an object at redshift z = 0 is a product of various non-linear structure formation processes, such as at what rate it can accrete mass from the surrounding structure and how e ffi ciently it is fed from outside by connected filaments. Such details of the structure around clusters cannot be recovered from reconstructed linear initial conditions. In the mass function of the 30 Mpc / h data zone (Figure 4) we see that BOX160 has a very specific distribution: two objects (Hydra and Centaurus) in the highest-mass bin, one object (Virgo) in the next bin, and no objects in the bin after that. The average mass function of the constrained realizations does not follow this peculiar shape, but instead tends towards a distribution that is statistically more likely to occur.", "pages": [ 7, 8, 9 ] }, { "title": "3.5 Filaments and voids", "content": "Apart from massive dark matter haloes, another characteristic of the large-scale distribution of matter is the presence of filaments and voids. An example in the selected BOX160 data zone is the filament parallel to the Y -axis, which goes upward from Virgo towards the BOX160's Coma cluster (outside of the map); it is not recovered consistently in the RZA resimulations. We saw already in the RZA reconstruction of the displacement field (Paper I), that the RZA reconstruction struggles to recover such filamentary features accurately. This could be due to the non-linear structure formation in such regions: in general, structure formation first proceeds along one dimension, forming sheets, followed by a collapse in the next dimension, forming filaments, and finally the material in these filaments falls into haloes (Shandarin & Zeldovich 1989). The peculiar velocity field at z = 0 only retains information about this last stage. In the particular case of the filament that connects the Virgo cluster to the Coma cluster in BOX160, the input data contains only the infall of objects along the filament toward Virgo, but not the initial velocity distribution that led to the formation of that filament, and therefore our reconstruction procedure (WF + RZA)cannot recover it. In the case of the C30 10 II A resimulation, this filament is instead replaced by another filament that is imposed by the random modes and aligned di ff erently. This behaviour is repeatedly seen in the resimulations at other locations as well. At the other end of the filament there is a flow towards the Coma cluster seen at the upper edge of the slice in Figure 5 (around X = 80; Y > 100). This flow is missed by all resimulations that use the C30 10 mock, since the BOX160 Coma cluster is too far away from the data zone. On the other hand, in the E60 10 II resimulations, this flow is recovered accurately because of the enlarged data zone, but this is not su ffi cient to also faithfully reproduce the alignment of the filament. It is not entirely clear at this point why the alignments of filaments are not always correctly reproduced in constrained simulations. Besides the inherent non-linearity of filament formation and therefore the inability to extrapolate this process back in time with RZA another possible reason could be the fact that the input data consist of only radial peculiar velocities and the accuracy of filament reconstruction could depend on how the filament is aligned compared with the line of sight. We therefore cannot generally trust that the alignment of filamentary structure in the Local Universe can be su ffi ciently reproduced in constrained simulations, except if their formation is strongly constrained by massive objects inside the data zone, as is the case for the thick filament hosting the Virgo cluster. The presence of the latter is generally reproduced in all resimulations with RZA, although again its exact alignment is unconstrained. We did not compare the alignment of voids in the resimulations in detail, but we see that the presence and alignment of the most prominent voids in the data zone is generally recovered well. The largest voids develop by expansion of underdensities in the initial conditions that are above the minimum scale of initial conditions reconstruction and are therefore su ffi ciently constrained. The reconstruction of the displacement field using RZA helps track the expansion of voids back in time and generate appropriate initial conditions for them. Since it is believed that in the Local Universe, the outward push caused by the Local Void has an important influence on shaping the Local Flow (Tully et al. 2008), it would be interesting to study in detail to what extent this behaviour can be seen in the constrained simulations. This will be the subject of future work.", "pages": [ 9, 10, 11 ] }, { "title": "3.6 The re-simulated peculiar velocity field", "content": "To conclude the analysis of our re-simulations, we want to mention the reproduction quality of the BOX160 peculiar velocity field at z = 0. We see that within the datazone, the peculiar velocity field is recovered exceptionally well in the RZA resimulations, especially for the better E60 10 mock. We already saw that in this case the deviation of the resimulations from the original BOX160 peculiar velocity field is only on the level of 1 / 4 of the total standard deviation of the field. Comparing in Figure 5 the peculiar velocity field of the original BOX160 (upper left) and the E60 10 II resimulation (bottom right), the structure of these two velocity fields is practically identical. The flows towards the major overdense regions and outwards of the large voids are all present with a correct alignment and amplitude. We can now argue that the limited ability to recover correct masses for particular clusters or the correct alignment of filaments plays a lesser role and does not substantially limit the overall meaningfulness of the constrained simulation, since the large-scale velocity field is recovered accurately by the constrained resimulations. If we imagine that a Local Group-like object would be placed at the centre of the E60 10 II resimulation in Figure 5, then it would experience practically the same large-scale flow and would be embedded in a very similar large-scale environment compared to the original BOX160, so that we can study its dynamics. This is precisely the main motivation of running constrained simulations, where of course the reference universe is the actually observed Local Universe.", "pages": [ 11 ] }, { "title": "4 SUMMARYAND CONCLUSION", "content": "In this paper, we investigated cosmological simulations that are designed to reproduce the large-scale structure of the Local Universe. Such simulations can be produced by constraining their initial conditions with the constrained realizations (CR) algorithm of Ho ff man & Ribak (1991), using radial peculiar velocity data as input. On top of the previously established CR method we have added the RZA reconstruction, which is a novel Lagrangian reconstruction scheme designed for peculiar velocity data. This new method allows to produce significantly more accurate constraints for the initial conditions. To quantify the accuracy of the method, we perform tests using mock data extracted from a previous simulation serving as the 'reference universe'. We use di ff erent mock catalogues with realistic observational errors that mimic the real data used for constrained simulations. After a reconstruction of the initial conditions from the mock data, we evolve them forward again until z = 0 in a series of re-simulations and compare their outcome to the original reference simulation at z = 0. We do so for both the previous method (without RZA) and our new method (with RZA). We find that with the previous method of generating constrained initial conditions, if we use a sparse mock limited to 30 Mpc / h and do not employ a Lagrangian reconstruction scheme (as in previous constrained simulations within the CLUES project), the threshold for robustly recovering structures is around ≈ 5 × 10 14 M /circledot / h , and the accuracy on their positions is typically at the scale of 10 Mpc / h . Adding the RZA reconstruction, we can lower this threshold to under 3 × 10 14 M /circledot / h , enough to robustly recover an object in our test simulation that is similar to the Virgo cluster. If we use the more complete E60 10 dataset, which covers a larger volume and approximates the accuracy of upcoming new datasets, the quality of reconstruction without using RZA does not increase much; but if we use RZA, we achieve a significantly better resolution that allows us to robustly recover structure down to mass scales of about 1 × 10 14 M /circledot / h . Additionally, the RZA method reduces the errors on the positions where the resimulated structures appear to a scatter within about 5 Mpc / h around the true position. The average position of the di ff erent realization is even only 2 Mpc / h from the true position, meaning that there is practically no systematic shift of structures; this is a significant improvement over previous constrained simulations from peculiar velocities, which featured shifts of the order of 10 Mpc / h and more due to a failure to account for the cosmological displacement field. We find that even with RZA and good input data, the method generally cannot reliably recover structures on mass scales below ≈ 1 × 10 14 M /circledot / h and also struggles to reproduce other large-scale features like the alignment of filaments. On the other hand, the peculiar velocity field at z = 0 is reproduced exceptionally well by the re-simulations. This may be due to the fact that these velocities were used to constrain the realizations in the first place, and that they are less susceptible to non-linear e ff ects than the cosmic density distribution. The ability to reproduce the peculiar velocity field makes constrained simulations constructed with the RZA method an ideal laboratory to study velocity flows in the Local Universe when applied to actual observational data. We expect that with real data, the same degree of accuracy can be obtained with our technique as we showed in the test presented here. Such an application of the method to the newest observational peculiar velocity data is a main focus of our future work. We expect that these simulations will present a significant methodical improvement over the currently available CLUES simulations and will provide a very useful framework for studying the dynamics of the Local Universe.", "pages": [ 11 ] }, { "title": "ACKNOWLEDGMENTS", "content": "TD would like to thank Ste ff en Knollmann for his help with computing the Tinker et al. (2008) mass function and generating Figure", "pages": [ 11 ] }, { "title": "12 Doumler et al.", "content": "4. YH and SG acknowledge support by DFG under GO 563 / 21-1. YH has been partially supported by the Israel Science Foundaction (13 / 08). TD and SG acknowledge support by DAAD for the collaboration with H.M. Courtois and R.B. Tully. We would like to thank the referee of this series of papers for her / his very careful and fast reading of the manuscripts and the many constructive comments which improved the three papers substantially.", "pages": [ 12 ] }, { "title": "REFERENCES", "content": "This paper has been typeset from a T E X / L A T E X file prepared by the author.", "pages": [ 12 ] } ]

[ { "title": "ABSTRACT", "content": "We use the HiZELS narrow-band H α survey in combination with CANDELS, UKIDSS and WIRDS near-infrared imaging, to investigate the morphologies, merger rates and sizes of a sample of H α emitting galaxies in the redshift range z = 0 . 40 -2 . 23 , an epoch encompassing the rise to the peak of the star formation rate density. Merger rates are estimated from space- and ground-based imaging using the M 20 coefficient. To account for the increase in the specific star-formation rate (sSFR) of the star forming 'main-sequence' with redshift, we normalise the star-formation rate of galaxies at each epoch to the typical value derived from the H α luminosity function. Once this trend in sSFR is removed we see no evidence for an increase in the number density of star-forming galaxies or the merger rate with redshift. We thus conclude that neither is the main driver of the enhanced star-formation rate density at z ∼ 1 -2 , with secular processes such as instabilities within efficiently fuelled, gas-rich discs or multiple minor mergers the most likely alternatives. However, we find that ∼ 40 -50% of starburst galaxies, those with enhanced specific star formation at their epoch, are major mergers and this fraction is redshift independent. Finally, we find the surprising result that the typical size of a star-forming galaxy of a given mass does not evolve across the redshift range considered, suggesting a universal size-mass relation. Taken in combination, these results indicate a star-forming galaxy population that is statistically similar in physical size, merger rate and mass over the ∼ 6 Gyr covered in this study, despite the increase in typical sSFR. Key words: galaxies: star formation, galaxies: evolution, galaxies: interactions", "pages": [ 1 ] }, { "title": "The merger rates and sizes of galaxies across the peak epoch of star formation from the HiZELS survey.", "content": "John P. Stott 1 ∗ , David Sobral 2 , Ian Smail 1 , Richard Bower 1 , Philip N. Best 3 , James E. Geach 4", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "The peak in the volume averaged star formation rate for galaxies occurs in the redshift range z = 1 -3 (Lilly et al. 1996; Madau et al. 1996; Sobral et al. 2013). At this epoch, the star formation rate (SFR) in typical galaxies is an order of magnitude higher than in the local Universe (Reddy & Steidel 2009). This is the era when most of the stars in the Universe were formed and represents the peak in black hole activity. The task is now to address 'how' and 'why' the Universe was so different then. A picture is emerging in which the dominant mode of star formation at this earlier epoch is very different to that in the local Universe. Rather than the quiescent formation of stars that is the norm in today's Universe, violent episodes of star formation are dominated by the formation of super-star clusters (e.g. Swinbank et al. 2010b). However, the origin of these differences is somewhat controversial: one picture, which has some observational support, is that they are driven by an increase in the galaxy merger rate (e.g. Somerville et al. 2001; Hopkins et al. 2006; Conselice et al. 2003, 2008), but other theories have suggested that it is the result of the higher rate of gas accretion expected in the high-redshift Universe (Kereˇs et al. 2005; Dekel et al. 2009). It is therefore important to study the SFR, merger fractions and gas content of these galaxies in order to identify the processes responsible for driving this epoch of enhanced activity. In recent years the presence of a star forming 'main-sequence' seen in the local Universe (e.g. Brinchmann et al. 2004) has been confirmed at increasingly high redshift (Elbaz et al. 2007, 2011; Daddi et al. 2007; Rodighiero et al. 2011; Sargent et al. 2012). This is a relation between SFR and stellar mass for star forming galaxies, with a typical specific star formation rate (sSFR, the ratio of the star formation rate to the stellar mass of the galaxy) found to increase with redshift (Elbaz et al. 2011). Galaxies that lie off this relation with sSFRs too high to be in the typical star-forming population are often described as 'starbursts' and are thought to be triggered by violent events such as major mergers (Hopkins et al. 2006; Elbaz et al. 2007; Rodighiero et al. 2011). From a theoretical perspective, in the Λ Cold Dark Matter ( Λ CDM) paradigm dark matter halos merge hierarchically from the bottom up, with the largest halos created at later times (e.g. Lacey & Cole 1993; Cole et al. 2000; Springel et al. 2005). As the galaxies trace the underlying dark matter we therefore expect those to merge hierarchically also. However, it has been known for sometime that the most massive galaxies appear to have older stellar populations than their less massive counterparts (Cowie et al. 1996; Bower et al. 2006; Gilbank et al. 2010). Environment also plays a key role with massive quiescent galaxies typically living in denser environments than lower mass star-forming galaxies (Dressler 1980). There are several ways to reconcile these observations with hierarchical merging which are implemented in phenomenological, semi-analytic models that seek to reproduce observations of galaxy evolution by populating dark matter halos from N-body simulations with mock galaxies (e.g. Bower et al. 2006; Croton et al. 2006). A reasonable match is achieved through interactions and feedback mechanisms that cease star formation in massive galaxies within massive dark matter halos, requiring that these galaxies build up their stellar mass at late times by so called 'dry' mergers which trigger no significant new star formation due to the lack of available cold gas (De Lucia & Blaizot 2007). In the high-redshift Universe the cold gas fraction in galaxies is higher than at low-redshift and thus there is more fuel for star formation (e.g. Tacconi et al. 2010; Geach et al. 2011). It is therefore possible to more easily trigger significant star-forming events during mergers (Somerville et al. 2001) or through high gas accretion rates and disk instabilities in isolated galaxies (Kereˇs et al. 2005; Bower et al. 2006; Dekel et al. 2009; Forster Schreiber et al. 2011; Cacciato et al. 2012). The latter process leads to the intriguing possibility of the enhanced star-formation rates at high redshift being dominated by secular evolution rather than mergers. In fact while some observations suggest an increase in the merger fraction with redshift (Conselice et al. 2003) others seem to prefer in-situ galactic processes over galaxy-galaxy merging, or at least a mixture of these processes (Lotz et al. 2008; Elbaz et al. 2007) To test whether it is galaxy mergers or secular processes that dominate and drive galaxy evolution at the peak era for star formation, a method to distinguish between galaxy mergers and nonmergers needs to be implemented. The two main methods of estimating the merger fraction are counting close pairs of galaxies, under the assumption that they will subsequently merge (e.g. Le F'evre et al. 2000; Lin et al. 2008; Bluck et al. 2009), and using a method of identifying galaxies with a merging morphology (e.g. Conselice et al. 2003; Lotz et al. 2004; Conselice et al. 2008; Lotz et al. 2008; Conselice et al. 2009). The results of both of these methods often suggest that the merger fraction increases with redshift and, depending on the mass range considered, the merger fraction at z /greaterorsimilar 1 , where the star formation rate density peaks, is roughly ∼ 0 . 1 -0 . 3 on average (but with some systematic offsets between studies) compared to a fraction /lessorsimilar 0 . 1 in the local Universe. A third, potentially more reliable, method is to employ detailed integrated field unit observations of z = 1 -2 galaxies to look for merger signatures in the dynamics of the galaxies. Such studies, although generally smaller in sample size, also find a merger fraction of ∼ 0 . 3 (e.g. Forster Schreiber et al. 2009; Shapiro et al. 2008). In order to study the star-forming population, an excellent tracer of ongoing star formation is the H α emission line which is less affected by dust obscuration than shorter wavelength starformation tracers (e.g. UV continuum light or [OII]3727). Unfortunately beyond z = 0 . 4 , H α is redshifted out of the optical window, thus high redshift studies of star formation have been limited to either using the obscuration-effected short wavelength tracers or studying small samples of H α emitters using conventional near-infrared spectrographs. However, in the last few years panoramic narrow-band surveys have started to provide large samples of H α -selected galaxies (e.g. the High-redshift (Z) Emission Line Survey, HiZELS, Geach et al. 2008, 2012; Garn et al. 2010; Sobral et al. 2009, 2010, 2012, 2013 and the studies of Villar et al. 2008 and Ly et al. 2011). Narrow-band surveys provide a well understood, volume-selected sample of star-forming galaxies allowing for straight-forward analysis of trends with SFR, mass and size etc. They provide emission line information over large areas of the sky and are thus able to probe a significant range of the H α luminosity and stellar mass functions for star-forming galaxies, required for an unbiased analysis of the star formation rate density (SFRD, e.g. Geach et al. 2008; Sobral et al. 2009, 2012, 2013). This selection method has also been shown to be extremely effective at detecting intrinsically faint galaxies, helping to overcome the bias towards massive galaxies associated with photometric redshift selection. In this study we use the z = 0 . 4 -2 . 23 HiZELS sample presented in Sobral et al. (2013), to not only analyse the merger rate as a function of redshift and stellar mass but also as a function of the well-determined SFR. We can therefore test whether it is major mergers that drive the rise to enhanced activity seen at these epochs. In contrast to earlier studies, which analyse Hubble Space Telescope ( HST ) rest frame UV morphologies, with the advent of the WFC3 camera we can also study the rest-frame optical bands for a subsample of our galaxies that lie within the CANDELS region of our survey and use this to calibrate morphologies derived from deep, wide-field, ground based near-infrared imaging, better matched to the extent of the full HiZELS fields. We also analyse the size-mass relation for star-forming galaxies over this epoch in order to study the size evolution which may also indicate the merger history of these systems. The structure of this paper is as follows. In § 2 we describe the HiZELS narrow band sample and the imaging data. We then derive SFR for the sample and analyse the evolution of the number density of galaxies above a given SFR. The size-mass relation is then studied in order to look for an evolution. A method for automating morphological classification is defined and this is used to study the merger rates of the galaxies in our sample and how they evolve and depend on SFR and mass. Finally, we discuss our findings in the context of understanding the physical processes that occur within galaxies, that lead to the rapid downturn in the global volume averaged SFR below z ∼ 1 . A Λ CDM cosmology ( Ω m = 0 . 27 , Ω Λ = 0 . 73 , H 0 = 70 kms -1 Mpc -1 ) is used throughout this work and all magnitudes are AB.", "pages": [ 1, 2 ] }, { "title": "2.1 The HiZELS survey", "content": "HiZELS (Geach et al. 2008; Sobral et al. 2013) is a Campaign Project using the Wide Field CAMera (WFCAM, Casali et al. 2007) on the United Kingdom Infra-Red Telescope (UKIRT) and exploits specially designed narrow-band filters in the J and H bands (NBJ and NBH), along with the H 2 S1 filter in the K band, to undertake panoramic, moderate depth surveys for line emitters. HiZELS targets the H α emission line redshifted into the nearinfrared at z = 0 . 84 , 1 . 47 and 2 . 23 using these filters. In addition, the UKIRT data are complemented by deeper narrow band observations with Subaru Suprime-Cam NB921 imaging (Sobral et al. 2012, 2013) to obtain H α emitting galaxies at z = 0 . 4 and the [OII] emission from the z = 1 . 47 H α sample, as well as deeper WFCAMand Very Large Telescope near-infrared imaging through the H 2 S1 filter in selected fields. The survey is designed to trace star-formation activity across the likely peak of SFR density and provide detailed information about a well-defined statistical sample of star-forming galaxies at each epoch (see Best et al. 2010). In this study we concentrate on the main HiZELS sample of z = 0 . 4 , 0 . 84 , 1 . 47 and 2 . 23 H α emitters in both the UKIRT Infrared Deep Sky Survey, Ultra Deep Survey (UKIDSS UDS, Lawrence et al. 2007, Almaini et al. in prep.) and The Cosmic Evolution Survey (COSMOS, Scoville et al. 2007) fields as described in Sobral et al. (2013) and we refer the reader to that paper for full details of the catalogues used. These data cover areas of 0 . 