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[ { "title": "Finite-time future singularities models in f ( T ) gravity and the effects of viscosity", "content": "M. R. Setare a 1 and M. J. S. Houndjo b,c 2 01 BP 613 Porto-Novo, B'enin", "pages": [ 1 ] }, { "title": "Abstract", "content": "We investigate models of future finite-time singularities in f ( T ) theory, where T is the torsion scalar. The algebraic function f ( T ) is put as the teleparallel term T plus an arbitrary function g ( T ). A suitable expression of the Hubble parameter is assumed and constraints are imposed in order to provide an expanding universe. Two parameters β and H s that appear in the Hubble parameter are relevant in specifying the types of singularities. Differential equations of g ( T ) are established and solved, leading to the algebraic f ( T ) models for each type of future finite time singularity. Moreover, we take into account the viscosity in the fluid and discuss three interesting cases: constant viscosity, viscosity proportional to √ -T and the general one where the viscosity is proportional to ( -T ) n/ 2 , where n is a natural number. We see that for the first and second cases, in general, the singularities are robust against the viscous fluid, while for the general case, the Big Rip and the Big Freeze can be avoided from the effects of the viscosity for some values of n . Pacs numbers:", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Recent observations of type Ia supernova (SNIa) and WMAP [1, 2] indicate that our universe is currently undergoing an accelerating expansion, which confront the fundamental theories with great challenges and also make the researches on this problem a major endeavour in modern astrophysics and cosmology. Missing energy density - with negative pressure - responsible for this expansion has been dubbed dark energy. Wide range of scenarios have been proposed to explain this acceleration while most of them can not explain all the features of universe or they have so many parameters that makes them difficult to fit. In this direction we can consider theories of modified gravity [3], or field models of dark energy. The field models that have been discussed widely in the literature consider a cosmological constant [4], a canonical scalar field (quintessence)[5], a phantom field, that is a scalar field with a negative sign of the kinetic term [6, 7], or the combination of quintessence and phantom in a unified model named quintom [8]. In the other hand modified models of gravity provides the natural gravitational alternative for dark energy [9]. Moreover, modified gravity present natural unification of the early time inflation and late-time acceleration thanks to different role of gravitational terms relevant at small and at large curvature. Also modified gravity may naturally describe the transition from non-phantom phase to phantom one without necessity to introduce the exotic matter. Among these theories, scalar-tensor theories [10], f ( R ) gravity [11], DGP braneworld gravity [12] and string-inspired theories [13] are studied extensively. Recently a theory of f ( T ) gravity has been received attention. Models based on modified teleparallel gravity were presented, in one hand, as an alternative to inflationary models [14, 15], and on the other hand, as an alternative to dark energy models [16]. New spherically symmetric solutions of black holes and wormholes are obtained with a constant torsion and for the cases for which the radial pressure is proportional to a real constant, to some algebraic functions f ( T ) and their derivatives, or vanishing identically [21]. In the same way, an algebraic function f ( T ) is obtained through the reconstruction method for two cases and the study of a polytropic model for the stellar structure is developed [22]. Moreover, f ( T ) gravity is reconstructed according to holographic dark energy is explicitly presented in [23] and latter an anisotropic fluid for a set of non-diagonal tetrads in f ( T ) gravity explored generating various classes of new black hole and wormhole solutions [24]. Also, many works have been done in order to check whether f ( T ) gravity can present results consistent with the many advances in cosmology and astrophysics [25]. Recently, Bamba et al investigate the reconstruction of power law model, exponential model and logarithmic model, able to reproduce some of the future finite time singularities and also discuss the thermodynamics near these singularities [26] (For other works about future finite time singularities, see [27]). Also, future singularities with the presence of a viscous fluid as well as other interesting properties, have been already studied (in the context of General Relativity and f(R) gravity)[28] . In the present paper we investigate the f ( T ) gravity models that are able to reproduce the four types of finite time future singularities from a suitable choice of Hubble parameter. A parameter β in the expression of the Hubble parameter plays an important role in specifying these singularities. The algebraic function f ( T ) is assumed as the sum of the teleparallel term T and an algebraic function g ( T ) with which all the task is done. According to some values of the parameter β , differential equation of g ( T ) are established and solved in some ways. The algebraic function f ( T ) for each type of singularity is obtained from each expression of g ( T ). On the other hand, we notify that the presence of finite-time future singularities may cause serious problems in the black holes or stellar astrophysics [29]. A way to probe the possible avoidance of these singularities is considering that the fluid possesses viscosity. This is the second purpose of this work. As previously mentioned, the models that lead to future finite time singularities have been reconstructed considering a non viscous fluid. Thus, we introduce the viscosity and investigate its effects on the singularities. We see that when the constant viscosity or the viscosity proportional to √ -T is considered, in general, the singularities are robust against the viscosity. However, when the viscosity is proportional to ( -T ) n/ 2 , for some values of the parameter n , the viscosity may cure the Big Rip and the Big Freeze. The paper is organized as follows. In Sec. 2, the f ( T ) gravity formalism and the field equations are presented. The Sec. 3 is devoted to the presentation of the Hubble parameter, the classification of future finite time singularities. Suitable scale factor coming from the Hubble parameter are presented according to the values of the parameter β for obtaining the algebraic f ( T ) function. The viscosity is introduced in the Sec. 4 and its effect is investigated as the singularities are approached. The conclusion and perspectives are presented in Sec. 5.", "pages": [ 1, 2, 3 ] }, { "title": "2 f ( T ) gravity and field equations", "content": "Let us define the notation of the Latin subscript as those related to the tetrad fields, and the Greek one related to the spacetime coordinates. For a general specetime metric, we can define the line element as The projection of this line element can be described in the tangent space to the spacetime through the matrix called tetrad as follows: where η ij is the metric on Minkowski's spacetime and e µ i e i ν = δ µ ν , or e µ i e j µ = δ j i . The action for the theory of modified gravity based on a modification of the teleparallel equivalent of General Relativity, namely f ( T ) theory of gravity, coupled with matter L m is given by [15, 18, 19, 20] where e = det ( e i µ ) = √ -g . Here, G is the gravitational constant and c the speed of the light. From now, we will use the units 8 πG = c = 1. The teleparallel Lagrangian T is defined as follows where and K µν ρ is the contorsion tensor The field equations are obtained by varying the action with respect to vierbein e i µ as follows where g T = g ' ( T ) and g TT = g '' ( T ) and T the energy momentum tensor. Now, we take the usual spatially-flat metric of Friedmann-Robertson-Walker (FRW) universe, in agreement with observations where a ( t ) is the scale factor as a one-parameter function of the cosmological time t . Let us assume first that the background is a non-viscous fluid. Using the Friedmann-Robertson-Walker metric and the perfect fluid matter in the Lagrangian (4) and the field equations (9), one obtains where ρ eff and p eff denote respectively the effective energy density and pressure of the universe and defined by where H is the Hubble parameter and defined by H = ˙ a/a . Using (14) and (15), and combining (12) and (13), one gets where we used the barotropic equation of state p = ωρ . Then, for a given scale factor corresponding to a future finite time singularity, the action may explicitly be reconstructed by solving the differential equation (16).", "pages": [ 3, 4 ] }, { "title": "3 Future finite time singularities", "content": "We propose to find in f ( T ) gravity, models that reproduce the four types of finite time future singularities from the Hubble parameter [32] where h , C and t s are positive constants and t < t s . These constraints are imposed to the parameter for providing an expanding universe. The parameter β can be a positive constant or a negative non-integer number. Then, as the singularity time t s is approached, H or some of its derivatives and therefore, the torsion, diverge. C is essentially relevant near the singularity only when β < 0 (where we denote it as C = H s , the Hubble parameter at the singularity time), and then, we can assume C = 0 when β > 0. The finite-time singularities are classified in the following way [30, 31] Let us now investigate the f ( T ) gravity models for which the finite time future singularities could occur, when (17) is assumed.", "pages": [ 4, 5 ] }, { "title": "3.1 Big Rip singularity models without viscosity", "content": "This sort of singularity may appear for β = 1 and β > 1. Let us treat the cases separately. The case β = 1 In this case, the corresponding scale factor can be written as and then, the first derivative of the Hubble parameter reads Hence, Eq. (16) becomes In the other hand, using (11), with the scale factor (18), one can write Thus, Eq. (20) takes a new form as The general solution of (22) reads where C 1 and C 2 are integration constants, and A and B defined respectively as The corresponding algebraic function f ( T ) reads Initial condition may be applied for finding the respective values of the constants C 1 and C 2 . We can follow the same process as in [24]. We assume that at the early time, that we denote t 0 , the corresponding value (the initial value) of the torsion scalar is T 0 such that The initial conditions imposed to the functions f read Making use of these initial conditions, (27), one gets Then, the algebraic function (25), with the constants (28), leads to the Big Rip when the fluid is free of viscosity.", "pages": [ 5, 6 ] }, { "title": "The case β > 1", "content": "In this case, the corresponding expression of the scale factor is and the first derivative of the Hubble parameter reads Thus, the Eq. (16) becomes Using Eq. (11) and the scale factor (29), one gets and the differential equation (31) can be written as Note that this equation is more complicated and cannot be solved trivially. However, it can be solved as the singularity is approached, i.e., when t → t s . Then, as the singularity is approached, the trace diverges ( T →-∞ ), and (33) becomes whose solution is where C 3 and C 4 are integration constants. Consequently, f ( T ) is written as In this case, the initial value of the first derivative of the torsion with respect to the cosmic time reads Making use of (37) and the initial conditions, (27), one gets Then the corresponding algebraic f ( T ) is This model also leads to the Big Rip in a universe dominated by a non-viscous fluid.", "pages": [ 6, 7 ] }, { "title": "3.2 The Big Freeze models without viscosity", "content": "The Big Freeze appears for 0 < β < 1, and the corresponding scale factor is the same as in (29), which leads to (33). However, as we are dealing with the singularity of type III, Eq. (33) becomes whose general solution is where C 5 and C 6 are integration constants. Then, f ( T ) is written as By using (26) and the initial conditions (27), the constant C 5 and C 6 are determined as Then, the algebraic function (42) is characteristic of the singularity of type III.", "pages": [ 7, 8 ] }, { "title": "3.3 The Sudden singularity models without viscosity", "content": "In this case, -1 < β < 0, and the corresponding scale factor reads and the first derivative of the Hubble parameter remains the same as in (30), and Eq. (16) becomes By using (17), with C substituted by H s , as explained above and (11), one obtains where T s denotes the torsion scalar at the singularity time, and may not be infinity. Using (46), Eq.(45) becomes This equation is not trivial and cannot be solved analytically. However, as the singularity time is approached, with -1 < β < 0, it becomes whose general solution reads where C 7 and C 8 are integration constants, and A previously defined in (24). The algebraic function f ( T ) is then written as The constants C 7 and C 8 can be determined in the same way as in the previously cases. In this case, the first derivative of the torsion scalar with respect to the time, at initial time, is written as and (27) is valid. Then, we obtain the constants Then, the algebraic function (50), with the constants C 7 and C 8 respectively in (52) and (53), is the f ( T ) models that may produce the singularity of type II.", "pages": [ 8, 9 ] }, { "title": "3.4 The Big Brake models without viscosity", "content": "Here, the scale factor is the same as in the case of the Sudden singularity, just that in this case, β < -1. Then, it can be easily observed that the differential equation (48) is also valid in this case. Consequently, the algebraic function (50) can lead to the Big Brake. The difference which appears here, with respect to the sudden singularity, is the values of the parameter β .", "pages": [ 9 ] }, { "title": "4 Analysing the possible avoidance of the singularities in a viscous fluid", "content": "Let us assume the fluid equation of state in the following form where ζ ( ρ ) is the bulk viscosity and in general depends on ρ . According to the thermodynamics grounds, the quantity ζ ( ρ ) has to be positive in order to guarantee the positive sign of the entropy change in an irreversible process. In this case, the stress-energy tensor T µν is written as The equations of motion (12) and (13) can be written as where the modified gravity part is formally included into the modified energy density ρ T and the modified pressure p T as follows If we assume that the ordinary and the dark fluids of the universe do not interact, the equation of continuity related to the ordinary viscous fluid reads As we have seen in the section 3, when the fluid does not possess viscosity, the four type of future finite-time singularities may appear. Now by introducing the viscosity, we analyse its effect near the singularities. Let us start with the constant viscosity case.", "pages": [ 9, 10 ] }, { "title": "4.1 The constant viscosity case: ζ ( ρ ) = ζ 0", "content": "In order to analyse the effect of the viscosity on the Big Rip, Sudden, Big Freeze and Big Brake singular models, it is worth considering the behaviour of the conservation law in Eq. (60) near the singularities. By using Eq. (17), one can write Eq. (60) as Solutions of Eq. (61) can be found according to different values of the parameter β as /negationslash where we used C = 0 for the Big Rip and the Big Freeze as previously discussed, since this constant is not relevant in these cases, while for the Sudden and the Big Brake, the constant C is taken equal to the Hubble parameter H s at singularity time. Since H s plays a crucial role in these two cases, it has to be different from zero. We conclude that in f ( T ) gravity and with the equation of state (54) for a fluid with constant viscosity, all the singularities are robust against the viscosity.", "pages": [ 10, 11 ] }, { "title": "4.2 The viscosity proportional to √ -T", "content": "From Eq. (11) one can observe that this case is that in which the viscosity is proportional to the Hubble parameter. Let us then consider ζ = τ √ -6 T/ 2 = 3 τH , where τ is a positive constant. The different solutions of (60) according to the values of the parameter β read /negationslash Let us now discuss the effects of the viscosity near each type of singularity.", "pages": [ 11, 12 ] }, { "title": "4.3 More general case: ζ proportional to ( -T ) n/ 2", "content": "In this general case, we consider the bulk viscosity as where n is a natural number, and τ a non null positive constant. Hence, the energy conservation leads to from which we obtain the behaviour of the energy density of the viscous fluid as the singularity is approached, H 2 diverges more than ρ vis . Then, the Big Rip cannot be avoided by the viscous fluid. However, for n > 1, the ρ vis diverges more than H 2 and then, the Big Rip can be avoided. As conclusion, we see that for a viscous fluid those viscosity is proportional to ( -T ) n/ 2 , with the equation of state (54), the Big Rip and the Big Freeze can be eliminated for some values of n . However, in the case of the Sudden and the Big Brake, for large values of n , H s and τ , the viscous fluid influences the singularities but does not necessary avoid they.", "pages": [ 12, 13 ] }, { "title": "5 Conclusion", "content": "We considered in this work the modified teleparallel theory, known as f ( T ) theory, where T is the torsion scalar. In a specific way, the algebraic function f ( T ) is taken as the teleparallel term T plus the algebraic function g ( T ). The equation of motion of the theory is used and differential equation is established with the algebraic g ( T ). The expression (17) is assumed for the Hubble parameter where the parameter β plays an important role in specifying the type of singularity. Besides the parameter β which is sufficient for characterizing the singularities, the Big Rip and the Big Freeze, we need to introduce the parameter H s , which substitutes the constant C , for specifying the Sudden singularity and the Big Brake. Then, the differential equation is solved in each case and the corresponding algebraic function f ( T ) which may lead to each type of singularity is obtained. All this is done considering a fluid without the viscosity. In order to probe the possible avoidance of the singularities, we introduce the bulk viscosity ζ ( ρ ) in three ways. The first case is when the viscosity is constant and then we observe that in general, the viscosity is inefficient against to the singularities. In the second second where the viscosity is proportional to √ -T , we see that for small values of the parameter τ the singularities are robust again the viscosity, while for large values of the this parameter, just the Big Rip and the Big Freeze may be cured by the viscosity (the Sudden and the Big Brake remain robust in this case). In the third case where the viscosity is proportional to ( -T ) n/ 2 , we observe that independently of the values of τ , the Big Rip and the Big Freeze may be avoided for some values of the parameter n . However, the Sudden and the Big Brake could be cured only for large values of n , τ and H s .", "pages": [ 13, 14 ] }, { "title": "Acknowledgements", "content": "M.J.S.Houndjo thanks IMSP for the Hospitality during the elaboration of this first version of this work and CNPq/FAPES for financial support. We also thank Dr. Manuel E. Rodrigues and prof. S. D. Odintsov for useful suggestions.", "pages": [ 14 ] }, { "title": "References", "content": "B 703 , 223 (2011) [arXiv:1108.5908 [gr-qc]]; S. H. Chen, J. B. Dent, S. Dutta and E. N. Saridakis, Phys. Rev. D 83 , 023508 (2011) [arXiv:1008.1250 [astro-ph.CO]]; G. R. Bengochea, Phys. Lett. B 695 , 405 (2011) [arXiv:1008.3188 [astro-ph.CO]]; P. Wu and H. W. Yu, Eur. Phys. J. C 71 , 1552 (2011) [arXiv:1008.3669 [gr-qc]]; R. J. Yang, Europhys. Lett. 93 , 60001 (2011) [arXiv:1010.1376 [gr-qc]]; arXiv:1007.3571 [grqc]; J. B. Dent, S. Dutta and E. N. Saridakis, JCAP 1101 , 009 (2011) [arXiv:1010.2215 [astroph.CO]]; T. Wang, Phys. Rev. D 84 , 024042 (2011) [arXiv:1102.4410 [gr-qc]]; Y. Zhang, H. Li, Y. Gong and Z. H. Zhu, JCAP 1107 , 015 (2011) [arXiv:1103.0719 [astroph.CO]]; C. Deliduman and B. Yapiskan, arXiv:1103.2225 [gr-qc]; B. Li, T. P. Sotiriou and J. D. Barrow, Phys. Rev. D 83 , 104017 (2011) [arXiv:1103.2786 [astro-ph.CO]]. [30] S. Nojiri, S. D. Odintsov and S. Tsujikawa, Phys. Rev. D 71 , 063004 (2005).", "pages": [ 17 ] } ]

[ { "title": "Holger R. Dullin", "content": "School of Mathematics and Statistics, The University of Sydney Friday 28 th September, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "We construct an explicit reversible symplectic integrator for the planar 3-body problem with zero angular momentum. We start with a Hamiltonian of the planar 3-body problem that is globally regularised and fully symmetry reduced. This Hamiltonian is a sum of 10 polynomials each of which can be integrated exactly, and hence a symplectic integrator is constructed. The performance of the integrator is examined with three numerical examples: The figure eight, the pythagorean orbit, and a periodic collision orbit. Keywords: geometric integration explicit symplectic integration numerical integration 3-body problem symmetry reduction hamiltonian system regularisation", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "It is well known that the flow Ψ t H of a Hamiltonian H of the form H = T ( p ) + V ( q ) with conjugate variables q and p can be approximated by splitting it into the integrable flow Ψ t T of T ( p ) and the integrable flow Ψ t V of V ( q ) and observing that Ψ t H = Ψ t T · Ψ t V + O ( t 2 ), see, e.g. [4, 8, 14, 12]. Thus a first order explicit symplectic integrator is obtained, and higher order methods can be constructed along similar lines [26]. In an integrable and separable Hamiltonian system of the form H = H 1 ( q 1 , p 1 ) + H 2 ( q 2 , p 2 ) such splitting gives the exact identity Ψ t H = Ψ t H 1 · Ψ t H 2 . If instead the Hamiltonian is a product H = H 1 ( q 1 , p 1 ) H 2 ( q 2 , p 2 ) again the system is integrable with integrals H 1 and H 2 and the flow can be written as In the superscript tH i denotes multiplication of t by the (constant) value of H i . A monomial Hamiltonian is a special case that has the same structure. There are obvious generalisations to more degrees of freedom. Hence any polynomial Hamiltonian is a sum of integrable monomial Hamiltonians, and thus a splitting integrator can be constructed. Symplectic integration of polynomial Hamiltonians has been discussed in [20, 6, 4, 1, 19]. In a recent paper by [3] various methods for time step control in geometric integrators are constructed and discussed. These methods could be useful in order to implement variable time stepping on top of our method, see the discussion in section 3.4. In this paper we apply these methods to the polynomial Hamiltonian of the globally regularised and symmetry reduced 3-body problem at angular momentum zero. For a review of numerical and regularisation methods in the n -body problem we refer to [10]. It is well known that binary collisions in the 3-body problem can be regularised. Regularisation consists of a canonical transformations which essentially extracts a square root near collision, and of a scaling of time so that the approach to the collision is slowed down. The classical simultaneous regularisation of the (spatial) 3-body problem is due to [9]. This increases the dimension of phase space from 18 to 24. Instead we would like to decrease the dimension of phase space by using reduction at the same time as regularisation. The simultaneous regularisation of the planar 3-body problem is due to [13], and we use a version due to [25]. This is a symmetric simultaneous regularisation of the symmetry reduced planar 3-body problem and has the smallest possible 6-dimensional phase space. The resulting Hamiltonian is a polynomial of up to degree 6 in the canonical variables. A modern extension of these regularising transformations has recently been given by [16], however, their Hamiltonians are not polynomial but rational. Our paper applies the methods for construction of an explicit symplectic integrator to Waldvogel's Hamiltonian with angular momentum zero. We also describe how a similar integrator could be constructed for Heggie's Hamiltonian, which works for non-zero angular momentum and in the spatial problem.", "pages": [ 1, 2 ] }, { "title": "2 The 3-body Hamiltonian", "content": "The classical 3-body problem has long been studied, but still many open questions regarding its dynamics remain. For many questions, e.g. the study of relative periodic orbits, it is useful to reduce by translational and rotational symmetries, so that the absolute rotation of an orbit can be separated from shape dynamics in the centre of mass frame. Moreover, to study collision or near-collision orbits it is essential to perform (global) regularisation of the binary collisions. Following [25] we are going to do both. If the position and momentum of mass m j , for j = 1 , 2 , 3, are given by complex Cartesian coordinates X j and P j respectively, we can transform into symmetry-reduced coordinates such that where a j = | X l -X k | is the length of the triangle's side opposite to m j , φ j is the angle of that side in the original coordinate system (in the direction of m k to m l ), as illustrated in figure 1, and ( j, k, l ) represents cyclic permutations of (1 , 2 , 3). This reduction results in coordinates a j and φ = 1 3 ( φ 1 + φ 2 + φ 3 ), which represents the orientation angle of the triangle with respect to the original m choice of Cartesian coordinates, and corresponding canonical momenta p j and p φ . The Hamiltonian rewritten in these coordinates is independent of φ , so p φ is a constant of motion. Hamilton's equations for ( a j , φ, p j , p φ ) give the reduced dynamics, including a differential equation for φ which may be integrated along to be able to recover the unreduced position of the triangle. The globally regularising transformation, illustrated in figure 2, goes from symmetry-reduced to regularised coordinates, simultaneously regularising all the binary collisions. Define α j for j = 1 , 2 , 3 such that a j = α 2 k + α 2 l . In this way α 2 j is the distance from m j to the point where the incircle of the triangle touches the sides adjacent to m j . The space of coordinates ( a j ) is the space of all triangles, not accounting for orientation. Orientation is taken to be positive if, going clockwise around the triangle, the masses are encountered in a cyclic permutation of (1 , 2 , 3) or negative otherwise. The space of all possible oriented triangles is called the shape space, and the space of ( α j ) is a four-fold covering of this space, in which the sign of the product α 1 α 2 α 3 determines the orientation of the triangle. Thus the triangle formed by ( α 1 , α 2 , α 3 ) is the same as the ones formed by ( α 1 , -α 2 , -α 3 ), ( -α 1 , α 2 , -α 3 ) and ( -α 1 , -α 2 , α 3 ). Canonically conjugate momenta π j are introduced using a generating function. Finally the time scaling together with Poincar'e's trick to make this Hamiltonian yields the regularised and symmetry reduced polynomial Hamiltonian where H is the original Hamiltonian written in the new coordinates and h = H ( α 0 , π 0 ) is the energy corresponding to the initial conditions ( α 0 , π 0 ), so only those solutions for which K ≡ 0 are physically meaningful. The Hamiltonian of the zero-angular momentum 3-body problem in regularised coordinates is where in which where The sum in (3) (and any hereafter where the index of summation is unspecified) is over cyclic permutations of (1 , 2 , 3), so that ( j, k, l ) is replaced by (1 , 2 , 3), (2 , 3 , 1), and (3 , 1 , 2) in turn, and then the three corresponding terms are added together. When there is no summation the indices ( j, k, l ) take on the three possible cyclic permutations in turn, as, e.g., in the definition of A j and B k above. The new Hamiltonian is a polynomial in α and π , and thus Hamilton's equations of motion for this system can be integrated with an explicit symplectic integrator obtained by splitting into monomials. As we are going to show in the next section it is more efficient to split into certain polynomials whose flow can be exactly solved.", "pages": [ 3, 4, 5 ] }, { "title": "3 Construction of the Symplectic Integrator", "content": "glyph[negationslash] The basic building blocks of the integrator are the exact solutions for monomial Hamiltonians H mn = q m p n in one degree of freedom. The flow of this Hamiltonian for m = n is while for m = n it is For the Hamiltonian we are studying the cases that occur are n = m = 1, n = m = 2, and m = 3, n = 1. We also recall that if the Hamiltonian is a function of positions or momenta only (with any number of degrees of freedom) the flows are", "pages": [ 6 ] }, { "title": "3.1 Integrable Polynomial Hamiltonians", "content": "The basic building blocks just mentioned are now combined to form integrators for the terms that appear in the Hamiltonian K . The main observation is that if the Hamiltonian is a product of factors that depend on disjoint groups of degrees of freedom, then each factor is a constant of motion. Each of the factors in our case is either depending on momenta or positions only (denoted by T ( p ) or U ( q )) or it is a single monomial in one degree of freedom (denoted by H mn ) or a sum of monomials of disjoint degrees of freedom (denoted by G ). We now list the cases that are relevant in our case (recall that each of the factors depends on disjoint groups of degrees of freedom): where G is a Hamiltonian which is the sum of Hamiltonians depending on disjoint degrees of freedom G = H 1 ( q 1 , p 1 ) + H 2 ( q 2 , p 2 ) and thus ψ G = ψ H 1 · ψ H 2 . Note that all these formulas are exact, and that the order of composition is irrelevant since the flows commute and the individual factors are constants of motion.", "pages": [ 6 ] }, { "title": "3.2 Splitting", "content": "Let us now explain how to split K (2) into such terms. It is a polynomial Hamiltonian of degree 6 in α and π with 34 monomials. There are 13 monomials, dependent only on α , of degrees 6 and 4, which may be treated as a single stage. The remaining 21 terms may be grouped such that only 9 more stages are necessary to approximate the flow of the full Hamiltonian to first order in the time step in 10 stages. Let K = ∑ 9 i =0 H i , where we set M j = m k m l and N j = 1 m k + 1 m l . Then the splitting is where each subindexed function C i is a constant of motion in its associated Hamiltonian. There are clearly four groups in equation (8), which we shall enumerate 0: { 0 } , 1: { 1,2,3 } , 2: { 4,5,6 } and 3: { 7,8,9 } . H 0 depends on coordinates only, so can be integrated by (6). Group 1 can be integrated by (7b), group 2 can be integrated by (7a), and finally group 3 can be integrated by (7c) where G is a sum of H mm Hamiltonians.", "pages": [ 7 ] }, { "title": "3.3 Higher order methods", "content": "An important ingredient in constructing higher order reversible methods is the adjoint ( φ t ) ∗ of a method φ t which is defined to be ( φ -t ) -1 . If φ t = ψ t 1 · ψ t 2 · · · · · ψ t n and each ψ t i is self-adjoint, then the adjoint is obtained by reversing the order of composition. This follows from the definition of the adjoint: In our case the self-adjointness of the individual steps ψ t i follows from the fact that they are exact solution of Hamilton's equations. Channell & Neri [4] offer a basic derivation of a reversible, symplectic map that is accurate to second order in the time step. When the splitting is of the form H = T ( p ) + U ( q ) this leads to the symplectic leapfrog integrator, by composing symplectic Euler with its adjoint. This construction also applies to the more complicated case with a first order integrator composed of 10 self-adjoint maps as in our case. Given φ t = ψ t 1 · ψ t 2 · · · · · ψ t n as above a reversible second order method is found as [26] gives a general method by which one may obtain integrators of arbitrary even order, if only one has, to start with, a reversible even-order integrator φ t 2 such as the symplectic leapfrog-or, more generally, symplectic midpoint. One can compose φ t 2 to obtain a fourth order integrator φ t 4 , and compose this to obtain φ t 6 and so on. In general, given φ t 2 n , where we define z 0 = -2 1 / (2 n +1) 2 -2 1 / (2(2 n +1)) , z 1 = 1 2 -2 1 / (2 n +1) to adjust the step size of the lower order method. This method is easy to construct and implement, but quickly becomes unwieldy. When n = 2 (order 4), there are three evaluations of the second order method, but at orders 6 and 8 there are, respectively, nine and twentyseven. As noted by [26], there are better methods, and he gives coefficients for a sixth order method and several sets of coefficients for eighth order methods. The construction of higher order methods is discussed extensively in [8] and [14]. We will assess in section 4 which methods give good results for our problem comparing the methods whose coefficients are given in [8] and those constructed by [26].", "pages": [ 7, 8 ] }, { "title": "3.4 Regularisation and variable time stepping", "content": "There is a well known restriction on symplectic integration that such integrators must use a constant step size, or the benefits of these methods for large integration times are lost due to the introduction of new secular error terms. Various authors have discussed methods of achieving adaptive step size in symplectic integration that avoids this problem; for example, [15, 18, 2, 3]. In particular, [3] explore the use of Sundman and Poincar'e transformations and give a good overview of the problem. In general the Sundman transformation is non-symplectic, though with care the transformation can be made to respect geometric structure. In the their framework, the time scaling d t d τ = a 1 a 2 a 3 is called the monitor function . Our situation is special because the regularisation transformation consists of two intimately related steps. First there is the canonical extension of the transformation of coordinates from distances a j to their 'roots' α j (space regularisation), and second there is the time scaling (time regularisation). The time scaling up to a constant factor is achieved using the square of the Jacobian determinant of the transformation of the coordinates. Only the combination of the two achieves global regularisation. Treating the time transformation separately as a monitor function would mean to integrate singular equations, since the original equations are singular at collision, and they are still singular after the spatial regularisation alone. Slight modifications of the time scaling are possible, see the remark at the end of the next section. In order to achieve variable time stepping a monitor function could be used in the way described by [3] by integrating another equation on top of the regularisation (in space and time) we have done. This may be particularly useful when integrating orbits with large distances between the bodies.", "pages": [ 8, 9 ] }, { "title": "3.5 Finite time blowup", "content": "glyph[negationslash] It must be noted that the solution of the Hamiltonian H = q m p n given in (4) can (for n = m ) reach infinity in finite time. This occurs when This obviously makes step sizes comparable to this threshold risky when this form of solution is used in the integrator. This singularity could be reached if the denominator is negative and large during forward timesteps, or if the denominator is positive and large for 'backward' timesteps (as during the middle stage of Yoshida's trick). This possibility arises in equation (2) in the group 4 of the splitting (8), which have terms of the form α 3 j π 1 j . It may appear that this finite time blowup is an artefact of the integrator. However, after the time scaling the Hamiltonian K does have finite time blow up when particles escape to infinity. In this light it seems less unexpected that a stage of the corresponding symplectic integrator shows the same behaviour. [1] provides a means by which to avoid such singularities, by way of rewriting the polynomial in terms of sums of binomials in the coordinates and momenta and finding coefficients such that the two expressions are equal. We did apply this to the Hamiltonian H 31 in our problem, and found a way to replace this with a Cremona map. However, it turned out that the overall error of the method was worse than without this modification. Our method is more expensive, since it needs to compute rational powers, but this additional cost is worth it. A way to possibly avoid finite time blowup when the configuration of the system becomes large would be to consider a rational-rather than polynomialtime scaling function as in [16]. One could consider, for example, d t d τ = a 1 a 2 a 3 α 2 γ (recalling α 2 = ∑ α 2 j ), which tends to 0 for γ = 3 or to a j / 4 for γ = 2 as α j → ∞ . For negative energy, the only possible escape to infinity is of one single mass and a hard binary; in regularised coordinates this is exactly one coordinate tending to infinity while the other two remain bounded. Such a time scaling would inevitably require that the Hamiltonian be split differently, possibly with more stages and complexity. In principle the methods described in this paper apply as long as exact solutions can be found for the partial Hamiltonians. Unfortunately we have not been able to solve all of the resulting rational Hamiltonians.", "pages": [ 9, 10 ] }, { "title": "3.6 Other polynomial globally regularised Hamiltonians", "content": "Our main concern in this paper is the zero-angular momentum reduced and regularised planar 3-body problem, which has 3 degrees of freedom. Other well known globally regularised polynomial Hamiltonians are due to [24] for the planar 3-body problem and to [9] for the spatial 3-body problem. These Hamiltonians are not fully symmetry reduced and have 4 degrees of freedom (planar arbitrary angular momentum) and 12 degrees of freedom (spatial arbitrary angular momentum). Heggie's simultaneously regularised Hamiltonian for the spatial 3-body problem [9] in canonical variables Q ji and conjugate P ji , j = 1 , 2 , 3, i = 1 , . . . , 4 has the form where µ jk = m j m k / ( m j + m k ), and a j = ∑ 4 i =1 Q 2 ji , | p j | 2 = ∑ 4 i =1 P 2 ji . In addition P l = ( P l 1 , P l 2 , P l 3 , P l 4 ) T and A l is the KS-matrix [11] of the form The terms H 0 and H 4 , 5 , 6 are analogous to the previous ones. The terms in H -1 can be split into 9 terms of the form a j f k g l where the functions f k and g l only depend on the degrees of freedom k and l , respectively. These terms are somewhat similar to the Hamiltonians H 1 , 2 , 3 in Waldvogel's case. Thus the Hamiltonian can be split into 13 polynomials of degree up to 6 each of which is integrable. When setting Q ji and P ji with i = 3 , 4 equal to zero Heggie's Hamiltonian describes a planar problem. However, this still has 6 degrees of freedom. We can reduce the number of degrees of freedom to 4 by instead using Waldvogel's Hamiltonian [24, 7]. This Hamiltonian is a polynomial of degree 12 and can be split into 15 terms in a way similar to the two cases discussed above.", "pages": [ 10, 11 ] }, { "title": "4 Numerical examples", "content": "In this section we will show some numerical results achieved using our integrator in a selection of orbits ranging from far from collision to close encounters to a collision orbit. Figure 3 shows the energy error for various integration methods in a 'work-precision' diagram. The error is averaged over several different initial conditions integrated over a fixed time interval. The error is displayed as a function of the computational cost. The methods compared are the base method of order 2 1 , the integrators of [26] (4 3 , 6 9 , and 8 27 ) and other higher order symmetric compositions of symmetric methods of various authors, whose coefficients are given in [8], section V.3.2, also see the references therein. The subscript with each method's order indicates the number of second order substeps in the evaluation of a single time step, indicating the cost of each method, where the second order method is given the base cost of 1. A close look at the graph reveals that integrator 8 17 achieves the lowest error with a step size of about 0 . 0027, though it is a close call between any of the three best methods 8 15 , 8 17 and 10 35 . Reducing the step size further creates larger round-off errors. All of the following examples are calculated with the 8 17 integrator and step size 0 . 0027, unless otherwise mentioned. Energy error vs computational cost Consider the figure-8 choreography, discovered by [17], proved to exist by [5] and explored by [22, 21]. We choose initial conditions in regularised coordinates, with h = -1 and equal unit masses. In scaled time, the figure-8 has a period of 2 . 221813718; in physical time its period is 9 . 2371333. The trajectory in regularised coordinates is shown in figures 4a and b and the energy error over 25 orbits with large time steps given by the period divided by 200 is shown in figure 4c. Figure 5 shows the trajectory of this orbit in the 3-dimensional space ( α j ). Note that each crossing of a plane α j = 0 corresponds to a syzygy with m j in the middle of the configuration. Next we look at the Pythagorean orbit [23] for m 1 = 3, m 2 = 4, m 3 = 5, with initial conditions as given in [7], which, in regularised coordinates, are α τ τ (a) Regularised coordinates. (c) Energy error for 25 periods with 200 time steps per period. π τ (b) Regularised momenta. This orbit has a close encounter between masses 1 and 3 at around t = 15 . 8 in physical time (about τ = 1 . 52 in scaled time). Waldvogel's analysis regularises the system, albeit slightly differently, and his integration is not symplectic. The final motions of this orbit compare well with other studies; plotting the orbit in physical space produces results indistinguishable from [23], [7]. The regularisation of the 3-body problem allows our integrator to cope well when the distances between any two masses are small. The result of the time scaling is that the regularised system has a finite time blowup for any escape orbit. If one continues to integrate the Pythagorean orbit past τ = 8 . 105, the error in the energy grows exponentially and the results become inaccurate. α τ Finally, we show results in a periodic collision orbit, discovered during a search for periodic orbits in the reduced space, with initial conditions equal unit masses and h = -1. This orbit has two collisions between masses 1 and 2, as can be seen by α 1 = α 2 = 0 at τ = 1 . 9362 and τ = 5 . 062 in figure 7a. This orbit is periodic in full phase space and is shown in figures 7, 8 and 9. Its scaled period is 6 . 2520511, corresponding to a physical period of 29 . 6117209. Note in figure 9 that the collisions happen on the α 3 -axis, when α 1 = α 2 = 0. Because Hamiltonian systems are time-reversible, it is desirable to have an integrator with the same property. The second order map φ t 2 is constructed as such, and Yoshida's formula for higher order integrators constructs them π τ α τ τ Error between forward and backward integrations to be reversible as well. That means that φ t 2 n φ -t 2 n = Id up to roundoff error. Figures 4d, 6d and 7d show how closely this integrator returns to its initial condition after a certain number of time steps in one direction, followed by the same number of iterations with a negative time step. By this measure, the integrator has the most trouble with Pythagorean orbit, which clearly shows signs that it exists within a chaotic region of phase space by the exponential growth of error. However; energy is well preserved in this and the other cases. π τ", "pages": [ 11, 12, 13, 15, 16 ] }, { "title": "5 Conclusion", "content": "We have constructed a symplectic integrator for the reduced and regularised planar 3-body problem at zero angular momentum. The method works well, but it is not very efficient, because each (first order) time step involves the computation of 10 individual maps. Our interests is the computation of relative periodic orbits including collision orbits, and for this task the method is appropriate. The detailed results about relative periodic orbits and their geometric phase will be reported in a forthcoming paper.", "pages": [ 19 ] }, { "title": "A Integrator stages", "content": "Subsection 3.1 described how to integrate a monomial Hamiltonian, and subsection 3.2 described the splitting of equation (2) into a minimal number of solvable parts and those solutions. Here we use those solutions to build an explicit first order symplectic composition method for (2). Let the timestep be ∆ τ , let µ j = ( m k + m l ), let the values of the system before and after one timestep respectively be z 0 = ( α 1 , 0 , . . . , π 3 , 0 ) T and z 1 = ( α 1 , 1 , . . . , π 3 , 1 ) T and intermediate steps be ξ i = ( α 1 ,.i -1 , . . . , π 3 ,.i -1 ) T . Now", "pages": [ 19 ] } ]

[ { "title": "ABSTRACT", "content": "Light curves of solar flares in the impulsive phase are complex in general, which is expected given complexities of the flare environment in the magnetic field dominant corona. With GOES observations, we however find that there are a subset of flares, whose impulsive phases are dominated by a period of exponential growth of the emission measure. Flares occurring from Jan. 1999 to Dec. 2002 are analyzed, and results from observations made with both GOES 8 and 10 satellites are compared to estimate instrumental uncertainties. The frequency distribution of the mean temperature during this exponential growth phase has a normal distribution. Most flares within the 1 σ range of this temperature distribution belong to GOES class B or C with the frequency distribution of the peak flux of the GOES low-energy channel following a log-normal distribution. The frequency distribution of the growth rate and the duration of the exponential growth phase also follow a log-normal distribution with the duration covering a range from half a minute to about half an hour. As expected, the growth time is correlated with the decay time of the soft X-ray flux. We also find that the growth rate of the emission measure is strongly anti-correlated with the duration of the exponential growth phase and increases with the mean temperature. The implications of these results on the study of energy release in solar flares are discussed at the end. Subject headings: Acceleration of particles - Plasmas - Radiation mechanisms: thermal - Sun: flares - Sun: X-rays", "pages": [ 1 ] }, { "title": "Exponential Growth of the Emission Measure in the Impulsive Phase Derived from X-ray Observations of Solar Flares", "content": "Feiran Han 1 , and Siming Liu 1", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Most observational studies of solar flares focus on detailed analysis of individual events (e.g., Masuda et al. 1994; Lin et al. 2003; Hannah et al. 2008; Raftery et al. 2009; Loncope et al. 2010; Bak-Steslicka et al. 2011). Statistical studies usually treat the flare population as a whole to extract a few quantities and analyze their occurrence frequency distribution (e.g., Aschwanden 1994; Veronig et al. 2002a,b; Su et al. 2006). Although these two approaches are complementary to each other, relations between results of these two studies remain obscure. For example, it is not clear how specific physical processes revealed from studies of individual flares may lead to power-law distributions of the flare occurrence rate with respect to many of their observed characteristics. It is possible that statistical properties of flares are mostly determined by the environment and may not depend on the detailed physical processes (Lu & Hamilton 1991). However, even for flares with similar peak value of the soft X-ray (SXR) flux, an important quantity characterizing the flare amplitude, their appearance can be drastically different, which reflects the intrinsic complexity of flares as a macroscopic phenomena with an enormous degree of freedom. The dominant physical process in one flare may be distinct from that of the other. To better understand the flare phenomena, it is therefore necessary to classify flares based on some of the observed prominent characteristics and explore their physical origin accordingly. A classification of large-scale coronal EIT waves has recently resolved some related controversies (Warmuth & Mann 2011). One of the key aspects of solar flare study is to explore the energy release processes, and it is generally accepted that the impulsive phase dominates the overall energy release (Hudson et al. 2011). However, in contrast to the relatively smooth decay of X-ray fluxes in the gradual phase of most flares, which is associated with coronal loops and has been studied extensively and better understood (Rosner et al. 1978; Antiochos & Sturrock 1978; Serio et al. 1991; Cargill et al. 1995; Kilmchuk et al. 2008), the X-ray light curves in the impulsive phase are generally complex and there appears to be a variety of physical processes involved. Many of the observed complexities of flares originate from the complex magnetic field structure carrying the flaring plasma. To better understand the basic physical processes related to the energy release, one may focus on studying flares with relatively simple structure, in particular those associated with single loops. Raftery et al. (2009) carried out a detailed analysis of a flaring loop, and we also notice that the impulsive phase of this flare is dominated by a period of exponential growth in both the SXR fluxes and the derived emission measure (EM) (Left panel of Fig. 4). These relatively simple behaviors of flaring loops may reflect some elementary processes in the flare energy release (Grigis & Benz 2004; Liu et al. 2010). Motivated by this observation, in this paper we analyze GOES observations from Jan. 1999 to Dec. 2002 to identify flares with the impulsive phase dominated by a period of exponential growth of the EM. These flares are a subset of flares. Detailed studies of them may help to reveal the physics of energy release in the impulsive phase in general. In § 2, we present the analysis of GOES data and the flare selection criteria. The results are shown in § 3. These results are discussed in § 4, where we also draw the conclusions.", "pages": [ 1, 2, 3 ] }, { "title": "2. Data Analysis", "content": "Background Selection and Peak Time: Both GOES 8 and 10 satellites cover the previous maximum of solar activity. We focus on a 4 year period of the activity peak from Jan. 1999 to Dec. 2002. RHESSI was launched into orbit in Feb. 2002. Some of these flares were observed by RHESSI as well (Lin et al. 2003). To derive the temperature and EM of the flaring plasma with the GOES data, it is essential to subtract the pre-flare background fluxes properly (Bornmann 1990). GOES satellites measure SXR fluxes from the Sun in two wave bands - 1-8 ˚ A and 0.5-4 ˚ A - with a cadence of 3 seconds (Garcia 1994). We make use of the GOES flare list from http://umbra.nascom.nasa.gov/sdb/ngdc/xray events/, where the flare onset time, the peak time of the flux in the lower energy channel, and the end time are given for 10511 flares. The difference between the former two may be called the duration of the flare rise phase: t r . We extract data for these flares from GOES 8 and 10 observations. For each flare, we extend the range of data analysis both before the flare onset time and after the flare end time by t r 1 . For some flares, the peak time in the flare list does not correspond to the maximum of the flux in the low energy channel between the onset and end time. We redefine the flare peak time as the time when the flux reaches its maximum value between the onset and end time. The background fluxes are chosen as that of a period before the flare peak time with a relatively low and constant flux level and are selected independently for the two energy channels 2 . In the following, we will mostly use results from the low-energy channel, where the background flux is high and the signal relatively weak, to define the flare characteristics. The left panels of Figure 1 show the frequency distribution of the background subtracted 2 In practice, we fit the light curve with a set of line segments. The error is assumed to be the same as the measured flux and the critical value of the χ 2 is set at 0.001. The fit starts from the first three data points and the corresponding χ 2 is calculated. If the χ 2 is less than the critical value, we include one more data point following this period for a new linear fit. This process is repeated. A new segment starts whenever the χ 2 of the current segment reaches this critical value. We then calculate the mean value of the flux for each line segment. For the two segments with the lowest mean fluxes before the flare peak time, we set the flux of the segment with a lower gradient as the background flux. s e r a l F f o n o i t u b i r t s i D s e r a l F f o n o i t u b i r t s i D 1000. 100. 10. 1. 1000. 100. 10. 1. 10 10 /Minus /Minus 7 7 10 10 /Minus 6 10 Peak Flux /LParen1 W /DotMath (a) 10 /Minus 6 Peak Flux /LParen1 W (c) 10 /Minus 6 Background Flux /LParen1 W (d) peak flux of all flares. The shaded histograms are for flares with a low level of pre-flare background flux ( ≤ 10 -6 W M -2 ). The other histograms are for all other flares with a high level of pre-flare background. The distributions of flares with low and high pre-flare background flux agree with each other at high peak fluxes, which is consistent with the prediction of the self-organized criticality model of Lu & Hamilton (1991). The instrumental bias is important at low peak fluxes, and the difference between GOES 8 and 10 observations is obvious. The right panels of Figure 1 show the correlation between the pre-flare background flux and the flux at the peak time. For a flare to be identified in the data, the one-minute averaged peak flux needs to exceed the pre-flare background flux by at least 40%, which explains why /Minus 5 /Minus 5 M M /DotMath /Minus /Minus 2 2 /RParen1 /RParen1 10 10 /Minus /Minus 4 10 GOES10 4 10 GOES8 /Minus /Minus 3 3 10 10 10 /Minus /Minus 5 6 /Minus 7 /RParen1 2 /Minus M /DotMath W /LParen1 x u l F 10 /Minus 7 /DotMath M /Minus GOES10 2 /RParen1 10 /Minus 5 there are no flares in the low-right side 3 . The seeming correlation between these two fluxes therefore is mostly caused by this flare identification procedure. The distribution in the left panels show that the occurrence chance of flares of a given amplitude does not depend on the pre-flare background flux. Actually the pre-flare background flux is mostly caused by decay of earlier flares (Aschwanden 1994). These results show that most flares studied here are independent from each other. The obvious horizontal strip for GOES 8 and horizontal and vertical strips for GOES 10 observations are caused by the digitalization process of the instrument (Garcia 1994). Onset Time: With the background fluxes selected, we redefine the onset time to better quantify the rise phase. The background subtracted flux (in the low energy channel) needs to exceed 2 . 1 × 10 -8 W m -2 to obtain reliable temperature and EM (Garcia 1994). For a background subtract flux below this critical value, the GOES software gives a default value of 4 MK and 0 . 01 × 10 49 cm -3 for the temperature and EM, respectively (See Fig. 4) 4 . The background subtracted flux therefore needs to exceed this critical value after the flare onset. Similarly, we require that the background subtracted flux in the high energy channel should be greater than 1 . 0 × 10 -10 W m -2 5 . We use the Coronal emission model version 6.0.1 to derive the temperature and EM. In the early rise phase, the signal may be weak so that the obtained EM and temperature can fluctuate significantly. We require that after the flare onset, the difference of the logarithm of the EM between two neighboring data points should not exceed 15% of the difference between the maximum and minimum values of the logarithm of the EM of the flare data range. Segments of Exponential Growth of the EM: We are mostly interested in the rise phase. To identify periods of exponential growth, we fit the time variation of the logarithm of the EM with a set of line segments. Specifically, starting from the peak time of the SXR flux, we do a linear fit to the logarithm of the EM of a period ending at the peak time and calculate the corresponding reduced χ 2 . The error has been assumed to be 1. We adjust the duration of this period until the reduced χ 2 reaches a value just below a prior chosen critical value, which gives a segment of approximately exponential growth phase right before the peak time of the SXR flux. Following a similar procedure, we identify the next segment of exponential growth before the first segment and all other segments before the peak time. In this study, this critical value of the reduced χ 2 is taken as 9 . 3 × 10 -4 , and we exclude flares, whose rise phase can be fitted with a single line segment. Such flares are usually weak and have a short rise phase, the corresponding signals are not reliable. Instrumental and Software Effects: With the background flux and the flare onset time determined, we can show all flares on the parameter space of the background flux and the background subtracted flux at the flare onset. Figures 2(a) and 2(b) show the results for observations with GOES 8 and 10 satellites, respectively. The upper panels show the results with the GOES default software. There are 10198 and 10230 flares for GOES 8 and 10 observations, respectively. Flares with poor signals and a few outliers outside the range of parameter space shown in these figures have been excluded. It is evident that there are some artifacts caused by either instrumental or software effects. When the background flux exceeds 1 . 5 × 10 -6 W m -2 , the default GOES software sets a high threshold of 5 × 10 -7 W m -2 for calculation of the temperature and EM 6 , which causes the sharp cut at these background and background subtracted flux levels. Not surprisingly there is a sharp cut at the critical value of 2 . 1 × 10 -8 Wm -2 , below which the temperature and EM are set to the default values. There are a few outliers, some of which correspond to data gaps. The few outliers in the low-right corner correspond to flares, whose difference in the logarithm of the EM between two neighboring points are always less than 15% of the difference between the maximum and minimum values of that of the flare data range. The flare onset is therefore triggered by the requirement that the background subtracted flux exceeds the critical value of 2 . 1 × 10 -8 Wm -2 . The lower panels are obtained by removing the threshold at 5 × 10 -7 W m -2 in the software. There are 10251 and 10275 flares from the GOES 8 and 10 observations, respectively. It is evident that after removing some artifact caused by the software, the instruments have at least two prominent states of response in the low energy channel, which are divided roughly by the background flux level. Moreover, when the background flux is high, the GOES 10 observations show that flares distribute in a few strips, which is likely caused by the digitalization process. A similar plot for the high energy channel, however, shows a more or less continuous distribution (See lower panels of Fig. 3). The two states of response therefore only exist for the low energy channel. Flare Selection: We identify simple flares, whose rise phase is dominated by an exponential growth segment of the EM with the following criteria: 1) Since the duration of the exponential growth phase is an important quantity to extract, we focus on flares with the background flux in the low energy channel not exceeding 1 × 10 -6 Wm -2 , which is indicated by the dashed line in the lower panels of Figure 2. When the background flux is high, the signal in the early rise phase may be too low to give reliable temperature and EM measurement so that the observed duration of the dominant segment of exponential growth in the EM becomes shorter. This constraint also addresses the instrumental effects discussed above. 2) The duration of the longest line segment of the logarithm of the EM must exceed 30 seconds and longer than the half length of the rise phase from the flare onset time to the flare peak time. The former criterium ensures that the period of exponential growth is prominent, and the latter ensures its dominance in the rise phase. 3) The increase of the logarithm of the EM during this line segment of exponential growth must exceed 40% of the difference of the maximum to minimum value of the logarithm of the EM during the rise phase. These two criteria define the dominance of the exponential growth phase. 4) To ensure the simplicity of the selected flares, we also have a linear fit (with an error of 1) to the logarithm of the background subtracted flux in the low-energy channel between the flare peak time and the time when the flux decreases to one half of the peak flux in the decay phase. The reduced χ 2 of this linear fit must be less than 10 -4 for a flare to be selected. Flares with higher values of the reduced χ 2 have more complicated decay phase and are likely associated with multiple loops. Figure 4 shows two selected flares. The bottom panels show RHESSI light curves of these two flares. It is unfortunate that RHESSI did not cover the early rise phase of both flares. The flare in the left side is studied in detail by Raftery et al. (2009), who showed that the flare is associated with a loop structure with prominent looptop and footpoint sources seen at different UV and EUV wavebands. The EM grows exponentially through the major part of the SXR rise phase. The temperature derived from GOES observation is nearly a constant in the rise phase. There is no evidence of prominent impulsive hard X-ray emission near the flare peak time. The flare in the right side panel is very similar to the one in the left side except that there is evidence of impulsive emission above 25 keV near the flare peak time. Figure 5 shows two flares with slightly complicated light curves especially in the high energy channel during the dominant period of exponential growth in the EM. These complicated light curves lead to complicated behaviors in the inferred temperature evolution. These complexities may be attributed to fluctuations in the dominant process of exponential growth in the EM, and therefore these flares are considered to be similar to those in Figure 4. However, the dominant period of exponential growth in the EM of these two flares extend to the flare peak time, which is different from the two flares in Figure 4, where the dominant exponential growth period ends before the flare peak time. From these analyses and for each selected flare, one can obtain the duration of the dominant exponential growth phase in the rise phase, the growth rate of the EM, the mean plasma temperature of this dominant exponential growth phase, the peak flux in the low energy channel, and the decay rate of the SXR flux in the low energy channel. In the following section, we will present the statistical properties of these quantities and their correlations.", "pages": [ 3, 4, 5, 6, 7, 8, 9 ] }, { "title": "3. Results", "content": "With the above flare selection criteria, a total number of 620 and 522 flares are selected from GOES 8 and 10 observations, respectively. There are 316 flares selected from both satellite data. Figure 6 shows the occurrence frequency distribution of the mean temperature of the dominant exponential phase of the EM of these 316 flares. These distributions can be fitted with a normal distribution. The temperature measurement in the tails of this distribution may not be reliable. In the following, we will focus on flares within 1 σ range of these Gaussian distributions. There are 210 and 209 flares for GOES 8 and 10 observations, respectively, and 192 of them are identified from both satellite data. Table 3 lists the characteristics of these 192 flares derived from GOES 8 observations. Here the decay time is defined as the time it takes for the background subtracted SXR flux in the low energy channel to decrease by a factor of 2 from the peak value divided by ln(2). There are good agreements between GOES 8 and 10 observations. We therefore will not list the characteristics of these flares derived with GOES 10 observations. Figure 7 shows the correlation between the decay time of the SXR flux t d and the rise time of the dominant exponential growth period of the EM t e defined as the time required for the EM to grow by a factor of e /similarequal 2 . 72. For most flares, especially those with long decay time, the rise time is shorter than the decay time. Only for a few very short flares, the decay time is shorter than the rise time. The rise time increases slowly with the decay time. A linear fit to the correlation of the logarithm of these two timescales gives t e = 4 . 0( t d /s ) 0 . 60 s and t e = 3 . 5( t d /s ) 0 . 61 s for GOES 8 and 10 observations, respectively. The most unexpected finding of this study is a strong anti-correlation between the growth rate Gr = t -1 e and the duration of the dominant exponential growth period Du as shown in the left panels of Figure 8. The result also indicates that Gr increases with the increase of the mean temperature T . A linear fit to the correlation among log( Gr ), log( Du ) and log( T ) gives and for GOES 8 and 10 observations, respectively. Such an anti-correlation suggests that the EM stop to increase exponentially after reaching certain level. Indeed, the occurrence frequency distribution of the peak flux of the selected flares shows a relatively narrow log-normal distribution 7 as shown in the right panels of Figure 8. Since the temperature covers a narrow range, the SXR peak flux gives a rough measurement of the EM at the peak time. The observed anti-correlation between Gr and Du therefore is consistent with the relatively narrow distribution of the SXR peak flux. All of these selected flares belong to GOES class B or C. Big flares are likely more complex and therefore have less chance to meet our selection criteria. However, there is a slight excess relative to the log-normal distribution at high values of the peak flux. The correlation between Gr , Du and T is much weaker. Based on the fitting result of their correlation above, Figure 9 shows the correlation between Du 0 . 7 Gr and T . Although Du 0 . 7 Gr tends to increase with the increase of T , the spread of the correlation is big. Therefore quantitative dependence of the results on T may not be trust worthy.", "pages": [ 10, 11, 18, 19, 20 ] }, { "title": "4. Discussion and Conclusions", "content": "To uncover the dominant physical processes in flaring loops and give more quantitative modeling, we have obtained a sample of flares with relatively simple SXR light curves from GOES observations. The complexity of the flare phenomena caused by the complex coronal environment is partially suppressed via our selection of flares with relatively simple time evolution. Specifically, we have focused on a class of flares whose SXR rise phase is dominated by a period of exponential growth of the EM. Detailed multi-wavelength studies show some of these flares are associated with single loops. The rise time ranges from 30 s to more than 10 minutes suggesting a (magneto-) hydrodynamical process. There are two possible mechanisms that can lead to a period of exponential growth of the EM. If the loop structure is relatively simple and stable, the increase of the EM has to be caused by the evaporation of plasmas from the chromosphere. The exponential growth of the EM implies exponential growth of the thermal energy and therefore a heating rate proportional to the thermal energy density. The latter suggests a feedback of the heated plasma on the energy dissipation processes. Since it is commonly accepted that the flare energy release happens in the corona, the evaporation has to be driven by energy fluxes from the corona to the chromosphere. The fact that the heating rate is proportional to the thermal energy density implies a saturated energy flux from the corona to the footpoints. Such a saturated energy flux may be caused by the saturated conduction in a low density plasma caused by the non-local transport of energetic particles in the loop (Jiang et al. 2006; Battaglia et al. 2009). In such a scenario, we would expect strong emission from the footpoints. If the topological structure of the magnetic fields in the loop is complex, for example, the magnetic field lines may be twisted and braided (Wilmot-Smith et al. 2010), the flare may be associated with a filament structure (Liu & Alexander 2009). The exponential growth of the EM can be caused by an exponential growth of the volume filling factor of the heated plasma in the filament. Strong evaporation from the chromosphere is not necessary in such a case. More detailed multi-wavelength studies of individual flare are necessary to distinguish the two scenarios. From the smooth distribution of the characteristics of the selected flares, the two scenarios are not distinguishable from the SXR light curves alone. We find a strong anti-correlation between the growth rate of the EM and the duration of the dominant exponential growth period, which suggests that the exponential growth phase ends when the EM reaches certain level. The ending of the exponential growth phase implies a new phase of energy release in the impulsive phase. According to the two mechanisms proposed above, the ending can be caused by either a high density of evaporated plasmas or the volume filling factor reaching a saturation level. Although there is no evident reason why the EM stops to grow exponentially once reaching certain level, the observed anti-correlation is consistent with the relative narrow log-normal distribution of the peak flux of the selected flares. Most of the selected flares belong to GOES B with a small fraction belonging to GOES C. The relatively small amplitude of these elected flares can be partially attributed to more complexity of bigger flares. The peak flux of flares in general follows a power-law distribution (Veronig et al. 2002a). The distribution of the peak flux of the selected flares, the duration of the exponential growth period, and the growth rate, however, follow a relatively narrow log-normal distribution. While a power-law distribution implies a lack of characteristic scales in the system. Our relatively narrow distribution of selected flares suggests that they may represent a particular class of flares with a characteristic peak flux of ∼ 4 × 10 -7 WM -2 . The selected flares have relative simple light curves and are likely associated with single loops. If these results are further confirmed with a larger flare sample, we can describe flares in general as a set of flaring loops. The power-law distribution of the flare peak flux is mostly caused by dramatic variation of the number of loops in different flares. For an X class flare, thousands of loops should be activated. Therefore the flare study may be separated into two aspects: 1) physical processes in a flaring loop; 2) the topological structure of the flare region that determines the number of loops to be activated during a flare. The flares selected here show relatively gradual evolution in general and the impulsive hard X-ray (HXR) emission is relatively weak, similar to the slow long-duration events studied by Bak-Steslicka et al. (2011). The more gradual evolution of the SXR may be intimately connected to the lack of impulsive HXR emission. Earlier studies by Su et al. (2006) suggest that the impulsive HXR emission is better correlated with the SXR growth rate than with the SXR flux. Since the emission is dominated by the gradual emission component in the rise phase, the process of particle acceleration may be unimportant. Flares with prominent particle acceleration may correspond to a class of events distinct from the flares selected here. Most observed characteristics of the selected flares should be explained in the context of magneto-hydrodynamic evolution of flaring loops (Wilmot-Smith et al. 2010). We thank Youping Li, Hugh Hudson, Lyndsay Fletcher, and Peter J. Cargill for helpful discussions. This work is supported by the National Natural Science Foundation of China via grants 11143007 & 11173064.", "pages": [ 20, 21, 22, 23 ] }, { "title": "REFERENCES", "content": "Antiochos, S. K., & Sturrock, P. A. 1978, ApJ, 220, 1137 Aschwanden, M. J. 1994, Sol.Phys. 152, 53 Bak-Steslicka, U., Mrozek, T., & Kolomanski, S. 2011, Sol.Phys., 271, 75 Battaglia, M., Fletcher, L., & Benz, A. O. 2009, A&A, 498, 891 Bornmann, P. L. 1990, ApJ, 356, 733 Cargill, P. J., Mariska, J. T., & Antiochos, S. K. 1995, ApJ, 439, 1034 Garcia, H. A. 1994, Sol. Phys., 154, 275 Grigis, P. C., & Benz, A. O. 2004, A&A, 426, 1093 Hannah, I. G., Krucker, S., Hudson, H. S., Christe, S., & Lin, P. R. 2008, A&A, 481, L45 Hudson, H., S. 2011, Spa. Sci. 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[ { "title": "The Stability of a Shearing Viscous Star with Electromagnetic Field", "content": "M. Sharif 1 ∗ and M. Azam 1 , 2 † 1 Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore-54590, Pakistan. 2 Division of Science and Technology, University of Education,", "pages": [ 1 ] }, { "title": "Abstract", "content": "We analyze the role of electromagnetic field for the stability of shearing viscous star with spherical symmetry. Matching conditions are given for the interior and exterior metrics. We use perturbation scheme to construct the collapse equation. The range of instability is explored in Newtonian and post Newtonian (pN) limits. We conclude that the electromagnetic field diminishes the effects of shearing viscosity in the instability range and makes the system more unstable at both Newtonian and post Newtonian approximations. Keywords: PACS: 04.20.-q; 04.25.Nx; 04.40.Dg; 04.40.Nr. Gravitational collapse; Electromagnetic field; Instability.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Charged self-gravitating objects have the tendency of undergoing many phases during gravitational collapse resulting a charged black holes or naked singularities [1]. The stability of these exact solutions is an interesting subject under the perturbation scheme. It is believed that these stars with huge charge cannot be stable [2, 3]. However, the electric charge has great relevance during the structure formation and evolution of astrophysical objects [4]-[11]. A stellar model may be stable in one phase and later becomes unstable in another phase. The stability of this model is subjected against any disturbance. Dynamical instability of self-gravitating objects is interrelated with structure as well as evolution of astrophysical objects. In this scenario, Chandrasekhar [12] investigated the problem of dynamical instability for the isotropic perfect fluid of a pulsating system and found the instability range in terms of adiabatic index Γ < 4 3 . It is well discussed in literature that the instability range of the fluid would be decreased or increased through different physical properties of the fluid. In this context, the dynamical instability for adiabatic, non-adiabatic, anisotropic and shearing viscous fluids have been explored [13]-[17]. Chan [18] found that both pressure anisotropy and effective adiabatic index are increased by the shearing viscosity in a collapsing radiating star. Horvat et al. [19] have used the quasi-local equation of state [20, 21] to explore the stability of anisotropic stars under radial perturbations. Sharif and Kausar [22] investigated stability of expansion-free fluid in f ( R ) gravity and found that stability of the fluid is constrained by energy density inhomogeneity, pressure anisotropy and f ( R ) model. The study of charged self-gravitating objects in the context of coupled Einstein-Maxwell field equations leads to the evolution of black hole [23][25]. The physical aspects of the electromagnetic field has a significant role in general relativity. Stettner [26] investigated the stability of a pulsating sphere with a constant surface charge. Glazer [27] generalized this result by taking an arbitrary distribution of charge and found that the Bonner's charged dust model is dynamically unstable. The stability limit for charged spheres has been proposed by many people [28]-[32] starting from the Buchdahl [33] work for neutral spheres. Recently, we have investigated the problem of dynamical instability of cylindrical and spherical systems with electromagnetic field at Newtonian and pN regimes [34]. Here we explore the role of electromagnetic field in the stability of collapsing fluid undergoing dissipation with shearing viscosity. Darmois matching conditions [35] are formulated for the continuity of interior general spherically symmetric solution to the exterior vacuum Reissner-Nordstro msolution. The paper is planned as: In the next section, we discuss some basic properties of the viscous fluid, Einstein-Maxwell equations and junction conditions. Sec-ti n 3 provides the perturbation scheme to form the collapse equation. In section 4 , we explore the collapse equation at Newtonian and pN regimes. Finally, we discuss our conclusion in section 5 .", "pages": [ 1, 2, 3 ] }, { "title": "2 Fluid Distribution, Field Equations and Junction Conditions", "content": "We consider a timelike three-space spherical surface, Σ, which separates the 4 D geometries into two regions interior M -and exterior M + . The M -is given by the general spherically symmetric spacetime in the comoving coordinates For the exterior metric, we take Reissner-Nordstro m metric describing the radiation field around a charged spherically symmetric source of the gravitational field where M and Q are the total mass and charge respectively. The fluid under consideration is locally dissipative in the form of shearing viscosity and the interior energy-momentum tensor of charged dissipative fluid is given by where ρ , ˆ p , ξ and u α are the energy density, isotropic pressure, coefficient of bulk viscosity and four-velocity associated with the fluid, respectively, F αβ is the electromagnetic field tensor. We can write the above equation with p = ˆ p -ξ Θ as The four velocity in the comoving coordinates is Here η > 0 is the coefficient of shearing viscosity, while the shear tensor σ αβ is defined as where a α = u α ; β u β is the four acceleration and Θ = u α ; α is the expansion scalar. The corresponding non-zero components turns out where dot and prime represent differentiation with respect to t and r , respectively. The non-vanishing components for the shear tensor are where", "pages": [ 3, 4 ] }, { "title": "2.1 The Einstein-Maxwell Field Equations", "content": "In four-vector formalism, the Maxwell equations are where φ α and J α are the four potential and current density vector, respectively. We consider the charge to be at rest resulting no magnetic field present in this local coordinate system. Thus φ α and J α can be written as follows where ζ ( t, r ) and Φ( t, r ) describe the charge density and the scalar potential, respectively. From Eqs.(10) and (11), the only non-vanishing component of electromagnetic field tensor is The corresponding Maxwell equations are Solving the above equations, it follows that where q ( r ) is the total amount of charge from center to the boundary surface of the star The electric field intensity is the charge per unit surface area of the sphere Using Eqs.(1) and (4), we have non-vanishing components of the EinsteinMaxwell field equations for (1) yields The Misner and Sharp [29] mass function m ( t, r ) with charge is given by The conservation equation, ( T -αβ ) ; β = 0, yields", "pages": [ 4, 5, 6 ] }, { "title": "2.2 Junction Conditions", "content": "We connect M -and M + metrics by considering the Darmois junction conditions. For the smooth matching of these geometries, it is required that the boundary is continuous and smooth. Thus the continuity of first fundamental form of the metrics provides, i.e., and the continuity of second fundamental form of the extrinsic curvature gives Considering the interior and exterior spacetimes with Eq.(25), we have From Eq.(26), the non-null components of the extrinsic curvature turn out to be Using Eqs.(27)-(31) and the field equations, we obtain where q ( r ) = Q has been used. The above equation shows that across the boundary, Σ, the masses of interior and exterior spacetimes are matched as well as momentum flux is conserved.", "pages": [ 6, 7 ] }, { "title": "3 Perturbation Scheme and Collapse Equation", "content": "Here we construct the collapse equation. For this purpose, we perturb the field equations, dynamical equations and the mass function upto first order in ε by using the perturbation scheme [22, 34]. We assume that initially all the physical functions and the metric coefficients depend on r , i.e., the fluid is unperturbed. Afterwards, all these quantities depend on time coordinate, which are given by where 0 < ε /lessmuch 1. For ¯ c = 0, we have shearfree metric. Using this scheme, the static configuration of Eqs.(18)-(21) are written as The corresponding perturbed quantities upto first order in ε with Eqs.(34)(40) become The static configuration for the dynamical equation (23) is trivially satisfied, while Eq.(24) implies that which yields The perturbed part of Eq.(23) becomes which on integration leads to For the second dynamical equation, we have The unperturbed and perturbed configurations of electromagnetic field are given by This shows that e ( r ) depends upon the static configuration of electromagnetic field. Similarly, we can write the static and perturbed configuration for Eq.(22) as From Eq.(55), it follows that The perturbed configuration for the shear scalar is written as The junction condition (33) with Eqs.(39) and (40) leads to Substituting the above relations in Eq.(47), we get temporal equation where The general solution of Eq.(60) yields where c 1 and c 2 are arbitrary constants. This provides two independent solutions. Here we take ψ Σ to be positive for physically meaningful result, i.e., when the system is in static position, it starts collapsing at t = -∞ when T ( -∞ ) = 0 and goes on collapsing diminishing its areal radius with the increase of t . The corresponding solution is found for c 2 = 0, also, we choose c 1 = -1, hence Next, we are interested to find instability range of the collapsing fluid in terms of adiabatic index Γ. Chan et al. [16] found a relationship between ¯ p and ¯ ρ with an equation of state of Harrison-Wheeler type [36] for the static configuration as where Γ describes the change in pressure for a given change in density (taken to be constant in the whole fluid distribution). The above equation with Eq.(52) leads to Using the above equation and (59) in Eq.(47), it follows that We can write the collapse equation by substituting Eqs.(52), (58), (65) and (66) in the perturbed configuration of the second dynamical equation as follows", "pages": [ 7, 8, 9, 10, 11 ] }, { "title": "4 Dynamical Instability of Charged Viscous Perturbation", "content": "This section deals with the dynamical instability of charged viscous fluid in the frame work of Newtonian and pN regimes. We see from Eq.(36) that the effects of shear explicitly appear only on metric function R ( t, r ). Using this fact, we can split Eq.(46) in two types of terms one with shearing viscosity and other without it. Thus, the first term as well as the second and third terms of Eq.(46) are taken identically equal to zero The first of the above equation provides while the second leads to Inserting Eqs.(50), (57), (63), (69) and (70) in the collapse equation, we have", "pages": [ 12, 13 ] }, { "title": "4.1 Newtonian Limit", "content": "Now we explore the instability range by applying the Newtonian limit, i.e., A 0 = 1 , B 0 = 1 and ρ 0 /greatermuch p 0 and discarding the terms of the order m 0 r in Eq.(71). It follows that We see from Eq.(58) that the perturbed configuration of the shear scalar depends on the velocity gradients. Also, it is known that the velocity of the particles for a self-gravitating star increases towards the center. This shows that ¯ σ has negative value inside the body and zero on the boundary surface. It is worth noticing that for a collapsing body ˙ T < 0 as t →-∞ , thus Eq.(58) implies that ¯ c > 0. Considering the physical requirement, i.e., p ' 0 < 0 and neglecting the terms like ρ 0 p 0 being relativistic terms, we have instability condition (independent of linear perturbation functions) for the charged viscous fluid as Here the critical value 4 3 corresponds to the spherical geometry and Newtonian gravity. In fact, 4 in the numerator corresponding to the weight of the envelop in Newtonian mechanics varying as r -2 which is distributed over the surface of sphere yielding another r -2 . The denominator 3 corresponds to the volume of the sphere r 3 . We note from the above equation that the electromagnetic field diminishes the impact of shearing viscosity on the dynamical instability and makes the system unstable at Newtonian approximation. We retain the Newtonian classical result, i.e., Γ < 4 3 , for the case of shearfree or when fluid is not charged viscous.", "pages": [ 14 ] }, { "title": "4.2 Post-Newtonian Limit", "content": "In view of pN limit, we consider A 0 = 1 -m 0 r , B 0 = 1+ m 0 r and the relativistic corrections of the order m 0 r in Eq.(71) so that we have Here we apply the same procedure as in the Newtonian limit and ignoring terms with higher order m 0 r , we get the instability range at pN limit as follows This equation shows that the positive terms diminish the relativistic effects of negative terms occurring from the shearing viscosity and electromagnetic field at pN approximation, making the fluid more unstable. It is mentioned here that instability condition depends upon the static configuration of the system, as perturbed variable e ( r ) and b ( r ) depend on static configuration given in Eqs.(54) and (69) respectively.", "pages": [ 14, 15 ] }, { "title": "5 Conclusion", "content": "We have investigated the role of electromagnetic field on the instability conditions (73) and (75) at Newtonian and pN approximations. Pinheiro and Chan [37] found that the collapsing stars with huge amount of charge ( Q ∼ 5 . 408 × 10 20 Coulomb) leads to the Reissner-Nordstro m black hole. They concluded that the models with charge to mass ratio Q m 0 ≤ 0 . 631 would form a black hole. Ernesto and Simeone [38] examined the stability of charged thin shell and found that Reissner-Nordstro m geometry for different values of charge either have an inner and outer event horizon or a naked singularity. This shows the relevance of charge on the evolution and structure formation of astrophysical objects. In general, the behavior of electromagnetic field is always positive being the Coulomb's repulsion force. Sharif and Abbas [1] explored that due to the weak nature of the electromagnetic field, the end state of collapsing cylinder results a charged black string. We see from Eq.(73) that shearing viscosity boost the stability of the fluid which is the consequence of the fact that the collapse with shear proceeds faster than without shear. Whereas the electromagnetic field being a positive quantity diminishes the effects of shearing viscosity and makes the fluid unstable at Newtonian approximation. This corresponds to the fact that charge delays the event horizon formation or even halts the complete contraction of the star [10]. Also, the electromagnetic field has the same impact for Eq.(75) at the pN approximation. It is worth mentioning here that our results for electromagnetic field are consistent with the results obtained in [34]. Finally, we would like to mention that we have made all the discussion on the hypersurface, where the areal radius is constant. The solution of the temporal equation (60) includes oscillating and non-oscillating functions correspond to stable and unstable systems. For the sake of instability conditions, we have confined our interest in the non-oscillating ones. We have investigated the role of physical quantities in the onset of dynamical instability of fluid during the collapse, hence the instability conditions contain those terms which have radial dependence.", "pages": [ 15, 16 ] }, { "title": "Acknowledgments", "content": "We would like to thank the Higher Education Commission, Islamabad, Pakistan, for its financial support through the Indigenous Ph.D. 5000 Fellowship Program Batch-VII . One of us (MA) would like to thank University of Education, Lahore for the study leave.", "pages": [ 16 ] } ]