6 -1 . 6 square degrees depending on the field and waveband. The narrow band excess sources are visually inspected to remove image artefacts and, to ensure the galaxies are at the desired redshift, spectral energy distribution (SED) fitting and optimised colour-colour selections are used to provide clean samples of H α emitters in the four redshift slices (Sobral et al. 2013). The excess narrow-band flux is then converted into an emission line luminosity. For the analyses in this paper we take these cleaned catalogues and introduce cuts to ensure that the data in each narrow-band filter are complete to the same flux limit across the entire area observed. These final catalogues contain: 428 H α emitters at z = 0 . 40 , 595 at z = 0 . 84 , 420 at z = 1 . 47 and 372 at z = 2 . 23 down to the SFR limits ∼ 0 . 2 , 3 . 0 , 12 . 0 and 25 . 0 M /circledot yr -1 respectively (assuming A H α = 1 . 0 ), to an H α equivalent width lower limit of 25 ˚ A. The star formation rates for the HiZELS sample are calculated from the H α luminosity and the relation of Kennicutt (1998) ( SFR(M /circledot yr -1 ) = 7 . 9 × 10 -42 L ( Hα )(erg s -1 ) ), assuming a dust extinction A Hα = 1 mag (see Sobral et al. 2013). Stellar masses are computed by fitting SEDs to the rest-frame UV, optical and near-infrared data available ( FUV,NUV,U,B,g,V,R,i,I,z,Y,J,H,K , 3 . 6 µ m , 4 . 5 µ m , 5 . 8 µ m , 8 . 0 µ m collated in Sobral et al. 2013, see references therein), following Sobral et al. (2011) and the reader is referred to that paper for more details. The SED templates are generated with the Bruzual & Charlot (2003) package using Charlot & Bruzual (2007, unpublished) models, a Chabrier (2003) IMF, and an exponentially declining star formation history with the form e -t/τ , with τ in the range 0.1 Gyrs to 10 Gyrs. The SEDs were generated for a logarithmic grid of 200 ages (from 0.1 Myr to the maximum age at each redshift being studied). Dust extinction was applied to the templates using the Calzetti et al. (2000) law with E ( B -V ) in the range 0 to 0.5 (in steps of 0.05), roughly corresponding to A H α ∼ 0 -2 . The models are generated with different metallicities, including solar; the reader is referred to Sobral et al. (2011) for further details. For each source, the stellar mass is computed as the median of stellar masses of the 1 σ best-fits over the range of parameters. In Figure 1 ( left ) we plot the number density of galaxies, from the combined UDS and COSMOS fields, above a stellar mass of 10 10 M /circledot and a given SFR, against redshift. From this we can see that for a given SFR limit the number density increases rapidly with redshift. This is a manifestation of the fact that a typical starforming galaxy has a greater sSFR at higher redshift, forming stars more rapidly for a given mass. In order to look for trends with redshift we therefore define a quantity which we term the epochnormalised star formation rate ENSFR which is the SFR of a galaxy divided by the SFR /star ( z ) . SFR /star ( z ) is the star-formation rate derived from the quantity L /star H α found by fitting a Schechter function to the H α luminosity function at a given redshift, which we take from Sobral et al. (2013). We note that normalising the SFR to SFR /star ( z ) accounts, to first order, for the increase in sSFR with redshift. However, significant evolution in either the slope of the SFR - stellar mass relation or the dust obscuration would invalidate this. The values of SFR /star essentially double for each HiZELS redshift interval considered with SFR /star ∼ 7 . 0 , 14 . 0 , 29 . 0 , and57 . 0 M /circledot yr -1 for z = 0 . 4 , 0 . 84 , 1 . 47 and 2 . 23 respectively. Interestingly, this same behaviour is seen in the evolution of the typical sSFR from Elbaz et al. (2011) with sSFR ∼ 0 . 2 , 0 . 4 , 0 . 8 , and2 . 0 yr -1 , again at these redshifts. We suggest that this is because the H α luminosity (and thus SFR) function evolves significantly more than the stellar mass function. In Figure 1 ( right ) we plot the number density of galaxies of a given mass above the thresholds SFR/SFR /star ( z ) = 0 . 6 , 1 . 2 , 2 . 4 . From this plot one can clearly see that the number of star-forming galaxies with their SFR normalised to the typical SFR at that epoch is broadly constant. This means that the number density of starforming galaxies of a given mass and ENSFR does not evolve significantly over the period studied here. This demonstrates that the star-forming population is constant with redshift but simply evolves in sSFR. This is similar to the result found in Sobral et al. (2013) in which there is no strong evolution in the Schechter parameterisation of the normalisation of the H α luminosity function, φ /star H α . We discuss the implications of this in § 5.", "pages": [ 2, 3 ] }, { "title": "2.2 Imaging data", "content": "In this study we analyse near-infrared imaging from the space-based HST /WFC3 Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS, Grogin et al. 2011; Koekemoer et al. 2011) and the ground-based UKIDSS UDS and the WIRCam Deep Survey (WIRDS, Bielby et al. 2012). The CANDELS imaging we use is from WFC3 F160W covering a 2-orbit depth over 720 sq. arcmin of the UDS. The CANDELS imaging has a pixel scale of 0.06 arcsec and a point spread function (PSF) with a FWHM of 0 . 18 '' . The CANDELS data are well suited to this project for which we require high resolution imaging in the rest-frame optical, however to obtain the wider area coverage needed to build up a statistical sample of rarer high-mass systems from HiZELS we also need to use ground-based near-infrared imaging. The UKIDSS UDS K -band imaging covers an area of 0.8 square degrees, to a depth of K = 24 . 6 ( 5 σ , AB) with a pixel scale of 0.13 arcsec and a PSF FWHM of 0 . 7 '' . The WIRDS K -band imaging covers a total effective area of 2.1 square degrees and reaches an AB 50% completeness limits of ∼ 24 . 5 across the COSMOS field, it has a pixel scale of 0.15 arcsec and a PSF FWHM of 0 . 7 '' and is thus comparable to the UKIDSS UDS. The combination of these three near-infrared imaging datasets allows us to probe the rest-frame optical morphologies and sizes of the HiZELS galaxies over a wide range in luminosity while at the same time providing a rest-frame optical view of the galaxies' stellar distribution.", "pages": [ 3, 4 ] }, { "title": "3.1 Sizes", "content": "Before studying the morphologies and the merger rates of the galaxies in the HiZELS sample, we first assess their typical sizes. This is interesting from a galaxy evolution perspective, as an increase in size with cosmic time may imply that mass is being built up either through mergers or accretion or that the mass is being redistributed somehow. If there is no direct evolutionary connection between the galaxy populations at each epoch then changes in typical size may suggest differing formation scenarios. Importantly, it will also help us to understand the reliability of the morphological classification as the smallest galaxies will be most affected by the resolution of our ground-based imaging. The surface photometry of galaxies is often described by a S'ersic profile (S'ersic 1968). where I ( r ) is the intensity, r is the radius from the centre of the galaxy, r e is the scale radius, I e is the intensity at r e , n in the exponent is a free parameter widely known as the S'ersic index and b n = 2 n -0 . 327 ; a coefficient chosen so that r e is the half-light radius defined as the radius which encircles half the light from the galaxy (e.g. Graham et al. 1996). To measure the sizes of the galaxies we fit a 2-dimensional S'ersic profile to the galaxy images using the GALFIT (version 3) software package (Peng et al. 2002). This software requires reasonable initial input parameters such as position, apparent magnitude and ellipticity, all of which are estimated by first running the SEXTRACTOR package (Bertin & Arnouts 1996) so that the iterative fitting process converges to the correct solution in the shortest possible time. GALFIT deconvolves the point spread function which is dominated either by the telescope itself, in the case of HST , or by the atmospheric seeing for the ground-based imaging. To this end we check that the effect of seeing has been correctly accounted for in the analysis of the ground-based imaging by comparing the CANDELS derived sizes to those from the UKIDSS UDS imaging for the same galaxies. Figure 2 shows this comparison of galaxy sizes for a sample drawn from a combination of all four HiZELS redshift slices and a sample of BzK (Daddi et al. 2004) galaxies in the UDS field (the photometry to select BzK galaxies is taken from the UDS catalogues, Almaini et al., in prep). These two independent size measurements are correlated and scattered around the 1-to-1 line with ∆ r e /r e ∼ 0 . 4 , which confirms that the sizes recovered are comparable, demonstrating that GALFIT is able to successfully account for the seeing. We note that there may be some selection effects and biases in size measurements, in that galaxies with large-sizes can be missed due to low surface brightnesses and compact galaxies may have sizes overestimated (Barden et al. 2005). The former is less likely as the HiZELS galaxies are selected on their H α emission. Also, Figure 2 demonstrates that there is no significant bias in size estimates between the ground and space-based analysis of the smallest galaxies so we take this as evidence that their sizes are not overestimated. shape of the size function of disc galaxies between 0 . 1 < z < 1 . 0 with just an evolution in the number density of discs. However, other groups have found evidence for a stronger size evolution for the most massive ( M /star > 10 10 M /circledot ) disc-like galaxies, with a 2 -4 fold increase in size since z ∼ 2 (Trujillo et al. 2007; Mosleh et al. 2011). By analysing the S'ersic index, n , which we obtain from the fitting process we divide our sample into disc-like and bulge-like galaxies where we define the former as having 0 . 5 /lessorequalslant n < 2 . 5 and the latter as 2 . 5 /lessorequalslant n < 5 . 0 . From this we find that the fraction of disc-like galaxies is > 80% in each redshift slice with no evidence for an evolution, which is not unexpected as star-forming galaxies such as those selected by HiZELS are in general found to be discs, consistent with Sobral et al. (2009). We note that this disc fraction also has no trend with SFR or stellar mass.", "pages": [ 4, 5 ] }, { "title": "3.2.1 Quantifying and calibrating morphology", "content": "To quantify the morphologies of the galaxies in this study we choose to use a combination of Gini and M 20 coefficients first proposed by Lotz et al. (2004). The Gini coefficient, developed by statistician Corrado Gini, measures the inequality among values of a frequency distribution. It was first applied to studies of galaxy morphology by Abraham et al. (2003). A Gini coefficient of zero expresses an equality where all values are the same (i.e. a galaxy with uniform surface brightness). A Gini coefficient of 1 expresses maximal inequality among values (i.e. where all of the flux is in one pixel/element). The M 20 coefficient, describes the second-order moment of the brightest 20% of pixels in the galaxy and is sensitive to merger signatures such as multiple nuclei (Lotz et al. 2004). The combination of Gini and M 20 can differentiate between 'normal' star forming galaxies and Ultra Luminous Infrared Galaxies (ULIRGs), as well as single galaxies and merging systems. However, there are some differences in the boundaries chosen to delineate these populations (e.g. see Lotz et al. 2006, 2008) and therefore we choose to perform our own tests and calibrate the Gini and M 20 coefficients by visual inspection. The Gini and M 20 coefficients are calculated using the Gini and M 20 components of the galVSM software (Huertas-Company et al. 2008). This software requires a segmentation map which tells galVSM which pixels are associated with the galaxy. We first cutout 10 '' postage stamp images, taken from the CANDELS mosaic, around each galaxy and generate a segmentation map using SEXTRACTOR (Bertin & Arnouts 1996). The Gini and M 20 codes are then run on the postage stamps and the corresponding segmentation maps. The sample we choose to run the initial visual inspection calibration analysis on is that of 167 star forming galaxies in the redshift range 1 . 4 < z < 2 . 5 selected via the BzK method (Daddi et al. 2004) which lie within the CANDELS survey region in the UDS. The F160W mosaic provides high resolution rest frame optical imaging of these galaxies. We choose this sample over the HiZELS narrow-band sample as it should consist of similar star forming galaxies but has a higher surface density and so a larger sample falls within the high resolution CANDELS imaging, key to testing the morphological classifications. From visual inspection of the BzK galaxy morphologies, the SEXTRACTOR parameters DEBLEND MINCONT=0.1 and DETECT MINAREA=5 and DETECT THRESH=1 σ are found to be relaxed enough to associate clear merging components of the same 'galaxy' with one segmentation map but still stringent enough so as to not produce clear false positives. We note that having a DEBLEND MINCONT set too high means that unrelated galaxies would be considered as mergers whereas when set to a low value separate features within the same galaxy separate into distinct objects and therefore this parameter has the most effect on the M 20 coefficient (see Appendix A for a discussion of this parameter). Setting the detection threshold to low sigma values includes fainter 'sky' pixels in the segmentation map and thus increases inequality, raising the Gini coefficient. In this way one can see that the way in which the segmentation map is created is the most important factor in determining the Gini and M 20 coefficients and differences between how this is done in different studies are the reason why we choose to calibrate our own definitions of mergers and non-mergers. Using the above method, fixing the SEXTRACTOR parameters to those found to give the best performance, the Gini and M 20 codes are run on the CANDELS imaging with the results for the BzK sample are displayed in Figure 4 ( upper ). Also included is a 0 . 35 < z < 0 . 45 photometric redshift sample with a similar magnitude range to the HiZELS z = 0 . 40 sample sourced from Williams et al. (2009) to demonstrate that this classification technique is not affected by redshift. By visually assigning the galaxies into two categories 'mergers' and 'non-mergers' with the former classification based on evidence of merging components either creating disturbed morphologies or very close potential mergers (on-sky separation /lessorsimilar 2 '' ). This information is included in Figure 4, with the delineation between mergers and non-mergers found to occur at an M 20 ∼ -1 . 5 for both high and low redshift regimes and thus the Gini coefficient does not seem to add any information. Using this method there is a contamination of ∼ 10% non-mergers in the mergers and < 5% mergers in the non-mergers. The simulations performed for Appendix A demonstrate that the M 20 coefficient is sensitive to merging components down to a luminosity (mass) ratio of ∼ 1 : 10 (in agreement with the simulations of Lotz et al. (2010)). As such we note that our analysis throughout this paper is a measure of major mergers only. The morphology codes are then run on the same galaxies but using the deep K -band ground-based UKIDSS UDS so we can compare the two independent measurements. We expect the higher resolution CANDELS imaging to be a truer reflection of a galaxy's intrinsic morphology. We also note here that it is difficult to measure the morphologies of the lowest luminosity galaxies in our sample as they tend to be smaller (see size-mass relations in § 3.1) and are thus more affected by the seeing of the ground-based near-infrared imaging. By performing tests we find that setting DEBLEND MINCONT=0.03 ensures that the M 20 parameter selects the same type of mergers in the ground-based data as that derived from the HST data (again see Appendix A). The ground-based versus HST Gini and M 20 values are plotted in Figure 5. By performing linear fits to these relations we can calibrate the ground-based Gini and M 20 values to those derived from HST . These fits are: and will now be applied to the HiZELS morphologies derived from the ground-based near-infrared imaging. One potential problem with measuring the morphologies of galaxies at different epochs, using the same near-infrared imaging, is that of morphological k correction. Galaxies look smoother at longer wavelengths, meaning that the lowest redshift galaxies in our sample may artificially appear less disturbed than those at high redshift. When we analyse the HST Advanced Camera for Surveys (ACS) F814W imaging data available in COSMOS, many of the galaxies are very low surface brightness and therefore it is difficult to assess whether the morphological classifications given by the M 20 coefficient are reliable. However, for the galaxies in the z = 0 . 4 sample with K AB < 22 . 5 , the same classifications as those derived from the near-infrared CANDELS imaging are recovered in ∼ 90% of the cases, so we conclude that our results are not significantly affected by this. An additional concern is that some disc galaxies at high redshift are found to contain large star-forming clumps (e.g. Swinbank et al. 2010b, 2012). There may therefore be a degeneracy between what we classify as 'mergers' and those galaxies that contain a small number of large star-forming clumps. It is practically impossible to differentiate between these two populations without dynamical information and thus we note with caution that so called 'clumpy disc' galaxies may make a up some fraction of our 'merger' sample, if the clumps are on scales of /greaterorsimilar 4 kpc . In fact when we run the sub-sample of nine HiZELS galaxies which, from dynamical analysis of integrated field unit data, are all found to contain clumps (see Swinbank et al. 2012 for a description of this sample) all of them have M 20 /greaterorsimilar -1 . 5 and thus we would classify them as 'mergers'. We note here that when visually classified not all of these clumpy galaxies appear as clear mergers which may explain the non-merger interlopers with M 20 /greaterorsimilar -1 . 5 in Figure 4.", "pages": [ 5, 6 ] }, { "title": "3.2.2 HiZELS morphologies", "content": "The number densities of galaxies in the HiZELS samples are lower than the BzK morphology calibration sample used in § 3.2.1 and therefore do not have the same level of overlap with the CANDELS imaging region in the UDS. We instead run the morphology codes on the CANDELS, UKIDSS UDS and COSMOS WIRDS imaging for the HiZELS samples at each of the four redshifts. The output Gini and M 20 values for the ground-based near-infrared imaging are calibrated to the CANDELS values using the fits found for the BzK sample in § 3.2.1. As a confirmation of the calibration of the ground-based morphologies to those derived from the HST data the UKIDSS UDS Gini and M 20 coefficients for those that line in the CANDELS sub-region are plotted in Figure 4 ( lower ) but with the visual classifications derived from the CANDELS data indicated. The result of analysing the morphologies of these calibrated data is that we now choose to delineate the difference between mergers and non-mergers at M 20 = -1 . 45 which minimises the visual contamination to 22 ± 12% non-mergers in the merger region and 15 ± 7% mergers in the non-merger region. We note that some of the contamination of visually classified non-mergers to the merger fraction may in fact be due to galaxies with clumpy discs (see § 3.2.1). Figure 6 displays a sub-sample of the HiZELS galaxies classified by the M 20 parameter as mergers ( left ) and non-mergers ( right ) for both ground and space-based imaging, with their SEXTRACTOR segmentation maps over-plotted. As the Gini coefficient is found to add little information, when using our particular analysis methods, Figure 7 presents a histogram of M 20 values, as measured from the ground-based imaging of the HiZELS population at all redshift slices.", "pages": [ 7 ] }, { "title": "4.1 Merger fractions", "content": "Here we define 'merger fraction' as the number of galaxies with a merger-like morphology (regardless of how many galaxies actually make up this merger) divided by the total number of galaxies in the redshift slice. The total fraction of mergers for the HiZELS galaxies in the redshift bins z = 0 . 40 , 0 . 84 , 1 . 47 , 2 . 23 are 0.33, 0.13, 0.18 and 0.32 respectively (see Figure 7), however these are not comparable as they are measured for different stellar mass and SFR ranges at the different redshifts. For comparison Sobral et al. (2009) find a higher merger fraction of 0.28 at z = 0 . 84 using the morphological classifications of Scarlata et al. (2007) and a visual classification that included mergers and close pairs (which explains the higher merger fraction), although this is from rest-frame B -band imaging. However, when we study the COSMOS HST ACS imaging used in that study we find that many of the galaxies appear as very low surface brightness meaning that their morphological classifications are more uncertain. In Figure 8 ( left ) the merger fraction is plotted against stellar mass, with the lowest mass galaxies progressively more likely to be classed as mergers with ∼ 5 -20% of the star-forming population being mergers at the highest stellar masses in each of our redshift slices. The z = 0 . 4 , 0.84 and 1.47 trends are all remarkably similar and in agreement but there is an increase in merger fraction at all masses to z = 2 . 23 . However, the HiZELS selection is dependent on SFR, not mass and as described in § 2 the typical sSFR for galaxies increases with redshift and therefore we need to investigate these effects too. Afraction of 10-20% mergers is seen in the most strongly starforming galaxies at each redshift (Figure 8, centre ). However, due to the flux-limited nature of the samples and the evolution of typical sSFRthere is little overlap between different redshifts. In this figure the combined SFR data for all of the HiZELS redshift bins taken at face value may actually hint at a trend in merger fraction with SFR rather than any evolution with redshift (at least out to z = 1 . 47 ). There is some evidence of an increase in merger rate at the same SFR when going from z = 1 . 47 to z = 2 . 23 but again this does not account for the evolution in typical sSFR. Combining the two results above we investigate the relative contribution of mergers to the range of sSFR covered by our sample, for galaxies with ENSFR > 0 . 2 to which we are complete at all redshifts, in Figure 8 ( right ). From this plot it is clear that the galaxies with the higher sSFR at all redshifts are increasingly more likely to have a merger-like morphology, with those with the highest sSFR having a merger fraction of ∼ 40 -50% . This strongly suggests that starbursts are more likely to be driven by major mergers when compared to the rest of the star-forming population. This is in agreement with the far infrared selected sample of Kartaltepe et al. (2012) who find that major mergers have, on average, a high sSFR compared to typical star forming galaxies. It is also in broad agreement with results from the mass selected sample of Kaviraj et al. (2012) who also find that major mergers tend to have high sSFR compared to undisturbed galaxies. We test whether the A H α = 1 . 0 dust correction we universally employ is reasonable and how it affects our results. This is by including both SED fit extinction values (Sobral et al. 2013) and those derived from the relation between stellar mass and extinction from Garn & Best (2010). We find that using these more sophisticated estimates makes little difference for the range of masses we study. The value of A H α is ∼ 1 mag at a mass of 10 10 M /circledot with this value increasing/decreasing to higher/lower mass, with the typical range being A H α = 0 . 5 -2 mags. In fact when this more sophisticated treatment of dust obscuration is included it acts to strengthen our conclusions by smoothing the relations in Figure 8. However, we choose to keep the extinction value at A H α = 1 . 0 as this is easier to compare to other works including the main results in Sobral et al. (2013) and to'epoch normalise' with the H α luminosity function. It is unlikely that HiZELS is missing a large population of 'typical' high redshift star forming galaxies with high dust obscurations, as Reddy et al. (2012) demonstrate that the dust content of typical star forming galaxies actually decreases with redshift. However, the HiZELS sample may miss extreme star forming and highly obscured galaxies such as sub-mm galaxies. Submm galaxies are found to have a spread in morphologies which is indistinguishable from that of typical star forming galaxies at high redshift (Swinbank et al. 2010a) and are relatively rare objects (12 × 10 -5 Mpc -3 , Wardlow et al. 2011), thus their omission would not affect our conclusions.", "pages": [ 7, 8, 9 ] }, { "title": "4.2 Merger rates", "content": "To calculate the merger rates (the number of mergers per Gpc 3 per Gyr) we follow the prescription outlined in Lotz et al. (2011): that the merger rate is simply the number of mergers per Gpc 3 divided by the average timescale over which the merger would be observed. In Lotz et al. (2011) this observed merger timescale is found, from simulations, to be ∼ 0 . 2 Gyr, when the Gini/ M 20 method is employed. We adopt this value for consistency with that study and with the data from other groups recalculated and used there. We note that as HiZELS is a narrow-band survey the volumes covered at each redshift slice are well defined with values of ∼ 1 -7 × 10 -4 Gpc 3 (Sobral et al. 2013). We now also assume that there are on average two galaxies per merger for consistency with other studies. For comparison with other surveys we initially cut our sample only on stellar mass. The merger rates for galaxies with M > 10 9 and10 10 M /circledot are plotted in Figure 9 ( left ). Also plotted are values from Conselice et al. (2003) ( M B < -19 which approximates M > 10 9 M /circledot ) and those with M > 10 10 M /circledot derived from Gini/ M 20 (Lotz et al. 2008), close pairs (Lin et al. 2008) and galaxy asymmetry from Conselice et al. 2009 and L'opez-Sanjuan et al. (2009). These merger rates are corrected to the timescales calculated by Lotz et al. (2011) using the galaxy evolution models of Somerville et al. (2008). Figure 9 ( left ) shows little evolution in merger rate with redshift and the results are generally in good agreement with those found in the studies of Conselice et al. (2003, 2009); Lin et al. (2008); L'opez-Sanjuan et al. (2009) where the redshift ranges overlap. The merger rates from Lotz et al. (2008) are systematically higher, which may be because that sample is mass selected and therefore includes a significant contribution from merging red sequence galaxies which would not have been included in the HiZELS sample. There could also be secondary effects due to a mismatch in the stellar mass calculation between the studies, a different way of defining mergers through the M 20 parameter or a differential in the timescales involved, so an offset is perhaps not unexpected. With the exception of the z = 0 . 4 data point, which is significantly higher, the HiZELS merger rates for galaxies with M > 10 9 M /circledot are also in good agreement with those of the only study with this approximate mass limit (Conselice et al. 2003). From this plot there is no strong evidence for an increase in the merger rate for mass-selected samples out to z ∼ 2 . However, this comparison does not account for the SFR limits of the different surveys or the increase in typical sSFR with z (Elbaz et al. 2011). The advantage of HiZELS over these earlier studies is that it is unbiased with respect to stellar mass and we derive the stellar mass and SFR from independent measurements, i.e. SED fitting and H α flux. We can therefore consider both SFR and stellar mass independently to split the population into sub-samples based on these properties. As defined in § 2 we account for the increase in the typical sSFR with redshift by employing the ENSFR. Figure 9 ( right ) shows the population split into three ENSFR bins > 0 . 6 , 1.2 and 2.4, for which the HiZELS observations are complete at all redshifts. The first obvious thing to notice is that undulating shape of the plot with just a mass cut (Figure 9, left ) has disappeared. Instead the trends are flat, showing no evidence for an increase in the merger rate with increasing redshift for all masses and ENSFR cuts. This mass and ENSFR selected sample is a cleaner sample than those in Figure 9 ( left ) and so we suggest that the peak in the merger rate at z ∼ 1 seen for some comparison samples may be due to the mixing of a mass limit with an SFR selection function which strongly effects photometrically-selected galaxies. From this merger analysis we can determine the total number of major mergers (with mass ratio > 1 : 10 ) a galaxy of a given mass will undergo during the epoch covered by our study. Using Equation 11 from Conselice (2006) we find that one would expect ∼ 3 mergers per star-forming galaxy with M ∼ 10 10 M /circledot between z = 2 . 23 and z = 0 . 4 , or a merger every 2 Gyrs on average. We note that these numbers depend on the value of τ the timescale over which mergers can be observed using the M 20 method (which we assume to be 0.2 Gyr, Lotz et al. 2011 ) and therefore more generally there are 0 . 6 τ 0 τ -1 mergers between z = 2 . 23 and z = 0 . 4 , corresponding to 0 . 1 τ 0 τ -1 mergers per Gyr, where τ 0 = 1 Gyr .", "pages": [ 9 ] }, { "title": "5 DISCUSSION & CONCLUSIONS", "content": "The HiZELS narrow-band H α survey selects star-forming galaxies within four well-defined volumes at z ∼ 0 . 4 -2 . 2 and flux limits with an SFR indicator which is unbiased in terms of stellar mass and is independent of its determination. In this paper we have used these properties to understand the star-forming population and its merger rate to help illuminate the processes responsible for the upturn in the SFRD with redshift. By defining the epoch-normalised star-formation rate ( ENSFR = SFR / SFR /star (z) ) we account for the increase in the typical star-formation rate of galaxies with redshift. In § 2 we demonstrate that the number of galaxies above a given mass and ENSFR does not evolve significantly over the 6 Gyr from z = 0 . 4 to 2.23. We also note that the HiZELS sample has already been shown to accurately trace the increase of the SFRD with redshift and that there is no strong evolution in the normalisation of the H α luminosity function (Sobral et al. 2013). Taken, in combination this means the increase in the SFRD with redshift is not due to an increase in the number of star-forming galaxies of a given mass but instead must result from an increase in the amount of star formation in these galaxies. This can be described as an increase in the average sSFR for star-forming galaxies (Rodighiero et al. 2010; Elbaz et al. 2011) without a significant increase in their number density. Also, we note that the SFR /star (derived from L /star H α ) evolves in the same way as the typical sSFR for star forming galaxies (Elbaz et al. 2011), which implies that the luminosity of the knee in the H α luminosity function is evolving significantly more rapidly than the characteristic mass of the stellar mass function. The size-mass relation for galaxies is assessed in § 3.1. In order to do this for a large sample we need to use wide-field ground-based imaging. Hence we confirm that we can reliably recover the galaxy size determined from the HST CANDELS imaging by deconvolving the affect of atmospheric seeing from the ground-based imaging. We find that the size-mass relation is surprisingly constant out to z = 2 . 23 , in agreement with the findings of Barden et al. (2005); Ichikawa et al. (2012) and at odds with the results of Trujillo et al. (2007); Mosleh et al. (2011). The lack of strong size evolution at a given mass and the universal size-mass relation for star-forming galaxies in the range 0 . 4 < z < 2 . 23 suggests that this population have not experienced significant size evolution, through mergers or star formation, during this period. Any evolution that does occur must thus act to move the galaxy along the locus of the relation. The slope of this relation is also shallow and thus low mass galaxies are not dramatically smaller than their higher mass counterparts. Even if there is no direct evolutionary connection between the galaxy populations at each epoch then this lack of change in typical size suggests a universal evolution scenario. In order to study the merger rates of the HiZELS galaxies we test the Gini and M 20 coefficients. By investigating these automated methods of determining merger classifications we find that SEXTRACTOR parameters that define the segmentation map employed in these analyses are the most important factor in how well the method performs (see Appendix A). We find that, for the segmentation maps generated by our set of SEXTRACTOR parameters, the best delineation between mergers and non-mergers is M 20 = -1 . 45 while the Gini coefficient provides no useful information. We acknowledge that other authors have found this not to be the case with M 20 and Gini being equally important in morphological classification (Lotz et al. 2004; Wang et al. 2012) but we assume this is due to the differences in the construction of the segmentation maps (see Appendix A) and potentially minor variations in the normalisation of the M 20 and Gini values, depending upon the exact nature of the morphological code used. The M 20 coefficient is found to be sensitive to mergers down to a mass ratio of ∼ 1 : 10 (in agreement with Lotz et al. (2010)). We note here that not all mergers are star forming and as such we will miss 'dry' mergers which do not induce activity, although obviously these will not be major contributors to the SFRD. As with the sizes we find that it is possible to use this morphological classification on ground-based data affected by atmospheric seeing, after applying a calibration derived from galaxies that are observed with both ground-based telescopes and HST . For the sample as a whole, without accounting for the H α flux (SFR) limit or the increase in the sSFR of the star forming galaxies with redshift, we find that the merger fraction anti-correlates with both stellar mass and SFR. By combining these two results we find that the merger fraction correlates strongly with sSFR. This suggests that the more rapid the star formation is, the more likely it is to be driven by violent major mergers than secular processes. In fact we find that, ∼ 50% , of starburst galaxies in our z ∼ 2 sample have major merger morphologies. Therefore to achieve such high sSFR, these galaxies are undergoing major merger driven and not 'main-sequence' star formation. Interestingly we see no evolution in the merger fraction of starbursts with a constant ∼ 40 -50% across all redshifts which suggests that merging is a universal process that can lead to a galaxy having enhanced sSFR for their epoch (Hopkins et al. 2006; Kartaltepe et al. 2012). Finally we consider the merger rates of star-forming galaxies initially only limiting our sample on stellar mass, where some previous studies have seen the characteristic merger rate increase to z ∼ 1 . However these other studies use photometrically selected samples where the method of determining stellar mass is directly linked to the determination of the SFR. As these two parameters are independent in HiZELS we can also select on SFR for a fair comparison across the redshift range, while also accounting for the increase in sSFR for typical star-forming galaxies with redshift. By applying these selections we see little evidence for an increase in the merger rates of typical galaxies over the redshift range considered. Therefore even though there is an order of magnitude increase in typical SFR across the redshift range of our study this is not reflected in the merger rate. This is strong evidence that it is not major mergers that drive the increase in the SFRD with redshift, in contrast to the models of Somerville et al. (2001) or Hopkins et al. (2006) and as observed in part by Conselice et al. (2003, 2008) and Lin et al. (2008) who find some evidence for an increase in merging. Our result agrees with Sobral et al. (2009) who find that the increase in SFRD between z = 0 and z = 0 . 84 was primarily due to regular (non-merging) galaxies. Depending on the timescale τ for which it is possible to view a galaxy undergoing a major merger using the M 20 parameter we find that star forming galaxies with mass > 10 10 M /circledot undergo ∼ 0 . 6 τ 0 τ -1 (3 if τ = 0 . 2 Gyr) mergers between z = 2 . 23 and z = 0 . 4 , corresponding to ∼ 0 . 1 τ 0 τ -1 (0.5 if τ = 0 . 2 Gyr) mergers per galaxy per Gyr, where τ 0 = 1Gyr . From analysis of the mass function of galaxies in COSMOS at z = 0 . 35 -0.75, Pozzetti et al. (2010) find merger rates of ∼ 0 . 1 -0 . 4 per galaxy per Gyr for galaxies with masses ∼ 10 10 . 5 -10 11 M /circledot , in reasonable agreement with our findings (both are very sensitive to the choice of τ ). From a theoretical point of view Hopkins et al. (2010) compile data from a number of simulations and models (see references therein). The predicted number of mergers per galaxy with mass ∼ 10 10 -10 11 M /circledot per Gyr is found to increase with redshift from a value of ∼ 0 . 05 at z = 0 . 4 to ∼ 0 . 25 at z = 2 . 2 apparently lower than the values we find. Again, this is dependent on τ so we are unable to provide solid constraints. In summary we find that the increase in SFRD is due to an increase in the sSFR of typical star-forming galaxies. The process responsible for this increase is not major mergers as we find that the merger rate does not increase in step with the SFRD. We therefore conclude that secular processes such as disc instabilities and/or an increase in the effective fuel for star formation are the main driver of the increase in the SFRD with redshift as predicted or observed by others (Kereˇs et al. 2005; Dekel et al. 2009; Bower et al. 2006; Forster Schreiber et al. 2011; Cacciato et al. 2012). Although we note that it could also be driven by an increase in the minor merger rate (mass ratios < 1 : 10 ) which this study is not sensitive to. We also find a constant merger fraction for starburst galaxies, in that around half are major mergers across all redshifts, demonstrating that extremely violent events are required for a galaxy to attain enhanced sSFR for their epoch and leave the 'main-sequence'. Bringing these results together along with the lack of size evolution since at least z = 2 . 23 we can say that many of the properties of star forming galaxies are surprisingly constant over the ∼ 6 Gyr covered in this study.", "pages": [ 9, 10, 11 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "Wefirst thank the anonymous referee for improving the clarity of this paper. J.P.S. acknowledges STFC for financial support. D.S. acknowledges the award of a NOVA fellowship. I.R.S. acknowledges support from STFC and the Leverhulme Trust. We also thank James Mullaney and Mark Swinbank for useful discussions. The United Kingdom Infrared Telescope is operated by the Joint Astronomy Centre on behalf of the Science and Technology Facilities Council of the U.K. Based on observations obtained with WIRCam, a joint project of CFHT, Taiwan, Korea, Canada, France, at the Canada-FranceHawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institute National des Sciences de l'Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii. This work is based in part on data products produced at TERAPIX, the WIRDS (WIRcam Deep Survey) consortium, and the Canadian Astronomy Data Centre. This research was supported by a grant from the Agence Nationale de la Recherche ANR-07-BLAN-0228. This work is based on observations taken by the CANDELS Multi-Cycle Treasury Program with the NASA/ESA HST, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.", "pages": [ 11, 12 ] }, { "title": "REFERENCES", "content": "Sobral D., Best P. N., Geach J. E., Smail I., Cirasuolo M., Garn T., Dalton G. B., Kurk J., 2010, MNRAS, 404, 1551 Sobral D. et al., 2009, MNRAS, 398, 75 Sobral D., Best P. N., Matsuda Y., Smail I., Geach J. E., Cirasuolo M., 2012, MNRAS, 420, 1926 Sobral D., Best P. N., Smail I., Geach J. E., Cirasuolo M., Garn T., Dalton G. B., 2011, MNRAS, 411, 675 Sobral D., Smail I., Best P. N., Geach J. E., Matsuda Y., Stott J. P., Cirasuolo M., Kurk J., 2013, MNRAS, 428, 1128 Somerville R. S., Hopkins P. F., Cox T. J., Robertson B. E., Hernquist L., 2008, MNRAS, 391, 481 Somerville R. S., Primack J. R., Faber S. M., 2001, MNRAS, 320, 504 Springel V. et al., 2005, Nature, 435, 629 Swinbank A. M. et al., 2010a, MNRAS, 405, 234 Swinbank A. M. et al., 2010b, Nature, 464, 733 Swinbank A. M., Sobral D., Smail I., Geach J. E., Best P. N., McCarthy I. G., Crain R. A., Theuns T., 2012, MNRAS, 426, 935 Tacconi L. J. et al., 2010, Nature, 463, 781 Villar V., Gallego J., P'erez-Gonz'alez P. G., Pascual S., N oeske K., Koo D. C., Barro G., Zamorano J., 2008, Astrophys. J., 677, 169 Wang T. et al., 2012, Astrophys. J., 752, 134 Wardlow J. L. et al., 2011, MNRAS, 415, 1479 Williams R. J., Quadri R. F., Franx M., van Dokkum P., Labb'e I., 2009, Astrophys. J., 691, 1879", "pages": [ 13 ] }, { "title": "APPENDIX A: M 20 SIMULATIONS", "content": "We discuss here the effect of the SEXTRACTOR property DEBLEND MINCONT on the segmentation map and derived M 20 value. In § 3.2.1 we find that by analysing the M 20 values and visual classification of a sample of z = 1 . 4 -2 . 5 star forming galaxies, a segmentation map generated with a value of DEBLEND MINCONT=0.1 provides a demarcation between major mergers and non-mergers, with a boundary found at M 20 ∼ -1 . 5 (we later fix this value to -1 . 45 ). To quantify how the M 20 value relates to a major merger we run some very basic simulations. The simulations comprise of moving one artificial galaxy towards another and plotting the variation of M 20 with distance. The artificial observations are created using the GALFIT software with both galaxies being face on discs (i.e. S'ersic index, n = 1 ) of the same magnitude and half-light radius. We first perform an analysis appropriate to the high redshift star forming galaxies for which DEBLEND MINCONT=0.1 is found to efficiently select mergers. For the simulations we use the appropriate values of the sky noise, magnitude zero point and PSF of the observation we are simulating. The results of this simulation are presented in the black points on Figure A1. This demonstrates that for a DEBLEND MINCONT=0.1 and CANDELS HST data the M 20 value remains low and consistent with being a nonmerger for galaxy separations down to ∼ 1 . 6 arcsec ( ∼ 13kpc at z = 0 . 84 -2 . 23 ), as the galaxies have individual segmentation maps. Once this separation drops below this value the two galaxies share the same segmentation map and thus the M 20 value jumps dramatically as the top 20% of the light is now spread over two locations rather than one. As the separation decreases further this value lowers until a point is reached where the top 20% of the light is essentially co-located at the centre of a single bright galaxy and as such the M 20 curve resembles a 'shark fin'. The distance over which this system would be classed as a merger is then ∼ 1 arcsec which at z = 0 . 84 -2 . 23 corresponds to a distance of ∼ 8 kpc . Potentially the most important factor influencing the M 20 value for a given galaxy is whether it is derived from the spacebased HST data as discussed above or from the ground-based imaging with a significantly larger PSF and different background characteristics. We first investigate this by using the original space-based value of DEBLEND MINCONT=0.1 on the ground-based data and find that this gives a factor of ∼ 2 larger range in separation over which the galaxies would be classified as a merger and would thus result in an increase in merger numbers relative to the HST imaging. A value of DEBLEND MINCONT=0.03 accounts for this difference, equalling the separation over which a 'merger' occurs with the results plotted as red squares in Figure A1. This plot confirms the slope in the relation seen between ground and space-based derived M 20 seen in Figure 5 and used to calibrate the ground-based values. For the lower redshift z = 0 . 4 sample, this angular distance range corresponds to 5 . 3 kpc and as such may miss some galaxies that would have been classed as mergers in the higher z samples. We therefore alter the value of DEBLEND MINCONT to 0.11 for the CANDELS and 0.04 for ground-based imaging to account for this so that the same separation in kpc is used at each redshift. By varying the relative magnitudes of the galaxies and assuming the flux is linearly proportional to the mass and the size is proportional to the square root of the mass we test what mass ratio of mergers can be seen with this method. The result is that the M 20 coefficient is sensitive to mergers with a luminosity (mass) ratio down to ∼ 1 : 10 (in agreement with the simulations of Lotz et al. 2010). For mass ratios less than this the M 20 coefficient does not increase significantly when the two galaxies share the same segmentation map.", "pages": [ 13 ] } ]