[ { "title": "Jingyi Zhang ∗", "content": "Center for Astrophysics, Guangzhou University, 510006, Guangzhou, China (Dated: April 20, 2021) In this paper, a canonical ensemble model for the black hole quantum tunneling radiation is introduced. With this model the probability distribution function corresponding to the emission shell is calculated. Comparing with this function, the statistical significance of the quantum tunneling radiation spectrum of black holes is investigated. Moreover, by calculating the entropy of the emission shell, a discussion about the mechanism of information flowing out from the black hole is given too. PACS number(s): 04.70.Dy Keywords: canonical ensemble model, black hole, quantum tunneling, information puzzle", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "In 2000, Parikh and Wilczek presented an approach to calculate the emission rate at which particles tunnel across the event horizon[1-3]. The main idea is that if the energy conservation is taken into account, the emission process will be a quantum tunneling, and the barrier will be determined by the self-gravitation interaction of the emission particle. In order to keep the spherical symmetry of the space time during the emission process, Parikh and Wilczek treat the tunneling particle as a spherical shell (or emission shell). In this way a corrected spectrum, which is accurate to a first order approximation, is given. Their result is considered to be in agreement with an underlying unitary theory, and support the information conservation during the emission process of the particles. Following this method, many static or stationary rotating black holes were studied, and similar results were obtained [4-26]. According to the Parikh-Wilczek tunneling framework, for a Schwarzschild black hole the corrected emission spectrum is where ω is the energy of the emitted particle (emission shell), M is the total mass of the black hole. If ω is so small that we can ignore the second order term, then we have here β = 1 T = -8 πM . On the other hand, the spectrum given by Hawking is[27, 28] where N 2 ω denotes the intensity of the positive energy particle flux outside and close to the event horizon, ω is the energy level of a single emitted particle. In the condition of classical limit, e βω /greatermuch 1, the Hawking radiation spectrum Since N 2 ω ∝ Γ, some people think that Eq.(2) and Eq.(4) are the same, and therefore the Parikh-Wilczek's tunneling framework and their tunneling radiation spectrum Eq.(1) give a semi-classical correction to the Hawking radiation spectrum Eq.(3). In fact, Eq.(1) and Eq.(3) are very different. In Eq.(3), ω is the energy level of a single emitted particle, whereas in Eq.(1) it denotes the energy of an emitted spherical shell. That is, in Parikh-Wilczek's tunneling framework, the ω is the total energy of a composite particle which contains a number of emitted particles and constructs a spherical shell. Thus, Eq.(1) and Eq.(3) reflect different statistical significance. Moreover, most people think that the greatest success of the Parikh-Wilczek's tunneling framework is the consistency with the conservation of information. However, what is the mechanism of information flowing out the black hole? How can an emission shell carry away information? In this paper, we first attempt to propose a canonical ensemble model to determine the statistical significance of the tunneling radiation spectrum. Then, with this model we calculate the entropy of the emission shell and discuss the mechanism of information flowing out from the hole. We use the Planck units c = G = /planckover2pi1 throughout the paper.", "pages": [ 1, 2 ] }, { "title": "II. CANONICAL ENSEMBLE MODEL CORRESPONDING TO THE BLACK HOLE QUANTUM TUNNELING RADIATION", "content": "As described in section I, in Parikh-Wilczek's tunneling framework, the tunneling particle should be a spherical shell (or S-wave) to keep the spherical symmetry of the space time during the emission process. It is actually equivalent to treat a tunneling spherical shell as a composite particle consisting of many particles. In this framework, the tunneling and the shrinking of the black hole take place at the same time. The tunnelling speed of a Spherical shell is[29] Obviously, in the vicinity of the horizon, the speed of the tunneling shell is very slow, infinitely close to zero, and therefore, we can think that it reaches a thermal equilibrium and has the same temperature with the black hole. Without loss of generality, we postulate that this composite particle consists of identical particles, and we can treat this spherical shell as a thermodynamical system. We imagine that the black hole acts as a large heat source, and together with the outgoing spherical shell constitutes an isolated system. Therefore, we can investigate the emission shell with the canonical ensemble theory. Suppose we have N identical black hole-emission shell systems, which constitute a mixture ensemble. let us define the statistical operator where | ψ i 〉 denotes the quantum states of the emission shell, P i is the probability of the state | ψ i 〉 . For a canonical ensemble system, we have and That is, we obtain This result tells us that if we treat the black hole together with the emission shell as an isolated system, with the canonical ensemble, we will obtain a probability distribution function P i , which is the same as the emission rate of a spherical shell, Γ. However, in our canonical ensemble model, ω denotes the total energy of the system, and the P i denotes the probability of the system staying at a macrostate with the total energy ω . Of course, if we consider the emission shell as an identical particles system, at the first order accuracy, we can obtain the same formula as that of Eq.(4). Moreover, since this emission shell contains lots of microstates, as it is emitted from the hole, it surely carrys away a lot of information. That is, there is an out-going information flux near the horizon during the emission process. Now, the most interesting question is: Is the total information of the black hole-emission shell system conservative? where E and E i denote the total energy of the isolated system and the energy of the emission shell, respectively. Ω BH ( E -E i ) is the number of microscopic states of the black hole, and Ω( E ) is that of the isolated system. Since E i /lessmuch E , in order to obtain the statistical operator ˆ ρ and compare it with the radiation spectrum of the Parikh-Wilczek tunnelling framework, we expand ln Ω BH ( E -E i ) into the form of Taylor series. Namely, Obviously, here C BH is the heat capacity of the black hole, T is the Hawking temperature. If we calculate the Eq.(8) to the second order approximation, we obtain where Therefore, we have where ˆ H is the Hamiltonian operator of the canonical system. For a Schwarzschild black hole, we have Substituting Eq.(15) into Eq.(11) we get", "pages": [ 2, 3 ] }, { "title": "III. ENTROPY AND INFORMATION OF THE EMISSION SHELL", "content": "In order to discuss the information puzzle during the emission process, we first calculate the entropy of the emission shell. Let us define a entropy operator Then, the mean value of the entropy is The exact computation about the entropy is very difficult. An alternative approach is treating the emission shell as an identical particles system and calculating its entropy with the modified brick-wall method-the thin film model [30, 31]. That is, we treat the emission shell as a thin film staying near the hole. Let us take the Schwarzschild black hole as an example, and review the thin film model of the entropy computation. The line element of the Schwarzschild black is The entropy of a thin film composed of identical particles near the hole can be expressed as an integral[31] where r H = 2 M is the location of the event horizon, and In Eq. (22), /epsilon1 is a cutoff and δ is the thickness of the thin film. Now we calculate the entropy of the emission shell staying outside the event horizon. The thickness of the emission shell approximately equal to r H ( M ) -r H ( M -ω ) = 2 ω . Moreover, considering the self-gravitation of emission particle, for a emission shell the effective space time should be[1-3] That is, when we calculate the entropy of the emission shell with the Eqs.(22) and (23), we should replace M with M -ω , and the entropy of the emission shell should be written as where r H ( M -ω ) is the radial coordinate of the event horizon, /epsilon1 is the coordinate distance between the emission shell and the horizon. By using the theorem of mean value we obtain where In general, /epsilon1 /greatermuch δ = 2 ω , r ξ ≈ 2 M . Substituting β = 1 T = 8 πM and r ξ ≈ 2 M into Eq.(26), we have In the following, We discuss Eq.(28) for three cases: The total entropy of the black hole-emission shell system is /negationslash It means that the total entropy of the black hole-emission shell system is greater than that of the black hole before emitting the shell. In fact, according to the definition of information[32] the information I decrease after emitting a shell. It means that information lose during the emission of the black hole. /negationslash It also means that the information I decrease. 3) If /epsilon1 > 1 √ 720 π , the shell is far away from the event horizon. In this case, the velocity of the shell can not be ignored, and the thermal equilibrium with the horizon does not exist, and therefore, the above method of calculating the entropy will no longer adapt.", "pages": [ 4, 5 ] }, { "title": "IV. CONCLUSION", "content": "We have introduced a canonical ensemble model corresponding to the Parikh-Wilczek's tunnelling framework. With this model, we can discuss not only the statistical significance of the quantum tunneling radiation spectrum but also the mechanism of information flowing out from the hole. We showed that the probability distribution function for the canonical ensemble model is the same as the tunnelling rate of the emission particles. It means that the quantum tunnelling rate is, in fact, equal to a probability that the black hole transits from one quantum state corresponding to the mass M to another state corresponding to the mass M -ω . We found that the emission shell contains information and when it is emitted out from the hole, there are information flowing out. However, according to the canonical ensemble model, the total information is not conservative. That is, information lose in the process of emission. It should be mentioned that our canonical ensemble model is different from the model presented in Ref.[33]. In Ref.[33] the authors treated the black hole as a canonical ensemble composed of a naked black hole and the two-dimensional thermodynamic surface (horizon of the black hole). By using the quantum statistical method, they also derived the energy spectrum of the black hole Hawking radiation.", "pages": [ 5 ] }, { "title": "Acknowledgments", "content": "This research is supported by the National Natural Science Foundation of China (Grant Nos. 11273009, 10873003, 10573005, 10633010) and the National Basic Research Program of China (Grant No. 2007CB815405).", "pages": [ 6 ] } ]

[ { "title": "Hui Li ∗", "content": "Department of Physics, Yantai University, 30 Qingquan Road, Yantai 264005, Shandong Province, P.R.China", "pages": [ 1 ] }, { "title": "Hongsheng Zhang †", "content": "Shanghai United Center for Astrophysics (SUCA), Shanghai Normal University, 100 Guilin Road, Shanghai 200234, P.R.China", "pages": [ 1 ] }, { "title": "Yi Zhang ‡", "content": "College of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing 400065, P.R.China We study the semi-holographic idea in context of decaying dark components. The energy flow between dark energy and the compensating dark matter is thermodynamically generalized to involve a particle number variable dark component with non-zero chemical potential. It's found that, unlike the original semi-holographic model, no cosmological constant is needed for a dynamical evolution of the universe. A transient phantom phase appears while a non-trivial dark energy-dark matter scaling solution keeps at late time, which evades the big-rip and helps to resolve the coincidence problem. For reasonable parameters, the deceleration parameter is well consistent with current observations. The original semi-holographic model is extended and it also suggests that the concordance model may be reconstructed from the semi-holographic idea. PACS numbers: 98.80.-k 95.36.+x 11.10.Lm", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The existence of dark energy is one of the most significant cosmological discoveries over the last century [1]. Various models of dark energy have been proposed [2], such as a small positive cosmological constant, quintessence, k-essence, phantom, quintom, etc., see Ref.[3] for a recent review. Stimulated by the holographic principle, it's conjectured that dark energy problem may be a problem of quantum gravity, although its characteristic energy scale is very low[4]. According to the holographic principle[5], the true laws of inside any surface are actually a description of how its image evolves on that surface. In the case of black hole, the event horizon is a proper holography. When applying the holographic principle to the universe, the event horizon is probably not a good candidate of holography screen at least for two reasons: the identification of the cosmological event horizon is causally problematic in that it depends on the future of the cosmic evolution; a decelerating universe would have no holographic description, since it has no event horizon at all! With reference to the seminal work of Jacobson[6], several arguments have been put forward that the apparent horizon should be a causal horizon and is associated with the gravitational entropy and Hawking temperature[7] [8], and hence the right holography of the universe. Apart from this key observation, other thermodynamical properties of the dark energy and dark matter have yet to be found in details[9]. Recently, the semi-holographic dark energy model[10] has been proposed to examine how the universe could evolve if a dark component of the universe strictly obeys the holographic principle. It was found that, based on the first law of thermodynamics, the existence of the other dark component (dark matter) is compulsory, as a compensation of dark energy. In that model, with a non-zero cosmological constant the standard cosmological kinematics is recovered and the cosmological coincidence problem is also alleviated with the existence of a stable Einstein-de Sitter scaling solution. For all that, the properties of the semi-holographic model should be clarified for further estimations from three points of view. First, only massless particles (photons) exchange are considered. Or we treat dark matter and dark energy as two closed systems. Surely, by nature they can be open systems, and thus the particle numbers of the two systems can be variables. Second, the EOS of dark energy is constrained to be ω de < -1. It certainly is in accord with the latest cosmological observations and phenomenologically reasonable. However, as a thermodynamics motivated dark energy model, the construction would be founded on a more solid basis of thermodynamics and statistical physics, and exactly in this sense, the well-known theoretical difficulties does matter. As a matter of fact, the case of phantom energy with ω < -1 is ruled out because the total entropy of the dark component is negative[11]. The equation of state (hereafter EOS) was restricted to the interval -1 ≤ ω < -1 / 2 and a fermionic nature to the dark energy particles is favored. Thermodynamics arguments[18] in favor of the phantom hypothesis had to resort to an unusual assumption that the temperature of a phantomlike fluid is always negative in order to keep its entropy positive definite (as statistically required) or by arguing that the scalar field representations of a phantom field has a negative kinetic term which quantifies the translational kinetic energy of the associated fluid system. Third, as is known, the possibility of a coupling between dark matter and dark energy is often considered with three types in the literature, and those interaction forms are always introduced phenomenologically or through the low energy effective action[15]: dark matter decaying into dark energy[12], dark energy decaying into dark matter[13] and interaction in both directions[14]. It's worth noting that, due to the non-adiabaticity of the holographic evolution of dark energy, the semi-holographic model obviously introduced a dark matter/dark energy non-gravitational interaction (A hypothetical dark energy property sometimes induces a non-gravitational coupling between dark matter and dark energy. See [16] for another interacting model of this kind.), and the interaction belongs to the last category: dark matter dominated in the past and transferred energy to dark energy; the latter begins to dominate at present, and in the near future, the transfer would be reversed and dark energy decays to dark matter with a stable scaling solution reached. The theoretical obstacle is that, provided that the chemical potential of both components vanish, it is the decay of dark energy into dark matter that was favored by the thermodynamical point of view and the second law. Owing to a series of systematic work[11][17][18][19][20], thermodynamics and statistical arguments on dark components and the cosmic evolution have come to several important conclusions which may kick off those dilemma. By considering the existence of a non-zero chemical potential, reanalysis of the thermodynamics and statistical properties of the dark energy scenario supports that the temperature of dark energy fluids must be always positive definite and that a negative chemical potential will recover the phantom scenario without the need to appeal to negative temperatures[19]. In addition, a bosonic nature of the dark energy component becomes possible. If the chemical potential of at least one of the cosmic fluids is not zero, the decay can occur from the dark matter to dark energy, without violation of the second law[20]. From this point of view, the decay process between dark components with non-zero chemical potential is not merely a theoretical tool, but should be seriously considered as a prerequisite to turn the phantom dark energy and the decay in both directions to be physically real hypotheses. More importantly, due to the insufficiency in discussing the particle number variable transferring process between dark components, the original semi-holographic model should not be regarded as a complete thermodynamical examination of the semi-holographic idea in cosmology. A general framework dealing with the decay of the dark component particles is still in lack and therefore well motivated. Particularly speaking, the first law of thermodynamics would be applied in the context of grand canonical ensemble rather than canonical ensemble describing the original semiholographic model. This paper is organized as follows: In the next section we will study the semi-holographic model in grand canonical ensemble with the dark energy chemical potential explicitly given. The dynamical analysis is left in section III. We find that there exists a stable dark matter-dark energy scaling solution at late time, which is helpful to address the coincidence problem. After a close analysis and comparison between model behavior and the observational facts, we present our conclusion and some discussions in section IV.", "pages": [ 1, 2, 3, 4 ] }, { "title": "II. THE MODEL", "content": "There is decisive evidence that our observable universe evolves adiabatically after inflation in a comoving volume, that is, there is no energy-momentum flow between different patches of the observable universe so that the universe keeps homogeneous and isotropic after inflation. That is the reason why we can use an FRW geometry to describe the evolution of the universe. In an adiabatically evolving universe, the first law of thermodynamics equals the continuity equation. In a comoving volume the first law reads, where U = Ω k ρa 3 is the energy in this volume, T denotes temperature, S represents the entropy of this volume, V stands for the physical volume V = Ω k a 3 , ˜ µ is the chemical potential of the energy component with particle number N . Here, Ω k is a factor related to the spatial curvature. For spatially flat case Ω 0 = 4 3 π , in this paper we only consider the spatially flat model, ρ is the energy density and a denotes the scale factor. The last term in the above equation indicates that the grand canonical ensemble instead of the canonical ensemble for the original semi-holographic model is considered. As a consequence, the framework allows decaying of the particles in consideration. The semi-holographic model concerns the possibility of a non-adiabatical dark energy, where the term TdS does not equal zero. Based on the investigations in [7, 8], the entropy in a comoving volume the entropy becomes, where H is the Hubble parameter, µ denotes the reduced Planck mass. The entropy has been reasonably assumed to be homogeneous in the observable universe. As our observable universe evolves adiabatically after reheating, the varying entropy of dark energy in a comoving volume should be compensated by an entropy change of the other component to keep the total entropy constant in a comoving volume. With the above supposition and conventions the entropy of the dark energy satisfies (2), Correspondingly, the entropy of dark matter in this comoving volume should be where C is a constant, representing the total entropy of the comoving volume. The Friedmann equation will determine the evolution of our universe where H is the Hubble parameter, ρ dm denotes the density of non-baryon dark matter, ρ de denotes the density of dark energy, and Λ is the cosmological constant (or vacuum energy). Partly because the partition of baryon matter is very small and does little work in the late time universe, and partly because the non-gravitational coupling between baryonic matter and dark energy is highly constrained, we just omit the baryon matter component in the discussion. The appearance of the cosmological constant in the expression is just for a general discussion, and as can be seen below, it is not a necessary component in the generalized semi-holographic model. As a comparison, the cosmological constant has to be present in the original model, or else the ratio of holographic dark energy and dark matter density would always remain constant which is obviously not realistic. Below we will assume the chemical potential of dark energy to be of the form in Ref.[20] with the efficient ˜ α a positive constant. As was shown by Pereira, if the coefficient ˜ α > 0, a negative chemical potential will make the phantom fluid thermodynamically consistent and a decay of dark matter into dark energy possible. This assumption may help us highly broaden the exploration of the cosmological parameter region motivated by thermodynamics.", "pages": [ 4, 5 ] }, { "title": "III. DYNAMICAL ANALYSIS", "content": "To investigate the evolution in a more detailed way, we take a dynamical analysis of the universe. The holographic principle requires that the temperature [7] and a dimensionless cosmological constant where H 0 denotes the present Hubble parameter. Then the equation set (9), (8) becomes respectively, where y = √ u + v + λ and α = ˜ αµH 0 are both dimensionless. We note that the time variable does not appear in the dynamical system (13) and (14) because time has been completely replaced by scale factor. Before presenting the numerical examples for special parameters we study the analytical property of this system. And we will below take Λ = 0 for simplicity. The critical points of the dynamical system (13) and (14) are given by which yields, By using (7), (5), (3), and using the particle number N to be proportional to the energy density, the first law of thermodynamics (1) becomes the evolution equation of dark energy, where a prime denotes the derivative with respect to ln a , w dm indicates the EOS of dark matter, w de represents the EOS of dark energy and Y = √ ρ de + ρ dm +Λ. Similarly, we derive the evolution equation of dark matter, Clearly, the above equations will degenerate to the equations (8) and (9) of the paper [10] when ˜ α = 0. For convenience we introduce two new dimensionless functions to represent the densities, and The trivial solution corresponds to a future infinitely diluted universe and is less interesting for the coincidence problem. On the other hand, the non-trivial fixed point satisfies the below equality: which is shared by the seminal model without chemical potential involved. So, finally the universe enters a de Sitter phase, and the ratio of dark matter over dark energy is independent of the chemical potential. Furthermore, there are two reasonable cases for the non-trivial scaling solution: case I, w de < -1 and w dm > -1; case II, w de > -1 and w dm < -1, since we should require that the final densities of dark matter and dark energy are both positive; as a matter of fact, the negative energy density may appear at most as a transient phenomenon, and it can not be a physically stable and permanent state. To study the stability property of the fixed points[21], the evolution equations should be perturbed to the first order and we list them in the Appendix. Then, the perturbation equations to the trivial critical point of 16 are simplified as And near this fixed point,the eigenvalues of the linearized system read Stability requires that all of real parts of the eigenvalues are less than zero. Combined this requirement with the observational fact of w de < -1 3 , we find A typical picture of the evolution is displayed in Figure 1. For illustration we set w de = -0 . 4, w dm = -0 . 1 and α = 0 . 9. As is seen to us, the orbits with initial values of energy densities lying in the first quadrant evolve to the second quadrant with the energy density of dark matter negative, and they will keep negative before the tracks reach the trivial critical point at the origin of coordinates. Evidently, this case is physically problematic and might be irrelevant to our universe. Now let's turn to the non-trivial fixed point. Analytic eigenvalues of the corresponding linearized system to this critical point have not been found, and we will explore the stability property of the scaling solution via numerical method. Here we plot Figure 2 to display the properties of evolution of the universe controlled by the dynamical system. In the figure, we set λ = 0, w de = -1 . 2, w dm = -0 . 2 and the positive coefficient α = 0 . 6 for illustration. The physically meaningful region is just the first quadrant and different initial value tracks evolve to the common non-trivial fixed point still in the first quadrant, which indicates the existence of the physically stable attractor. The most significant parameter from the viewpoint of observations is the deceleration parameter q , which carries the total effects of cosmic fluids. Using (5), (8), and (9) we obtain the deceleration parameter in this model For a numerical example, we take the present dark matter partition u 0 = 0 . 3 and the present dark energy partition v 0 = 0 . 7, which is favored by present observations[1]. Figure 3 illuminates the evolution of deceleration parameter. As a simple example we just set w dm = -0 . 2, w de = -1 . 2 and α = 1 . 0. The dark matter dominates in the past, and since its effective EOS in that period is larger than the assumed EOS, dark matter decays into dark energy. Only some time ago, this decay made the dark matter stiffer and with the accumulation of the dark energy, the universe endures a deceleration-acceleration transition. The acceleration goes through the present era and, if the dimensionless coefficient α is about 1 or less, it will march for a phantom phase in the near future. That is to say, the deceleration parameter can be smaller than -1. However, the super-acceleration will not last for ever. The decay process of dark matter into dark energy then reverses and dark energy begins to transfer its energy to dark matter component. This conversion impedes the super-acceleration and eventually the evolution of the universe is pulled back to the Einstein-de Sitter track; therefore, the troublesome cosmic big-rip is evaded[22]. From the figure one sees that current q ∼ -1 and at the high redshift region it goes to 0 . 5, which is consistent with current observations [3, 23]. As a comparison, the evolution of the deceleration parameter in a spatially flat ΛCDM is also plotted, in which the density parameter of dark matter Ω dm = 0 . 3 too. The density evolution of the cosmic fluid does not depend on the assumed EOS above, but the effective EOS. We define the effective EOS as the following procedure. Supposing the dark matter evolves adiabatically itself, we obtain its evolution from (1), where p eff denotes the effective pressure of dark matter. Then we obtain which is variable in the evolution history of the universe. Note that we have assumed both w dm and w de are constant from the beginning. Figure 4 displays the effective EOS of dark matter in which the same parameters are set as in Figure 3. It shows that, although its EOS is clearly negative, the dark matter behaves like cold dark matter until now. Only recently the dark matter has become stiff which relates to the current cosmic acceleration era. In Figure 5, different evaluations of the parameter α are taken and the corresponding deceleration evolution illustrated. A smaller α denotes a later deceleration-acceleration transition and a greater degree of acceleration in the near future. As the coefficient gets bigger, the cosmic evolution becomes closer to the concordant ΛCDM model and the transition happens more gently. For all that, the universe will go back to the same track of an everlasting de-Sitter phase with each of the energy densities unchanged thereafter. . The deceleration-acceleration transition happens abruptly in the original semi-holographic model, and the present model in grand canonical ensemble is prone to moderate the process. As the chemical potential coefficient changes, the deceleration parameter tracks fill in the intermediate region between the original semi-holographic model and the ΛCDM model. Therefore, in this sense we can say that the particle decay picture extends the original semi-holographic model moderately and reconstructs the concordance model with the semi-holographic idea. The tracks may have or may not have a super-acceleration phase which depends on the different values of α .", "pages": [ 5, 6, 7, 8, 9, 10 ] }, { "title": "IV. CONCLUSION AND DISCUSSION", "content": "Based on the semi-holographic model inspired by holographic principle, especially the previous studies of the relation between thermal dynamics and general relativity, we find that, the semiholographic dark energy model in the grand canonical ensemble identification recovers the standard expansion history and evolves to the Einstein-de Sitter final state through a possible transient super-acceleration. A phantom phase may appear in the near future without a big-rip in the end. Assuming the total matter in a comoving volume to be evolved adiabatically, a compensating dark matter component should exist and the non-gravitational coupling between dark energy and dark matter is also compulsory. As is shown in the above, this interaction yields a future attractor solution, which is a stable scaling solution for the dark matter-dark energy system in proper region of the parameters ( w dm , w de ). The final ratio of dark matter to dark energy only depends on w dm , w de , and is independent of the initial values of the densities of dark matter and dark energy and the coefficient α of chemical potential. This result is helpful to address the coincidence problem. The conversion of canonical ensemble context to grand canonical ensemble one seems to deform the original model to the well established ΛCDM model in the sense that different tracks roughly fill in the zone between the two models in the q -ln a plane. Therefore, the particle decay picture extends the original semi-holographic model moderately and reconstructs the concordance model with the semi-holographic idea. This grand canonical ensemble viewpoint opens the possibility to deepen our understandings of dark energy thermodynamics and statistical properties, and is closely related to previous systematic application of the cosmological thermodynamics. With numerical examples, it is illustrated that the deceleration parameter is well consistent with observations. As a phenomenological model, the parameters including α may be further constrained by observational data and detailed investigations on global fitting of cosmological parameters will also be interesting. Acknowledgments. H. Li is supported by National Natural Science Foundation of China under grant No. 10747155, H. Zhang is supported the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, Shanghai Municipal Pujiang grant No. 10PJ1408100, and National Natural Science Foundation of China under grant No. 11075106, and Y. Zhang is supported by the National Natural Science Foundation of China under grant Nos. 11005164, 11175270 and 10935013, the Distinguished Young Scholar Grant 10825313, CQ CSTC under grant No. 2010BB0408, and CQ MEC under grant No. KJTD201016.", "pages": [ 11, 12 ] }, { "title": "V. THE APPENDIX: THE PERTURBATION EQUATIONS", "content": "The perturbation equations for the combination ( ρ de , ρ dm ) are: with all the notations declared in the main text. [arXiv:hep-th/0506212]; X. Zhang, [arXiv: astro-ph/0504586]; B. Wang, C.Y. Lin and E. Abdalla, [arXiv:hep-th/0509107]; Z. Chang, F.Q. Wu and X. Zhang, [arXiv:astro-ph/0509531]; M. Li, X. D. Li, S. Wang, Y. Wang and X. Zhang, JCAP 0912 , 014 (2009); Z. H. Zhang, S. Li, X. D. Li, X. Zhang and M. Li [arXiv:1204.6135]; H. Wei and R.G. Cai, Phys. Lett. B660 , 113 (2008); Y. Zhang and H. Li, JCAP 1006 , 003 (2010) [arXiv:1003.2788 [astro-ph.CO]]. Jesus, R. C. Santos, J. S. Alcaniz and J. A. S. Lima, Phys. Rev. D78 , 063514 (2008), [arXiv:0806.1366]; J. C. Carvalho, J. A.S. Lima and I. Waga, Phys. Rev. D46 , 2404 (1992); P. Wang and X. Meng, Class. Quant. Grav. 22 , 283 (2005); J. F. Jesus, Gen. Rel. Grav. 40 , 791 (2008) [arXiv:astro-ph/0603142]; J. S. Alcaniz and J. A. S. Lima, Phys. Rev. D72 , 063516 (2005).", "pages": [ 12, 13, 14, 15 ] } ]

[ { "title": "Energy eigenfunctions of the 1D Gross-Pitaevskii equation", "content": "ˇ Zelimir Marojevi'c, Ertan Goklu, and Claus Lammerzahl ZARM Universitat Bremen, Am Fallturm, 28359 Bremen, Germany (Dated: July 27, 2018) We developed a new and powerful algorithm by which numerical solutions for excited states in a gravito optical surface trap have been obtained. They represent solutions in the regime of strong nonlinearities of the Gross-Pitaevskii equation. In this context we also shortly review several approaches which allow, in principle, for calculating excited state solutions. It turns out that without modifications these are not applicable to strongly nonlinear Gross-Pitaevskii equations. The importance of studying excited states of Bose-Einstein condensates is also underlined by a recent experiment of Bucker et al in which vibrational state inversion of a Bose-Einstein condensate has been achieved by transferring the entire population of the condensate to the first excited state. Here, we focus on demonstrating the applicability of our algorithm for three different potentials by means of numerical results for the energy eigenstates and eigenvalues of the 1D Grosss-Pitaevskiiequation. We compare the numerically found solutions and find out that they completely agree with the case of known analytical solutions.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "One of the most interesting problems in today's physics is the exploration of the quantum-gravity regime. This is due to the fact that General Relativity and quantum theory are not compatible which makes it necessary to search for a new theory called quantum gravity which at the end should lead to effective modifications of General Relativity and/or quantum theory. Another issue is that in some approaches gravity is regarded as a solution to the measurement problem in quantum theory. Therefore there are a lot of reasons showing that it is important to explore the interaction of quantum matter with gravity with better accuracy. One possibility to study the behavior of quantum matter in gravitational fields is neutron and atom interferometry [1-3]. One may even go further and investigate the energy eigenstates of quantum matter in a gravitational trap as has been pushed forward using ultracold neutrons at the ILL [4]. In this experiment the various eigenstates manifest themselves through a neutron flux which depends on the height in a step-like form. One difficulty in this experiment is that the steps are of the order of µ m which comes from the strength of the gravitational acceleration. With the recently developed technology of Bose-Einstein condensates (BEC) in microgravity condition [5] another physical system is available for investigating the quantum-gravity regime for a wider range of parameters. Is is feasible to perform similar experiments with ultracold atoms in a GravitoOptical Surface Trap (GOST) with a small and variable gravitational acceleration so that the density profile of the quantum states related to various energy levels can be measured with better resolution. The solution of the eingenvalue problem for the Schrodinger equation in such a GOST has been solved in terms of the Airy-functions in, e.g., [6-8]. In order to be able to describe also the eigenstates for a BEC, we are solving here the eigenvalue problem for the nonlinear Schrodinger equations, that is, for the Gross-Pitaevskii equation (GPE). For doing so we developed in this paper a new numerical algorithm which is capable to find solutions of the GPE which belong to saddle points of the action. This algorithm is applied to three different potentials, the box, the harmonic trap and the GOST. Our numerical solutions which, among others, correspond to excited states completely agree with the known analytic solutions for the box. For a first description of the method and in order to present first results we restrict at the moment to one-dimensional problems. Concerning a further physical motivation to study excited states it has to be mentioned that recently Bucker and coworkers [9, 10] demonstrated the vibrational state inversion of a Bose-Einstein condensate. This system is confined in an anharmonic trapping potential and the inversion can be achieved by controlled displacement of the trap center. By means of this procedure they transferred BECs to the first antisymmetric stationary state which is, in fact, an excited state. In this paper in section II we first state the problem and introduce the notation. In section III we describe the newly developed algorithm and apply this method in section IV for solving the energy eigenvalue problem of the GPE for three different physically relevant potentials. The paper closes in Section V with an outlook indicating further work in this direction. with the free energy where we assumed a real Ψ. The particle number is given by Throughout this paper we will restrict ourselves to one-dimensional problems in order to demonstrate our new algorithm. In order to facilitate the numerical calculations, as one usually does, we rescale and renormalize the coordinates and the wave function according to where Ψ( x ) is normalized to 1, leading to with the dimensionless quantities The length scale L is arbitrary and may depend on various physical parameters. It is chosen in such a way that the dimensionless quantities are convenient for numerical calculations. Note that in particular the nonlinearity parameter γ depends on the length scale L . Note that we do not restrict ourselves to functions that are normalized to one. Instead we are searching for solutions that are not normalized for a given pair γ, ε . From equation (7) it is evident that each found solution can be normalized to one by adjusting the nonlinearity γ .", "pages": [ 1, 2 ] }, { "title": "II. THE MODEL", "content": "We start with the time-dependent GPE which describes the dynamics of a BEC subject to two-particle interactions, given by the nonlinear term g S | Ψ( x , t ) | 2 , and to an external potential V ext where Ψ( x ), x = ( x, y, z ) is normalized to the total number of particles N = ∫ Ω | Ψ( x ) | 2 d 3 x . The GPE is valid for dilute condensates obeying the diluteness criterion, that is, the s-wave scattering length a and the average density of the gas ¯ n must fulfill ¯ n | a | 3 /lessmuch 1. The nonlinearity parameter g S is determined by the scattering length via g S = 4 π /planckover2pi1 2 a m , where m is the mass of the atom. Moreover, the scattering length can acquire both signs, having magnitudes of some nanometers. However, in this work we will focus on the case g S > 0 which describes repulsive two-particle interactions. The function Ψ( x , t ) has the meaning of an order parameter, is a classical field and is also interpreted as the wave function of the condensate. For the calculation of the ground states and higher modes of a BEC in a time-independent external potential one makes the ansatz Ψ( x , t ) = Ψ( x ) exp ( -iµt/ /planckover2pi1 ) leading to the stationary GPE where µ is the chemical potential. We also assume that the potential V ext ( x ) is bounded from below so that we can take V ext ( x ) ≥ 0. The stationary GPE can be derived from the action", "pages": [ 2 ] }, { "title": "A. The general setting", "content": "In computer numerics an attempt to solve nonlinear partial differential equations is to use some variant of the Newton method or the imaginary time propagation. The latter method is based on the the splitting and discretisation (e.g. Crank-Nicolson) of the unitary time evolution operator [11, 12]. This is reliable for ground state solutions. In this paper we will present a new Newton Method. Newton methods are gradient based algorithms that follow a descent direction until a local minimum of the action is reached. The direction depends on the choice of the inner product and/or of the preconditioning procedure. Solutions can be understood as critical points of some action A on the underlying dual space defined by the inner product. The type of a critical point related to a solution is determined by the eigenvalues of the Hessian [27] evaluated at the critical point: The number of negative eigenvalues is known as the Morse index. Solutions that belong to a local minimum of an action A are candidates for solutions which can be easily found by standard Newton methods, which searches in the whole L 2 space. Unfortunately, finding critical points of a certain saddle type depends on an educated guess. In order to have a straightforward method at hand it is necessary to confine the search on a subspace of our Hilbert space. For linear eigenvalue problems this is easy to do because of the orthogonality of eigenfunctions. The gradient at every iteration step is orthogonalized with respect to the previously found eigenfunctions using the Gram-Schmidt procedure. Therefore it is easy to find eigenfunctions in ascending order of eigenvalues or, equivalently, in ascending order of the Morse index. Unfortunately, in the nonlinear case the orthogonality no longer holds, so that an other approach is needed. The basic idea is to constrain the quest for a solution to a submanifold in the underlying function space. In the following we need the first variational derivative (Gˆateaux derivative) which via where is the search direction and O -1 denotes a preconditioning operator that improves the convergence behaviour. In this context k is the iteration index and τ the stepsize. The minus sign in front of τ denotes that the correction of the step φ k is performed in the negative direction of the preconditioned L 2 gradient. The stepsize can be a constant or can be determined at every iteration step using the linesearch or trusted region method. The solution then is given by φ sol := lim k →∞ φ k . In the nonlinear case the widely used Newton method is only capable to find Morse index zero solutions. Finding higher Morse index solutions for strong nonlinearities is a hard task and the standard Newton method is not able to do that. A von Neumann analysis applied to the standard Newton method (i.e. with the preconditioning O -1 = 1 ) leads to a convergence criterion like the famous Courant-Friedrich-Lewy condition which estimates a bound on the stepsize τ depending on the discretization lengths and other parameters of the differential equation. Therefore a bad choice for τ causes a failure of the Newton method. A too small τ decreases the convergence rate. In order to handle this issue the preconditioning O -1 is necessary. There are two well known methods, among others: Upon the choice of preconditioning the direction of the gradient is altered. For a descend direction we have 〈O -1 ∇ L 2 A , ∇ L 2 A〉 > 0 for some arbitrary action A . The stepsize control of classical Newton methods fails if this condition does not hold.", "pages": [ 3, 4 ] }, { "title": "C. Newer approaches", "content": "/negationslash From equation (4) it is evident that F [ φ k ] > 0 for any φ k = 0 and g S > 0 so that φ k = 0 is the only critical point of F . Therefore it is only the term µN in (3) by which new critical points can appear. Accordingly, the key idea for the existence of solutions of non linear differential equations is to have terms that are capable of balancing the non linearity and all other terms. In order to emphasize the idea of balancing the nonlinearities we consider for demonstration purpose a classical case [15, 16] for the situation without external potential and attractive two-particle interaction, thus ˜ V ext = 0 and γ < 0. Then the functional F reads can be identified with an L 2 gradient ∇ L 2 A [ φ k , µ ]. Here 〈· , ·〉 denotes the L 2 scalar product. For the GPE we have Therefore, if Ψ is a critical point of the action A then the L 2 gradient of A vanishes and, hence, Ψ is a solution of the GPE.", "pages": [ 4 ] }, { "title": "B. Review of the Newton method", "content": "The discrete Newton method is given by F [0] = 0 is a critical point and F is not bounded. Without further constraints a standard Newton method would fail. It is clear that there exists a t k = 0 that fulfils /negationslash which means that the kinetic part is balancing the interaction part. /negationslash A Newton method which calculates the L2 gradient at the point t k φ k defined by equation (15) generates a sequence { t k , φ k } where the φ k converge to a solution φ sol = 0 in L 2 , which is a local minimum of F . This is known as a minimization process restricted to the Nehari manifold For the k -th step, t k ref φ k is a reference point in the underlying function space where the L 2 gradient is calculated. The definition (16) is equivalent to F ' [ t k ref φ k ] φ k = 0. The sequence of functions φ k calculated this way will converge to the solution φ sol . With the restriction to the Nehari manifold it is possible to find Morse index one solutions of (14). (For a general functional this may not always work.) Therefore it is, in general, very useful to confine the search for solutions to a manifold where the critical point lies in a local minimum on this manifold so that classical Newton methods are able to find such solutions. If one finds no extremum then the method is not applicable. A generalization of this idea has been presented in [17]. There, in the k -th iteration step the function has been defined where /vector t := ( t k 1 . . . t k D , t k ) and the Υ i are the previously found solutions, which were calculated by the algorithm [17], and D is the dimension of the support that is spanned by the Υ i . The idea behind this is to find solutions in the order of their Morse index which is similar to linear problems. First find the global minimum, then use the ground state to define a solution manifold in order to stay away from the ground state. After the the first excited state is found, it is used again together with the ground state to define a new solution manifold in order to stay away from the first and second solution. This is repeated until the algorithm fails. The ground state has Morse index zero and the first excited state has Morse index one. Then, for some action J [ P D,k /vector t ] what can be interpreted as function of /vector t , the extrema of the J [ P D,k /vector t ] determine the vector /vector t which is taken to define the reference point If γ > 0 then the action may have minima. Zhou uses in [17] the so called active Lagrangian J := F -1 2 ε k ( N -1) for his algorithm in order to find normalized solutions. The eigenvalue term ε k is indispensable, but the final eigenvalue is not known from the beginning and has to be altered at every iteration step k . This includes the risk that at some iteration step k the solution is the trivial one /vector t ref = 0. Zhou demonstrated that for γ ∈ O (1) in a 2D GPE with a 2D harmonic trapping it is possible to find solutions in the order of their eigenvalues ε i where ε i > ε i -1 . Then the P D,k /vector t ref is the reference point at which the L 2 gradient ∇ L 2 J [ P D,k /vector t ref ] is calculated. A related way to define a solution manifold [18] is to require that the directional derivatives in direction of the D known solutions and the current iterations step vanishes. In order to find the reference point /vector t ref the following system of equations has to be solved: This is a system of D +1 equations for the D +1 unknown variables ( t k 1 , ..., t k D , t k ). The trivial solution is always a solution, but not the desired one. The numerical solution of this system depends on the initial guess of the vector ( t k 1 , ..., t k D , t k ). Due to the nonlinearity of ∇ L 2 J [ P 1 ,k /vector t ref ] this system has more than one solution for suitable ( ε k , φ k ). In order to find an optimal reference point the initial vector ( t k 1 , ..., t k D , t k ) has to be guessed systematically. We used all corners and all center points of the faces of a D +1 dimensional cube as guesses and took the initial guess where ∀ i ≤ D | t 0 i | < | t 0 | . Under this condition we always succeeded to find a new solution for nonlinearities of the order O (1). With increasing nonlinearity it becomes impossible to fulfil this condition when a solution with smaller non linearity is used as a guess. As a result of our numerical simulations, we like to add some remarks concerning the feasibility of the methods presented in [17] and [18]:", "pages": [ 4, 5, 6 ] }, { "title": "D. The new algorithm", "content": "In order to overcome these problems, we developed a modified algorithm (see Fig. (2)) which yields the following characteristics: That means that we do not need the previous D solutions of lower Morse index as in (19). Here r k s , q k s ∈ R and k and s are numerical counter variables which are only used in the algorithm. The quantum number n refers to the mode of the solutions we are interested in and is defined through the linear eigenfunctions. The functions ψ n,s and φ k n,s have the same nodal structure and φ k n,s can be viewed as a correction to the solution ψ n,s for the previous eigenvalue. we are using the inverse of the analytic Hessian evaluated at φ k n,s as the preconditioning operator O -1 . Our reference point defined by (20) together with the preconditioning (21) assures that we do not leave the subspace with same nodal structure. In contrast, a Sobolev preconditioning would lead to ground state solutions only. The algorithm is implemented in C++. The main part consists of two nested loops with the inner loop counter k and the outer loop counter s . The inner loop (STEP 1 to STEP 6) represents our Newton method. Within the outer loop µ n,s is increased and storage operations are conducted. For convenience we take the calculated γ n,s as starting point for the solution for the next eigenvalue µ n,s +1 . For the numerical derivatives a three point stencil is used. The integrals are evaluated with Simpson's rule and the differential equation from STEP 5 is solved in a finite difference setup with a Bi Conjugate Gradient solver. We used ∆ µ = 0 . 5 , τ = 0 . 01, dx = 0 . 05 and 1401 grid points. In the following we present the individual computational steps as depicted in Fig. (2). STEP 1 The functions ψ 0 and φ 0 n, 0 are initialized to Ψ n , where Ψ n is the analytic eigenfunction of the linear Schrodinger equation for the n -th quantum number. However, if ground state solutions are to be calculated, then one has to set ψ n ← 0. In this case the solution manifold reduces to the Nehari manifold. At the end, set γ n,s ← 1. STEP 2 The numerical algorithm which solves the system of the two equations (20) needs an initial guess for /vector t k ref . Due to the nonlinearity of ∇ L 2 A [ φ k n,s ; µ n,s ] with respect to the φ k n,s the solution is not unique. As guesses we used /vector t 0 ref = (0 , 1) and /vector t 0 ref = (1 , 1) and selected for convenience the final /vector t 0 ref with the larger vector norm. STEP 3 Calculate the L 2 gradient using (11). STEP 4 Test for the quality of convergence. We use the maximum norm to check for convergence (the L 2 norm could be used also since both norms are equivalent). For our numerical calculations we set η = 1 · 10 -5 . On expense of more iteration steps better results can be achieved for smaller η . STEP 5 In general, the numerical inversion of the operator O on the l.h.s. of (21) consumes much computer storage and calculation time. In order to avoid this, we solve the differential equation of STEP 5 in figure 2 and, thus, obtain the search direction d . The disadvantage of this procedure is that the differential operator O has to be assembled at every iteration step. STEP 6 Amodified Newton step is carried out. We set the stepsize to a constant value τ = 0 . 01. A classical linesearch or trusted region stepsize control is not applicable due the fact that d is not always a descent direction. Note that equation (20) is invariant under the simultaneous sign change of r k s and q k s . This reflects the invariance of the GPE under the transformation of the wave function ψ →-ψ . Thus, the factor sgn( q k s ) has to be introduced into the second term in the r.h.s. of (12) in order to have a unique notion of ascent and descent directions, respectively. Therefore, the search direction can be made unique by means of multiplying τd [ φ k n,s ; µ n,s ] with sgn( q k s ). STEP 7 First we calculate the particle number N of P 1 ,k /vector t ref according to (5). Then we save the solution P 1 ,k /vector t ref for the current eigenvalue µ n,s , the adjusted nonlinearity γ s , and the corresponding normalized solution N -1 / 2 P 1 ,k /vector t ref . After that we replace the function ψ n,s by the just calculated P 1 ,k /vector t ref . Before we proceed to the next inner loop s +1 we increase the eigenvalue µ n,s by ∆ µ , where ∆ µ has a typical value of 0 . 5. Go to STEP 2. n (b) = 5 for the GOST. In Figs. 3(a)-3(c) we show typical forms of the error estimate ‖∇ L 2 A [ φ k n,s ; µ n,s ] ‖ ∞ as a function of the inner loop counter k for each of the three problems discussed later. The number of iteration steps for this algorithm applied to these problems was of the order of O (10 2 ) -O (10 3 ) for the inner loop. From these graphs it is evident that the error estimate is not necessarily decreasing right from the first iteration step k = 0 as one might expect. Thus our algorithm permits that the search direction 〈∇ L 2 A [ φ k n,s ; µ n,s ] , d k 〉 ≶ 0 can be descending or ascending in contrast to the aforementioned algorithms (see Appendix A1). We have to emphasize that this depends on the preconditioning. In these logarithmic plots the linear behaviour reveals the exponential decay of the norm.", "pages": [ 6, 7, 8 ] }, { "title": "IV. SOLUTIONS FOR VARIOUS POTENTIALS", "content": "In this section we present analytical and numerical solutions for the energy eigenstates and the energy eigenvalues for the GPE for three potentials, that is, (i) for a box, (ii) gravitational surface trap, and (iii) the harmonic trap. While usually in experiments BECs are created in the ground state, excited states might emerge through an appropriate periodic motion of, e.g., the walls of a box potential. This is similar to the creation of waves of a viscous fluid in a box through the motion of walls. The explicit procedure of the creation of excited states of a BEC obeying the GPE will be discussed in a subsequent paper.", "pages": [ 8 ] }, { "title": "A. BEC in a box", "content": "In this section we present the numerical results of a BEC confined in a box of finite size L . With (6) and the natural length scale L = /planckover2pi1 / √ 2 mµ the dimensionless GPE reads The potential is given by As usually, we require the standard boundary conditions Ψ n (0) = 0 and Ψ n (1) = 0. For γ = 0 the eigenfunctions and energies are simply given by where n = 1 , 2 , 3 , . . . . For γ > 0 this problem can be solved analytically by means of the Jacobi elliptic function sn [19] where K is the complete elliptic integral of the first kind. The definitions of the elliptic integrals and functions are taken from [20]. 2 nK ( m ) is the real period of the Jacobi sn function. The modulus m of the Jacobi sn function is determined by the following equation which is derived from the normalization condition and E( m ) is the complete elliptic integral of the second kind. The energy spectrum then is The limiting case γ = 0 leads to m = 0, K (0) = π 2 , E (0) = π 2 , and the Jacobi elliptic function sn in equation (25) reduces to sin ( πnx ). 1 The knowledge of these analytically given solutions is very useful as a benchmark for our algorithm. In Figs. 4(a) and 4(b) the comparison between the analytical and numerical solution for the ground state and the first mode is shown. The solid red line is the numerical result and the dots are calculated with the analytic solution. The deviations are of the order of O (10 -4 ). For large nonlinearities the numerical calculation of modulus with equation (26) becomes difficult because the number of required decimals increases fast. In Fig. 5 the first six modes are plotted, starting with the strictly positive zeroth mode. The solid black lines show the eigenfunctions for the linear case γ = 0. The other two lines show the numerical solutions of the nonlinear problem for different eigenvalues µ n . All solutions are normalized to one. The corresponding eigenvalues µ n for a given γ n can be read off from table I or from Fig. 6. The relation between γ n and µ n in Fig. 6 is proportional but not linear. The amplitudes of the wave function shown in Fig. 5 decrease with increasing γ n and the maxima and minima become more and more flat. This can be easily understood from the fact that the repulsion becomes stronger for larger γ n so that the wave functions tend to a spatial equalization. As a consequence, the gradient of the wave functions at the boundary grows and with that the kinetic energy. Note that for increasing mode numbers and at fixed nonlinearity γ the broadening effect gets smaller. This can be seen from the solutions for µ 0 = 500 at mode zero and for the 5-th mode at µ 5 = 1000 (see Fig. 5).", "pages": [ 9, 10 ] }, { "title": "B. Gravitational Trap", "content": "Now we solve the GPE with a gravitational potential. With the potential we have the natural length scale L = ( /planckover2pi1 2 / 2 m 2 g ) 1 / 3 so that the dimensionless GPE reads In the linear case with γ = 0 the eigenfunctions are well known [7]. The general solution is a linear combination of the AiryAi and AiryBi functions where the AiryBi is omitted since it is not compatible with the boundary conditions Ψ n (0) = 0 and Ψ n ( ∞ ) = 0. The n -th eigenfunction is given by where x n is the n -th zero of the AiryAi function and of course the orthogonality relation 〈 Ψ n ( x ) , Ψ m ( x ) 〉 = δ nm holds. All zeros are negative so that the normalizable part of the general solution is shifted to the right. The n -th eigenvalue ε n is also given by the n -th zero. Unfortunately no analytic expression for the normalization factor A n exists. Hence, it is given by In the nonlinear case it is much more complicated to find solutions because equation (29) admits a huge number of not normalizable solutions. Most of them have poles on the real axis. For ε n = 0 and γ = 2 problem (29) is known as the Painlev'e II equation (PII). This equation possesses the Painlev'e property [21] which is a condition of integrability. These differential equations cannot be integrated by means of elementary functions. A possibility of finding solutions to PII is to solve the corresponding Riemann-Hilbert problem numerically [22]. Using this method many different solutions can be found, including nonphysical ones [28]. In Fig. 7 the first eight solutions of the GPE with linear potential are calculated for different values of µ n with our new method. The solid black lines depict the AiryAi solutions for the linear case. The corresponding nonlinearity factors can be found in table II. The difference here is that the mathematical domain is not finite and that for x →∞ the potential is diverging. However, in the numerical implementation the domain has finite size, i.e. of length L . On contrary to the standard treatment of boundary conditions it is only necessary to specify the value at Ψ(0) due to the diverging nature of the potential the value at Ψ( L ) adjusts itself automatically. We do not pose any boundary conditions explicitly for Ψ( L ) so that the finite size domain has no effect on the solutions. The first observation is that for a given mode number n > 0 higher nonlinearities cause a quenching of the region between the boundary and the outermost maximum in comparison with the linear case. This can be understood by taking into account that V ( x = 0) = ∞ limits the space on the left side for encountering particles. As a result, they move towards the outermost maximum causing a depletion of the particle number on the left. The second observation which seems to be surprising is that the bulk of the wave function for different modes appears not to be changing for fixed µ n . However the explanation is simple: with increasing modes the nonlinearity parameter γ n decreases at fixed µ n . This behaviour can be clearly seen in Fig. 8. Higher nonlinearities always enlarge the bulk of the solution. Fig γ n ( µ n ) in Fig. 8 shows the one to one correspondence between γ and µ .", "pages": [ 11, 12 ] }, { "title": "C. Harmonic Trap", "content": "As a last example we discuss the BEC in a harmonic potential given by With the natural length scale L = √ /planckover2pi1 /mω the dimensionless equation reads In the linear case γ = 0 the solutions are where are the weighted Hermite polynomes. /negationslash As far as we know there are no analytic solutions known for this potential for γ = 0. Numerical simulations basically focus on the zeroth mode [23, 24]. For the zeroth mode there is a rough approximation which can be obtained by neglecting the kinetic energy term in equation (33). Then we have an algebraic equation which can easily be solved for Ψ 0 ( x ) and is known as the Thomas-Fermi solution In Fig. 10 we compared the Thomas-Fermi solution with the numerical solution of the ground state. For large nonlinearities the Thomas-Fermi approximation agrees very well with the numerical results in the center region of the condensate. In Fig. 11 the first eight numerical solutions for the harmonic oscillator potential are depicted for different eigenvalues µ n calculated with our new method. The corresponding nonlinearities γ n can be found in table (III). The solid black lines correspond to the linear case with the weighted Hermite polynomial. In the numerical implementation there are no boundary conditions specified. The diverging nature of the potential forces the wave function to decay for x →∞ . The curves in Fig. 11 show that with increasing nonlinearity particles from the center region are pushed towards the outer region whereas the inner structures are squeezed. The harmonic potential has a much higher confinement so that the bulk remains relativity small compared to the gravitational potential.", "pages": [ 12, 14 ] }, { "title": "V. DISCUSSION AND OUTLOOK", "content": "In this article we presented a new algorithm that is capable to find higher Morse index solutions of the stationary GPE for large nonlinearities. Mathematically speaking, these are saddle point solutions. The three crucial points are (i) to start with a fixed eigenvalue µ , (ii) the reference point that contains only a function of the same Morse index in the support and (iii) the preconditioning of the L 2 gradient by using the analytic expression for the Hessian. Furthermore we demonstrated that we can find solutions for the GPE with large nonlinearity parameter in external potentials by starting only with the eigenfunction for the n-th mode of the corresponding linear problem. In summary we calculated the eigenfunctions and energies for the one dimensional GPE for three classical potentials: the box, the harmonic trap and the GOST. In the case of the GOST we obtained higher order modes up to order seven for large nonlinearity parameters in the range of γ = 336 -442. To our present knowledge this seems to be the first time that solutions to the GOST setup for such high nonlinearities and high modes have been calculated. Furthermore, we showed that in the case of the box the numerically found solutions completely agree with known analytical solutions which confirms our algorithm. Also, there is a good agreement between the numerical zero mode solutions for the trap and the Thomas-Fermi approximation in the central region of the BEC. The next logical step is to apply this algorithm in 2 D and 3 D setups of the aforementioned three cases so that more realistic physical systems will be modelled. Moreover, the physical stability may be checked by propagating the solutions in time. Another issue which may be treated in future is to include self-gravity effects. At first, for an efective equation as the GPE self-gravity should be considered. For very dilute gases one may expect no effects but for high density BECs corresponding effects should be estimated. Self gravity also is an idea stated by Penrose [25] to understand the collapse of the wave function. Therefore it might be of interest to investigate whether in this context such effects might be accessible to experiment. We also plan to adopt our method to the case of coupled many component GPEs. Finally, concerning the algorithm, in the future it may be interesting to find a new stepsize control that incorporates the ascent direction. This may improve convergence behaviour. For spatial dimensions larger than 1 we also would like to extend our algorithm in order to incorporate also different coordinate systems which are more adapted to the physical problem.", "pages": [ 15, 17 ] }, { "title": "Acknowledgement", "content": "We would like to thank S. Herrmann and V. Perlick for many fruitful discussions. E.G. and ˇ Z.M. gratefully acknowledge financial support from DLR, project number 50WM-0942 C.L. also thanks the cluster of excellence QUEST for support.", "pages": [ 17 ] }, { "title": "Appendix A: Search direction", "content": "For the action A we can write down the Taylor expansion around φ k up to first order: Using the Newton step (12) and the search direction (21) equation (A1) can be written as: If A [ φ k +1 ; µ s ] < A [ φ k ; µ s ] and 〈∇ L 2 A [ φ k ; µ s ] , d k 〉 > 0 then -d k is a descent direction. If A [ φ k +1 ; µ s ] > A [ φ k ; µ s ] and 〈∇ L 2 A [ φ k ; µ s ] , d k 〉 < 0 then -d k is a ascent direction. [10] R. Bucker, T. Berrada, S. van Frank, J.-F. Schaff, T. Schumm, and J. Schmiedmayer, Vibrational state inversion of a Bose-Einstein condensate: optimal control and state tomography , http://arxiv.org/abs/1212.4173 (2012), 1212.4173. [15] Y. Choi and P. McKenna, Nonlinear Analysis: Theory, Methods and Applications 20 , 417 (1993), ISSN 0362-546X.", "pages": [ 17 ] } ]