[ { "title": "ABSTRACT", "content": "We argue that one of the basic assumptions of the Bondi accretion process, that the accreting object has zero pressure, might not hold in many galaxies because of the pressure exerted by stellar winds of star orbiting the central super massive black hole (SMBH). Hence, the Bondi accretion cannot be used in these cases, such as in the galaxy NGC 3115. The winds of these highvelocity stars are shocked to temperatures above the virial temperature of the galaxy, leading to the formation of a hot bubble of size ∼ 0 . 1 -10 pc near the center. This hot bubble can substantially reduce the mass accretion rate by the SMBH. If the density of the hot bubble is lower than that of the interstellar medium (ISM), a density-inversion layer is formed. As the gas loses energy by X-ray radiation, eventually more mass of the cooling shocked stellar winds will be accreted to the SMBH. This accretion will be of cold clumps. After a period of millions of years of low AGN activity, therefore, a stronger AGN activity will occur that will heat and expel gas, much as in cooling flow clusters. Adding to other problems of the Bondi process, our results render the Bondi accretion irrelevant for AGN feedback in cooling flow in galaxies and small groups of galaxies and during galaxy formation.", "pages": [ 1 ] }, { "title": "SUPPRESSING HOT GAS ACCRETION TO SUPERMASSIVE BLACK HOLES BY STELLAR WINDS", "content": "Shlomi Hillel 1 and Noam Soker 1", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "It is widely accepted that feedback powered by active galactic nuclei (AGN) has a key role in galaxy formation and in cooling flows in galaxies and in clusters of galaxies. In galaxy formation AGN feedback heats and expels gas from the galaxy (e.g., Bower et al. 2008; Ostriker et al. 2010 and references therein), and by that can determine the correlation between the central super-massive black hole (SMBH) mass and some properties of the galaxy (King 2003, 2005; Soker 2009; Soker & Meiron 2011). In cooling flow clusters jets launched by the SMBH heat the gas and maintain a small, but non zero cooling flow (see review by McNamara & Nulsen 2007, 2012; Fabian 2012); this is termed a moderate cooling flow. There is a dispute on how the accretion onto the SMBH occurs, in particular in cooling flows. One camp argues for accretion to be of hot gas via the Bondi accretion process (e.g., Allen et al. 2006; Russell et al. 2010; Narayan & Fabian 2011), while the other side argues that the accretion is of dense and cold clumps in what is termed the cold feedback mechanism (Pizzolato & Soker 2005, 2010). The cold feedback mechanism has been strengthened recently by observations of cold gas and by more detailed studies (Revaz et al. 2008; Pope 2009; Wilman et al. 2009, 2011; Nesvadba et al. 2011; Cavagnolo et al. 2011; Gaspari et al. 2012a,b; McCourt et al. 2012; Sharma et al. 2012; Farage et al. 2012; Kashi et al. 2012). The Bondi accretion process, on the other hand, suffers from two problems. The first problem is that in cooling-flow clusters the Bondi accretion rate is too low to account for the AGN power (e.g., McNamara et al. 2011; Cavagnolo et al. 2011). The second is that there is no time for the feedback to work (Soker et al. 2009). This is because the time for cooling gas at distances of /greaterorsimilar few × kpc in the Bondi accretion process to be accreted and power jets that heat back the ISM, is much longer than the cooling time of the gas. This is already true for gas cooling at a moderate distance of ∼ 1 kpc from the center. In other words the gas at large distances has no time to communicate with the SMBH before it cools. In this paper we point out yet another problematic point with the Bondi accretion process. In a recent paper, Wong et al. (2011) resolved the region within the Bondi accretion radius of the S0 galaxy NGC 3115. If the density and temperature profile is interpreted as resulting from a Bondi accretion flow onto the M BH = 2 × 10 9 M /circledot central SMBH, the derived accretion rate is ˙ M B = 2 . 2 × 10 -2 M /circledot yr -1 . They note that for a radiation power of 0 . 1 ˙ M B c 2 , the expected accretion luminosity is six orders of magnitude above the observed upper limit. They attribute this to a process where most of the inflowing gas is blown away, or the gas is continuously circulating in convective eddies, or to that the region they resolve is not yet incorporated to the Bondi accretion flow. The idea of circulating eddies has some similarities to the density inversion layer behavior we discuss here. In any case, some AGN activity does take place in NGC 3115 (Wrobel & Nyland 2012). Wrobel & Nyland (2012) detected a radio nucleus in NGC 3115 with a radio power of L radio = 3 × 10 35 erg s -1 . This indicates the presence of a weak AGN, that might substantially reduce the accretion rate (Wrobel & Nyland 2012). As we discuss later, the feeding of the SMBH might be from the stellar winds rather than from the ISM. Several other processes were considered to reduce the accretion rate by a SMBH much below the Bondi accretion rate. Such processes include magnetic field reconnection (Igumenshchev & Narayan 2002), angular momentum (Proga & Begelman 2003a,b), magneto-thermal instabilities (Sharma et al. 2008), and instabilities due to self-gravitation of the infalling gas (Levine et al. 2010). Lack of spherical symmetry in realistic situations is an additional factor (Debuhr et al. 2011). Turbulent media can have higher than Bondi-Hoyle accretion rate, but due to vorticity, a lower accretion rate is also possible (Krumholz et al. 2005, 2006). Hobbs et al. (2012) claim that the Bondi-Hoyle solution is only relevant for hot virialized gas with no angular momentum and negligible radiative cooling. We take a different view on the suppression of the Bondi accretion. We argue that in many galaxies for a fraction of the time the Bondi accretion flow might not be relevant because one cannot assume a zero pressure at the center, either because of stellar winds or because of jets blown by the AGN.", "pages": [ 1, 2 ] }, { "title": "2. THE PRESSURE OF STELLAR WINDS", "content": "The pressure exerted by stellar winds of high velocity stars (i.e., moving much faster than the dispersion velocity in the galaxy) with an average mass loss rate per star of ˙ m ∗ can be calculated in two limits, which basically lead to the same result. First we calculate the pressure by considering the total outward momentum flux at radius r . Because the orbital velocities of stars around the SMBH are much larger than the typical velocities of the stellar winds (as most of the mass loss is during the asymptotic giant branch, AGB, phase), the relevant velocity in general is not that of the wind relative to the star, but rather the velocity of the star under the gravitational influence of the SMBH, This holds as long as the SMBH gravity dominates that of the galaxy. In NGC 3115 that we study in more detail in section 3, for example, the SMBH gravity dominates that of the galaxy to a distance of ∼ 30 pc as the black hole mass is M BH = 2 × 10 9 M /circledot . Let stellar winds from high-velocity stars dominate the pressure inside a sphere of radius R h . The pressure exerted by the wind on a surface of radius R h is approximately given by adding the ram pressures of winds from all stars inside the sphere of radius R h , where n ∗ is the stellar number density in the center of the galaxy, and η is the fraction of the mass lost by stars that is shocked and heats up. In all our expressions the stellar mass loss rate appears as η ˙ m ∗ . Some of the mass lost by stars will form dense clumps that will cool rapidly even if being shocked, or will not even be shocked. This is particularly true as most of the mass is being lost by AGB stars that have dense winds. The thermal pressure of the ISM in the center will cause part of the winds' gas to form dense clouds. Many of the cold clumps can be evaporated by heat conduction form the hot gas in the bubble. However, some clumps might flow inward and feed the SMBH, and explain the AGN activity observed by Wrobel & Nyland (2012). The average mass loss rate is calculated as follows. A solar-like star loses ∼ 0 . 5 M /circledot over ∼ 10 10 yr. Considering an old population of stars, the mass loss rate is lower even. More accurately, most of the mass loss is due to AGB stars, which live for ∼ 10 7 yr, and lose mass at an average rate of ∼ 10 -7 M /circledot yr -1 (Willson 2007). During the final stages of the AGB the evolution is faster and the mass loss rate is higher. If there is a young stellar population, the total mass loss rate can be much higher. The ram pressure will not increase much beyond few pc because the stellar density decreases. An alternative point of view would be to express the pressure as (roughly) the energy density of the shocked stellar wind. We also assume a constant pressure and density inside this sphere. This is justified because we are interested mainly in the outer part of the hot bubble, where density inversion might take place. Even a steep power law profile, say of ρ ∼ r -2 , will not change much the density from 0 . 5 R h to R h , which contains 0.875 of the volume of the bubble. We can calculate the rate of energy input and multiply by the time it takes the hot gas to leave the inner region where β /lessorsimilar 1 takes into consideration that the hot gas at the center escapes at velocity lower than the escape velocity. The stellar wind pressure in this case can be written as where the energy deposition rate is Scaling the different quantities and assuming a constant stellar density we find where the stellar density is scaled by the average stellar density within ∼ 3 pc from the center of NGC 3115 (Kormendy et al. 1996). Equations (4) and (6) are more accurate than equation (2) when the radiative cooling time of the colliding stellar winds is larger than the escape time τ esc , which is the case here due to the high-temperature low-density post-shock stellar winds. The radiative cooling time is τ c = (5 / 2) nkT/ ( n e n p Λ) /similarequal 10 7 -10 8 years, This is much longer than the escape time given in equation (3) τ esc /similarequal 10 2 -10 3 years. Here n e , n p , and n are the electron, proton, and total number density, respectively, and Λ is the cooling function. Therefore, from now on we will refer to the hot gas region formed by the shocked stellar winds as the hot bubble, and to its radius as R h . For a constant stellar density within radius r , we find P e ∗ = 3 2 β -1 P m ∗ . If the stellar density drops to zero at some radius r z (a nonrealistic ideal case), the pressure beyond r z will drop like ( r/r z ) -2 . The average density of the hot shocked stellar wind is given by The flow structure is schematically drawn in Fig. 1. Relevant to this flow structure is the simulations of Cuadra et al. (2008). They simulated the dynamics of stellar winds in the Galactic center and found the accretion rate to be highly variable, due in part to the stochastic nature of infalling cold clumps. Fryer et al. (2007) suggest that the inner ∼ 5 pc region surrounding Sgr A ∗ in our Galaxy can be approximated by a wind-blown hot bubble density structure.", "pages": [ 2, 3, 4 ] }, { "title": "3. THE CASE OF NGC 3115", "content": "At the Bondi radius R B /similarequal 210 pc of the galaxy NGC 3115 the ISM pressure is P ( R B ) = 2 × 10 -11 erg cm -3 , the electron number density is n e ( R B ) = 0 . 02 cm -3 , and the temperature is T ( R B ) = 3 . 5 × 10 6 K (Wong et al. 2011). The Bondi radius is given by where c s is the sound speed in the undisturbed gas. The temperature and electron density increase inward, reaching values of T 20 /similarequal 10 7 K and n e 20 /similarequal 0 . 3 cm -3 at r = 20 pc (Wong et al. 2011; no values are given at smaller radii). We also note that in NGC 3115 the BH gravity dominates that of the galaxy to a distance of ∼ 30 pc as the black hole mass is M BH = 2 × 10 9 M /circledot . The average density and pressure of the hot bubble according to equations (7) and (6), are drawn in Fig. 2 for a SMBH mass of M BH = 2 × 10 9 M /circledot , and a stellar density given by and for β = 1 (eq. 3) and η = 0 . 1 (eq. 2). The density within r = 3 pc is from Kormendy et al. (1996), while at r > 3 pc is our assumption. The particular form of the decline in stellar density at r > 3 pc has no significant consequences, and the particular power law was chosen for the sake of simplicity and definite calculations. The value of the mass loss efficiency, which is the fraction of the mass lost by stars that ends up as hot gas in the hot bubble, is chosen as η = 0 . 1 to more or less match the pressure and density of the ISM at r = 20 pc. It is a parameter of the model that should be typically in the range of ∼ 0 . 1 -1. The temperature that is calculated from the pressure is also drawn on Fig. 2. Beyond ∼ 30 pc the average temperature is only ∼ 2 times as large as the virial temperature of the cluster, and our assumptions of a hot bubble become inadequate. The following conclusions emerge from Fig. 2. (1) The pressure of the shocked stellar winds of the highvelocity circum-SMBH stars is larger than the ISM pressure near the center, even for a mass loss efficiency of only η ∼ 0 . 1. This accounts, we argue, for the accretion rate of NGC 3115 being much lower than the Bondi accretion rate (Wong et al. 2011). (2) At the center, r < 3 pc, the rate of mass loss into the hot gas per unit volume is ˙ χ ≡ ( n ∗ η ˙ m ∗ ) c = 5 × 10 -6 M /circledot pc -3 yr -1 . Even if this value is ten times lower, a hot bubble with pressure larger than the ISM pressure of NGC 3115 can still be formed. (3) For ˙ χ /lessorsimilar 10 -5 M /circledot pc -3 yr -1 the hot bubble's density is lower than that of the ISM. This structure is Rayleigh-Taylor (RT) unstable. This structure is analyzed below. We note that the structure presented here is a temporary one. Eventually, the gas in the center originated from stellar winds will radiatively cool and form cold clumps. Some will be accreted and amplify the AGN activity. Many other clumps will be evaporated by the hot bubble and by the new AGN activity. Accretion of clumps onto a SMBH in a turbulent medium was studied by Hobbs et al. (2011), and accretion of cold clumps onto Sgr A ∗ in the Milky Way was simulated by Cuadra et al. (2008).", "pages": [ 5 ] }, { "title": "4. A TENUOUS HOT BUBBLE FORMED BY STELLAR OR AGN WINDS", "content": "We found above that in some cases the hot bubble that formed by the stellar winds of circum-SMBH high-velocity stars can have a lower density than the ISM while its pressure is about equal to the ISM pressure P ISM . This situation is prone to RT instability. The same might hold for AGN winds. The power of the winds that is required to form a hot bubble that can support the ISM is where the escape time τ esc is given by equation (3), and V is the volume of the hot bubble. This implies that even a very weak AGN wind can form such a bubble. With an efficiency of 1%, namely, W wind = 0 . 01 ˙ M BH c 2 , the required accretion rate is ˙ M BH = 5 × 10 -8 M /circledot yr -1 . For comparison, we note that the Chandra upper limit on the luminosity of NGC 3115 is ∼ 10 38 erg s -1 (Diehl & Statler 2008), and the radio power is L radio = 3 × 10 35 erg s -1 (Wrobel & Nyland 2012). The luminosity of the hot bubble as studied here has a low X-ray luminosity compared with the external gas. First, the volume of the bubble is very small. Second, the density inside the bubble is lower than that of the surrounding gas, and hence its emissivity is lower. Even if more of the stellar wind incorporated to the bubble, the X-ray luminosity from the bubble is much below detection limits. The flow structure considered in this section has the following properties. The hot bubble is continuously supplied by hot gas from the shocked stellar winds or the AGN wind or jets. A pressure equilibrium is maintained between the hot bubble and the ISM, and a structure of a hot tenuous gas supporting a denser and cooler gas is achieved. This structure is RT unstable. Such a structure, we claim, is similar to the density inversion found in the outer atmosphere of red giant stars (e.g., Harpaz 1984; Freytag & Hofner 2008), but not identical. At the outer edge of the recombination zone of hydrogen in red giant stars the convection heat transfer becomes less efficient. The requirement to transfer energy leads to a steep temperature gradient that in turn causes a density inversion, i.e., the density increases outward (e.g., Harpaz 1984). This occurs in the convective region, which is already unstable. In the density-inversion layer in stars, therefore, cold convective cells fall and hot convective cells buoy outward. We suggest that the same process occurs in the flow structure discussed here. There are some basic differences in the properties of the density-inversion layers of stars and of the case studied here. The main differences are that the hot gas in our case buoys to large distances, and fresh gas from stellar wind or the AGN replaces it. Also, the entire region is optically thin, unlike stars where it is optically thick. In stars the width of the density-inversion region is determined by heat transfer requirements, whereas in our case it is determined by dynamics, mixing, and local heat conduction. In stars the density scale height is not much shorter than the pressure scale height l p . The size of the convective cells is taken to be of the order of the pressure scale height. In our case the density can change by an order of magnitude from the inner tenuous region to the denser outer ISM, and we expect the RT instability to break the cells to smaller cells. We therefore take the size of the rising and falling gas elements to be R c /lessmuch l p . We take the density-inversion zone to be of the order of the pressure scale height (in stars it can be much smaller). For a central gravity source the pressure scale height for a constant temperature is given by where C i is the isothermal sound speed, and u ∗ ( R h ) is the stellar velocity given in equation (1) and evaluated at the radius of the hot bubble R h . The shocked stellar wind will be heated to a temperature of T ≈ (3 / 16) mu 2 ∗ /k , where m is the mean mass per particle in the gas. The sound speed is [(5 / 3) kT/m ] 1 / 2 ≈ 0 . 6 u ∗ . Thus, we can take l p ∼ R h . Therefore, we assume first that the width of the density-inversion layer is ∆ r i ∼ R h . Consider then a spherical parcel of gas (a blob) of radius R c and density of ρ c moving with a terminal velocity v b through an external medium of density ρ e . The buoyancy force on the blob is where g is the gravitational acceleration. The drag force on the bubble is where C D /similarequal 0 . 75 (Kaiser 2003). Assuming ρ c /lessmuch ρ e and taking g = u 2 ∗ /R h , the terminal velocity of the bubble is where in the second equality we identify the terminal velocity as the velocity by which the hot gas escapes from the hot bubble outward, with Complex processes take place in the density-inversion layer. (1) Heat conduction time scale over a distance of ∆ r T = R c ∼ 0 . 1 pc and a temperature difference of ∆ T = 10 7 K, is few × 10 yr. This is shorter than the fall time of a dense clump from ∼ 1 pc. Therefore, the hot bubble gas heats the clump by heat conduction. Closer to the center, the clump will be shredded to smaller cells. Hence, before the dense ISM clumps can reach the center they will be evaporated. This is not true for denser and cooler blobs that fall inward, as in the cold feedback mechanism (Pizzolato & Soker 2005). (2) Because of the stellar motion and/or AGN activity, the density-inversion layer is expected to be more chaotic than just a RT-unstable region. There will be vortices that will increase mixing, namely, reduce the effective value of ∆ r T .", "pages": [ 5, 6, 7, 8 ] }, { "title": "5. DISCUSSION AND SUMMARY", "content": "We studied the pressure exerted by the winds of circum-SMBH high-velocity stars on the surrounding ISM. We found that in some cases this pressure is significant and can substantially suppress the inflow of the ISM relative to what a simple Bondi accretion would give. Our result can explain the finding of Wong et al. (2011) that the Bondi accretion rate calculated by them from the ISM density and temperature is six orders of magnitude above the observed upper limit on the accretion rate in the S0 galaxy NGC 3115. In section 3 we quantitatively examined the situation in the galaxy NGC 3115. Shocked winds of circumSMBH high-velocity stars form a bubble of hot gas whose pressure is significant, as evident from Fig. 2. The colliding winds heat up to very high temperatures, build significant pressure, and are not expected to be accreted by the SMBH even though they lose angular momentum. Cooler clumps that fall inward, from the ISM or from inhomogeneities within the hot bubble, will encounter the winds of fast-moving stars very close to the SMBH. This collision will heat such clumps, suppressing their accretion. Even if there is a small accretion rate, a very weak disc wind from the accretion disc might further lower the accretion rate. The study of the interaction of AGN winds with the gas near the SMBH is a subject of a future study using numerical simulations. There are some uncertainties in the model, such as the exact behavior of the stellar mass loss, trajectories of stars around the SMBH, and the stochastic behavior of the post-shock stellar winds. Some of these will be studied in future numerical simulations. However, the result that the stellar winds cannot be ignored is robust. For some values of the parameters we found that a situation might arise where the hot bubble's density is lower than the ISM density. In this case, Rayleigh-Taylor (RT) instability takes place, and a densityinversion layer is formed (see schematic description in Fig. 1). Although hot tenuous gas buoys outward and dense ISM gas moves inward, the density-inversion layer itself continues to exist. The ISM gas is heated near the center and accumulated into the hot bubble. While the scenario suggested here may explain the low X-ray luminosity observed in the galaxy NGC 3115, its properties have not yet been observed or affirmed directly. The size of the hot bubble described is below the resolution limit of the observations and cannot yet be observed. Alternative explanations for a below-Bondi accretion rate are mentioned in section 1. We note that in our scenario there can be no steady state over a very long time of ∼ 10 7 -10 8 yr. Over this time scale radiative cooling becomes important and more of the cooling gas will be accreted by the SMBH. This will lead to stronger AGN activity that will heat and expel gas, hence reducing back the accretion rate and AGN power. In addition stellar formation must occur from time to time. Most likely, there are local star-burst episodes when the accretion rate is much higher than the Bondi accretion rate. The high accretion rate is probably driven by cold clumps (filaments, streams). Indeed, the stellar-wind pressure cannot prevent accretion of very dense clouds. Our result is more general in showing that in many cases the Bondi accretion process does not work because one of its basic assumptions, that there is no central pressure, breaks down. This is one of several reasons why the Bondi accretion model may not apply in some cases (see section 1). Finally, we note that our model may be relevant for active galaxies where the hot bubble might be formed by the AGN jets or winds. For typical values of AGN jets and winds the hot bubble density will be low, and a density-inversion layer will be formed. We expect this process to be of high significance in the process of AGN feedback acting in young galaxies. Barring Bondi-like accretion, dense and cold clumps in the ISM can still flow inward and feed the SMBH. Namely, AGN feedback mechanisms require the feeding to be by cold clumps, i.e., a cold feedback mechanism. We thank an anonymous referee for many detail and very helpful comments that substantially improved the manuscript. This research was supported by the Asher Fund for Space Research and the E. and J. Bishop Research Fund at the Technion, and the Israel Science foundation.", "pages": [ 8, 9 ] }, { "title": "REFERENCES", "content": "Allen, S. W., Dunn, R. J. H., Fabian, A. C., Taylor, G. B., & Reynolds, C. S. 2006, MNRAS, 372, 21 Bower R. G., McCarthy I. G., & Benson A. J. 2008, MNRAS, 390, 1399 Cavagnolo, K. W., McNamara, B. R., Wise, M. W., Nulsen, P. E. J., Br¨uggen, M., Gitti, M., & Rafferty, D. A. 2011, ApJ, 732, 71 Cuadra, J., Nayakshin, S., & Martins, F. 2008, MNRAS, 383, 458 Debuhr, J., Quataert, E., & Ma, C.-P. 2011, MNRAS, 412, 1341 Diehl, S., & Statler, T. S. 2008, ApJ, 680, 897 Fabian, A. C. 2012, ARAA, (arXiv:1204.4114) Farage, C. L., McGregor, P. J., & Dopita, M. A. 2012, ApJ, 747, 28 Freytag, B., H¨ofner, S. 2008, A&A, 483, 571 Fryer, C. L., Liu, S., Rockefeller, G., Hungerford, A., & Belanger, G. 2007, ApJ, 659, 389 Gaspari, M., Brighenti, F., & Temi, P. 2012a, arXiv:1202.6054 Gaspari, M., Ruszkowski, M., & Sharma, P. 2012b, ApJ, 746, 94 Harpaz, A. 1984, MNRAS, 210, 633 Hobbs, A., Nayakshin, S., Power, C., & King, A. 2011, MNRAS, 413, 2633 Hobbs, A., Power, C., Nayakshin, S., & King, A. R. 2012, MNRAS, 421, 3443 Igumenshchev, I. V., & Narayan, R. 2002, ApJ, 566, 137 Kaiser, C. R. 2003, MNRAS, 343, 1319 Kashi, A., Nagamine, K., Proga, D., Ostriker, J. P., & Varghese, S. 2012, preprint King, A. 2003, ApJ, 596, L27 King, A. 2005, ApJ, 635, L121 Kormendy, J., et al. 1996, ApJ, 459, L57 Krumholz, M. R., McKee, C. F., & Klein, R. I. 2005, ApJ, 618, 757 Krumholz, M. R., McKee, C. F., & Klein, R. I. 2006, ApJ, 638, 369 Levine, R., Gnedin, N. Y., & Hamilton, A. J. S. 2010, ApJ, 716, 1386 McCourt, M., Sharma, P., Quataert, E., & Parrish, I. J. 2012, MNRAS, 419, 3319 McNamara, B. R. & Nulsen, P. E. J. 2007, ARA&A, 45, 117 McNamara, B. R., & Nulsen, P. E. J. 2012, New Journal of Physics Focus Issue on Clusters of Galaxies McNamara, B. R., Rohanizadegan, M., & Nulsen, P. E. J. 2011, ApJ, 727, 39 Narayan, R., & Fabian, A. C. 2011, MNRAS, 926 Nesvadba, N. P. H., Boulanger, F., Lehnert, M. D., Guillard, P., & Salome, P. 2011, A&A, 536, L5 Ostriker, J. P., Choi, E., Ciotti, L., Novak, G. S., & Proga, D. 2010, ApJ, 722, 642 Pizzolato, F. & Soker, N. 2005 ApJ, 632, 821 Pizzolato, F., & Soker, N. 2010, MNRAS, 408, 961 Pope, E. C. D. 2009, MNRAS, 395, 2317 Proga, D., & Begelman, M. C. 2003, ApJ, 582, 69 Proga, D., & Begelman, M. C. 2003, ApJ, 592, 767 Revaz, Y., Combes, F., & Salom'e, P. 2008, A&A, 477, L33 Russell, H. R., Fabian, A. C., Sanders, J. S., Johnstone, R. M., Blundell, K. M., Brandt, W. N., & Crawford, C. S. 2010, MNRAS, 402, 1561 Sharma, P., Quataert, E., & Stone, J. M. 2008, MNRAS, 389, 1815 Sharma, P., McCourt, M., Quataert, E., & Parrish, I. J. 2012, MNRAS, 420, 3174 Soker, N. 2009, MNRAS, 398, L41 Soker, N., & Meiron, Y. 2011, MNRAS, 411, 1803 Willson, L. A. 2007, in Why Galaxies Care About AGB Stars: Their Importance as Actors and Probes. ASP Conference Series, Vol. 378, Edited by F. Kerschbaum, C. Charbonnel, and R. F. Wing. San Francisco: Astronomical Society of the Pacific, 2007., p.211 Wilman, R. J., Edge, A. C., McGregor, P. J., & McNamara, B. R. 2011, MNRAS, 416, 2060 Wilman, R. J., Edge, A. C., & Swinbank, A. M. 2009, MNRAS, 395, 1355 Wong, K.-W., Irwin, J. A., Yukita, M., Million, E. T., Mathews, W. G., & Bregman, J. N. 2011, ApJ, 736, L23 Wrobel, J. M., & Nyland, K. 2012, AJ, 144, 160", "pages": [ 9, 10, 11 ] } ]

[ { "title": "ABSTRACT", "content": "We have performed stability analysis of axisymmetric accretion mounds on neutron stars in High Mass X-ray Binaries (HMXB) by 2-D MHD simulations with the PLUTO MHD code. We find that the mounds are stable with respect to interchange instabilities, but addition of excess mass destabilizes the equilibria. Our simulations confirm that accretion mounds are unstable with respect to MHD instabilities beyond a threshold mass. We investigate both filled and hollow mounds and for the latter also compute the expected profile of cyclotron resonance scattering features (CRSF). In comparison to the CRSF from filled mounds reported in our earlier work, hollow mounds display wider and more complex line profiles. Key words: accretion - magnetic fields - (stars:) binaries: general - X-rays: binaries line: formation - radiation mechanisms: non-thermal", "pages": [ 1 ] }, { "title": "Dipanjan Mukherjee 1 /star , Dipankar Bhattacharya 1 /star † and Andrea Mignone 2 ‡", "content": "1 IUCAA, Post Bag 4, Pune, India - 411007 2 Dipartimento di Fisica Generale, Universit degli Studi di Torino , Via Pietro Giuria 1, 10125 Torino, Italy Submitted to MNRAS on 29th June, 2012.", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Neutron stars in accreting X-ray pulsars accrete matter from the companion star either from stellar winds (Davidson & Ostriker 1973) or through disc accretion by Roche lobe overflow (Ghosh et al. 1977; Koldoba et al. 2002; Romanova et al. 2003). They can be broadly classified into two classes: 1) high mass X-ray binaries (HMXB) with companion stars of masses several times the solar mass and neutron stars with high surface magnetic field ∼ 10 12 G and 2) low mass X-ray binaries (LMXB) with companion stars of masses less than a solar mass and neutron star magnetic fields several orders lower in magnitude ∼ 10 7 G -10 9 G(see Bhattacharya & van den Heuvel (1991) for a review). In this paper we consider the effect of accretion on the evolution of surface magnetic field of HMXB sources by the formation of accretion mounds. The accreted matter in HMXB passes through a shock, gradually settling down on the polar cap to form an accretion mound. X-ray emission from such mounds show characteristic cyclotron resonance scattering features (CRSF) (Harding & Preece 1987; Araya & Harding 1999; Araya-G'ochez & Harding 2000; Becker & Wolff 2007). The CRSF depends on the magnetic field of the local emitting region, and hence serve as a tool to understand the structure of accretion columns. CRSF often show complex line features and characteristic variations with rotation phase and the luminosity of the neutron star (Coburn et al. 2002; Heindl et al. 2004; Mihara et al. 2007; Lutovinov & Tsygankov 2008). Explaining such features require appropriate modelling of the structure of the accretion column and the effect of accretion induced field distortion from the accretion mound. Also, several authors propose that diamagnetic screening of the magnetic field can lower the apparent dipole moment of the neutron star (Romani 1990; Cumming et al. 2001; Melatos & Phinney 2001; Choudhuri & Konar 2002; Konar & Choudhuri 2004). Some recent works on magnetic screening by accretion mounds (Payne & Melatos 2004; Payne & Melatos 2007; Vigelius & Melatos 2008; Vigelius & Melatos 2009) report that large mounds of mass ∼ 10 -5 M /circledot may form on the neutron star, which can then bury the field as the matter spreads on the surface. However several questions regarding the effects of MHD instabilities (Litwin et al. 2001; Cumming et al. 2001) remain to be addressed fully. Magneto-static solutions of accretion mounds have earlier been found by several authors including Hameury et al. (1983), Brown & Bildsten (1998), Payne & Melatos (2004) and Mukherjee & Bhattacharya (2012). It was shown in Mukherjee & Bhattacharya (2012) (hereafter MB12) that magneto-static solutions cannot be found for mounds beyond a threshold height (and mass), which may be indicative of the presence of MHD instabilities. Similar results were also reported in Payne & Melatos (2004) (hereafter PM04) where closed magnetic loops were seen to form beyond a threshold mound mass. In this paper we attempt to study the stability of the accretion mound by 2D axisymmetric MHD simulations with the PLUTO MHDcode (Mignone et al. 2007). The study of the full set of MHD instabilities in such mounds requires global 3D simulations. However, results from 2D simulations would help to identify modes that grow despite of the restrictive assumption of axisymmetry. This will be a stepping stone to future 3D simulations where many other modes may grow simultaneously. Here we investigate the presence of interchange instabilities as predicted for such mounds by Litwin et al. (2001), and also the physical cause of the threshold in mound mass obtained in MB12. To study the latter, we add a small amount of mass to an existing GS solution and dynamically evolve the system to see if it settles to a new equilibrium state. This is carried out for different mound sizes up to the threshold mass, at which one expects MHD instabilities to be triggered if the threshold happens to be due to a physical effect. Our approach differs from that of PM04 in various aspects. We consider a cylindrical geometry with strict containment of the accreted matter in the polar cap, while PM04 consider spherical geometry with mass loading on all field lines up to the equator. Also, we consider degenerate non-relativistic Fermi plasma near the polar cap surface instead of the isothermal equation of state used by PM04. As we consider densities as high as ∼ 10 8 g cm -3 inside the mound, a degenerate non-relativistic plasma is more appropriate (see MB12 for a discussion). Early models of accretion column formed by discmagnetosphere interaction proposed hollow ring-like accretion column on neutron star poles (Basko & Sunyaev (1976), Ghosh & Lamb (1978) and Ghosh & Lamb (1979)). Several authors have used hollow ring-like accretion columns to fit the pulse profiles of HMXBs (e.g. Shakura et al. (1991), Leahy (1991), Riffert et al. (1993)). Panchenko & Postnov (1994) and Klochkov et al. (2008) discuss effects of emission from two disconnected rings to explain shape of observed pulse profiles and nature of cyclotron features in the emission from Her X-1. Following the formalism of pulse profile decomposition developed by Kraus et al. (1995), ring-like columns have been inferred for sources like Her X-1 (Kraus 2001), 4U 1909+07(Furst et al. 2011), A0535+262 (Caballero et al. 2011) and V 0332+53 (Ferrigno et al. 2011). Even for LMXB sources, ring like polar cap models are preferred for fitting pulse profiles (Poutanen et al. 2009; Kajava et al. 2011). We therefore perform a study of the structure and stability of hollow accretion mounds to compare with results from filled mounds. We also perform simulations of CRSF emission from hollow mounds, following the method described in MB12. We structure the paper as follows: in Sec. 2 we outline the numerical set up involved in the problem. We discuss the solution of the Grad-Shafranov equation to determine the structure of the static mound. We also discuss details of the set up of the MHD simulations with PLUTO. In Sec. 3 we discuss the testing of the equilibrium solution with PLUTO. In Sec. 4 we discuss the method and results of the perturbation analysis with PLUTO to investigate the stability of the mounds. In Sec. 5 we discuss the results of the simulations of hollow mounds and we summarise the results in Sec. 6.", "pages": [ 1, 2 ] }, { "title": "2 NUMERICAL SET UP", "content": "To test the hydromagnetic stability of the confined mound we first evaluate the equilibrium solution to the Magneto Hydrostatic equations by solving the Grad-Shafranov (hereafter GS) equation. The solution of the GS equation is used as initial condition in PLUTO, where perturbation analysis is performed. In the following section we outline the solution of the GS equation and the set up of the simulation using PLUTO.", "pages": [ 2 ] }, { "title": "2.1 Equilibrium solution from Grad-Shafranov equation", "content": "For an axisymmetric system, one may write the magnetic field in terms of the flux function in cylindrical coordinates as Using eq. (1) in the static Euler equation and using separation of variables in cylindrical coordinates using method of characteristics (as in MB12) we get the GS equation for an adiabatic gas ( p = k ad ρ γ ) where g is acceleration due to gravity and density is given by the equation Z 0 ( ψ ) is the mound height function which determines the shape of the mound. For our work we use the equation of state for a degenerate non-relativistic zero temperature Fermi plasma with µ e = 2 :  Most of the mound will be dominated by degeneracy pressure except for a thin layer at the top ( ∼ 4 cm at 1keV plasma, see MB12 for a discussion). Thus effects of thermal stratification would play a limited role, and the zero temperature degenerate equation of state would be an adequate assumption. We solve the GS equation for an accretion mound of radius R p = 1 km, on the poles of a slowly spinning neutron star of mass 1 . 4 M /circledot and radius R = 10 km. The intrinsic field is assumed to be dipolar, which in the polar cap region can be approximated as an uniform field along ˆ z ( B p = B 0 ˆ z ). We consider Newtonian gravity with constant acceleration: Our set up is similar to that in Hameury et al. (1983) and Litwin et al. (2001). Following MB12, we carry out most of our analysis for the mound height profile: where Z c is the central height of the mound and ψ p = (1 / 2) B 0 R 2 p . This is a smoothly varying parabolic profile in ψ which describes a filled axisymmetric mound. We also discuss the GS solution for a hollow mound in Sec. 5, which is specified by the mound height function: The GS is a coupled non-linear elliptic partial differential equation. Wehave solved the GS equation by an iterative under-relaxation algorithm with an inner Successive Over-relaxation loop with Chebyshev acceleration (Press et al. 1993) as is outlined in MB12. For a given polar magnetic field ( B p ), the solutions to the GS equations are obtained up to a threshold height Z max , beyond which the numerical scheme does not converge to give an unique solution. Details of the numerical algorithm and convergence of the GS solutions have already been discussed in MB12.", "pages": [ 2 ] }, { "title": "2.2 PLUTO setup: Initialisation", "content": "We use the Godunov scheme based MHD code PLUTO (Mignone et al. 2007) to test the stability of the confined mound. The solutions of the GS equation are used as initial condition in PLUTO. The GS solutions are imported into PLUTO using bilinear interpolation. We use the MHD module of PLUTO to solve the full set of ideal magneto hydrodynamic equations: where the factor 1 / √ 4 π is absorbed in the definition of magnetic field and c 2 s is the speed of sound (which for adiabatic gas is c 2 s = γp/ρ ). The system is closed by an equation of state (hereafter EOS) which we choose to be either adiabatic ( ρ/epsilon1 = p/ ( γ -1) ) or barotropic for which p = p ( ρ ) . In the second case eq. (11) is redundant. To investigate the effects of pressure driven interchange modes and gravity driven modes, we perform perturbation analysis with the adiabatic EOS (see Sec. 4.1 and Sec. 4.2). PLUTO initialisation and boundary conditions are provided in terms of primitive variables ( ρ, v , p, B ) defined in eq. (8) -eq. (11). The computation is carried out in conservative variables ( ρ, ρ v , E, B ) , where E = ρ/epsilon1 + ρ v 2 / 2 + B 2 / 2 is the total energy density. We use the extended generalised Lagrangian multiplier (EGLM) scheme (Mignone & Tzeferacos (2010), Mignone et al. (2010)) to preserve the ∇· B = 0 constraint. The EGLM scheme preserves the divergence criterion by modifying the induction equation (eq. 10) with a scalar field function ψ GLM (Dedner et al. 2002) and also the energy momentum equations with extra source terms. This scheme transports the non-zero divergence errors to the boundary of the domain at the fastest possible characteristic speed, and damp them at the same time. For our problem, we have found that the HLL Riemann solver (Toro 2008), HLLD Riemann solver (Miyoshi & Kusano 2005) and TVD Lax-Friedrichs solver (Toro 2008) combined with EGLM scheme provide solutions free from numerical instabilities. Due to the presence of very sharp gradients in the physical quantities, higher order schemes need to be employed to reduce numerical errors. A third order Runge-Kutta scheme is used for time evolution and a third order accurate piece-wise parabolic interpolation scheme (PPM scheme as in Colella & Woodward (1984)) has been employed. The simulations were set up using square cells ( ∆ r /similarequal ∆ z ) to minimise numerical errors. The resolutions used were less than ∼ 0 . 5 m as listed in Table. 1 for some sample runs. The physical variables in PLUTO are scaled to non-dimensional forms before initialisation. For example for mounds with polar magnetic field B p = 10 12 G, we use ρ = 10 6 g cm -3 as the density unit, L 0 = 10 5 cm as the length unit, B 0 = 10 12 G as the magnetic field unit and V A 0 = B 0 / √ 4 πρ = 2 . 82 × 10 8 cms -1 as the velocity unit. In these units, time is measured in units of t A = L 0 /V A 0 = 3 . 55 × 10 -4 s, which can be taken as the mean Alfv'en time, while the scale velocity is the mean Alfv'en velocity. An unique Alfv'en vel ocity cannot be prescribed for the whole domain as the Alfv'en speeds will vary over the domain depending on local density and magnetic field.", "pages": [ 3 ] }, { "title": "2.3 Boundary Conditions", "content": "For stability studies, we run the simulations with either fixed boundaries where quantities are kept fixed to initial values ( Q = Q 0 ) or fixed gradients where the initial gradients are preserved. The fixed gradient boundary implies outflow of perturbed quantities as gradients of perturbations are set to zero ( ∇ Q = ∇ Q 0 + ∇ ˜ Q →∇ Q 0 , ∇ ˜ Q = 0 ). The standard outflow boundary condition ( ∇ Q = 0 ) is inapplicable for our problem as the initial solution has non-zero gradients at the boundaries of the domain. The fixed gradient boundary condition is applied to the upper and the rightmost boundary. For filled mounds, the inner left boundary is kept fixed as it is close to or equal to the axis of the column. For hollow mounds, the inner left boundary is kept at a fixed gradient to allow for inward flow of perturbed matter. The bottom boundary is kept fixed to simulate a hard crust. The set-up with fixed gradients on the outer sides and fixed crust gives numerically stable solutions, as tested from simulations of the equilibrium solutions obtained from GS-solver (see Sec. 3).", "pages": [ 3 ] }, { "title": "3 EQUILIBRIUM STUDIES", "content": "The GS solutions for adiabatic mounds have density profiles which go to zero beyond Z 0 ( ψ ) (see eq. (3)). To avoid unrealistic Alfv'en velocities, we restrict the computation domain inside the mound such that Alfv'en speeds in the mound are non-relativistic. A typical computation domain is depicted in Fig. 1 for a mound of height Z c = 65 m. We first evolve the initial equilibrium solution without applying perturbation in order to check the stability of the numerical schemes and also to study the effects of initial transients contributed by the numerical errors accumulated in interpolating the solution from GS grid to PLUTO domain. The solutions have been evolved to t ∼ 80 t A for different choices of schemes. For the set of schemes outlined in Sec. 2.2 and Sec. 2.3, the equilibrium solution remains intact, with very small build up of internal flow velocities. For example, for a mound of height Z c = 72 m, at t ∼ 80 t A , the maximum velocity is ∼ 7 . 5 × 10 -4 in normalised units ( ∼ 2 . 15 × 10 5 cms -1 , which is much smaller than typical scale velocities). This shows that the schemes used are free from artificial numerical effects and also verifies the validity of the equilibrium solution obtained from the GS solver.", "pages": [ 3, 4 ] }, { "title": "4 PERTURBATION ANALYSIS", "content": "We perturb the equilibrium solution by adding a normalised perturbation field ξ ( r, z ) to any of the physical quantities where η is a positive number signifying the perturbation strength. The perturbations are kept away from the boundaries on all sides. This is to preserve the equilibrium at the boundary layers and avoid spurious interaction with the boundary. For our studies we apply a random perturbation on the density inside the simulation domain, namely ξ is assigned a random value at each grid point within the perturbation zone. The edges of the perturbing region are smoothed with an exponential function to avoid sharp gradients which can lead to spurious effects. The lack of any preferred perturbation scale should allow the growth of the fastest growing modes. The perturbation analysis is performed for mounds of different heights up to the threshold height Z max beyond which the GS-solver does not converge, as has been found in MB12.", "pages": [ 4 ] }, { "title": "4.1 Zero-mean perturbations: interchange modes", "content": "Zero-mean random perturbation with 〈 ξ 〉 = 0 , implies rearranging of density from the equilibrium solution without adding any net mass. In this case, the system quickly converges to stable pockets of perturbations, irrespective of perturbation strength ( η in eq. 12). See Fig. 4 for the results of a run with perturbation strength η = 10% . The system settles down to an energy state close to the original equilibrium value (see Fig. 2). However, for larger perturbation strengths, a longer time is taken to relax into stable pockets of perturbed matter. For example a mound with B p = 10 12 G and Z c = 65 m stabilizes after t ∼ 1 t A for η = 2% and t ∼ 4 t A for η = 10% . The perturbation tests have been carried out for mounds of different heights and polar magnetic field strengths. No instabilities are seen at the threshold mound heights, e.g. Z c ∼ 72 m for B = 10 12 G and Z c ∼ 25 m for B = 10 11 G etc. The simulations show that the mounds are stable with respect to small departures from equilibrium resulting from rearrangement of flux tubes. Thus interchange or ballooning modes are not seen in 2-D axisymmetric simulations of the mounds.", "pages": [ 4 ] }, { "title": "4.2 Adding excess mass to equilibrium solution", "content": "In order to study the effect on the mound of the addition of matter which eventually descends due to gravity, we apply a positive definite random perturbation field: 〈 ξ 〉 > 0 on the density without any corresponding change in pressure. Such a change in density implies local departure of k ad from that in eq. 4. In this work we do not attempt to model the exact composition of the accretion mound. The perturbations were set up to ensure that the added matter is heavier than its surroundings and will descend due to gravity, thus triggering the gravity driven modes. However, a change in k ad can indeed occur due to changes in chemical composition e.g. η ∼ 5% local perturbation on a Z c ∼ 65 m mound would correspond to a change of mean molecular weight by ∆ µ e ∼ 0 . 1 . The added mass settles down along the field lines, dragging and distorting the equilibrium field configuration in the process. For small perturbation strengths ( η /similarequal 1% for mound of height Z c = 65 m) the matter quickly settles down to a new equilibrium, without appreciable distortion of the field lines. With an increase in η beyond a threshold, e.g. η T ∼ 3% for Z c = 65 m and B p = 10 12 G mound, magnetic Rayleigh-Taylor type instabilities are triggered by the descending heavier matter and results in the formation of closed loops due to reconnection of field lines (see Fig. 5). 1 Bunching of field takes place in the radial direction (e.g. Fig. 6) and the system eventually relaxes to a steady state. Further increase in perturbation strength, e.g. η ∼ 5% for Z c = 65 m, disrupts the equilibria completely. Several closed loops are formed across the perturbed region (see Fig 7 and Fig. 8). Individual closed loops merge to form larger knots without showing any signs of decay. From Fig. 3 we see that gravitational potential energy and internal energy decreases from initial value, whereas magnetic energy increases with time. This indicates that internal flows stretch and twist the field lines converting internal energy and gravitational energy to magnetic energy. The system does not relax to a steady state within the run time of the simulation ( t ∼ 50 t A ). Thus for a mound with Z c = 65 m and B p = 10 12 G, the threshold perturbation strength is η T ∼ 3% beyond which gravity and pressure driven modes disrupt the MHD equilibria. Convergence has been tested by running the simulations for successive higher resolutions: .e.g. for Z c = 65 m, B p = 10 12 G with positive random perturbation of strength η = 5% simulations were carried out for resolutions ( 1088 × 104 ), ( 2176 × 208 ) and ( 4352 × 416 ). It was seen that MHD instabilities persist on increase of resolution. Increase in resolution reduces numerical resistivity, thus decreasing cross field diffusion. The field lines are then more prone to be deformed by gravity driven modes triggered by the weight of the overlying matter. With an increase in mound height, it is easier to excite such unstable behaviour. The threshold perturbation strength is larger for mounds of smaller height: for Z c = 45 m and B p = 10 12 G, η T ∼ 7% . Mounds near the GS threshold height Z max ( ∼ 72 mfor B p = 10 12 G; ∼ 25 m for B p = 10 11 G) are only marginally stable at η T /similarequal 1% . Thus, mounds higher than a threshold (as previously obtained in MB12) are prone to gravity driven Rayleigh-Taylor and pressure driven instabilities on addition of excess mass, and stable magneto-static solutions cannot be obtained.", "pages": [ 4, 6, 7 ] }, { "title": "5.1 Grad-Shafranov for hollow mounds", "content": "For systems with magnetospheric accretion, mass loading at the accretion disc takes place over a finite range of accretion disk radii ( ∆ r ∼ 0 . 