[ { "title": "Optimization of the Neutrino Oscillation Parameters using Differential Evolution", "content": "Ghulam Mustafa ∗ , Faisal Akram † , Bilal Masud ‡ Centre for High Energy Physics, University of the Punjab, Lahore(54590), Pakistan.", "pages": [ 1 ] }, { "title": "Abstract", "content": "We combine Differential Evolution, a new technique, with the traditional grid based method for optimization of solar neutrino oscillation parameters ∆ m 2 and tan 2 θ for the case of two neutrinos. The Differential Evolution is a population based stochastic algorithm for optimization of real valued non-linear non-differentiable objective functions that has become very popular during the last decade. We calculate well known chi-square ( χ 2 ) function for neutrino oscillations for a grid of the parameters using total event rates of chlorine (Homestake), Gallax+GNO, SAGE, Superkamiokande and SNO detectors and theoretically calculated event rates. We find minimum χ 2 values in different regions of the parameter space. We explore regions around these minima using Differential Evolution for the fine tuning of the parameters allowing even those values of the parameters which do not lie on any grid. We note as much as 4 times decrease in χ 2 value in the SMA region and even better goodness-of-fit as compared to our grid-based results. All this indicates a way out of the impasse faced due to CPU limitations of the larger grid method.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The flux of solar neutrino was first measured by Raymond Davis Junior and John N. Bahcall at Homestake in late 1960s and a deficit was detected between theory (Standard Solar Model) and experiment [1]. This deficit is known as the Solar Neutrino Problem . Several theoretical explanations have been given to explain this deficit. One of these is neutrino oscillations, the change of electron neutrinos to an other neutrino flavour during their travel from a source point in the sun to the detector at the earth surface [2]. There was no experimental proof for the neutrino oscillations until 2002 when Sudbury Neutrino Observatory (SNO) provided strong evidence for neutrino oscillations [3]. The exact amount of depletion, which may be caused by the neutrino oscillations, however, depends upon the neutrino's mass-squared difference ∆ m 2 ≡ m 2 2 -m 2 1 ( m 1 and m 2 being mass eigen-states of two neutrinos) and mixing angle θ , which defines the relation between flavor eigen-states and mass eigen-states of the neutrinos, in the interval [0 , π/ 2]. The data from different neutrino experiments have provided the base to explore the field of neutrino physics. In the global analysis of solar neutrino data, we calculate theoretically expected event rates with oscillations at different detector locations and combine it with experimental event rates statistically through the chi-square ( χ 2 ) function, as defined below by Eq.(1), for a grid of values of the parameters ∆m 2 and tan 2 θ . The values of these parameters with minimum chi-square in different regions of the parameter space suggest different oscillation solutions. The names of these solutions, found in the literature, along with specification of the regions in the parameter space are: Small Mixing Angle (SMA: 10 -4 ≤ tan 2 θ ≤ 3 × 10 -2 , 3 × 10 -7 eV 2 ≤ ∆m 2 ≤ 10 -4 eV 2 ), Large Mixing Angle (LMA: 3 × 10 -2 ≤ tan 2 θ ≤ 2 , 2 × 10 -6 eV 2 ≤ ∆m 2 ≤ 10 -3 eV 2 ), Low Probability Low Mass (LOW: 3 × 10 -2 ≤ tan 2 θ ≤ 2 , 10 -8 eV 2 ≤ ∆m 2 ≤ 2 × 10 -6 eV 2 ) and Vacuum Oscillation (VO: 0 . 1 ≤ tan 2 θ ≤ 1 , 10 -11 eV 2 ≤ ∆m 2 ≤ 10 -8 eV 2 ) [4]. Extensive work has been done on the global analysis of solar neutrino data [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] and now is the era of precision measurement of the neutrino oscillation parameters [16, 17]. Traditionally, the whole parameter space (10 -4 ≤ tan 2 θ ≤ 10, 10 -13 eV 2 ≤ ∆m 2 ≤ 10 -3 eV 2 ) is divided into a grid of points by assigning a variable to each parameter and varying its logarithm uniformly. The chi-square values are calculated for each point in the parameter space either by using 8 B flux constrained by the Standard Solar Model, e.g., BS05(OP) [18] in our case, or by using unconstrained 8 B flux [9] where it is varied about the value predicted by the Standard Solar Model. The global minimum chi-square value χ 2 min is found and 100 β % C.L. (Confidence Level) contours are drawn in the tan 2 θ -∆m 2 plane by joining points with χ 2 = χ 2 min +∆ χ 2 for different confidence levels. From the chi-square distribution one can easily find that ∆ χ 2 = 2 . 28 , 4 . 61 , 5 . 99 , 9 . 21 , 11 . 83 for 68%, 90%, 95%, 99% and 99.73% C.L. for two degrees of freedom. Minimum chi-square values are found in all the regions and the goodness-of-fit, corresponding to each of the minimum chi-square, is calculated. To find the each goodness-of-fit the chi-square distribution is used and confidence level 100(1 -β )%, corresponding to the minimum chi-square in the region and the degree of freedom of the analysis, is calculated [4, 9]. In our analysis we used total event rates of chlorine (Homestake), Gallax+GNO, SAGE, Superkamiokande, SNO CC and NC experiments. So the number of degrees of freedom was 4 (6(rates)-2(parameters: tan 2 θ and ∆m 2 )). When we use the Differential Evolution (DE), the parameters are randomly selected in the given range and checked for a decrease of chisquare, in contrast with the traditional grid based method as described in the above paragraph. Thus we selected the vectors with least chi-square values, in different regions of the selected grid, as starting points and used DE for the fine tuning of the parameters by exploring region around the selected vectors in the parameter space. Here in section 2, we define the chi-square ( χ 2 ) function for the solar neutrino oscillations. We use the same χ 2 function definition in the algorithm of DE as well as in the traditional method. In section 3, we describe algorithm of Differential Evolution along with its salient features. In section 4 and 5, we describe results of global analysis by grid and those obtained using Differential Evolution respectively. Our conclusions are given in section 6.", "pages": [ 1, 2 ] }, { "title": "2 Chi-square ( χ 2 ) Function Definition", "content": "In our χ 2 analysis, we used the updated data of total event rates of different solar neutrino experiments. We followed the χ 2 definition of ref. [19] and included chlorine (Homestake) [20], weighted average of Gallax and GNO [21], SAGE [22], Superkamiokande [23], SNO CC and SNO NC [24] total rates. The expression for the χ 2 is given as: where R th j is the theoretically calculated event rate with oscillations at detector j and R exp j is the measured rate. For chlorine, Gallax+GNO and SAGE experiments R th and R exp are in the units of SNU (1 SNU=10 -36 captures/atom/sec) and for Superkamiokande, SNO CC and SNO NC these are used as ratio to SSM Eq.(8) below. V j 1 j 2 is the error matrix that contains experimental (systematic and statistical) errors and theoretical uncertainties that affect solar neutrino fluxes and interaction cross sections. For the calculation of the error matrix V j 1 j 2 we followed ref. [19] and for updated uncertainties we used ref. [25]. For the calculation of theoretical event rates, using Eqs.(4-7) below, we first found the time average survival probabilities, over the whole year, of electron neutrino 〈 P k ee ( E ν ) 〉 ( E ν is the neutrino energy in MeV) at the detector locations for the k th neutrino source and for the grid of 101 × 101 values of ∆m 2 E and tan 2 θ following the prescriptions described in ref. [9]. For the uniform grid interval distribution we used the parameters ∆m 2 E and tan 2 θ as exponential functions of the variables x 1 and x 2 as: and so that discrete values of x 1 and x 2 from 0 to 100 cover the entire tan 2 θ -∆m 2 parameter space. We used the expression for the average expected event rate in the presence of oscillation in case of Chlorine and Gallium detectors given as: Here E j th is the process threshold for the j th detector ( j =1,2,3 for Homestake, Gallax+GNO and SAGE respectively). The values of energy threshold E j th for Cl, Ga detectors are 0.814, 0.233 MeV respectively [26]. φ k are the total neutrino fluxes taken from BS05(OP) [18]. For Gallium detector all fluxes contribute whereas for Chlorine detector all fluxes except pp flux contribute. λ k ( E ν ) are normalized solar neutrino energy spectra for different neutrino sources from the sun, taken from refs. [27, 28], and σ e,j is the interaction cross section for ν e in the j th detector. Numerical data of energy dependent neutrino cross sections for chlorine and gallium experiments is available from ref. [27]. Event rates of Chlorine [20] and Gallium [21, 22] experiments and those calculated from Eq.(4) directly come in the units of SNU. Superkamiokande and SNO detectors are sensitive for higher energies, so φ k are the total 8 B and hep fluxes for these detectors respectively. The expression of the average expected event rate with oscillations for elastic scattering at SK detector is as below: Here σ e and σ µ are elastic scattering cross sections for electron and muon neutrinos that we took from ref. [29]. For the SNO charged-current (CC) reaction, ν e d → e -pp , we calculated event rate using the expression: Here σ CC is νd CC cross section of which calculational method and updated numerical results are given in refs. [30] and [31] respectively. The expression for the SNO neutral-current (NC) reaction, ν x d → ν x p n ( x = e, µ, τ ), event rate is given as: Here σ NC is νd NC cross section and 〈 P k ea ( E ν ) 〉 is the time average probability of oscillation into any other active neutrino. We used updated version of CC and NC cross section data from the website given in ref. [31]. In case of oscillation of the ν e into active neutrino only, 〈 P k ee ( E ν ) 〉 + 〈 P k ea ( E ν ) 〉 = 1 and N th NC is a constant. For Superkamiokande [23] and SNO [24] experiments, the event rates come in the unit of 10 6 cm -2 s -1 . We converted these rates into ratios to SSM predicted rate. We also calculated theoretical event rates as ratios to SSM predicted rate in order to cancel out all energy independent efficiencies and normalizations [8]. Here N SSM j ( j =4,5,6 for SK, SNO CC and SNO NC respectively) is the predicted number of events assuming no oscillations. We used the Standard Solar Model BS05(OP) [18] in our calculations. Theoretical event rates, so calculated, were used in Eq.(1) to calculate the chisquare function for different points in the tan 2 θ -∆m 2 parameter space.", "pages": [ 2, 3, 4 ] }, { "title": "3 Differential Evolution", "content": "Differential Evolution (DE) is a simple population based, stochastic direct search method for optimization of real valued, non-linear, non-differentiable objective functions. It was first introduced by Storn and Price in 1997 [32]. Differential Evolution proved itself to be the fastest evolutionary algorithm when participated in First International IEEE Competition on Evolutionary Optimization [33]. DE performed better when compared to other optimization methods like Annealed Nelder and Mead strategy [34], Adaptive Simulated Annealing [35], Genetic Algorithms [36] and Evolution Strategies [37] with regard to number of function evaluations (nfe) required to find the global minima. DE algorithm is easy to use, robust and gives consistent convergence to the global minimum in consecutive independent trials [32, 38]. The general algorithm of DE [39] for minimizing an objective function carries out a number of steps. Here we summarize the steps we carried out for minimizing the χ 2 function defined in section 2. We did optimization of the χ 2 function individually for different regions of the parameter space to do one fine tuning in each region. The results of the optimization are reported in the section 5 below.", "pages": [ 4 ] }, { "title": "Step I", "content": "An array of vectors was initialized to define a population of size NP =20 with D =2 parameters as The parameters, involved here, are x 1 and x 2 of Eqs.(2) and (3) on which ∆m 2 / E and tan 2 θ depend. Upper and lower bounds ( b j,U and b j,L ), individually for different regions of the parameter space described in the introduction section, for the x values were specified and each vector i was assigned a value according to where rand j ∈ [0 , 1] is j th evaluation of a uniform random number generator. The χ 2 function was calculated for each vector of the population and the vector with least χ 2 function value was selected as base vector x r o .", "pages": [ 4 ] }, { "title": "Step II", "content": "Weighted difference of two randomly selected vectors from the population was added to the base vector x r o to produce a mutant vector population v i of NP trial vectors. The process is known as mutation . Here the scale factor F ∈ [0 , 2] is a real number that controls the amplification of the differential variation. The indices r 1 , r 2 ∈ [1 , NP ] are randomly chosen integers and are different from r o . Different variants of DE mutation are denoted by the notation ' DE/x/y/z ', where x specifies the vector to be mutated which can be 'rand' (a randomly chosen vector) or 'best' (the vector of the lowest χ 2 from the current population), y is the number of difference vectors used and z is the crossover scheme. The above mentioned variant Eq.(11) is DE/best/1/bin , where the best member of the current population is perturbed with y =1 and the scheme bin indicates that the crossover is controlled by a series of independent binomial experiments. The two variants, reported in the literature [32, 38], very useful for their good convergence properties, are DE/rand/1/bin and DE/best/2/bin For our problem, we used the variant DE/best/2/bin Eq.(13) for DE mutation, where 2 difference vectors were added to the base vector. The values of F we used are reported in section 5 below.", "pages": [ 4, 5 ] }, { "title": "Step III", "content": "The parameters of mutant vector population Eq.(13) were mixed with the parameters of target vectors Eq.(9) in a process called uniform crossover or discrete recombination. After the cross over the trial vector became: Here Cr ∈ [0 , 1] is the cross over probability that controls fraction of the parameters inherited from the mutant population (the values of Cr we used are given in section 5), rand j ∈ [0 , 1] is the output of a random number generator and j rand ∈ [1 , 2] is a randomly chosen index.", "pages": [ 5 ] }, { "title": "Step IV", "content": "The χ 2 function was evaluated for each of the trial vector u i obtained from Eq.(14). If the trial vector resulted in lower objective function than that of the target vector x i , it replaced the target vector in the following generation. Otherwise the target vector was retained. (This operation is called selection .) Thus the target vector for the next generation became: The processes of mutation, crossover and selection were repeated until the optimum was achieved or the number of iterations (generations) specified in section 5 were completed.", "pages": [ 5 ] }, { "title": "4 Analysis from the Selected Grid", "content": "Figure 1 and Table 1 show our best fit oscillation parameters, in different regions, calculated using a grid of 101 × 101 points of the parameter space. The symbol of star shows the best fit points in the respective regions of the parameter space. Calculations of goodness-of-fit and confidence level are described in the introduction section. We used chi-square function definition of section 2. We used 8 B flux constrained by the Standard Solar Model BS05(OP). We saw that the point with global minimum or the best fit point in the parameter space lies in the LMA region with ∆m 2 = 2 . 512 · 10 -5 eV 2 and tan 2 θ = 3 . 981 · 10 -1 that is consistent with the results found in the literature where SNO data is included in the analysis [9, 10, 12]. Before including the SNO data the best fit was found in the SMA region [26]. A selection of a fine grid with larger number of points in the parameter space, of course, will give better results but limitations of the CPU time restricted us, like others, to a grid with a small number of points. But we point out that even without increasing the number of points in the grid we can get lower χ 2 and better g.o.f. by fine tuning of the oscillation parameters using DE. We describe what we mean by fine tuning and report our improvements obtained this way in the next section.", "pages": [ 5, 6 ] }, { "title": "5 Optimization of the Chi-square Function using DE", "content": "We have described algorithm of the Differential Evolution in detail in section 3. We wrote the subroutine of the chi-square function, denoted by χ 2 , following the definition of chi-square in section 2, that depends on x 1 and x 2 and used it as objective function of the DE algorithm. We combined the traditional grid-based method with DE in two aspects: First, we used the survival probabilities 〈 P k ee ( E ν ) 〉 already calculated for the discrete values of x 1 and x 2 for our grid of 101 × 101 points of the parameter space and interpolated them to the continuous values of x 1 and x 2 to calculate event rates and chi-square function in DE algorithm. We used cubic polynomial fit for the interpolation purpose to fit the data. Second, we used the points with minimum chi-square in different regions of the selected grid Table 1 as the starting points (and members of the respective population array) and explored the space around them for the fine tuning . That is, we searched for the points with smaller χ 2 values and better goodness-of-fit of the oscillation parameters. In our analysis, the values of DE control variables F and CR were taken as 0.4 and 0.9 respectively for the LMA, SMA and VAC regions. For the LOW region F and CR were both taken as 0.3 for better convergence. Maximum number of iterations were taken to be 50 for all regions. We took the best point in a region of the 101 × 101 grid in Table 1 as the first member of the population in the first iteration and used the strategy DE/best/2/bin for DE mutation in all the remaining iterations/generations. The steps of DE algorithm, described in section 3, are repeated for the number of iterations specified. Table 2 and Figure 2 show the results in different regions during and after fine tuning of the oscillation parameters using Differential Evolution. The value of χ 2 min persisted, rejecting all the mutations, for the iterations mentioned in column 2 of Table 2. Accepted mutations resulted in new vectors whose components are given in column 3 and 4 of the following rows. χ 2 best is the minimum chi-square value we obtained in the region specified. In comparison to the results of Table 1, we note here as much as 4 times decrease in the χ 2 min of the SMA region after fine tuning using DE along with a small decrease in all the other regions. Different vectors in Figure 2 show the track of DE algorithm for optima in different regions during iterations specified in Table 2.", "pages": [ 7, 8 ] }, { "title": "6 Conclusions", "content": "Fine tuning of the neutrino oscillation parameters using Differential Evolution has been introduced as a solution to the impasse faced due to CPU limitations of the larger grid alternative. We can explore the parameter space deeply due to real nature of the parameters x 1 and x 2 using DE in contrast to discrete nature of these parameters in the traditional grid based method. We conclude that combination of Differential Evolution along with traditional method provides smaller chi-square values and better goodness-of-fit of the neutrino oscillation parameters in different regions of the parameter space. We also note a significant change in the results of χ 2 min and g.o.f. in the SMA region after the fine tuning using DE. Even though it is a local decrease, it indicates importance of the exploration of the points within the grid and the efficiency that can be achieved through DE.", "pages": [ 9 ] }, { "title": "Acknowledgements", "content": "We are thankful to the Higher Education Commission (HEC) of Pakistan for its financial support through Grant No.17-5-2(Ps2-044) HEC/Sch/2004.", "pages": [ 9 ] } ]

[ { "title": "Thermodynamics of the apparent horizon in massive cosmology", "content": "Department of Physics, Yantai University, 30 Qingquan Road, Yantai 264005, Shandong Province, P.R.China", "pages": [ 1 ] }, { "title": "Yi Zhang †", "content": "College of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing 400065, P.R.China Applying Clausius relation with energy-supply defined by the unified first law of thermodynamics formalism to the apparent horizon of a massive cosmological model proposed lately, the corrected entropic formula of the apparent horizon is obtained with the help of the modified Friedmann equations. This entropy-area relation, together with the identified internal energy, verifies the first law of thermodynamics for the apparent horizon with a volume change term for consistency. On the other hand, by means of the corrected entropy-area formula and the Clausius relation δQ = TdS , the modified Friedmann equations governing the dynamical evolution of the universe are reproduced with the known energy density and pressure of massive graviton. The integration constant is found to correspond to a cosmological term which could be absorbed into the energy density of matter. Having established the correspondence of massive cosmology with the unified first law of thermodynamics on the apparent horizon, the validity of the generalized second law of thermodynamics is also discussed by assuming the thermal equilibrium between the apparent horizon and the matter field bounded by the apparent horizon. It is found that, in the limit H c → 0 which recovers the Minkowski reference metric solution in the flat case, the generalized second law of thermodynamics holds if α 3 +4 α 4 < 0. Apart from that, even for the simplest model of dRGT massive cosmology with α 3 = α 4 = 0, the generalized second law of thermodynamics could be violated. PACS numbers: 98.80.-k 95.36.+x 11.10.Lm", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "SNIa observations support a present accelerating universe[1]. With regard to general relativity(GR), a hypothetic dark energy component is necessary to meet the remarkable observations[2]. Cosmological constant is a simplest resolution in the framework of classical field theory; however, the surprisingly small value of the cosmological constant seems unnatural in light of quantum gravity, which is believed to take over the UV region of quantum fluctuations, remove the singularity problem and unify general relativity and quantum field theory at short distance. That means an infrared peculiarity to some extent is entangled with the UV divergence and the IR region should also be modified. Most of dark energy models have reasonable motivations and observational expectations; in the meantime, due to that cosmological constant problem[3] and moreover the so-called cosmological coincidence problem, they only acquire limited success and are still far from satisfactory. A second approach to understand the acceleration phenomenon relies on the modified gravity theories, such as theories of extra dimensions such as DGP models[4] and massive gravity. Different from theories of extra dimensions where gravitons acquire mass through dimensional reduction to four dimensions, a tiny mass is endowed to the graviton simply by hand[5]. Interestingly enough, this deformation of general relativity can effectively give rise to a small cosmological constant term within, for instance, the simplest bimetric models of massive gravity[6]. It turns out that the graviton mass not only reproduces a cosmological term, but at the same time can manifests itself as other types of matter content with different equations of state[7]. In the linear model of massive gravity with Fierz-Pauli mass, the longitudinal graviton maintains a finite coupling to the trace of the source stress tensor even in the massless limit. This incurs the problem of vDVZ discontinuity[8] which means the Fierz-Pauli model can not reduce to GR in the massless limit m → 0 and therefore directly contradicts experiments on the solar system. By way of the Vainshtein mechanism[9] in the classical framework, the neglected non-linearity may be strong and the nature of non-linear instability helps to restore continuity with GR below the Vainshtein radius. The Lagrangian for the helicity-0 component generically contains nonlinear terms with more than two time derivatives; the latter give rise to the sixth degree of freedom on local backgrounds[10]. The presence of the Boulware-Deser (BD) ghost notoriously hinders us from constructing a healthy theory of Lorentz invariant massive gravity which recovers GR. Recently, de Rham, Gabadadze and Tolley (dRGT) have successfully constructed a non-linear model[11] of massive gravity which is ghost-free in the decoupling limit to all orders and furthermore at the complete non-linear level[12]. Therefore, dRGT gravity is well under investigations theoretically and observationally[13] as well. To inspect a gravitational theory thermodynamical analysis has becoming a powerful tool. As pivotal events, blackhole thermodynamics[14] and recent AdS/CFT correspondence[15] show explicit significance and strongly suggest the deep connection between gravity and thermodynamics. A recent landmark of the identification of gravity theories and thermodynamics is the seminal work of Jacobson where the inverse problem of reproducing gravity theories from thermodynamical systems was seriously dealt with and successfully realized[16]. By assuming the Clausius relation δQ = TdS holds for all local Rindler causal horizons through each space-time point, Einstein field equations are deduced with the well-known entropy formula S = A/ (4 G ). The variation of heat flow δQ is measured by an accelerated observer just inside the horizon and correspondingly T denotes its Unruh temperature. Although the formulas were deduced in the null directions, it is suggested that the results may also be applied to all other directions in the tangential of the space-time. More recently, Eling and his collaborators discussed corresponding thermodynamical implications of f ( R ) theories by means of similar method[17]. To reproduce the correct equations of motion of f ( R ) gravity, an entropic generating term should be added to the Clausius relation δQ = TdS as well as the substitution of S = αf ' ( R ) A to the entropy formula S = A/ (4 G ). It infers that f ( R ) gravity is a non-equilibrium thermodynamics in essence. (See [18] for a different viewpoint.) Along with this direction, various gravity theories have been checked and it is found that scalar-tensor theory of gravity also corresponds to non-equilibrium thermodynamics and an appropriate entropy production term is needed to derive the dynamic equations of motion space-times[19]. This theoretical complication entails examining the correspondence for different contexts besides for different gravity theories. For specific space-times in various gravitational theories, different strategies have been developed in the past few years. In the case of Einstein-Hilbert gravity, Einstein equations for a spherically symmetric space-time can be interpreted as the thermodynamic identity dE = TdS -PdV [20] with S and E being the entropy and energy derived by other approaches. What's more, the field equations for Lanczos-Lovelock action in a spherically symmetric space-time can also be expressed as the above form. As the modified terms could emerge in quantum pictures, it is remarkable to find thermodynamics can profile gravity beyond the classical level in this way. Another progress more relevant to our present work is the method of Hayward for dynamical blackholes[21, 22]. In dealing with thermodynamics of a dynamical black hole in 4-dimensional Einstein theory, the associated trapping horizon is introduced for spherically symmetric space-times. In this formalism, Einstein field equations can be recast into the so-called unified first law and the first law of thermodynamics for the dynamical black hole is thus obtained by projecting the unified first law along a vector tangent to the trapping horizon[23]. The change of local Rindler horizon to a topologically different trapping horizon which is globally geometric seems crucial for the thermodynamical reformulation of non-stationary space-times in various gravity theories[24]. Our universe is also a non-stationary gravitational system which should be cautiously handled while carrying out thermodynamical analysis. In cosmological settings, the homogeneous and isotropic Friedmann-Robertson-Walker (FRW) metric is often assumed and the expanding 3-space is characterized by the cosmic scale factor which evolves with time. At a first glance, it appears that the FRW universe as one kind of dynamical spherically symmetric space-times can be easily dealt with by the method of unified first law. The subtlety occurs when we notice that, in the FRW universe, the (out) trapping horizon is absent. But fortunately, an inner trapping horizon still exists in cosmology. In the context of FRW metric, this horizon coincides with the apparent horizon and therefore, the apparent horizon is a natural choice in the foundation of thermodynamics. It stimulates a series of work on the foundation and discussion of the associated gravitational thermodynamics on this apparent horizon. In cosmology, apart from the apparent horizon, there exist many other special surfaces which are the Hubble horizon, the particle horizon, and the cosmological event horizon, etc.. And in certain cases they could coincide with one another. Therefore, it is interesting but difficult to know which one is appropriate for the formulation of the first law of thermodynamics. Due to a radical speculation that any surface in any space-time should have an entropy related to its area concurs with the entanglement entropy approach to dynamical blackholes[21], it is believed that the question deserves deep investigations; however, it is not the point we discuss below and we will not step further on this in the present work. When focusing on the apparent horizon and the associated thermodynamics, in the setting of FRW universe, some authors investigated the relation between the first law and the Friedmann equations describing the dynamic evolution of the universe[25]. By applying the fundamental relation δQ = TdS to the apparent horizon of the FRW universe, Cai and Kim derived the Friedmann equations[26] with arbitrary spatial curvature. The Friedmann equations for the dynamical spherically symmetric space-times were also derived in the Gauss-Bonnet gravity and more general Lovelock gravity, where the actions of gravity theories are beyond Einstein theory with only a linear term of scalar curvature. When using the cosmological event horizon other than the apparent horizon in the calculation, the Friedmann equations describing the dynamics of the universe could only be obtained for the flat universe with k = 0 FRW metric where the cosmological event horizon coincides with the apparent horizon (see also Ref.[27]). In Ref. [28], the derivation of the corresponding Friedmann equations by way of the first law of thermodynamics with a volume change term on the apparent horizon was also implemented for Einstein gravity, Gauss-Bonnet gravity and Lovelock gravity. For the scalartensor gravity and f ( R ) gravity, the possibility to derive the corresponding Friedmann equations in those theories was investigated in [19, 29]. In study of having established the first law of thermodynamics, it is usually propelled to test the validity of the second law of thermodynamics. In the accelerating universe, for instance, the dominant energy condition may be violated and the second law of thermodynamics ˙ S h > 0 does not hold any more. It is at this point a tentative version of the generalized second law of thermodynamics is proposed. The key idea is to assume that the thermal system bounded by the apparent horizon remains equilibrious and the temperature of the whole system is uniform; then the total entropy of the apparent horizon and the entropy of the matter fields inside the apparent horizon can be calculated with the well-founded settings of the first law of thermodynamics. The generalized second law of thermodynamics is often examined in this sense for the accelerating phase, viscous fluid and other exotic matter dominating[31] universe and extended gravity theories such as Gauss-Bonnet gravity, Lovelock gravity[32], scalar-tensor theories[33], f ( R ) theories[34], f(T) gravity[35], Horava-Lifshitz cosmology[36], modified f(R) Horava-Lifshitz gravity[37], GaussBonnet braneworld[38], warped DGP braneworld[39] and loop quantum cosmology[40]. Cosmological solutions of massive gravity with self-acceleration feature have been widely studied[41] and it becomes appealing to explore the dark energy and dark matter problems in the framework of dRGT massive gravity. For the dRGT model, spatially open and flat de-Sitter solutions with an effective cosmological constant proportional to the graviton mass have been found. With certain evaluation of model parameters, the solutions with any spatial curvature also exist. Cosmological consequences have also been discussed in details; nevertheless, all those work assumes a Minkowski reference metric. Langlois and his collaborator proposed a slightly modified version of the original dRGT massive gravity in which the a priori arbitrary reference geometry is chosen to be de Sitter instead of Minkowski. Apart from the first two de-Sitter branches which were founded with the Minkowski reference metric, a third branch of self-accelerating solution has also been obtained[42] and is subsequently studied in details in the literature[43]. In this paper, we will examine the thermodynamical properties of such a cosmological model of dRGT massive gravity by the strategy elaborated in the work of Cai and Cao[19][30]. In this paper, dRGT massive cosmology with de-Sitter reference metric is introduced. Then, the Clausius relation with energy-supply defined by the unified first law of thermodynamics formalism is employed on the apparent horizon. With the help of the Friedmann equations, the corrected entropic formula of the apparent horizon is obtained. This entropy-area relation, together with the identified internal energy, verifies the first law of thermodynamics with a volume change term for consistency; secondly, by means of this corrected entropy-area formula and the Clausius relation δQ = TdS , where the temperature of the apparent horizon for energy crossing during the time interval dt is 1 / (2 π ˜ r A ) and the energy-supply of pure matter and the effective graviton energy density and pressure are expressed in terms of the Hubble parameter, the modified Friedmann equations governing the dynamical evolution of the universe are reproduced. The integration constant is found to correspond to a cosmological term which could be absorbed into the energy density of matter. Then, having established the correspondence of massive cosmology with the unified first law of thermodynamics on the apparent horizon, the validity of the generalized second law of thermodynamics is also discussed by assuming the thermal equilibrium between the apparent horizon and the matter field bounded by the apparent horizon. The temperature of the thermal system is therefore uniform and it could be appropriately handled to calculate the total entropy of the apparent horizon and the matter fields inside the apparent horizon. Finally we give the conclusion and discussions. We start with the massive cosmology of dRGT gravity applying to homogeneous and isotropic space-time. The ghost free theory of massive gravity proposed by [11] is of the form where m g is the mass of graviton, the nonlinear higher derivative terms for the massive graviton is As dRGT construction points out, no higher order polynomial terms in K would exist and thus the most general Lagrangian density has only three free parameters, m g , α 3 and α 4 . The tensor K µ ν is and Σ µν is defined by four Stuckelberg fields φ a as Usually the reference metric η ab is taken to be Minkowski. Recently, a different approach by choosing the priori arbitrary reference metric as de-Sitter instead of Minkowski has been proposed and in addition to the cosmological constant solutions, a new branch with much more sophisticated behavior has also been found. Specifically speaking, by varying the action with respect to the lapse function and scale factor, the Friedmann equations are obtained to be: where for the spatially flat k = 0 case, the effective energy density ρ g and pressure p g for the massive graviton are[43], It is interesting to note that, when H ( z ) = H c , ρ g equals zero, the energy density from massive graviton vanishes at this point.", "pages": [ 1, 2, 3, 4, 5, 6, 7 ] }, { "title": "II. FROM CLAUSIUS RELATION AND THE MODIFIED FRIEDMANN EQUATIONS TO THE CORRECTED ENTROPY-AREA RELATION", "content": "In this part, we will employ the Clausius relation with energy-supply defined by the unified first law of thermodynamics formalism on the apparent horizon ro obtain the corrected entropic formula of the apparent horizon. The modified Friedmann equations of dRGT cosmology with the spatially flat FRW metric will also be utilized. By regarding the introduction of the massive graviton as the deformation of Einstein gravity to dRGT massive gravity, it is appropriate to identify the contribution of massive graviton to be an effective energy-momentum part. Therefore, it can be reduced to the unified first law of thermodynamics of Einstein gravity and the energysupply projecting along a vector ξ tangent to the trapping horizon contains both the ordinary matter and the effective part from the massive graviton. After re-splitting the energy-supply term and presuming the heat flow of the Clausius relation to be the variation of heat flow δQ , the entropy of the apparent horizon can be obtained. As is implicitly meant in the unified first law of thermodynamics, the first law of thermodynamics for the apparent horizon still holds with a volume change term for consistency, and this point will also be checked with the resulting entropy-area relation and the identified internal energy. Choosing g µν to be n -dimensional FRW metric: where ˜ r = a ( t ) r , x 0 = t, x 1 = r , h ab = diag ( -1 , a 2 / (1 -kr 2 )) with k = -1 , 0 and 1 for open, flat and closed spatial geometry respectively. The dynamical apparent horizon is defined to be the marginally trapped surface with vanishing expansion, and can be determined by the equality h ab ∂ a ˜ r∂ b ˜ r = 0. Therefore, we can get the radius of the apparent horizon: where H = ˙ a/a is the Hubble parameter and the dot denotes the derivative with respect to cosmic time t . Differentiating the above equation with respect to the cosmic time t , it is obtained For the spatially flat k = 0 case investigated bellow, the horizon radius and its evolution equations degenerate to be of the form and That is, the apparent horizon coincides with the Hubble horizon in the spatially flat FRW case. Suppose that the energy-momentum tensor T µν of matter as well as the graviton has the form of a perfect fluid T µν = ( ρ + p ) U µ U ν + pg µν , where ρ and p are the corresponding energy density and pressure respectively. The energy conservation law is valid for the matter and graviton separately, and for the former it leads to the continuity equation where the subscript denotes the quantities of matter in the universe throughout the paper by default. Following Ref.[21], the energy -supply vector Ψ and the work density can be defined as where T ab is the projection of the (3 + 1)-dimensional energy-momentum tensor T µν in the normal direction of 2-sphere of the FRW universe. For the present case, it is easy to find By means of the geometrical quantities of the area and volume of the ( n -2)- sphere A n -2 = Ω n -2 ˜ r n -2 and V n -2 = A n -2 ˜ r/ ( n -1), the Misner-Sharp energy in n dimensions inside the apparent horizon of the FRW universe is written as [19] with ˜ r A the radius of the apparent horizon. Putting the (00)-component of the equations of motion into the unified first law form, it reads Thus, the true first law of thermodynamics of the apparent horizon is obtained by projecting the above formula along a vector ξ = ∂ t -(1 -2 /epsilon1 ) Hr∂ r with /epsilon1 = ˙ ˜ r A / (2 H ˜ r A )[23], Note that κ = -(1 -˙ ˜ r A / (2 H ˜ r A )) / ˜ r A is just the surface gravity of the apparent horizon. We will derive an entropy expression associated with the apparent horizon of an FRW universe described by the modified Friedmann equations by using the method proposed in Ref. [23]. The energy-supply vector can be split into two parts: with and The projection of the pure matter energy-supply A Ψ m on the apparent horizon supplies the heat flow δQ in the Clausius relation δQ = TdS . By using the unified first law of thermodynamics on the apparent horizon, there is From Equations Eqns. (10), (11) and (25), we obtain Assuming the temperature of the apparent horizon to be the above equation can be recast into ˜ ˜ Compared with the Clausius relation δQ = TdS , it is easy to accomplish the integration and obtain the corresponding entropy scaling which deviates from the usual S = A/ (4 G ); that is, we reach for the first time a corrected entropy-area relation in massive gravity: Note that we have introduced some new symbols of parameters for clarity and all of them are determined by the two free parameters of massive gravity: Therefore, the entropy of massive gravity does not observe the usual area law and the correction terms are all proportional to the square of the graviton mass. Once the mass of graviton approaches zero, the entropy-area relation reproduces the well-known result of Einstein gravity. Notice that, in all the terms of the entropy formula, the power exponents are positive integers which is clearly different from those of the Gauss-Bonnet gravity and the more general Lovelock gravity. For the latter cases, the blackhole entropy reads [44]: where A = n Ω n r n -1 + is the horizon area of the black hole and c i are some coefficients. Reasonably, by adopting the form of the total energy inside the apparent horizon to be E m = ρ m V , it is not difficult to verify the first law of thermodynamics for the apparent horizon with the entropy formula Eqn. ( 30), Once again, we refer to the work density W m = ( ρ m -p m ) / 2 and volume of the apparent horizon V = 4 / 3 π ˜ r 3 A .", "pages": [ 7, 8, 9, 10, 11 ] }, { "title": "III. FROM THE CORRECTED ENTROPY FORMULA TO MODIFIED FRIEDMANN EQUATIONS", "content": "In the above paragraph we have obtained the corrected entropy-area formula Eqn. (30). Let me refer to the assumptions to proceed: the heat flow δQ is the energy-supply of pure matter projecting on the vector ξ tangent to the apparent horizon and should be looked on as the amount of energy crossing the apparent horizon during the time interval dt ; the temperature of the apparent horizon for energy crossing during the same interval dt is 1 / (2 π ˜ r A ). After reckoning on the substantial form of energy density and pressure of massive graviton in spatially flat dRGT FRW cosmology with de-Sitter reference metric, the modified Friedmann equations governing the dynamical evolution of the universe will be reproduced by way of the Clausius relation δQ = TdS . Assuming the radius of apparent horizon ˜ r A constant, the amount of energy crossing the apparent horizon during the time internal dt is approximately[26] where A = 4 π ˜ r 2 A is the area of the apparent horizon. Moreover, suppose that the apparent horizon has an associated corrected entropy S obtained above and temperature T = 1 / (2 π r A ), the first law of thermodynamics of the above equation gives With the help of Eqn. (16), it leads to As was stated above, the matter density ρ m satisfies the continuity equation individually, Therefore, the Clausius relation yields Integrating this equation yields where C is the integral constant. Compared with the Friedmann Eqn. (8) of massive gravity the constant should be Clearly the integration constant corresponds to a cosmological term and could be absorbed into the energy density of matter. That fulfils the derivation of the modified Friedmann equations from the Clausius relation with the corrected entropy-area formula in massive cosmology.", "pages": [ 11, 12 ] }, { "title": "IV. THE GENERALIZED SECOND LAW OF THERMODYNAMICS", "content": "Together with previous systematic research on identifying the gravitational field equations with the first law of thermodynamics on the apparent horizon in various space-times, the calculation presented above once again indicates that the universality of the connection between gravity and thermodynamics can be enlarged to the case of massive gravity. It is of great interest to take a further step on the exploration of other thermodynamical aspects such as the tentative formulation of the thermodynamical second law in the settlings of massive cosmology. Having established the correspondence of massive cosmology with the unified first law of thermodynamics on the apparent horizon, it is not hard to compute the derivative of the entropy of the apparent horizon with respect to cosmic time. Recall the modified Friedmann equations of massive gravity Combined with Eqns.( 15) and ( 16), it is found that With respect to the continuity equation ( 17) of the matter density ρ m , Eqn. (45) gives On the other hand, the associated temperature on the apparent horizon can be expressed in the form ˜ ˜ where ˙ ˜ r A / (2 H ˜ r A ) < 1 to ensure the positivity of the temperature. Recognizing the entropy S h of the apparent horizon to be deduced through the connection between gravity and the first law of thermodynamics, we know that therefore The two Friedmann equations (43) and (44) can be recast into the following form As a result, Eqn. ( 48) bcomes The positivity of the apparent horizon temperature requires ˙ ˜ r A < 2 in the spatially flat FRW case, and then the result means, without exotic matter components violating weak energy condition, the apparent horizon entropy always increases with time and the second law of thermodynamics holds in the whole history of cosmic expansion. However, in the accelerating universe the dominant energy condition is violated and the second law of thermodynamics ˙ S h > 0 does not hold any more. It is at this point a tentative version of the generalized second law of thermodynamics is proposed. The key idea is to assume that the thermal system bounded by the apparent horizon remains equilibrious so that the temperature of the system is uniform across the boundary and then to consider the total entropy of the apparent horizon and the matter fields inside the apparent horizon. This requires that the temperature T m of the energy inside the apparent horizon should be the same as that of the apparent horizon; that is, T m = T h throughout the whole evolution of the universe. A possible difference of the two temperatures would measure the spontaneous heat flow between the horizon and the matter inside it, which will not be dealt with in the present work. The entropy of matter fields inside the apparent horizon, S m , can be obtained by the Gibbs equation where E = ρ m V is its energy and p is its pressure in the horizon and With regard to Eqn. ( 50), we have and the last two terms of the right hand side of Eqn.( 53) read Therefore, or Considering Eqn. (46), the evolution of the total entropy can be obtained: It is not hard to find that the generalized second law of thermodynamics with this setting does not always hold and its validity clearly depends on the signature of the denominator in Eqn. (58) or all the free parameters α 3 , α 4 , m g and H c . When the deforming parameter m g is vanishing, the present model naturally degenerates to the FRW cosmology in general relativity and the generalized second law of thermodynamics clearly holds. The last free parameter H c is an additional parameter which does not appear in the original dRGT massive cosmology. The de-Sitter reference metric brings about such a mass scale and in the limit H c → 0, one recovers the Minkowski reference metric solution in the flat case. In this limit, δ = α 3 + 4 α 4 should be negative for the generalized second law of thermodynamics to be valid. As for other cases, and even for the minimal massive cosmological model with α 3 = α 4 = 0, the interplay of m g and other parameter(s) makes the situation complicated and the Higuchi bound may deeply involve in the discussion since there exists an absolute minimum for the mass of a spin-2 field set by such a bound in de Sitter space-time[45]. It seems that, at present no such version of the generalized second law of thermodynamics in massive cosmology could be verified.", "pages": [ 12, 13, 14, 15 ] }, { "title": "V. CONCLUSION AND DISCUSSION", "content": "Jacobson found that the connection between gravity and thermodynamics can be materialized by identifying Einstein field equation with the Clausius relation δQ = TdS . By assuming the space-time to be spherically symmetric, the thermodynamical relation which holds pointwise can be transferred to be associated with a globally geometric horizon. While applying this program to cosmological settings, the cosmological apparent horizon is employed as well as the unified first law of thermodynamics which primarily aims at the description of dynamical blackholes. These two theoretical elements have been incorporated into a systematic formulation which is elaborated in the work of Cai et.al.. In this paper, by means of that strategy, starting from the modified Friedmann equations in dRGT massive cosmology with de-Sitter reference metric, an entropy expression associated with the apparent horizon of an FRW universe is obtained by use of the unified first law projecting on the dynamical horizon and splitting the energy-supply term into the pure matter part and the effective energy-supply part. The form of the corrected entropy-area formula is clearly different from that of general relativity and more general Lovelock gravity. With this apparent horizon entropy-area formula, the first law of thermodynamics dE = TdS + W m dV is naturally satisfied in terms of the identified total energy E and the work term in the unified first law of thermodynamics. On the other hand, applying the Clausius relation δQ = TdS to apparent horizon of a spatially flat FRW universe, and assuming that the apparent horizon has the temperature of T = 1 / (2 π ˜ r A ), the observation that the pure matter energy-supply A Ψ m (after projecting along the apparent horizon) gives the heat flow δQ in the Clausius relation directly leads to the modified Friedmann equations governing the dynamical evolution of the universe. The integration constant as well as other terms in the right hand side of the derived modified Friedmann equation reflects the fact that our present cosmological model of dRGT massive gravity exhibits a richer dynamical behavior. The effective gravitational fluid generated by the graviton mass not only contains a cosmological constant, but manifests itself as other types of matter content with different equations of state. On the footing of the first law of thermodynamics which is verified, a further step is also taken for the comprehensive understanding of the thermodynamical properties of dRGT massive cosmology. As is well-known to us, the second law of thermodynamics is not always satisfied for different fluids in various gravitational theories or in the accelerating universe. Together with the matter fields' entropy inside the apparent horizon, the generalized second law of thermodynamics was proven to hold in Gauss-Bonnet braneworld and in warped DGP braneworld and so on. By inspecting the evolution of the apparent horizon entropy deduced through the connection between gravity and the first law of thermodynamics, for massive gravity, we follow the strategy of the generalized second law and as a rudimentary calculation, we adopt their hypothesis that the thermal system remains equilibrious between the apparent horizon and the matter field inside the horizon. We found that the total entropy can decrease with time and this version of the generalized second law of thermodynamics seems invalid in some parameter space. To extract appropriate parameter evaluation scope deeply involves the complicated interplay of all the four free parameters in massive cosmology[45] and no definite results exists at present. Lately, it is found that all homogeneous and isotropic backgrounds, as well as most of known spherically-symmetric inhomogeneous solutions, have an intrinsic instability which is irrelevant to the BD ghost[46]. Whether the violation of the generalized second law of thermodynamics should be attributed to the incompleteness of the massive gravity theory or the absence of some new principles is thoroughly unclear and may be worth further investigations. Acknowledgments. H. Li is supported by National Foundation of China under grant Nos. 11205131 and 10747155. Y. Zhang is supported by the Ministry of Science and Technology of the National Natural Science Foundation of China key project under grant Nos. 11175270, 11005164, 11073005 and 10935013, CQ CSTC under grant No. 2010BB0408. Lett. B634 , 101 (2006); H. Wei and R. G. Cai, Phys. Lett. B660 , 113 (2008).", "pages": [ 15, 16, 17 ] } ]

[ { "title": "ABSTRACT", "content": "Title : will be set by the publisher Editors : will be set by the publisher EAS Publications Series, Vol. ?, 2018", "pages": [ 1 ] }, { "title": "FOUR OPEN QUESTIONS IN MASSIVE STAR EVOLUTION", "content": "Georges Meynet 1 , Patrick Eggenberger 1 , Sylvia Ekstrom 1 , Cyril Georgy 2 , Jos'e Groh 1 , Andr'e Maeder 1 , Hid'eyuki Saio 3 and Takashi Moriya 4 Abstract. We discuss four questions dealing with massive star evolution. The first one is about the origin of slowly rotating, non-evolved, nitrogen rich stars. We propose that these stars may originate from initially fast rotating stars whose surface has been braked down. The second question is about the evolutionary status of α -Cygni variables. According to their pulsation properties, these stars should be post red supergiant stars. However, some stars at least present surface abundances indicating that they should be pre red supergiant stars. How to reconcile these two contradictory requirements? The third one concerns the various supernova types which are the end point of the evolution of stars with initial masses between 18 and 30 M /circledot , i.e. the most massive stars which go through a red supergiant phase during their lifetime. Do they produce types IIP, IIL, IIn, IIb or Ib supernovae or do they end without producing any SN event? Finally, we shall discuss reasons why so few progenitors of type Ibc supernovae have yet been detected?", "pages": [ 1 ] }, { "title": "1 Puzzle 1: What is the origin of the N-rich, non-evolved, slowly rotating stars?", "content": "The authors would like to present their greetings to Sylvie Vauclair whose scientific activity has produced so wonderful achievements in the field of stellar physics. The work done by Sylvie Vauclair very well reflects the qualities that Jean Rostand (1894-1977), a french biologist, writer and to some extent philosopher, see to be in the word researcher: 'Beau mot que celui de chercheur, et si pr'ef'erable 'a celui de savant ! Il exprime la saine attitude de l'esprit devant la v'erit'e : le manque plus que l'avoir, le d'esir plus que la possession, l'app'etit plus que la sati'et'e 1 . In the recent years, many efforts have been made to study the impact of rotation on massive star evolution (see e.g. Langer 2012, Maeder & Meynet 2012, Chieffi & Limongi 2013 and references therein). One of the main effects of rotation is to transport some elements abundant in the core to the surface and inversely some other elements, abundant in the envelope, into the convective core. This rotational mixing process has many consequences for the evolutionary tracks in the HR diagram, the lifetimes, the massive star populations, the nucleosynthesis, the properties of the final stellar collapse and of the stellar remnants. The understanding of the physics of rotation is not only important for the modeling of single stars but also of close binaries. One way of checking the physics included in these models is to observe surface abundances and surface velocities of B-type stars and to see whether the observations agree with the predictions of the theoretical models (see e.g. Hunter et al. 2009; Brott et al. 2011, Przybilla et al. 2010). For making relevant comparisons, the observed sample must be composed of stars with different surface velocities but with all other characteristics being as similar as possible ( i.e. same initial mass, initial composition, age, same magnetic field if any, no close companion). When such precautions are taken, general good agreement is obtained between theory and observations (Maeder et al. 2009). There is however a small group of stars which does not appear to follow the general trend predicted by the rotating stellar models. They present three properties which are difficult to reconcile: they show 1.- low υ sin i values, where υ is the surface equatorial velocity and i , the angle between the direction along the line of sight and the rotational axis, 2.- strong nitrogen enrichment at the surface and 3.- they are non-evolved stars. This last condition implies that the enrichment should have occurred fast. The problem is then how to obtain strong N-rich stars, in a short time with a low surface velocity? Are all these stars fast rotators seen pole on? This may be the case for a small fraction of them 2 but it would be unreasonable to invoke this explanation for the whole observed sample. Another possibility is the following: the large nitrogen enrichment results from a strong internal differential rotation produced by a braking mechanism acting on the surface layers. The braking can be induced by a surface magnetic field strong enough to couple with the stellar wind. A nice example of this process is provided by the star σ Ori E, whose rotation period (1.19 days) increases of 77 milliseconds per year (Townsend et al. 2010). Very interestingly, this observed increase of the period is well reproduced by the model of magnetic wind braking proposed by Ud Doula et al. ( 2009). To know, whether such a mechanism could indeed be interesting to provide an explanation for these stars, we have computed 10 M /circledot rotating models for solar metallicity, accounting for the magnetic braking of the surface according to the recipe suggested by Ud-Doula et al. (2009; see Meynet et al. 2011). In Fig. 1, the evolution of the surface nitrogen abundance is plotted as a function of the surface equatorial velocity. Two cases are shown. In the first one (look at the continuous and dashed lines), the interior of the star is rotating differentially and the mixing is mainly driven by the shear, while in the second case (dotted lines), the interior is rotating as a solid body and the mixing is driven by meridional currents. In differentially rotating star, we note that the magnetic braking can produce main-sequence stars which are slowly rotating and nitrogen enriched. At the moment very few of these stars are observed at solar metallicity, two of them are plotted in Fig. 1 (left panel) 3 . The error bars are quite large and the comparison is therefore not very constraining. But we can see that in principle the mechanism could indeed work. It is interesting to mention that one of the star plotted (the pentagon) has a detected surface magnetic field of about 335+120 -65G (Neiner et al. , 2003). In a solid body rotating star, on the other hand, the braking mechanism damps the source of the chemical mixing (meridional circulation) before it can produce any effect. Thus, only slow rotators would be expected in that case with no nitrogen enrichment. These stars would be similar to truely initially slow rotators, except that they can present a very slight depletion of fragile elements like Boron (Meynet et al. 2011). In case the present explanation would be the correct one, it would mean that non-evolved stars with high surface N content, low υ sin i stars would be magnetic stars. Unless the magnetic field would have decayed or be no long sustained by a dynamo process due to the slowing down of the star, one would expect to observe significant surface magnetic field for these stars. Another interesting point is that, these stars should not be solid body rotating star, at least during the braking period because otherwise, no nitrogen enrichment would be expected. The fact that only a small fraction of stars belong to these group 2 stars, indicate that most of the massive stars should not have a very strong surface magnetic fields. From the left panel of Fig. 1, we see that a minimum surface magnetic field of about 500 G is needed to produce group 2 stars. In Fig. 1 (right panel), the minimum surface magnetic field for obtaining a magnetic braking is indicated for various initial mass stars (Georgy et al. 2012). We see that for stars with masses below about 30 M /circledot , already a surface magnetic field above 100 G would already be sufficient to exert some coupling. It means that if a large majority of stars in this mass range would have a surface magnetic field of the order of 100 G then most of the stars would be very slow rotators with no or with strong N-enrichments (depending on internal rotation profile) which is not observed. This does appear quite in line with the results of the recent MIMES (Magnetism in Massive Stars, see e.g. Grunhut et al. 2012) indicating that only 6-7% of OB stars show surface magnetic fields with intensities above 0.1-2.0 kG (the detectability depends on the spectral type).", "pages": [ 1, 2, 3, 4 ] }, { "title": "2 Puzzle 2: What is the origin of the pulsating properties of alpha Cygni stars?", "content": "The pulsation properties of blue supergiants may be an interesting way to discriminate the blue supergiants which are evolving for the first time from the Main-Sequence to the red supergiant phase (hereafter called BSG1) from the blue supergiants which have already evolved through a red supergiant phase (hereafter called BSG2). Actually Saio et al. (2013) have shown that BSG2 present many more excited pulsation modes than BSG1, when comparisons are made between stars of the same luminosity. The main reason for this is that BSG2 have lower actual masses since large amounts of mass have been lost by stellar winds in the red supergiant phase. This increases the L/M ratio and thus allows many more pulsation modes to be excited. In order to validate this conclusion, we need to answer at least two questions: 1) are there observed blue supergiants showing excited modes as predicted by models for BSG1 and BSG2? 2) Is there any other observational features indicating that BSG1 stars are actually on their first crossing of the HR gap and indicating that BSG2 have already evolved through a red supergiant phase? To the first questions, we can answer yes. There are indeed observed blue supergiants showing pulsations with frequencies predicted by the models for respectively the BSG1 and BSG2. Typically, the star HD 62150 presents pulsations compatible with those predicted by BSG1 models, while a star like Rigel shows modes compatible with the predicted modes of BSG2 (see references and further examples in Saio et al. 2013). Thus, the modes predicted by the models do appear quite in agreement with the observed pulsations in these cases. The answer of the second question first needs to find some criteria allowing to differentiate BSG1 from BSG2 other than the pulsation properties. We can list the following characteristics: Let us now examine what these three criteria tell us about a star like Rigel whose initial mass is about 25 M /circledot . According to its pulsation properties, this star should be a BSG2 (Saio et al. 2013). According to Przybilla et al. (2006), the N/C ratio is around 12 times the initial one quite in agreement with the value predicted for a BSG1. Using spectroscopically determined gravity (log g =1.75) and estimating the radius (78.9 R /circledot ) from interferometry and Hipparcos measurement, a mass of 13 M /circledot is obtained (see references in Saio et al. 2013). Models indicate a mass between 10 and 13 M /circledot for a 25 M /circledot along the blue loop, supporting thus the idea that Rigel is a BSG2. Typical wind velocity for red supergiant winds is 10-15 km s -1 . If a few 100 thousand years separate the red supergiant phase from the blue supergiant one, then at least part of the matter ejected at the end of the red supergiant phase would be at about 6 10 5 stellar radii from the BSG2 (taking 79 R /circledot for the radius of the BSG2). Recent interferometric observations of Rigel (Kaufer et al. 2012) show a complex circumstellar environment around Rigel at distance inferior to 1-2 solar radii. Therefore, the region investigated is much too near the star to probe mass loss episodes which might have occurred during a previous red supergiant phase. Thus this characteristic cannot be used to infer the past history of Rigel. To conclude, in case of Rigel, two criteria, the one using pulsation and the one concerning the actual mass of the star support the idea that this star is a BSG2, while surface abundances favour a BSG1 solution. At the moment, no obvious solution can be proposed to level off the apparent contradiction between the abundance and mass and pulsation properties of Rigel. If Rigel was initially a truly 25 M /circledot , how can the surface abundances not be changed after losing mass down to 13 M /circledot ? In Fig. 2, we can see that the loss of only 4 M /circledot would be sufficient to increase significantly the N/C ratio well above the observed enhancement factor of 12. At face, the observed feature for Rigel points towards less mixing in the radiative envelope, allowing thus to remove larger amount of mass without too much altering the surface abundances. Of course, such tests should be applied to more stars in order to see whether some general trends appear.", "pages": [ 4, 5, 6, 7 ] }, { "title": "3 Puzzle 3: How do the most massive stars going through a Red Supergiant phase end their life?", "content": "The final state of star in the mass-range 15-25 M /circledot is strongly dependent on the very uncertain mass-loss rates during the RSG phase, and can be as diverse as RSG, YSG, and LBV (Georgy 2012, Groh et al. 2013a). Type IIP supernovae are characterized by light curves showing a plateau for periods which lasts typically 100 days. This plateau comes from the fact that the pseudophotosphere formed at the recombination front of hydrogen remains more or less at the same position due to two counteracting effects: 1) as radiation cools the photosphere, the ionization front recedes inwards in mass 2) at the same time, due to expansion, mass moves outwards. This implies that the radius of the pseudophotosphere remains constant for a while and at a temperature corresponding to that of H recombination, i.e. around 6000 K. This produces the plateau in luminosity (see e.g. more details for instance in Kasen & Woosley 2009). This kind of light curve occurs when a star with an extended H-rich envelope explodes and thus their progenitors are expected to be red supergiants. The most direct way to constrain the type IIP progenitors is to search for progenitors in archive images of the region where such supernovae have been observed. This has been the strategy by Smartt et al. (2009). They studied 20 type IIP events for which images before explosion was available. In five cases they indeed found a red supergiant progenitor. The other 15 cases are inconclusive because the events occurred in too crowded regions or only an upper limit for the luminosity of the progenitor was obtained. Using stellar tracks, they provided masses or upper mass limits for these 20 supernovae. The lowest initial mass they found is 8+1-1.5 M /circledot and the maximum initial mass is around 16.5+-1.5 M /circledot . We focus here on discussing the upper limit for the mass they found. Note that Smartt et al. (2009) consider this upper mass limit as statistically significant at 2.4 σ confidence. If stars above, let us say 17-18 M /circledot do not produce type IIP SNe, then of course the question is what other kinds of core collapse supernova do occur? If we look at Fig. 2 (right panel), one sees first that red supergiants are observed well above the empirical upper limit for the progenitors of type IIP given by Smartt et al. (2009). Second, one sees also that the tracks plotted have a different behavior for what concerns the red supergiant stage depending whether the initial mass of the star is below or above that empirical upper limit. Indeed, the tracks below that mass limit end their evolution as red supergiants, while stars with higher initial masses do not end their evolution as a red supergiant. Due to heavy mass loss, they evolve back in the blue part of the HR diagram. It happens that the 20 and 25 M /circledot present at their presupernova stage a spectrum similar to Luminous Blue Variable stars (Groh et al. 2013a). Due to the low mass of hydrogen at their surface, they will probably not produce a type IIP SN event but more likely explode as a type IIb SN. Very interestingly, the present 20 M /circledot ends its lifetime when its effective temperature is near one of the critical limits obtained by Vink et al. (1999) where the stellar winds show rapid and strong changes (bistability limit 4 ). It happens that this model, due to this effect, presents strong variations of its mass loss just before the explosion and this results in bumps in the radio light curves due to interactions between the ejecta and the matter released by these mass loss episode (Moriya et al. 2013). Interestingly such bumps in radio light curves are indeed observed in the case for instance of SN IIb 2001ig (Ryder v 2004). This gives support to the present models and indicates that single star evolutionary scenarios exist for producing type IIb supernovae.", "pages": [ 7, 8 ] }, { "title": "4 Puzzle 4: What is the nature of the SN Ibc progenitors?", "content": "Type Ibc supernovae are produced by the collapse of stars having lost their H-rich envelope. This loss may have occurred through stellar winds and/or through a Roche Lobe Overflow (RLOF) event in close binaries. One can wonder whether it is important to know how this mass loss occurred, i.e. through stellar winds or a RLOF event? This has some importance indeed, since these two mechanisms will make different predictions about the mass of the core at the time of the SN explosion, they will also linked these types of SNe events to different initial mass ranges and thus make different predictions concerning their frequency and the dependance of their frequency on the metallicity. Twelve Ibc supernovae with deep pre-explosion images are discussed in Smartt (2009 and see further references therein) and in Eldridge et al. (2013). In none of these cases, a progenitor has been detected. Comparing the upper limits for the progenitor's luminosity (in right panel of Fig. 3, the lowest of this upper limit is indicated) with the observed luminosity of WR stars suggests at 90% confidence level that the hypothesis according to which the WR stars we observe in the local group are the only progenitors of type Ibc supernovae is false (Smartt 2009). However, the observed WN and WC stars may not be at their pre-supernova stage. Even if time is short before the explosion, still some changes of the effective temperature and luminosity may make them much less easy to detect. In Groh et al. (2013b), the spectrum of the presupernova models at solar metallicity given by Ekstrom et al (2012) have been computed using the wind-atmosphere code CFMGEN (Hillier & Miller 1998). It happens that at the presupernova stage, the stars progenitors of type Ic supernovae are very hot, and have a typical spectrum of a WO star. They would have luminosities in the different bands well below the upper limit determined by Smartt (2009). This goes exactly in the same direction as the conclusion by Yoon et al. (2012) and can be a reasonable explanation for the non-detection mentioned above. Thus the non-detection mentioned above cannot be taken at the moment as an argument for invoking other stars than the observed WR stars as progenitors for this type of supernovae. It will be interesting to push down the dectability limit and to see whether a star will appear with a luminosity predicted by the models of single stars or not. It seems that such a star may have been detected in the case of a SN Ib (Cao et al. 2013; see also discussion in Groh et al. 2013c).", "pages": [ 8, 9 ] }, { "title": "5 Conclusion", "content": "In case positive answers would be given to the four questions below, 1.- Do slow, non-evolved rotators which are N-rich have a surface magnetic field and rotate differentially in their interior? 2.- Do most of Alpha-Cygni variables show strong N-enrichments? 3.- Do most stars with masses between 18 and 30 M /circledot end their lifetimes as blue supergiants and/or explode as type IIL, or IIb supernova? 4.Are the progenitors of type Ic Sne, WO type stars with properties characteristic of single star evolution? then this would provide some support to the present models. If negative answers are obtained, we can remind ourselves this sentence of Jean Rostand: One must credit an hypothesis with all that has been discovered in order to demolish it.", "pages": [ 9 ] } ]

[ { "title": "ABSTRACT", "content": "Title : will be set by the publisher Editors : will be set by the publisher EAS Publications Series, Vol. ?, 2018", "pages": [ 1 ] }, { "title": "OBSERVATIONAL CONSTRAINTS FROM BINARY STARS ON STELLAR EVOLUTION MODELS", "content": "Guillermo Torres 1 Abstract. Accurate determinations of masses and radii in binary stars, along with estimates of the effective temperatures, metallicities, and other properties, have long been used to test models of stellar evolution. As might be expected, observational constraints are plentiful for mainsequence stars, although some problems with theory remain even in this regime. Models in other areas of the H-R diagram are considerably less well constrained, or not constrained at all. I summarize the status of the field, and provide examples of how accurate measurements can supply stringent tests of stellar theory, including aspects such as the treatment of convection. I call attention to the apparent failure of current models to match the properties of stars with masses of 1.1-1.7 M /circledot that are near the point of central hydrogen exhaustion, possibly connected with the simplified treatment of convective core overshooting.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Stellar evolution theory represents the backbone of much of modern Astrophysics. For decades astronomers have worked to gather observations of many different kinds to constrain and test various physical ingredients of the models, and to calibrate a number of free parameters. These include the helium abundance, mass loss rates, and convective quantities such as the mixing length parameter ( α ML ) and the amount of overshooting from the convective core ( α ov ). In the last decade or two, accurate observations have revealed several shortcomings in our knowledge of stellar physics. One example is the difficulty in reproducing the radii and effective temperatures of late-type stars, which are larger and cooler than predicted by current standard models (see, e.g., Torres 2013a, and also the contribution by Greg Feiden in these Proceedings). Below I will describe another problem that is possibly related to the treatment of convection. Perhaps one of the best known ways of constraining stellar evolution theory is by means of color-magnitude diagrams (CMDs) of star clusters, which have been compared extensively with model isochrones to infer other interesting properties of the clusters such as age, distance, or chemical composition. Comparisons like these are powerful probes of stellar physics, but are not without their difficulties. Contamination of the CMDs by field stars, or unresolved binaries, can complicate or bias the analysis, as can stellar variability and reddening. An additional source of uncertainty is the color-temperature transformations used to convert models from the theoretical plane to the observational plane. The technique of asteroseismology provides very different but highly complementary observational constraints, through the measurement of oscillation frequencies that give us access to properties of the stellar interiors. These are challenging measurements, however, which typically require high-precision, continuous, and long-term observations, and are generally best done in bright stars with luminosities similar to the Sun or higher. A third, important way to test models that is again complementary to the previous two is through the observation of detached binary systems, which enable the model-independent measurement of fundamental stellar properties such as the mass and radius, and also effective temperature, luminosity, etc. While simple in principle, this technique requires special configurations and is not always easy for all types of stars. In this paper I will focus on how binary stars can help to test aspects of stellar evolution theory. It is useful to begin by reviewing the status of fundamental mass and radius determinations in eclipsing binaries, as recorded in the handful of 'critical' reviews that have appeared in the literature. These are compilations that pay special attention not only to the formal precision of the measurements, but also to the quality of the data and the analysis, particularly regarding systematic errors. The first critical review by Popper (1967) listed only two systems with mass determinations (but no radii) having relative errors under 3%. A subsequent compilation by Popper (1980) increased this to 7 systems with masses and radii good to the same accuracy. Andersen (1991) brought the total to 45 systems, and Torres et al. (2010) more than doubled it, to 95 systems. The masses and radii for some of these systems, along with other measured properties, allow for very stringent tests of models, as illustrated in the latter two references. Here I will concentrate on the phenomenon of overshooting from the convective core.", "pages": [ 1, 2 ] }, { "title": "2 Convective core overshooting: how binaries can help", "content": "Overshooting can be understood as mixing beyond the boundary of the convective core as given by the classical Schwarzschild (1906) criterion: rising convective elements 'overshoot' into the radiative zone. There are a number of important consequences of overshooting that affect the later stages of evolution. Enhanced mixing prolongs core hydrogen burning by feeding more H-rich material into the core. This changes the ages predicted by models. Access to a larger hydrogen reservoir during the H-burning phase enhances the mass of the helium core left behind, and this alters the global characteristics of the giant phases. In particular, it shortens the shell H-burning phase, reduces the lifetime of the core He-burning phase, and affects the luminosities in the giant stages. The effect on the main sequence portion of the evolutionary tracks in the H-R diagram is to extend the tracks toward cooler temperatures and higher luminosities, as illustrated, e.g., by Schroder et al. (1997). All giant phases occur at higher luminosities than they would without overshooting. For massive stars even the pre-main sequence (PMS) phases are affected, as shown by Marques et al. (2006): the evolutionary track for a 4 M /circledot PMS star develops an extra loop near the zero-age main sequence that is completely absent if overshooting is not considered. Even though there has been considerable progress in understanding turbulent convection, the sizes of convective cores in stars still cannot be predicted from first principles (VandenBerg et al. 2006). The most common approach in stellar evolution models is to parametrize the effect of overshooting in terms of a single variable representing the length of overshooting as a function of H p , the local pressure scale height: l ov = α ov H p . This formulation is easy to implement, but the overshooting parameter α ov must be calibrated using observations. This can be done in several ways. One is to use CMDs of clusters. In this approach one tries to match the detailed shape and extent of the 'blue hook' region by adjusting α ov , as illustrated by Demarque et al. (2004). This method works quite well, though it is somewhat vulnerable to the uncertainties mentioned earlier. Another technique takes advantage of accurate measurements of the masses, radii, and temperatures of eclipsing binaries that are near the end of their main-sequence life. It is based on the premise that the likelihood of finding a random field star in the region of the H-R diagram corresponding to the shell H-burning phase is small, because evolution across this so-called 'Hertzprung gap' is very rapid. Therefore, if a star appears to be slightly beyond the point of hydrogen exhaustion as marked by an evolutionary track, it is usually possible to increase the amount of overshooting in the models so that the track 'reaches out' to the star, bringing it onto a location still on the main sequence that is a priori much more likely. An example of this procedure is seen in the study of the eclipsing binary GX Gem by Lacy et al. (2008), shown in Figure 1. Caveats are that this procedure can be sensitive to errors in the measured temperatures, and that there is a certain amount of degeneracy with metallicity, if it is not known observationally for the system. Typical values of α ov are in the range 0.1-0.2. The treatment of overshooting in the transition region where stars begin to develop convective cores (approximately the mass range 1.1-1.7 M /circledot ) is particularly difficult. The ways in which different models ramp up the overshooting from zero to some maximum value varies from model to model, but they are all rather arbitrary and therefore a source of concern. For example, in the Yonsei-Yale models (Yi et al. 2001) the overshooting parameter is increased in steps of 0.05 starting at some mass value M conv crit at which stars develop cores, which is metallicity-dependent. It is then held constant at the value α ov = 0 . 2 for masses above M conv crit +0 . 2 M /circledot (Demarque et al. 2004). The VictoriaRegina models (VandenBerg et al. 2006) use a somewhat different prescription for overshooting that is equivalent to the single-parameter formalism used in the Yonsei-Yale models, and ramps up the strength of the overshooting in a smoother but different way, which also depends on metallicity. A persistent question has been whether and exactly how α ov depends on stel- ar mass. Schroder et al. (1997) used accurate measurements for binary systems containing giant or supergiant primaries to estimate the degree of overshooting in the same way described above over the mass range 2-8 M /circledot , and concluded that α ov increases from about 0.2 to 0.3 over this interval. A similar study by Ribas et al. (2000) used eight main-sequence eclipsing binaries with components ranging from 2 to 12 M /circledot , and also found a systematic increase in α ov , consistent with the previous results. However, the more recent study by Claret (2007) based on a larger number of main-sequence eclipsing binaries (13) with masses between 2 and 30 M /circledot found a much shallower dependence of α ov on mass that is also considerably more uncertain, so the question of the mass dependence of overshooting, if any, remains.", "pages": [ 2, 3, 4 ] }, { "title": "3 Constraints on overshooting from the eclipsing binary AQ Ser: indications of a new problem with stellar evolution models", "content": "Another illustration of how eclipsing binaries can help to calibrate models is provided by the F-star system AQ Ser, which has component masses of about 1.42 and 1.35 M /circledot , and relative errors smaller than 2% in both the masses and radii (Torres et al. 2013b). This is a highly evolved system presumably at the very end of its main-sequence phase, which makes it uniquely sensitive for testing convective core overshooting. A similar exercise as described earlier for GX Gem constrains α ov to be between 0.2 and 0.3 (see Figure 2, left panels), using the Granada stellar evolution models by Claret (2004). However, the models are unable to match the well measured temperature difference between the components (∆ T eff = 90 ± 20K), even in sign: they predict the primary star to be hotter, while the observations indicate the reverse. This discrepancy manifests itself in other ways, independently of the model con- sidered. For example, the top-right panel of Figure 2 shows the best-fit isochrone from the Victoria-Regina series. In these models the strength of overshooting is fixed and cannot be changed by the user. The models seemingly match the ob- servations very well for an age of 2.9 Gyr at a metallicity of [Fe / H] = -0 . 20, and suggest the stars are slightly beyond the point of hydrogen exhaustion. However, the predicted location of the stars on this isochrone from their nominally measured masses, which is indicated with the asterisks, is very far from the actual locations marked by the filled circles and error bars. In other words, the models would predict a mass ratio much closer to unity ( q ≡ M 2 /M 1 = 1 . 00083) than that measured spectroscopically ( q = 1 . 054 ± 0 . 011). The difference is highly significant, at nearly the 5σ level. Yet another way to visualize the disagreement is presented in the lower-right panels of Figure 2, particularly in the diagram of radius versus mass. It is seen that the models predict the more massive primary star to be younger than the secondary, by about 0.3 Gyr (10%). Similar discrepancies in the same direction are obtained with the Yonsei-Yale and Granada models (0.45 Gyr and 0.5 Gyr, respectively). Experiments with several different trial values of [Fe/H] in which α ov and also α ML are allowed to vary independently for each star do not improve the age agreement, pointing to a fundamental problem with the models. As it turns out, similar discrepancies have been reported by Clausen et al. (2010) for a handful similarly evolved F stars: GX Gem, V442 Cyg, BW Aqr, and BKPeg. All yield predicted ages for the primary stars that are younger than the secondaries, no matter which model is used. Two other systems showing the same anomaly have been identified more recently, although the problem was not noted in the original publications: CO And (Lacy et al. 2010) and BFDra (Lacy et al. 2012). A common property of these binary systems is that the components all have masses in the range of 1.1-1.7 M /circledot , which is precisely where models ramp up the importance of overshooting, and they are all considerably evolved (i.e., near the end of the main-sequence phase). This is also the mass range in which stars transition from having their energy production dominated by the p-p cycle to the CNO cycle. Figure 3 displays all binary systems with well measured properties that have masses in the range indicated above. The shaded area represents the region of the blue hook, according to the Yonsei-Yale models, and an increase in α ov would shift this region upwards. AQ Ser is seen to be the most evolved system in this regime, and is also the one showing the most pronounced discrepancy with theory. Given that convective core overshooting has a direct impact on evolution timescales, especially for main-sequence stars in the more advanced stages, it is natural to suspect that the simplified treatment of this phenomenon in current stellar models has something to do with their failure to reproduce the observed properties of well-measured eclipsing binaries at a single age. However, from the experiments with AQSer described above, the explanation does not appear to be a simple difference in α ov for the two components, and may be more complex. At the very least, it may involve a dependence of overshooting on the state of evolution, in addition to mass and metallicity. This problem has not received much attention in the literature beyond the work of Clausen et al. (2010), and is a good example of the usefulness of accurate measurements of eclipsing binaries for testing our knowledge of stellar evolution. This work was supported in part by grant AST-1007992 from the US National Science Foundation.", "pages": [ 4, 5, 6, 7 ] }, { "title": "References", "content": "Schr¨oder, K.-P., Pols, O. R., & Eggleton, P. P. 1997, MNRAS, 285, 696", "pages": [ 7 ] } ]