03 R A , R A ≡ Alfv'en radius e.g. Ghosh & Lamb (1978, 1979)). The inner edge of the polar cap ring 2 for such systems will 2 which corresponds to the outermost radius in the accretion disc ∼ R A + ∆ r , where mass loading begins. be while the outer edge of the polar cap radius is ( R s /R A ) 1 / 2 R s (Poutanen et al. 2009), R s being the neutron star radius. For small values of ∆ r the columns would be hollow and thin walled. On the surface of the star this would create an accretion ring around the polar cap instead of a filled mound. To model such an accretion ring, we choose the mound height function to give a hollow mound in which the density falls off to zero both at the axis and at the polar cap radius. For the solution presented in Fig. 9 we use a mound height profile as in eq. (7) with Z c = 45 mand B p = 10 12 G. The solution shows considerable distortion of field lines on both sides of the apex ( r ∼ 698 m). This is in contrast to the case of filled mounds, where curvature of field lines occur towards the outer edge. Larger curvature of field lines allow larger mass to be accumulated per flux tube, as compared to that of filled mounds. Hence, although the central part is hollow, the total mass contained in the hollow mound ( M ∼ 5 . 87 × 10 -13 M /circledot ), is comparable to that of a filled mound of the same height and field ( M ∼ 5 . 09 × 10 -13 M /circledot for Z c ∼ 45 m and B p ∼ 10 12 G and a parabolic profile as in eq. 6). The family of GS solutions for hollow mounds behave similarly as for filled mounds. With increase in maximum mound height Z c , the GS solutions show larger curvature of field lines on both sides of ridge apex. The GS solutions fail to converge for mounds greater than a threshold height for a given magnetic field. For the mound height profile of eq. (7), the threshold height is around Z max ∼ 47 mfor a polar magnetic field B p = 10 12 G.", "pages": [ 7 ] }, { "title": "5.2 Stability analysis of Hollow mounds", "content": "Using the GS solutions for hollow mound, we perform stability analysis with PLUTO. The results are similar to that of a filled mound. Zero-mean density perturbations do not show growth of the perturbed region, indicating that the mounds are intrinsically stable with respect to interchange modes. For positive perturbations in density, closed loops are formed after a threshold perturbation strength. See Fig. 12 and Fig. 13 for the results of a run with η = 5% . The closed loops form quickly within a few Alfv'en times and migrate away from the center, on both sides of the central height. This results in the formation of alternate regions of enhanced and reduced magnetic field due to the bunching of field lines, which have considerable departure from equilibrium solution. The field knots dissipate gradually as they migrate outwards.", "pages": [ 8 ] }, { "title": "5.3 Cyclotron lines from hollow mounds", "content": "Following the algorithm outlined in MB12, we have simulated the cyclotron resonance scattering features (hereafter CRSF) that will be observed in the emitted spectrum from a hollow mound. The spectra have been calculated by integrating the emission from different parts of the mound towards a given line of sight (hereafter los). We assume a Gaussian absorption profile whose depth and width are evaluated from interpolated results of Schonherr et al. (2007) for the slab 1-0 geometry. As in MB12, the line centre of the CRSF is obtained from the expression where n = 1 , 2 , 3 ... is the order of the harmonic, E c 0 = 11 . 6 B 12 in keV, θ αb is the angle between the direction of emission and local magnetic field and u = r s /r , r s being the Schwarzschild radius. Emission from the inner part of the hollow mound may be blocked", "pages": [ 8 ] }, { "title": "c © 2012 RAS, MNRAS 000 , 1-??", "content": "by the walls on the opposite side. In Appendix. (A) we explain the scheme we follow to account for such shielding. For the simulated spectra shown in Fig. 11, we consider emission from a single pole at inclination angle η p = 10 · and a los at ı = 60 · , both measured from the spin axis. The spectrum shows multiple absorption features due to the large variation of field strength at the top of the mound (see Fig. 10). The different absorption features correspond to emission from different locations on top of the mound, with different magnetic field values. The nature of this spectrum is significantly different from that expected from a filled parabolic mound of the same height (see Fig. 11). When convolved with a Gaussian of standard deviation ∼ 10% of the local energy, to simulate the finite resolution of a detector (see MB12 for details), the spectrum becomes a broad absorption feature.", "pages": [ 9 ] }, { "title": "6 DISCUSSION AND SUMMARY", "content": "(i) Absence of interchange mode instabilities: In this paper we have tested for the stability of magneto-static accretion mounds by MHD simulations using the PLUTO MHD code. From perturbation analysis we conclude that mounds are stable with respect to interchange or ballooning modes in 2-D axisymmetric simulations 3 . Linear stability analysis by Litwin et al. (2001) predict the onset of ballooning modes for a threshold plasma β ( β = p/ ( B 2 / 8 π ) ). However such modes are inherently multi-dimensional in nature, with finite toroidal and zero poloidal wave vectors, normal to the local magnetic field (see Freidberg (1982) for a review of MHD instabilities in confined plasma). Hence such modes cannot be excited in an axisymmetric 2D simulation. Litwin's approximate analytical estimates give a threshold β T ∼ 11 . 7( R p /Z c ) for γ ∼ 5 / 3 , beyond which MHD instabilities will set in. For GS solution of a filled mound with Z c = 45 m and B p = 10 12 G, we get maximum β ∼ 293 , which is close to 3 Note that in this paper we consider a T = 0 · K Fermi gas. However, finite plasma temperature can induce additional thermal modes (Cumming et al. 2001) which have not been explored here. Litwin's threshold for the same mound β T ∼ 260 . For higher mounds, β T decreases with increase in Z c , and is much smaller than the maximum β obtained from our GS solutions. For example, for a filled mound with Z c = 65 m and B p = 10 12 G, β T ∼ 180 whereas maximum β ∼ 911 from the GS solution (see Fig 14). Hence results from 2-D simulations cannot rule out the presence of such modes in a 3-D set up. Also, interchange mode instabilities (Chen 1984) can be excited in 3D simulation runs, as is seen in other examples of confined plasmas, e.g. in tokamak reactors. Work on 3-D stability analysis of accretion mounds is currently underway and will be addressed in a forthcoming publication (Mukherjee, Bhattacharya and Mignone in preparation). (ii) Instabilities due to excess mass: From our 2-D simulations we have found that addition of excess mass destabilizes the equilibrium due to gravity driven magnetic Rayleigh-Taylor type instabilities. For mounds with higher mass, the GS solutions have large radial (horizontal) component of magnetic field, which being perpendicular to gravity are also prone to Parker type instabilities (Cumming et al. 2001; Melatos & Phinney 2001). Topologically disconnected closed loops are formed beyond a threshold perturbation strength η T . From the expression of the energy integral for linear perturbations (Litwin et al. 2001) on an adiabatic plasma ( p = kρ γ ), we have where ξ is the plasma displacement, ˜ B = ∇ × ( ξ × B ) is the perturbed magnetic field, κ c = ( b ·∇ ) b is the magnetic field curvature vector, c s is the sound speed and φ the gravitational potential. B φ is zero for our case. Instabilities will develop if the negative contribution from any (or all) of the terms containing field curvature, pressure gradient and gravity overcomes the stabilizing effects of the magnetic and pressure compression terms. 4 Hence, it is not a surprise that the closed loops are formed in regions with the largest curvature in field lines. This also corresponds to the regions with high plasma β , e.g. the red region in the middle of Fig. 14 where β ∼ 911 . Pressure driven instabilities typically lead to a threshold plasma β beyond which instabilities are triggered (e.g. Freidberg (1982), Litwin et al. (2001)). For mounds near the stability threshold, e.g. Z c ∼ 72 mat B p = 10 12 G, the maximum plasma β is as high as ∼ 1 . 26 × 10 4 . The magnitude of η T decreases with increase in mound height, with η T → 0 as Z c → Z max , indicating inherent unstable nature of the mound for the modes under investigation. This corroborates the result of MB12 that GS solutions do not converge beyond a threshold height. The tests involving addition of mass are not meant to reflect realistic accretion rates. Although the amount of excess mass added in our simulations is small ( ∼ 7 . 6 × 10 -15 M /circledot for η = 5% perturbation on 65m mound), in a real system such mass will be accumulated slowly as mounds of larger mass are built. Effects of such inflow of material on an initially static mound have not been addressed here. However, from our current 2-D simulations we conclude that for large mound masses, gravity and pressure driven modes result in the onset of MHD instabilities and no static equilibrium solution can be found beyond a threshold Z max . Buoyancy related instabilities due to the formation of topologically disconnected closed loops have previously been reported in the static mound simulations of Hameury et al. (1983) and Payne & Melatos (2004), and also dynamic MHD simulations by Vigelius & Melatos (2008) (hereafter VM08). However the threshold mass of the mound for the formation of closed loops in PM04 and VM08 is M ∼ 10 -5 M /circledot , which is much larger than the mass of the mounds in our present work. This may be due to the following differences in approach: However density perturbations behave differently in barotropic and adiabatic simulations. For a barotropic simulation, positive density perturbations on an initial static equilibrium create regions of excess pressure. The perturbed regions with high local pressure overcome the downward gravitational force and are quickly transported vertically upwards. Hence to study the effect of gravity driven modes due to the descent of added matter, adiabatic simulations have been performed in this work. The fixed gradient boundary condition can induce artificial stability as it results in line tying type boundaries, which are known to give extra stability. In a real system, the plasmoids will be eventually ejected from the system. Plasma travelling inwards on closed loops may then eventually fill up the hollow. However there was no significant mass loss seen in our 2-D simulations. Thus CRSF from hollow mounds will be characterised by broad line widths and complex structures in the line shape, which may be observed with improved detector resolution. Thus we conclude from this work that accretion mounds on neutron stars in HMXB are stable up to a threshold height and mass, beyond which MHD instabilities will disrupt the equilibria. Structure and stability of hollow mounds have been explored. It is shown that CRSF from such mounds will be characterised by broad features with a complex line shape. More work needs to be done to explore the 3-D stability of such systems and the effect of nonaxisymmetric modes on the field structure and cyclotron emission from such mounds.", "pages": [ 9, 10 ] }, { "title": "7 ACKNOWLEDGEMENT", "content": "We thank CSIR India for Junior Research fellow grant, award no 09/545(0034)/2009-EMR-I. We thank Dr. Petros Tzeferacos for his help and suggestions in setting up the boundary conditions in the PLUTO simulations. We also thank Dr. Kandaswamy Subramanian, Dr. Ranjeev Misra and Sandeep Kumar from IUCAA, for useful discussions and suggestions during the work, and IUCAA HPC team for their help in using the IUCAA HPC where most of the numerical computations were carried out. We also the thank the anonymous referee for the detailed comments which have greatly helped in improving the work. DB acknowledges the hospitality of ISSI, Berne and discussions with the Magnet collaboration which have benefited the paper.", "pages": [ 10, 11 ] }, { "title": "APPENDIX A: SHIELDING OF RADIATION FROM INNER WALLS OF HOLLOW MOUNDS", "content": "In HMXBs an accretion column is formed by the infalling matter after it passes through a shock which may be several kilometres from the surface of the star, depending on the accretion rate (e.g. Basko & Sunyaev (1976), Becker & Wolff (2007), Becker et al. (2012)). In this work we consider the spectra generated from the mound without incorporating the effects of scattering from the overlying accretion column. This is valid for systems with low accretion rates and optically thin columns. The emission from the mound will then be directly visible and effects of overlying column will be small. However for systems with optically thick columns and large accretion rates, the emission from the mound will be obscured by scattering and absorption in the column. A proper Monte-Carlo simulation of the radiative transfer through the column must be carried out to address such cases, which will be reported in a future work (Kumar, Bhattacharya and Mukherjee in preparation). The rays of light coming from the hollow region can be blocked by the inner walls of the mound on the opposite side. Such rays will not contribute to the total spectra. The path of the emitted ray lies in the plane defined by the radius vector from the origin (centre of the neutron star) to the point of emission ( r ) and the unit vector along the line of sight ˆ n ψ (see for e.g. Beloborodov (2002), Poutanen & Beloborodov (2006) and Mukherjee & Bhattacharya (2012)). To exclude rays that may be blocked by the inner walls of the hollow mound, we first find the point where the plane defined by r and ˆ n ψ passes through the top of the mound r c as in Fig. A1. The radial and vertical coordinate of r c ( r c and z c respectively) are found by fitting a polynomial to the top of the mound obtained from the GS solution and evaluating the coordinate where z is maximum. Since the three vectors r , ˆ n ψ and r c lie in the same plane, the angular coordinate φ c of r c is found from the condition Following MB12 we use the following definitions for the vectors ˆ n ψ ≡ ( n ψx , n ψy , n ψz ) ≡ (sin i sin ω, sin i cos ω, cos i ) and r ≡ ( x, y, z ) ≡ { ρ cos φ, ρ cos η p sin φ + ( ξ + R s ) sin η p , ( ξ + R s ) cos η p -ρ sin η p sin φ } where i is the azimuthal angle of the observer's line of sight with respect to the spin axis, ω is the spinphase angle, ( ρ, φ, ξ ) are coordinates of the emitting region in the polar cap frame with cylindrical coordinate system, R s is the neutron star radius and η p is the azimuthal angle of the centre of the polar cap. Using the above, we can rewrite eq. (A1) as where Eq. A2 is solved using a modified Newton-Raphson scheme following Press et al. (1993). After finding the coordinate of r c we evaluate the angle θ lc (see Fig. A1) between the local normal ( ˆ n l ) and the radius vector from the point of emission to the top of the mound on the other side. The normal vector is found as outlined in MB12 by evaluating the slope m s = dξ top /dρ of the function ξ top = f ( ρ ) ( ρ being the radial coordinate) that fits the top profile of the mound obtained from the GS solutions: ˆ n l ≡ {-sin θ s cos φ, -sin θ s cos η p sin φ + cos θ s sin η p , cos θ s cos η p + sin θ s sin η p sin φ } , where sin θ s = m s √ 1+ m 2 s and cos θ s = 1 √ 1+ m 2 s . Using the above definitions of the vectors, one can write Any ray with emission angle larger than θ lc will not contribute to the total spectra. This implicitly assumes that light will travel in a straight line and curvature effects from bending due to gravity is ignored for such short paths. More accurate methods should be used to calculate the tangent vector from the point of emission to the mound surface on the other side. However, this involves more computation, and for sharp profiles of the hollow mound used, approximating the tangent point as the top of the mound will result in only a small correction.", "pages": [ 11 ] }, { "title": "REFERENCES", "content": "Araya R. A., Harding A. K., 1999, ApJ, 517, 334 Araya-G'ochez R. A., Harding A. K., 2000, ApJ, 544, 1067 Basko M. M., Sunyaev R. A., 1976, MNRAS, 175, 395 Becker P. A., Klochkov D., Schonherr G., Nishimura O., Ferrigno C., Caballero I., Kretschmar P., Wolff M. T., Wilms J., Staubert R., 2012, A&A, 544, A123 Becker P. A., Wolff M. T., 2007, ApJ, 654, 435 Beloborodov A. M., 2002, ApJ, 566, L85 Bernstein I. B., Frieman E. A., Kruskal M. D., Kulsrud R. M., 1958, Royal Society of London Proceedings Series A, 244, 17", "pages": [ 11 ] } ]

[ { "title": "ABSTRACT", "content": "Many supernova remnants (SNRs) are characterized by a knotty ejecta structure. The Vela SNR is an excellent example of remnant in which detached clumps of ejecta are visible as X-ray emitting bullets that have been observed and studied in great detail. We aim at modelling the evolution of ejecta shrapnel in the Vela SNR, investigating the role of their initial parameters (position and density) and addressing the e ff ects of thermal conduction and radiative losses. We performed a set of 2-D hydrodynamic simulations describing the evolution of a density inhomogeneity in the ejecta profile. We explored di ff erent initial setups. We found that the final position of the shrapnel is very sensitive to its initial position within the ejecta, while the dependence on the initial density contrast is weaker. Our model also shows that moderately overdense knots can reproduce the detached features observed in the Vela SNR. E ffi cient thermal conduction produces detectable e ff ects by determining an e ffi cient mixing of the ejecta knot with the surrounding medium and shaping a characteristic elongated morphology in the clump. Key words: Hydrodynamics - Shock waves - Methods: numerical - ISM: supernova remnants - ISM: kinematics and dynamics - ISM: individual object: Vela SNR", "pages": [ 1 ] }, { "title": "M. Miceli 1 , 2 /star , S. Orlando 2 , F. Reale 1 , 2 , F. Bocchino 2 , G. Peres 1 , 2", "content": "1 2 Dipartimento di Fisica, Universit'a di Palermo, Piazza del Parlamento 1, 90134 Palermo, Italy INAF - Osservatorio Astronomico di Palermo, Piazza del Parlamento 1, 90134 Palermo, Italy Accepted . Received ; in original form", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "The ejecta in supernova remnants (SNRs) drive the exchange of mass and the chemical evolution of the galactic medium. The structure of SNR ejecta has been proved to be knotty, and several clumps have been observed at di ff erent wavelegths in remnants of core-collapse supernovae, as G292.0 + 1.8 (Park et al. 2004), Puppis A (Katsuda et al. 2008), and Cas A, where knots have been observed also beyond the main shock front (Fesen et al. 2006, Hammell & Fesen 2008, DeLaney et al. 2010). The Vela SNR, being the nearest SNR, represents a privileged target for this kind of studies, since it is possible to observe fine structures down to small physical scales. Despite the bulk of the X-ray emission of the Vela SNR is associated with the post-shock interstellar medium, X-ray emitting ejecta have also been observed. In particular, six protruding features, with characteristic boomerang morphology, (labelled Shrapnel A-F) have been identified in the Vela SNR by Aschenbach et al. (1995), who argued an ejecta origin for these structures which appear to be detached from the remnant. The association with ejecta fragments has been supported by more recent observations performed with Chandra , XMM-Newton , and S uzaku . The analysis of the XMM-Newton observation of Shrapnel D, has revealed that O, Ne, and Mg abundances are significantly larger than solar (Katsuda & Tsunemi 2005). A similar abundance pattern has been observed with S uzaku in Shrapnel B (Yamaguchi & Katsuda 2009), but in this case the overabundances of the lighter elements are less prominent, suggesting more e ff ective mixing with the interstellar medium (ISM). A Chandra observation of Shrapnel A, whose projected distance from the center of the remnant is larger by ∼ 20% than Shrapnel D, reveals instead oversolar Si:O ratios (Miyata et al. 2001). Significant Si overabundance (Si ∼ 3) has been confirmed by Katsuda & Tsunemi (2006), who analyzed an XMM-Newton observation, finding solar or subsolar values for the O, Ne, Mg, and Fe abundances. These results show di ff erences in the chemical composition between Shrapnel A and Shrapnel B and D. In the northern rim of the Vela shell, Miceli et al. (2008) discovered new X-ray emitting clumps of ejecta whose projected position is behind the main shock front. The relative abundances (O:Ne:Mg:Fe) of these new shrapnel are in good agreement with those observed in Shrapnel D. Similar abundance pattern have been observed also by LaMassa et al. (2008), who found ejecta-rich plasma in the direction of the Vela X pulsar windnebula. The present day morphology of SNRs and the structure of ejecta are believed to reflect the physical characteristics of the SN explosion (e. g., intrinsic asymmetries of the explosion, interaction of the early blast with the inhomogeneities of the circumstellar medium, physical processes in the aftermath of the explosion, etc.) and their detailed study promises to contribute to our understanding of the SN explosion physics. In the light of these considerations, it is then interesting to model the evolution of the ejecta knots to understand how the current position and chemical properties of the shrapnel in the Vela SNR depend on the physical conditions at the supernova explosion and on the dynamics of the explosion itself. The evolution of dense, supersonic clumps of SN ejecta running in a uniform medium has been studied by Anderson et al. (1994) and Jones et al. (1994), who identified three main stages of evolution: a bow-shock phase, an instability phase and a dispersal phase. However, these models do not describe in detail the interaction of the clump with the remnant (post-shock medium, main shock, reverse shock) and do not include important physical e ff ects (as thermal conduction and radiative cooling). Cid-Fernandes et al. (1996) included radiative losses in their 2-D models, but they focussed on the interaction of a knot with a very small supernova remnant ( ∼ 10 17 cm, i. e. more than 100 times smaller than the Vela SNR) evolving in an extremely dense medium (10 7 cm -3 ). A hydrodynamic model (without thermal conduction and radiative cooling) specifically tuned for the Vela SNR has been developed by Wang & Chevalier (2002) (hereafter WC02) who followed the evolution of a shrapnel by using 2-D simulations in spherical coordinates (because of the geometry of their simulations, the shrapnel are modeled as toroidal structures with very large masses). WC02 did not model the early evolution of the ejecta knot, but started their simulations at the time trev , corresponding to the first interaction of the shrapnel with the reverse shock front. They explored di ff erent values of trev and of the density contrast between the shrapnel and the surrounding ejecta, χ , and found that, in order to produce an observable protrusion on the shock front (like that observed in Shrapnel A-F), a very high density contrast ( χ ∼ 1000) is necessary. With lower density contrasts ( χ < 100), the shrapnel are rapidly decelerated and fragmented by hydrodynamic instabilities and the observed features cannot be reproduced (for the e ff ects of hydrodynamic instabilities on shocked clouds see also Klein et al. 1994 and Orlando et al. 2005). Large density inhomogeneities in the clumps are di ffi cult to explain in a core-collapse SN explosion. WC02 argued that a model that includes the e ff ects of radiative cooling may show that lower values of χ are needed to match the observed protrusions. Also, WC02 do not include in their model the e ff ects of thermal conduction that, as shown by Orlando et al. (2005), can e ffi ciently suppress the hydrodynamic instabilities, thus allowing the shrapnel to overcome the main shock-front without being disrupted. Recently, the evolution of knotty ejecta in a Type Ia SNR has been modelled by Orlando et al. (2012) (hereafter O12), who found that small clumps with initial χ < 5 can reach the SNR shock front after ∼ 1000 yr. Nevertheless, these ejecta knots are then rapidly eroded and do not produce significant protrusions in the SNR shock front, thus being unable to reproduce the features observed in the Vela SNR. Here we present a set of 2-D hydrodynamic simulations of the evolution of an (initially spherical) ejecta shrapnel in the Vela SNR. We include in our model both thermal conduction and radiative cooling and explore di ff erent values of χ and of the initial position of the shrapnel in the ejecta profile. We aim at addressing the role of thermal conduction and radiative cooling and at understanding how the initial properties of the shrapnel influence its evolution. We also aim at evaluating whether values of χ lower than 100 ( χ = 10 -50) can reproduce the observed features. The paper is organized as follows: the hydrodynamic model is described in Sect. 2, the results of the simulations are shown in Sect. 3, and our conclusions are discussed in Sect. 4.", "pages": [ 1, 2 ] }, { "title": "2.1 Initial conditions and model equations", "content": "We model the evolution of a shrapnel in a SNR by performing a set of 2-D simulations in a cylindrical coordinate system ( r , z ), assuming axial symmetry. The system setup consists of a spherically symmetric distribution of ejecta with initial kinetic energy K = 10 51 erg (the initial thermal energy is only the 0 . 