[ { "title": "Affine Quantization and the Initial Cosmological Singularity", "content": "Micha¨el Fanuel 1, ∗ and Simone Zonetti 1, † 1 Centre for Cosmology, Particle Physics and Phenomenology (CP3), Institut de Recherche en Math´ematique et Physique, Universit´e catholique de Louvain, Chemin du Cyclotron 2, bte L7.01.01, B-1348, Louvain-la-Neuve, Belgium", "pages": [ 1 ] }, { "title": "Abstract", "content": "The Affine Coherent State Quantization procedure is applied to the case of a FRLW universe in the presence of a cosmological constant. The quantum corrections alter the dynamics of the system in the semiclassical regime, providing a potential barrier term which avoids all classical singularities, as already suggested in other models studied in the literature. Furthermore the quantum corrections are responsible for an accelerated cosmic expansion. This work intends to explore some of the implications of the recently proposed 'Enhanced Quantization' procedure in a simplified model of cosmology. PACS numbers: 02.60.Cb, 03.65.Sq, 04.20.Dw, 04.60.Bc", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The initial singularity problem is a long standing problem in modern cosmology. It is often believed that the effects of quantum gravity should provide an answer to this question. Renowned candidate theories for quantum gravity are loop quantum gravity and superstring theories. Loop quantum cosmology and gauge-gravity duality are possible avenues of exploration (see for example [1] and [2], respectively). However other alternative or complementary approaches could be conceived. Among them, affine quantization has been recently put forward in order to quantize gravity [3-5], but has also been studied previously in [6, 7], while this approach was used to study a strong coupling limit of gravity in [8-10]. It is certainly interesting to examine the implications of this proposal. In this work, we apply the Affine Coherent State Quantization program to the dynamics of the scale factor in the FLRW (Friedman-Lemaˆıtre-Robertson-Walker) framework for cosmology, inclusive of a cosmological constant. We notice that the quantum corrections provide in an natural way a potential barrier term and we analyse the semiclassical behaviour using the 'Weak Correspondence Principle'. In Section II we introduce the classical model and calculate the classical equations of motion. Section III introduces the Affine Coherent State Quantization scheme and discusses the derivation of the Extended Hamiltonian. The equations of motion are also calculated and the effect of the quantum corrections is briefly discussed. In Section IV the numerical solutions for the classical and semiclassical cases are studied and compared, while in Section V we present our conclusions.", "pages": [ 2 ] }, { "title": "II. THE CLASSICAL MODEL", "content": "In an earlier work [11], a toy model for gravitation was studied from the affine perspective and it was argued that the singularities of the classical solutions were regularized because of the quantum effects. In a more recent article [3] the semiclassical behaviour of the onedimensional Hydrogen atom was analysed and it was shown a potential barrier emerges at the scale of the Bohr radius resolving the Coulomb singularity. In a similar way, we suggest here a simple model of a FRLW universe with a cosmological constant and discuss the consequences of the Affine Quantization on the classical singularities. We shall consider the action: where a ( t ) is the scale factor, Λ is the cosmological constant, k is the geometric factor and the scale α ensures that the action has the right dimensions. In the following we will set α = 1 for simplicity. The explicit choice of a time coordinate t is emphasized by the presence of the lapse function N ( t ). As it is well known, classical solutions to this model, because of the constraints produced by diffeomorphism invariance, depend on the value of the factor k and the sign of the cosmological constant. In particular de Sitter solutions (Λ > 0) are available for all values of k , while Anti-de Sitter solutions (Λ < 0) are only possible with κ = -1. A vanishing cosmological constant, on the other hand, does not allow a solution for κ = 1.", "pages": [ 2, 3 ] }, { "title": "A. Hamiltonian Formulation", "content": "In what follows we will relabel a ( t ) = q ( t ). Given the Lagrangian density in (1) the corresponding Hamiltonian, in the gauge N ( t ) = 1, reads where p is the conjugate momentum of q . The equations of motion are easily calculated as: The Hamiltonian is constrained to vanish as per effect of the diffeomorphism invariance. The symplectic structure is given by the Poisson bracket { q, p } = 1. The configuration space variable q is constrained to stay strictly positive: q > 0. At this stage quantizing the phase space with canonical commutation relations [ Q,P ] = i glyph[planckover2pi1] would lead to difficulties of interpretation if the spectrum of the self-adjoint operator Q is the real line, i.e. including negative eigenvalues. Actually it is possible to define the operators P and Q so that [ Q,P ] = i glyph[planckover2pi1] and Q > 0, however in this instance the operator P will only be hermitian (symmetric) but not self-adjoint, namely P may not be made self-adjoint by any choice of boundary conditions. Hence the exponential exp i qP/ glyph[planckover2pi1] will then not be an unitary translation operator, as can be shown easily [11]. Thus the canonical operators are not suitable, and a new set of kinematical self-adjoint operators are needed. We will see that quantizing another algebra of operators constitutes an interesting alternative.", "pages": [ 3, 4 ] }, { "title": "A. Construction of Affine Coherent States", "content": "Along time ago Affine Coherent States have been claimed to be useful in order to quantize gravity in its ADM (Arnowitt-Deser-Misner) formulation [6, 7, 12]. These states rely on the quantization of the ' ax + b ' affine algebra rather than the Heisenberg algebra. The major advantage for their use in a quantization of gravity is that they appropriately implement the condition of positive definiteness of the spatial metric. In the problem at hand we have a similar condition on the 'scale factor': q > 0. In order to define the affine coherent states we introduce the affine variables ( q, d ) by defining d = qp , which reparametrize the phase space ( q, p ). The affine coherent states form an overcomplete basis of the Hilbert space and µ is a scale with dimension of length. The fiducial vector | η 〉 satisfies the polarization condition with β a free dimensionless parameter. In particular one has: It is worth to notice that the condition (6) is built by analogy with the canonical coherent states construction and provides a differential equation for the wave function of the fiducial state. Because the state | η 〉 satisfies 0 < 〈 η | Q -1 | η 〉 < ∞ , the associated coherent states (5) admit a resolution of identity: where C = µ 〈 η | Q -1 | η 〉 . Subsidiarily, it should be underlined that a canonical coherent state construction would not be meaningful here, because the momentum operator P may not be made self-adjoint on the half line, as it is well known from the von Neumann theorem and the deficiency indices theory. We may now proceed to the affine quantization of the Hamiltonian formulation. In terms of the affine variables q and d = qp the Hamiltonian takes the form which is suitable to apply to correspondance principle q → Q and d → D . The classical affine commutation relations { q, d } = q are quantized as [ Q,D ] = i glyph[planckover2pi1] Q . The operators D and Q are conveniently represented in x -space by so that the interpretation of the algebra in terms of dilatations is now completely intuitive. For the sake of the consistency of the coherent state definition (5), the operator D represented above should be self-adjoint, while there is no difficulty to define properly the operator Q and its domain. In order to thoroughly specify D , we require that the boundary term, originating from vanishes. Because the wave functions ψ ( x ) ∈ Dom D and φ ( x ) ∈ Dom D † have to be square integrable on the half line, they should both verify, in particular, the condition which means that, if | ψ ( x ) | diverges at zero, one can find glyph[epsilon1] > 0 so that | ψ ( x ) | diverges slowlier than x -1 / 2+ glyph[epsilon1] close to zero. Thanks to (13), we can actually notice that the domains of D and D † indeed coincide. The self-adjoint operators Q and D appropriately realize the algebra [ Q,D ] = i glyph[planckover2pi1] Q . The representation theory of such algebra guarantees the existence of a unitary irreducible representation with the spectrum Q > 0 [13]. The fiducial state is then described by the wave function with N = (2 -2 β β -2 β Γ[2 β ] µ ) -1 / 2 . It is easy to interpret the role of µ from (15) in a comparison of affine and canonical coherent states: in the case of the latter a parameter λ q sets the width of the Gaussian fiducial vector, as in which satisfies: [ P/λ p -i Q/λ q ] | Ω 〉 = 0, with λ 0 = λ p /λ q . For simplicity both scales are usually used to define a unit system so that λ p /λ q = 1. Therefore the parameter µ of affine coherent states can be interpreted as the analogue of λ q , since it sets the width of the fiducial wave function and the average value of Q in the affine coherent states. Besides, if we wish to extend further the comparison, β glyph[planckover2pi1] is the analogue of λ p λ q as we may guess from (11): D = Q 1 / 2 PQ 1 / 2 . The existence of β can be understood as an artifact of the representation. Different values of β lead to different representations of the same physical states. However it is possible to see that there is a lower bound on the value of β : if we require the matrix element 〈 η | Q -1 DQ -1 D... | η 〉 (containing a number n (resp. n -1) of Q -1 (resp. D ) operators) and 〈 η | Q -n | η 〉 to be finite, we are forced to have β > n/ 2. We emphasize that this lower bound on the value of β is dictated by mathematical consistency and not by physical arguments. Besides this constraint, no other requirement is set on β at this stage, hence it will be considered as a free parameter. We will see that the specific value of β is irrelevant in determining the qualitative cosmological behaviour in the semiclassical regime.", "pages": [ 4, 5, 6 ] }, { "title": "B. Quantization and the semiclassical regime", "content": "One proceeds to quantization of the classical Hamiltonian (10) by defining the quantum Hamiltonian as The choice of operator ordering taken here is the one consistent with the Coherent State Quantization 'rule', also called 'anti-Wick quantization'. In order to have a self-adjoint operator, the conditions on the domain of K = Q -1 DQ -1 DQ -1 have to be specified. We should require that the boundary term vanishes. Hence we can choose to ask that ψ ( x ) ∈ Dom K verifies The functions φ ( x ) ∈ Dom K † , as may be seen from (18), have to satisfy the same conditions (19). As a result, the domains of D and D † coincide. Let us point out that the affine coherent states belong to the domain of K , whenever β > 3 / 2. Namely the wave function of an affine coherent state reads which verifies (19) when β > 3 / 2. The Hilbert space of states and the domain of the relevant operators being thoroughly identified, we may now try to take advantage of the coherent states to understand the dynamics. The quantum dynamics of the model may be described by a Coherent State Path Integral but in a first stage of this work we are interested in the classical limit of the system as viewed by a macroscopic observer. The notion of geometry being difficult to interpret in a purely quantum theory of gravity we are tempted to consider a semiclassical quantity that could emerge from the quantum theory and be interpreted in a geometrical context. The Extended Classical Hamiltonian provides such a description. It is associated to a Coherent State | p, q 〉 as and should take into account quantum corrections while describing a semiclassical behaviour. We follow here the 'Weak Correspondence Principle' as advocated by Klauder [14]. Intuitively we would like that classical and quantum mechanics coexist as they do in the physical world. The weak correspondence principle allows us to consider quantum effects in a classical description of the world where we know that glyph[planckover2pi1] takes a non-vanishing finite value. The fundamental reason why (21) is believed to incorporate quantum corrections is that it originates from the variational principle implementing the Schrodinger equation but where the 'restricted' quantum action is varied only on the set of (affine) coherent states | p ( t ) , q ( t ) 〉 rather than the full space of quantum states. Because of their semiclassical features, we believe that the coherent states may be the only ones accessible to a classical observer. Consequently, this restricted action principle gives a motivation for considering the equations of motion of h ( p, q ) as a meaningful semiclassical approximation of the dynamics of the quantum system [5]. Finally, we underline here the noticeable result that, starting from an affine quantized theory, the restricted action leads to a canonical theory (23). Making use of we obtain The required matrix elements can be easily calculated using (15) and (11), and read Note that both quantities are independent of the scale µ and finiteness of these matrix ele- where ments requires a lower bound on the value of β 1 . Notwithstanding we have to emphazise that, once β is chosen so that all matrix elements are finite, the value of γ = || Q -1 / 2 DQ -1 | η 〉|| 2 may never be negative. We stress that the value of the matrix elements given above should only be evaluated for β > 3 / 2, while other values of β lead to inconsistent results. To summarize, we find that the Extended Hamiltonian takes the form Once again diffeomorphism invariance will require h ( p, q ) = 0 to be enforced by the dynamics. It is remarkable that the dependency on the scale µ has been completely simplified. The classical limit (10) of the Extended Hamiltonian is readily reproduced by taking simultaneously glyph[planckover2pi1] → 0 and β →∞ , while their product is kept constant glyph[planckover2pi1] β → ˜ β . In this way we obtain that Z → 1 and δ → 1, while γ → 0.", "pages": [ 6, 7, 8, 9 ] }, { "title": "C. Qualitative analysis of the dynamics", "content": "Interestingly the quantum corrections generate one unique new dynamical term in the Hamiltonian, proportional to q -3 . This contribution will naturally affect the dynamics for small values of the scale factor q . We can infer its behaviour by looking at the equations of motion for (34), which read: As it is known in General Relativity the large scale gravitational dynamics, i.e. q glyph[greatermuch] 0, is dominated by the cosmological constant term: Λ > 0 generates a repulsive force and determines an accelerated expansion while Λ < 0 is responsible for an attractive force that, for example, can slow down cosmic expansion. In the same way, as we can see from (35), the small scale dynamics, i.e. q glyph[lessmuch] 1, will be dominated by the second term, proportional to γ . This quantity is always positive for β > 3 / 2, hence it behaves as a small scale equivalent of a positive cosmological constant, generating a repulsive force when the universe contracts. In particular, as we will see by solving numerically the equations of motion, this quantum correction is able to keep the scale factor from vanishing, avoiding to reach big crunch singularities. Furthermore the large scale behaviour is also modified: the constant δ , defined in (33), multiplies Λ and it is strictly greater than 1 for finite β , so that the effects of the cosmological constant are amplified for finite β and the effective cosmological constant is δ Λ. Finally we can see also that Z > δ > 1 for all β . Therefore even if different β 's label distinct quantum theories, the qualitative effects, as the avoidance of the classical singularity and the increased expansion rate, are universal.", "pages": [ 9, 10 ] }, { "title": "IV. NUMERICAL SOLUTION OF THE EQUATIONS OF MOTION", "content": "To illustrate the claims of the previous section it is possible to consider numerical solutions to the classical and semiclassical equations of motion, comparing the behaviour of the scale factor with the same (or close enough) set of initial conditions for both regimes. This however is non trivial due to the presence of the diffeomorphisms constraints: to be able to have a meaningful comparison both (1) and (34) have to vanish simultaneously at any given time t . Ideally we would like to solve the system of equations H ( p 0 , q 0 ) = h ( p 0 , q 0 ) = 0 to (possibly) determine a unique set of initial conditions ( p 0 , q 0 ) as functions of the parameters Λ , κ, β : this turns out to be possible only for the case of a de Sitter universe (Λ > 0) with κ = 1. In a more pragmatic approach we will apply the following procedure in all possible combinations of Λ glyph[lessequalgreater] 0 and κ = ± 1 , 0: has the purpose of providing initial conditions that are as close as possible for the two regimes. We can now look at the effects of the quantum corrections in all possible cases. The specific form of the classical Hamiltonian, as mentioned earlier, allows the Hamiltonian constraint to be enforced only in the case κ = -1, in which the classical scale factor has a sinusoidal behaviour and it reaches q = 0. With the inclusion of quantum corrections the singularity is avoided and the scale factor enters an infinite cycle of contractions and expansions, by effect of the γ term. The frequency of these oscillations is increased with respect to the frequency of classical sinusoidal solution, due to the multiplicative constant δ > 1 which amplifies the effect of the cosmological constant. In Figure 1a are plotted in blue the classical solution (continuous line), as a reference for the frequency, and the semiclassical solution (dashed line). Note how the minima of the semiclassical scale factor appear at earlier and earlier times with respect to the singularities of the classical q for effect of the modified dynamics. From the plot of q ( t ) (red dashed line in Fig. 1a) it is possible to visualize the important contribution to the cosmic acceleration provided by the γ term. The phase space trajectory for different initial values q 0 , and therefore different values of β and p a 0 , is plotted in Figure 1b as an example of the independence of the behaviour from the specific value of β . With initial values for q ranging from q 0 = 0 . 8 to q (0) = 1 . 3 the required values for β range from β ∼ 13 to β ∼ 40. In all cases the orbits are closed and the universe is bouncing.", "pages": [ 10, 11, 12 ] }, { "title": "· Open ( κ = -1 ) de Sitter universe", "content": "No classical singularity is present and the scale factor grows indefinitely, determining an accelerated expansion of the universe. The quantum correction however affects the dynamics speeding up the expansion and increasing the acceleration. Figure 2a shows the plot for q ( t ) and q ( t ). Note again how the behaviour close around the minimum of q ( t ) sees a substancial contribution from the γ term.", "pages": [ 12 ] }, { "title": "· κ = 0 de Sitter universe (Figure 2b)", "content": "At the classical level the scale factor decreases rapidly in a first phase and then slowly approaches q = 0 asymptotically. In the semiclassical case the quantum correction is dominant after the initial rapid contraction and determines an highly accelerated expansion which avoids the classical singularity.", "pages": [ 12 ] }, { "title": "· Closed ( κ = 1 ) de Sitter universe", "content": "The classical behaviour is singular, with a scale factor that reaches the singularity at finite times. The quantum correction once more avoids reaching q = 0 and determines an accelerated expansion. However the extended hamiltonian h is non-vanishing for any real set of allowed initial conditions, due to the presence of the positive constant γ : glyph[negationslash]", "pages": [ 12, 13 ] }, { "title": "· κ = -1 Λ = 0 universe (Figure 2d)", "content": "Again the classical behaviour is singular and is determined by the negative geometric factor κ , resulting in a linear, i.e. constant velocity, approach of q = 0. Also in this case quantum corrections are responsible for avoiding the singularity and inducing an expansion Independence of these behaviours from the specific value of the parameter β can be and has been tested successfully by repeating the analysis for different initial conditions.", "pages": [ 13, 14 ] }, { "title": "V. CONCLUSIONS", "content": "In this work we have studied the possibility of applying the Affine Coherent States Quantization scheme to a model of FRLW cosmology, in the presence of a cosmological constant. We considered a semiclassical regime, motivated by the 'Weak Correspondence Principle' formulated by Klauder [14]. We found that the additional terms and multiplicative constants arising from the quantization of the dilation algebra profoundly changes the dynamics, independently from the specific value of the parameter β > 3 / 2, which labels different quantum theories: the large scale dynamics is modified by an increased absolute value of the effective cosmological constant and the small scale dynamics is affected by a potential barrier generated by quantum corrections. In the case of an open de Sitter universe, which already at the classical level exhibits no singularity and expands eternally, expansion is accelerated by a combination of small and large scale effects. The possibility of a connection with Dark Energy is worth investigating. More interestingly all cases that possess a classical singularity, i.e. q → 0, exhibit a non singular behaviour in the semiclassical regime and enter an expansion phase after reaching a minimal length at which the quantum dynamics is dominant. In the case of a closed Anti-de Sitter universe, in addition, the scale factor enters an infinite cycle of expansions and contractions. These results, although limited to semiclassical considerations, provide additional support to the proposal of applying the affine quantization procedure in the approach of quantum gravity and quantum cosmology. Further investigations should be put forward to fully understand the role of affine coherent states and the potential of this approach: for instance it would be interesting to see whether alternative choices for the fiducial vector are available and provide a similar behaviour; alternative ordering prescriptions can also be employed and their consistency has to be checked.", "pages": [ 14 ] }, { "title": "Acknowledgments", "content": "John Klauder is warmly thanked for his accurate comments and constant encouragements. It is a pleasure to acknowledge Jan Govaerts for his stimulating remarks and Christophe Ringeval for helpful discussions. The work of MF is supported by the National Fund for Scientific Research (F.R.S.-FNRS, Belgium) through a 'Aspirant' Research fellowship and SZ benefits from a PhD research grant of the Institut Interuniversitaire des Sciences Nuclaires (IISN, Belgium). This work is supported by the Belgian Federal Office for Scientific, Technical and Cultural Affairs through the Interuniversity Attraction Pole P6/11.", "pages": [ 15 ] } ]

[ { "title": "Synthesized grain size distribution in the interstellar medium", "content": "Hiroyuki Hirashita 1 and Takaya Nozawa 2 1 Institute of Astronomy and Astrophysics, Academia Sinica, P.O. Box 23-141, Taipei 10617, Taiwan 2 Kavli Institute for the Physics and Mathematics of the Universe, Todai Institutes for Advanced Study, the University of Tokyo, Kashiwa, Chiba 277-8583, Japan (Received November 7, 2011; Revised March 15, 2012; Accepted March 16, 2012; Online published Xxxxx xx, 2008) We examine a synthetic way of constructing the grain size distribution in the interstellar medium (ISM). First we formulate a synthetic grain size distribution composed of three grain size distributions processed with the following mechanisms that govern the grain size distribution in the Milky Way: (i) grain growth by accretion and coagulation in dense clouds, (ii) supernova shock destruction by sputtering in diffuse ISM, and (iii) shattering driven by turbulence in diffuse ISM. Then, we examine if the observational grain size distribution in the Milky Way (called MRN) is successfully synthesized or not. We find that the three components actually synthesize the MRN grain size distribution in the sense that the deficiency of small grains by (i) and (ii) is compensated by the production of small grains by (iii). The fraction of each contribution to the total grain processing of (i), (ii), and (iii) (i.e., the relative importance of the three contributions to all grain processing mechanisms) is 30-50%, 20-40%, and 10-40%, respectively. We also show that the Milky Way extinction curve is reproduced with the synthetic grain size distributions. Key words: cosmic dust, interstellar medium, grain size distribution, extinction, Milky Way.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Dust grains are important in some physical processes in the interstellar medium (ISM). For example, they dominate the absorption and scattering of the stellar light, affecting the radiative transfer in the ISM. The extinction (absorption + scattering) by dust in the ISM as a function of wavelength is called extinction curve (Wickramasinghe, 1967; Hoyle and Wickramasinghe, 1991; Draine, 2003 for review). Extinction curves are important not only in basic radiative processes in the ISM but also in interpreting observational data: part of stellar light in a galaxy is scattered or absorbed by dust grains within the galaxy in a wavelength-dependent way according to the extinction curve. Therefore, to derive the intrinsic stellar spectral energy distribution of a galaxy, we always have to correct for dust extinction by considering the extinction curve (Calzetti, 2001). Extinction curves generally reflect the grain composition and the grain size distribution. Mathis et al. (1977, hereafter MRN) show that a mixture of silicate and graphite dust, as originally proposed by Hoyle and Wickramasinghe (1969), with a grain size distribution (number of grains per grain radius) proportional to a -3 . 5 , where a is the grain radius ( a ∼ 0 . 001 -0.25 µ m), reproduces the Milky Way extinction curve. Pei (1992) shows that the extinction curves in the Magellanic Clouds are also explained by the same power-law grain size distribution (i.e., ∝ a -3 . 5 ) with different abundance ratios between silicate and graphite. Kim et al. (1994) and Weingartner and Draine (2001) have applied more detailed fit to the Milky Way extinction curve in order to obtain the grain size distribution. Although their grain size distributions deviate from the MRN size distribution, the overall trend from small to large grain sizes roughly follows a power law with an index near to -3 . 5 . Therefore, the MRN grain size distribution is still valid as a first approximation of the interstellar grain size distribution in the Milky Way. What regulates or determines the grain size distribution? There are some possible processes that actively and rapidly modify the grain size distribution. Hellyer (1970) shows that the collisional fragmentation of dust grains finally leads to a power-law grain size distribution similar to the MRN size distribution (see also Bishop and Searle, 1983). In fact, Hirashita and Yan (2009) show that such a fragmentation and disruption process (or shattering) can be driven efficiently by turbulence in the diffuse ISM. However, they also show that grain velocities are strongly dependent on grain size; as a result, the grain size distribution does not converge to a simple power-law after shattering. Moreover, dust grains are also processed by other mechanisms. Various authors show that the increase of dust mass in the Milky Way ISM is mainly governed by grain growth through the accretion of gas phase metals onto the grains (we call elements composing dust grains 'metals'). (Dwek, 1998; Inoue, 2003; Zhukovska et al. , 2008; Draine, 2009; Inoue, 2011; Asano et al. , 2012). In the dense ISM, coagulation also occurs, making the grain sizes larger (e.g., Hirashita and Yan, 2009). In the diffuse ISM phase, interstellar shocks associated with supernova (SN) remnants (simply called SN shocks in this paper) destroy dust grains, especially small ones, by sputtering (e.g., McKee, 1989). Shattering also occurs in SN shocks (Jones et al. , 1996). Modeling the evolution of the grain size distribution in the ISM is a challenging problem because a variety of processes are concerned as mentioned above. Those processes are also related to the multi-phase nature of the ISM. Liffman and Clayton (1989) calculate the evolution of grain size distributions by taking into account grain growth and shock destruction. However, their method could not treat disruptive and coagulative processes (i.e., shattering and coagulation). O'Donnell and Mathis (1997) also model the evolution of grain size distribution in a multi-phase ISM, taking into account shattering and coagulation in addition to the processes considered in Liffman and Clayton (1989). They use the extinction curve and the depletion of gas-phase metals as quantities to be compared with observations. Although their models are broadly successful, the fit to the ultraviolet extinction curve is poor, which they attribute to the errors caused by their adopted optical constants. They also show that inclusion of molecular clouds in addition to diffuse ISM phases improves the fit to the observed depletion, but they did not explicitly show the effects of molecular clouds on the extinction curve. Yamasawa et al . (2011) have recently calculated the evolution of the grain size distribution in the early stage of galaxy evolution by considering the ejection of dust from SNe and subsequent destruction in SN shocks. Since they focus on the early stage, they did not include other processes such as grain growth and disruption (shattering), which are important in solar-metallicity environments such as in the Milky Way (Hirashita and Yan, 2009). Comparing theoretical grain size distributions with observations is not a trivial procedure. In a line of sight, we always observe a mixture of grains processed in various ISM phases. Therefore, a 'synthetic' grain size distribution, which is made by summing typical grain size distributions in individual ISM phases with certain weights, is to be compared with observations. In this paper, we first formulate a synthetic way of reproducing the grain size distribution. Then, we carry out a fitting of synthetic grain size distributions to the observational grain size distribution, in order to obtain the relative importance of individual grain processing mechanisms. We do not model the multi-phase ISM in detail, but our fitting contains the information on the weights (i.e., relative importance) of different grain processing mechanisms, which depend on the ISM phase. This paper is organized as follows. In Section 2, we explain our synthetic method to reconstruct the grain size distribution in the ISM. In Section 3, we fit our synthetic models to the observational grain size distribution in the Milky Way and examine if the fitting is successful or not. In Section 4, after we discuss our results, we calculate the extinction curves to examine if our synthesized grain size distributions are consistent with the observed extinction or not. In Section 5, we give our conclusions.", "pages": [ 1, 2 ] }, { "title": "2. Synthetic grain size distribution", "content": "As explained in Introduction, we 'synthesize' the observational grain size distribution (here, the MRN size distribution) by summing some representative grain size distributions in various ISM phases. These representative grain size distributions are explained in Section 2.1. First, the ISM is divided into two parts: one is the part where the grain processing is occurring (called 'grainprocessing region'), and the other is the area where the grains already processed in the various grain-processing regions are well mixed (called 'mixing region'). The mass fractions of the former and the latter regions are, respectively, f proc and 1 -f proc . It is reasonable to assume that the grain size distribution in the mixing region should be the mean grain size distribution in the ISM (Section 2.2). We assume that all grains are spherical with material density s ; thus, the grain mass m is expressed as m = 4 3 πa 3 s . Although coagulation may produce porous grains (e.g., Ormel et al. , 2009), we neglect the effects of porosity and assume all grains to be compact. Two grain species are treated in this paper; silicate and graphite. To avoid complexity arising from compound species, we treat these two species separately. This separate treatment is also practical in this paper as we (and other authors usually) assume that the observed extinction curve can be fitted with the two species (Section 4.4). Before being processed, the grain size distribution is assumed to be MRN: a power-law function with power index -r ( r = 3 . 5 ), and upper and lower bounds for the grain radii (whose values are determined below) a min and a max , respectively: for a min ≤ a ≤ a max . If a < a min or a > a max , n MRN ( a ) = 0 . The grain size distribution is defined so that n MRN ( a ) da is the number of grains whose sizes are between a and a + da per hydrogen nucleus. The dust mass density, ρ d , is related to the metallicity Z (the mass fraction of elements heavier than helium in the ISM) and the hydrogen number density n H as (Hirashita and Kuo, 2011) where m X is the atomic mass of the key element X (X = Si for silicate and C for graphite), f X is the mass fraction of X in the dust, ξ is the fraction of element X in gas phase (i.e., the fraction 1 -ξ is in dust phase), and (X/H) /circledot is the solar abundance relative to hydrogen in number density. The metallicity is assumed to be solar ( Z = Z /circledot ). We fix the maximum grain radius as a max = 0 . 25 µ m (MRN). Although the lower bound of the grain size is poorly determined from the extinction curve (Weingartner and Draine, 2001), we assume that a min = 0 . 3 nm, since a large number of very small grains are indeed necessary to explain the mid-infrared excess of the dust emission in the Milky Way (Draine and Li, 2001). For the other parameters, we follow Hirashita (2012). We assume that 0.75 of Si is condensed into silicate (i.e., ξ = 0 . 25 ) while 0.85 (i.e., ξ = 0 . 15 ) of C is included into graphite. Those values are roughly consistent with the observed depletion (e.g., Savage and Sembach, 1996), and reproduce the Milky Way extinction curve (Section 4.4). We adopt the following abundances for Si and C: (Si / H) /circledot = 3 . 55 × 10 -5 and (C / H) /circledot = 3 . 63 × 10 -4 . We assume that Si occupies a mass fraction of 0.166 ( f X = 0 . 166 ) in silicate while C is the only element composing graphite ( f X = 1 ). We adopt s = 3 . 3 and 2.26 g cm -3 for silicate and graphite, respectively. By using the MRN size distribution as the initial condition, we calculate the evolution of grain size distribution by the various processes treated in Section 2.1. In the numerical calculation, the grains going out of the radius range between a min and a max are removed from the calculation (the removed mass fraction is < 1 %). The processes considered are (i) 'grain growth' - grain growth by accretion and coagulation in dense medium, (ii) 'shock destruction' - destruction by sputtering in SN shocks, and (iii) 'grain disruption' grain disruption by shattering in interstellar turbulence. Shattering in SN shocks (Jones et al. , 1996) could be included as a separate component, but in our framework, it is not possible to separately constrain the contributions from the two shattering mechanisms because both shattering mechanisms (turbulence and SN shocks) selectively destroy grains with a /greaterorsimilar 0 . 03 µ m and increase smaller grains, predicting similar grain size distributions. Thus, we simply assume that the size distribution of shattered grains, whatever the shattering mechanism may be, is represented by the one adopted in Section 2.1.3. Although our fitting procedures are based on grain size distributions, we should keep in mind that observational constraints on the grain size distribution is mainly obtained by extinction curves. Weingartner & Draine (2001) performed a detailed fit to the Milky Way extinction curve. However, the grain size distributions derived by them broadly follow an MRN-like power law, although there are bumps and dips at some sizes. We will examine the consistency with the extinction curve later in Section 4.4.", "pages": [ 2, 3 ] }, { "title": "2.1 Processes considered", "content": "2.1.1 Grain growth Grain growth occurs in the dense ISM, especially in molecular clouds, through the accretion of metals (called accretion) and the sticking of grains (called coagulation). The change of grain size distribution by grain growth has been considered in our previous paper (Hirashita, 2012). The grain size distribution after grain growth is denoted as n grow ( a, t grow ) , where t grow is the duration of grain growth. We adopt t grow = 10 and 30 Myr based on typical lifetime of molecular clouds (e.g., Lada et al. , 2010). The metallicity in the Milky Way is high enough to allow complete depletion of grain-composing materials onto dust grains in ∼ 10 Myr. Thus, the total masses of silicate and graphite become 1.33 ( = 1 / 0 . 75 ) and 1.18 ( = 1 / 0 . 85 ) times as large as the initial values, respectively (recall that the dust mass becomes 1 / (1 -ξ ) times as much if all the dust grains accrete all the gas-phase metals). The difference in the grain size distribution between t grow = 10 and 30 Myr is predominantly caused by coagulation rather than accretion. 2.1.2 Shock destruction We calculate the change of grain size distribution by SN shock destruction in a medium swept by a SN shock, following Nozawa et al . (2006). All SN explosions are represented by an explosion of a star which has a mass of 20 M /circledot at the zero-age main sequence, and the SN explosion energy is assumed to be 10 51 erg. For the ISM, we adopt a hydrogen number density of 0.3 cm -3 (since the destruction is predominant in the diffuse ISM; McKee, 1989), and solar metallicity. The calculation of grain destruction is performed until the shock velocity is decelerated down to 100 km s -1 ( 8 × 10 4 yr after the explosion). We apply the material properties of Mg 2 SiO 4 and carbonaceous dust in Nozawa et al. (2006) for silicate and graphite, respectively. We denote the grain size distribution after shock destruction by n shock ( a ) . The destroyed mass fractions of silicate and graphite are 0.38 and 0.27, respectively. 2.1.3 Disruption Grain motions driven by interstellar turbulence lead to grain disruption (shattering) in the diffuse ISM (Yan et al., 2004; Hirashita and Yan, 2009). Among the various ISM phases, dust grains can acquire the largest velocity dispersion in a warm ionized medium (WIM). We recalculated the results of earlier workers based on our assumed initial conditions. We adopt the same grain velocity dispersions and hydrogen number density ( n H = 0 . 1 cm -3 ) in the WIM as adopted in Hirashita and Yan (2009). The fragments are assumed to follow a power-law size distribution with a power index of -3 . 3 (Jones et al. , 1996; Hirashita and Yan, 2009). We denote the grain size distribution after disruption as n disr ( a, t disr ) , where t disr is the duration of shattering in the WIM. The lifetime of WIM is estimated to be a few Myr from the recombination timescale and the lifetime of ionizing stars (Hirashita and Yan, 2009). Thus, we adopt t disr = 3 and 10 Myr for our calculation in causing moderate and significant disruption.", "pages": [ 3 ] }, { "title": "2.2 Synthesizing the grain size distribution", "content": "In the beginning of this section, we introduced the mass fraction ( f proc ) of ISM hosting grains which are now being processed ('grain-processing region'). The mean grain size distribution over all the grain-processing region, n synt ( a ) , can be synthesized with the processed grain size distributions, n grow ( a, t grow ) (grain size distribution after grain growth with a growth duration of t grow ), n shock ( a ) (grain size distribution after shock destruction), and n disr ( a, t disr ) (grain size distribution after disruption with a shattering duration of t disr ): where f grow , f disr and f shock are the mass fractions of medium hosting, respectively, grain growth, disruption, and grain destruction in the grain-processing region. We call n synt ( a ) 'synthetic grain size distribution'. If both species are spatially well mixed, they would have common values for f grow , f shock , and f disr . The mean grain size distribution in the ISM is denoted as n mean ( a ) and expressed as since it is assumed that the grain size distribution in the mixing region has already become the mean grain size distribution. By assumption, the mean size distribution is MRN: n mean ( a ) = n MRN ( a ) . This condition is equivalent to In the Milky Way ISM, since the grain mass is roughly in equilibrium between the growth in clouds and the destruction by SN shocks (Inoue, 2011), we apply f grow R 1 = f shock R 2 , where R 1 is the fraction of dust mass growth in clouds (0.33 and 0.18 for silicate and graphite, respectively; Section 2.1.1), and R 2 is the destroyed fraction of dust in a SN blast (0.38 and 0.27 for silicate and graphite, respectively; Section 2.1.2). Thus, we put a constraint, We approximately adopt R 1 /R 2 = 0 . 8 as a mean value between silicate and graphite. As mentioned above, if the two species (silicate and graphite) are spatially well mixed, both species would have common values for f grow , f shock , and f disr . Thus, we adopt a single value for R 1 /R 2 . Using the above constraints, Eq. (3) is reduced to where n g , s ( a ) ≡ n grow ( a, t grow ) + ( R 1 /R 2 ) n shock ( a ) . Thus, we treat f grow and f disr as free parameters. We define the sum of all the fractions as If the grain size distribution is predominantly modified by the three processes considered in this paper, we expect that f tot = 1 . The deviation of f tot from 1 is an indicator of goodness of our assumption that the grain size distribution is modified by the three processes.", "pages": [ 3, 4 ] }, { "title": "2.3 Best fitting parameters", "content": "We search for a set of parameters, ( f grow , f disr ) , which minimizes the square of the difference: where a i is the grain size sampled by logarithmic bins (i.e., log a i +1 -log a i is the same for any i ), and N is the number of the sampled grain radii ( N = 512 in our model, but the results are insensitive to N ). The individual components of processed grain size distributions [ n grow ( a, t grow ) , n shock ( a ) , and n disr ( a, t disr ) ] as well as n g , s ( a ) for silicate and graphite are shown in Fig. 1. The MRN size distribution, which should be fitted, is also presented. Our fitting procedure is first applied separately for silicate and graphite, although we discuss a possibility that both species have common values for ( f grow , f disr ) later in Section 4.", "pages": [ 4 ] }, { "title": "3. Results", "content": "In Table 1, we show the best-fitting values of f grow and f disr . We examine t grow = 30 and 10 Myr, and t disr = 3 and 10 Myr as mentioned in Section 2.1. We observe that f grow = 0 . 16 -0.55 and f disr = 0 . 06 -0.57 fit the MRN grain size distribution. The sum of all the fractions, f tot (Eq. 8) is unity with the maximum difference of 15% (see the column of f tot in Table 1). In order to show how the synthetic grain size distributions reproduce the MRN size distribution, we present Fig. 2, where we only show Models A and D for the smallest and the largest residuals δ 2 . We observe that the best-fitting results are fairly consistent with the MRN size distribution. In particular, the enhanced and depleted abundances of small grains at a /lessorsimilar 0 . 001 µ min n disr and n g , s , respectively, cancel out very well, especially in Model A. In Model D, the synthesized size distribution slightly fails to fit the MRN around a ∼ 0 . 001 -0 . 002 µ mbecause both n g , s and n disr (which are used for the fitting) show an excess around this grain radius range; thus, the excess around these sizes in the synthesized grain size distribution inevitably remains in Model D. However, the Milky Way extinction curve is reproduced even by Model D within a difference of ∼ 10 %(see Section 4.4). In order to see the details of the fitting, we show the ratio between the synthesized grain size distribution and the MRN distribution in Fig. 3. There is a general trend of excess around a ∼ 0 . 001 -0.003 µ m, which is due to grain growth (see Fig. 1). The excess is stronger in Models B and D than in Models A and C, which is why the fit is worse in Models B and D than Models A and C (Table 1). Since the bump comes from grain growth, the fit tends to suppress f grow in the presence of a strong bump. As a result, f tot is smaller in Models B and D than Models A and C (Table 1). We also observe in Fig. 3 that the bump appears at different grain radii between Models A/C and B/D because of the difference in the duration of grain growth. This bump may disappear if coagulation is more efficient than assumed here: more efficient coagulation may be realized if t grow /greatermuch 30 Myr and/or coagulation also occurs in denser regions (Section 4.2). Fig. 3 also indicates that the synthetic grain size distributions tend to be deficient at the largest grain sizes ( a /greaterorsimilar 0 . 1 µ m). This is because shattering tends to process large grains into small sizes (see Fig. 1). The deficiency of large grains may be overcome if we include the supply of large grains by stellar sources of efficient coagulation as discussed in Section 4.2. Because of significant grain growth in Models A and C, the deficiency of large grains is recovered by grain growth at a ∼ 0 . 01 -0 . 03 µ m for silicate. In Models A and C of graphite, shattering causes a dip feature around a ∼ 0 . 03 µ m as seen in Fig. 1, which also appears in Fig. 3. In the WIM, where shattering is assumed to occur in this paper, grains with a /greaterorsimilar a few × 10 -2 µ m are accelerated up to velocities larger than the shattering threshold by turbulence. Shattering efficiently destroys small grains because of their large surface-to-volume ratios. Thus, the shattering efficiency is the largest for the smallest grains that attain a velocity above the shattering threshold. This is the reason why the grains around a ∼ 0 . 03 µ mare particularly destroyed by shattering. This dip feature would be smoothed out in reality since the grain velocity driven by turbulence has a dependence on grain charge, gas density, magnetic field, etc., all of which have a wide range within a galaxy.", "pages": [ 4 ] }, { "title": "4.1 Derived parameters", "content": "The obtained values of the parameters f grow and f disr reflect the fraction of individual grain processing mechanisms. In other words, these two quantities show the relative importance of grain growth and disruption. Note that the efficiency of shock destruction is automatically constrained by the balance with the mass growth by grain growth (Eq. 6). Table 1 shows that the best-fitting parameters are not very sensitive to t grow (duration of grain growth) but that they are sensitive to t disr . For larger t disr , only a smaller f disr is necessary because the grain size distribution is more modified. As expected, f disr t disr is less sensitive to t disr ; note that f disr t disr is the mean duration of disruption per processed grain. Seeing all the models, we find f grow ∼ 0 . 2 -0.6 and f disr ∼ 0 . 06 -0.6 (or f disr t disr ∼ 0 . 6 -1.7 Myr). As mentioned in Section 2.2, if silicate and graphite are well mixed in the ISM, they are expected to have the common values for f grow , f shock , and f disr . In this sense, Models C and D work better than Models A and B. In summary, 20-60% of processing occurs in dense clouds (i.e., grain growth), while a processed dust grain experiences disruption for ∼ 1 Myr on average (or disruption accounts for 6-60% of processing). From the equilibrium constraint of the total dust mass (Eq. 6), the fraction of shock destruction to all the processing is f shock = 0 . 8 f grow ∼ 0 . 1 -0.4. The sum of all the fractions, f tot (Eq. 8), is unity with the maximum deviation of 15%. In other words, we cannot reject other processing mechanisms, which could contribute to the grain processing with /lessorsimilar 15 %. We have shown that the grain size distributions after (i) grain growth, (ii) shock destruction, and (iii) grain disruption can synthesize the MRN size distribution. It is also likely that we can say the opposite; that is, to realize the MRN size distribution, those three processes are crucial. Without (i) Note: f shock = 0 . 8 f grow from Eq. (6). the grain mass just decreases; without (ii) the grain mass just increases; without (iii) there is no mechanism that produce the large abundance of small grains.", "pages": [ 4, 5, 6 ] }, { "title": "4.2 Stellar sources?", "content": "In this paper, we did not consider dust supply from stars, because the dust mass in the Milky Way is governed by the equilibrium between grain growth in molecular clouds and grain destruction by SN shocks (e.g. Inoue, 2011). Dust grains supplied from stars may be biased to large sizes. Production of large grains from asymptotic giant branch (AGB) stars is indicated observationally (Groenewegen, 1997; Gauger et al. , 1998; Hofner, 2008; Mattsson & Hofner, 2011). The dust ejected from SNe is also biased to large grain sizes because small grains are selectively destroyed by the shocked region within the SNe (Bianchi and Schneider, 2007; Nozawa et al. , 2007). Coagulation associated with star formation is also a source of large grains if coagulated grains in circumstellar environments are somehow ejected into the ISM. For this possibility, Hirashita and Omukai (2009) have shown that dust grains can grow up to micron sizes by coagulation in star formation (see also Ormel et al. , 2009). As mentioned in Section 3, efficient coagulation may also solve the bump problem around a ∼ 0 . 001 -0.003 µ m. These possible sources of large grains may be worth including in dust evolution models in the future.", "pages": [ 6 ] }, { "title": "4.3 Fitting under other constraints", "content": "In Section 2.2, we adopted the balance between the dust mass growth by accretion and the dust mass loss by shock destruction (Eq. 6) as a constraint. Although this constraint is reasonable for the dust content in the Milky Way (e.g., Inoue, 2011), it may be useful to apply other constraints without using Eq. (6), to see how the best-fitting parameters have been controlled by Eq. (6). First we try to fit the MRN size distribution with the three Note: f shock = 1 - f grow - f shock . components under the condition that the sum of all the fractions is unity: f grow + f shock + f disr = 1 . The results with this fitting are shown in Table 2. The best-fitting values of f disr vary from those in Table 1 within a difference of 10% except for Model B of silicate (25% less). However, the bestfitting values of f grow is broadly 1/2-2/3 of those in Table 1. As a result, Eq. (6) is not satisfied, and the total dust mass decreases. Next, we perform fitting to the MRN size distribution with the parameters f grow , f shock , and f disr free. The results are shown in Table 3. Again, the values of f disr differ by only /lessorsimilar 10 %except for Model B of silicate (18% less). However, f grow is only ∼ 1 / 3 -2/3 of the values in Table 1, and f shock is made large to compensate for the decreased f grow . This means that the fitting is practically dominated by the balance between the decreased small grains in shock destruction and the increased small grains in disruption (shattering). Because of the dominance of f shock , Eq. (6) is not satisfied, and the graphite relative to hydrogen nuclei are already inherent in the models through the abundances of Si and C and ξ (Section 2). First we show the extinction curves of the individual components, which are used to fit the MRN size distribution, in Fig. 4. Grain growth does not make the extinction curve flatter in spite of the increase of the mean grain size. The reason is already explained in Hirashita (2012): Accretion predominantly occurs at the smallest sizes. Since the extinction at short wavelengths is more sensitive to the increase of the mass of small grains than that at long wavelengths, the extinction curve becomes rather steeper. Although coagulation flattens the extinction curve, the flattening due to coagulation does not overwhelm the steepening due to the above effect of accretion. Shock destruction makes the extinction curve flatter because small grains are more easily destroyed than large grains. Grain disruption steepens the extinction curve because of the production of a large number of small grains. The 0.22 µ m bump created by small graphite grains in this model becomes also prominent by grain disruption. We also show the extinction curve for the grain size distribution n g , s ( a ) (The component n g , s has a total dust mass 1.8 times as large as the initial value. To see the difference of the extinction curve, it would be helpful to compare under the same dust mass, so n g , s / 1 . 8 is compared with the MRN in Fig. 4.) In Fig. 5, we show the extinction curves calculated for Models A-D. First of all, we confirm that the MRN size distribution reproduces the observed extinction curve (some small deviations can be fitted further if we adopt a more detailed functional form of the grain size distribution, which is beyond the scope of this paper; see Weingartner and Draine, 2001, for a detailed fitting). Comparing the extinction curve for the MRN size distribution and those for Models A-D, we observe that the extinction curve is reproduced within a difference of ∼ 10 %. Models A and C are successful, while Models B and D systematically underproduce the MRN extinction curve (although the difference is small). The underprediction by ∼ 10 %in Models B and D occurs because f tot is ∼ 0 . 9 . In the above, we adopted different values for f grow and f disr between silicate and graphite. As mentioned in Section 2.2, if both species are well mixed in the ISM, they would have common values of these parameters. In Fig. 6, we show the extinction curve by taking the average of the values for silicate and graphite (for example, f grow = 0 . 35 and f disr = 0 . 35 for Model A). We find that the difference between Figs. 5 and 6 is small. Therefore, the mean values work to reproduce the Milky Way extinction curves. The mean values are in the range of f grow = 0 . 3 -0.5 and f disr = 0 . 1 -0.4. We conclude that the synthetic grain size distributions with these parameter ranges reproduce the Milky Way extinction curve.", "pages": [ 6, 7, 8 ] }, { "title": "5. Conclusion", "content": "In our previous papers (Nozawa et al. , 2006; Hirashita and Yan, 2009; Hirashita, 2012), we showed that dust grains are quickly processed by shock destruction, disruption, and grain growth. In this paper, thus, we have examined if the MRN grain size distribution, which is believed to represent total dust mass decreases.", "pages": [ 8 ] }, { "title": "4.4 Extinction curve", "content": "The MRN grain size distribution is originally derived from the Milky Way extinction curve. Therefore, in order to check if our fitting by synthetic grain size distributions is successful or not, it is useful to calculate extinction curves. Extinction curves are calculated by using the same optical properties of silicate and graphite as those in Hirashita and Yan (2009). The grain extinction cross section as a function of wavelength and grain size is derived from the Mie theory, and is weighted for the grain size distribution per hydrogen nucleus to obtain the extinction curve per unit hydrogen nucleus (denoted as A λ /N H ). The abundances of silicate and the grain size distribution in the Milky Way, can be reproduced by the processed grain size distributions. We 'synthesized' the grain size distribution by summing the processed grain size distributions under the condition that the decrease of dust mass by shock destruction is compensated by grain growth. We have found that the synthetic grain size distribution can reproduce the MRN grain size distribution in the sense that the deficiency of small grains by grain growth and shock destruction can be compensated by the production of small grains by disruption. The values of the fitting parameters indicate that, among the processed grains, 30-50% is growing in dense medium, 20-40% is being destroyed by shocks in diffuse medium, and 10-40% is being shattered in diffuse medium (the percentage shows the rel- ative importance of each process). The extinction curves calculated by the synthesized grain size distributions reproduce the observed Milky Way extinction curve within a difference of ∼ 10 %. This means that our idea of synthesizing the grain size distribution based on major processing mechanisms (i.e., grain growth, shock destruction, and disruption) is promising as a general method to 'reconstruct' the extinction curve. Acknowledgments. We are grateful to C. Wickramasinghe and an anonymous reviewer for helpful comments in their review of this paper. HH has been supported through NSC grant 99-2112-M-001006-MY3. TN has been supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, and by the Grant-in-Aid for Scientific Research of the Japan Society for the Promotion of Science (20340038, 22684004, 23224004).", "pages": [ 8, 9 ] }, { "title": "References", "content": "Asano, R. S., Takeuchi, T. T., Inoue, A. K., and Hirashita, H., Dust formation history of galaxies: a critical role of metallicity for the grain growth by accreting metals in the interstellar medium, Earth Planets Space , submitted, 2012. lar dust during the chemical evolution of a two-phase interstellar medium, Astrophysical Journal , 340 , 853-868, 1989.", "pages": [ 9, 10 ] } ]

[ { "title": "Vacuum spherically symmetric solutions in f ( T ) gravity", "content": "K. Atazadeh ∗ and Misha Mousavi † Department of Physics, Azarbaijan Shahid Madani University , Tabriz, 53714-161 Iran October 29, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "Spherically symmetric static vacuum solutions have been built in f ( T ) models of gravity theory. We apply some conditions on the metric components; then the new vacuum spherically symmetric solutions are obtained. Also, by extracting metric coefficients we determine the analytical form of f ( T ).", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "It is difficult to obtain explicit solutions of modified gravitational field equations on account of their nonlinear character, for instance the equations of motion in General Relativity (GR) and f ( R ) theory, respectively, are of order 2 and 4. However, there are a reasonable number of static spherically symmetric exact solutions which have a measure of physical interest, particularly solutions of f ( R ) gravity theories in vacuum space [1, 2] and for global static sphere with finite radius at which the pressure vanishes, called stellar model [3]. Clearly, we are interested in analyzing such a kind of results for vacuum space, in solar system tests; then we can get to know with the exterior gravitational field surrounding some massive spherical object such as a star. We can use the extracted metrics to investigate the physics in the vicinity of a spherical object, in particular the trajectories of freely falling massive particles and photons. A new group of models which has been added recently to the candidate class of models for explaining the present day acceleration of our universe is known as f ( T ) gravity theories [4]-[6]. The idea of f ( T ) gravity theory refers to 1928 when Einstein was trying to redefine the unification of gravity and electromagnetism by means of the introduction of a tetrad (vierbein) field together with the idea of absolute parallelism [7]. In the teleparallel gravity (TG) theories the dynamical object is not the metric g µν but a set of tetrad fields e a ( x µ ) and rather than the well-known torsionless Levi-Civita connection of GR, a Weitzenbok connection is used to define the covariant derivative; torsion plays the role of curvature in TG [8]-[10]. The degree of non-linearity in f ( T ) field equations is the same as the order of GR. A crucial point about the f ( T ) is that it does not respect local Lorentz symmetry [15, 17]. From a theoretical perspective this is a rather undesirable feature and experimentally there are stringent constraints. A Lorentz-violating theory is only attractive if the violations are small enough to avoid detection and it leads to some other significant achievements. So far, the only pay-off that has been suggested is that f ( T ) gravity might provide an alternative to conventional dark energy in general relativistic cosmology. In accordance with the very recent attention to spherically symmetric space-time in f ( T ) gravity, a large number of vacuum and non-vacuum solutions have been built in this theory [11, 12]. In searching for solutions of the field equations of f ( T ) gravity models, considering vacuum solutions of nonlinear second-order field equations of f ( T ) gravity theory comes first. In [13] exact spherically symmetric solutions of f ( T ) theories by very different methods are studied. The usual way of finding the complete vacuum model with exact solution necessitates us to start with the form of f ( T ), then replace it in modified field equations to find the metric coefficients. In this paper we follow a different strategy and construct modified field equations thus rewriting them in such a manner as to make the nonlinear differential two reduced nonlinear differential equations somewhat easy, to obtain a variety of explicit solutions by means of inserting some constrains in the metric coefficients to make considerable simplification. This method is exactly the same as Tolman strategy which put varied relations for the metric components to solve the reduced Einstein equations in an easy way. As a result we introduce three relations for X ( r ), F ( r ) and A ( r ) as additional constraints on field equations and solve the equations, thereafter we can extract the form of f ( T ). It should be noted that this it is not guaranteed that all these constraints produce f ( T ) in a physically analytical form. This paper is organized as follows: in the section 2, we consider some basic concepts of f ( T ) theory, and in the presence of the locally Lorentz violation we do some calculations on the field equations to convert those into the covariant version according to the approach that has been introduced in [16]. In the section 3, by using the covariant version of the field equations we discuss different situations for the spherically symmetric metric coefficients, by taking X = X 0 , X = X 0 r m and A = A 0 r m , F = F 0 r n and in the sub-subsections 3 . 2 . 1, 3 . 2 . 2 and 3 . 2 . 3 we find the Schwarzschild-de Sitter exterior solution, de Sitter solution and asymptotic solution, respectively. Finally we will finish with the conclusions.", "pages": [ 1, 2 ] }, { "title": "2.1 Field equations", "content": "To consider teleparallelism, one employs the orthonormal tetrad components e A ( x µ ), where an index A runs over 0 , 1 , 2 , 3 to the tangent space at each point x µ of the manifold. Their relation to the metric g µν is given by where µ and ν are coordinate indices on the manifold and also run over 0 , 1 , 2 , 3, and e µ A forms the tangent vector on the tangent space over which the metric η AB is defined. Instead of using the torsionless Levi-Civita connection in General Relativity, we use the curvatureless Weitzenbock connection in teleparallelism [14], whose non-null torsion T ρ µν and contorsion K ρ µν are defined by respectively. Here Γ ρ µν is the Levi-Civita connection. Moreover, instead of the Ricci scalar R for the Lagrangian density in general relativity, the teleparallel Lagrangian density is described by the torsion scalar T as follows: where The modified teleparallel action for f ( T ) gravity is given by [5] where | e | = det ( e A µ ) = √ -g and the units have been chosen so that c = 16 πG = 1. Varying the action in equation (6) with respect to the vierbein vector field e µ A , we obtain the equation [4] where a subscript T denotes differentiation with respect to T and θ ν A is the matter energy-momentum tensor.", "pages": [ 2, 3 ] }, { "title": "2.2 Covariant field equations", "content": "The field equation (7) is written in terms of the tetrad and partial derivatives and to be appear very different from Einsteins equations. In this subsection, following [16], we obtain an equation relating T with the Ricci scalar of the metric R . These will make the equivalence between teleparallel gravity and general relativity clear. On the other hand, the tetrad cannot be eliminated completely in favor of the metric in equation (7), because of the lack of local Lorentz symmetry, but we will show that the latter can be brought in a form that closely resembles Einsteins equation. This form is more suitable for constructing spherical summitry solutions in the f ( T ) theory. To start writing the field equations in the covariant version, we must replace partial derivatives in the tensors by covariant derivatives compatible with the metric g µν , i.e. ∇ σ where ∇ σ g µν = 0. Thus, equations (2), (3), and (5) can be written as where we have used the fact that Γ σ µν is symmetric in the subscripts µ, ν : respectively. On the other hand, from the relation between and Weitzenbock connection and the Levi-Civita connection given by equation (3), one can write the Riemann tensor for the Levi-Civita connection in the form whose associated Ricci tensor can then be written as Now, by using K ρ µν given by equation (5) along with the relations K ( µν ) σ = T µ ( νσ ) = S µ ( νσ ) = 0 and considering that S µ ρµ = 2 K µ ρµ = -2 T µ ρµ one has [15, 16, 17, 18] This last equation implies that the T and R differ only by a covariant divergence of a space-time vector. Therefore, the Einstein-Hilbert action and the teleparallel action ( i.e. S = ∫ d 4 x | e | T ) will both lead to the same field equations and are dynamically equivalent theories. In Ref. [16] the authors have shown that this equivalence is directly at the level of the field equations. By using the equations listed above and after some algebraic manipulations, one can gets where G µν = R µν -(1 / 2) g µν R is the Einstein tensor. Finally, by using equation (14), the field equations for f ( T ) gravity equation (7) can be rewritten in the form /negationslash where F ( T ) = df ( T ) dT , F T ( T ) = dF ( T ) dT , B µν = S νµ σ ∇ σ T and θ µν is the matter energy-momentum tensor. Equation (15) can be taken as the starting point of the f ( T ) modified gravity model, and it has a structure similar to the field equation of f ( R ) gravity. Note that in the more general case with f ( T ) = T , the field equations are covariant form. Nevertheless, the theory is not local Lorentz invariant. In case of f ( T ) = T and constant torsion, f ( T 0 ), GR is recovered and field equations are covariant and the theory is Lorentz invariant.", "pages": [ 3, 4 ] }, { "title": "3 Spherically symmetric static solutions of f ( T ) gravity", "content": "In this section, we are looking for time-independent vacuum spherically symmetric solutions, henceforth the line element has the following form: where d Ω 2 = dr 2 + sin 2 θdϕ 2 and where A , B , and R are three unknown functions. One possible tetrad field (we can make arbitrary Lorentz transformations to the tetrads without changing the metric) can be written as to which we refer to as the diagonal gauge. In vacuum we have Following [1] under the assumption (18), the new following form of equation (15) (with fixed indices) is obviously index-independent: It is clear that for all µ and ν we can write K [ µ ] -K [ ν ] = 0; therefore, having considered this result, the line element (16) and equation (19), we obtain two equations which include two unknown parameters A and B ; thus the first equation arises by rewriting K [ t ] -K [ r ] = 0; note that X ( r ) = A ( r ) B ( r ) and also we have defined ' The second equation will result from K [ t ] -K [ θ ] = 0; ≡ d dr . 8 F T XA 2 +4 F T X ' A 2 r -4 FAX 2 r 2 +2 FXX ' Ar 3 -4 F T XAA ' r +2 F T A ' X ' Ar 2 -4 r 2 F T XAA '' (21) +4 Fr 2 X 3 -Fr 4 XA ' X ' -2 F T r 3 A ' 2 X ' +2 Fr 4 X 2 A '' +2 F T A ' r 3 XA '' = 0 . The corresponding torsion scalar is this equation describes that T depends on A ( r ) and B ( r ), meanwhile since the metric only depends on r , one can say that F ( T ) behaves like F ( r ). Thus it will be certain that function F ( r ) and it's derivative F ' ( r ) depend on the metric coefficients in a nonlinear and complicated way leading, to missing the analytical solutions of (20) and (21) in a closed form. The only way to deal with this situation is use of the numerical method. It should be noted that we are searching for physically acceptable solutions which includes analytical functions. So as to obtain analytical solutions for empty space in the f ( T ) gravity model we are supposed to consider one or two assumptions about the form of metric coefficients ( X or A ) or F for each solution. As a first step in this direction, we consider X = X 0 , which simplifies the equations.", "pages": [ 4, 5 ] }, { "title": "3.1 Solutions with X = X 0", "content": "According to this assumption, both equations (20) and (21) will be simplified. By unifying these equations, the corresponding metric coefficient will be given by where c 1 and c 2 are integration constants. This equation directs us at F ( T ) = F ( T 0 ) with constant torsion T 0 = 6 c 1 X 0 . This metric is a solution for any form of f ( T ) for which there exists a constant T 0 such that -T 0 F ( T 0 ) + 2 f ( T 0 ) = 0, and T 0 should be real. Because the dependence of X on r , obviously we can also generalize the last constraint and for the next step we consider X = X 0 r m .", "pages": [ 5 ] }, { "title": "3.2 Solutions with X = X 0 r m", "content": "By solving the equations (20) and (21) for arbitrary m , A ( r ) and F ( r ) are given by Also we easily can find T It has to be noted here that the equation F = F 0 r m has a result exactly the same as the result of the assumption X = X 0 r m and again A and T will be given by equations (24) and (26), and also equation (24) shows that m = 2. To continue, we proceed to obtain f ( T ) by choosing m = 1 to simplify the equations, specially the relation between T and r . Thus for T we have /negationslash Having done a bit simplification, we lastly find f ( T ): and assuming 9 c 2 1 4 TX 2 0 << 1, we expand f ( T ) as follows: Equation (28) is a square quantity if F 0 > 0, thus we can say that in equation (29) the positive terms are more effective than negative terms.", "pages": [ 5, 6 ] }, { "title": "3.2.1 Schwarzschild-de Sitter solution", "content": "Comparing the solution of equation (24), to the well-known Schwarzschild-de Sitter metric, we have to set m = 0, X 0 = 1 and require c 1 = -Λ 3 , c 2 = -2 M , therefore we can directly find A ( r ) and f ( r ): here M represents the total mass which is result of a part of gravitational energy inside the sphere radius r m , while the total energy also includes vacuum energy caused by a positive cosmological constant. We have hence where λ is the integration constant.", "pages": [ 6 ] }, { "title": "3.2.2 de Sitter solution", "content": "The only difference between this solution and Schwarzschildde Sitter solution is because of the value of c 2 . Here if we assume m = 0 and c 2 = 0 then we will have Note that the values of f ( T ) and T scalar have no change and they are thoroughly the same as their values in Schwarzschildde Sitter solution. As you can see in equations (30) and (33) the Schwarzschild-de Sitter Solution; survives in the range ( r → 0) and the de Sitter solution survives in the range ( r →∞ ).", "pages": [ 6 ] }, { "title": "3.2.3 Asymptotic solution", "content": "Finding asymptotic solution needs the matching equation (24) with the asymptotically flat solution, for which A ( r ) → 1 at large r . Due to this condition we should fix m = 0, c 1 = 0, c 2 = -2 M , and X 0 = 1, so finally we gain the asymptotic metric coefficient It obvious that we have not gained any new results that tend to constant scalar torsion ( T → 0) in the large r limit along with A ( r ), showing that any new solutions will be thoroughly different from the Schwarzschild (-de Sitter) solution.", "pages": [ 6 ] }, { "title": "3.3 Solutions with A = A 0 r m , F = F 0 r n", "content": "By means of this assumption and simplifying equations (20) and (21), the unknown quantity X ( r ) is given by By taking m = -1, A and X are given by Surprisingly, by substituting the above equations in the (22) the torsion scalar vanishes and we cannot get any physically acceptable solutions. One can try these results for other values of m .", "pages": [ 7 ] }, { "title": "4 Conclusions", "content": "In this article we have obtained explicit solutions in a spherically symmetric space-time in f ( T ) theory by choosing a diagonal form of the tetrad associated to the spherically symmetric metric. We started with modified field equations of f ( T ) and rearranged them to have a class of equations involving some terms but not all terms of the field equations. Thereafter We went through just three special cases and determined the relations of metric coefficients of F . As a result we became able to reduce our unknown parameters and gain the metric coefficients. We extended our discussion and compared our results with the well-known Schwarzschild-de Sitter and de Sitter solutions and considered the asymptotic solution as well. Taking notice of tetrad rules, the existence of tetrad in the f ( T ) field equations causes this theory not to respect local Lorentz transformation symmetry. Thus, a different field equations which in turn might have different solutions. Some of these solutions do not have a valid GR counterpart, while others tend to their GR counterparts in the appropriate limit. Therefore, special attention has to be given to the choice of tetrad. In Ref. [19] the authors have shown that there are two tetrad groups, bad tetrads and good tetrads. A good tetrad is the one that gives rise to field equations which do not constrain the functional form of f ( T ). In such cases one can always consider the limit f ( T ) → T where the correct general relativistic limit is recovered. Otherwise we will talk of a bad tetrad. We have studied spherical symmetric solutions in f ( T ) theory by means of a diagonal tetrad, which are compatible with GR results; for an example refer to equation (30). ,", "pages": [ 7 ] } ]