2% of K ) and mass Mej = 12 M /circledot (representing the initial blast wave), where we place a dense, spherical, knot (the shrapnel) in pressure equilibrium with the surrounding ejecta and with central coordinates (0 , Rs ). The radial density profile of the ejecta consists of two powerlaw segments ( ρ ∝ r -m on the inside and ρ ∝ r -b on the outside), in agreement with the density structure in a core-collapse SN described by Chevalier (2005). For our simulations, we use m = 1 and b = 11 . 2, and the position of the transition between the flat and steep regimes is derived by following Chevalier (2005). The initial velocity of the ejecta increases linearly (up to 6 × 10 8 cm / s) with their distance from the center. The maximum velocity is reached at the initial radius of the ejecta, i. e., R 0 ej = 4 . 5 × 10 18 cm. These values correspond to ∼ 240 yr after the explosion, appropriate for the relatively late stages of the SNR evolution that we address in this study. In fact, the starting time of our simulations corresponds to only ∼ 2% of the Vela SNR age and the shrapnel reaches the reverse shock ∼ 2500 -3000 yr after the explosion in all our simulations (i. e., the system has enough time to evolve, before the interaction of the shrapnel with the SNR reverse shock occurs). We then conclude that our simulations can provide a realistic description of the actual conditions in the Vela SNR. The initial mass of the shrapnel is 0 . 05 Mej (1 . 19 × 10 33 g), its density is χ times larger than that of the surrounding ejecta at distance Rs from the center 1 , and its velocity is the same as that of the ejecta at distance Rs from the center. We aim at showing that the detached shrapnel observed in the Vela SNR can be the result of moderately overdense clumps of ejecta originating in relatively internal layers. We explored di ff erent values of χ and of Rs , namely χ = 10 , 20 , 50, and Rs = 1 / 6 R 0 ej , 1 / 3 R 0 ej . Figure 1 shows the initial density and temperature conditions for the case with χ = 20 and Rs = 1 / 3 R 0 ej . The simulation setups discussed in this paper are summarized in Table 1. Vela SNR is the result of a core-collapse SN and we expect the ambient medium to be 'perturbed' by the wind residuals of the massive progenitor star. However, we assume for simplicity a uniform ambient medium as in WC02, since here we are not interested in modeling the details of the remnant evolution. The final (i. e. after 11000 yr, the age of the Vela SNR, Taylor et al. 1993) radius of the remnant strongly depends on the choice of the ambient density value, nISM . We set nISM = 0 . 5 cm -3 , because with this value (and with the chosen values of K , Mej , nISM , b , and m ), the radius of the shell after 11000 yr is Rshell ∼ 5 . 4 × 10 19 cm, in good agreement with the observed radius of the Vela SNR, that ranges between 5 × 10 19 cm and 5 . 5 × 10 19 cm (by assuming a distance of 250 pc, in agreement with Bocchino et al. 1999, Cha et al. 1999). Our model solves the time-dependent compressible fluid equations of mass, momentum, and energy conservation. In three cases we ran the same simulation with / without thermal conduction and radiative cooling inside our system, as shown in Table 1. As for thermal conduction, we considered both the Spitzer and the saturated regimes, while radiative losses (that can play an important role in the Vela SNR, as shown by Miceli et al. 2006) were computed for an optically thin thermal plasma. The model equations are described in Miceli et al. (2006) (equations 1-5 therein) and were solved by using the FLASH code (Fryxell et al. 2000). The computational domain extends over 8 × 10 19 cm in the r and z directions. We use axisymmetric boundary conditions at r = 0, reflection boundary conditions at z = 0, and zero-gradient (outflow) boundary conditions (for v, ρ , and p ) elsewhere. We trace the motion of the ejecta material and of the shrapnel with passive tracers 2 . Considering the large range in spatial scales of our simulations, we exploited the adaptive mesh capabilities of the FLASH code by adopting up to 10 nested levels of resolution (the resolution increases by a factor of 2 at each level). The refinement / derefinement criterion (Lohner 1987) follows the gradients of density, temperature, and tracers. The finest spatial resolution is 1 . 95 × 10 16 cm at the beginning of the simulation, therefore there are 230 computational cells per initial radius of the ejecta, and ∼ 10 cells per initial radius of the shrapnel (that varies in the range 1 . 8 -2 . 9 × 10 17 cm). Because of the expansion of the system, the resolution is reduced by a factor of 2 after 2500 yr. We verified that by changing the resolution of our simulations by a factor of 2, the results do not change significantly (see Appendix A for further details).", "pages": [ 2, 3 ] }, { "title": "3.1 Evolution of the system", "content": "Wefirst focus on simulation R 1 / 3 -CHI 20. Figure 2 shows the 2-D cross-sections through the ( r , z ) plane of temperature and density at di ff erent evolutionary stages of the R 1 / 3 -CHI 20 simulation. The left panel shows the system 5000 yr after the beginning of the simulation 3 , when the shrapnel interacts with the inter-shock region. Rayleigh-Taylor and Richtmyer-Meshkov instabilities are visible as finger-like structures both in the density and temperature maps. At this stage, the knot is partially eroded by the hydrodynamic instabilities and evolves toward a core-plume structure. The core of the knot, however, is still significantly overdense with respect to the surrounding shocked ejecta. The right panel of Fig. 2 shows the shrapnel at t = 11000 yr, with its characteristic supersonic bow shock protruding beyond the SNR main shock. WC02 found that ejecta knots with density contrast χ /lessorequalslant 100 are rapidly fragmented and decelerated in the intershock region and do not even reach the main shock front (these e ff ects being more dramatic for small clumps). Nevertheless, we notice that the value of χ in WC02 refers to the onset of the interaction between the knot and the reverse shock and that χ is not constant during the evolution of the system. In the 'free' expansion phase, the density of the shrapnel does not drop down uniformly (as that of the other ejecta does) and the shrapnel undergoes both di ff usion and expansion. Figure 3 presents a close-up view of the shrapnel density structure at t = 2500 yr, showing that, while the outer parts of the knot di ff use and mix with the expanding ejecta, its central core remains much denser. The density of the core of the clump drops down much more slowly than that of the spherically expanding ejecta. Therefore, the inhomogeneous rarefaction of the knot makes the density contrast between the core of the shrapnel and the expanding ejecta higher, and χ rapidly increases until the shrapnel reaches the reverse shock. We computed χ during the expansion phases, by calculating the shrapnel density, ρ s , as the average of the density in all the computational cells where the shrapnel content is > 90% 4 . We then divided ρ s by the ejecta density (along the r axis) at the same distance from the origin as the shrapnel cen- Figure 4 shows the evolution of χ as a function of time for the R 1 / 3 -CHI 20 simulation. The figure shows that the ejecta knot reaches χ > 100 as it approaches the reverse shock, hence our results are in agreement with those of WC02. Our model shows that a knot that was only 20 times denser than the surrounding ejecta (at the beginnig of the simulations) can reach the SNR main shock without being fragmented in the intershock region and can produce protrusions that are similar to those actually observed in the Vela SNR.", "pages": [ 3, 4 ] }, { "title": "3.2 E ff ects of thermal conduction and radiative cooling", "content": "Figure 5 shows the 2-D cross-sections through the ( r , z ) plane of temperature and density at t = 5000 yr and t = 11000 yr for the R 1 / 3 -CHI 20 -TR simulation (same parameters as R 1 / 3 -CHI 20, but including radiative cooling and thermal conduction). The di ff usive thermal conduction completely suppresses the formation and the development of hydrodynamic instabilities and smoothes the temperature and density profiles. This result is in agreement with expectations, as shown below. The characteristic amplitude growth rate, da / dt of a single-mode perturbation of Richtmyer-Meshkov instabilities can be calculated as (see Richtmyer 1960) where k is the perturbation wavenumber, ∆ v is the velocity jump at the instability and A = ( ρ 1 -ρ 2) / ( ρ 1 + ρ 2) is the Atwood number. The characteristic time-scale, τ inst = a / ( da / dt ), for the growth of the perturbation is therefore where l is the structure size. As for the thermal conduction (see Spitzer 1962), where κ ( T ) = 5 . 6 × 10 -7 T 5 / 2 erg s -1 K -1 cm -1 is the Spitzer's coe ffi cient and l is the characteristic length of temperature variation. therefore, the thermal conduction time-scale is: For a characteristic structure with size l ∼ 4 × 10 18 cm, particle density n = 0 . 8 cm -3 , Atwood number A ∼ 0 . 44, ∆ v ∼ 2 × 10 7 cm / s, T ∼ 1 . 3 × 10 7 K (similar to that shown in Fig. 2), τ inst ∼ 1 . 4 τ cond ∼ 2000 yr. Therefore, the thermal conduction di ff usive processes develop faster than the hydrodynamic instabilities and density and temperature inhomogenieties are smoothed out before they can grow. The evolution of the position of the shrapnel head and the protrusion it produces to the remnant shock front are similar to those obtained without including thermal conduction and radiative cooling. Nevertheless, the shrapnel evolution is remarkably di ff erent from that obtained in the pure HD simulations. In particular, as shown by the blue contours in Fig. 5, the ejecta knot is elongated along its direction of motion and rapidly assumes a cometary shape, characterized by a prominent tail which is rich in shrapnel material. After t = 11000 yr, shrapnel material is present at ∼ 10 pc away from the shrapnel head. Moreover, the shrapnel material is e ffi -ciently heated by thermal conduction with the surrounding shocked ejecta. Let MXshra be the mass of the plasma in the computational cells consisting of the original shrapnel material by more than 90% and having a temperature higher than 10 6 K (and therefore emitting thermal X-rays). Figure 6 shows the evolution of MXshra as a function of time for the simulations R 1 / 3 -CHI 20 -TR and R 1 / 3 -CHI 20. When thermal conduction is at work, ∼ 90% of the original shrapnel mass is heated up to X-ray emitting temperature at t = 11000 yr (i. e., at the age of the Vela SNR), while if thermal conduction in inhibited, only 50% of the original mass has temperature higher than 10 6 K. Figure 6 also shows the amount of X-ray mass beyond the shock front. In the pure HD simulation the hot shrapnel material is all beyond the SNR shock front. In the R 1 / 3 -CHI 20 -TR simulation, only part of the X-ray emitting shrapnel is beyond the shock front and there is a significant fraction of the ejecta knot material (the shrapnel tail) that is inside the SNR shell and that is expected to emit thermal X-rays (see Sect. 4). We point out that in SNRs the e ffi ciency of thermal conduction can be significantly reduced by the presence of the magnetic field (which is not taken into account in our model). If we assume an organized ambient magnetic field, the thermal conduction is anisotropic, because the conductive coe ffi cient in the direction perpendicular to the field lines is several orders of magnitude lower than that parallel to the field lines, which coincides with the Spitzer's coe ffi cient κ ( T ). The e ff ects of the magnetic-field-oriented thermal conduction in the interaction between shocks and dense clump have been investigated in detail in Orlando et al. (2008). Because of the high beta of the plasma, the magnetic field lines are expected to envelope the hydrodynamic fingers thus hampering the thermal conduction with the surrounding material. At the same time, the magnetic field is expected to be trapped at the top of the ejecta clumps, and this yields to an increase of the magnetic pressure and field tension which limits the growth of hydrodynamic instabilities (see O12 and Sano et al. 2012). The basic physics of the interaction between the ejecta knots and the SNR shocks is similar to that for the interaction of planar shocks with an interstellar cloud (e. g. WC02) and it has been shown that, in this case, sim- lations including thermal conduction in an unmagnetized plasma and pure HD simulations are limiting cases that encompass the results obtained with di ff erent configurations of the magnetic field (Orlando et al. 2008). We can then conclude that our simulations provide the two extreme cases that bracket all the possible intermediate scenarios.", "pages": [ 4, 5 ] }, { "title": "3.3 E ff ects of the initial conditions", "content": "We study the e ff ects of the initial conditions on the shrapnel evolution with di ff erent simulations, as shown in Table 1. In particular, we explored two di ff erent initial positions of the shrapnel in the ejecta profile ( Rs = 1 / 6 R 0 ej , 1 / 3 R 0 ej ) and three di ff erent density contrasts ( χ = 10 , 20 , 50). In agreement with WC02, we found that shrapnel formed in the inner ejecta layers (i. e., those that reach the reverse shock later) produce smaller protrusions. In particular, an ejecta knot originating at Rs = 1 / 6 R 0 ej , does not even reach the SNR shock in the time spanned by our simulations. Therefore, our model indicates that shrapnel A-F (all protruding well beyond the Vela main shock) originated in more external layers. Left panel of Fig. 7 shows the 2-D cross-sections through the ( r , z ) plane of temperature and density at the age of the Vela SNR for the R 1 / 6 -CHI 50 simulation. In this case, the knot is well within the intershock region, even though its initial density contrast ( χ = 50), was higher than that of the R 1 / 3 -CHI 20 run. However, our models of knots originating at Rs = 1 / 3 R 0 ej clearly show that denser shrapnel produce deeper protrusions and are more stable against the fragmentation and the deceleration induced by the hydrodynamic instabilities in the intershock region (as in WC02) 5 . Fig. 7, right panel, shows the 2-D cross-sections through the ( r , z ) plane of temperature and density at the age of the Vela SNR for the R 1 / 3 -CHI 50 simulation. As expected, the shrapnel head is much further away from the SNR shell than in the R 1 / 3 -CHI 20 case (shown in the right panel of Fig. 2). We computed the time evoulution of χ for the R 1 / 3 -CHI 50 run by following the procedure described in Sect. 3.1. We found that, in this case, the ejecta knot reaches the reverse shock with a very high density contrast ( χ > 1000, see Fig. 8), thus producing a very prominent protrusion in the SNR. Figure 9 shows the position of the shrapnel head (in units of the shell radius) as a function of time for all our simulations. Figure 9 also shows the projected positions of Shrapnel A-D (Aschenbach et al. 1995) and of the ejecta knots FilE and RegNE (Miceli et al. 2008) with respect to the position of the shock front in the Vela SNR. These values were calculated by approximating the Vela SNR as a circular shell with angular radius 211 ' and center with coordinates α J 2000 = 8 h 36 m 19 . 8 s , δ J 2000 = -45 · 24 ' 45 '' . Models including the e ff ects of radiative cooling and thermal conduction (blue curves in Fig. 9) do not provide significant di ff erences with respect to pure HD models (black curves) in terms of the shrapnel position. By considering all the simulations with Rs = 1 / 3 R 0 ej , we find that the position of the head of the knot, Rh , after ∼ 11000 yr ranges between ∼ 1 . 1 Rshell (for χ = 10) and ∼ 1 . 4 Rshell (for χ = 50). These values are similar to those observed for Shrapnel B, C, and D. Shrapnel A, E, and F (Shrapnel E, F are not shown in Fig. 9) instead, have Rh = 1 . 57 Rshell , Rh = 1 . 88 Rshell , and Rh = 1 . 91 Rshell , respectively. These values are much larger than those obtained in our simulations. Our results clearly suggest that these large distances from the shell can be produced by ejecta knots originating in outer ejecta layers. An alternative possibility is that the original density contrast for these shrapnel was > 50. Nevertheless, Fig. 9 shows that the final position of an ejecta clump is more sensitive to its initial position and depends only weakly on χ . In fact, by varying the initial position of the knot by a factor of two, we found that its final position varies approximately by the same factor, while a variation of the initial density contrast by a factor of five, only determines a 30% variation in the final position. Therefore it is more likely that Shrapnel A, E, and F were produced at Rs > 1 / 3 R 0 ej . The two simulations with Rs = 1 / 6 R 0 ej show that the ejecta knots originating in the inner ejecta layers do not reach the forward shock, even for the highest density contrast ( χ = 50). The position of the head of the knot predicted by our simulations is in agreement with that observed for FilE (Miceli et al. 2008). X-ray emitting ejecta knots have been observed, in projection, inside the Vela SNRshell (e. g., RegNE and FilE, Miceli et al. 2008). However, we point out that the actual position of these 'internal' shrapnel can be well outside the SNR shell, and therefore the values reported in Fig. 9 might be considered as lower limits.", "pages": [ 5, 6 ] }, { "title": "4 DISCUSSIONS AND CONCLUSIONS", "content": "The structure of the ejecta in a SNR contains the imprint of the metal-rich layers inside the progenitor star, and may help to understand the processes occurring in the latest stage of stellar evolution. We performed a set of hydrodynamic simulations to study the evolution of the ejecta knots in the Vela SNR. We found that moderately overdense clumps (initial density contrast χ ∼ 10) can produce protrusions in the SNR shell similar to those observed for the Vela shrapnel. WC02 found that only clumps that reach the reverse shock with density contrast χ /greaterorequalslant 100 can reach the main shock front and produce significant protrusions. This criterium is fulfilled in all our simulations. In fact, the (initially) moderately overdense clump experiences di ff usion in the 'free' expansion phase, and, while its outer parts mix with the surrounding ejecta, its central core remains much denser. This inhomogeneous rarefaction makes the density contrast between the core of the shrapnel and the expanding ejecta higher as the remnant evolves, and χ reaches values ∼ 100 -1000 at the interaction with the reverse shock. In particular, a knot with initial χ = 20 and Rs = 1 / 3 R 0 ej (simulations R 1 / 3 -CHI 20 and R 1 / 3 -CHI 20 -TR ) can explain the observed features associated with the Vela Shrapnel D. Figure 10 shows the the ROSAT All-Sky Survey image of the Vela Shrapnel D in the 0 . 1 -2 . 4 keV energy band, compared with a synthesis of the X-ray emission in the 0 . 1 -2 . 4 keV band derived from the R 1 / 3 -CHI 20 -TR simulation at 11000 yr. The synthesized X-ray map has been obtained as in Miceli et al. (2006): we produced the 3D map of the emission measure and temperature in the cartesian space ( x ' , y ' , z ' ), where the y ' axis corresponds to the direction of the line of sight and is perpendicular to the ( r , z ) plane. We derived the distribution EM ( x ' , z ' ) vs. T ( x ' , z ' ) by considering all the contrbutions along the line of sight, for each ( x ' , z ' ). We then synthesized the map of the X-ray emission using the MEKAL spectral code (Mewe et al. 1985, Mewe et al. 1986, Liedahl et al. 1995), assuming a distance of 250 pc, and an interstellar column density NH = 2 × 10 20 cm -2 . Finally, we degraded the spatial resolution of the synthesized X-ray map to match the resolution of the ROSAT image and randomized the map by assuming Poisson statistics for the counts in each image bin. As shown in Sect. 3.3, our model suggests that Shrapnel B, C, and D were all originated at ∼ 1 / 3 R 0 ej . This conclusion is in agreement with the results of X-ray data analysis that show that Shrapnel B and D have similar abundance patterns (Katsuda & Tsunemi 2005, Yamaguchi & Katsuda 2009). The fragment RegNE, which appears inside the Vela shell, has similar abundances as Shrapnel D (Miceli et al. 2008) and its position is compatible with an origin in the same layer where Shrapnel B, C, and D were generated 6 . In Sect. 3.3, we also pointed out that it is highly unlikely that Shrapnel A originated in the same ejecta layer as Shrapnel D. Indeed, the abundance patterns observed in Shrapnel A are remarkably di ff erent from that observed in Shrapnel B and D (Katsuda & Tsunemi 2006), thus suggesting a di ff erent location of this knot in the ejecta profile. Nevertheless, the Si:O ratio is much higher in Shrapnel A than in Shrapnel D and this indicates that Shrapnel A comes from a region of the progenitor star below that of the Shrapnel D (e. g. Tsunemi & Katsuda 2006), at odds with our predictions. As explained above, an unrealistically high initial χ is required for an inner shrapnel to overcome outer knots, if we assume that the initial velocity profile of the ejecta increases linearly with their distance from the center. A possible solution for this puzzling result is that the Si burning layer (or part of it) has been ejected with a higher initial velocity, e. g., as a collimated jet. It is noteworthy to remark that in other core-collapse SNRs the Sirich ejecta may show a very peculiar jet-counterjet structure. The well known case of Cas A has been studied in detail thanks to a very long Chandra observation (Hwang et al. 2004) showing a jet (with a weaker counterjet structure) composed mainly of Si-rich plasma. Laming et al. (2006) have performed X-ray spectral analysis of several knots in the jet and concluded that the origin of this interesting morphology is due to an explosive jet and it is not arising because of an interaction with a cavity or other ISM / CSMpeculiar structure. Therefore a jet origin for the Si-rich knots is sound. Finally, we investigated the e ff ects of thermal conduction, finding that it determines an e ffi cient 'evaporation' of the ejecta knot and accelerate its mixing with the surrounding medium. Moreover, it a ff ects the shrapnel morphology, producing the formation of a long, metal-rich tail. Enhanced metallicities have been observed in the tails of Shrapnel A, B, and D and the abundance analysis performed on the X-ray spectra clearly suggests an e ffi cient mixing of the ejecta knots with the surrounding medium (Katsuda & Tsunemi 2005, 2006, and Yamaguchi & Katsuda 2009). These results are in qualitative agreement with our findings. A quantitative comparison between models and observations requires a forward modeling approach, consisting in the synthesis of the X-ray spectra from the simulations and a detailed comparison between the synthesized observables and the corresponding observations (e. g., through a spatially resolved spectral analysis, as in Miceli et al. 2006, e. g.). Moreover it will also be important for future models to include some seed magnetic fields in the simulations to study how tangled the field becomes and what this implies for thermal conduction. These further studies are beyond the scope of this paper. In conclusion, our hydrodynamic modelling of ejecta knots in the Vela SNR allowed us to find that: i) the observed shrapnel can be the results of moderate density inhomogeneities in the early ejecta profile; ii) the evolution of a shrapnel in the SNR is very sensitive to its initial position and depends much less (but does depend) on the initial density contrast; iii) thermal conduction plays an im- portant role and explains the e ffi cient mixing of the ejecta knots observed in X-rays.", "pages": [ 7, 8 ] }, { "title": "ACKNOWLEDGMENTS", "content": "We thank the anonymous referee for their constructive suggestions to improve the paper. The software used in this work was in part developed by the DOE-supported ASC / Alliance Center for Astrophysical Thermonuclear Flashes at the University of Chicago. The simulations discussed in this paper have been performed on the HPC facility at CINECA, Italy, and on the GRID infrastructure of the COMETA Consortium, Italy.", "pages": [ 8 ] }, { "title": "References", "content": "2E-23 4E-23 6E-23 8E-23 Orlando S., Bocchino F., Reale F., Peres G., Pagano P., 2008, ApJ, 678, 274 Orlando S., Peres G., Reale F., Bocchino F., Rosner R., Plewa T., Siegel A., 2005, A&A, 444, 505 Park S., Hughes J. P., Slane P. O., Burrows D. N., Roming P. W. A., Nousek J. A., Garmire G. P., 2004, ApJ, 602, L33 Richtmyer R. D., 1960, Comm. Pure Appl. Math., 13, 297 Sano T., Nishihara K., Matsuoka C., Inoue T., 2012, ArXiv eprints Wang C.