[ { "title": "Intermediate-Generalized Chaplygin Gas inflationary universe model", "content": "Ram'on Herrera, ∗ Marco Olivares, † and Nelson Videla ‡ Instituto de F'ısica, Pontificia Universidad Cat'olica de Valpara'ıso, Avenida Brasil 2950, Casilla 4059, Valpara'ıso, Chile. (Dated: December 10, 2017)", "pages": [ 1 ] }, { "title": "Abstract", "content": "An intermediate inflationary universe model in the context of a generalized Chaplygin gas is considered. For the matter we consider two different energy densities; a standard scalar field and a tachyon field, respectively. In general, we discuss the conditions of an inflationary epoch for these models. We also, use recent astronomical observations from Wilkinson Microwave Anisotropy Probe seven year data for constraining the parameters appearing in our models. PACS numbers: 98.80.Cq", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The inflationary universe was introduced [1, 2] as a manner of addressing pressing problems (horizon, flatness, monopoles, etc.) that were eating away at the bases of the otherwise rather prosperous Big-Bang model. The most significant feature of the inflationary universe model is that it provides a causal interpretation of the origin of the observed anisotropy of the cosmic microwave background radiation (CMB) and the structure formation in the universe[3, 4]. Exact solutions exist for power-law and de-Sitter inflationary universes and they are created by exponential and constant scalar potentials, see Ref.[1, 5]. Exact solutions can also be obtained for the scenario of intermediate inflation, where the scale factor, a ( t ), increases as /negationslash in which A and f are two constants; A > 0 and 0 < f < 1 [6]. The expansion of this inflationary scenario is slower than de-Sitter inflation, but faster than power law inflation, this is the denotation why it is called 'intermediate'. The intermediate model was originally formulated as an exact solution, but it may be best inspired from the slow-roll approximation. From the slow-roll approximation, it is possible to have a spectrum of density perturbations which presents a spectral index n s ∼ 1 and also in particular n s = 1 (Harrizon-Zel'dovich spectrum) for the value f = 2 / 3 [7]. However, the value n s = 1 is disfavored by the current Wilkinson Microwave Anisotropy Probe (WMAP) observational data[3, 4]. Also, the tensor perturbations which could be present in this model, through of the parametrized by the tensor to scalar ratio r , which is significantly r = 0[8, 9]. On the other hand, the motivation to study this expansion becomes from string/M-theory, indicates that in order to have a ghost-free action high order curvature invariant corrections to the Einstein-Hilbert action must be relative to the Gauss-Bonnet (GB) term[10], where this expansion appears to the low-energy string effective action[11, 12] (see also, Ref.[13]). On the other hand, the generalized Chaplygin gas (GCG) is other aspirant for explaining the acceleration of universe. The exotic equation of state of the GCG is given by [14] where ρ Ch and p Ch are the energy density and pressure of the GCG, β is a constant in which β ≤ 1, and α is a positive constant. In particular, when β = 1 corresponds to the original Chaplygin gas [14]. Replacing, Eq.(2) into the stress-energy conservation equation, the energy density results Here, a is the scale factor and B is a positive integration constant. In this way, the GCG is characterized by two parameters, B s = α/ρ 1+ β Ch 0 and β . These parameter has been confronted by observational data, see Refs.[15, 16]. In particular, the values of B s = 0 . 73 +0 . 06 -0 . 06 and β = -0 . 09 +0 . 15 -0 . 12 was obtained in Ref.[16]. Also, in Ref.[17] the values 0 . 81 < ∼ B s < ∼ 0 . 85 and 0 . 2 < ∼ β < ∼ 0 . 6 were found from the observational data arising from Archeops for the location of the first peak, BOOMERANG for the location of the third peak, supernova and high-redshift observations. Recently, the values of B s = 0 . 775 +0 . 0161+0 . 037 -0 . 0161 -0 . 0338 and β = 0 . 00126 +0 . 000970+0 . 00268 -0 . 00126 -0 . 00126 was obtained from Markov Chain Monte Carlo method[18]. The Chaplygin gas arises as an effective fluid of a generalized d-brane the space time, in a Born-Infeld action [17] and these models have been extensively analyzed in Ref.[19]. In the model of Chaplygin inspired in an inflationary scenario commonly the standard scalar field drives inflation, in which the energy density given by Eq.(3), can be extrapolate in the Friedmann equation for archiving an appropriate inflationary period [20]. However, also a tachyonic field in a Chaplygin inflationary universe model was considered in Ref.[21]. The possibility of having Chaplygin models with scalar field and tachyon field has been considered in Ref.[22]. The modification of the Friedmann equation is realized from an extrapolation of Eq.(3), where we identifying the density matter with the contributions of the density energy associated to the standard scalar field or tachyonic field [17, 21]. In this way, the GCG model may be viewed as a variation of gravity and there has been great interest in the elaboration of early universe scenarios motivated by string/M-theory[23]. It is well known that these modifications can lead to significant changes in the early universe. In this paper we would like to study intermediate-GCG inflationary universe model in which different types of energy densities are taken into account. In particular, (i) when the energy density is a standard scalar field, and (ii) when the energy density is a tachyon field. We will investigate the dynamic in both models and also we shall utilize to the seven-year data WMAP to restrict the parameters in our models. The outline of the paper is as follows. The next section presents the dynamic of the intermediate-GCG Inflationary scenario for our two models. Section III deals with the calculations of cosmological perturbations. Finally, in Sect.IV we conclude with our finding.", "pages": [ 2, 3, 4 ] }, { "title": "II. INTERMEDIATE-GCG INFLATIONARY UNIVERSE MODEL", "content": "It is well know that the GCG model can also be used to describe the early universe. During inflation the gravity dynamics may give rise to a modified Friedmann equation[17] where κ = 8 π/m 2 p , m p is the reduced Planck mass, ρ φ is the energy density of the scalar field and H = ˙ a/a is the Hubble parameter. This modification in the Friedmann equation is the so-called Chaplygin inspired inflation scenario[17]. Following this idea, lately, some work has been done, this involves Chaplygin inflationary universe model, see Refs.[21, 24]. In the following, we will considers two matter fields for ρ φ ; the standard scalar field and tachyonic field, respectively. For convenience we use units in which c = ¯ h = 1.", "pages": [ 4 ] }, { "title": "A. Standard scalar field", "content": "In the following, we will consider the power spectra of scalar and tensor perturbations to the metric in Chaplygin inflation. We introduce the gauge invariant quantity[27, 28] where ψ is the gravitational potential. On slices of uniform density ζ reduces to the curvature perturbation. A fundamental characteristic attribute of ζ is that it is nearly constant on super-horizon scales. This feature, result to be a consequence of stress-energy conservation and does not depend on the gravitational dynamics[29] (see also, Ref.[30]). In this context, it continues unchanged in Chaplygin inflation[20, 31]. In this form, the power spectrum related to curvature spectrum, could be written as P R /similarequal 〈 ζ 2 〉 . It can be shown that on super-horizon scales, the curvature perturbation on slices of uniform density is equivalent to the comoving curvature perturbation. Therefore, for the spatially flat gauge, we have ζ = H δφ ˙ φ , in which | δφ | = H/ 2 π [32]. In this way, the power spectrum considering Eq.(7), is given by or equivalently in terms of the standard scalar field φ The power spectrum P R , also can be expressed in terms of the number of e-folds N , as Numerically from Eq.(20) we obtained a constraint for the parameter A . In fact, we can obtain the value of the parameter A for given values of f , α and β parameters when number N and the power spectrum P R are given. In particular, for the values P R = 2 . 4 × 10 -9 , N = 60, f = 1 / 2 and κ = 1, we obtained that for the pair ( α = 0 . 775, β = 0 . 00126)[18], which corresponds to the parameter A /similarequal 8 . 225 × 10 -2 , for the pair ( α = 0 . 81, β = 0 . 2)[17], corresponds to A /similarequal 2 . 635 × 10 -2 and for the pair ( α = 0 . 85, β = 0 . 6)[17], which corresponds to A /similarequal 8 . 407 × 10 -5 . On the other hand, the scalar spectral index n s is given by the expression n s = d ln P R / ln k and considering Eq.(19), we get where the constants α 0 = α ( κ 3 A 2 f 2 ) 1+ β and γ = 2(1 -f )(1 + β ), respectively. /negationslash From Eq.(21), we clearly see that n s = 1, for f = 2 / 3 (recall that 1 > f > 0). However, as occurs in Ref.[9], n s = 1 for the value f = 2 / 3, where the scale factor increases as a ( t ) ∼ e t 2 / 3 . Also, we noted that in the limit α → 0, the scalar spectral index n s , given by Eq.(21), coincides with that corresponding to intermediate-inflationary model, where n s = 1 -C 1 /φ 2 with C 1 = 8(1 -f )(2 -3 f ) /f 2 , see Ref.[9]. The scalar spectral index n s in terms of the number of e-folds N , becomes On the other hand, the generation of tensor perturbations during the scenario inflationary would produce gravitational wave [29]. The corresponding spectrum is /s32 An important observational quantity is the tensor to scalar ratio r , which is defined as /s32 r = ( P g P R ) . From Eqs.(19) and (23) we write the tensor to scalar ratio as Combining Eqs.(11) and (24), we can write the tensor-scalar ratio r in terms of the number N , as In Fig.(1), the upper panel shows the evolution of the scalar spectrum index n s versus the number of e-folds N , and the lower panel shows the contour plot for the parameter r as a function of the n s at lowest order, for different values of the parameters-GCG, α and β in the case of the standard scalar field. In particular, the dotted, dashed, solid and dotdashed lines are for the pairs ( α = 0 . 81, β = 0 . 2) see Ref.[17], ( α = 0 . 775, β = 0 . 00126)[18], ( α = 0 . 85, β = 0 . 6)[17], and the standard intermediate model ( α = 0)[9], respectively. Here, we have used the value ρ Ch 0 = 1, then the parameter B s = α/ρ 1+ β Ch 0 = α . From the upper panel, we noted that the n s graphs for the pair ( α = 0 . 775, β = 0 . 00126) present a small displacement with respect to the number of e-folds N , when compared to the results obtained in the standard intermediate model, in which α = 0. On the other hand, from Ref.[3], two-dimensional marginalized constraints (68% and 95% confidence levels) on inflationary parameters r and n s , the spectral index of fluctuations, defined at k 0 = 0.002 Mpc -1 . In order to write down values that relate the tensor to scalar ratio and the spectral index we numerically solved Eqs. (21) and (24). Also, we have used the values f = 1 / 2, κ = 1 and for the parameter A the values A /similarequal 2 . 635 × 10 -2 ; 8 . 225 × 10 -2 ; 8 . 407 × 10 -5 , respectively. We noted that the pairs ( α = 0 . 81, β = 0 . 2) and ( α = 0 . 775, β = 0 . 00126), the model is well supported by the data as could be seen from Fig.(1). Also, we noted that the pair ( α = 0 . 85, β = 0 . 6) given by solid line, becomes disfavored from observational data, since the spectral index n s > 1. Also, we noted that for this pair r ∼ 0 (solid line). We have found that the pair ( α = 0 . 775, β = 0 . 00126), present a small displacement in relation to the standard intermediate model that corresponds to α = 0, as could be seen from the Fig.(1). In this way, we have shown that the intermediate-GCG inflationary model is less restricted than analogous ones standard intermediate inflationary models due to the introduction of new parameters, i.e., α and β parameters.", "pages": [ 7, 8, 9, 10, 11 ] }, { "title": "B. Tachyon field", "content": "For a tachyonic field the power spectrum of the curvature perturbations is given by P R = ( H 2 2 π ˙ φ ) 2 1 Z S [33], where Z S = V (1 -˙ φ 2 ) -3 / 2 ≈ V [34]. Following Ref.[34], the power spectrum P R is approximated to be P R /similarequal ( H 2 2 π ˙ φ ) 2 1 V . From Eq.(13) and considering Eq.(15), we write the power spectrum in terms of the tachyonic field in the following way The scalar spectral index n s , using Eq.(15), is given by Again, as the case of the standard scalar field from Eq.(27), we see that n s = 1, for the case f = 2 / 3. /negationslash On the other hand, the amplitude of tensor perturbations P g , is given by From expressions (26) and (28) we write the tensor to scalar ratio as Again, we noted that when α → 0 and considering Eqs.(27) and (29) the consistency relations at lowest order, n s = n s ( r ), reduced to standard tachyonic model, where n s = 1 -2 -3 f 16(1 -f ) r , see Ref.[26]. We noted numerically from Eqs.(27) and (29) that the trajectories in the n s -r plane between standard field and tachyon field can not be distinguished at lowest order. This coincidence in the consistency relations, n s = n s ( r ) between standard field and tachyon field, has already been noted in Ref.[35]. Nevertheless, the tachyon field inflationary leads to a deviation at second order in the consistency relations, where the spectral index at second order n (2) s , becomes[35] where the product ηγ = (9 m 4 p / 2)[2 V '' V ' /V 4 -10 V '' V ' 2 /V 5 + 9 V ' 4 /V 6 ], the constant C 1 is a numerical constant approximately C 1 /similarequal -0 . 72 and the constant C 2 ; is C 2 = 0 in the case of the standard scalar field and C 2 = 1 / 6 for tachyon field, respectively. Following Ref.[35], the expression for the tensor to scalar ratio at second order r (2) , in the tachyon field is given by In Fig.(2), we show the dependence of the tensor to scalar ratio r on the spectral index n s , for the pair ( α = 0 . 81, β = 0 . 2) in the case of the tachyonic field. Solid and dotted lines are for the trajectories at lowest order and at second order, respectively. In order to write down values that relate the tensor to scalar ratio and the spectral index, we numerically solved Eqs.(27), (29), (30) and (31). Again as before, we have used the values A /similarequal 2 . 635 × 10 -2 , ρ Cho = 1, f = 1 / 2 and κ = 1. We observed numerically, that the trajectories in the n s -r plane for the tachyonic field, when we used the second-order corrections to our analysis at first-order in slow roll, are small and this correction can be neglected to a very good approximation.", "pages": [ 11, 12, 13 ] }, { "title": "III. COSMOLOGICAL PERTURBATIONS", "content": "In this section we will analyze the scalar and tensor perturbations for our models, where the matter content of the universe are the standard scalar field and the tachyonic field, respectively.", "pages": [ 7 ] }, { "title": "IV. CONCLUSIONS", "content": "In this paper we have investigated the intermediate inflationary model in GCG. In the slow-roll approximation we have found solutions of the Friedmann equations for a flat universe containing a standard scalar field or a tachyonic field, respectively. In particular, for both scalar fields and from the scenario of intermediate inflation, we have obtained explicit expressions for the corresponding, effective potential, power spectrum of the curvature perturbations, tensor to scalar ratio and scalar spectrum index. For the scalar field, we have considered the constraints on the parameters of the GCG, from the WMAP seven year data. Here, we have taken the constraint r -n s plane at lowest order in the slow roll approximation. In order to write down values that relate the tensor to scalar ratio and the spectral index we numerically solved Eqs. (21) and (24). We noted that the pairs ( α = 0 . 81, β = 0 . 2) and ( α = 0 . 775, β = 0 . 00126), the model is well supported by the data as could be seen from Fig.(1). Also, we noted that the pair ( α = 0 . 85, β = 0 . 6) given by solid line, becomes disfavored from observational data, since the spectral index n s > 1. We have found that the pair ( α = 0 . 775, β = 0 . 00126), present a small displacement in relation to the standard intermediate model that corresponds to α = 0, as could be seen from the Fig.(1). In particular, we have used the values ρ Ch 0 = 1, f = 1 / 2, κ = 1 and A /similarequal 2 . 635 × 10 -2 ; 8 . 225 × 10 -2 ; 8 . 407 × 10 -5 , respectively. For the tachyonic field, we noted numerically from Eqs.(27) and (29) that the trajectories in the n s -r plane between standard field and tachyon field can not be distinguished at lowest order. However, we have obtained the dependence of the tensor to scalar ratio r on the spectral index n s at second order. In order to write down values that relate the tensor to scalar ratio and the spectral index at second order, we numerically solved Eqs.(27), (29), Finally, we have shown that the intermediate-GCG inflationary models are less restricted than analogous ones standard intermediate inflationary models due to the introduction of new parameters, i.e., α and β parameters. The incorporation of these parameters gives us a freedom that allows us to modify the standard intermediate model by simply modifying the corresponding values of the parameters α and β .", "pages": [ 13, 14 ] }, { "title": "Acknowledgments", "content": "R.H. was supported by COMISION NACIONAL DE CIENCIAS Y TECNOLOGIA through FONDECYT grants N 0 1090613, N 0 1110230 and by DI-PUCV grant 123.703/2009. M.O. was supported by Proyecto D.I. PostDoctorado 2012 PUCV. N.V. was supported by Proyecto Beca-Doctoral CONICYT N 0 21100261. R. Herrera and E. San Martin, Eur. Phys. J. C 71 , 1701 (2011); R. Herrera and M. Olivares, Mod. Phys. Lett. A 27 , 1250101 (2012); R. Herrera and M. Olivares, Int. J. Mod. Phys. D 21 , 1250047 (2012).", "pages": [ 14, 15 ] } ]

[ { "title": "Stellar equilibrium in Einstein-Chern-Simons gravity", "content": "C.A.C. Quinzacara 1, ∗ and P. Salgado 1, † 1 Departamento de F'ısica, Universidad de Concepci'on, Casilla 160-C, Concepci'on, Chile", "pages": [ 1 ] }, { "title": "Abstract", "content": "We consider a spherically symmetric internal solution within the context of Einstein-ChernSimons gravity and derive a generalized five-dimensional Tolman-Oppenheimer-Volkoff (TOV) equation. It is shown that the generalized TOV equation leads, in a certain limit, to the standard five-dimensional TOV equation. PACS numbers: 04.50.+h, 04.20.Jb, 04.90.+e", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Some time ago was shown that the standard, five-dimensional General Relativity can be obtained from Chern-Simons gravity theory for a certain Lie algebra B [1], which was obtained from the AdS algebra and a particular semigroup S by means of the S-expansion procedure introduced in Refs. [2, 3]. The five-dimensional Chern-Simons Lagrangian for the B algebra is given by [1] where l is a length scale in the theory (see [1]), R ab = dω ab + ω a c ω cb is the curvature two-form with ω ab the spin connection, and T a = De a with D the covariant derivative with respect to the Lorentz piece of the connection. From (1) we can see that [1]: It is the purpose of this letter to find the stellar interior solution of the Einstein-ChernSimons field equations, which were obtained in Refs. [5, 6]. We derive the generalized five-dimensional Tolman-Oppenheimer-Volkoff (TOV) equation and then we show that this generalized TOV equation leads, in a certain limit, to the standard five-dimensional TOV equation.", "pages": [ 2 ] }, { "title": "II. EINSTEIN-CHERN-SIMONS FIELD EQUATIONS FOR A SPHERICALLY SYMMETRIC METRIC", "content": "In Ref. [5] it was found that in the presence of matter described by the Lagrangian L M = L M ( e a , h a , ωab ), we see that the corresponding field equations are given by For simplicity we will assume T a = 0 and k ab = 0. In this case the field equations (2) can be written in the form [6] where α = α 3 /α a , κ E = κ/ 2 α 3 , T ab = /star ( δL M /δe a ), ' /star ' is the Hodge star operator (see Appendix B) and T ab is the energy-momentum tensor of matter fields (for details see Ref. [6]). Since we are assuming spherical symmetry the metric will be of the form /negationslash where d Ω 2 3 = dθ 2 1 +sin 2 θ 1 dθ 2 2 +sin 2 θ 1 sin 2 θ 2 dθ 2 3 and η ab = diag( -1 , +1 , +1 , +1 , +1). The two unknown functions f ( r ) and g ( r ) will not turn out to be the same as in Ref. [6]. In Ref. [6] was found a spherically symmetric exterior solution, i.e. a solution where ρ ( r ) = p ( r ) = 0. Now f ( r ) and g ( r ) must satisfy the field equations inside the star, where ρ ( r ) = 0 and p ( r ) = 0. For this we need the energy-momentum tensor for the stellar material, which is taken to be a perfect fluid. /negationslash Introducing an orthonormal basis, Taking the exterior derivatives, using Cartan's first structural equation T a = de a + ω a b e b = 0 and the antisymmetry of the connection forms we find the non-zero connection forms. The use of Cartan's second structural equation permits to calculate the curvature matrix R a b = dω a b + ω a c ω c . Introducing these results in (3) and considering the energy-momentum tensor as the energy-momentum tensor of a perfect fluid at rest, i.e., T TT = ρ ( r ) and T RR = T ii = p ( r ), where ρ ( r ) and p ( r ) are the energy density and pressure (for the perfect fluid), we find [6]", "pages": [ 3, 4 ] }, { "title": "III. THE GENERALIZED TOLMAN-OPPENHEIMER-VOLKOFF EQUATION", "content": "Since when the torsion is null, the energy-momentum tensor satisfies the following condition (see Appendix B): we find that, for a spherically symmetric metric, (8) yields an expression known as the hydrostatic equilibrium equation. Following the usual procedure, we find that (5) has the following solution: where the Newtonian mass M ( r ) is given by On the other hand, from (6) we find that Introducing (12) into (9) we find and introducing (10) into (13) we obtain the generalized five-dimensional TolmanOppenheimer-Volkoff equation From (14) we can see that in the case of small l 2 limit, we can expand the root to first order in l 2 . In fact Introducing (15) into (14) we find From (16) we can see that, in the limit where l -→ 0, we obtain which is the standard five-dimensional Tolman-Oppenheimer-Volkoff equation (see Eq. (A4)) (compare with the four-dimensional case shown in Ref. [7]). To solve the generalized TOV equation (14), an equation of state relating ρ and p is needed. This equation should be supplemented by the boundary condition that p ( R ) = 0 where R is the radius of the star. Given an equation of state p ( ρ ), the problem can be formulated as a pair of first-order differential equations for p ( r ), M ( r ) and ρ ( r ), (14) and with the initial condition M (0) = 0. In addition, it is necessary to provide the initial condition ρ (0) = ρ 0 . Let us return to the problem of calculating the metric. Once we compute ρ ( r ), M ( r ), and p ( r ), we can immediately obtain g ( r ) from (10) and f ( r ) from (12) where we have set f ( ∞ ) = 0, a condition consistent with the asymptotic limit from the exterior solution. It should be noted that if r > R , i.e., out of the star, the following conditions are satisfied: Integrating (19) we find so that which coincides with the outer solution.", "pages": [ 4, 5, 6 ] }, { "title": "A. Constant Density: ρ ( r ) = ρ 0", "content": "We will now consider the solution of (14) in the case where the energy density is constant, ρ ( r ) = ρ 0 , inside the star. In this case the hydrostatic equilibrium equation (9) can be directly integrated, where C is an integration constant. On the other hand, from (18) M ( r ) is given by Introducing (24) into (10) we have Now, let us add the field equations (5) and (6) Using now (23), multiplying by e -g and integrating we have where C 0 is the corresponding integration constant. Since we find where √ √ Then, we proceed to adjust the constants C , C 0 , and C 1 , so that the interior solution and exterior must match at r = R . In addition one should require that the pressure vanishes at r = R . The calculations give and", "pages": [ 6, 7 ] }, { "title": "IV. SUMMARY AND OUTLOOK", "content": "We have considered a spherically symmetric internal solution within the context of Einstein-Chern-Simons gravity. We derived the generalized five-dimensional TolmanOppenheimer-Volkoff (TOV) equation and then we proved that this generalized TOV equation leads, in a certain limit, in the standard five-dimensional TOV equation.", "pages": [ 7 ] }, { "title": "Acknowledgments", "content": "This work was supported in part by Direcci'on de Investigaci'on, Universidad de Concepci'on through Grant # 212.011.056-1.0 and in part by FONDECYT through Grant N 1130653. One of the authors (C.A.C.Q) was supported by grants from the Comisin Nacional de Investigacin Cient'ıfica y Tecnol'ogica CONICYT and from the Universidad de Concepci'on, Chile.", "pages": [ 8 ] }, { "title": "Appendix A: The standard Tolman-Oppenheimer-Volkoff equation in 5D", "content": "Let us recall that the energy-momentum tensor satisfies the condition If T TT = ρ ( r ) and T RR = T ii = p ( r ) we find so that an expression known as the hydrostatic equilibrium equation . From Eqs. (A10) and (A22) of Ref. [6] we find Introducing (A2) into (A3) we obtain the standard five-dimensional Tolman-OppenheimerVolkoff equation This may be compared with the four-dimensional case shown in equation (1.11.13) of reference [7].", "pages": [ 8 ] }, { "title": "Appendix B: Energy-momentum tensor", "content": "It is known that if the torsion is null, then the energy-momentum tensor is divergence-free, ∇ µ T µν = 0. The 1form energy-momentum is given by Theorem. If the energy-momentum tensor T µν and the 1-form energy-momentum ˆ T a are related by equation (B1), then in a torsion-free space-time Proof. After some algebra, we find and using the Weyl's lemma we obtain", "pages": [ 8, 9 ] }, { "title": "1. The Hodge star operator", "content": "The Hodge star operator for a p -form P = 1 p ! P α 1 ··· α p dx α 1 · · · dx α p in a d -dimensional manifold with a non-singular metric tensor g µν is defined as where ε α 1 ··· α d is the total antisymmetric Levi-Civita tensor density of weight -1.", "pages": [ 9 ] }, { "title": "2. Hydrostatic equilibrium equation", "content": "Let us consider a spherically and static-symmetric metric in five dimensions. The 1-form energy-momentum is given by where T ab is the energy-momentum tensor in a comoving orthonormal frame. So, if the matter is a perfect fluid then Computing the conservation equation we have where we have used the torsion-free condition D ω e a = 0. Therefore The calculations give from which we get the so-called hydrostatic equilibrium equation", "pages": [ 9, 10 ] }, { "title": "Appendix C: Dynamic of the field h a", "content": "We consider now the field h a . Expanding the field h a = h a µ dx µ in their holonomic index we have [6] For the space-time to be static and spherically symmetric, the field h µν must satisfy the Killing equation L ξ h µν = 0 for ξ 0 = ∂ t and the six generators of the sphere S 3 must be ξ 1 = ∂ θ 3 , ξ 2 = sin θ 3 ∂ θ 2 +cot θ 2 cos θ 3 ∂ θ 3 , ξ 3 = sin θ 2 sin θ 3 ∂ θ 1 +cot θ 1 cos θ 2 sin θ 3 ∂ θ 2 +cot θ 1 csc θ 2 cos θ 3 ∂ θ 3 (C2) ξ 4 = cos θ 3 ∂ θ 2 -cot θ 2 sin θ 3 ∂ θ 3 , ξ 5 = sin θ 2 cos θ 3 ∂ θ 1 +cot θ 1 cos θ 2 cos θ 3 ∂ θ 2 -cot θ 1 csc θ 2 sin θ 3 ∂ θ 3 , ξ 6 = cos θ 2 ∂ θ 1 - cot θ 1 sin θ 2 ∂ θ 2 . Then, we have From Eq. (3) we know that the dynamic of the field h a is given by with where Introducing (C6 - C8) into (C4) we have To find solutions to (C9, C10, C11), we assume that h t ( r ) depends on r only through f ( r ), namely Introducing (C12) into (C11) we have from which we obtain the following linear differential equation, which is of first order and inhomogeneous: where ˙ h t := dh t ( f ) df . The homogeneous solution is given by where A is a constant to be determined. The particular solution depends on the shape of h r . If we assume a functional relationship h with f , then the linearity of differential equation suggests the following ansatz: where B n and C m are real constants. So that the particular solution is given by Therefore the general solution is of the form From (C10) we find where D is an integration constant. This means where A , B n , and C m are arbitrary constants, and -e 2 f ( r ) is the metric coefficient g 00 .", "pages": [ 10, 11, 12 ] }, { "title": "1. Field asymptotically constant", "content": "Consider the simplest case where in this case (C19) leads and Since the vielbein is regular at r = 0 (center of the star), h a should also be regularly at r = 0, i.e. we should have D = 0. Note that the coefficient e f ( r ) is regular at r = 0 as can be seen from (C19). From (C20) we can see that the asymptotic behavior of the metric coefficients is given by Thus the asymptotic behavior of the field h a is given by", "pages": [ 12, 13 ] }, { "title": "2. Constant density", "content": "If the density is constant then the inner solution is given by (25) and (29). In this case the solution for the field h a is given by and where", "pages": [ 13 ] } ]

[ { "title": "Newtonian Noise Limit in Atom Interferometers for Gravitational Wave Detection", "content": "Flavio Vetrano a,1,2 , Andrea Vicer'e b,1,2 2 INFN, Sezione di Firenze, INFN, I-50019 Sesto Fiorentino, Italy Received: date / Accepted: date Abstract In this work we study the influence of the newtonian noise on atom interferometers applied to the detection of gravitational waves, and we compute the resulting limits to the sensitivity in two different configurations: a single atom interferometer, or a pair of atom interferometers operated in a differential configuration. We find that for the instrumental configurations considered, and operating in the frequency range [0 . 1 -10] Hz, the limits would be comparable to those affecting large scale optical interferometers. Keywords Atom interferometry · Newtonian noise · Gravitational waves PACS 04.80.Nn · 95.55.Ym · 03.75.Dg · 37.25.+k", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The direct detection of Gravitational Waves is one of the most exciting challenges of current scientific research. The first generation of ground-based optical interferometric detectors, including Virgo [1] and GEO600 [3] in Europe, and the LIGO [2] interferometers in USA, achieved design sensitivity and carried out several science runs, which set interesting upper limits on several classes of astrophysical sources [4,5,6,7]. The construction of a 'second generation' of optical interferometers, Advanced LIGO [8] and Virgo [9], and the new Japanese detector KAGRA [10], is well underway; thanks to the implementation of several technical upgrades, the advanced detectors are expected to come on line with a sensitivity about ten times better than first generation instruments. In the meanwhile, the conceptual design of third generation detectors, like the Einstein Telescope [11, 12], has started. For all these optical ground based detectors the sensitivity in the low frequency band, below 10 Hz, is ultimately limited by the so called 'gravity gradient', or Newtonian Noise (NN) [13,14], whose source is the direct coupling of the test masses with any mass-density change in the environment, especially of seismic or atmospheric origin. Atom interferometers (see [15] for a review) have been proposed recently as GW detectors [16,17,18,19,20,21], on the basis of previous general ideas [22]. These instruments promise to be less sensitive to some of the noise sources affecting optical instruments: for instance, being the atoms in free fall, no direct seismic noise should be present. The effect of gravitational waves is a change in the phase accumulated by atoms' wave functions, which can be detected by observing the interference of two atom beams. However, also the non-radiative gravitational fields of terrestrial origin affect the phase, in a different way as we will show: the question arises then, if the 'low frequency wall' due to NN is relevant also for these new proposed detectors. In this paper we consider only the NN of seismic origin and we carry out a detailed calculation of its contribution to the sensitivity curve of an atom interferometer both in the 'single detector' configuration and in the 'coupled differential' configuration. It is worth underlining that this study is motivated by the different way in which gravitational fluctuations couple to atom interferometers and to optical interferometers, related to the fact that in the first case the test masses are atoms freely traveling across the instrument. We anticipate our conclusions: the atom interferometers are subject to NN in a degree similar to optical interferometers, and therefore will require appropriate technical solutions to overcome this noise limit in the frequency band below 10Hz. The paper is organized as follows: in Sec. 2 we consider a definite atom interferometer and we compute its response to a fluctuating gravity field; in Sec. 3 we apply the formulas to the case of a single detector, deriving the limits on sensitivity; finally in Sec. 4 we consider two atom interferometers operated in differential configurations.", "pages": [ 1, 2 ] }, { "title": "2 Newtonian noise of seismic origin in atom interferometers", "content": "In optical interferometric GW detectors the test masses are suspended mirrors: a pendular suspension is indeed the best approximation on Earth for a freely falling test mass. In atom interferometers instead the role of test masses is played by atoms in free fall, hence our intent is to determine the influence of the Newtonian coupling to an external, time-varying mass distribution, on freely falling masses. Some general considerations are possible: if the effect originates from seismic noise, it is driven by an external masses displacement field, whose linear power spectral density will generally have the form ˜ W ( ω ) ∼ ω -2 , mediated by a transfer function from the seism to the test masses motion behaving also as ω -2 [14,23,24], where ω is the angular frequency. Therefore the effect on test masses is expected to be of the form θ ( ω ) Γω -4 , hence more relevant at low frequencies, where θ ( ω ) is a kind of reduced transfer function, depending on the detection device, and Γ is a scale factor depending on the model of seismic waves (it is recognized that the role of main source is played by Rayleigh surface waves, especially the fundamental mode and few overtones [23,24]). To derive the actual expression of θ ( ω ) for NN in an atom interferometer, we use the ABCD formalism for matter waves, described elsewhere in detail [20,25]. Assume that the Hamiltonian of the motion for the atoms is at most quadratic in momentum and position operators where p n ( r ) and q n ( r ) are vectors of momentum and position, respectively, whereas α, β, γ, δ are suitable square matrices (note that δ = -α T , with T indicates the transposed matrix), and M is the atom rest mass. The last term in the Hamiltonian represents the response to the local, fluctuating gravitational field g ( t ): in the following, we will consider only the component along the direction of motion of the atoms, as in the paraxial approximation all transverse effects are neglected. The γ term allows to model the response to gravitational waves: in the Fermi gauge, and considering Fourier components, one can show that ˆ γ = ω 2 2 ˆ h ( ω ), where ˆ h ( ω ) is the gravitational wave strain tensor (see for instance [20]). Consider an atoms' beam (a Gaussian packet under paraxial approximation [20, 25,26,27,28]) which is divided and recombined through a sequence of R light-field beam splitters, supplied by the same laser: from the first beam splitter to the last one (the output port) we may identify two paths, conventionally labeled s and i . By exploiting the ttt theorem [25] for the atoms/beam splitter interactions, and the mid-point property of Gaussian beams [29], the phase difference at the output port of the interferometer can be written as: where k s ( i ) j is the momentum transferred to the atoms by the j -th beam splitter along the s ( i ) arm, ω s ( i ) j is the angular frequency of the laser beam and θ s ( i ) j is the phase of the laser beam at the j -th interaction, q s ( i ) j is the distance of j-th interaction point from the laser source; equal masses are assumed for the atoms along the s and i paths. The expression in Eq. 2 is manifestly gauge-invariant [20, 25], and the evolution of the wave packets can be obtained, by means of the Ehrenfest theorem, from Hamilton's equations for the vector χ ( t ) [20,25,28] where in the form where here τ represents the time-ordering operator, and an appropriate perturbative expansion can be used to evaluate the time-ordered exponential in Eq. 6 [20,25, 28]. As a simple reference configuration let us consider a 'Ramsey-Bord'e' atom interferometer, with a Mach-Zehnder geometry, as outlined in Fig. 1 [15,20,25]. In the following, we will also assume that the instrument is crossed by a plane GW with '+' polarization and amplitude h , propagating along the x 3 = z axis, perpendicular to the plane of the interferometer; we adopt in the following a description in Fermi coordinates, which represents the best approximation to the Laboratory Cartesian system [30]. Assuming the same 'stable' frequency for the laser beams and neglecting the steady proper laser phases, the phase shift formula in Eq. 2 becomes Let us assume that atoms are subjected only to a fluctuating gravitational field g ( t ). Considering Eq. 1, Eq. 3, Eq. 4 and Eq. 7 we have /negationslash and we obtain We are interested in the low frequency range, where the newtonian noise is expected to be the limiting factor on account of its ω -4 shape. We will therefore assume that the single atom interferometer has a linear dimension smaller than the wavelength of seismic surface waves, which we will assume to set also the coherence length. Introducing the Fourier transform ˆ g ( ω ) of the fluctuating field we can also write and we assume, in the long wavelength approximation, that ˆ g ( ω ) is the same at any point of the interferometer. Therefore the solution of the Hamilton equations Eq. 5 can be written as this expression allows to compute the values of the coordinates and momenta of the atoms at the interaction points with the laser: by iterating the relation in Eq. 5 to the four interaction points of the interferometer in Fig. 1, setting t 3 = t 2 and defining T = t 4 -t 3 = t 2 -t 1 , we finally obtain the phase shift at the output port of the interferometer: this is the fundamental formula to estimate the effect of the fluctuating field ˆ g . We recall that k is the unperturbed wave vector of the laser beam, corresponding to the impulse (in units of the reduced Planck constant ¯ h ) transferred to the atom at each interaction point. Note also that in the limit ω → 0 the expression in Eq. 13 corresponds to the well known static result [29,31].", "pages": [ 2, 3, 4, 5 ] }, { "title": "3 Newtonian-Noise limit on sensitivity: the single detector case", "content": "In the weak field approximation, to first order in the amplitude h of an impinging gravitational wave, the phase shift at the output of the interferometer in Fig. 1 has been already obtained in a fully covariant way [20]. Indicating with q 1 the unperturbed distance of the first interaction point from the laser, and with p 1 the unperturbed momentum of the atoms, just before the first interaction with the laser beam, we recall that the Fourier transform of the phase shift, as a function of the Fourier transformed amplitude ˆ h of the GW, can be written as in which the proper laser phases have been neglected. Comparing with the expression of the response to a fluctuating local gravity field Eq. 13, we note that the second term of Eq. 14 corresponds to it, with the substitution ˜ g → q 1 2 ω 2 ˜ h : however, the overall response to GWs includes also a dynamic term depending on the atom momentum p 1 and on the momentum k transferred to the atoms: hence the effects of the local gravitational field and of the gravitational waves are in principle distinguishable. For a single interferometer with the laser source close to the device, actually the last term can be neglected and the more relevant one is the term proportional to p 1 , since we can also generally neglect the recoil term k ¯ h 2 M . This expression can be directly translated into a relation between linear power spectral densities (LPSD), that we denote by a tilde, defined in terms of the two-point correlation functions as 〈 ˆ g ( ω ) ˆ g ( ω ' )〉 = 2 πδ ( ω -ω ' ) ˜ g 2 ( ω ) (15) in which the angular brackets represent the statistical average. From Eq. 13 and Eq. 14 we obtain where the distance L = 2 Tp 1 /M travelled by the atoms in the interferometer of Fig. 1 has been introduced; combining the two equations, we deduce the expression for the equivalent strain ˜ h NN induced by the fluctuating field ˜ g ( ω ). It is useful to discuss here the scale of the ˜ g ( ω ) LPSD, referring to typical values measured at the site of the Virgo interferometers; we recall indeed that we are considering the effect of an external fluctuating gravity field on freely falling test masses, which is the same situation experienced by the test masses of optical interferometers [14,23,24]; even though the detailed shape of the NN affecting a instrument like Virgo depends on the model for the seismic sources and the superficial Earth layers, similar results are obtained in different cases, which can be summarized as follows where L V = 3000m is the length of Virgo arms, ˜ X ( ω ) is the displacement LPSD for a single suspended mirror, and ˜ x seism ( ω ) is the measured LPSD of the ground seism [32]; the factor √ 4 takes into account that in Virgo the noise due to the four end-station mirrors adds in quadrature. Considering the relation between the mirror motion and its acceleration, due to the fluctuating gravitational field, ˜ g ( ω ) = ω 2 ˜ X ( ω ), we obtain we further assume that the seismic noise measured at the Virgo site is well approximated by [33] Following [28], let us assume very ambitious parameters for the single RamseyBord'e atom interferometer: a length L ∼ 200m, which could result in interesting sensitivities to gravitational waves, and a time of flight T = 0 . 4s, in order to have not too small a bandwidth; obviously the choice implies atom speeds of the order of 250 m/s, and we underline that such choices are probably beyond the limits of current technologies. Anyway, we obtain as an estimate of the scale of the fluctuating gravitational field seen by the atom interferometer. To appreciate the result, we show in Fig. 2 a example of the newtonian noise of Eq. 17 assuming the expression in Eq. 21 for the LPSD of the fluctuating gravitational field; in the same figure we plot, for comparison, the corresponding newtonian noise for the Virgo detector 1 . The zeroes represent frequencies at which the atom interferometer is insensitive both to the gravity gradient noise and to GW; note that the one shown is not a complete noise budget, to which other noises would contribute, particularly the atom shot noise which would exhibit peaks at those frequencies, not differently from an optical interferometer in a Michelson configuration and without FabryPerot cavities. Apart this specific feature, the comparison with a large optical interferometer shows a similar behavior as a function of the frequency, with a different noise scale dictated by the different linear dimensions of the instruments. We underline that for this type of atom interferometer, it could be unrealistic to increase the linear size L even further: to this end, a differential configuration appears more promising.", "pages": [ 5, 6, 7 ] }, { "title": "4 Two detectors operated in a differential configuration", "content": "Let us now consider the second term in Eq. 14, proportional to the position q 1 . This term, already introduced in a different context [34], is a sort of 'clock' term which takes into account the influence of the GW on the laser beam, along its path from the source to a well defined physical point. Its role was discussed in recent papers [35,36,37] and the most relevant new property is the introduction of q 1 (path of laser beam) in place of L (path of atom beam); so, in order to improve the sensitivity, enlarging q 1 seems in principle easier than enlarging L . This solution requires measuring the distance from the laser, and carries additional requirements on the coherence and stability of the laser beam, while maintaining it at a sufficient power density: it is therefore premature to draw too optimistic conclusions about the practicality of the configuration. However, the idea of adopting a two-interferometers differential configuration [21] appears very appealing in order to render the system independent from the laser position, and may furthermore yield a good common-modes rejection. Under the hypothesis of a common laser source for two identical Mach-Zehnder atom interferometers in differential configuration, for which the relative distance D satisfies the condition ωD/c /lessmuch 1 (with c the speed of light in vacuum), from Eq. 14 the overall difference between the two partial phase differences at the output ports can be formally obtained as where D ≡ q II 1 -q I 1 as anticipated. Considering also Eq. 13 we obtain for the differential configuration where the difference in the right hand side requires some discussion. In a given frequency band, if the two fluctuating gravity fields ˆ g 1 , 2 act upon sufficiently distant atom interferometers, they will be uncorrelated, and we will obtain for the LPSD simply a sum in quadrature displaying no conceptual difference with respect to the limits obtained for optical interferometers with long arms [32]. Considering instead a low-frequency, longwavelength approximation, it may be appealing the situation in which, even with two separated interferometers, the residual correlation leads to a partial noise cancellation in Eq. 23. We recall that the signals ˆ g 1 , 2 ( ω ) are assumed to be stochastic acceleration fields in positions 1 and 2, projected along the direction specified by the segment D as in Fig. 3. We further assume to model the stochastic noise in the simplest possible way, namely as due to uncorrelated fluctuations in the density of the material surrounding the detector [14]. In other words a density fluctuation ∆M ( t ) will contribute to the acceleration field in points 1 and 2 as Considering only the component acting along the direction separating the two points 1 and 2, we obtain as the contribution to the fluctuation of the acceleration field due to a single mass element. To obtain the total fluctuation, we need now to sum over the space. We first assume for simplicity that the space around the two stations with atom interferometers can be considered homogeneous: this could be the case for instance if the instrumentation is placed in a deep mine, at a depth much larger than D . We are therefore interested in the quantity which should be summed over the volume. It is convenient to evaluate the spectral density where, following again Saulson [14], we assume the sum to be extended over volume elements of linear size λ/ 2 , with ∆M fluctuating coherently inside these regions, and totally uncorrelated otherwise: We obtain therefore If we additionally assume that the mass fluctuations do not depend on r , we can further simplify, obtaining where we have approximated the sum with an integral, normalizing by the volume element of the coherent region ( λ/ 2) 3 . If we were to retain only the first term, we would obtain the same result as in [14], corrected for a factor 2 which is wrong in the original paper. The integration over the angular functions is directly carried out, resulting in a lengthy expression: which, as expected, displays double poles in r = 0 and in r = D . Both divergences are artefacts, which should be regulated introducing cutoffs r ≥ λ 4 and at | r -D | ≥ λ 4 . However, it is now necessary to distinguish two cases Short wavelength If the distance D /greatermuch λ , then the integral over r gives and we obtain Long wavelength In the long wavelength approximation the integral in Eq. 33 can be carried out assuming r ≥ λ 4 /greatermuch D , obtaining hence it seems at first surprising that the dependence on D cancels out in the long wavelength approximation, whereas one could have expected to retain a dependence, which could lead to zero the noise in the D → 0 limit case. However, we are actually in a situation in which the instrument is sensitive to the gradient of the gravity acceleration (see Eq. 23), and therefore, barring other sources of noise, the sensitivity is independent on the baseline D . We can now use Eq. 12 of [24] to relate the mass fluctuations with the measured seism where ρ 0 is the density of the medium. We finally obtain where we have used the relation λω = 2 πc L , with c L the speed of longitudinal seismic waves. Comparing with Eq. 18 for the gravity gradient noise affecting the Virgo interferometer, we see that in the short wavelength limit, represented by Eq. 39, the frequency dependence (as expected) is the same. Instead, in the long wavelength limit Eq. 40, the NN affecting the atom interferometer has a slower growth for ω → 0, reflecting the presence of correlated noise at the two stations, that partially cancels out in Eq. 23. We underline that this cancellation is not specific of a dual atomic interferometer: the same effect would occur in optical interferometers like Virgo, for shorter baselines. However, in optical interferometers long baselines are motivated by the need to reject the mirror position noise, which scales inversely with the distance: in atom interferometers some position noises, like the thermal noise, are instead expected to be absent, hence the baseline could be shorter. In order to assess the significance of the cancellation effect, we choose favorable, yet realistic parameters: for the medium surrounding our hypothetical instrument, we assume a large c L = 5000 m/s, characteristic of compact rock, and a density ρ 0 /similarequal 2 . 7 × 10 3 kgm -3 , a typical value for the continental crust; we also assume, on the basis of measurement taken in underground environments (for instance in the Kamioka mine [38] which will host KAGRA) a seismic noise ˜ x seism 10 times lower than the one measured at the Virgo site (Eq. 20). We also assume to build a relatively large instrument, taking for the distance between the atom interferometers a value D /similarequal 1km as proposed in [39]: we obtain The resulting limit to the atom interferometer sensitivity is displayed in Fig. 4, over a frequency range which runs from the long to the short wavelength regimes; for comparison we display also the NN affecting the Virgo instrument; in the high frequency regime, the two curves differ just by a small scale, reflecting the different size of the instruments and the lower seismic noise anticipated for an underground atom interferometer. In the low frequency regime the residual correlation of the newtonian noise which affects the two atom interferometers, thanks to the shorter baseline, contributes to a milder growth as ω → 0, and therefore leads to a sizable, though not dramatic, reduction of the noise over the Virgo case.", "pages": [ 8, 9, 10, 11, 12 ] }, { "title": "5 Conclusions", "content": "In this work we have evaluated the effect of fluctuations of the gravity field on the sensitivity of atom interferometers, thus providing an estimate of the so-called newtonian (or gravity gradient) noise for this kind of instruments. We have seen that a mid-scale atom interferometer, with a baseline L ∼ 200m, is subject to a noise essentially equivalent to the one affecting a large scale optical interferometer, as Virgo. We have also found that operating two small-scale atom interferometers, linked by a laser, at a larger distance D ∼ 1 km, in differential configuration, as proposed for instance in [39], there is an advantage at low frequency thanks to the residual newtonian noise correlation and the resulting partial cancellation. However, the noise reduction is not dramatic and the newtonian limit remains very significant: it is worth reminding that in order to detect a binary neutron star inspiral (say, at z ∼ 1) sensitivities better than 10 -22 would be required at 1 Hz; even for larger systems, say a 1000 M /circledot binary black-hole coalescence, sensitivities of the order of 10 -20 should be achieved, as discussed for instance in [40]. We conclude that, similarly to what is foreseen for future optical interferometers [11], operating successfully atom interferometers in the [0 . 1 , 10] Hz frequency window will require mitigating the gravity gradient noise; not just by choosing very quiet, underground sites, but also devising clever noise subtraction strategies. We acknowledge that this study has a limitation in the model for the gravity fluctuations, which is approximate; however, as it has been the case for similar studies carried out for optical interferometers [24,23], we believe that the use of more refined models will change the numerical results only by small factors, which would not alter our conclusions.", "pages": [ 12, 13 ] } ]

[ { "title": "Gravitational Waves in a Spatially Closed de-Sitter Spacetime", "content": "Amir H. Abbassi ∗ and Jafar Khodagholizadeh † Department of Physics, School of Sciences, Tarbiat Modares University, P.O. Box 14155-4838, Tehran, Iran. Amir M. Abbassi ‡ Department of Physics, University of Tehran, P.O. Box 14155-6455, Tehran, Iran.", "pages": [ 1 ] }, { "title": "Abstract", "content": "Perturbation of gravitational fields may be decomposed into scalar, vector and tensor components. In this paper we concern with the evolution of tensor mode perturbations in a spatially closed de-Sitter background of Robertson-Walker form. It may be thought as gravitational waves in a classical description. The chosen background has the advantage of to be maximally extended and symmetric. The spatially flat models commonly emerge from inflationary scenarios are not completely extended. We first derive the general weak field equations. Then the form of the field equations in two special cases, planar and spherical waves, are obtained and their solutions are presented. The radiation field from an isolated source is calculated. We conclude with discussing the significance of the results and their implications. PACS numbers: Keywords: de-Sitter space, Gravitational waves, Tensor mode", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Here we first investigate the freely propagating gravitational field requiring no local sources for their existence in a particular background. As an essential feature of the analysis of general theory of small fluctuations, we assume that all departures from homogeneity and isotropy are small, so that they can be treated as first order perturbations. We focus our analysis on an unperturbed metric that has maximal extension and symmetry by taking K = 1 and presence of a positive cosmological constant. The background is de-Sitter spacetime in slicing such that the spatial section is a 3-sphere. In the previous works mostly the case K = 0 wre investigated extensively [1-5]. Even though in some works K is not fixed for demonstrating the general field equations, but for solving them usually K = 0 is imposed [6,7]. The study of this particular problem is interesting and relevant to present day cosmology for the following. Seven-year data from WMAP with imposed astrophysical data put constraints on the basic parameters of cosmological models. The dark energy equation of state parameter is -1 . 1 ± 0 . 14, consistent with the cosmological constant value of -1. While WMAP data alone can not constraint the spatial curvature parameter of the observable universe Ω k very well, combining the WMAP data with other distance indicators such as H 0 , BAO, or supernovae can constraint Ω k . Assuming ω = -1, we find Ω Λ = 0 . 73 + / -0 . 04 and Ω total = 1 . 02 + / -0 . 02. Even though in WMAP seven-year data it has been concluded as an evidence in the support of flat universe, but in no way the data does not role out the case of K = 1 [8]. In the nine-year data from WMAP the reported limit on spatial curvature parameter is Ω k = -0 . 0027 +0 . 0039 -0 . 0038 [9]. There is much hope Planck data reports release makes the situation more promising to settle down this dispute. But Planck 2013 results XXVI ,merely find no evidence for a multiply-connected topology with a fundamental domain within the last scattering surface. Further Planck measurement of CMB polarization probably provide more definitive conclusions[10] . In the analysis of gravitational waves commonly Minkowski metric is taken as the unperturbed background. According to mentioned observational data, the universe is cosmological constant dominated at our era. So in the analysis of gravitational waves we should replace the Minkowski background with de-Sitter metric. The essential point is that spatially open and flat de-Sitter spacetime are subspaces of spatially closed de-Sitter space. The first two are geodesically incomplete while the third is geodesically complete and maximally extended. From the singularity point of view the issue of completeness is crucial for a spacetime to be non-singular. Taking the issue of completeness seriously, we have no way except to choose K = 1. By choosing the maximally extended de-Sitter metric as our unperturbed background we include both cosmological and curvature terms in discussion of gravitaional waves [11-15]. We begin by deriving the required linear field equations. Then the solution of the obtained equation are discussed. At the end an attempt is done to solve the field equation by source.", "pages": [ 2, 3 ] }, { "title": "II. LINEAR WEAK FIELD EQUATIONS", "content": "Supposed unperturbed metric components in Cartesian coordinate system are [16]: with the inverse metric where K is curvature constant and α = √ 3 Λ . The non-zero components of the metric compatible connections are: Dot stands for derivative with respect to time. Since we are working in a holonomic basis, then the connection is torsion-free or symmetric with respect to lower indices. Let us decompose the perturbed metric as: where ¯ g µν is defined by eq.(1) and h µν is small symmetric perturbation term. The inverse metric is perturbed by with components Perturbation of the metric produces a perturbation to the affine connection [15] Thus eq.(7) gives the components of the perturbed affine connection as: δ Γ 0 i 0 = ˙ a a h i 0 - 1 2 ∂ i h 00 (12) The tensor mode perturbation to the metric can be put in the form where D ij s are functions of /vector X and t , satisfying the conditions The perturbation to the affine connection in tensor mode are: δ Γ i jk = 1 2 [ ∂ k D ij + ∂ j D ik - ∂ i D jk - 2 K ( D im - Kx x D lm ) × i l The Einstein field equation without matter source for the tensor mode of perturbation gives where Scale factor a ( t ) satisfies the Friedmann equation, so we get Inserting eq.(22) in eq.(21) and eq.(21) in eq.(20) , we would have It is straightforward to show that It remains to put eq.(24) in eq.(23), then we obtain the final equation. Our first task to establish the field equations is fulfilled. Next we look for special solutions of this field equation analogue to plane and spherical waves. For the plane wave like solutions, using Cartesian coordinate systems is suitable while for the spherical waves, polar coordinates ( χ, θ, φ ) are convenient.", "pages": [ 3, 4, 5 ] }, { "title": "III. PLANE-WAVE ANALOGUE", "content": "In the case of flat models i.e. K = 0 condition (15) reduces to Looking for a wave propagating in z-direction, eq.(26) simply gives with two independent modes where D ( z, t ) satisfies /square 2 D ( z, t ) = 0 with the well-known plane wave solution. In the case of K = 1 an analogue solution for eq.(15) exists. Following a lengthy calculation due to non-diagonal components of ˜ g ij we obtain where X 2 = x 2 + y 2 + z 2 and By inserting eqs.(29) and (30) in eq.(25) with some manipulation we conclude that each mode, × and +, satisfies We use method of separation of variables to find the solutions of eq.(31). Then we may write Using eq.(32), eq.(31) leads to Eq.(33) may hold merely if each side is equal to a constant, i.e. we have: where q 2 is an arbitrary positive constant. We should take it positive since we are looking for a periodic wave. Eqs.(34) and (35) can be written as and To solve eq.(36) and finding D q ( z ) we define inserting eq.(38) in eq.(36) we get the following equation for U q ( z ) We notice that the solutions of eq.(39) may be written as Chebyshev polynominal of type I provided we take, q 2 = n 2 -3, where n is integer and U n ( z ) is So we have and Next we examine the temporal dependence of this mode. Putting eq.(41) in eq.(37) gives It is convenient to define the conformal time τ by: Integrating eq.(44) gives: Notice that t = -∞ , 0 , + ∞ corresponds to τ = 0 , π/ 2 , π respectively. So while the domain of coordinate time is -∞ < t < + ∞ , the domain of conformal time is 0 < τ < π . We may recast eq.( 43) in terms of conformal time as where ˆ D n ( t ) = ˘ D n ( τ ). To solve eq.(46), let us define a new parameter Y = cos τ with domain -1 < Y < +1 and t = -∞ , 0 , + ∞ correspond to Y = 1 , 0 , -1 respectively. In terms of the new parameter Y , eq.(46) becomes where ˘ D n ( τ ) = ˜ D n ( Y ). If we define W n ( Y ) = d dY ˜ D n ( Y ) and differentiate eq.(47) with respect to Y , this gives Again eq.(48) is Chebyshef of first kind and its solutions are: For last step we should solve We have previously defined Y = cos τ , so we have τ = arccos Y . Let us recast eq.(50) in terms of τ , it becomes with We may write  The first mode n = 1 is pure gauge mode and should be excluded from the acceptable solutions. This is the analogue of a plane wave moving in z-direction for a closed model. We may find a similar solution for the waves that are analogue to plane wave in x and y directions. In this case we would have: where D ( x, t ) and D ( y, t ) are given by: and /negationslash /negationslash  It is important to notice that eq.(58) and Eq.(59) may be achieved from eq.(29) and eq.(30) respectively by the coordinate transformation x → z, y → y, and z →-x . This leads us to write the solution of waves moving in an arbitrary direction. Let us assume this arbitrary direction is The result can be obtained from eqs.(29) and (30) by the coordinate transformation introduced by We get /negationslash where the explicit forms of A + × ij ( /vector X, ˆ n ) are listed in the appendix. This result may be used to expand a general function as linear superposition of these eigenfunctions, i.e. it should be replaced with exp ( ik µ x µ ) in Fourier transformations.", "pages": [ 6, 7, 8, 9, 10 ] }, { "title": "IV. SPHERICAL WAVE ANALOGUE", "content": "To consider this case it is more suitable to work in polar coordinates, x i = ( χ, θ, φ ). In this basis the non-zero components of the unperturbed metric are: with the inverse The non-zero components of the unperturbed connections are In this case ˜ g ij is diagonal and the conditions (15) for a transverse wave give We may distinguish two independent polarizations as Inserting eqs.(64) and (65) in eq.(25) and expressing ∇ 2 in polar coordinates, with a rather lengthy but straightforward calculation it can be shown that both D + ( χ, t ) and D × ( χ, t ) must satisfy the same equation as To solve eq.(69) we may assume that Then we have Eq.(71) holds provided that each side is equal to a constant, i.e. where q 2 is an arbitrary positive constant. So we have To solve eq.(74) for D q ( χ ) we may define a new parameter X = cos χ and D ( χ ) = ˆ D ( X ), then eq.(74) gives: Eq.(76) has a solution as where U q ( X ) satisfy the following equation Now we take V q ( X ) = d dX U q ( X ) which satisfies Eq.(79) is a Chebyshef type I provided that we take q = n . Then we have and The temporal part is the same as plane wave analogue eq.(52) and we have  It is interesting to note that in the case of flat models, i.e. K = 0, eq.(74) in the ( r, θ, φ ) bais takes the form which has the solution where q can be any arbitrary real number. If we consider the ratio h 22 g 22 we get Eqs.(85) and (86) both decrease by increasing the radial coordinate. The radiation fields(EM or GW)always decreases as inverse of radial coordinate regardless it is dipole or quadrupole field. its dependence on the source properly appears as a factor which could be dipole or quadrupole.", "pages": [ 10, 11, 12, 13 ] }, { "title": "V. THE EFFECT OF GRAVITATIONAL WAVES", "content": "To obtain a measure of the waves effect in this background ,we consider the rotation of the nearby particles as described by the geodesic deviation equation. For some nearly particles with four-velocity U µ ( x ) and sepration vector S µ , we have Since the Riemann tensor is already first order for test particles that are moving slowly we may write U µ = (1 , 0 , 0 , 0) in eq.(87). So in computing eq.(87)we only need R i 00 j which is For slowly-moving particles we have τ = ˙ x = t to lowest order so the geodesic deviation equation becomes The first term in eq.(88) will came a exponential expansion which is a general characteristic of de Sitter space .of course this is not the effect of gravitational waves and we may ignore it . Making use of eq.(43) and eq.(52) and noticing that n /greatermuch 1 . The contribution of last term in eq.(89) vanishes, so we have: To be specific we chose eq.(29), as a wave moving in z -direction , so we have The eigenvalues of matrix eq.(91) are Certainly the z -component of the separation vector is not affected by this gravitational waves. As the case of flat space a circle of particles hit by a gravitational wave has a oscillatory motion but in closed spaces the normal axis of oscillation rotate at different location of spacetime.", "pages": [ 13, 14 ] }, { "title": "VI. GENERATION OF GRAVITATIONAL WAVES", "content": "With presenting plane-wave solutions to the linearized vacuum field equation, it remains to discuss the generation of gravitational radiation by source. For this purpose it is necessary to consider the equation coupled to matter. Making use of completeness relation of the eigenfunction of vacuum equation we may write the solution of the field equation with source as We assume the source is isolated , far away and slowly moving. This implies | /vector x ' |/lessmuch| /vectorx | . Also we take the distance to the source is not of cosmic scale so that | /vectorx |/lessmuch 1. With these approximations eq.(93)may be written as Performing summation on m gives Integrating on ˆ n and putting eq.(95)in eq.(94)gives Since the source is localized and spacetime locally looks flat , we have : Putting µ = 0 in eq.(97)and differentiating with respect to x 0 gives: Then putting µ = k in eq.(97) gives: Combining eq.(98) and eq.(99) leads to Now we multiply both sides of eq.(100)by x n x m and integrate by parts two times over all space, ignoring the surface terms, finishing up with where I mn is the quadrupole moment of the mass distribution of the source. Inserting eq.(101) in eq.(96) gives The result of summation on n gives the retarded solution as: Therefore we have: The result depends on the first time derivative of moment at retarded time.", "pages": [ 14, 15, 16 ] }, { "title": "VII. DISCUSSION", "content": "Our investigations show that in analysis of gravitational waves the background of deSitter with K = +1 fundamentally differs from the scale- free de-Sitter with K = 0. We found the wave numbers should be discrete as already it has been realized that the spectrum of the Laplacian in spherical space is always discrete [17]. Another relevant feature is the existence of cut off on the long-wavelength of gravitational waves. This may be tested by the measurement of dipole and higher multi-pole moments of the CMBR anisotropy which contains information about the long-wavelength portion of the spectrum of energy density produced the large scale galactic structure of the universe. These are sensitive to the presence of long-wavelength perturbations. The obtained eigenmodes are the fundamental tools of analysis of cosmic evolution of perturbations in spatially closed models. In the formation of large scale structure and study of anisotropies of CMBR we should use these eigenmodes to expand the perturbations. Another feature of the obtained eigenmodes is that they are effectively transverse in zone near to origin and at far distances, in contrast to flat spacetimes, this is not so. For example a wave moving in z direction its 3 j components are not vanishing exactly but in near zone may be approximated to zero. It has been shown that h ij = a 2 D ij ∝ D n ( χ,t ) sin 2 τ . This means that in a collapsing phase i.e. in the time interval -∞≤ t ≤ 0 corresponding to conformal time 0 ≤ τ ≤ π/ 2, the perturbation is decaying while in the expanding phase, i.e. in the time interval 0 ≤ t ≤ + ∞ corresponding to conformal time π/ 2 ≤ τ ≤ π , it is growing. Of course always the smallness conditions of the perturbation with respect to the unperturbed metric holds. The amplitude of the gravitational wave changes with time as h ij ∝ √ 1+( n 2 -1) sin 2 τ ( n 2 -1) sin 2 τ . So its growth is significant for the modes that satisfy the condition √ n 2 -1 sin 2 τ /lessmuch 1, where it changes as | h ij | ∝ 1 ( n 2 -1) sin 2 τ . For those modes that satisfy the condition √ n 2 -1 sin 2 τ /greatermuch 1,we have | h ij | ∝ 1 √ n 2 -1 sin τ and changes are relatively smooth. Singularities appear in the solutions are of coordinate type, where we may see them in the unperturbed metric too. At last the obtained results are crucial to expand any perturbation appears in different contexts of the spatially closed cosmological models in terms of them.", "pages": [ 16, 17 ] }, { "title": "VIII. APPENDIX", "content": "The components of the amplitude of gravitational waves moving in an arbitrary direction: where D + × ij ( /vector X, ˆ n ) are as follows:", "pages": [ 18, 19 ] } ]

[ { "title": "ABSTRACT", "content": "Noname manuscript No. (will be inserted by the editor)", "pages": [ 1 ] }, { "title": "Resonant scattering of light in a near-black-hole metric", "content": "Y. V. Stadnik · G. H. Gossel · V. V. Flambaum · J. C. Berengut Received: date / Accepted: date Abstract We show that low-energy photon scattering from a body with radius R slightly larger than its Schwarzschild radius r s resembles black-hole absorption. This absorption occurs via capture to one of the many long-lived, densely packed resonances that populate the continuum. The lifetimes and density of these meta-stable states tend to infinity in the limit r s → R . We determine the energy averaged cross-section for particle capture into these resonances and show that it is equal to the absorption cross-section for a Schwarzschild black hole. Thus, a non-singular static metric may trap photons for arbitrarily long times, making it appear completely 'black' before the actual formation of a black hole. Keywords black hole · resonant scattering · photon scattering PACS 04.62.+v, 04.70.Dy, 04.70.-s", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "In this work, we consider the scattering of photons by the gravitational field of a non-rotating, finite-sized body. Such finite-sized bodies have radius R that slightly exceeds their Schwarzschild radius r s = 2 GM/c 2 . We show that lowenergy photon scattering from such objects resembles black hole absorption. The absorption arises due to the existence of a dense spectrum of narrow resonances (meta-stable states) whose lifetime and density tend to infinity in the limit r s → R . Photons captured to such a resonant state are trapped on the interior of the body for a time t ∼ /planckover2pi1 /Γ n , where Γ n is the width of a given resonance. For r s → R , both the energy spacing D and width Γ n tend to zero (resonance lifetime t →∞ ), while their ratio remains finite. This allows us to define the School of Physics, University of New South Wales, Sydney 2052, Australia E-mail: [email protected] total cross-section for particle capture into these long-lived states using the optical model [1], which is calculated by averaging over a small energy interval containing many resonances. At low energy the resonance capture cross-section is σ a = 4 πε 2 r 4 s / 3 where ε is the energy of the incident particle. The absence of a longitudinal mode for photons means there is no state with total angular momentum equal to zero, hence the above cross-section tending to zero for zero energy. This cross-section exactly matches the absorption cross-section for a Schwarzschild black hole calculated by assuming complete absorption at the event horizon (see Refs. [2,3,4,5,6,7,8]) derived previously for massless spin-1 particles [9,10]. Note, however, that our calculation does not impose any special conditions at the boundary. Thus by considering the purely elastic cross-section of low-energy incident photons, we find that the absorption properties of a body with r s → R resemble those of a black hole. The gravitational field of these near-black-hole objects is described using a suitable metric to model the interior, which is then joined to the standard Schwarzschild exterior metric at the boundary of the body. The above result is shown to be valid for any interior metric satisfying the following conditions: continuity with the Schwarzschild exterior at r = R , a potential that deepens and develops a singularity in the black hole limit (allowing the particle to be treated semi-classically) and that this potential can be modeled as a harmonic oscillator in some finite region around the origin. We expect these conditions to be met by a large class of metrics; in this paper we present two metrics for which these conditions hold. Additionally, the results presented include the spin-0 (scalar) case considered previously (for j = 0) [11], but given here for arbitrary angular momentum j . In this case taking r s → R once again yields the black hole absorption cross-section given in [4]. For a more detailed discussion of previous calculations involving scattering of various particles in the black hole spacetime we turn the readers attention to the introductions in [11,7] and references therein. As in previous work [11] we perform both analytical and numerical calculations, with good agreement between the two.", "pages": [ 1, 2 ] }, { "title": "2 Wave equations", "content": "The Klein-Gordon equation for a massless spin-0 particle on a curved manifold (with /planckover2pi1 = c = 1) reads For massless spin-1 particles (photons) Maxwell's equations on a curved manifold free of charges read where F αβ = ∂ α A β -∂ β A α and A µ is the contravariant electromagnetic 4potential. One may write a general static, spherically symmetric metric in the form By applying the separation of variables to (1) and (2) respectively in the metric (3), one arrives at the radial differential equations for spin-0 and spin-1 respectively. In the latter we have implemented a gauge such that A ϕ is the only non-zero component [12]. Defining s as the particle spin, either 0 or 1, the radial equations for both spins can be combined to yield where j the total angular momentum such that j -s ≥ 0.", "pages": [ 2, 3 ] }, { "title": "3 Interior Solution", "content": "Equation (4) can be transformed into a Schrodinger-like equation by making the substitution φ ( r ) = χ ( r ) /r and then mapping the radial coordinate to the Regge-Wheeler 'tortoise' coordinate r ∗ defined by dr ∗ = e ( λ -ν ) / 2 dr . This gives us In the subsections below we construct a solution to the above equation by dividing the interior into two regions. This solution is valid under certain constraints placed on the metric coefficients e ν and e λ . The specific metrics considered in later sections are entirely consistent with the restrictions imposed. It is worth noting that in the s-wave scalar case s = j = 0 there is only one interior region akin to the one defined in Sec. (3.2), albeit with different constraints. As such, this case is not treated here and we instead refer the reader to our previous treatment of this case in [11]. For the purposes of the discussion below we note that for the interior metric of near-black-hole object, e ν ( r ) → 0 for 0 ≤ r ≤ R , as time slows down in the limit r s → R .", "pages": [ 3 ] }, { "title": "3.1 Interior Region I", "content": "In the vicinity of the origin spacetime, and indeed the potential, must be locally flat. This necessitates that both e ν ( r ) and e λ ( r ) be approximately constant, and furthermore e λ ( r ) ≈ 1, in the area around the origin. Under these conditions we may ignore the second term in square brackets in Eqn. (5) (moreover, it is zero for spin s = 1), and we may approximate the tortoise coordinate by r ∗ 0 ≈ e -ν 0 / 2 r , where ν 0 is a constant. For the metrics we consider, the second bracketed term in Eqn. (5) is always smaller than either the centrifugal term or ε 2 throughout Region I. Therefore in Region I we re-write Eqn. (5) as the regular solution of which is the Bessel function", "pages": [ 3, 4 ] }, { "title": "3.2 Interior Region II", "content": "This region is defined as the area of the interior where the energy term ε 2 dominates over other terms in square brackets in Eqn. (5). Note that these terms are suppressed by a factor of e ν , which tends to zero in the black hole limit. Therefore the wave equation is simply which has the solution where φ is a phase to be determined by matching to the solution in Region I in an appropriate overlap region. For the metrics we consider, Region II is valid up to a point near the boundary beyond which ε 2 is no longer the dominant term. However, for the metrics we consider it may be shown that in the black hole limit the size of this region decreases as a function of R -r s such that it does not appreciably alter the phase of the wavefunction from Region II to the boundary r = R .", "pages": [ 4 ] }, { "title": "3.3 Matching in the overlap region", "content": "It can be shown that for both the Florides and Soffel metrics there exists an overlap region where the combined conditions defining regions I and II are satisfied. This overlap region exists for r such that: Lastly, the condition allows us to take the asymptotic form of the solution in Region I given by (7), √ εr ∗ J j +1 / 2 ( εr ∗ ) → sin( εr ∗ -jπ/ 2). Matching to this solution in the overlap region defined above gives the final Region II solution as where Furthermore, at the boundary we have Φ ' ( R ) = εe ( λ -ν ) / 2 ∣ ∣ R = εR/ ( R -r s ). The latter is computed by imposing continuity of the metric at the boundary (where all but the energy term in Eqns. (4) and (5) are suppressed). Defining the logarithmic derivative of the interior wavefunction at r = R can be written as ∣ In the black-hole limit e ν → 0, therefore from Eqn. (14) we see that as r s → R , Λ ( r s ) tends to infinity. This is because Λ ( r s ) is related (but not equal) to the total phase accumulated by the particle on the interior. This phase is large due to the wave function oscillating many times on the interior as the particle moves rapidly in the strong field. However this integral also gives the classical time that a massless particle ( ds 2 = 0) spends on the interior, which goes to infinity in the black hole limit.", "pages": [ 4, 5 ] }, { "title": "4 Exterior Solution", "content": "The metric on the exterior ( r ≥ R ) of a static massive body is given by the standard Schwarzschild metric, which yields the following radial differential equation", "pages": [ 5 ] }, { "title": "4.1 Exterior Region I", "content": "For ε /lessmuch √ r -r s (later verified for resonance energies ε n ), Eqn. (16) becomes which has the exact solution where P ( a,b ) n ( x ) represent the Jacobi polynomials and 2 F 1 ( a, b, c ; z ) is the Gaussian hypergeometric function. This solution is valid from the boundary r = R (provided ε /lessmuch √ R -r s ) and while ε 2 r 2 /lessmuch j ( j +1). 4.2 Exterior Region II Taking r /greatermuch r s in Eqn. (16) we make the substitutions φ ( r ) = χ ( r ) /r and r = ρ/ε . This yields The corresponding solution in terms of the regular and irregular Coulomb wave functions [13] reads There exists an overlap between the two regions described above when r /greatermuch r s but ε 2 r 2 /lessmuch j ( j +1) (which automatically satisfies ε /lessmuch √ r -r s ). In this overlap region we can use the asymptotic form ( εr /lessmuch 1) of the Coulomb wavefunctions in (20). Matching with (18) we arrive at the following relationship between the coefficients of the solutions in the two exterior regions:", "pages": [ 6 ] }, { "title": "5 S-Matrix", "content": "The solution to Eqn. (16) at large distances can also be written in terms of outgoing and incoming waves as where z = εr + εr s ln(2 εr ) + δ C j -jπ/ 2 and δ C j = arg[Γ( j +1+ iεr s )] is the Coulomb phase shift. This allows us to write the scattering matrix as [1] Imposing the condition εr /greatermuch 1 in Eqn. (20), we have the following asymptotic form of the wavefunction in region II which, upon matching to (22), gives the scattering matrix as Note that δ C j is small compared to the total phase accumulated on the interior, given by Λ ( r s ), and slowly varying for ε /lessmuch 1.", "pages": [ 7 ] }, { "title": "6 Matching of wavefunctions at boundary and resonance energies", "content": "Matching the logarithmic derivatives of the exterior (18) and interior (15) wavefunctions at r = R and taking the black hole limit r s → R gives Resonances occur at energies where the absorption cross-section is maximized, i.e. S = -1. This is achieved in Eqn. (25) when α 2 /β 2 = 0, which by Eqn. (21) is equivalent to α 1 /β 1 = 0. Setting α 1 /β 1 to zero in (26) results in the resonance condition for the energy where n = 1 , 2 , . . . . Note that these resonance energies strongly depend on the value of j , but in this approximation do not depend on s .", "pages": [ 7 ] }, { "title": "7 Resonance Widths", "content": "The full resonance is obtained by extending ε into the complex plane. The scattering matrix has a pole at complex energy ε = ε n -iΓ n / 2 which corresponds to the resonance condition We may also express α 1 /β 1 in the vicinity of a resonance as ∣ As detailed previously, on resonance α 1 /β 1 = 0. At the complex pole of the scattering matrix, ε = ε n -iΓ n / 2, this gives ∣ Let α 2 /β 2 = f ( ε ) α 1 /β 1 [Eqn. (21)], then using the above expression we may write Eqn. (28) as ∣ Taking the derivative of α 1 /β 1 given by Eqn. (26) and solving for Γ n in (31) gives the resonance widths where we have assumed ε n Λ ( r s ) /greatermuch jπ/ 2, i.e. large n (see (27)), which is already assumed when deriving the interior solution.", "pages": [ 8 ] }, { "title": "8 Absorption cross-section", "content": "As discussed in Section 3, in the black hole limit we find that Λ ( r s ) tends to infinity. Thus by Eqns. (27) and (32) both ε n and Γ n tend to zero in the limit r s → R for any fixed, finite values of n and j . However, the ratio Γ n /D remains constant, where D = ε n +1 -ε n /similarequal π/Λ ( r s ) is the spacing between adjacent levels. This allows us to use the optical-model (energy-averaged absorption cross-section) [1]. This is obtained by averaging over a small energy interval containing many resonances and reads Substituting (27) and (32) into (33), gives which is independent of Λ ( r s ) and thus of the interior metric. Therefore, in the low energy limit the cross-sections for massless scalar particles and photons are The above expressions exactly match the cross sections for these particles incident on a Schwarzschild black hole [4,9,10].", "pages": [ 8, 9 ] }, { "title": "9 Specific interior metrics", "content": "In this section we present calculations involving two specific interior metrics that allow the r s → R limit to be taken: the Florides [14] and Soffel [15] metrics. Specifically, we verify our analytic solutions with numerically calculated resonance widths and energies via the short range phase shift δ ( ε ). To calculate δ ( ε ) we solve the second-order differential equation (4) numerically, for given e ν and e λ , with the boundary condition φ ( r → 0) ∼ r j using Mathematica [16]. This solution provides a real boundary condition for the exterior wave function at r = R . (We set R = 1 in the numerical calculations). Equation (16) is then integrated outwards to large distances r /greatermuch r s . In this region Eq. (16) takes the form of a non-relativistic Schrodinger equation for a particle with momentum ε and unit mass in the Coulomb potential with charge Z = -r s ε 2 . Hence, we match the solution with the asymptotic form [1] and determine the short-range (numeric) phase shift δ . It is found that this phase possesses steps of height π at the resonance positions ε n . We fit the step profile of an individual resonance to the Breit-Wigner function where δ n is a constant, from which we extract the numeric resonance widths and positions Γ n and ε n . s", "pages": [ 9, 10 ] }, { "title": "9.1 Florides Interior", "content": "The Florides metric is characterized by The leading term of Λ ( r s ) is The resulting resonance energies and widths are compared with their numeric counterparts in Figures (1) and (2) respectively.", "pages": [ 10 ] }, { "title": "9.2 Soffel Interior", "content": "The Soffel metric is characterized by [15] with e λ ( r ) equal to that of the Florides case. The leading term of Λ ( r s ) Analytic and numeric ε n and Γ n for the Soffel metric are compared in Figures (3) and (4) respectively. The widths in Fig. (4) illustrate that the exponential suppression gives rise to numerical instabilities as r s → R (and thus why the r s /R values used in the Soffel case are much lower than those in the Florides case).", "pages": [ 10, 11 ] }, { "title": "10 Conclusions", "content": "The problems of the scattering of low-energy, massless spin-0 and spin-1 particles from a massive, static, spherical body have been considered. We have shown that such scattering is characterized by a dense set of long lived resonances. Capture to these long-lived states gives rise to effective absorption in a purely potential scattering setting. In the black hole limit the cross-section for absorption exactly equals the cross-section in the pure black hole case (for low energy). Thus scattering of photons (and massless scalars) by a near-black-hole object resembles black hole absorption. We thank G. F. Gribakin for useful discussions. This work is supported by the Australian Research Council. s", "pages": [ 11, 12 ] } ]

[ { "title": "Different representations of the Levi-Civita Bertotti Robinson solution", "content": "Øyvind Grøn ∗ and Steinar Johannesen ∗ Abstract The Levi-Civita Bertotti Robinson (LBR) spacetime is investigated in various coordinate systems. By means of a general formalism for constructing coordinates in conformally flat spacetimes, coordinate transformations between the different coordinate systems are deduced. We discuss the motion of the reference frames in which the different coordinate systems are comoving. Furthermore we characterize the motion of the different reference frames by their normalized timelike Killing vector fields, i.e. by the four velocity fields of the reference particles. We also deduce the formulae in the different coordinate systems for the embedding of the LBR spacetime in a flat 6-dimensional manifold. In particular we discuss a scenario with a spherical domain wall having LBR spacetime outside the wall and flat spacetime inside. We also discuss the internal flat spacetime using the same coordinate systems as in the external LBR spacetime with continuous metric at the wall. Among the different cases one represents a Milne-LBR universe model with a part of the Milne universe inside the wall and an infinitely extended LBR universe outside it. In an appendix we define combinations of trigonometric and hyperbolic functions that we call k-functions and present a new k-function calculus.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Conformally flat spacetimes have vanishing Weyl tensor. The line element of such spacetimes can in general be given the form of a conformal factor times the Minkowski line element. The coordinates in which the line element takes this form are called conformally flat spacetime (CFS) coordinates. The FRW universe models are conformally flat. We have recently given a systematic description of these universe models in CFS coordinates [1-3]. In the present article we shall give a similar treatment of the LBR spacetime which was found by T. Levi-Civita [4,5] already in 1917, and was rediscovered by B. Bertotti [6] and E. Robinson [7] in 1959. It was proved by N. Tariq and B. O. J. Tupper [8] and by N. Tariq and R. G. McLenaghan [9], and later emphasized by H. Stephani et al. [10] that the LBR spacetime is the only conformally flat solution of the Einstein-Maxwell equations which is homogeneous and has a non-null Maxwell field. The physical interpretation of the solution has been discussed by D. Lovelock [11,12], P. Doland [13] and the present authors [14]. Our article is organized as follows. In section 2 we present a new method for finding different coordinates of the LBR spacetime. We give a general formalism in section 3 for finding coordinate transformations between the canonical CFS system and an arbitrary coordinate system. Section 4 is the main part of the article. Here we find the different coordinate systems and give a thorough discussion of their properties and of the reference frames in which they are comoving. In section 5 we discuss a particularly interesting example, a Milne-LBR universe model where there is LBR spacetime outside a charged domain wall with a radius equal to the distance corresponding to its charge, and there is a part of the Milne universe inside the domain wall. The motions of the reference frames are further characterized in section 6, where we calculate the four-acceleration of the reference particles from the Killing vectors. In section 7 we present embedding parametrizations for the different coordinate representations of the LBR spacetime in a 6-dimensional, flat spacetime. Our results are summarized in section 8. We define k-functions, which are combinations of trigonometric and hyperbolic functions, in an appendix where we also present the k-calculus of these functions.", "pages": [ 1, 2 ] }, { "title": "2. A new method for finding different representations of the LBR spacetime", "content": "By the Levi-Civita Bertotti Robinson (LBR) spacetime we shall mean a conformally flat and static spacetime which is a solution of the Einstein-Maxwell equations with an electromagnetic field having a constant energy momentum tensor. This solution has usually been called the Bertotti Robinson solution, but it was actually discovered by T. LeviCivita already in 1917 [4,5]. Hence we shall call it the Levi-Civita Bertotti Robinson solution. In a previous paper [14] we have given a new interpretation of the LBR solution. According to our interpretation this solution describes a static, spherically symmetric and conformally flat spacetime with a radial electrical field outside a charged domain wall. There is Minkowski spacetime inside the wall. It is well known that the LBR solution can be represented by a spacetime which is the product of a 2-dimensional anti de Sitter space and a spherical surface [13]. Hence the line element may be written in a spherically symmetric form with an angular part which is K 2 d Ω 2 , where K is a constant. According to our interpretation [14] the constant K is equal to the radius R Q of the domain wall. Also the radius of the domain wall is determined by its charge Q so that R Q = [ G/ (4 π/epsilon1 0 c 4 )] 1 / 2 Q , i.e. R Q is the length corresponding to the charge Q . The line element may then be given the form This line element is rather general, and only in the case where the Weyl tensor vanishes does it describe the LBR spacetime. M. Gurzes and O. Sario˘glu [15] have shown that a D -dimensional conformally flat LBR spacetime, which is a product of a 2-dimensional anti de Sitter spacetime and a ( D -2)-dimensional spherical surface, permits a cosmological constant proportional to 1 -( D -3) 2 . Hence in the 4-dimensional LBR spacetime the cosmological constant vanishes, which has earlier been noted by V. I. Khlebnikov and ' E. Shelkovenko [16] and by J. Podolsk'y and M. Ortaggio [17]. Using the radial coordinate ˇ r in the line element invites the interpretation of the spacetime as a spherically symmetric space in the spacetime R 4 . An alternative interpretation is also possible. Neglecting the time dimension in the 2-dimensional anti de Sitter space and a spatial dimension in the spherical surface, replacing it by a circle, the spacetime can be interpreted as a cylinder. Then the electrical field is directed along the axis of the cylinder. We here want to consider both physical interpretations. The spacetime with a domain wall will be called the WLBR spacetime, and the spacetime with a product of a 2-dimensional anti de Sitter space and a spherical surface will be called the PLBR spacetime. We use LBR in statements concerning both WPBL and PLBR. Note that in the PLBR interpretation the coordinate ˇ r shall not be interpreted as a radial coordinate. In the present case it follows from the geodesic equation that a free particle instantaneously at rest has an acceleration where the dot denotes differentiation with respect to the proper time of the particle. Hence there is attractive gravity, i.e. the acceleration of gravity points in the negative e ˇ r -direction, if α is an increasing function of ˇ r and repulsive gravitation if α is a decreasing function of ˇ r . With the line element (1) the condition that the Weyl tensor vanishes takes the form Calculating the components of the Einstein tensor from the line element (1) and using Einstein's field equations it follows that when equation (3) is fullfilled, the mixed components of the energy momentum tensor reduce to which represents a constant radial electric field, as is the case in the LBR spacetime. This shows that the LBR spacetime does not allow a non-vanishing cosmological constant. In the section 4 equation (3) will be solved under different coordinate conditions. The solutions found is the subsections 4.Ia and 4.Ib will turn out to be special cases of the line element where x 0 is a time coordinate, x 1 is a radial coordinate, G ( x 0 , x 1 ) is a function of x 0 and x 1 , and d Ω 2 is a solid angle element. In the next section we shall develop a formalism for finding transformations between the coordinates where the line element takes the form (5) and the CFS coordinates.", "pages": [ 2, 3 ] }, { "title": "3. Conformally flat spacetime coordinates for the LBR spacetime", "content": "We want to write the line element (5) of a spacetime with spherically symmetric space in terms of conformally flat spacetime (CFS) coordinates ( T, R ). Then the line element takes the form of a conformal factor C ( T, R ) 2 times the Minkowski line element, In order to perform this we shall generalize the method developed in reference [1]. We then use transformations of the form where f and g are functions that must satisfy an identity deduced below. A transformation of this form can be described as a composition of three simple transformations. The first transforms from the coordinates x 0 and x 1 in the line element (5) to light cone coordinates (null coordinates) associated with a Minkowski diagram referring to the ( x 0 , x 1 ) coordinate system In the Minkowski diagram this rotates the previous coordinate system by -π/ 4 and scales it by a factor √ 2. The scaling is performed for later convenience. The coordinate u is constant for light moving in the negative x 1 -direction, and v in the positive x 1 -direction. The second transforms u and v to the coordinates Finally, we scale and rotate with the inverse of the transformation (8), The inverse of the transformation (10) is showing that U and V are light cone coordinates associated with a Minkowski diagram referring to the CFS coordinate system. The coordinate U is constant for light moving in the negative R -direction and V in the positive R -direction. Note that Taking the differentials of T and R we get Comparing the expressions (5) and (6) for the line element and using the previous formula, we find where x 0 , x 1 , u and v are functions of T and R , and From equations (14) and (15) it follows that By (7) and (8) equation (16) may be written as Substituting v = u we get the condition As shown in reference [1] if G ( u, 0) = 0, the line element (6) can be written in the form (5) with G ( x 0 , x 1 ) = S k ( x 1 ), where the function S k is defined in equation (A.1). Then equation (17) reduces to Substituting v = u and utilizing that S k (0) = 0, this equation gives g ( u ) = f ( u ). Hence equation (19) may be written With the function [1] where a , b , c , d are arbitrary constants and the function I k ( x ) is defined in equation (A.4), the transformation (21) leads from (5) with G ( x 0 , x 1 ) = S k ( x 1 ) to (6) with C ( T, R ) given by equation (15) in the case of the LBR spacetime. It follows from equations (1), (6) and (15) that the line element of the Minkowski spacetime inside the domain wall in the different coordinate systems takes the form The equations (1) and (23) give the general connection between the form of the line element of the WLBR spacetime outside the domain wall in an arbitrary coordinate system and the form of the line element of the flat spacetime inside the domain wall in the same coordinate system.", "pages": [ 3, 4, 5 ] }, { "title": "4. The LBR spacetime in different coordinate systems", "content": "Equation (3) will now be solved under different coordinate conditions. Ia. Static metric and coordinates ( η, χ ) with β ( χ ) = α ( χ ) . In this case equation (3) reduces to where the prime means differentiation with respect to the radial coordinate. This equation may be written Integration gives where c > 0 is an integration constant and k takes the values 1, 0 or -1. The general solution of (26) is given by where S k ( x ) is the function defined in equation (A.1) in Appendix A. Here χ 0 is an integration constant and c = 1 when k = 0. The value k = 0 is a very important special case. Then one can introduce CFS coordinates simply by putting χ 0 = 0. The line element with ( η, χ ) replaced by ( T, R ) then takes the form /negationslash with -∞ < T < ∞ , R > R Q for the WLBR spacetime, and with -∞ < T < ∞ , -∞ < R < ∞ , R = 0 for the PLBR spacetime. This form of the line element is in agreement with equations (6) and (15). Note that the metric is static. This means that the coordinate clocks go with the same rate at all positions. The line element has the Minkowski form at the domain wall at R = R Q . At this surface g TT = -1, meaning that the coordinate clocks of the CFS system show the same time as standard clocks at rest at the domain wall. The fact that there exists a coordinate system so that the metric is static means that the LBR spacetime is static, although we will show later that there exist coordinates so that the metric of this spacetime is time dependent. This time dependence is due to the motion of the reference frame in which the coordinates are comoving. As has been noted by O. J. C. Dias and J. P. S. Lemos [18] there is an interesting connection between the WLBR spacetime and the Reissner-Nordstrom spacetime, which is usually described by the line element where R S = 2 GM/c 2 is the Schwarzschild radius, and R Q is the length corresponding to the electric charge Q . The extremal Reissner-Nordstrom spacetime has R S = 2 R Q , and then the line element takes the form A Taylor expansion of f ( r ) = (1 -R Q /r ) 2 about r = R Q gives to 2. order in r , f ( r ) ≈ ( r -R Q ) 2 /R 2 Q . Hence, the near-horizon limit of the line element for the extremal ReissnerNordstrom spacetime takes the form where the angular part is correct only to 0. order in r . Introducing coordinates leads to the form (28) of the line element. Hence the line element of the near-horizon limit of the Reissner-Nordstrom spacetime has the same form as the line element of the LBR spacetime. But the coordinates R and r in equation (32) increase in opposite directions. If this is forgotten, gravity seems to be repulsive in the near-horizon limit of the Reissner-Nordstrom spacetime as expressed in terms of the CFS coordinate R , since α is a decreasing function of R . However, gravity is attractive in the near-horizon limit of the Reissner-Nordstrom spacetime. This is a coordinate independent property of the spacetime. In the LBR spacetime the CFS coordinate R increases in the direction away from the symmetry center, and there is repulsive gravity. The LBR spacetime is therefore very different from the near-horizon limit of the Reissner-Nordstrom spacetime. We shall define the acceleration of gravity in a coordinate system with an arbitrary radial coordinate ˇ r as the acceleration of a free particle instantaneously at rest and measured with standard measuring rods and clocks. Hence it is the component along the unit radial basis vector of the second derivative of the radial coordinate with respect to the proper time of the particle, In the present case ¨ ˇ r = R where R is given by the geodesic equation For the WLBR spacetime this gives i.e. in the CFS system the acceleration of gravity is constant and directed away from the domain wall. We will show that the solutions (27) with k = 1 and k = -1 represent the same spacetime as the solution with k = 0. This will be shown by demonstrating that there exists a coordinate transformation that transforms the line elements of the solutions (27) with k = 1 and k = -1 to the form (28). Putting c = 1 and χ 0 = 0, the line element (1) with the solution (27) takes the form In the case k = 1 the coordinate clocks showing η go at the same rate as a standard clock at χ = π/ 2, scaled by the factor R Q . It may be noted that radially moving light has a coordinate velocity dχ/dη = ± 1 for all values of k , which is due to the condition α = β . Note that the form (36) of the line element is valid for all values of k . In the case k = 0 the line element reduces to form (28) with ( T, R ) replaced by ( η, χ ). In order to find a coordinate transformation between the ( η, χ )-coordinates and the CFS coordinates we apply the formalism in section 3. By choosing a = 0, b = 0, c = B and d = 0 in equation (22) we obtain the generating function where T k ( x ) is defined in equation (A.3) and B is a positive constant satisfying Hence B equals 2 when k = 0, and has an arbitrary positive value when k = 1 and k = -1. Using the generating function (37) as shown in Appendix B, the transformation (21) between the ( η, χ )-system and the CFS system takes the form where C k ( x ) is defined in equation (A.2). We have shown in Apppendix B how the inverse transformation is obtained from the generating function giving the result /negationslash when T = 0, where I k ( x ) is defined in equation (A.4). In the case T = 0 we have that η = 0. Note that the formulae (36) - (41) are valid for all values of k . A special case of the line element (36) with k = -1 has been used by A. C. Ottewill and P. Taylor [19] in connection with quantum field theory on the LBR spacetime. The world lines of points on the domain wall are given by the second of equations (39) with R = R Q , which leads to Introducing the constant equation (42) takes the form which can also be written as The point of intersection (0 , χ 0 ) with the χ -axis, where χ 0 is the coordinate radius of the domain wall in the ( η, χ )-system at the point of time η = 0, is found by inserting η = 0 in equation (45). Using that C k (0) = 1 for all values of k we then obtain a physical interpretation of the constant χ Q , From equation (43) it then follows that When k = 1 equation (42) takes the form which is plotted in Figure 1 as the left hand boundary of the hatched region. It follows that in the case k = 1 the WLBR spacetime is represented in the ( η, χ )-plane by the hatched region in Figure 1, which is given by We want to find the corresponding region in the ( η, χ )-system representing the PLBR spacetime. From equation (21) we obtain Hence η + χ and η -χ must belong to the domain ( -π, π ) of the generator function in equation (37) with k = 1. This gives the region /negationslash when k = 1, as illustrated in Figure 1. η When k = -1 the region representing the PLBR spacetime is the whole ( η, χ ) coordinate space except the η -axis, but T + R and T -R must belong to the range ( -B,B ) of the generator function in equation (37) with k = -1, which gives the region /negationslash Hence in this case the ( η, χ )-system does not cover the whole PLBR spacetime, but the constant B secures the possibility of choosing the region given in (52) to be arbitrarily large. The WLBR spacetime for k = -1 is given by The world lines of fixed particles χ = χ 1 in the ( η, χ )-system as described in the CFS system is found from equations (41), which gives when k = 1 and k = -1. For k = 0 we get R = χ 1 . The corresponding simultaneity curves η = η 1 are given by when k = 1 and k = -1. For k = 0 we get T = η 1 . Note that the ( η, χ )-coordinates and the CFS coordinates are comoving in the same reference frame when k = 0. Using the transformation (41) the line element (36) is given the form (28). This shows that the solution (27) represents the LBR spacetime for all values of k . With the line element (36) the coordinate acceleration of a free particle instantaneously at rest is since ˙ η = | g ηη | -1 / 2 for such a particle. Calculating the Christoffel symbol Γ χ ηη from the line element (36) we obtain The acceleration of gravity in the ( η, χ )-system is defined as the component of a χ e χ along the unit basis vector e ˆ χ , giving Hence for k = 1 the acceleration of gravity in the ( η, χ )-system is a ˆ χ = (1 /R Q ) cos χ so that a ˆ χ > 0 for 0 < χ < π/ 2 and a ˆ χ < 0 for π/ 2 < χ < π . This is different from the situation in the CFS system, where the acceleration of gravity is directed away from the domain wall everywhere according to equation (35). However, in the ( η, χ )-system there is a region π/ 2 < χ < π where the acceleration of gravity is directed towards the domain wall. This is due to the motion of the reference frame in which ( η, χ ) are comoving coordinates, as will be explained below. The charged domain wall is at rest in the ( T, R )-system. Hence the CFS coordinates are those of a static, but not inertial, reference frame. In this case the world lines are given by equation (54) with k = 1 which represents the hyperbolae shown in Figure 2. From this figure it seems that the ( η, χ )-system covers only a part of the WLBR spacetime. The worldlines of fixed particles in ( η, χ )-system are hyperbolae which never enter the future region above the asymptotes. This is however not the case because R 1 can have different values depending on χ 1 . If R 1 is moved to the left towards R = 0 the hyperbolae are straightened out. Hence the ( η, χ )-system covers all of the WLBR spacetime. It is known from the description of the WLBR spacetime with CFS coordinates, where the metric is static and the domain wall is at rest, that there is repulsive gravitation outside the domain wall. Nevertheless in the region R > √ B 2 + T 2 in Figure 3 the acceleration of gravity is directed towards the wall in the ( η, χ ) coordinate system. This apparent contradiction will be explained by comparing the acceleration of fixed points in the ( η, χ )-system with the acceleration of free particles, both measured relative to the CFS coordinate system. The world line of a particle at rest in the ( η, χ )-system is given by equation (54). From this it follows that the velocity and the acceleration of the particle in the CFS system are at an arbitrary point ( T 2 , R 2 ). We now consider a free particle with Lagrangian function where the dot denotes differentiation with respect to the proper time of the particle. Since the metric is static, is a constant of motion. Together with the four-velocity identity this leads to (for a particle moving outwards) where F means that the particle is free. We now demand that the free particle passes through the point ( T 2 , R 2 ) with the same velocity as the particle with χ = χ 1 . Using the equations (63) and (54) we obtain Integrating equation (63) we find the equation for the world line of the particle, where and T 0 is a constant of integration. With the boundary condition R ( T 2 ) = R 2 it follows that Note that ( dR/dT ) F = 0 for R = R 0 . Hence the particle falls from rest at R = R 0 at the point of time T = T 0 . The fact that T 0 depends upon R 1 , i.e. on χ 1 , means that different reference particles in the ( η, χ )-system are instantaneously at rest relative to the CFS system at different points of time. Differentiating we find that the acceleration of the free particle at R = R 2 is From equations (59) and (68) it follows that the ratio between the acceleration of a fixed particle in the ( η, χ )-system and a free particle is From the definition of R 1 in equation (54) and the transformation (41) for k = 1 it follows that This implies that Hence N > 1 for R 2 2 -T 2 2 > B 2 , i.e. to the right of the hyperbola in Figure 3, which means that the reference particles of the ( η, χ )-system have a greater outwards acceleration than a free particle. This is the reason why an observer at rest in the ( η, χ )-system experiences that the acceleration of gravity is directed in the negative χ -direction. The wall has a decreasing radius in the ( η, χ )-system. This is, however, a coordinate effect. In reality the wall is static and the ( η, χ ) coordinate system is comoving in an expanding reference frame. The acceleration of gravity vanishes in the ( η, χ )-system on the hyperbola χ = π/ 2 in Figure 3. This leads to the following physical interpretation of the constant B . As seen from equation (54) the point (0 , B ) in the CFS system corresponds to the point (0 , π/ 2) in the ( η, χ )-system where the acceleration of gravity vanishes. From equation (58) it follows that a particle with χ = π/ 2 moves freely. Equation (54) gives R 1 = 0 for this particle. The coordinate acceleration of this particle at the point of time T = 0 as given by equation (68) is 1 /B . Hence B is the inverse of the coordinate acceleration of a free particle at (0 , B ). We shall now consider the case k = -1. The world lines of reference particles with χ = χ 1 are given by equation (54) and are shown in Figure 4. We see that the reference points in the WLBR spacetime accelerate in the negative R direction. Hence in this reference frame the acceleration of gravity is directed outwards and is larger than in the static CFS system. This is verified by the expression (58) which implies that in this case g = (1 /R Q ) cosh χ ≥ 1 /R Q . Wenow consider the flat spacetime inside the shell. The line element of the Minkowski spacetime in this region has the following form in the CFS coordinate system, Inserting e α = R Q /S k ( χ ) from the line element (36) and the expression (39) for R into the line element (23) we find the form of the line element (72) in the ( η, χ )-system. Comparing with the line element (36) and using equation (42) we see that the metric is continuous at the domain wall. Note that the line element (73) reduces to (72) for k = 0 replacing ( η, χ ) with ( T, R ). In this case a spherical surface with radius R has area 4 πR 2 in the region inside the domain wall. Outside the domain wall, on the other hand, a spherical surface with radius R has area 4 πR 2 Q which is independent of R . The reason for this strange result is that the space T = constant is curved outside the domain wall. Calculating the acceleration of gravity inside the shell as experienced by an observer at rest in the ( η, χ )-system in the same way as in equation (58), we find In order to find the discontinuity of the acceleration of gravity in the ( η, χ )-system at the domain wall, it is sufficient to consider the point of time η = 0. Then the domain wall has the position χ = 2 χ Q , where χ Q is given in equation (43). Inserting this into equation (74) and using equations (A.18) and (A.34) we find the acceleration of gravity just inside the domain wall, We see that the acceleration of gravity depends on the value of k . There is no acceleration if k = 0 because in this case the ( η, χ )-system is comoving in a static reference frame in flat spacetime. When k = 1 there is an acceleration of gravity towards the point χ = 0. In this case the ( η, χ )-system is comoving in a reference frame accelerating in the outwards direction. In the case k = -1 we must have B > R Q in order that the WLBR spacetime shall exist outside the domain wall as seen in Figure 4. Then the acceleration of gravity points outwards, meaning that the reference frame of the ( η, χ )-system is accelerating inwards. The acceleration of gravity just outside the domain wall as given by equation (58) is found in a similar way using equations (A.19) and (A.34) with the result When k = 0 the ( η, χ )-system is comoving in the same reference frame as the CFS coordinates, which is at rest relative to the domain wall. In this case the acceleration of gravity in the ( η, χ )-system just outside the domain wall is equal to 1 /R Q just as in the CFS system. When k = 1 and B > R Q , the acceleration of gravity is directed away from the domain wall. But when B < R Q it is directed towards the domain wall. If B = R Q the acceleration of gravity vanishes. This behaviour can be understood by considering Figure 3. In the case B = R Q the hyperbola χ = π/ 2 touches the domain wall at R = R Q when T = 0, corresponding to η = 0. For B > R Q the hyperbola moves to the right, and for B < R Q to the left. For all values of k and B the discontinuity of the acceleration of gravity at the domain wall is This shows that the domain wall produces repulsive gravity. In spite of the fact that the LBR spacetime is static, it may be described in terms of coordinates comoving with a reference frame expanding in such a way that the line element takes a time dependent form. Assuming that the metric functions are independent of the radial coordinate, equation (3) reduces to which may be written This equation has the general solution where a > 0 is an integration constant. The general solution of (80) is given by where τ 0 is an integration constant. Choosing a = 1 and τ 0 = 0 the line element (1) takes the form where -∞ < τ < ∞ and -∞ < ρ < ∞ . The form of this line element when the proper time of the reference particles is used as a time coordinate is given in equation (189). We want to investigate whether particles with constant ρ are free, and hence whether their world lines fullfill the geodesic equation. The radial component of this equation then reduces to Calculating the Christoffel symbol from the line element (82) we find that Γ ρ τ τ = 0. Hence a particle with constant ρ has vanishing acceleration. It is a free particle. Accordingly the ( τ, ρ )-system is comoving with free particles. From equation (82) it follows that the coordinate clocks of the ( τ, ρ )-system go at a rate which is increasing relative to the rate of standard clocks at rest in the reference frame where ( τ, ρ ) are comoving coordinates. Note that the coordinate time τ is not equal to the proper time t of the reference particles with constant ρ . The relationship between τ and t will be treated in section IIIb where the proper time will be used as coordinate time. In this reference frame the physical distances in the radial direction are extremely small when τ →-∞ . However the space expands in the radial direction and the radial scale factor has a maximal value equal to R Q when τ = 0. Then space contracts in the radial direction towards vanishingly small distances in the infinitely far future. /negationslash We shall find the transformation relating the line elements (28) and (82). In this case G ( x 0 , x 1 ) = cosh( x 0 ) in the line element (5) so that G ( x 0 , 0) = 0. Hence we need two generating functions. We introduce the generating functions using the same constant B as in equation (37) when k = 1 and k = -1 in order to simplify the transformations. By means of equations (7), using the procedure shown in Appendix B, we find the following transformation from the ( τ, ρ )-coordinates to the CFS cordinates, This transforms the region τ + ρ < 0 in the ( τ, ρ )-system to the region T + R > B , | T -R | < B in the CFS system, and the region τ + ρ > 0 in the ( τ, ρ )-system to the region T + R < -B , | T -R | < B in the CFS system. As shown in Appendix B the inverse transformation is found from the generating functions which gives From the second of equations (86) with R = R Q it follows that the charged domain wall which represents the inner boundary of the WLBR spacetime, moves according to in the ( τ, ρ )-system. It follows that the WLBR spacetime is represented in the ( τ, ρ )-plane by the hatched region in Figure 5, which is given by The coordinate velocity of the domain wall is Using equation (91) and looking at Figure 5(a) we see that in the case B > R Q the domain wall initially has velocity in the positive ρ -direction with decelerating motion. It stops at an event given by and moves in the negative ρ -direction. This means that the ( τ, ρ )-system accelerates in the positive R -direction. From equation (91) it follows that /negationslash when B = R Q . Hence we see that when B > R Q the domain wall has initially a velocity close to the velocity of light in the positive ρ -direction, and finally the same velocity in the negative ρ -direction. Next we consider the case B = R Q . Then the expression for the velocity of the domain wall can be written From this equation it follows that In this case the domain wall has initially a vanishing coordinate velocity, but it accelerates slowly in the negative ρ -direction and ends up approaching the velocity of light. We finally consider the case 0 < B < R Q . In this case the motion of the domain wall is more complicated. In the limit that τ → -∞ the domain wall moves nearly with the velocity of light in the negative ρ -direction. Then it decelerates and obtains a minimal velocity -√ 1 -( B/R Q ) 2 when it passes ρ = 0. Afterwards it accelerates again and approaches the velocity of light in the infinite future. Since the domain wall is at rest in the CFS system, all of this reflects the motion of the reference frame in which the ( τ, ρ )-coordinates are comoving. We shall find the transformation relating the line elements (36) and (82). Combining the generating functions (85) with the inverse of the generating function (37) with k = 1, we obtain the generating functions which give the transformation as shown in Appendix B. This transforms the region τ + ρ < 0 in the ( τ, ρ )-system to the region π/ 2 < η + χ < π , | η -χ | < π/ 2 in the ( η, χ )-system, and the region τ + ρ > 0 in the ( τ, ρ )-system to the region -π < η + χ < -π/ 2 , | η -χ | < π/ 2 in the ( η, χ )-system. The inverse transformation is found in a similar way using the generating functions which give the transformation In the present case the transformation (97) and its inverse can also be found from the equations (39), (41), (86) and (88). Combining the first equation in (88) and (41) for k = 1 and substituting for R/T from (86) we get Note that the hyperbola χ = π/ 2 in Figure 3 corresponds to ρ = 0. As shown above a free particle has constant ρ , say ρ = ρ 1 . Hence it follows from the second of the transformation equations (99) that the world line of a free particle as described in the ( η, χ )-system is given by We will now show that this is a solution of the Lagrangian equation for a free particle moving radially. With the line element (36) and k = 1 the Lagrangian is The conserved momentum conjugate to the time coordinate is giving The 4-velocity identity then takes the form The last two equations lead to We now transform from differentiation with respect to the proper time of the particle to differentiation with respect to the coordinate time η by means of equation (104), and find that the solution of this differential equation with the initial condition χ (0) = π/ 2 is This is in accordance with equation (101) if the conserved energy of the particle is Inserting the expression (81) for e α with a = 1 and τ 0 = 0, and (86) for R into the line element (23), we obtain the form of the line element for the Minkowski spacetime inside the domain wall in the ( τ, ρ ) coordinates, It follows from equation (82), the last of equations (86) with R = R Q , and the line element (109) that the metric is continuous at the shell. In the ( τ, ρ )-system the acceleration of gravity is which is positive. Hence an observer at rest in this coordinate system experiences an acceleration of gravity in the outwards direction in the flat spacetime inside the wall, which means that the reference frame of these coordinates is accelerating in the inwards direction. IIa. Static metric and coordinates ( ˜ t, ˜ r ) with β (˜ r ) = -α (˜ r ) . In this case equation (3) reduces to which may be written The general solution of this equation can be written as where D and ˜ r 0 are constants. The line element (1) then takes the form This form of the line element with R Q = 1, ˜ r 0 = 0 and D = ± 1 has been used by Ottewill and Taylor [19] and by V. Cardoso, O. J. C. Dias and J. P. S. Lemos [20]. It may further be noted that A. S. Lapedes [21] has studied particle creation in the LBR spacetime. He then considered three forms of the line element (114) with ˜ r 0 = 0 and D = -1 , 1 , 0 respectively, with a rescaling of ˜ t and ˜ r by R Q , and constructed a Penrose diagram for the LBR spacetime. The coordinate clocks showing ˜ t go at the same rate as a standard clock at ˜ r = ˜ r 0 scaled by the factor √ D when D > 0. For D = 1 and ˜ r 0 = 0 the components g ˜ t ˜ t and g ˜ r ˜ r have the same form as the corresponding components of the anti De Sitter metric in static coordinates, while the angular part of the line element represents a spherical surface. Writing D = kA 2 where k = sgn( D ) and /negationslash the transformation between the ( ˜ t, ˜ r )- and the ( η, χ )-system used in the line element (36) is given by The transformation has been chosen so that χ and ˜ r increases in the same direction. The transformation from χ to ˜ r is It follows that that the ( ˜ t, ˜ r )-system is comoving with the reference particles of the same reference frame as the ( η, χ )-system. Inserting equation (117) into equation (113) and using the relation (A.13) we obtain Note that this expression in consistent with the line element (36) due to the relation between η and ˜ t in the transformation (116). Differentiating equation (117) we get Using (118) and (119) we see that the line element (114) takes the form (36). It follows from equations (49) and (117) for k = 1 that in this case the WLBR spacetime is represented by the hatched region in Figure 6 given by and where η = ( A/R Q ) ˜ t and χ Q = arccot( B/R Q ). In the PLBR spacetime all the values k = -1 , 0 , 1 are allowed. For k = 1 the PLBR spacetime is represented by the region between the horizontal lines in Figure 6 given by The part to the left of the hatched region corresponds to 0 < R < R Q in the CFS system, while the part to the right corresponds to R < 0. For k = -1 the WLBR spacetime is represented in the ( ˜ t, ˜ r )-system by a region given by where η = ( A/R Q ) ˜ t and χ Q = arccoth( B/R Q ). For k = 0 it is represented by the region We shall now give a physical interpretation of the constants ˜ r 0 and D valid for all values of k by considering the motion of a free particle. The acceleration of a free particle instantaneously at rest is here given by Using the line element (114) we get where τ is the proper time of the particle. Furthermore This gives for the acceleration of gravity in the ( ˜ t, ˜ r )-system This means that ˜ r = ˜ r 0 is the position where the acceleration of gravity vanishes in the ( ˜ t, ˜ r )-system. In the WLBR spacetime a free particle is falling towards the domain wall in the region ˜ r > ˜ r 0 and away from the domain wall in the region ˜ r < ˜ r 0 . In section 5 we shall show that this is due to the motion of the reference frame in which ( ˜ t, ˜ r ) are comoving coordinates. In the case k = 1 the constant ˜ r 1 in Figure 6 represents the position of the shell in the ( ˜ t, ˜ r )-system at the point of time T = 0. The constant A = √ D has the following physical interpretation. The coordinate clocks ˜ t go at a constant rate equal that of the standard clocks at the domain wall at the point of time T = 0 scaled by the factor A . The line element (114) can now be written as In the previous cases with k = 1 the PLBR spacetime corresponds to only a part of the coordinate region in the ( ˜ t, ˜ r )-system. However, when k = -1 the ( ˜ t, ˜ r )-system covers only a part of the PLBR spacetime as shown in Figure 4. Combining the transformations (116) and (39), and using the identities (A.34), we obtain the coordinate transformation from ( ˜ t, ˜ r ) to ( T, R ) in the following form In the cases k = 0 and k = -1 we use plus when ˜ r < ˜ r 0 and minus when ˜ r > ˜ r 0 . In the case k = 1 we use plus for all values of ˜ r . This generalizes and modifies the corresponding transformation for k = 1, ˜ r 0 = 0 and A = B = D = 1 given by Griffiths and Podolosky [22]. When k = 1 this transformation maps the region -πR Q /A < ˜ t < πR Q /A , ˜ r < ˜ r 0 + R Q A cot( A | ˜ t | /R Q ) in the PLBR spacetime shown in Figure 6 onto the right half plane in the CFS system, and the region -πR Q /A < ˜ t < πR Q /A , ˜ r > ˜ r 0 + R Q A cot( A | ˜ t | /R Q ) in the PLBR spacetime onto the left half plane in the CFS system. When k = -1 the transformation maps the region -∞ < ˜ t < ∞ , ˜ r < ˜ r 0 -R Q A in the PLBR spacetime onto the triangle 0 < R < B , | T | < B -R in the CFS system, and the region -∞ < ˜ t < ∞ , ˜ r > ˜ r 0 + R Q A onto the triangle -B < R < 0, | T | < B + R in the CFS system. The WLBR spacetime described by the inequalities (123) is mapped onto the triangle R Q < R < B , | T | < B -R in the CFS system. Using the relationship (A.11) we find that the inverse transformation can be written /negationslash when T = 0. In the case T = 0, we have t = 0. From the last one of the transformation equations (132) it follows that for k = 1 the hyperbola of Figure 3 where the acceleration of gravity vanishes is given by ˜ r = ˜ r 0 , in agreement with equation (128). In the case k = 0 the transformation (130), (131) reduces to which has been chosen so that e ˜ r points in the same direction as e R . This transformation maps the region -∞ < ˜ t < ∞ , ˜ r < ˜ r 0 onto the right half plane in the CFS system, and the region -∞ < ˜ t < ∞ , ˜ r > ˜ r 0 onto the left half plane in the CFS system. The inverse transformation is In this case the line element (114) takes the form which was considered in reference [19] with ˜ r 0 = 0. In these coordinates we shall write down the form of the line element in the flat spacetime inside the domain wall only for the case k = 0 when the external metric is given by the equation (135). Inserting the expression (133) for R in the line element (23) then leads to It follows from the transformation (134) that (˜ r -˜ r 0 ) /R Q = 1 when R = R Q , showing that the metric is continuous at the domain wall. Calculating the Christoffel symbols from the line element (136) shows that there is vanishing acceleration of gravity inside the domain wall in this coordinate system. The reason is that for k = 0 the ( ˜ t, ˜ r ) coordinates are comoving in a static reference frame in this region. IIb. Time dependent metric and coordinates ( t, r ) with β ( t ) = -α ( t ) . In this case equation (3) reduces to which may be written The general solution of this equation can be written as where D and t 0 are constants. A special case of this solution with t 0 = 0 and D = 1 has earlier been found by N. Dadhich [23]. The line element (1) then takes the form From equation (139) we see that the constant D must be positive, and we introduce the constant A = √ D as in section IIa. Here the standard measuring rods have a time dependent length, and the coordinate rods have a constant length equal to the length of the standard rods at the point of time t = t 0 scaled by the factor A . The allowed range of the time t is The transformation between the ( t, r )- and the ( τ, ρ )-system used in the line element (82) is given by The transformation has been chosen so that τ and t increase in the same direction. The second of these equations shows that the ( t, r )-system is comoving with the same reference frame as the ( τ, ρ )-system. Hence particles with r = constant are moving freely. The transformation from τ to t is Inserting equation (143) into equation (139) we obtain Differentiating equation (143) we get Using (144) and (145) we see that the line element (140) takes the form (82). Combining the transformations (86) and (142) we obtain the coordinate transformation from ( ˜ t, ˜ r ) to ( T, R ) in the following form The inverse transformation is given by IIIa. Static metric and coordinates ( ˆ t, ˆ r ) with α = α (ˆ r ) and β = 0 . In this case the radial coordinate ˆ r is equal to physical distance in the radial direction. Equation (3) then reduces to The general solution of this equation is or alternatively where c i , i = 1 , 2 , 3 , 4 are constants. Here the coordinate clocks go with a position independent rate equal to that of the standard clocks at ˆ r = 0 scaled by the factor c 3 . This solution (151) was found already in 1917 by T. Levi-Civita [4,5], and was later mentioned in [23] and in [24] with c 3 = 0. We are now going to find the coordinate transformation between the physical coordinates ( ˆ t, ˆ r ) and the CFS coordinates ( T, R ). In this connection we will also deduce the transformation between χ and ˆ r and between ˜ r and ˆ r . Since ˆ r represents the physical radial distance we have from equation (36) that which may be written Integration using the identities (A.10), (A.18) and (A.33) gives ∣ ∣ where ˆ r 0 is a constant. With a suitable scaling of the time coordinate the transformation between the ( ˆ t, ˆ r )-system and the ( η, χ )-system is given by where A is defined in equation (115). Comparing with equation (116) we see that ˆ t = ˜ t . In the case k = -1 we use the upper sign when ˆ r < ˆ r 0 and the lower sign when ˆ r > ˆ r 0 . In the cases k = 0 and k = 1 we use the upper sign for all ˆ r . These rules mean that ˆ r increases in the same direction as χ . Using the identity (A.31) with x = χ/ 2 combined with equations (116) and (155) we obtain with the same rule for choosing the signs as above, meaning that ˆ r and ˜ r increases in the same direction. This implies that The inverse transformation is given by Since the relationship between ˜ r and ˆ r is time-independent, the ˆ r -coordinate is comoving in the same reference frame as the ˜ r -coordinate. It follows from equations (49) and (156) for k = 1 that in this case the WLBR spacetime is given by and ˆ r 0 -R Q arcsinh(cot ( χ Q +arcsin (sin χ Q cos η ))) < ˆ r < ˆ r 0 + R Q arcsinh(cot | η | ) , (160) η = ( A/R Q ) ˆ t and χ Q is given in equation (46). where In the PLBR spacetime all the values k = -1 , 0 , 1 are allowed. For k = 1 the PLBR spacetime is represented by For k = -1 the WLBR spacetime is represented in the ( ˆ t, ˆ r )-system by a region given by -∞ < ˆ t < ∞ , ˆ r 0 -R Q arccosh(coth ( χ Q +arcsinh (sinh χ Q cosh η ))) < ˆ r < ˆ r 0 , (162) where η = ( A/R Q ) ˆ t . For k = 0 it is represented by the region From equations (156), (A.10), (A.13) and (A.35) we obtain Using the formulae (164) and (155) it follows that the line element (36) takes the form The connection between the general solution (151) and the form (165) of the line element is given by Inserting the relations (164) into equations (39) we obtain the transformation between the physical coordinates ( ˆ t, ˆ r ) and the CFS coordinates, Using equation (156) and the identity (A.35) we see that this transformation is consistent with equations (130) and (131). The inverse transformation is given by In the case k = 0 the transformation (167), (168) reduces to which has been chosen so that e ˆ r points in the same direction as e R . The inverse transformation is Then the line element (165) takes the form The line elements (28) and (172) are related by the transformation (170) with ˆ r 0 = R Q . In this case the ˆ r -coordinate and the R -coordinate are comoving in the same reference frame. Although different choices of c i , i = 1 , 2 , 3 , 4 all represent conformally flat solutions of the field equations with the same energy momentum tensor representing a constant, radial electrical field, the physical properties of the solutions are different. This may be most clearly seen by utilizing the geodesic equation. Inserting the solution (151) in the line element (1) we find that a free particle instantaneously at rest has a coordinate acceleration The acceleration of gravity in the ( ˆ t, ˆ r )-system is the component of a ˆ r e ˆ r along the unit basis vector e ˆ ˆ r . Since g ˆ r ˆ r = 1, we have that a ˆ ˆ r = a ˆ r . A reference particle with given values of ˆ r, θ, φ is at rest relative a reference particle with given values of χ, θ, φ . Hence the transformation from the ( η, χ )-system to the ˆ t, ˆ r -system is a so called internal transformation, i.e. a coordinate transformation inside a single reference frame. In addition, the unit radial vector in the ( η, χ )-system is identical to the unit radial vector in the ˆ t, ˆ r -system, e ˆ r = e χ . These two conditions mean that a ˆ ˆ r = a ˆ χ . Using equation (164) we obtain This expression for the acceleration of gravity can be positive or negative. We shall here discuss these possibilities for the WLBR spacetime. The reason for these differences is found in the different motions of the ( ˆ t, ˆ r )-systems relative to the CFS system for different values of k and ˆ r . The world line of a reference point ˆ r = ˆ r 1 as described with reference to the ( T, R )-system is given by equation (54). By means of equation (156) the constant R 1 in equation (54) can be expressed in terms of the coordinate ˆ r 1 as R 1 = ∓ kBa -k ((ˆ r 0 -ˆ r 1 ) /R Q ) . The world line is shown for k = 1 in Figure 1. The form (172) of the line element for the WLBR spacetime in a uniform electric field outside a charged domain wall shows that the time does not proceed infinitely far from the domain wall. The coordinate velocity of light moving radially outwards is Hence lim ˆ r →∞ d ˆ r/d ˆ t = 0. There is, however, no horizon at a finite distance from the wall. Again we shall write down the form of the line element in the flat spacetime inside the domain wall only for the case k = 0. Inserting e α from the external line element (172) and the expression (170) for R we obtain the internal line element in the ( ˆ t, ˆ r ) coordinates, It follows from equation (171) that the ˆ r coordinate of the shell is showing that e 2(ˆ r -ˆ r 0 ) /R Q = R 2 Q at ˆ r = ˆ r Q . Inserting this in the line elements (172) and (176) shows that metric is continuous at the shell. As in the ( ˜ t, ˜ r ) coordinates there is vanishing acceleration of gravity inside the domain wall in the ( ˆ t, ˆ r ) coordinate system for the case that k = 0 because then it is comoving in a static reference frame in this region. IIIb. Time dependent metric and coordinates ( t, r ) with α = 0 , β = β ( t ) . With e β ( t ) = a ( t ) the line element then takes the form Here t corresponds to the cosmic time of the FRW universe models, i.e. it is the proper time of clocks with fixed spatial coordinates, and a ( t ) is a scale factor describing the expansion of space in the radial direction. There is no expansion in the directions orthogonal to the radius vector. Calculating the Christoffel symbols from this line element we find that From the geodesic equation it follows that a free particle instantaneously at rest will remain at rest in this coordinate system. Hence the coordinates r, θ, φ are comoving with free particles. Therefore ( t, r, θ, φ ) are the coordinates of an inertial reference frame. They may be called inertial coordinates in the PLBR spacetime. These coordinates are analogous to the standard coordinates used in the FRW universe models. The Einstein-Maxwell equations then take the form where the dots denote differentiation with respect to the proper time of the reference particles. With the line element (178) this is also the condition that the Weyl tensor vanishes. The general solution of equation (180) is where d 1 and d 2 are constants. We shall find the transformation relating this line element to the line element (28) of the LBR spactime in CFS coordinates. In this connection we will also deduce the transformation between τ and t and between t and t . Since t represents the proper time of clocks with fixed spatial coordinates it follows from the line element (82) that Integration gives where t 0 is a constant. With a suitable scaling of the radial coordinate the transformation from the ( t, r )-system to the ( τ, ρ )-system is given by transforming the region t 0 -R Q π/ 2 < t < t 0 + R Q π/ 2 , -∞ < r < ∞ in the inertial system to the region -∞ < τ < ∞ , -∞ < ρ < ∞ in the ( τ, ρ )-system. It also follows that Combining this with equation (142) we obtain which implies that The inverse transformation is given by Using the formulae (182), (185) and (184) it follows that the line element (82) takes the form In these coordinates the line element has a form similar to that of a Friedmann Robertson Walker universe model with radial scale factor a ( t ) = A cos(( t -t 0 ) /R Q ). The coordinate time t corresponds to the cosmic time as measured by clocks comoving with free particles. There is initially an expansion in the radial direction, turning to contraction at the point of time t = t 0 . Hence t 0 is the point of time with maximal physical distances. The connection between the general solution (181) and the form (189) of the line element is given by Inserting the relations (184) and (185) into equations (86) we obtain the transformation between the inertial coordinates ( t, r ) and the CFS coordinates, transforming the region t 0 -R Q π/ 2 < t < t 0 + R Q π/ 2 , -∞ < r < ∞ in the inertial system to the region | T + R | > B , | T -R | < B in the CFS system (see Figure 7). Using equation (186) we see that this transformation is consistent with equations (146) and (147). The inverse transformation is given by Note that the denominators cannot vanish in the regions specified above. The world lines of fixed points r = r 1 in the inertial frame with reference to the CFS system are given by which represents hyperbolae as shown in Figure 7. This form of the world line of a fixed point r = r 1 is in accordance with equation (65) for the world line of a free particle. Differentiating equation (194) we find the coordinate velocity of a particle with r = r 1 in the CFS system. The initial velocity of the particle at T = 0, R = B is We want to find the region in the ( t, r )-system corresponding to WLBR spacetime. This region is given by R > R Q . From equation (192) it follows that this corresponds to which gives and The WLBR spacetime is shown as the hatched region in Figure 8. The part to the left of the hatched region corresponds to 0 < R < R Q in the CFS system, while the part to the right corresponds to R < 0. Let us consider a particle falling freely with outwards directed initial velocity from R = R Q at the event P 2 . This particle follows the world line r = r 1 < 0 as shown in Figure 7. As observed in the CFS system it accelerates away from the wall as seen from equation (35). Hence there is repulsive gravitation. However, it follows from the line element (189) that as observed by freely falling observers, the 3-space t = constant first expands and then contracts in the radial direction. This strange behaviour can be understood by considering the equation of geodesic deviation. In comoving coordinates with tangent vector u = (1 , 0 , 0 , 0) for the geodesic curves the equation takes the form With the line element (189) this equation reduces to having the solution which is equal to the scale factor in the line element (189). This then provides an explanation for the surprising contraction of the space between the events ( t 0 , r 1 ) and P 3 in Figure 8. The transition from expansion to contractions happens at t = t 0 , corresponding to the simultaneity curve T 2 -R 2 = B 2 as seen from the first of the equations (193). The world line of an observer with r = r 1 will intersect this simultaneity curve only when r 1 < 0. Hence only these observers will experience contraction. A simple special case of the line element (189) is obtained by choosing t 0 = R Q π/ 2 and A = 1, giving A deeper understanding of the t coordinate may be obtained by giving a parametric description of a free particle in the PLBR spacetime with the proper time t of the particle as parameter. We consider a particle with r = r 1 in the inertial coordinate system, with world line given in equation (194). The particle is instantaneously at rest at the point P with CFS coordinates ( T 1 , R 1 ) where R 1 = √ B 2 -T 2 1 . We shall now apply Lagrangian dynamics in the CFS system to this particle. Putting the velocity (63) equal to zero at the point P shows that the constant of motion p T for this particle is where the minus sign has been chosen in order that The first equality follows from equation (61). Inserting this into the four-velocity identity (62) and integrating leads to where t 1 is a constant of integration. Inserting the expression (205) into (204) and integrating gives where T 2 is a new constant of integration. Demanding that equations (205) and (206) is a parametric representation of the hyperbola (194) gives T 2 = T 1 . From equations (205) and (206) we then have The constant t 1 is now determined by eliminating t from equations (207) and the first of the transformation formulae (193) at the point P . This gives The equations (205) to (208) give a parametric representation of the world lines of free particles with r = r 1 as shown in Figure 7. We shall now show that this parametric description of the path of a free particle with the proper time of the particle as parameter is in agreement with the transformation (193) from the CFS coordinates to the comoving coordinates of the particle. We have that Inserting equations (205) and (207) gives From equation (194) and the second of the equations (193) it follows that Inserting this into equation (210) we finally obtain the first of the transformation equations (193). In Figure 7 we have drawn the world lines of particles with different values of r 1 . All of the particles come from the point ( B, 0) and move so that T and R approach infinity when t increases towards t 0 as seen from equations (205) and (206). When t passes t 1 the values of T and R switch to minus infinity, and all particles approach the point ( -B, 0) when t increases towards t 0 + πR Q / 2. This highly surprising behaviour may be understood by noting that according to the line element (28) the physical distances in the PLBR spacetime approach zero when | R | approaches infinity. The reference particles of the ( t, r )-system are freely falling. Their world lines are hyperbolae corresponding to particles with constant proper acceleration. This is in accordance with the fact that the acceleration of gravity as given in equation (35) is constant in the LBR spacetime. In the final part of this section we shall present the transformations between the previous coordinate systems and the inertial one. Combining equation (186) with the transformation (142) we obtain The inverse transformation is Hence which means that the coordinate clocks showing t go at an increasingly slower rate than the standard clocks showing t . We shall find the transformation relating the line element (1) of the LBR spacetime in inertial and physical coordinates respectively. Here α in the line element (1) is given by equation (151) and β = 0. Combining the transformation (97) with the equations (184) and (185) we obtain the transformation Now using η = ( A/R Q ) ˆ t and (156) we obtain the transformation The inverse transformation is found in a similar way by combining the transformation (99) with equation (186) which gives Introducing ˆ t and using the equations (164) we obtain the transformation Hence the world line of a free particle with r = r 1 as described with reference to the ( ˆ t, ˆ r )-system is given by /negationslash The coordinate transformations from the CFS system to the inertial system and the ( ˆ t, ˆ r )-system with k = -1 are defined on the disjoint domains | T + R | > B , | T -R | < B and | T + R | < B , | T -R | < B , R = 0 respectively. There is therefore no coordinate transformation from the inertial system to the ( ˆ t, ˆ r )-system in this case. For k = 0 we have We can also find the transformation between the ( ˜ t, ˜ r )-system and the inertial system. Combining the transformations (215) and (116) we obtain The inverse transformation is given by Equation (128) gives the following equation of motion for a free particle in the ( ˜ t, ˜ r )-system, where t is the proper time of the particle. This is the equation of harmonic motion about the position ˜ r = ˜ r 0 as noted by Dadhich [23]. His interpretation is that a free particle would execute harmonic oscillation about ˜ r = ˜ r 0 . He has given an explanation of this motion in terms of electrostatic energy filling the LBR spacetime. In our opinion, however, there is another explanation for this motion. Equation (225) has the general solution where A 1 and t 0 are integration constants. According to equation (222) a fixed point r = r 1 in a freely moving reference frame has a radial coordinate given by the above equation with A 1 = R Q A sinh( Ar 1 /R Q ). Hence ( ˜ t, ˜ r ) are comoving coordinates in a reference frame that performs harmonic motion relatively to a freely falling reference frame. This is the reason for the oscillating motion of a free particle in the ( ˜ t, ˜ r )-system which was noted by Dadhich, assuming that -∞ < t < ∞ . In the context of the LBR as interpreted in the present article, the situation is different. The coordinate region in ( t, r )-system of the LBR spacetime is given by the inequalities (197) and -∞ < r < ∞ . This restriction of the time interval means that the oscillating character of the motion of a free particle in the ( ˜ t, ˜ r )-system as given by equation (226) vanishes. Choosing A = 1, B = R Q , t 0 = 0 and introducing the coordinates ˜ τ = t , x = r , y = R Q φ and z = R Q ( θ -π/ 2) in equation (189), the PLBR line element takes the form [19] Using the formulae (191) and (192) we see that this form of the line element is obtained from (28) by the transformation Note that x , y and z are not to be interpreted as Cartesian coordinates.", "pages": [ 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36 ] }, { "title": "IV. A new type of coordinate systems for the case k = -1 .", "content": "When k = -1 in equation (36) there exist different types of coordinates for the LBR spacetime obeying the coordinate conditions β = α , β = -α and β = 0. Here we will introduce coordinates ( η ' , χ ' ) with β = α , ( ˜ t ' , ˜ r ' ) with β = -α and ( ˆ t ' , ˆ r ' ) with β = 0 different from the coordinates ( η, χ ), ( ˜ t, ˜ r ) and ( ˆ t, ˆ r ) respectively. In order to find the transformation between the line element (28) and the line element (36) with marked coordinates, we replace the generating function (37) by Like the function (37) it satisfies the condition (20). It is obtained from equation (22) with a = 0, b = -1, c = 2 B and d = B . This leads to the transformation The inverse transformation is In the ( T, R )-system, each reference particle with χ ' = constant in the coordinate system has a constant velocity which is less than 1. According to this equation χ ' is the rapidity of a reference particle with radial coordinate χ ' . Figure 9 shows the ( η ' , χ ' )-system in a Minkowski diagram referring to the CFS system of the observer at χ ' = 0. It follows from equations (234) that the world lines of the reference particles with χ ' = constant are straight lines, and the curves of the space η ' = constant are hyperbolae with centre at the origin as shown in the diagram in Figure 9. One rather sublime point should be noted. Since the line element of the LBR spacetime has similar coordinate expressions in (36) and (231), one might think that for instance the kinematics of free particles are identical in these coordinate systems. Calculating the acceleration of a free particle from the geodesic equation, one finds identical coordinate expressions for the Christoffel symbols. Hence it seems that the coordinate acceleration of a free particle at a given event is the same in the marked and unmarked coordinate systems. This is, however, not the case since the transformations (41) and (234) imply that the χ - and χ ' -coordinates of an event with coordinates ( T 1 , R 1 ) are different. Inserting the expression e α = R Q / sinh χ ' from the line element (231) and the expression (233) for R , we find the line element of the flat spacetime inside the domain wall in the ( η ' , χ ' ) coordinates, From equations (231), (233) with R = R Q and (236) it is seen that the metric is continuous at the domain wall. Combining the transformations (233) and (116) we obtain the transformation with inverse transformation given by Combining the transformations (233) and (164) we obtain the transformation with inverse transformation given by This transformation has earlier be considered by Zaslavskii [30]. In this section we shall introduce new coordinates (ˆ η, ˆ χ ) which will be useful when we introduce lightlike coordinates in section VII. These coordinates are assumed to obey the coordinate condition Introducing the function f (ˆ χ ) = e α (ˆ χ ) the differential equation (3) takes the form In order to solve this differential equation we introduce a function y (ˆ χ ) defined by This transforms the equation (243) to The general solution of this differential equation is where a and ˆ χ 0 are integration constants. There are two special cases where the solution can be expressed in terms of elementary functions. The first is a = 0. Choosing ˆ χ 0 = -π/ 2 we obtain giving /negationslash where -∞ < ˆ η < ∞ and -π/ 2 < ˆ χ < π/ 2, ˆ χ = 0. With these coordinates the line element of the LBR spacetime takes the form The transformation between the (ˆ η, ˆ χ )- and the ( η, χ )-system used in the line element (36) is given by This transformation shows that (ˆ η, ˆ χ ) and ( η, χ ) are comoving coordinates in the same reference frame. The transformation represents a rescaling of the radial coordinate such that the infinite interval -∞ < χ < ∞ is transformed into the finite interval -π/ 2 < ˆ χ < π/ 2, where χ = 0 and ˆ χ = 0. Combining the transformation (250) and (39) with k = -1, we obtain the coordinate transformation from (ˆ η, ˆ χ ) to ( T, R ) in the following form /negationslash /negationslash The inverse transformation is On the other hand, choosing ˆ χ 0 = 0 we obtain giving /negationslash where -∞ < ˆ η < ∞ and 0 < ˆ χ < π , ˆ χ = π/ 2. With these coordinates the line element of the LBR spacetime takes the form The transformation between the (ˆ η, ˆ χ )- and the ( η, χ )-system used in the line element (36) is given by From the line element (255) it follows that the coordinate velocity of light moving in the radial direction is Integrating we obtain the equation of the world line of light in the (ˆ η, ˆ χ )-system where ˆ η 0 is a constant. Combining the transformation (256) and (39) with k = -1, we obtain the coordinate transformation from (ˆ η, ˆ χ ) to ( T, R ) in the following form The inverse transformation is The second case is a = 1. Choosing ˆ χ 0 = 0 the solution of the differential equation (246) can then be written giving With these coordinates the line element of the LBR spacetime takes the form The transformation between the (ˆ η, ˆ χ )- and the CFS system used in the line element (28) is given by From the line element (263) it follows that the coordinate velocity of light moving in the radial direction is Integrating with the initial condition ˆ χ (0) = 0 we obtain the equation of the world line of light in the (ˆ η, ˆ χ )-system According to equation (264) this corresponds to R = ± T , which is the equation of radially moving light in the CFS system as seen from the line element (28). Combining the transformation (264) and (41), we obtain the coordinate transformation from (ˆ η, ˆ χ ) to ( η, χ ) in the following form when ˆ η = 0. In the case ˆ η = 0 we have that η = 0. The inverse transformation is /negationslash", "pages": [ 36, 37, 38, 39, 40, 41 ] }, { "title": "VI. Cylindrical coordinates.", "content": "We shall now consider an axially symmetric space using cylindrical coordinates ρ , θ , z , assuming that the line element has the form where f = f ( ρ, z ). Demanding that the Weyl tensor vanishes, we find that where a , b and c are constants. In general the energy momentum tensor has the following physical interpretation. Since the tensor is symmetrical, the eigenvectors of the tensor can be chosen to be orthonormal with one timelike and three spacelike vectors. These vectors will then represent an orthonormal basis that may be associated with an observer with four velocity equal to the timelike eigenvector u = e 0 . The eigenvalue λ 0 is interpreted as the energy density measured by this observer, and the eigenvalues λ i are interpreted as the stresses he measures. For a = 1, b = c = 0 the vectors of the observer's orthonormal basis are The corresponding eigenvalues of the energy momentum tensor are These eigenvalues are recognized as those of the energy momentum tensor of an electrical field. The line element then takes the form As shown by D. Garfinkle and E. N. Glass [25] this may also be found by transforming the line element (28) to cylindrical coordinates by means of or inversely Note that the charged domain wall defining the inner boundary of the WLBR spacetime according to our interpretation is now given by ρ 2 + z 2 = R -2 Q . From equations (23), (273) and (275) we find that in the cylinder coordinates the line element of the Minkowski spacetime inside the domain wall takes the form It follows from equations (273), (275) with R = R Q and (276) that the metric is continuous at the domain wall. There is no acceleration of gravity in the reference frame in which these coordinates are comoving. The unusual form of the spatial part of the line element is a coordinate effect. The space is defined by T = constant just as in the CFS coordinate system. Hence it is a Euclidean space described by using non-standard coordinate measuring rods that are related to the standard rods by the transformation (274). In these coordinates the space looks like a curved, but conformally flat space. In Cartesian and spherical coordinates, respectively, this line element takes the form VII. Light cone coordinates. In spherical coordinates the line element on a 2-sphere with radius R Q has the form One can project the spherical surface from the north pole onto the equatorial plane by means of stereographic coordinates given by with inverse transformation given by Taking the differentials and inserting into equation (278) we find the line element of the 2sphere parametrized with the stereographic coordinate ζ representing two real coordinates, i.e. the real and imaginary part of ζ , where ζ is the complex conjugate of ζ . We shall now deduce a corresponding form for the line element of the 2-dimensional anti de Sitter spacetime. For this purpose we introduce new coordinates U and V for the anti de Sitter part of the LBR spacetime in analogy with stereographic coordinates for the spherical part, where (ˆ η, ˆ χ ) are the coordinates introduced in section 4.V. From equation (266) it follows that U = constant for light moving in the positive ˆ χ -direction, and V = constant for light moving in the negative ˆ χ -direction. Hence ( U, V ) are light cone coordinates. The inverse of the transformation (282) is given by In the same way as for the spherical part we find In terms of the light cone coordinates ( U, V ) and the stereographic coordinates the line element of the LBR spacetime takes the form This form of the line element has earlier been considered by M. Ortaggio [26] and later mentioned by Ortaggio and Podolsk'y [27] and Griffiths and Podolsk'y [22] with a different scaling of the coordinates. In order to find the coordinate transformation between the light cone coordinates ( U, V ) and the CFS coordinates, we will utilize the transformation (39) between the ( η, χ ) and the CFS coordinates. The ( U, V ) coordinates are related to the (ˆ η, ˆ χ ) coordinates by the transformation (282). Using the transformation (256) we get Inserting this in equation (282) we obtain the transformation from the ( η, χ )- to the ( U, V )-system, The inverse transformation is given by From this we also obtain and Inserting these expressions into the transformation (39) we find The inverse transformation is We now introduce coordinates (˜ u, ˜ v ) by the coordinate transformation The inverse transformation is Taking the differentials of U and V and substituting into the line element (285) gives This line element has earlier been studied by Podolsk'y and Ortaggio [17] with a different scaling of the coordinates. Inserting the formulae (294) for U and V into (291), we obtain the transformation between the (˜ u, ˜ v ) coordinates and the CFS coordinates, This transformation corresponds to the transformation immediately preceding (A1) in the appendix of reference [17], but with a different scaling of the coordinates. The inverse transformation is found by inserting the exressions for U and V in (292) into equation (293), giving", "pages": [ 41, 42, 43, 44 ] }, { "title": "5. A Milne-LBR universe model", "content": "We consider the flat spacetime inside the domain wall. In the ( η ' , χ ' )-system the line element is given by (236). Introducing the proper time τ ' of the reference particles in the ( η ' , χ ' )-system as a time coordinate we have Integrating with the initial condition τ ' (0) = 0, we obtain and the line element of the flat spacetime inside the domain wall takes the form where the scale factor is This line element represents the Milne spacetime, which is simply the flat Minkowski spacetime as described from a uniformly expanding reference frame. The coordinates ( τ ' , χ ' ) will here be called the Milne coordinates. The transformation between the CFS coordinates and the Milne coordinates is obtained from equation (233) with the substitution Be η ' = τ ' , giving The inverse transformation is In these coordinates the line element of the WLBR spacetime outside the domain wall takes the form It follows from the transformation (303) that the world lines of particles with χ ' = constant are straight lines both inside and outside the domain wall as illustrated in Figure 9. Imagine observers with constant value of χ ' . The coordinate time τ ' both inside and outside the domain wall is equal to the proper time of these observers. Note from the form of the line element (304) that τ ' is not equal to the proper time of standard clocks with χ ' = constant outside the domain wall. The rate of the proper time of these clocks is given by This formula shows that a standard clock outside the domain wall with χ ' = constant goes at an increasingly slower rate than a standard clock inside the domain wall. The reason is that the clocks with χ ' = constant move in the outwards direction in the static CFS system. This does not change the rate of the clocks inside the domain wall because they have constant velocity and there is no gravitational field in this region. However, outside the domain wall there is an outwards directed gravitational field. Hence a clock with constant χ ' comes lower in this field, and therefore its rate decreases. We will now consider clocks in the WLBR region with a fixed physical distance from the domain wall, R = R 1 . It follows from the transformation (302) that the χ ' coordinate of this clock is given by τ ' sinh χ ' = R 1 . Hence equation (305) shows that these clocks go at a constant rate. Observers comoving with the reference particles inside the domain wall, χ ' = constant, will observe that the domain wall collapses towards them. The physical distance from an observer at the origin to an object with coordinate χ ' is The physical velocity of the object relative to the observer is where H = ˙ a/a is the Hubble parameter. The first term is the velocity of the Hubble flow as given by Hubble's law, i.e. in the present case the velocity of 'the river of space' [28] in the Milne universe, and the second term represents the so-called peculiar velocity of the object, i.e. its velocity through space. The physical velocity of the domain wall is found by inserting R = R Q in the second of the transformation equations (302), which gives Hence the coordinate velocity of the domain wall is and its physical velocity is Surprisingly the domain wall has a non-vanishing physical velocity in the Milne universe inside the wall which is even infinitely great initially, and then decreases to zero in an infinitely far future. Hence the Hubble flow dominates over the peculiar motion all the time. Integrating with the initial condition l (0) = 0 we find the physical distance from the observer at the center to the domain wall, The chosen initial condition is necessary in order to obtain a result in accordance with the expression for l in equation (306). Taking the limit when τ ' → ∞ we find that the final distance of the domain wall from the observer at the center is l = R Q . The physical velocity of the domain wall in the CFS system inside the wall vanishes. Hence, as described by an observer at the center of these coordinates, which coincides with the center of the Milne coordinates, the domain wall is at rest. Since we talk about physical velocity and physical distance one might think that these quantities should be coordinate invariant. The reason that this is not so, is that the spaces of the CFS system and the Milne universe are different simultaneity spaces.", "pages": [ 44, 45, 46 ] }, { "title": "6. The Killing vector field defining the motion of the reference frames", "content": "The LBR spacetime has a timelike Killing vector field which is most easily seen in the coordinate systems in which the metric is static. Then the timelike coordinate basis vector is a timelike Killing vector [29]. From the line element (36) it follows that K = e η = ∂/∂η is a Killing vector. In order to make it explicit that there are three different cases we define unit vectors in the direction of K by These timelike unit vectors V can be interpreted as the 4-velocity of reference particles following trajectories of the Killing vector field K , i.e. it is the 4-velocity of the reference particles defining the reference frame in which the ( η, χ )-coordinates are comoving. The vectors V given in equation (312) for k = -1 , 0 , 1 may be distinguished by the magnitude of the 4-accelerations of the particles. The 4-acceleration of a particle with a world line having V as a unit tangent vector is Since the only non-vanishing component of V is V η , this expression reduces to Using the expression (57) for the Christoffel symbol we obtain The square of the acceleration scalar of this reference particle is The physical meaning of the acceleration scalar of an arbitrary particle is that it represents the ordinary acceleration of the particle as measured with standard clocks and measuring rods relative to a local inertial frame in which the particle is instantaneously at rest. In other words it represents the acceleration of the particle relative to a free particle. This is called the proper acceleration of the particle and will here be denoted by A k for reasons that will be apparent below. It follows that the acceleration of gravity as defined in equation (33) is equal to minus the proper acceleration of the reference particles defining the motion of a reference frame. The acceleration of a free particle instantaneously at rest in the ( η, χ )-system is given by equation (58). This acceleration is due to the non-inertial character of the reference frame. Hence the proper acceleration of a reference particle in the ( η, χ )-system is given by for k = 1 , 0 , -1. In the ( ˜ t, ˜ r )-system the Killing vector is K = ( R Q /A ) e ˜ t = ( R Q /A ) ∂/∂ ˜ t . Using the transformation (116) and the formula (A.13) we find that the proper acceleration of the reference particle is given by This expression has earlier been deduced by Lapedes [21] with ˜ r 0 = 0 and A = 1 /R Q . We shall here use equation (318) to discuss the motion of the reference frame in which the ( η, χ ), ( ˜ t, ˜ r ) and ( ˆ t, ˆ r ) coordinate systems are comoving in the WLBR spacetime. The equation shows how the reference particles move in the radial direction. We first consider the case k = 0. Equation (134) implies that ˜ r < ˜ r 0 -R Q in the WLBR spacetime. Hence ˜ r < ˜ r 0 in this region. Equation (318) shows that in this case the reference particles have a constant acceleration A 0 = -1 /R Q which is directed towards the domain wall relative to a free particle, with just the magnitude that keeps it at rest relative to the domain wall. We then consider the case k = 1. At ˜ r = ˜ r 0 the reference particles have vanishing proper acceleration, i.e. they are freely falling. When ˜ r < ˜ r 0 in the region given by the inequalities (121) we then have -1 /R Q ≤ A 1 < 0. This means that a reference particle in this region accelerates away from the domain wall, but with a smaller acceleration than that of a free particle. Hence in this region the reference frame accelerates inwards relative to a local inertial frame. When ˜ r > ˜ r 0 in the region given by the inequalities (121) we have that 0 < A 1 ≤ 1 /R Q , and the reference frame accelerates outwards relative to a local inertial reference frame. Finally we consider the case k = -1. In this case the proper acceleration of the reference particles is directed towards the domain wall and has a magnitude greater than 1 /R Q , i.e. greater than that of a free particle. Thus the reference frame accelerates towards the domain wall. The proper acceleration of the reference particles depends upon k in the following way This gives a physical meaning of the constant k . It tells whether the magnitude of the proper acceleration of the reference particles is smaller than, equal to, or greater than that of a free particle. If k = 1 the reference frame accelerates away from the domain wall, if k = 0 it is at rest relative to the domain wall, and if k = -1 it accelerates towards the domain wall. Finally, in the ( ˆ t, ˆ r )-system the Killing vector is K = ( R Q /A ) e ˆ t = ( R Q /A ) ∂/∂ ˆ t . From equation (317) and the transformation (156) we get which is consistent with the expression (174) for the acceleration of gravity in the ( ˆ t, ˆ r )-system.", "pages": [ 46, 47, 48 ] }, { "title": "7. Embedding of the LBR spacetime in a flat six-dimensional manifold", "content": "In order to exhibit the topological structure of the LBR spacetime Dias and Lemos [18] considered the embedding of the LBR spacetime in a flat six-dimensional manifold M 2 , 4 . We shall here show how LBR spacetime is parametrized in M 2 , 4 in the six main coordinate systems that we have considered in this paper. The coordinates in M 2 , 4 are denoted by ( z 0 , z 1 , z 2 , z 3 , z 4 , z 5 ), and for k = ± 1 the line element of M 2 , 4 has the form Note that z 1 and z 2 are exchanged when k changes sign. The LBR 4-submanifold is determined by the two constraints The first of these constraints defines the AdS 2 hyperboloid, and the second defines the 2-sphere of radius R Q . From equation (1) it follows that the spherical part of the LBR submanifold is invariant. Hence the parametrization of the 2-sphere takes the same form in all the coordinate systems, This satisfies the constraint (323) and gives the last three terms of equation (321). We shall now consider the different parametrizations of the AdS hyperboloid satisfying the constraint (322) and giving the first three terms at the right hand side of equation (321) using the coordinate systems mentioned above. In CFS coordinates the parametrization takes the form giving Note that the cases k = 1 and k = -1 give the same parametrization, but with the coordinates z 1 and z 2 exchanged. A special case of this parametrization has earlier been considered by O. B. Zaslavskii [30]. From the transformation (41) it follows that with the ( η, χ )-coordinates the parametrization of the AdS hyperboloid in M 2 , 4 takes the form Using equations (A.11), (A.13), (A.32) and (A.33) one may show that this parametrization fullfills equation (321) and the constraint (322). Using the transformation (88) we find that the parametrization that transforms between the line element (82) with ( τ, ρ )-coordinates and the first three terms of (321) with k = -1 is With the ( ˜ t, ˜ r )-coordinates equation (132) gives From equation (148) it follows that the parametrization with the ( t, r )-coordinates has the form corresponding to k = -1 in equations (321) and (322). With ( ˆ t, ˆ r )-coordinates equation (169) leads to the parametrization In order to show that this parametrization fullfills the constraint (322), one has to use equation (A.35). Equation (193) leads to the following parametrization in ( t, r )-coordinates, again corresponding to k = -1 in equations (321) and (322). The parametrization of the AdS hyperboloid in ( η ' , χ ' )-coordinates takes the form With ( ˜ t ' , ˜ r ' )-coordinates the parametrization of the AdS hyperboloid is With ( ˆ t ' , ˆ r ' )-coordinates the parametrization is We shall now consider the case k = 0. Then the line element of M 2 , 4 has the form The LBR 4-submanifold is determined by the constraint (323) and The ( η, χ )-system with k = 0 coincides with the CFS system, and the line element takes the form (28). In this case the parametrization of the AdS hyperboloid is given by (325) with T = η , R = χ and k = -1, The reason for inserting k = -1 instead of k = 0 is that the case k = 0 concerns the type of coordinate system which we consider, while the k = -1 value in equation (325) concerns the parametrization. These parametrizations of the AdS hyperboloid in the M 2 , 4 manifold makes it clear that the line elements (28), (36), (82), (114), (140), (165) and (189) describe the same LBR spacetime. With (ˆ η, ˆ χ )-coordinates the parametrization of the AdS hyperboloid is From the embedding parametrization (327) and equations (289) and (290) we obtain the following embedding parametrization of the LBR spacetime in the coordinates introduced in section 4.VII, corresponding to k = -1 in equations (321) and (322). This is in agreement with the embedding parametrization of the LBR spacetime used by Ortaggio and Podolsk'y [27] with a different scaling of the coordinates.", "pages": [ 48, 49, 50, 51 ] }, { "title": "8. Conclusion", "content": "The LBR solution of Einstein's field equations was found more than 90 years ago by T. Levi-Civita [4,5] and rediscovered in 1959 by B. Bertotti [6] and I. Robinson [7]. The solution was interpreted physically as a spacetime with an electric or a magnetic field with constant field strength. However the source of the electrical field remained rather obscure. We recently used Israel's formalism [31] for describing singular shells in general relativity to investigate the physical properties of a shell with LBR spacetime outside the shell and flat spacetime inside it, and found [14] that the source then had to be a charged domain wall with a radius equal to the distance corresponding to its charge. From equation (44) in reference [14] we see that the radius of the shell is one half of its Schwarzschild radius. We have found different coordinate representations of the LBR spacetime by taking a general form (1) of a spherically symmetric line element as our point of departure, permitting the metric functions to depend upon the radial and the time coordinate. The differential equation (3) obtained from the requirement that the spacetime is conformally flat, i.e. that the Weyl curvature tensor vanishes, was then solved under different coordinate conditions. Remarkably, with the general form (1) of the line element and the requirement that the Weyl tensor vanishes, Einstein's field equations restrict the energymomentum tensor to be of a form (4) representing a constant electric or magnetic field. In the present article we have only discussed the case of an electric field. Next we have given a general prescription for finding coordinate transformations between the 'canonical' CFS coordinate system in which the line element of the LBR spacetime is equal to a conformal factor times the Minkowski line element, and the coordinate representations obtained by the method based on solving equation (1). In sections 4 and 6 of this article we have given a detailed discussion of the kinematical properties of the reference frames both outside and inside the domain wall, in which the coordinate systems are comoving. We have found that in several coordinate systems there are three cases which we have parameterized by the constant k having the values 1, 0 or -1. The corresponding reference frames have different motions. In the case k = 0 the ( η, χ ) coordinate system is comoving in the same referenc frame as that of the CFS coordinates. The domain wall at R = R Q of the WLBR spacetime is static in this reference frame, and the acceleration of gravity is constant and equal to 1 /R Q . In the case k = 1 the ( η, χ ) coordinate system is comoving with a reference frame that accelerates away from the domain wall in the WLBR spacetime. Then the acceleration of gravity is smaller that that in the static case ( k = 0), and even directed towards the domain wall for R > √ B 2 + T 2 . In the case k = -1 the ( η, χ ) coordinate system is comoving with a reference frame that accelerates towards the domain wall in the WLBR spacetime. Hence observers in this reference frame will experience an acceleration of gravity directed away from the domain wall larger than 1 /R Q . In section 5 we have presented a Milne-LBR universe model with a part of the Milne universe inside the domain wall and an infinitely extended LBR spacetime outside it. Finally we have considered embedding of the LBR spacetime in a flat, 6-dimensional manifold, M 2 , 4 , and deduced the parameterizations of this embedding for the main coordinate systems considered in the present article.", "pages": [ 51, 52 ] }, { "title": "Appendix A. Calculus of k-functions", "content": "In this appendix we shall define functions which we call k-functions and deduce their main properties. Motivated by the angular part of the Robertson-Walker line element in standard coordinates it is natural to introduce the function In the present paper we shall need several functions of similar type defined by and Motivated by the scale factor of the DeSitter line element we also introduce and Note that The series expansions for the function S k ( x ) and C k ( x ) are and Furthermore and which implies that and We have the following addition formulae and With y = x this gives and From equations (A.16) and (A.17) we also obtain and Furthermore and T k ( - x ) = - T k ( x ) , I k ( - x ) = - I k ( x ) . (A.25) Combining equations (A.14) - (A.17) we also have that and Using (A.10), (A.18) and (A.19) we obtain The derivatives of the k-functions are and The following identities will also be needed From the definition (A.5) it follows that and", "pages": [ 52, 53, 54, 55 ] }, { "title": "Appendix B. From generating functions to transformations", "content": "We shall here show how the transformation (39) is deduced from the generating function (37). From equation (7) with g = f , x 0 = η and x 1 = χ and using equations (A.10), (A.26) and (A.28) it follows that In the same way we find which gives the transformation (39). Next we show how the transformation (41) is deduced from the generating function (40). Using this generating function and equation (7), replacing T by η , x 0 by T and x 1 by R , we obtain From equation (A.22) it then follows that Hence we obtain the first of equations (41). The second is found in the same way. We shall now deduce the transformation (86) between the ( τ, ρ )-koordinates and the CFS coordinates. From equation (7) with x 0 = τ and x 1 = ρ and using the generating functions (85), it follows that which gives the transformation (86). Next we show how the transformation (88) is deduced from the generating functions (87). Using these generating functions and equation (7), replacing T by τ , x 0 by T and x 1 by R , we obtain Hence we find Hence we obtain the first of equations (88). The second equation is found in the same way. We shall now deduce the transformation (97) between the ( τ, ρ )- and the ( η, χ )-coordinates. From equation (7) with x 0 = τ and x 1 = ρ and using the generating functions (96), it follows that which may be rewritten as Multiplication and using the addition formulae for hyperbolic functions give Hence we obtain the first of equations (97). The second equation is found in a similar way.", "pages": [ 55, 56 ] }, { "title": "Acknowledgement", "content": "We would like to thank Marcello Ortaggio for providing us with the references 16, 17, 26 and 27. In the same way we find", "pages": [ 56 ] } ]