-Y., Chevalier R. A., 2002, ApJ, 574, 155 Yamaguchi H., Katsuda S., 2009, ApJ, 696, 1548", "pages": [ 9 ] }, { "title": "APPENDIX A: TEST ON SPATIAL RESOLUTION", "content": "As explained in Sect. 2.1, the spatial resolution of our simulations is reduced by a factor of 2 at t > 2500 yr. To check whether this resolution is su ffi cient to capture the basic evolution of the system, we repeated simulation R 1 / 3 -CHI 20 by mantaining the finest level of resolution at its initial value (1 . 95 × 10 16 cm, hereafter simulation R 1 / 3 -CHI 20 -hiRES ) for the whole run. We verified that simulations R 1 / 3 -CHI 20 and R 1 / 3 -CHI 20 -hiRES yield very similar results and all the quantities discussed in the paper (position of the shrapnel as a function of time, X-ray emitting mass of the shrapnel, etc.) do not change significantly. Figure A1 shows the 2-D cross-sections of the density through the ( r , z ) plane obtained for simulations R 1 / 3 -CHI 20 and R 1 / 3 -CHI 20 -hiRES at t = 5000 yr. This evolutionary stage is the most critical, since the system is still relatively small and the ejecta knots interact with the hydrodynamic instabilities in the intershock region. Figure A1 clearly prove that simulation R 1 / 3 -CHI 20 already provides a very accurate description of the system and run R 1 / 3 -CHI 20 -hiRES does not introduce any major di ff erences.", "pages": [ 9 ] } ]

[ { "title": "ABSTRACT", "content": "We present tests carried out on optical and infrared stellar spectra to evaluate the accuracy of different types of interpolation. Both model atmospheres and continuum normalized fluxes were interpolated. In the first case we used linear interpolation, and in the second linear, cubic spline, cubic-Bezier and quadratic-Bezier methods. We generated 400 ATLAS9 model atmospheres with random values of the atmospheric parameters for these tests, spanning between -2 . 5 and +0 . 5 in [Fe/H], from 4500 to 6250 K in effective temperature, and 1.5 to 4.5 dex in surface gravity. Synthesized spectra were created from these model atmospheres, and compared with spectra derived by interpolation. We found that the most accurate interpolation algorithm among those considered in flux space is cubic-Bezier, closely followed by quadratic-Bezier and cubic splines. Linear interpolation of model atmospheres results in errors about a factor of two larger than linear interpolation of fluxes, and about a factor of four larger than high order flux interpolations.", "pages": [ 1 ] }, { "title": "On the Interpolation of Model Atmospheres and High-Resolution Synthetic Stellar Spectra", "content": "Sz. M´esz´aros 1 , 2 , C. Allende Prieto 1 , 2", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Even within the framework of classical LTE 1D models, the calculation of a stellar model atmosphere takes a finite amount of time, which can be significant when complex opacities are involved. Massive spectroscopic surveys require large numbers of models spanning a wide range of parameters and can become very time consuming. Depending on the algorithm used, even the analysis of a single star may require many model atmospheres to evaluate the performance of different combinations of parameters. In practice, the need for model spectra for many parameter combinations is satisfied by taking one or several shortcuts that avoids the actual calculation of self-consistent models. The most wideused strategy is some sort of interpolation, either in the model atmosphere (the run with height of the main thermodynamical variables), or in the emerging radiative fluxes or intensities. Different recipes have been used, and codes circulate among researchers, but few have been published and thoroughly tested. In this paper we perform a battery of tests in order to quantify the typical errors incurred when interpolating model atmospheres or the model fluxes calculated from them. Section 2 describes our calculations and § 3 our results, with a summary provided in § 4.", "pages": [ 1, 2 ] }, { "title": "2. Calculations", "content": "We generated a regular grid of ATLAS9 model atmospheres with [Fe/H] from -2 . 5 to +0 . 5 in steps of 0.5 dex, T eff from 4500 K to 6250 K in steps of 250 K, and log g from 1.5 to 4.5 dex in steps of 0.5 dex. We also calculated 400 additional models with random parameters within the boundaries of the regular grid. The microturbulence was chosen to be constant at 2 km s -1 in all our calculations. The relatively small range of T eff is to ensure that all ATLAS9 models are fully converged throughout the entire atmosphere. Since ATLAS9 start to experience convergence problems in the outer layers of the atmosphere below 4250 K and above logg = 4, especially at low metallicities, we chose to omit that region from the calculations. Stars warmer than 6250 K have spectra dominated mainly by hydrogen lines, thus it is expected that interpolation errors will decrease as temperatures increase, and the examples of spectra with 6250 K are good representation of warmer stars. After careful consideration we decided that the above values gave the largest range in all three parameters combined. Two representative wavelength regions were chosen, one in the optical, and including both weak and strong spectral lines (the Mg I b triplet) between 516.5 and 519.5 nm, and one in the near-infrared, and in particular the H-band window, targeted by the Apache Point Galactic Evolution Experiment (APOGEE, Allende Prieto et al. 2008; Eisenstein et al. 2011; Wilson et al. 2012), spanning between 1509.1 and 1699.5 nm. We carried out two types of tests, one involving interpolation in the model atmospheres directly, and a second one involving interpolation in the emergent fluxes. In the first type of test, we interpolated models with the chosen random parameters from the grid, and synthesized the emergent spectra. The comparison was made between these and the fluxes calculated from the models with the randomly generated parameters. In the second case, we calculated the emergent spectra for the models generated from the random parameters (true flux), and the evenly distributed models. Then, we interpolated the spectra from the evenly spaced models to the sets of random parameters and examined the differences in continuum normalized flux relative to the true flux. The flux interpolation tests were performed with linear (F(L)), cubic spline (F(CS)), cubic-Bezier (F(CB)), and quadratic-Bezier (F(QB)) methods, while in the case of the model atmosphere interpolations (MA) we only explored the linear algorithm.", "pages": [ 2 ] }, { "title": "2.1. Model atmosphere calculations", "content": "Our regularly spaced ATLAS9 model atmospheres were calculated as described by Meszaros et al. (2012). However, there are some differences between these models and those presented by Meszaros et al. (2012) 1 : those used here correspond to different versions of the line data, and an older version of the code, plus a number of other differences regarding the configuration of the input for the ATLAS9 code. The calculations used here are older, and the updated ones are to be preferred, but since these details are irrelevant for the evaluation of the interpolation accuracy, we have chosen to retain the custom-made calculations. The opacity distribution functions (ODFs) and Rosseland opacities needed as input were calculated using the DFSYNTHE and KAPPA9 codes, while the model atmospheres were generated with the linux version of the ATLAS9 code (Kurucz 1979, 1993; Sbordone 2004, 2005). The ODF calculations followed the method described by Castelli & Kurucz (2003); Castelli (2005), while the model atmosphere calculations are detailed in Meszaros et al. (2012). ATLAS9 gives excellent convergence in the parameter range chosen above. In addition to the regular grid, we made calculations for 400 additional random models. The parameters for these were drawn from random uniform distributions across the chosen ranges. These calculations are fully consistent with the models in the grid.", "pages": [ 3 ] }, { "title": "2.2. Model atmosphere interpolation", "content": "We interpolated model atmospheres for the 400 sets of random parameters. For each target model, we identified the 8 immediate neighbors with higher and lower values for each parameter in the grid, calculated by numerical integration the Rosseland optical depth for each, re-sampled all the thermodynamical quantities in the atmosphere (temperature, gas pressure, and electron density) on a common optical depth scale for all models by linear interpolation, and then interpolated, linearly, all the thermodynamical quantities to the parameters ( T eff , log g , and [Fe/H]) of the target model. Other quantities included in the models (Rosseland opacities, radiative pressure, etc.) were also interpolated in the same way. The interpolations were carried out using the kmod code. This code has already been used in a number of investigations (Reddy et al. 2003, 2006; Yong et al. 2013). It is written in IDL and it is publicly available 2 .", "pages": [ 3 ] }, { "title": "2.3. Calculation of model fluxes", "content": "We calculated model fluxes for all model atmospheres using the ASS /epsilon1 T spectral synthesis code (Koesterke et al. 2008; Koesterke 2009) with detailed continuum opacities by Allende Prieto et al. (2003) and updates from Allende Prieto (2008). Line data come mainly from the calculations and compilations by Kurucz (available from his website 3 ), enhanced with damping constants from Barklem (2007) when available. We adopted solar reference abundances as in Asplund et al. (2005), and the compositions used in the calculations of the model atmospheres were consistent with those adopted in the spectral synthesis. We underline, however, that the interpolation tests performed here are fairly insensitive to the particular choices for the reference solar composition and the atomic and molecular data, as long as these choices are reasonable and lead to spectra that resemble approximately the modeled stars. The most critical aspect, is to ensure that the calculations for the models in the grid and those with random parameters used for tests, are completely consistent.", "pages": [ 4 ] }, { "title": "2.4. Flux Interpolations", "content": "Flux interpolations are consecutively performed in surface gravity, effective temperature, and metallicity. Quadratic and cubic Bezier interpolation are implemented with control values that make the algorithm identical to Hermite interpolation, and therefore both the interpolating function and its derivatives are continuous. In the case of cubic Bezier interpolation, quadratic Bezier interpolation was forced for those dimensions for which the target parameters where in the intervals adjacent to the edges of the grid. The Bezier interpolations were implemented afresh in the FORTRAN spectral fitting code FERRE (Allende Prieto 2006) following the description by Auer (2003) and references therein. Cubic splines interpolation was included calling a subroutine from the library provided with the book by Chapman (2004), which follows the discussion given in Press et al. (1992).", "pages": [ 4 ] }, { "title": "3. Discussion", "content": "Before examining the errors in the fluxes, we compared interpolated model atmospheres with the calculated ones. The linear interpolation does a good job at estimating the real atmospheric structure; the temperature differences were between 2 -3%, the pressure and electron density differences were between 1 -2%. This translates to 10 -30 K average differences in temperature in case of cool atmospheres above log τ Ross < 1, from where all of the spectral lines form. This difference slightly increases as the effective temperature gets higher. To measure the amount of error each interpolation method makes, we calculated the same statistics for each case. In the case of absolute fluxes, we only examined the average differences throughout the whole wavelength range, to track how much the continuum level changes. In the case of optical spectra, the atmosphere interpolation gives 1 -5% smaller continuum levels than the direct calculations, and we find that the linear flux interpolation gives -0.3 -+0.5% relative differences, usually shifted slightly higher than the true continuum in case of the optical spectrum, but not in the infrared. The errors are smaller in the IR region, 0.5 -+1% in the continuum level when atmospheres are interpolated and 0.01 -0.1% with linear interpolation in flux. Two examples of the differences between interpolated and direct calculations of continuum normalized spectra are given in Figures 1 and 2. Figure 1 shows a metal-poor G-type giant, while Figure 2 shows a metal rich, K-type dwarf - these two cases represent the extremes of our parameters space. The metal-poor spectra only show large differences in the line cores. The average error can be up to 1% in the optical and infrared metal-poor warm spectrum, while in case of the metal-rich cool spectrum this goes up to 5 -10% in the optical and 1 -2% in the infrared. The differences in the line profile for the cooler atmosphere increase by a factor of 4 in both wavelength regions compared to the metal-poor case due to weak atomic metal and molecular lines very sensitive to small temperature changes in the atmosphere. In the infrared spectra of the G-tpe metal-poor star, the differences are dominated by the hydrogen Brackett lines, while in the optical these are associated with the magnesium triplet lines. This picture changes dramatically in case of the metalrich, cool dwarf (Figure 2), where the Brackett lines disappear and the 'noise' increases greatly in the continuum of the IR spectrum due to weak atomic and molecular lines (CN, OH, and CO). Also, relative errors in the line cores are significantly higher than near the continuum. To measure the overall performance of the interpolation methods across the parameter range, the differences respect to the true fluxes were calculated at each wavelength in the continuum normalized spectra. The average, standard deviation and maximum deviation above and below the continuum in the whole wavelength range were determined. The last two of these three parameters track different aspects of the error distribution. The standard deviation gives an estimate of the overall changes in the full flux range, while the maximum deviation shows mainly the differences in the line cores, where errors tend to be largest. The differences vary depending on where one samples the spectrum: they are small in the continuum, but typically larger in the line cores. Figure 3 illustrates how the average differences depend on metallicity, effective temperature and gravity. The errors, both the average errors and the standard deviation, grow with increasing metallicity and decreasing temperature, but they do not depend so much on surface gravity. These correlations are easily understood as a result of an increased number of molecular and atomic absorption lines appearing in the spectrum at high metallicity and low temperature. Both in the model atmosphere and linear flux interpolation methods, it is clearly visible that errors are significantly enhanced for values of the metallicity half way between the nodes of the grid. The evenly distributed models spanned -2 . 5 to +0 . 5 in [Fe/H] with steps 0.5, and in the vicinity of these values the linear interpolation gives significantly lower errors. Using higher order interpolations makes this effect to disappear. None of the methods is sensitive to this issue along the axis for effective temperature (nodes spaced every 250 K) or log g (nodes every 0.5 dex). The overall results for each test are listed in Table 1 and illustrated in Figure 4. In the optical, the average differences are about 0.17 ± 0.19% for the case of MA interpolation, and smaller than 0.07% for all the interpolations in flux. The standard deviation of the relative differences paints a more dramatic picture. The MA method clearly shows the largest deviation up to 1% near the continuum level in the optical and, while F(CB) is the most accurate method with only 0.1% errors. In the infrared all interpolation methods perform well giving much smaller errors than in the optical, close to an average of 0 -0.02% differences. The scatter is about four times larger in the case of MA than in any other methods in both optical and infrared. Figure 5 shows the maximum deviation of differences below and above the continuum level in the case of the cubic-Bezier interpolation. At optical wavelengths, the differences are generally higher than in the infrared, but they are usually smaller than 0.01 in ∆ F (1 -2%). While a reduction of the errors near the grid nodes was not visible in the average differences, here it is apparent as a function of metallicity below [Fe/H]= -1, and the effect increases as [Fe/H] decreases. The interpolation gives the highest errors in the line cores, i.e. in the highest atmospheric layers the spectrum is sensitive to.", "pages": [ 4, 5, 6 ] }, { "title": "4. Conclusions", "content": "We conclude that if model spectra need to be interpolated, the best way to proceed, among those explored, is to use high-order interpolation in continuum-normalized fluxes. Linear interpolation of model atmospheres leads to about a factor of 3 -5 higher error than high order interpolations in flux, at about -0.17 ± 0.19% average differences in the optical, and 0.0004 ± 0.038% in the infrared. Linear interpolation in flux space gives a factor of two smaller errors than using model atmospheres, but there is still a factor of two improvments found using high order functions. The most accurate flux interpolation method is cubic-Bezier with average errors of 0.011 ± 0.047% in the optical and 0.0011 ± 0.0075% in the infrared for our grids and random parameters. The cubic spline and quadratic-Bezier interpolations lead to only marginally higher errors than this. We conclude that interpolation errors become visible at S/N =10 -20 in the line cores in case of model atmosphere interpolations in the optical, and at S/N =20 -40 in the infrared, while the errors from cubic-Bezier flux interpolation will only show up above S/N ∼ 100 -200 in both wavelength ranges. We must stress that only one code for the interpolation of model structures has been tested. Other codes with somewhat different strategies could provide somewhat different performances. We have allowed our high-order interpolations in flux to get to any value, even if artificial extrema are created (see the discussion by Auer 2003). Our tests are restricted to stars with spectral types roughly between F5 and K5. Nevertheless, we expect that the tendencies seen in our study will persist outside this domain. For example, we expect that errors will continue to reduce for the interpolation of stars with warmer temperatures, and to increase at temperatures under 4500 K. Our analysis includes only linear and polynomial interpolation, but more sophisticated schemes are available and their performance could be somewhat different. Further work dealing with principal component analysis (PCA), neural networks, and other algorithms would be a natural followup of this investigation. We would like to thank our collegues at the SDSS-3 APOGEE survey for providing continuous discussions on various interpolation techniques. We greatly appreciate the contribution of Lars Koesterke on spectral synthesis with the ASS /epsilon1 T code. We are also grateful for the referee for his helpful suggestions.", "pages": [ 6, 7 ] }, { "title": "REFERENCES", "content": "Allende Prieto, C., Beers, T. C., Wilhelm, R., Newberg, H. J., Rockosi, C. M., Yanny, B. & Lee, Y. S. 2006, ApJ, 636, 804 Allende Prieto, C. 2008, Physica Scripta Volume T, 133, 014014 Allende Prieto, C., Majewski, S. R., Schiavon, R., Cunha, K., Frinchaboy, P., Holtzman, J., John- ston, K., Shetrone, M., Skrutskie, M., Smith, V., & Wilson, J. 2008, AN, 329, 1018 Allende Prieto, C., Lambert, D. L., Hubeny, I., & Lanz, T. 2003, ApJS, 147, 363 Asplund, M., Grevesse, N. & Sauval, A. J. 2005, ASPC, 336, 25 Auer, L. 2003, in Stellar Atmosphere Modeling, I. Hubeny, D. Mihalas and K. Werner, eds., ASP Conf. Series, 288, p3-15 Barklem, P. S. 2007, A&A, 466, 327 Castelli, F. 2005, MSAIS, 8, 344 Castelli, F., & Kurucz, R. L. 2003, New Grids of ATLAS9 Model Atmospheres, IAUS, 210, 20P Chapman, S. J. 2004, Fortran 90/95 for scientist and engineers, 2nd edition, Mcgraw-Hill Eisenstein, D.J., et al. 2011, AJ, 142, 72 Koesterke, L. 2009, American Institute of Physics Conference Series, 1171, 73 Koesterke, L., Allende Prieto, C. & Lambert, D. L. 2008, ApJ, 680, 764 Kurucz, R. L. 1979, ApJS, 40, 1 Kurucz, R. L. 1993, ATLAS9 Stellar Atmosphere Programs and 2 km s -1 grid. Kurucz CD-ROM No. 13. Cambridge, Mass.: Smithsonian Astrophysical Observatory, 1993, 13 Meszaros, Sz., Allende Prieto, C., Edvardsson, B., Castelli, F., Garc'ıa P'erez, A. E., Gustafsson, B., Majewski, S. R., Plez, B., Schiavon, R., Shetrone, M., & de Vicente, A. 2012, AJ, 144, 120 Press, W. H., Teukolsky, S. A., Vetterling, W. T., Flannery, B. P. 1992, Numerical Recipes in Fortran 77: the art of scientific computing, 2nd edition, Cambridge Univ. Press Reddy, B. E., Lambert, D. L. & Allende Prieto, C. 2006, MNRAS, 367, 1329 Reddy, B. E., Tomkin, J., Lambert, D. L. & Allende Prieto, C. 2003, MNRAS340, 304 Sbordone, L. 2005, MSAIS, 8, 61 Sbordone, L., Bonifacio, P., Castelli, F., & Kurucz, R. L. 2004, MSAIS, 5, 93 Wilson, J. C. et al. 2012, SPIE, 8446, 0 Yong, D. et al.2013, ApJ, 762, 26 Note. - OP: optical, 516.5 -519.5 nm, IR: infrared, 1500 -1700 nm. The type of interpolations are the following: MA: model atmosphere, F(L): linear flux, F(CS): cubic spline flux, F(CB): cubic-Bezier flux, F(QB): quadraticBezier flux", "pages": [ 7, 8, 14 ] } ]