[ { "title": "Red giant seismology: observations", "content": "Benoˆıt Mosser 1 , a LESIA, CNRS, Universit'e Pierre et Marie Curie, Universit'e Denis Diderot, Observatoire de Paris, 92195 Meudon cedex, France Abstract. The CoRoT and Kepler missions provide us with thousands of red-giant light curves that allow a very precise asteroseismic study of these objects. Before CoRoT and Kepler, the red-giant oscillation patterns remained obscure. Now, these spectra are much more clear and unveil many crucial interior structure properties. For thousands of red giants, we can derive from the seismic data precise estimates of the stellar mass and radius, the evolutionary status of the giants (with a clear di ff erence between clump and RGB stars), the internal di ff erential rotation, the mass loss, the distance of the stars... Analysing this mass of information is made easy by the identification of the largely homologous red-giant oscillation patterns. For the first time, both pressure and mixed mode oscillation patterns can be precisely depicted. The mixed-mode analysis allows us, for instance, to probe directly the stellar core. Fine details completing the red-giant oscillation pattern then provide further information for a more detailed view on the interior structure, including di ff erential rotation.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The CNES CoRoT mission (Michel et al., 2008) and the NASA Kepler mission (Borucki et al., 2010) have opened a new era in red giant asteroseismology (De Ridder et al., 2009), with thousands of highprecision photometric light curves. This amount of data has motivated collaborative working in dedicated working groups, in an organisation very profitable for promoting e ffi cient work and impressive results (e.g. Hekker et al., 2009; Bedding et al., 2010; Huber et al., 2010; Mosser et al., 2010). This paper has benefitted from all this work. Before space-borne observation, ground-based observations have revealed that red giants, with an outer convective envelope, show solar-like oscillations (e.g. Frandsen et al., 2002). Owing to their low gravity, oscillations in red giants are excited at low frequency. Owing to their low mean density, their oscillation pattern show frequency di ff erences at low frequency, with the so-called large separation of the order of a few microhertz. Limitations due to both too short observing runs (even if the longest lasted about two months) and a poor duty cycle have hampered a rich output of these ground-based observations but raised crucial questions concerning the degree of the observed modes and the mode lifetimes. Frandsen et al. (2002) explicitly state that 'a most important and exciting result of [their] study is the confirmation of the possibility, suggested by the results reported on α UMa and Arcturus, to observe solar-like oscillations in stars on the red giant branch'. These questions were not answered by observations with the microsatellite MOST (e.g. Barban et al., 2007), with time series limited to one month. However, the pioneering role of these observations was highly valuable, so that red giants were considered as valuable asteroseimic targets. Without them, both CoRoT and Kepler should have missed an impressive harvest. In Section 2, we first present results obtained when considering global seismic parameters only. Such parameters allow us to perform ensemble asteroseismology. The tools for identifying the individual frequencies are then presented in Section 3. The identification of the dipole mixed-mode pattern, not as easily identifiable as the radial pressure mode pattern, is developed in Section 4. These mixed modes unveil unique properties of the core. Open questions and upcoming work are presented in Section 5. An introduction to red giant seismology, by Christensen-Dalsgaard (2011) and Montalb'an et al. (2012) for theoretical aspects or by Bedding (2011) in an observational perspective, can be useful for setting the scene.", "pages": [ 1, 2 ] }, { "title": "2 Scaling relations", "content": "Alarge amount of scaling relations have been recently derived in asteroseismology (e.g. Hekker et al., 2009; Kallinger et al., 2010; Huber et al., 2011). Such relations are based on global seismic parameters used to sum up the mean properties of a solar-like oscillation spectrum. They allow us to perform ensemble asteroseismology, since they monitor the evolution of various parameters for a large population of stars.", "pages": [ 2 ] }, { "title": "2.1 Global seismic parameters", "content": "Most scaling relations involve the large separation ∆ν and / or the frequency ν max corresponding to the maximum oscillation signal. The determination of these global seismic parameters can be done diversely (e.g. Hekker et al., 2011). Here, we use essentially the data analysis provided by the method of Mosser & Appourchaux (2009), called envelope autocorrelation function (EACF). Deriving the large separation from the autocorrelation of the time series is physically e ffi cient, since it corresponds to measure the delay between any oscillation signal first seen directly, then after propagation throughout the stellar diameter back and forth. Achieving this autocorrelation of the time series of the oscillation signal by computing the Fourier transform of its Fourier transform of the oscillation signal is computationally very e ffi cient. Considering a windowing of the spectrum, as proposed by Roxburgh (2009), allows us to select a given frequency range, or to investigate the variation of the frequency separations with frequency, or to study independently the frequency separations of the even and odd ridge (Mosser, 2010). With a filter width corresponding to the frequency range around ν max where solar-like oscillations are excited, the method provides the mean value of the observed large separation. Last but not least, the methodology used by the EACF method provides a test for determining the reliability of the detection, based on the H0 hypothesis. Scaling relation in asteroseismology is an old story, when Eddington (1917) noted that the pulsation of cepheids are related to their mean density. This can be expressed by the scaling relation where ∆ν is the mean large separation, and M and R are the stellar mass and radius. ∆ν is usually defined as the mean frequency di ff erence between consecutive radial modes (Fig. 1). In fact, this definition is misleading: frequency di ff erences yield the observed value of the large separation, which is di ff erent from the asymptotic value that verifies Eq. (1). The link between the large separation and the mean stellar density has been addressed by White et al. (2011) for di ff erent stellar masses and evolutionary stages. The relation between the observed and asymptotic values of the large separation is established by Mosser et al. (2013): with ζ = 0 . 038 (red giant regime: n max ≤ 15) , (4) where n max = ν max /∆ν measures the frequency of maximum of oscillation signal in a dimensionless manner. The relation between ν max and the acoustic cuto ff frequency ν c proposed by Belkacem et al. (2011) introduces the Mach number M in the uppermost convective layers so that ν max ∝ ν c M 3 . The variation of this number with stellar type and evolution is limited but remains unknown.", "pages": [ 2, 3 ] }, { "title": "2.2 Seismic mass and radius", "content": "The importance of the measurements of ∆ν and ν max is emphasized by their ability to provide relevant estimates of the stellar mass and radius The reference value ∆ν ref /similarequal 3106 µ Hz and ν ref /similarequal 138 . 8 µ Hz have been determined by Mosser et al. (2013), relying on the exact use of the second-order asymptotic expression and on the calibration with modeled stars. Unbiased estimated of R and M are provided only if the asymptotic value of the large separation is used. The use of the observed large separation induces significant bias, of the order of 3% for the radius and 6 % for the mass. Even if the calibration e ff ort is not complete, the scaling relation give relevant estimates. Mosser et al. (2013) have shown that the correct use of the scaling relations with the asymptotic large separation provides estimates of R and M with uncertainties of about 4 and 8 %, respectively, for low-mass stars. Uncertainties are twice larger when M ≥ 1 . 3 M /circledot or for red giants.", "pages": [ 3 ] }, { "title": "2.3 Ensemble asteroseismology", "content": "Scaling relations on global parameters allow us to perform ensemble asteroseismology.", "pages": [ 3 ] }, { "title": "3 Frequency pattern", "content": "Any person involved in the data analysis of red giants rapidly gets the impression that all red giant spectra are very similar. This has to be related to the fact that red giants have necessarily very similar interiors. Before evolving into a red giant, the star has undergone the exhaustion of hydrogen in its core, the contraction of its helium core, the separation of the continuously contracting core from the continuously growing envelope, with a thin hydrogen-burning shell at the interface, and the growth of a large convective envelope. All these steps, mostly governed by the properties of the hydrogen-burning shell (equation of state, power supply rate), have erased most of the original characteristics of the stars. After the tip of the RGB, all low-mass red giants pass through the helium flash. As a consequence, they gain a new opportunity to reach almost the same interior structure, as shown by the mass-radius relation of clump stars (Fig. 2). Mosser et al. (2011b) have capitalized this necessary similarity to set up a method for measuring very precisely the large separation and for identifying in an automated way red-giant oscillation spectra. Assuming that these oscillations obey to a universal pattern, they have proposed that the o ff set ε of the asymptotic relation (Tassoul, 1980) is a function of the large separation. They have expressed the second-order term of the asymptotic relation with a quadratic term that relates the curvature of the ridges observed in the 'echelle diagrams: We use here the subscript obs to emphasize the di ff erence with the asymptotic value. The di ff erent d 0 /lscript terms indicate the small spacings of non-radial modes (Mosser et al., 2011b). Mosser et al. (2012b) have shown that α 0 is also function of the large separation. This implies that all curvatures α/lscript depend on ∆ν obs. This method has proven to be e ffi cient for all red giants, with a large separation in the range [0.4 - 40 µ Hz], especially for oscillation spectra recorded with a low signal-to-noise ratio. The univocal relation between ε obs and ∆ν obs, updated by Corsaro et al. (2012), is insured if the large separation is observed in a large frequency range. When determined in a limited frequency range, the small di ff erence of the o ff set ε between RGB and clump stars allows us to determine the evolutionary status of the giant (Kallinger et al., 2012). Mosser et al. (2013) have recently shown that the relation ε ( ∆ν obs) is an aretefact, so that the radial modes of red giants follow the pattern: based on the asymptotic value ∆ν as of the large separation. This equation is fully equivalent to Eq. 7, with the relation between the asymptotic and observed values of the large separation provided by Eq. 2. Departures to such a regular spectrum are due to rapid structure discontinuities. They induce socalled glitches in the oscillation spectrum, as due to the second ionisation of helium (Miglio et al., 2010). Provost et al. (1993) have shown that an asymptotic development can be used for addressing the signature of such discontinuity. However, the red giant oscillation spectrum is also much more complex, due to the presence of other oscillation modes than pure pressure modes.", "pages": [ 4, 5 ] }, { "title": "4.1 Stellar evolution", "content": "Beck et al. (2011) have identified mixed modes in an RGB star. Such mixed modes result from pressure waves propagating in the envelope coupled with gravity waves trapped in the core. Due to the contraction of the inert helium core, the Brunt-Vaisalafrequency reaches much higher values than in main-sequence stars, so that the coupling between the di ff erent waves in the envelope and in the core is e ffi cient (e.g. Montalb'an et al., 2012). This coupling permits the information of gravity modes to percolate to the surface. Hence, Bedding et al. (2011) could show that the mixed-mode frequency separation depends on the evolutionary status of the star and allows us to distinguish helium-burning stars in the red clump from shell hydrogen-burning stars in the RGB. Mosser et al. (2011a) have proposed an alternative method, based on the EACF with narrow filters centered on the dipole modes. These first approaches only deliver the bumped period spacing, significantly perturbed by the coupling of the pressure and gravity waves and quite di ff erent from the period spacing ∆Π 1 of gravity modes.", "pages": [ 5 ] }, { "title": "4.2 Asymptotic development of the mixed mode pattern", "content": "Measuring the period spacing ∆Π 1 is derived from the asymptotic development for mixed modes exposed by Mosser et al. (2012c), based on the method exposed by Unno et al. (1989). Observations of red giant with a large number of dipole mixed modes give rise to this development. The mixed-mode frequencies related to the pure pressure dipole mode of radial order n are solutions of the implicit equation: where ν n ,/lscript = 1 is the pure pressure mode frequency previously determined, q is a dimensionless coupling factor, ∆Π 1 is the period spacing of pure gravity modes and ε g is a constant fixed to 0. For each pressure radial order n , one obtains N + 1 solutions, with N /similarequal ∆ν∆Π -1 1 ν -2 max . The value of ∆Π 1 is derived from a least-squares fit of the observed values to the asymptotic solution. As shown by Mosser et al. (2012c), the observation of high gravity mode orders insures a precise description of the mixed-mode pattern", "pages": [ 5 ] }, { "title": "EPJ Web of Conferences", "content": "Corsaro, E., Stello, D., Huber, D., et al. 2012, ApJ, 757, 190 De Ridder, J., Barban, C., Baudin, F., et al. 2009, Nature, 459, 398 Dziembowski, W. A. & Soszy'nski, I. 2010, A&A, 524, A88 Jiang, C., Jiang, B. W., Christensen-Dalsgaard, J., et al. 2011, ApJ, 742, 120 Unno, W., Osaki, Y., Ando, H., Saio, H., & Shibahashi, H. 1989, Nonradial oscillations of stars, ed. Unno, W., Osaki, Y., Ando, H., Saio, H., & Shibahashi, H. (Tokyo: University of Tokyo Press) White, T. R., Bedding, T. R., Stello, D., et al. 2011, ApJLetters, 742, L3", "pages": [ 10 ] }, { "title": "4.3 Rotational splittings", "content": "Beck et al. (2012) have shown that gravity-dominated mixed modes revealed the core rotation in red giant. They analysed the rotational splittings of three red giant oscillation spectra, in the early stages of the RGB. These splittings reveal a significant di ff erential rotation, with a core rotating at least ten times faster than the surface. Mosser et al. (2012b) have developed a method for analysing rotation splittings in an automated way, based on the EACF function with ultra-narrow filters. This method has provided splittings in more than 260 red giants observed with Kepler . A direct identification of the rotational splittings, provided by the method proposed by Mosser et al. (2012c), was also used for more than 100 red giants (Fig. 4, 5). Under the hypothesis that a linear analysis can provide the mean core rotation from the rotational splittings of the gravity-dominated mixed modes, the evolution of this mean core rotation indicates a significant spin down of the core rotation occurs in red giants. This spin down, observed on the RGB but much more marked for clump stars, requires an significant angular momentum transport between the di ff erent regions of the star.", "pages": [ 6 ] }, { "title": "5 From asteroseismic observations to stellar physics", "content": "The analysis of the thousands of red giant oscillation spectra has just started. The description of these spectra with the combination of the universal red giant oscillation pattern, the asymptotic development of mixed modes and an empirical description of the rotational splittings has proven to be fruitful. As shown by Table 1, four parameters are enough to identify all modes. Refined fits are obtained with eight free parameters, to be compared to the number of fitted modes (in the range 40 - 140) and to the complexity of some spectra (Fig. 5). Undoubtedly, the high-quality asteroseismic constrains, especially those sounding the stellar core, is promoting large progress in stellar physics.", "pages": [ 7 ] }, { "title": "5.1 Standard candles", "content": "The precise asteroseismic constraints on red giants, and especially the precise estimate of the radius from scaling relations, completed with the more precise determination derived from stellar modeling, allows us to use red giants as standard candles (Miglio et al., 2009, 2012). According to Eq. (5), this requires the use of reliabl e ff ective temperatures T e ff , determined from photometry and colourT e ff", "pages": [ 7 ] }, { "title": "5.2 Modeling", "content": "Modeling e ff ort has been achieved for a limited number of red giants with seismic constraints (Carrier et al., 2010; Miglio et al., 2010; Jiang et al., 2011; di Mauro et al., 2011; Baudin et al., 2012). If not based on grid computing, this e ff ort is time consuming, as it allows to address the physical input in the modeling. Then, it makes the best of the seismic constraints. In some stars, the lifetimes of the gravity-dominated mixed modes is so long that it yet exceeds the total duration of the observation run (31 months at the time this article is written), so that the accuracy of the frequency determination is equal to the frequency resolution ( /similarequal 12 nHz), much better than the current performance of modeling (di Mauro & al., 2013). As a consequence, future developments are very promising.", "pages": [ 8 ] }, { "title": "5.3 Low-amplitude dipole mixed modes", "content": "Most red giants spectra show a complex spectrum, with short-lived pressure-dominated and long-lived gravity-dominated mixed modes. A family of red giants shows non-standard spectra, with depressed dipole modes (Mosser et al., 2012a). Such red giants are found at all evolutionary stage from the early RGB to the red clump (Fig. 6). The coupling between the two cavities in the envelope and in the core certainly obeys to specific conditions that govern such a behaviour. Clarifying the situation of these stars will greatly help our understanding of the mixed modes in red giants.", "pages": [ 8 ] }, { "title": "5.4 Upper red giant branch; asymptotic giant branch", "content": "Red giants ascending the RGB or the AGB have such large radii that their oscillation occur at very low frequencies, as shown by the analysis of the upper RGB from OGLE observations (Dziembowski & Soszy'nski, 2010). By extrapolation of the current results, the extension of the Kepler mission can provide us with the observation of giants with large separation as low as 0.20 µ Hz. If the scaling relations are still valid, this corresponds to radii of about 80 R /circledot , maybe not enough for investigating the tip of the RGB at all masses, but useful for combining with OGLE results.", "pages": [ 9 ] }, { "title": "5.5 Differential rotation and angular momentum transport", "content": "The observation of the rotational splittings implies that angular momentum is, as expected, significantly redistributed between the di ff erent regions of the stars. A thorough analysis of this redistribution has just started. This will take time, but we are confident that the new constraints provided by asteroseismic observation will be translated by theoreticians into highly valuable information.", "pages": [ 9 ] }, { "title": "References", "content": "Barban, C., Matthews, J. M., De Ridder, J., et al. 2007, A&A, 468, 1033 Baudin, F., Barban, C., Goupil, M. J., et al. 2012, A&A, 538, A73 Beck, P. G., Bedding, T. R., Mosser, B., et al. 2011, Science, 332, 205 Beck, P. G., Montalban, J., Kallinger, T., et al. 2012, Nature, 481, 55 Bedding, T. R. 2011, ArXiv e-prints 1107.1723 Bedding, T. R., Huber, D., Stello, D., et al. 2010, ApJLetters, 713, L176 Bedding, T. R., Mosser, B., Huber, D., et al. 2011, Nature, 471, 608 Belkacem, K., Goupil, M. J., Dupret, M. A., et al. 2011, A&A, 530, A142 Borucki, W. J., Koch, D., Basri, G., et al. 2010, Science, 327, 977 Carrier, F., De Ridder, J., Baudin, F., et al. 2010, A&A, 509, A73 Christensen-Dalsgaard, J. 2011, ArXiv e-prints 1106.5946", "pages": [ 9 ] } ]

[ { "title": "GAUFRE: a tool for an automated determination of atmospheric parameters from spectroscopy", "content": "M. Valentini 1 ; a , T. Morel 1 , A. Miglio 2 , L. Fossati 3 , and U. Munari 4 Abstract. We present an automated tool for measuring atmospheric parameters (Te GLYPH<11> , log g ), [Fe / H]) for F-G-K dwarf and giant stars. The tool, called GAUFRE, is written in C ++ and composed of several routines: GAUFRE-RV measures radial velocity from spectra via cross-correlation against a synthetic template, GAUFRE-EW measures atmospheric parameters through the classic line-by-line technique and GAUFRE-CHI2 performs a GLYPH<31> 2 fitting to a library of synthetic spectra. A set of F-G-K stars extensively studied in the literature were used as a benchmark for the program: their high signal-to-noise and high resolution spectra were analyzed by using GAUFRE and results were compared with those present in literature. The tool is also implemented in order to perform the spectral analysis after fixing the surface gravity (log g ) to the accurate value provided by asteroseismology. A set of CoRoT stars, belonging to LRc01 and LRa01 fields was used for first testing the performances and the behavior of the program when using the seismic log g .", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Spectroscopy is one of the most powerful tool that astronomy possesses in order to derive atmospheric parameters and abundances of stars. From the stellar spectrum it is possible to measure the e GLYPH<11> ective temperature (Te GLYPH<11> ), the surface gravity (log g ) and the abundance of iron ([Fe / H]) and of several other elements. The classic method consists in measuring the equivalent widths (EW) of a species in two di GLYPH<11> erent ionization states, usually FeI and FeII. By imposing excitation and ionization equilibrium through stellar atmosphere models, it is possible to derive Te GLYPH<11> , log g and to infer elemental abundances from the curves of growth. This method is precise and it is widely adopted [1]. In spite of the precision, the line-by-line classical analysis is time consuming: usually EWs are measured by hand using the IRAF 'splot' routine (where the choice of the continuum and the position of the line is completely manual) and the procedure for finding the best Te GLYPH<11> and log g requires a large number of iterations. The growing amount of stellar data due to large surveys (i.e. RAVE, SEGUE, LAMOST, HERMES and Gaia ESO) and the availability of dedicated telescopes and multi-object spectrograph, requires the development of automated pipelines, methods and programs that fasten the analysis process. For example, DAOSPEC [2] and ARES [3] are useful and free codes that performs an automated EW measurement; the MOOG code [4] is a valid collection of routines useful for determining atmospheric parameters and abundances. In particular, MOOG abfind and synth tasks are widely used in literature [5] [1]. Another method for the estimation of atmospheric parameters and elemental abundances is to compute", "pages": [ 1 ] }, { "title": "EPJ Web of Conferences", "content": "a set of synthetic spectra and to find the best match between the synthetic and the observed spectrum. This method is adopted by surveys as RAVE [6] and HERMES [7] as well as in small surveys as ARCS [8] [9]. In this paper we present a new automatic code, GAUFRE, that can perform both type of analysis. In section 2 we give a short description of the idea behind the program and the two types of analysis that it performs. In subsection 2.4 we present the first results of tests using spectra of objects well known in literature. In section 3 we present an additional tool of GAUFRE that consists in using the asteroseismic gravity as a fixed value for log( g ) in order to refine the measurement of Te GLYPH<11> , microturbulence velocity ( GLYPH<24> mic ) and elemental abundances. Section 4 discusses the future perspectives for the code.", "pages": [ 2 ] }, { "title": "2 The GAUFRE code", "content": "GAUFRE is a collection of several C ++ routines. It is written in order to measure fast and precisely radial velocity (Vrad) and atmospheric parameters (Te GLYPH<11> , log g , [Fe / H]) of a star starting from a onedimensional normalized spectrum. The radial velocity is measured with a cross-correlation technique (routine GAUFRE-RV), atmospheric parameters can be measured adopting a GLYPH<31> 2 fitting over a library of synthetic spectra (GAUFRE-CHI2 routine) or with the classic FeI-FeII lines technique (GAUFREEWroutine). GAUFRE is an updated and extended version of the code written for the Asiago Red Clump Spectroscopic survey (ARCS) [8]. GAUFRE was created because of the need of introducing new libraries of synthetic spectra, adapting the code for di GLYPH<11> erent resolutions and implementing the classic line-by-line technique. The code was written in C ++ and does not need any particular library, in order to avoid any software license problems and to be executable in di GLYPH<11> erent platforms. The only additional program needed for running GAUFRE is MOOG, which can be easily installed and downloaded from its homepage (http: // www.as.utexas.edu / GLYPH<24> chris / moog.html). The user is also supposed to download the libraries of synthetic spectra and model atmospheres required by GAUFRE. So far GAUFRE is tested for F-G-K stars (both giants and dwarfs) and works in the 3500 - 9000 Å spectral range. The program has currently been tested for spectral resolutions of the Asiago Echelle Spectrograph (R = 20,000), ESO-FLAMES-GIRAFFE HR10, HR15N, HR21, HR14 setups (R = 16,000 - 20,000), ESO-FLAMES-UVES U580 setup (R = 52,000), ESO-FEROS (R = 48,000) and AAOmega 2dF 580V and 385R (R = 1,300). Details about the adopted libraries can be found on subsections 2.3 and 2.4.", "pages": [ 2 ] }, { "title": "2.1 Radial Velocities: GAUFRE-RV", "content": "The radial velocity subroutine (GAUFRE-RV) measures the radial velocity Vrad by cross-correlating the observed spectrum with a synthetic spectral library [10]. The procedure is the same as described in [8]. The procedure starts from the continuum normalized spectrum in a 2-column ASCII format. First, the synthetic normalized spectra of the selected library are renormalized, following the same parameters as adopted for the normalization of the observed spectrum (same function, same order and same high and low rejection values). To lower the impact of the di GLYPH<11> erent noise level in the observed and synthetic spectra, they are scaled to match their geometric mean. This procedure is needed in order to improve the accuracy of the cross-correlation and of the GLYPH<31> 2 fitting and it is performed by the GAUFRE-LIB routine (see also section 2.2). The result of the GAUFRE-RV subroutine is a file containing the value of Vrad and an ascii file containing the continuum normalized spectrum corrected for the radial velocity. To validate the GAUFRE-RV routine we measured the radial velocity of a set of Red Giants stars that are IAU standard radial velocity stars. These stars were observed with the Asiago Echelle Spectrograph (INAF-OAPd); we then downloaded, when available, the spectrum from the ESO-Archive. For this test, we adopted the synthetic spectra library provided by L. Fossati (cf. section 2.2). Results are summarized in Table 1. The mean di GLYPH<11> erence between our values and those present in the literature is GLYPH<1> RV GLYPH<12> = 0 : 10 km s GLYPH<0> 1 , with a rms of 0.3 km s GLYPH<0> 1 .", "pages": [ 3 ] }, { "title": "2.2 GLYPH<31> 2 on synthetic spectra libraries: GAUFRE-CHI2", "content": "Atmospheric parameters (Te GLYPH<11> , log g , [Fe / H], [ GLYPH<11> / Fe], Vrot sin i ) are obtained by GAUFRE-CHI2 via GLYPH<31> 2 fitting of the continuum normalized spectrum against a synthetic spectral library. The choice of the spectral library is up to the user, available libraries are: a library based on Kurucz model atmosperes [11], the library provided by L. Fossati (built adopting the spectral synthesis code Synth3 described in [12]) and the AMBRE library [13]. Characteristics of di GLYPH<11> erent libraries are summarized in Table 3. Libraries are provided at di GLYPH<11> erent resolutions and they cover a wide spectral range (usually 3,500 - 10,000 Å). Before starting GLYPH<31> 2 analysis, the desired library must be cut, normalized and degraded at the same wavelength interval, normalization function and resolution of the real spectra. For this purpose a routine GAUFRE-LIB has been created. The degradation of the spectra at the desired resolution is performed through deconvolution with a Gaussian profile.", "pages": [ 3 ] }, { "title": "2.3 EW and MOOG: GAUFRE-EW", "content": "GAUFRE-EW automatically performs the classical line-by-line analysis for deriving atmospheric parameters and abundances. The procedure starts from a continuum normalized spectrum (ASCII 2 columns format), a file containing a list of lines to measure and their parameters (as requested by MOOG) and a file containing the parameters of the spectrum like the wavelength coverage, resolution and, if any, guessed values for Te GLYPH<11> and log g . The EW of every line present in the input file is measured (except when it is not detectable). The program selects an area of 3-4 Å around the wavelength of the line (this parameter is selected by the user). The spectrum is then fitted with a polynomial function in order to determine the continuum and point with the lowest intensity. In order to test the automated EW measurement of the line we compared the EW values obtained by measuring features by hand (using IRAF-splot) and the correspondent EW measured by GAUFRE. The test was performed on the spectrum of Arcturus, taken with the ESO-FLAMES-UVES instrument, with the U580 setup (5770-6825 Å). The agreement is quite good and rms is of GLYPH<24> 3 mÅ (see Figure 1). Atmospheric parameters were computed by using MOOG abfind driver (the program uses its noniinteractive version, MOOGSILENT), the measured EW of FeI and FeII lines and a family of model atmospheres (MARCS [14] or Kurucz [15]). Te GLYPH<11> is calculated by assuming the excitation equilibrium and minimizing the trend of the Fe abundance versus the excitation potential. The surface gravity, log g is derived by assuming the ionization equilibrium: log n (FeI) = log n (FeII). The procedure is iterative and the program will converge to Te GLYPH<11> and log g that satisfy both the ionization and excitation equilibria. The value of the microturbulence GLYPH<24> mic is derived by minimizing the trend of the FeI abundance versus the FeI lines EW.", "pages": [ 3, 4 ] }, { "title": "2.4 Atmospheric parameters validation", "content": "We took spectra of 7 F-G-K stars taken with FLAMES-UVES U580 or FEROS from the ESO-archive (http: // archive.eso.org). We selected spectra of targets very well known in literature: GLYPH<11> Cen A, GLYPH<22> Cas A, GLYPH<12> Vir, Arcturus, GLYPH<22> Leo, GLYPH<24> Hya and GLYPH<13> Sge. As a reference we used an average value of the most recent entries of the PASTEL catalog [3]. We analyzed spectra with both GAUFRE-EW (Kurucz model atmospheres) and GAUFRE-CHI2 (Fossati library of synthetic spectra) and we compared our results with those present in literature. The agreement is quite good as showed in Tab.3.", "pages": [ 4 ] }, { "title": "3 Asteroseismic constraints", "content": "As extensively discussed, for example, in [16], the frequency of maximum power, GLYPH<23> max can be used for deriving a precise estimate of the surface gravity: The precision of the gravity derived from Eq. 1 is generally expected to be below 0.05 dex, thanks to the weak sensitivity of the scaling relation to the assumed Te GLYPH<11> and the high precision usually achieved for the measurement of GLYPH<23> max. The scaling relation has been demonstrated to be reliable [17] [18] [19] [20] and it can be used for refining atmospheric parameters and abundances. As a matter of fact one can improve the spectroscopic analysis by fixing the log g to the seismic value: this largely increases the accuracy of the derived T e GLYPH<11> , [Fe / H] and hence chemical abundances. Aset of spectra of 111 RG stars belonging to the LRc01 and LRa01 fields of CoRoT have been used as benchmark for testing the log g values derived by GAUFRE. Spectra are taken with the ESO-FLAMES GIRAFFE 9 setup, centered in the MgI triplet (5278 Å). Spectra have already been analyzed [21] by using MATISSE tool [22]. First, we compared log g values of GAUFRE-CHI2 (using the synthetic library provided by L. Fossati, see Tab3) with those provided by asteroseismology (see Fig2, top panels), then we compared the values of Te GLYPH<11> and [Fe / H] derived by fixing log g to the seismic value (see Fig2, middle and bottom panels). The set of spectra taken from the work of Gazzano has very poor SNR (see Fig2), of GLYPH<24> 30 on average. In addiction the spectral range covered by spectra is a GLYPH<11> ected by the presence of MgH molecular bands. This complicates even more the continuum normalization, leading to several systematics in the atmospheric parameters determination. As shown in Fig. 2, the seismic gravity can be used to enhance the accuracy of the atmospheric parameters. In the case of Gazzano et al. dataset [21] with a poor SNR, the seismic gravity helped in providing a more precise determination of Te GLYPH<11> and [M / H] reducing average errors from 120 K and 0.20 dex to 75 K and 0.11 dex respectively.", "pages": [ 5 ] }, { "title": "4 Conclusions", "content": "We present the GAUFRE program, a versatile tool developed for measuring radial velocities and atmospheric parameters from optical spectra. The program is composed by di GLYPH<11> erent routines: GAUFRE-RV, GAUFRE-CHI2, GAUFRE-EW. We performed some preliminary tests in order to check the performances of the tool. These tests show that the program is reliable and that it can be used to process spectra of F-G-K dwarfs and giants with a SNR above 40. At the moment it is adopted by the Li'ege node within the Gaia-ESO Survey (PI: G. Gilmore and S. Randich) and further applications are planned. We also showed an interesting and useful extension of GAUFRE that uses, when avaiable, the seismic log g as a fixed value for the surface gravity: the precise and also likely more accurate values given by asteroseismology allow us to greatly refine the values of Te GLYPH<11> and [Fe / H]. The GAUFRE tool is continously in development: we plan to implement new libraries of synthetic spectra and to extend the analysis to the infrared region. New and detailed tests are planned as well, in order to better investigate the performances of GAUFRE at di GLYPH<11> erent SNR. In the next future we plan to make the GAUFRE code avaiable through the web and to create an userfriendly graphical interface. MV aknowledges financial support from Belspo for contract PRODEX COROT. TM acknowledges financial support from Belspo for contract PRODEX GAIA-DPAC. Paper based on observations collected at Asiago Observatory and from data recovered from the ESO Archive.", "pages": [ 5, 6 ] }, { "title": "Conference Title, to be filled", "content": "18. D. Stello, D. Huber, T. Kallinger, S. Basu, B. Mosser, S. Hekker, S. Mathur, R.A. Garc'ıa, T.R. Bedding, H. Kjeldsen et al., ApJ 737 , L10 (2011), 1107.0490", "pages": [ 7 ] } ]

[ { "title": "Extensive Air Showers: from the muonic smoking guns to the hadronic backbone", "content": "L. Cazon 1 1 LIP, Av. Elias Garcia 14-1, 1000 Lisboa June 22, 2021", "pages": [ 1 ] }, { "title": "Abstract", "content": "Extensive Air Showers are complex macroscopic objects initiated by single ultra-high energy particles. They are the result of millions of high energy reactions in the atmosphere and can be described as the superposition of hadronic and electromagnetic cascades. The hadronic cascade is the air shower backbone, and it is mainly made of pions. Decays of neutral pions initiate electromagnetic cascades, while the decays of charged pions produce muons which leave the hadronic core and travel many kilometers almost unaffected. Muons are smoking guns of the hadronic cascade: the energy, transverse momentum, spatial distribution and depth of production are key to reconstruct the history of the air shower. In this work, we overview the phenomenology of muons on the air shower and its relation to the hadronic cascade. We briefly review the experimental efforts to analyze muons within air showers and discuss possible paths to use this information.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Our understanding of high energy physics is supported by experiments up to the TeV scale. Beyond such high energy frontier we must rely on extrapolations of our theories in terrains which might hide unexpected phenomena. The only direct processes surpassing these high energies and which might represent a challenge in particle physics are the reactions initiated by Ultra High Energy Cosmic Rays (UHECR) high up in the atmosphere. They reach up to ∼ 10 20 eV in the lab system, which corresponds to a center of mass energy of about ∼ 400 TeV. The origin and nature of UHECR remains a mystery. Our current understanding says that the vast majority of these particles are hadronic (atomic nuclei [1, 2]), excluding neutrinos [3] and photons [4]. The final solution to the UHECR puzzle must put together different pieces: the astrophysical mechanisms that allow the acceleration to such gigantic energies, the propagation through the intergalactic space filled with magnetic fields, and last, the subject of this paper, the interaction with the Earth's atmosphere, which creates Extensive Air Showers (EAS). EAS encode the information of the primary among millions of secondaries by means of high energy interactions which lie on kinetic regions never accessed by experiments before. Among all secondaries, muons can travel many kilometers from the hadronic backbone almost unaffected, carrying valuable information. Understanding this information is key to break the degeneracy between the uncertainties on the extrapolation of the hadronic interaction models to the highest energies and the composition of the UHECR beam . This paper is organized as follows: In section 2 we overview certain aspects of air showers and discuss the energy balance between the hadronic and elec- tromagnetic cascade. In section 3 we discuss how muons are produced in the hadronic cascade inheriting valuable information from it. In section 4 we illustrate the ground distributions of muons. In section 5 we briefly discuss the experimental efforts to use this information. In section 6 we conclude.", "pages": [ 1, 2 ] }, { "title": "2 Extensive Air Shower dynamics", "content": "Extensive Air Showers are complex phenomenon initiated by a single particle with an enormous energy. The collision with an air nucleus generates typically thousand of secondaries, which can interact again, creating a multiplicative process which is referred as cascade, and that can reach up to 10 11 particles at ground level for 10 20 eV showers. Depending on the kind of particles driving the multiplicative process, there are two main subtypes of cascades. The ones initiated and driven by photons or electrons, and the ones originated and driven by hadrons. The study of the cascade can be done by means of the cascade equations, assuming some simplifications, or by means of full Monte Carlo simulations that include many important details difficult to account for otherwise. On the other hand, Heitler models offer a simplified version of the main multiplicative process of a cascade and serves to qualitatively understand the most important features, giving approximated values for relevant variables of the cascade. See for instance [5] for more details on the hadronic and EM cascade.", "pages": [ 2 ] }, { "title": "2.1 The electromagnetic and the hadronic cascades", "content": "When a high energy photon is injected into matter, the most likely process to occur is an electronpositron pair production. Each of the new particles suffers bremsstrahlung, producing new photons. This multiplicative process repeats itself n times originating the so called electromagnetic (EM) cascade. The total number of particles grows as 2 n . The energy of secondaries decreases as E = E 0 2 n to eventually reach the so called critical energy ( E c ∼ 80 MeV) at which electrons are more likely to lose their energy through ionization. At this point the cascade reaches the maximum. After that, the multiplicative process stops, and the number of particles declines. The EM cascade practically keeps all the energy flowing within the EM channel, and does not leak into the hadronic cascade except for a small fraction by photopion production. On a hadronic reaction at high energies, ∼ 80% of the produced particles are pions, ( π + , π -and π 0 ) in a ∼ 1:1:1 ratio, and ∼ 8% are kaons, ( K 0 L , K 0 S , K + and K -) also with a ∼ 1:1:1:1 ratio. Neutrons/protons are produced with an overall probability of ∼ 4-5%, and the rest is shared among other particles at the subpercent level as given by QGSJET-II.03 [6][7]. Neutral pions feed the EM cascade almost immediately, whereas charged pions either interact, sustaining the hadronic cascade, or decay into muons (99.988%). In the same way, kaons interact feeding the hadronic cascade untill they reach their critical energy, which is of the same order of magnitude compared to pions. K 0 S has a shorter lifetime ( cτ = 4 cm) compared to the rest of kaons ( cτ ∼ few meters), which implies a higher probability of decay before interacting. A few hadronic generations after the first interaction, K 0 S decays, 31% of the times into π 0 π 0 , and 69% into π + π -. This means that kaons go from a ∼ 0% contribution to the EM cascade in the first generations up to ∼ 8% in higher generations, compared to the steady ∼ 33% contribution of pions to the EM cascade. Finally, neutrons and protons keep interacting hadronically with no direct feeding into the EM cascade. The most relevant features of hadronic shower can also be approximately described by a Heitler model. After each hadronic generation n , there are created m particles which subdivide in two main categories: those which continue to feed the hadronic cascade, and those which feed the EM cascade, leaving the hadronic channel. They typically correspond to charged and neutral pions, in a 2/3 m and 1/3 m proportion. Thus, total number grows with the hadronic generation as (2 / 3 m ) n whereas the energy decreases as E π = E 0 m n . The energy fraction f carried by the sum of all charged pions in generation n to the total shower energy E 0 is That is, in each generation, the energy carried by charged pions ∑ E π is reduced by a factor 2 3 . In a more realistic approach, we can include an effective factor κ ∈ [0 , 1] that modifies the amount of energy flowing to the EM cascade through π 0 decay as: κ can account for different aspects of the hadronic reactions. For instance, if a leading baryon takes (1 -κ ) E 0 , κ accounts for the inelasticity, being the fraction of energy going into pion production, and therefore 1 3 κE 0 goes into the EM channel, as explained in [5]. There might be other mechanisms that could effectively reduce the feeding to the EM channel, for instance, increasing the amount of kaon production [6].", "pages": [ 2, 3 ] }, { "title": "2.2 The energy balance between cascades", "content": "The energy share between both cascades evolves with the hadronic generation as showed in Fig. 1. In the beginning all the energy is in the hadronic sector. After 3 generations, ( κ = 1) 70% of the energy has been transfered to the EM sector. This means that the evolution of the EM cascade is rapidly decoupled from the hadronic cascade. Also shown is the case for κ = 0 . 5, where the transfer from the hadronic to the electromagnetic cascade is slower. The energy balance affects the longitudinal developement and the muon content of the shower, see for instance [6]. The factor κ can change with the energy of the hadronic reaction, and thus change with the generation n . The energy at which the first and second generation reactions occur might be out of reach of the current man made accelerators. Fig. 1 also shows a case where κ changes from a value κ = 0 . 2 to κ = 1 after the first generation. It can be seen how the energy balance of the whole shower is affected.", "pages": [ 3 ] }, { "title": "3 The production of muons in EAS", "content": "Most muons in the shower come from the decay of pions, which are 10 times more numerous than kaons. Kaon decay can lead directly to muons (20%) or to charged pions (40%). Simple kinematics shows that the maximum transverse momentum p t that muons can obtain is just the center of mass momentum of the outgoing particles, which is 29.8 MeV. Given that the total momentum of the parent particles is of the order of a few tens of GeV, the direction of motion hardly varies, with deviation angle ∆ θ πµ ∼ 0 . 01 · . The experimental data of hadronic collisions available up to a few hundreds of GeV per nucleon in the center of mass show a p t distribution that decreases exponentially dN 2 πp t dp t ∝ exp( -pt Q ) where Q changes slowly with the energy of the collision and the rapidity region. Q is of the order of tenths of GeV/c, that compared to the muon maximum transverse momentum available from the pion decay ( ∼ 0.03 GeV) gives a 10% correction. This makes the p t distribution of the outgoing muons very similar to that of their parents. This is a very important feature responsible for many of the observed characteristics of the hadronic and muonic showers. In [8], it was argued that the transverse position of the production of muons, thus of the parent mesons decay, is confined to a relatively narrow cylinder: as the angle with respect to the shower axis goes as sin α = cp t E , the average traveled distance before the pion decay is l = E m π c 2 cτ π , where τ π and m π are the lifetime and mass of the charged pions. The perpendicular distance to the shower axis of the pion decay is r π = l sin α = τ π 2 Q/m π ∼ 22 m. Note that after each interaction n , the p t increases as p t ∼ Q √ n . The outgoing angle goes as a geometrical progression with n as α i glyph[similarequal] sin α n = Q √ N (3 / 2 N ch ) n . The total outgoing angle ∑ n i =1 α i is then dominated by the last interaction Every dX 2 along the shower axis, dN muons are produced within a given energy and transverse momentum interval dE i and dp t . Their overall distribution at production can be described in general with a 3-dimensional function, as: The projection into the X (or z ) axis becomes and it is the so called total/true Muon Production Depth (Distance) distribution, or MPD-distribution for short. It does not depend on the observational conditions since it does not contain any propagation effects of muons through the atmosphere. A detailed study of its shape is done in [10]. Notice that this is different from the MPD-distributions of detected muons at a given position on ground dN dX | ( r,ζ ) , which includes the effects of propagation, as it will be explained later. This distribution is sometimes referred to as apparent MPD-distribution. The total number of muons produced in a shower is It should be noted that this number is intrinsically different from the number of surviving muons, which is affected by the fluctuations of the depth of the first interaction, and thus change the distance traveled by muons to the ground. Some of the techniques used by experiments like Auger [11] use a fixed distance to the shower core, so they can also be affected by the lateral spread of the parent mesons. Eq. 4 can be factorized and expressed as the product where the function f X ( E i , cp t ) = F ( X,E i ,cp t ) h ( X ) becomes the normalized E i and cp t distribution at a given production depth X . In the approximations made in [8, 12, 13], f X did not depend on X and it was factorized in 2 independent distributions on E i and cp t . This allowed analytical approximations of the distributions at ground. In [14] we have included these correlations, improving the accuracy of the energy, production depth, and time distributions at ground, and allowing for a proper description of the muon lateral distribution at ground. The function h ( X ) tracks the longitudinal development of the hadronic cascade and represents the production rate of muons per g cm -2 . Its shape and features are extensively discussed in [10]. The depth at which h ( X ) reaches the maximum is denoted as X µ max . X µ max correlates with the first interaction point X 1 which corresponds to the first interaction of the primary in the atmosphere and the start of the cascading process [10]. The most important source of fluctuations in air showers corresponds to the fluctuations of X 1 , which causes an overall displacement of the whole cascade at first approximation. The amount X ' ≡ X -X µ max defines the amount of traversed matter with respect to the shower maximum. The distributions can be expressed in terms of X ' , where the most important source of fluctuations has been eliminated, and only the remaining effects are present. In Fig. 2 (right panel) h ( X ) is shown for a sample of 50 showers. The fluctuations on the normalization and on X µ max are clearly observed. In [14] it was shown that both the energy and the transverse momentum show similar features when referred to the same distance to the shower maximum, X ' . In [8, 12, 13] the muon spectrum at production was approximated by a power law, E -2 . 6 i , following the high energy tails of the pion production on hadronic reactions. A more accurate description of this the spectrum was done in [14]: at low energies the single power law clearly does not work and, in addition, the energy spectrum evolves with X ' by becoming softer, and stabilizing the shape after the shower maximum. In Fig. 3, left panel, the average energy spectrum of all muons at production is displayed for proton showers at 10 19 eV in different X ' layers. The transverse momentum distributions are responsible for most of the lateral displacement of muons with respect to the shower axis. In [8, 12, 13], the p t distributions were approximated by an unique function, dN/dp t = p t /Q 2 exp( -p t /Q ), independent of the energy of the muon and its production depth, primary mass and zenith angle. In [14], we uncover in detail all the dependencies. As the shower evolves, the p t spectrum becomes softer (Fig. 3, left panel shows the evolution as a function of X ' ). Besides this dependence on X ' , the p t distributions also depend on the energy of the muons, as discussed in [14]. The low energy muons display a smaller p t , and at high energies, the p t distribution prefers higher p t values. We have found that the different correlations of the p t with E i and X must be included into the model in order to properly predict the muon lateral distribution at ground. In [14] it is shown that there are mild dependencies of both the energy and p t distributions on the energy and zenith angle of the primary. In addition, the photon initiated showers display quite different distributions due to the different nature of the processes that lead to the muon production, through photopion production. Proton and iron showers, and different hadronic models also display mild differences among them.", "pages": [ 3, 4, 5, 6 ] }, { "title": "4 Propagation and ground distributions", "content": "In [14] it was shown that a few simple considerations are enough to account for most of the features observed in the muon distributions at ground. Firstly, muons exit the shower axis with an angle α determined by the energy and transverse momentum of the muon at production (sin α = cp t E i ). The polar angle is distributed symmetrically over 2 π . Once the muon is produced, the trajectory is extrapolated in a straight line to the ground, and the arrival time due to geometric path is calculated. Once the main trajectory is defined, the energy loss, decay probability, multiple scattering and effects of the magnetic field are accounted for and the impact point on ground and arrival time delay are corrected. Table 4 summarizes different effects for 5 GeV and 10 GeV muons produced at z = 10 km and arriving at a distance from the core r = 1000 m. The most important propagation effects that shape the ground distributions are, in this order: geometry, decay and energy loss. The magnetic effects become more important in showers with zenith angle above 60 degrees. On the other hand, the multiple scattering effects are negligible at distances to the core above 100 m.", "pages": [ 6 ] }, { "title": "4.1 The energy distribution", "content": "The energy at ground E f was analyzed as a function of the impact point on ground ( r, ζ ). Typically, the muon energy is not directly measured by cosmic ray detectors since it would require carpeting extensive areas with particle detectors like those used in accelerator experiments. Nevertheless, the spectrum of muons has an impact on other quantities that are measured by current air shower detector arrays, like the muon lateral distribution at ground, the arrival angle, and the arrival time delay. Fig. 4 displays the normalized energy spectra of a 60 deg shower, at different distances from the shower core. The energy of muons decreases as ∼ 1 /r and increases with the zenith angle [8, 12, 13], being the details determined by the p t , z and E i distributions. Low energy muons dominate at large distances from the core.", "pages": [ 6, 7 ] }, { "title": "4.2 Apparent production depth distribution", "content": "The shape of the production depth distribution of the detected muons, the apparent MPD-distribution, changes with the observation position. The angular position of the observation point respect to the production point z , selects particular ( E i , p t ) regions which can be more or less populated. In addition, the propagation effects, specially the decay, modulate the apparent MPD-distribution depending on the energy spectrum of muons and also the path traveled from production to ground, l . Fig. 5 displays the apparent MPD-distributions for a 40 deg shower at different distances from the core, where the distortions introduced in the dN/dX | ( r,ζ ) distributions when compared to h ( X ) can be clearly observed. The dN/dX | ( r,ζ ) distribution is never directly observed, but reconstructed from the arrival time or the arrival angle at ground. The correct inference of the total/true MPD-distribution, h ( X ), requires the knowledge of the exact dependence of dN/dX | ( r,ζ ) with the observation point coordinates and detection energy threshold. dN/dX | ( r,ζ ) explores different kinematic regions at production when reconstructed at different distances from the core. For instance, the algorithm proposed in [12] and [15] requires the conversion of each dN/dX | ( r,ζ ) observed in each station to an universal distribution in order to sum up the contributions of all detectors in a single shower.", "pages": [ 7 ] }, { "title": "4.3 Time distributions", "content": "The total time delay is the sum of four different contributions t = t g + t glyph[epsilon1] + t B + t MS where t g is the geometric delay, t glyph[epsilon1] is the kinematic delay, t B is the contribution produced by the geomagnetic field, and finally t MS includes the delay due to multiple scattering. Fig. 6, left panel, displays the different contributions to the total delay for 60 degrees zenith angle. At large distances from the core, the geometric delay is the most important. At distances typically from a few hundred meters to 1 km, the kinematic delay has a large impact. As we increase the zenith angle, the geometric delay looses importance relatively to the other contributions. At 500 m from the core, the geometric delay represents glyph[similarequal] 60% of the total. Fig. 6, right panel, displays the overall time distributions at 1300 m from the shower core for a 60 deg shower. Filled histograms show the contributions of different muon energies at ground. High energy muons arrive earlier at ground. This is so because they are produced higher up in the atmosphere, and therefore have less geometric delay, but also because they have less kinematic delay. The muon arrival time distributions can be used to extract relevant information. Far from the core, the time distributions are to a very good extent a one to one map of the apparent MPD-distributions. They can be determined by converting each muon time into a production distance, being the kinematic time a second order correction. Since the energy of each muon is typically not known, it is approximated by the mean value, taken from the energy spectrum at each observation point as it was explained in [8, 12, 13]. The energy would also determine the parameters of the multiple scattering delay distribution, although its concrete value follows a random distribution. The geomagnetic delay can take only two possible values depending on the charge of the muon. In general this technique will require a stringent r cut for those regions where the geometric delay is a large fraction of the total delay, in order to avoid distortions of the reconstructed dN/dX | ( r,ζ ) . A more promising method consists in fitting the time distributions at once leaving a set of shape parameters on h ( X ) free. Close to the core, the geometric delay is not dominant and the arrival time is mostly determined by the energy of each muon. This opens to possibility to measure, or at least constrain, the shape of the muon energy spectrum. A global fit would also allow to extract parameters from the p t distributions.", "pages": [ 7, 8 ] }, { "title": "4.4 Muon lateral distribution at ground", "content": "The number of muons per surface area unit is ρ ( r, ζ ) = d 2 N rdrdζ . As it was shown in [14], low energy muons have a major impact on the fine details of the muon lateral distribution at ground. In vertical showers the number of muons per surface area does not depend much on ζ . As we increase the zenith angle, asymmetries appear because of the different propagation effects, mainly decay and geometry. The effects of the magnetic field become important above 60 degrees, and they completely dominate the distributions at very inclined showers, typically between 80 and 90 degrees [16]. Fig. 7 displays the muon density as a function of r for 3 different polar angles ζ on a 70 deg shower. 10 10 The shape of the ground distributions is fully determined by the distributions at production, h ( X ) and f X ( E i , p t ). A change in the overall muon content of the shower, N 0 , produces a change in the muon density at ground, and therefore in the normalization of all distributions. The other main source of fluctuations comes from the depth of the first interaction, which directly affects h ( X ) by changing its maximum, X µ max . The position of X µ max directly influences all distributions at ground since it changes the total distance traveled by muons to ground.", "pages": [ 8, 9 ] }, { "title": "4.5 Average energy and transverse momentum distributions", "content": "One of the main applications of the present model is to be used in a global fit to extract information on the total number of muons in the shower N 0 , and the total/true production depth distribution, h ( X ), and its maximum, X µ max . In order to do so, a f X ( E i , p t ) distribution must be assumed. The energy and transverse momentum distributions display more universal features when they are expressed in terms of X ' = X -X µ max , once the effects of the fluctuations induced by the first interaction point are removed. The average energy and transverse momentum distributions do not change when changing the energy of the primaries, whereas they show mild differences between proton and iron primaries, and between hadronic interaction models. If we substitute f X ( E i , cp t ) of a given shower by an average over showers of the same hadronic interaction model, primary, and zenith angle, 〈 f X ' ( E i , cp t ) 〉 , leaving only h ( X ) from the original shower, the ground density displays differences of about ∼ 2% at 1000 m compared to the prediction if we used f X ( E i , cp t ), whereas the rest of the ground distributions remained unchanged. It is thus possible to use an universal energy and traverse momentum distribution that depends only on X ' , where the position of X µ max is naturally accounted for through X = X ' + X µ max . The systematics of any concrete application, including a global fit, are to be studied and accounted for in each particular method and/or experimental setup. The effects of the choice of hadronic inter- (t/ns) (t/ns) action model on 〈 f X ' ( E i , cp t ) 〉 might introduce some systematics that should be also accounted for. On the contrary, those differences might be used to constrain f X ' itself when compared to data which is very promising. One could also think of a method to experimentally constrain the energy and transverse momentum spectrum based on simultaneous observations of the ground distributions in different conditions. For instance, the ground muon distributions of inclined showers contain valuable information about the energy spectrum due to the spectrographic effect of the geomagnetic field.", "pages": [ 9, 10 ] }, { "title": "5 Experimental efforts", "content": "In this section I will illustrate some of the experimental efforts to reconstruct the muon distributions with a few selected examples. KASCADE has recently published [17] the apparent MPD-distributions for showers between E ∼ [10 15 , 10 17 . 7 ] and zenith angle [0,18] deg at distances to the core [40,80] m. KASCADE uses a combination of different detectors which can separate the components of the shower, being possible to individually tag single muons. It also has a muon telescope, able to track the trajectory of the muon back to the shower axis and thus determine the production height. The back-tracking technique can be used in combination with the time-to-X technique in the Time-Track Complementarity method [18], which is able to separate high energy muons from low energy muons, opening new observables. As a drawback, these heavily instrumented observatories are hardly scalable to the high energy end of the spectrum, at energies around 10 19 eV. The Pierre Auger observatory has recently published the maximum of the apparent -MPD using the time-to-X technique. Although the water Cherenkov tanks were not specifically designed to distinguish muons from electrons and photons, a fiducial cut can remove those stations close to the core and keep the muon richness sufficiently high to reconstruct the MPD-distributions. KASCADE-Grande and Auger have also published in [19] and [20] the number of muons as a function of the energy, in two different energy ranges. Auger sees an excess in the number of muons when compared to simulations. It is still unclear whether the number of muons measured by the two experiments match due to the gap region around 10 18 eV. The shower-to-shower distribution of the number of muons contains also valuable information that it is not yet fully exploited. The RMS of the number of muons adds valuable information to help break the degeneracy between hadronic models and composition. To achieve this goal, it is important to enhance the muon capabilities at the highest energies and gain precision in the muon reconstruction. MARTA (Muon Auger RPC Tank Array) is one of the efforts in this direction. It envisages to add highly robust and autonomous RPC detectors [21] to the Auger Cherenkov tanks to enhance the muon capabilities.", "pages": [ 10 ] }, { "title": "6 Conclusions", "content": "Particle reactions beyond the energy achieved by man made accelerators are continuously happening in the hadronic backbone of EAS. Muons are true smoking guns of the hadronic cascade. In fact, we have shown that the relation between the distributions directly inherited from hadrons and the distributions observed at ground are well understood and that ground distributions can be used to reconstruct and constrain the distributions at production. We must study muons to enhance the sensitivity to the hadronic phenomena in the cascade. This is the path to solve the problem of composition of the UHECR and might uncover new particle physics phenomena at high energies. The future cosmic ray experiments above LHC energies need precise muon dedicated detectors added to the EAS arrays.", "pages": [ 10, 11 ] }, { "title": "7 Acknowledgments", "content": "I would like to thank R. Concei¸c˜ao, C. Espirito-Santo, M. Pimenta and M. Oliveira for careful reading of this manuscript and their constructive comments. This work is partially funded by Funda¸c˜ao para a Ciˆencia e Tecnologia (CERN/FP11633/2010), and fundings of MCTES through POPH-QREN Tipologia 4.2, Portugal, and European Social Fund.", "pages": [ 11 ] } ]

[ { "title": "Measurement of Cosmic Ray Spectrum and Anisotropy with ARGO-YBJ", "content": "G. Di Sciascio 1 , a on behalf of the ARGO-YBJ Collaboration 1 INFN,SezionediRomaTorVergata,VialedellaRicercaScientifica1,Roma,ItalyI-00133. Abstract. The combined measurement of the cosmic ray (CR) energy spectrum and anisotropy in their arrival direction distribution needs the knowledge of the elemental composition of the radiation to discriminate between di ff erent origin and propagation models. Important information on the CR mass composition can be obtained studying the EAS muon content through the measurement of the CR rate at di ff erent zenith angles. In this paper we report on the observation of the anisotropy of galactic CRs at di ff erent angular scales with the ARGO-YBJ experiment. We report also on the study of the primary CR rate for di ff erent zenith angles. The light component (p + He) has been selected and its energy spectrum measured in the energy range (5 - 200) TeV for quasi-vertical events. With this analysis for the first time a ground-based measurement of the CR spectrum overlaps data obtained with direct methods for more than one energy decade, thus providing a solid anchorage to the CR spectrum measurements carried out by EAS arrays in the knee region. Finally, a preliminary study of the non-attenuated shower component at a zenith angle θ > 70 · (through the observation of the so-called horizantal air showers) is presented.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The understanding of CRs origin at any energy is made di ffi cult by the poor knowledge of the elemental composition of the radiation. The determination of the CR arrival direction does not depend on knowledge of the mass of the primary particle, however, the use of combined data on the energy spectrum and arrival direction distribution requires the knowledge of the primary mass distribution to discriminate between di ff erent origin and propagation models. Inclined showers ( θ > 60 · ) induced by very highenergy CRs are mainly produced by secondary muons, in contrast to the vertical ones dominated by photons and electrons stemming from π 0 decays. Measurements of the CR rate at di ff erent zenith angles gives information on the relative number of muons in a shower, which is dependent on the CR elemental composition, thus providing an important tool to probe the CR mass distribution. As CRs are mostly charged nuclei, their arrival direction is deflected and highly isotropized by the action of galactic magnetic field (GMF) they propagate through before reaching the Earth atmosphere. However, di ff erent experiments [1-4] observed an energy-dependent \"large scale\" anisotropy (LSA) in the sidereal time frame with an amplitude of about 10 -4 - 10 -3 , suggesting the existence of two distint broad regions, one showing an excess of CRs (called \"tail-in\"), distributed around 40 · to 90 · in Right Ascension (R.A.). The other a deficit (the so-called \"loss cone\"), distributed around 150 · to 240 · in R.A.. In the last years the Milagro [5] experiment reported evidence of the existence of a medium angular scale anisotropy (MSA) contained in the tail-in region. The observation of similar small scale anisotropies has been recently claimed by the Icecube experiment in the Southern hemisphere [4]. So far, no theory of CRs in the Galaxy exists which is able to explain the origin of these di ff erent anisotropies leaving the standard model of CRs and that of the local GMF unchanged at the same time. A joint analysis of concurrent data recorded by di ff erent experiments in both hemispheres, as well as a measurement of energy spectrum and elemental composition of the anisotropy regions, should be a high priority to clarify the observations. The ARGO-YBJ experiment, located at the YangBaJing Cosmic Ray Laboratory (Tibet, P.R. China, 4300 m a.s.l., 606 g / cm 2 ), is an air shower array able to detect the cosmic radiation with an energy threshold of a few hundred GeV. The full detector is in stable data taking since November 2007 with a duty cycle larger than 85%. The trigger rate is 3.6 kHz. The detector characteristics are described in [6-8]. Details on the analysis procedure (e.g., reconstruction algorithms, data selection, background evaluation, systematic errors) are discussed in [9, 10]. The performance (angular resolution, pointing accuracy, energy scale calibration) and the operation stability are monitored on a monthly basis by observing the Moon shadow, i.e., the deficit of CR detected in its direction [10]. The last results obtained by ARGO-YBJ are summarized in [11]. In this paper the observation of CR anisotropy at di ff erent angular scales with ARGO-YBJ is reported for di ff erent primary energies. We report also on the measurement of the primary CR spectrum for di ff erent zenith angles.", "pages": [ 1 ] }, { "title": "2 Cosmic Ray Anisotropy", "content": "To study the anisotropy at di ff erent angular scales the isotropic background of CRs has been estimated with two well-known methods: the equi-zenith angle method [12] and the direct integration method [13]. The equi-zenith angle method, used to study the LSA, is able to eliminate various spurious e ff ects caused by instrumental and environmental variations, such as changes in pressure and temperature that are hard to control and tend to introduce systematic errors in the measurement. The direct integration method, based on time-average, relies on the assumption that the local distribution of the incoming CRs is slowly varying and the time-averaged signal may be used as a good estimation of the background content. Timeaveraging methods (TAMs) act e ff ectively as a high-pass filter, not allowing to inspect features larger than the time over which the background is computed (i.e., 15 · / hour × ∆ t in R.A.) [14]. The time interval used to compute the average spans ∆ t = 3 hours and makes us confident the results are reliable for structures up to ≈ 35 · wide.", "pages": [ 2 ] }, { "title": "2.1 Large Scale Anisotropy", "content": "The observation of the CR large scale anisotropy by ARGO-YBJ is shown in Fig. 1 as a function of the primary energy up to about 25 TeV. The so-called 'tail-in' and 'loss-cone' regions, correlated to an enhancement and a deficit of CRs, respectively, are clearly visible with a statistical significance greater than 20 s.d.. The tail-in broad structure appears to break up into smaller spots with increasing energy. In order to quantify the scale of the anisotropy we studied the 1-D R.A. projections integrating the sky maps inside a declination band given by the field of view of the detector. Therefore, we fitted the R.A. profiles with the first two harmonics. The resulting amplitude and phase of the first harmonic are plotted in Fig. 2 and Fig. 3 where are compared to a full compilation of measurements as a function of the energy. The ARGO-YBJ results are in agreement with those of other experiments, suggesting a decrease of the anisotropy first harmonic amplitude at energies above 10 TeV.", "pages": [ 2 ] }, { "title": "2.2 Medium Scale Anisotropy", "content": "Fig. 4 shows the ARGO-YBJ sky map in equatorial coordinates containing about 2 · 10 11 events reconstructed with a zenith angle ≤ 50 · . The zenith cut selects the declination region δ ∼ -20 · ÷ 80 · . According to simulations, the median energy of the isotropic CR proton flux is E 50 p ≈ 1.8 TeV (mode energy ≈ 0.7 TeV). The most evident features are observed by ARGO-YBJ around the positions α ∼ 120 · , δ ∼ 40 · and α ∼ 60 · , δ ∼ -5 · , positionally coincident with the excesses detected by Milagro [5]. These regions, named '1' and '2', are observed with a statistical significance of about 14 s.d.. The deficit regions parallel to the excesses are due to a known e ff ect of the analysis, that uses also the excess events to evaluate the background, overestimating this latter [14]. The left side of the sky map is full of few-degree excesses not compatible with random fluctuations (the statistical significance is more than 6 s.d.). The observation of these structures is reported here for the first time and together with that of regions 1 and 2 it may open the way to an interesting study of the TeV CR sky. To figure out the energy spectrum of the excesses, data have been divided into five independent shower multiplicity sets. The number of events collected within each region are computed for the event map (Ev) as well as for the background one (Bg). The relative excess (Ev-Bg) / Bg is computed for each multiplicity interval. The result is shown in the Fig. 5. Region 1 seems to have spectrum harder than isotropic CRs and a cuto ff around 600 shower particles (proton median energy E 50 p = 8 TeV). On the other hand, the excess hosted in region 2 is less intense and seems to have a spectrum more similar to that of isotropic CRs. As a reference value, the upper horizontal scale reports the median energy of isotropic CR protons for each multiplicity interval obtained via MC simulation.", "pages": [ 2, 3 ] }, { "title": "2.3 The Compton-Getting effect", "content": "The origin of CR anisotropies is still unknown therefore, the observation of an expected anisotropy is im- portant to check the reconstruction algorithms, the background calculation and the stability of the detector performance. A well-known expected anisotropy is the socalled Compton-Getting (CG) e ff ect, a dipole anisotropy in the local solar frame, due to the Earth's motion around the Sun [16]. A significant signal compatible with CG is seen by ARGO-YBJ in solar time above ∼ 8 TeV to avoid additional e ff ects due to heliospheric magnetic field and solar activity. In fact, we found that including lower energy events results in much larger modulation amplitudes than those obtained when these events were excluded. Fig. 6 shows the solar variations observed by ARGO-YBJ together with the sinusoidal curve best fitted to the data. The fair agreement between data and calculations ( φ = 6:00 hr, A = 9.7 · 10 -5 ) make us confident about the capability of ARGO-YBJ in detecting anisotropies at a level of 10 -4 .", "pages": [ 3 ] }, { "title": "3.1 Light component (p+He) spectrum of CRs", "content": "Requiring quasi-vertical showers ( θ < 30 · ) and applying a selection criterion based on the particle density, a sample of events mainly induced by protons and helium nuclei, with shower core inside a fiducial area (with radius ∼ 28 m), has been selected. The contamination by heavier nuclei is found negligible. An unfolding technique based on the Bayesian approach has been applied to the strip multiplicity distribution in order to obtain the di ff erential energy spectrum of the light component (p + He nuclei) in the energy range (5 - 200) TeV [17]. The spectrum measured by ARGO-YBJ is compared with other experimental results in Fig. 7. Systematic e ff ects due to di ff erent hadronic models (Corsika 6.710 with QGSJet-II and SYBILL) and to the selection criteria do not exceed 10%. The ARGOYBJ data agree remarkably well with the values obtained by adding up the p and He fluxes measured by CREAM both concerning the total intensities and the spectral index [18]. The value of the spectral index of the power-law fit to the ARGO-YBJ data is -2.61 ± 0.04, which should be compared with γ p = -2.66 ± 0.02 and γ He = -2.58 ± 0.02 obtained by CREAM. The present analysis does not allow the determination of the individual p and He contribution to the measured flux, but the ARGO-YBJ data clearly exclude the RUNJOB results [19]. We emphasize that for the first time direct and ground-based measurements overlap for a wide energy range thus making possible the crosscalibration of the di ff erent experimental techniques.", "pages": [ 3, 4 ] }, { "title": "3.2 Horizontal Air Showers", "content": "At zenith angles θ > 60 · an excess of events (the socalled HAS, horizontal air showers) is observed above the rate of EAS as expected from the exponential absoprtion (with Λ EAS ≈ 220 g / cm 2 ) of the air shower electromag- component in the large atmospheric depth (see Fig. 8), which implies a decrease of the EAS counting rate with Λ c ≈ 130 g / cm 2 . The physical nature of these showers is confirmed by the absence of events from the direction of the sky shaded by the mountains around the ARGO-YBJ detector, as can be seen in the Fig. 9 where the shower rate as a function of the reconstructed azimuthal angle is compared to the shadow angle due to the surrounding mountains. The expected anti-correlation is clearly visible and the mountain profile is reproduced quite well. Moreover, the dependence of the barometric e ff ect on the zenith angle, shown in Fig. 10, clearly shows a deviation from the sec θ behaviour for sec θ > 2. In fact, the barometric coe ffi cient β = 1 n dn dx ( n = counting rate, x = atmospheric pressure) is related to zenith angle as: β ( θ ) = β (0 · ) sec θ . This can be explained by the presence of a \"non-attenuated\" EAS component that dominates for angles larger than 70 · . Due to the small ARGO-YBJ e ff ective area at large zenith angles, we expect that the observed HAS are due to high energy single muons which interact through bremmstrahlung (which dominate 10:1) or deep inelastic scattering and initiate showers at the appropriate depth (few hundreds g / cm 2 ) for detection, as shown in Fig. 11 where some typical events observed by ARGO-YBJ are displayed. The characteristic elliptical shape of the showers, well contained in the central carpet, is clearly visible. Such showers are essentially electromagnetic, since the remnant muons from the initial showers are dispersed over a very large area. In Fig. 12 the shower rate measured by ARGO-YBJ is shown, as a function of the fired strips number, for different primary zenith angles. The spectra soften with increasing angle up to about 70 · , as can be appreciated in Fig. 13 where the best-fit spectral indices are plotted. In the zenith angle region 50 · - 70 · a quick transition to a value of about -3.6, characteristic of the EAS muon component, is observed. Detailed simulations to reproduce the observed size spectrum of HAS are under way.", "pages": [ 4, 5 ] }, { "title": "4 Conclusions", "content": "The ARGO-YBJ detector exploiting the full coverage approach and the high segmentation of the readout is imaging the front of atmospheric showers with unprecedented resolution and detail. The digital and analog readout will allow a deep study of the CR phenomenology in the wide TeV - PeV energy range. In this paper the observation of CR anisotropy at different angular scales is reported for di ff erent primary energies. The large scale CR anisotropy has been clearly observed up to about 25 TeV. The existence of di ff erent fewdegree excesses in the Northern sky is showed by ARGOYBJ for the first time. We reported also on the measurement of the primary CR rate for di ff erent zenith angles. The light component (p + He) has been selected and its energy spectrum measured in the energy range (5 - 200) TeV for quasi-vertical events. A preliminary study of HAS with ARGO-YBJ is also presented. More than 10 7 well-contained horizontal events have been analyzed, thus providing a 'well shielded' sample useful to study the production and interaction of high energy CR muons and neutrinos.", "pages": [ 5 ] } ]

[ { "title": "the search for a prompt component", "content": "Thomas K. Gaisser Bartol Research Institute and Department of Physics and Astronomy University of Delaware, Newark, DE USA The flux of high-energy ( ≥ GeV) neutrinos consists primarily of those produced by cosmic-ray interactions in the atmosphere. The contribution from extraterrestrial sources is still unknown. Current limits suggest that the observed spectrum is dominated by atmospheric neutrinos up to at least 100 TeV. The contribution of charmed hadrons to the flux of atmospheric neutrinos is important in the context of the search for astrophysical neutrinos because the spectrum of such 'prompt' neutrinos is harder than that of 'conventional' neutrinos from decay of pions and kaons. The prompt component therefore becomes increasingly important as energy increases. This paper reviews the status of the search for prompt muons and neutrinos with emphasis on the complementary aspects of muons, electron neutrinos and muon neutrinos.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Before experimental discovery of charmed hadrons at accelerators in the mid-1970s there was intense interest in using atmospheric muons to find evidence for production of heavy, short-lived hadrons. For example, a highlight of the 1973 cosmic-ray conference at Denver was an update of measurements made over several years with the underground muon spectrometer at Park City, Utah [1]. At first, the observed angular dependence of multi-TeV muons had appeared to be more isotropic than could be explained solely by production through the pion and kaon channels, which are strongly enhanced at large zenith angles. With an improved understanding of the overburden and the detector response, however, it was finally concluded that the Park City data were consistent a 'conventional' origin from decay of pions and kaons. An isotropic 'prompt' component was not manifest for energies below 10 TeV. Production of charmed hadrons has now been measured over a large range of energy at accelerators. The production cross section increases significantly from approximately ∼ 1 µ b at √ s ≈ 10 GeV to several mb at √ s ≈ 7 TeV [2]. There is still not full coverage of phase space for charm production, however. In particular, the level of 'intrinsic' charm [3] production is still uncertain. The SELEX measurement [4] shows a large asymmetry in the ratio of charm to anti-charm baryons produced by baryon beams on a fixed target, while little or no asymmetry is observed in a pion beam. This observation indicates some level of intrinsic charm in which the valence quarks of the projectile pick up a charmed quark. Charmed hadrons produced as fragments of the incident nucleon beam will contribute disproportionately to the spectrum of atmospheric leptons because of the steep cosmic-ray energy spectrum. Thus, even if intrinsic charm contributes less to the total cross section for producing charm than production via QCD processes, it may have a significant effect on the prompt contribution to atmospheric muons and neutrinos. In addition to the intrinsic interest in identifying the charm contribution to the fluxes of atmospheric muons and neutrinos, there is another, perhaps more important, reason for trying to measure this component. That is because of the relevance of prompt neutrinos in the search for neutrinos of astrophysical origin. Like a flux from unresolved extra-galactic neutrino sources, the prompt contribution is isotropic for E ν < 10 7 GeV. It is also harder by one power of energy than the spectrum of conventional atmospheric neutrinos. For these reasons, prompt neutrinos constitute an important background for neutrino astronomy. The paper begins in § II with a discussion of the ingredients needed to calculate fluxes of atmospheric muons and neutrinos, including relevant analytic approximations and the primary cosmic-ray spectrum. Section III reviews models for charm production and corresponding predictions for fluxes of muons and neutrinos. In § IV we calculate the fluxes of conventional atmospheric muons and neutrinos and compare them with the charm contribution. The effect of the knee of the primary spectrum is included. The predictions are illustrated in § V with approximate calculations of the event rates for detector with a gigaton target volume like IceCube [5-7]. The concluding Section VI comments on the current status and prospects for detection of prompt leptons in the near future.", "pages": [ 1 ] }, { "title": "II. ATMOSPHERIC MUONS AND NEUTRINOS", "content": "The two main ingredients in the calculation of atmospheric neutrinos are the primary spectrum and the hadronic physics of meson production in hadronic interactions. Because production of pions, kaons and charmed hadrons occurs at the nucleon level, what is most relevant is the primary spectrum of nucleons per GeV/nucleon. Composition comes in through the ratio of protons to total nucleons, which determines the charge ratio of muons and particle/anti-article ratio for neutrinos. 5 E N 10 6 10 (GeV/nucleon)", "pages": [ 1, 2 ] }, { "title": "A. Primary spectrum", "content": "For illustration I use a phenomenological model of the primary spectrum with three populations of particles and five nuclear components [8], as shown in Fig. 1 (Left). There are two basic assumptions. First, it is assumed that all energy dependence (whether from acceleration or propagation) depends only on how particles are affected by their magnetic environment. As a consequence, each nuclear component (mass number Z and total momentum per particle P ) depends on magnetic rigidity ( R ) in the same way, where R = Pc/Ze . Peters [9] pointed out the consequence of this assumption for the primary composition in the region of the knee of the spectrum, namely, that, when expressed in terms of total energy per particle, protons would steepen first followed by helium and then by nuclei with successively higher charge. The other assumption, following Hillas [10], is that three populations of particles are sufficient to characterize the entire cosmic-ray spectrum. This is almost certainly an oversimplification. A more realistic picture would likely involve many individual sources injecting particles at various distances and times, as in the model for galactic cosmic rays of Blasi and Amato [11]. Thus the three populations represent three classes of sources: The contribution of nuclei of mass A i to the all-particle spectrum is given by where E is the total energy per nucleus. The spectral indices for each group and the normalizations are given explicitly in Table I. The parameters for Population 1 are based on fits to spectra of nuclear groups measured by CREAM [12, 13], which we assume can be extrapolated to a rigidity of 4 PV to describe the knee. This is an unverified simplifying assumption that needs to be checked by measurements in the PeV range. In Eq. 1 φ i is d N/ dln E and γ i is the integral spectral index. The subscript i = 1 , 5 runs over the standard five groups (p, He, CNO, Mg-Si and Fe), and the all-particle spectrum is the sum of the five. The spectrum of nucleons as a function of energy per nucleon corresponding to Eq. 1 is given by Because of the steep cosmic-ray spectrum, protons are relatively more important and heavy nuclei less important in the spectrum of nucleons (as a function of E N = E tot /A ) than in the all particle spectrum. The spectrum of nucleons is plotted for several assumptions in Fig. 1 (right). The straight solid line shows a simple E -2 . 7 spectrum of nucleons to guide the eye. The straight dotted line shows the spectrum below 100 TeV recommended in 2001 [14] as a standard for use in calculating fluxes of atmospheric leptons up to 10 TeV. At low energy the fit was based on measurements of BESS [15] and AMS [16]. The spectral index used for protons at the time was rather steep (2.74), based on the measurements of BESS and AMS below 200 GeV. Recent results ) 1.5 GeV -1 s -1 sr -2 (m dN/dE 2.5 E 10 10 10 10 4 3 2 1 10 3 10 4 10 TG-H3C fit -2.7 Polygonato 2001 as in Honda06 TIG NSU 7 10 8 10 9 of PAMELA [17] show that the spectrum of protons hardens above 200 GeV. Two options were given for helium, which contributes of order 25% of the spectrum of nucleons. The high option for helium (with an integral spectral index of γ = 1 . 64) suggested by emulsion chamber measurements in the 10 TeV range at the time [18, 19] has since been confirmed by ATIC [20], CREAM [12] and PAMELA [17]. A version of the spectrum of Ref. GHLS is used in the standard calculations of the flux of atmospheric neutrinos by Honda et al. [21] and by the Bartol group [22]. The spectrum of Honda et al. (as described in [23]) uses a harder spectrum for hydrogen (2.71 instead of 2.74) above 100 GeV. Their overall spectrum is nearly constant at γ = 1 . 69 from 200 GeV to 50 TeV with a fraction of helium that increases from 20% to 25% in the same region. The spectral index of the spectrum of nucleons in the model of Ref. [8] is nearly constant at γ = 1 . 63 from 200 GeV to 50 TeV with a corresponding increase in the contribution of helium from 22% to 30%. The contribution from heavier nuclei is at the level of 10%. The other lines in the spectrum of nucleons all include the effect of the knee in the cosmic-ray spectrum in one way or another. The heavy solid curve is the nucleon spectrum corresponding to the model in Table I. The nearby pink dash-dot is an analytic approximation to that model, which is described below in Eq. 13. The strong knee around 1 PeV is the consequence of the increasing fraction of heavy nuclei in the model. In addition to the model of Ref [8], Fig. 1 also shows the polygonato model [24] without any contribution from nuclei heavier than iron. Each nuclear component in the model steepens by δ = 1 . 9 at a rigidity of 4 . 49 PV. The effect of the knee begins to show up in the nucleon spectrum already somewhat below one PeV. Using the rule of thumb that there is on average a factor of ten between the parent cosmic ray energy and the secondary leptons, taking account of the steepening of the spectrum will be important for muon and neutrino energies of 100 TeV and above, which we is discussed in § IV. The double dotted line that steepens from a differential index of -2 . 7 to -3 . 0 at 5 × 10 6 GeV is the primary spectrum used in the charm calculation of Ref. TIG.", "pages": [ 2, 3 ] }, { "title": "B. Hadron production and decay kinematics in the atmosphere", "content": "The phenomenology of atmospheric leptons depends on the production of pions, kaons and heavier hadrons by interactions of cosmic-rays in the atmosphere and on the kinematics for the relevant decay channels. Production and subsequent decay occur through generation by a steep spectrum of primary and secondary cosmic rays in the atmosphere. The competition between reinteraction and decay of unstable hadrons depends on density and altitude. In the framework of a set of analytic approximations for solution of the cascade equations, the essential dependence on energy and zenith angle comes through the critical energy defined as where the index ι indicates the hadron ( π ± , K ± , K L or charmed hadron), τ ι is the meson lifetime, h 0 is the scale height of the atmosphere and θ ∗ is the zenith angle ( ∗ corrected for curvature of the Earth for θ ≥ 70 · ). Values of the important characteristic energies are given in Table II. For a power-law spectrum of primary nucleons, the expression for the lepton spectrum factorizes into a product of the primary spectrum and an expression that reflects the properties of production of secondary hadrons by the cosmic-ray spectrum and their subsequent decay to muons and neutrinos. The A -factors in Eq. 4 are a product of the spectrum weighted moments for production of mesons by nucleons times the spectrum weighed moments of the meson decay distributions, which include both the decay kinematics and the branching ratios. For a power-law spectrum of decaying pions with a differential spectral index α the decay factor is for muons and neutrinos respectively. In the low energy limit, the spectral index is the same as that of the primary spectrum of nucleons, but at high energy the spectrum of the decaying pions is one power steeper because of the competition between decay and reinteraction. Low and high are with respect to the critical energy /epsilon1 π / cos( θ ). The ratio r π = m 2 µ /m 2 π = 0 . 5731. The forms for two-body decay of charged kaons are the same, but the mass ratio factor is much smaller: r K = 0 . 0458. The larger critical energy for charged kaons leads to an increase in the contribution of kaons with increasing energy for both muons and neutrinos. The differences between the kinematic factors for two-body decay to neutrinos and muons amplifies the importance of the kaon channel for neutrinos. At high energy, in the TeV range and above, charged kaons account for about 80% of muon neutrinos as compared to 25% of muons. Each term in Eq. 4 is a form that combines the low energy and high energy solutions to the cascade equation respectively for pions, kaons and charmed hadrons in the atmosphere. The numerator is a product of the spectrum weighted moment for meson production and the spectrum weighted moment of the decay distribution to ν µ [25] with α = γ +1, where γ is the integral spectral index of the spectrum of primary nucleons. The denominator governs the transition between the low and the high energy regimes. The forms for muons are similar. For low energy, meson decay dominates over reinteraction and the resulting lepton spectrum has the same shape as the primary spectrum of nucleons. Charmed hadrons are in the low-energy regime for E lepton < 10 7 GeV. For high energy ( E lepton > /epsilon1 α / cos θ ), reinteraction of the hadron is more likely and the lepton spectrum becomes one power of energy steeper than the primary spectrum. In the high energy limit, the spectrum weighted moment for meson decay has to be evaluated on the steeper spectrum, and the attenuation lengths for reinteraction come into play. The B ij quantities in the denominators are the product of the ratios of low-energy to high energy decay distributions combined with the function of attenuation lengths that accounts for cascading of the mesons [26]. Explicitly, for neutrinos and for muons The forms for kaons are the same as functions of r K and Λ K . The dependence of γ on energy in the case of a non-power law primary spectrum needs to be accounted for. For a power-law primary spectrum of nucleons and assuming Feynman scaling for hadron production, the cascade equations can be solved analytically as in Eq. 4. The primary spectrum can always be described locally as a power law, and similarly the hadronic interactions can be written in terms of the scaled energy ( x = E α /E N ) for a given primary energy per nucleon. In both cases the dependence on energy is sufficiently gradual that the approximate analytical forms can be used for quantitative calculations if the slow variation with energy is accounted for. This approach is taken in the calculation of Thunman, Ingelmann and Gondolo (TIG) [27], which I follow here. They define energy-dependent Z-factors as in the following example for nucleons producing charged pions: Here λ N ( E ) is the nucleon interaction length and d n ± π is the number of charged pions produced in d E by nucleons of energy E ' , and φ N ( E ) is the spectrum of nucleons. The energy-dependent Z-factors are then used in applicable version of Eq. 4 to evaluate the lepton spectrum. This approximation is valid to the extent that the energy dependences are gradual. They showed that the numerical approximation based on the spectrum weighted moments taken from the interaction model used in their Monte Carlo produced similar results to the full Monte Carlo. The advantage is that the analytic approximations can be tuned to match well to a particular model of hadronic interactions. They can then be used to extend a Monte Carlo calculation to arbitrarily high energies without the statistical problems that arise from the fact that mesons usually interact rather than decay at high energy. This method has been used recently in Ref. [28] to complement Monte Carlo calculations of atmospheric leptons in the region of the knee of the cosmic-ray spectrum.", "pages": [ 3, 4 ] }, { "title": "III. EXPECTATIONS FOR PROMPT LEPTONS", "content": "Limits on the prompt contribution to atmospheric muons are conventionally expressed in terms of the ratio R c of charm production to pion production by rewriting Eq. 4 as This form applies for lepton energy < 10 7 GeV where the charm contribution is isotropic. The current upper limit from LVD [29] is R c ≤ 2 x 10 -3 assuming a differential primary spectrum ∝ E -2 . 77 . The primary spectrum of nucleons used in this analysis is less steep by about 0 . 1, so ν the LVD limit at 10 TeV (for example) would be reduced by a factor of two for comparison with the models as discussed here. In what follows I will compare the current models for prompt leptons to R c /similarequal 10 -3 considered as an experimental upper limit.", "pages": [ 4, 5 ] }, { "title": "A. Calculations of charm production", "content": "Reference [27] uses a simple Monte Carlo to generate the distribution of hadronic interactions and decays in the atmosphere and Pythia [30] to generate the secondary hadrons at each interaction point. Charm production is calculated within Pythia using first order QCD matrix elements to calculate c ¯ c production by gluons and by quarks. A renormalization factor K = 2 is used to represent higher order QCD effects. In addition to their Monte Carlo calculation, They parameterized their results for the charm contribution in a form similar to the third term of Eq. 4 as with A ≈ 3 × 10 7 GeV and γ = 1 . 77 below and ≈ 2 . 0 above 10 6 GeV. The value of R c for this model is /similarequal 2 × 10 -5 at 10 TeV. The reference calculation used for evaluation of the atmospheric neutrino background in IceCube at high energy is that of Enberg, Reno & Sarcevic (ERS) [31]. It is a QCD calculation that gives results somewhat higher than some previous calculations [27, 32] and lower than an earlier NLO-QCD calculation [33]. The ERS calculation assumes the same primary spectrum of nucleons as in TIG [27], which is shown in the right panel of Fig. 1. The shape of the ERS calculation is similar to that of TIG, and its central value is approximately a factor of two higher. ERS assign an uncertainty range of approximately ± 50% to their calculation. The value of R c for this model is /similarequal 10 -4 at 10 TeV. The 'Recombination quark-parton model' (RQPM) [34] embodies the idea of intrinsic charm. The underlying concept is that-also for heavy quarksthere is a process of associated production in which the c ¯ c pair produced when the projectile proton fragments can recombine with valence quarks (di-quarks) and with sea quarks to produce charmed hadrons, including charmed hyperons. For example, in this model the process p → Λ + c + ¯ D 0 would be expected at a level ( m s /m c ) 2 relative to associated production of strangeness, p → Λ K + . The parameters of the model are adjusted to fit the then available data on charm production in Ref. [35], and a parameterization of the prompt muon flux in the energy range from 5 TeV to 5000 TeV is given. The value of R c for this model is /similarequal 8 × 10 -4 at 10 TeV. The RQPM model is close to the LVD upper limit, but still allowed by it. It is interesting that the recent IceCube limit on the prompt contribution to ν µ -induced upward muons [36] is just at the level of the RQPM model.", "pages": [ 5 ] }, { "title": "IV. ATMOSPHERIC LEPTONS INCLUDING THE KNEE", "content": "My presentation at ISVHECRI-2012 showed a series of figures in which the contribution of charm was shown separately from the 'conventional' atmospheric leptons from decay of pions and kaons. In those figures, a simple power law spectrum was assumed for evaluating the fluxes of the conventional leptons, and the charm contributions were taken directly from the models. Here I take account of the knee in the spectrum, and I make a preliminary effort to present a consistent representation of the prompt component by rescaling their fluxes to the same spectrum of nucleons used for the conventional leptons, as described in the previous paragraph. Since the primary spectrum is no longer a power law, the spectrum-weighted moments depend on energy. To evaluate energy-dependent Z-factors, Eq. 8 is used. For simplicity here only the energy dependence of the spectrum is considered, and standard values of nucleon interaction and attenuation lengths are used. In addition, a scaling form for meson production is assumed. The goal here is to demonstrate the effect of the knee in the cosmic-ray spectrum on the lepton fluxes.", "pages": [ 5, 6 ] }, { "title": "A. Approximation for pion and kaon production", "content": "Explicit approximations for scale-independent meson production forms are given in Ref. [38]. The approximation for nucleons to produce charged pions is with a similar form for production of kaons. With these scaling forms for particle production, the integral in Eq. 8 can be rewritten as", "pages": [ 6 ] }, { "title": "B. Approximation for nucleon spectrum", "content": "The nucleon spectrum with the knee as parameterized in Table I and Eq. 2 can be approximated well (better than 10% to 30 PeV) with the standard two-power-law form of Ref. [39] to describe the knee, a form which is also used in Ref. [24]. Specifically, with γ = 1 . 64, δ = 0 . 67, E ∗ = 9 .E + 5 GeV and /epsilon1 = 3 . 0. This approximation locates the knee in the nucleon spectrum just below a PeV, and the slope steepens from an integral spectral index of 1 . 64 below the knee to 2 . 31 after the knee. The normalization constant is 10290 . m -2 sr -1 s -1 . The steepening in the nucleon spectrum is a consequence of the steepening of the all-particle spectrum amplified by the increasing fraction of nuclei in the all-particle spectrum. Equations 12, 12 and 13 are combined and integrated numerically to obtain energydependent Z-factors.", "pages": [ 6 ] }, { "title": "C. Atmospheric leptons including the knee", "content": "The fluxes of µ ± , ν µ + ¯ ν µ from decay of pions and kaons are obtained using the energy-dependent Z-factors to evaluate Eq. 4. The energy-dependence of the spectral index in the meson decay factors that appear in Eq. 4 are also accounted for by using the local (energy-dependent) integral spectral index of the nucleon spectrum, which steepens gradually from γ = 1 . 64 to 2 . 31 through the knee region. Calculation of the flux of electron neutrinos requires tracing the contributions of Ke3 decays of both charged and neutral kaons. This has been done by taking account of the neutron/proton ratio of the primary nucleons and tracking separately the production of K ¯ K pairs and production of kaons in association with Λ and Σ hyperons. An approximate value of n/p = 0 . 54 has been used [8]. As noted in the previous section, associated production of kaons by dissociation of an incident nucleon into a kaon and a hyperon is a prototype for intrinsic charm. Results for the lepton fluxes are shown in Figs. 3 and 4. The vertical muon fluxes are shown in Fig. 3, comparing the ERS model for prompt muons (left) to the RQPM model (right). Below the knee the RQPM prompt flux is rescaled down by about 15%, while the ERS prompt muon flux is rescaled up by about 20%. The rescaled ERS prompt muons are repeated on the right panel for comparison. In the region between 100 TeV and 1 PeV the rescaled RQPM prompt flux is a factor of 5 to 6 higher than the rescaled ERS flux. The RQPM prompt component crosses the conventional flux around 300 TeV, as compared to a crossover at 2 PeV for ERS (left panel). Figure 4 compares the situation for ν µ (left panel) with that for ν e (right panel). The ERS model is shown for the prompt component. The RQPM flux is also repeated on both plots for reference. The conventional atmospheric ν e flux is approximately a factor 20 lower than the conventional flux of ν µ , so the electron neutrino component is dominated by the prompt component at quite low energy.", "pages": [ 6, 7 ] }, { "title": "V. EXPECTED RATES", "content": "The fluxes described above can be used to estimate the rate of events in a kilometer scale detector. For atmospheric muons the rate per year is simply the flux multiplied by 3 × 10 7 seconds/yr and by 10 10 cm 2 /km 2 . The corresponding integral rate of events I µ ( > E µ ) is shown in Fig. 5. The rate of neutrino-induced muons can be obtained in a similar way, with one additional step. It is necessary to calculate the effective area to convert the rate of neutrinos with trajectories passing through the detector to a rate of neutrino-induced muons. Effective area is defined in such a way that φ ( E ν , θ ) × A eff ( E ν , θ ) is the rate of neutrino-induced muons per second per sr at zenith angle θ . Explicitly where P ( E ν ) is the probability that a neutrino converts and produces a muon that reaches the detector with enough energy to be reconstructed. Absorption of neutrinos in the Earth becomes significant in the 10 to 1000 TeV range, first for vertically upward trajectories and for neutrinos with zenith angles ∼ 120 · around a PeV. An accurate calculation of A eff requires a detector simulation. Here I use an estimate for an ideal km 2 detector from Ref. [40] and estimate the rate of neutrinoinduced muons in the zenith angle range from horizontal to -120 · . The result is shown for the ERS assumption in the left panel of Fig. 6. Electron neutrinos must interact in the detector to be identified as cascades in the detector. Such cascades are virtually indistinguishable from neutral current interactions of muon or electron neutrinos of energy E ν ∼ E ν e /y , where y is the inelasticity of the neutral current neutrino interaction. The neutral current interactions of atmospheric ν µ make a comparable contribution to cascades for the conventional atmospheric neutrinos because the flux of ν µ is significantly higher than that of ν e . For the charm component, however, the neutral cur- ribution is relatively unimportant because of the equality of the fluxes of prompt ν e and ν µ . The integral rate of ν e interactions from all directions is where φ ( E ν ) is the spectrum of ν e +¯ ν e averaged over all directions and σ cc is the charged current cross section, taken here from Ref. [41]. N is the number of nucleons per cubic kilometer of ice and T=1 year. The resulting estimate of the rate of atmospheric ν e interactions is shown in the right panel of Fig. 6. The plot includes the effect of neutrino shadowing by the Earth in the upward hemisphere [42]. This amounts to a suppression for the whole sky of 9% at 100 TeV and 23% at a PeV. The plot shows about 5 electron neutrino interactions above 100 TeV per km 3 of ice per year assuming ideal (full) efficiency. The spectrum is steeply falling so that less than one such event in ten years is expected above a PeV.", "pages": [ 7, 8 ] }, { "title": "VI. SUMMARY COMMENTS", "content": "In view of the recent observation by IceCube of two cascade-like events with observed energy just above a PeV [43], together with the observation of several high=energy cascades reported at this meeting [44], it is important to achieve a good understanding of the background from atmospheric neutrinos in the energy region around 100 TeV and above. The work outlined in this paper is just a start. There are large uncertainties in the primary spectrum in the TeV range and above that need to be assessed. The model used here is just one possibility. A model for charm production that includes recent data from LHC [45, 46] in its fits needs to be developed. As an example, the framework for including charm within SIBYLL [48] exists [49], but the parameters need to be tuned to data over a wide range of energy and phase space. An updated model could then be used for a full Monte Carlo simulation including charm. The analytic approximations will remain an important tool to supplement the Monte Carlo, for example to track the consequences of uncertainties in the input spectrum and hadronic interactions for the expected fluxes [47], as well as to parameterize and extrapolate the limited statistics of the full Monte Carlo [28].", "pages": [ 9 ] }, { "title": "Acknowledgments", "content": "I am grateful to colleagues in the IceCube collaboration for discussions in various contexts that stimulated this work. My research related to IceCube is supported in part by NSF-PHY-1205809. My research related to hadronic interaction models is supported in part by the U.S. Department of Energy, DE-SC0007893.", "pages": [ 9 ] } ]

[ { "title": "Disentangling the Air Shower Components Using Scintillation and Water Cherenkov Detectors", "content": "Javier G. Gonzalez, Markus Roth, and Ralph Engel Institute for Nuclear Physics, Karlsruhe Institute of Technology (KIT) Abstract. We consider a ground array of scintillation and water Cherenkov detectors with the purpose of determining the muon content of air showers. The di ff erent response characteristics of these two types of detectors to the components of the air shower provide a way to infer their relative contributions. We use a detailed simulation to estimate the impact of parameters, such as scintillation detector size, in the determination of the size of the muon component.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The measurement of the mass composition of ultra-high energy cosmic rays is one of the keys that can help us elucidate their origin. Using a surface detector array to disentangle the contributions to the detector signal from the di ff erent components of an air shower is a way to accomplish this. Up to energies around 10 15 eV, the Galaxy is believed to be the source of cosmic rays. Several acceleration mechanisms are certainly at play but it is widely expected that the dominant one is first order Fermi acceleration at the vicinity of supernova remnant shock waves. These Galactic accelerators should theoretically become ine ffi cient between 10 15 eV and 10 18 eV. The KASCADE experiment has measured the energy spectra for di ff erent mass groups in this energy range and found that there is a steepening of the individual spectra at an energy that increases with the cosmic ray mass [1]. As a result, the mass composition becomes progressively heavy. It is also thought that extra-galactic sources can start to contribute to the total cosmic ray flux at energies above 4 × 10 17 eV. The onset of such an extra-galactic component would probably produce another change in composition. The X max measurements from the HiRes-MIA experiment have been interpreted as a change in composition, from heavy to light, starting at 4 × 10 17 eV and becoming proton-dominated at 1.6 × 10 18 eV [2,3]. Both HiRes and the Pierre Auger Observatory has measured a suppression of the flux of cosmic rays at the highest energies [4,5] and the X max measurements hint at a light or mixed composition that becomes heavier beyond 2 × 10 18 eV [7]. Roughly speaking, the techniques for inferring the mass composition of cosmic rays can be split in two categories, depending on whether they exploit the sensitivity to the depth of shower maximum ( X max) or to the ratio of the muon and electromagnetic components of the air shower [8]. Direct measurements of the fluorescence emission fall in the first category, and so do the various measurements of the Cherenkov light produced by air showers. Most ground-based detector observables depend one way or another on the number of muons in the air shower. However, the arrival time profile of shower particles has been used as an observable mostly sensitive to X max, in particular the so-called rise-time , the time it takes for the signal to rise from 10% to 50% of the integrated signal [9]. The measurement of the number of muons and electrons in an air shower can be done directly, for example, the way it was done with the KASCADE detector [10]. It is important that we disentangle the contributions from the di ff erent components of the air shower, as this relates to the primary mass as well as possible systematic uncertainties arising from the use of Monte Carlo hadronic interaction generators.", "pages": [ 1 ] }, { "title": "EPJ Web of Conferences", "content": "The Pierre Auger Observatory is developing a series of enhancements that aim at measuring showers in the energy range between 10 17 eV and 10 19 eV [11,12]. In particular, the objective of the AMIGA enhancement [11] is the measurement of the muon component of the air showers using scintillators shielded by several meters of soil. In the same spirit, we are considering a combined surface array, consisting of two super-imposed ground arrays, a Water Cherenkov Detector (WCD) array and a scintillation detector array. The purpose of the scintillation detectors is to increase the sensitivity to the electromagnetic component of air showers. In this article we consider the possibility of studying extensive air showers induced by cosmic rays using a combined detector consisting of water Cherenkov and scintillation detectors. In order to do this, we have developed a detailed simulation and reconstruction chain whose characteristics will be briefly described in Section 2. We will then look at the general features that allow us to gain sensitivity to the mass composition of cosmic ray primaries at energies around 10 18 eV and conclude, in Section 4, by considering the possibility of determining the contribution from the muon component of a shower from a pair of scintillation and WCD detectors.", "pages": [ 2 ] }, { "title": "2 Studying the Characteristics of a Combined Surface Detector", "content": "The detector setup we are considering consists of an array of WCDs and an array of scintillators, both covering the same area. Each WCD is like a typical Pierre Auger detector. That is: it is made of a 12 ton cylindrical water tank with 10 m 2 top surface area and the light is collected by three 9 inch photomultipliers placed at the top of the tank facing down. The light collected on the PMTs is then digitized by a 40 MHz Flash ADC. The array of scintillation detector stations is arranged in a regular grid. We have considered different grid configurations and these will be specified in Sections 3 and 4. Each scintillation detector station is made of 3 cm thick plastic scintillator tiles. In order to enhance the signal from gamma-rays in air showers, we have studied the e ff ect of adding a certain amount of lead on top of the scintillators. For conversion of around 80% of the high energy gamma-rays one normally needs a shielding of about 2 radiation lengths. One radiation length corresponds to 0.56 cm of lead. In order to study such a combined detector, we have implemented a simulation and reconstruction chain based on the Pierre Auger Observatory o ffl ine framework [13]. All showers were generated using CORSIKA [14], with QGSJET II for high energy hadronic interaction simulations [15]. The specific zenith angles and energies studied will be mentioned in Sections 3 and 4. The simulation of the interactions of the shower particles with the detector is done using the Geant4 package [16,17]. The scintillation e ffi ciencies used in the simulation correspond to the specifications for Bicron's BD-416 scintillators: Polyvinyl Toluene scintillators with a nominal light yield of about 10 4 photons / MeVand a density of 1.032 g / cm 3 . The resulting scintillation photons are sampled with a 100% e ffi ciency at a frequency of 100 MHz to produce one FADC trace per station. The signal in each station is measured in units of Minimum Ionizing Particle equivalent, or MIP , where a MIP is given by the position of the peak of the Landau distribution for vertical muons. We then consider only stations with signals between 1 and 2000 MIP in order to simulate a limited dynamical range. The arrival direction and core position of each event are estimated using only the WCDs. The arrival direction is determined by fitting a spherical shower front to the signal start times of the stations in the event. The core position is determined by adjusting a lateral distribution function (LDF) of the form to the total signal in the stations in the event. The r 0 parameter depends on the WCD array grid spacing. It will be 450 m when the stations are separated by 750 m and 1000 m when they are separated by 1500m. Correspondingly, S 450 and S 1000 are the usual energy estimators for a WCD array with these grid spacings as they are close to the optimum distance for determining the signal in each case [18, 19].", "pages": [ 2 ] }, { "title": "3 Measuring Composition at Energies around 10 18 eV", "content": "We have considered various configurations in order to estimate the cosmic ray primary composition at energies around 10 18 eV. In a previous contribution we showed that the addition of lead converters on top of the scintillator stations to enhance the contribution from photons in the shower does not increase the sensitivity to the primary mass [20]. We have also considered various spacings between scintillator detectors, while keeping the total array area as well as the total collecting area constant. We have considered three such arrangements for the scintillator array, corresponding to three di ff erent spacings: 433, 612, and 750 m regular triangular grid. The area of the scintillator stations in these configurations are 3.2, 6.4, and 9.7 m 2 respectively. One can see in Figure 1 that the average LDF from light primaries is steeper than that from heavier primaries. In order to be sensitive to the mass of the primary, one needs to sample the LDF at points away from the crossing point of the two LDFs. This crossing point depends on energy as well as zenith angle, as displayed in Figure 2. We need therefore to measure the signal either close to or far away from the shower axis. Since we are considering this scintillator array to be placed around a base configuration consisting of a fixed-size WCD array, it follows that the scintillators must probe the region close to the axis. The signals from the scintillation detectors are used to estimate an LDF on an event-by-event basis using a function of the form Asample of a few reconstructed LDFs is depicted in Figure 3. In this figure one can see that there will be an optimum distance from the shower axis to measure the scintillator signal. The optimum radial distance will be the one where the spread of the reconstructed signal relative to the total reconstructed signal is minimal. This is shown in Figure 4, where we display the relative signal fluctuations for the di ff erent configurations used. For reference, in both figures, we display the radius at which the local trigger is 95% e ffi cient, r 95. At this point it becomes clear that trading o ff individual detector size for a denser array will increase the composition sensitivity, since the optimum distance will be displaced closer to the axis. In a similar way, the optimum distance for composition sensitivity is found by minimizing the signal fluctuations in relation to the average separation of the proton and iron LDFs. For this reason one can choose slightly smaller distances to the shower axis. We can then correlate the recorded scintillator signal at a specified distance with the WCD at 450 m. This correlation, with the scintillator signal measured at 400 m, can be seen in Figure 5. From the log 10 ( S sci) log 10 ( S WCD) correlation it is possible to provide an estimator of the primary mass composition.", "pages": [ 3, 4 ] }, { "title": "4 Estimating the Muon Signal At the Highest Energies", "content": "In the previous Section we discussed how to use an array of scintillators to measure the electromagnetic (EM) component of air showers and found that decreasing the spacing between the scintillators increases the sensitivity to the primary mass composition. The LDF of the electromagnetic component is steeper, therefore we need a denser array in order to reconstruct it. We now turn to the other extreme. We will consider how well we can determine the contributions from the EM and muon components in a single pair of WCD / scintillator stations, one next to the other. At the highest energies, the electromagnetic LDF will extend to larger distances and, while it will then not be possible to reconstruct it using an array of scintillators separated the same distance as the WCD detectors, a significant fraction of events will contain at least a pair of WCD / scintillator stations with which to determine the contribution from the EM and muon components. We are then interested in the fraction of events that will have a station with a signal at a short distance to the axis. In Figure 6 we show the radial distribution of the non-saturated station closest to the axis for each event in a collection of 10 19.5 eV protons arriving at a zenith angle of 38 · . One can see that 30% of the events will have at least one station within 800 m of the axis (the region marked in blue). The peak around 1000 m corresponds to events where the closest station is saturated. The radial distribution shows two distinct peaks. The peak around 950 m corresponds to events where the WCD station closest to the axis is saturated and therefore the station that enters the distribution is the closest WCD station that is not saturated. If we consider the closest WCD station, regardless of its saturated status, the fraction raises to 95%. The relation between the scintillator and WCD signals should provide, in principle, a way to estimate the contribution from the muon component, as shown in Figure 7, where we depict a collection of 10 19.5 eV protons arriving at 38 · with the vertical where we have scaled the number of muons by an arbitrary factor between 0 and 3. We use the linear correlation between the scintillator and WCD signals for each component: to determine s WCD µ , the contribution from the muon component to the WCD signal. In Figure 8 we show the statistical uncertainty with which we can measure s WCD µ when we use scintillator stations with an area of 1.6 m 2 . This uncertainty is between 40% and 50% and could be reduced by increasing the detector size. For 10 m 2 detectors it would be around 30%. This calculation was done for stations at fixed radii but the uncertainty in the core position introduces another source of uncertainty that is related to slight changes in α EM in equation 3. The impact of this e ff ect can be estimated and is shown in Figure 9. From this we can conclude that an uncertainty of 30 m in the core position would produce an uncertainty of less than 5% in the reconstructed muon signal.", "pages": [ 4, 5 ] }, { "title": "5 Summary", "content": "We have conducted a detailed study of the sensitivity of a combined scintillator / WCD array to primary cosmic-ray mass composition. Studying the response of this array to 10 18 eV showers, we have concluded that adding photon converters on top of the scintillators to enhance the signal from the electromagnetic component of the shower does not increase the sensitivity. We have also concluded that having many small scintillation detectors is better than having few detectors of larger size since the optimum distance is closer to the shower axis. Specifically, an array of 3.2 m 2 detectors separated by 375m is better than an array of 9.7 m 2 separated by 750 m. We have applied these ideas to the highest energies and determined that, for 10 19.5 eV primaries, more than 30% of the events will have a pair of scintillator / WCD station within 800m of the axis. Using this pair stations it would be possible to estimate the signal from the muon component in the WCDtanks with an uncertainty of 50% when the WCD station has an area of 10 m 2 and the scintillator has an area of 1.6 m 2 . This uncertainty can be reduced by increasing the scintillator size.", "pages": [ 5, 6 ] } ]

[ { "title": "ABSTRACT", "content": "EPJ Web of Conferences will be set by the publisher DOI: will be set by the publisher c © Owned by the authors, published by EDP Sciences, 2018", "pages": [ 1 ] }, { "title": "Self Sustained Traversable Wormholes Induced by Gravity's Rainbow and Noncommutative Geometry", "content": "Remo Garattini 1 , 2 , a 1 Università degliStudidiBergamo,FacoltàdiIngegneria, VialeMarconi,524044Dalmine(Bergamo)ITALY 2 I.N.F.N.-sezione diMilano,Milan,Italy Abstract. We compare the e ff ects of Noncommutative Geometry and Gravity's Rainbow on traversable wormholes which are sustained by their own gravitational quantum fluctuations. Fixing the geometry on a well tested model, we find that the final result shows that the wormhole is of the Planckian size. This means that the traversability of the wormhole is in principle, but not in practice.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "In 1988 on the American Journal of Physics, M. S. Morris and K. S. Thorne published a paper entitled ' Wormholes in spacetimes and their use for interstellar travel: A tool for teaching general relativity '[1]. Although the subject of the paper could be regarded as an argument of Science Fiction more than of Science, its impact on the scientific community was so amazing to open the doors to new investigations in Astrophysics, General Relativity and Quantum Gravity. In practice a traversable wormhole is a solution of the Einstein's Field equations, represented by two asymptotically flat regions joined by a bridge: roughly speaking it looks like a short-cut in space and time. To exist, traversable wormholes must violate the null energy conditions, which means that the matter threading the wormhole's throat has to be ' exotic '. Classical matter satisfies the usual energy conditions. Therefore, it is likely that wormholes must belong to the realm of semiclassical or perhaps a possible quantum theory of the gravitational field. Since a complete theory of quantum gravity has yet to come, it is important to approach this problem semiclassically. On this ground, the Casimir energy on a fixed background. has the correct properties to substitute the exotic matter: indeed, it is known that, for different physical systems, Casimir energy is negative. However, instead of studying the Casimir energy contribution of some matter or gauge fields to the traversability of the wormholes, we propose to use the energy of the graviton on a background of a traversable wormhole. In this way, one can think that the quantum fluctuations of the traversable wormholes can be used as a fuel to sustain traversability. Di ff erent contexts can be invoked to study self sustained traversable wormholes. In this paper, we review some aspects of self sustained traversable wormholes fixing our attention on Noncommutative geometry and Gravity's Rainbow.", "pages": [ 1 ] }, { "title": "2 Self-sustained Traversable Wormholes", "content": "In this Section we shall consider the formalism outlined in detail in Refs. [3, 4], where the graviton one loop contribution to a classical energy in a wormhole background is used. The spacetime metric representing a spherically symmetric and static wormhole is given by where Φ ( r ) and b ( r ) are arbitrary functions of the radial coordinate, r , denoted as the redshift function, and the form function, respectively [1]. The radial coordinate has a range that increases from a minimum value at r 0, corresponding to the wormhole throat, to infinity. A fundamental property of a wormhole is that a flaring out condition of the throat, given by ( b -b ' r ) / b 2 > 0, is imposed [1, 2], and at the throat b ( r 0) = r = r 0, the condition b ' ( r 0) < 1 is imposed to have wormhole solutions. Another condition that needs to be satisfied is 1 -b ( r ) / r > 0. For the wormhole to be traversable, one must demand that there are no horizons present, which are identified as the surfaces with e 2 Φ → 0, so that Φ ( r ) must be finite everywhere. The classical energy is given by where the background field super-hamiltonian, H (0) , is integrated on a constant time hypersurface. R is the curvature scalar, and using metric (1), is given by We shall henceforth consider a constant redshift function, Φ ' ( r ) = 0, which provides interestingly enough results, so that the curvature scalar reduces to R = 2 b ' / r 2 . Thus, the classical energy reduces to A traversable wormhole is said to be ' self sustained ' if where E TT is the total regularized graviton one loop energy. Basically this is given by where τ denotes a complete set of indices and E 2 i ( τ ) > 0, i = 1 , 2 are the eigenvalues of the modified Lichnerowicz operator acting on traceless-transverse tensors of the perturbation and where /triangle L is the Lichnerowicz operator defined by with /triangle = -∇ a ∇ a . For the background (1), one can define two r-dependent radial wave numbers where   are two r-dependent e ff ective masses m 2 1 ( r ) and m 2 2 ( r ). When we perform the sum over all modes, E TT is usually divergent. In Refs. [3, 4] a zeta regularization and a renormalization have been adopted to handle the divergences. In this paper, we will consider the e ff ect of the Noncommutative geometry and Gravity's Rainbow on the graviton to one loop.", "pages": [ 2, 3 ] }, { "title": "3 Noncommutative Geometry and Gravity's Rainbow at work on a Traversable Wormhole Background", "content": "One of the purposes of Eq.(3) is the possible discovery of a traversable wormhole with the determination of the shape function. Nevertheless, another strategy can be considered if we fix the wormhole shape to be traversable, at least in principle. One good candidate is which is the prototype of the traversable wormholes[1]. Plugging the shape function (9 a ) into Eq.(3), we find that the left hand side becomes while the right hand side is divergent. To handle with divergences, we have several possibilities. In this paper we adopt and compare two schemes: the Noncommutative scheme and the Gravity's Rainbow procedure. Beginning with the Noncommutative scheme, we recall that in Ref.[6], we used the distorted number of states to compute the graviton one loop contribution to a cosmological constant. The distortion induced by the Noncommutative space time allows the right hand side of Eq.(3) to be finite. Indeed, plugging dni into Eq.(3), one finds that the self sustained equation for the energy density becomes where we have used the shape function (9 a ) to evaluate the e ff ective masses (8). If we define the dimensionless variable Eq.(12) leads to ( G = l 2 P ) 3 π 2 2 P l θ = F ( x ) , and where we have also fixed g 1( E / EP ) = exp ( -α E 2 / E 2 P ) with α variable. Now, in order to have only one solution with variables α and rt , we demand that which takes the following form after the integration ( G -1 = E 2 P ) where F ( x ) has a maximum for ¯ x = 0 . 24, where This fixes θ to be and As regards Gravity's Rainbow, as shown in Ref.[8], the self sustained equation (3) becomes where and   Eq.(18) is finite for appropriate choices of the Rainbow's functions g 1 ( E / EP ) and g 2 ( E / EP ). Fixing the shape function as in Eq.(9 a ) and assuming g 2 ( E / EP ) = 1, to avoid Planckian distortions in the classical term, we find where x = r 0 EP and where K 0( x ) and K 1( x ) are the modified Bessel function of order 0 and 1, respectively. Even in this case, to have one and only one solution, we demand that the expression in the right hand side of Eq. (24) has a stationary point with respect to x which coincides with the constant value 1. For a generic but small α , we can expand in powers of α to find which has a root at Substituting ¯ x into Eq. (24), we find fixing therefore α /similarequal 0 . 242. It is interesting to note that this value is very close to the value α = 1 / 4 used in Ref.[7] inspired by Noncommutative analysisof Ref.[6] . As in Refs. [4, 5], it is rather important to emphasize a shortcoming in the analysis carried in this section, mainly due to the technical di ffi culties encountered. Note that we have considered a variational approach which imposes a local analysis to the problem, namely, we have restricted our attention to the behavior of the metric function b ( r ) at the wormhole throat, r 0. Despite the fact that the behavior is unknown far from the throat, due to the high curvature e ff ects at or near r 0, the analysis carried out in this section should extend to the immediate neighborhood of the wormhole throat. Nevertheless it is interesting to observe that in Ref.[3] the greatest value of the wormhole throat was fixed at r 0 /similarequal 1 . 16 / EP using a regularizationrenormalization scheme. From Eq.(27), one immediately extracts r 0 /similarequal 1 . 46 / EP which is slightly larger. We have to remark that in the Noncommutative case we have only one parameter to be fixed: θ = 7 . 43 × 10 -2 l 2 P . On the other hand, in Gravity's Rainbow, we have more flexibility, because of the unknown functions g 1 ( E / EP ) and g 2 ( E / EP ). Therefore we conclude that Gravity's Rainbow o ff ers a wider variety of examples which deserve to be explored.", "pages": [ 3, 4, 5 ] } ]

[ { "title": "ABSTRACT", "content": "The Journal's name will be set by the publisher DOI: will be set by the publisher c © Owned by the authors, published by EDP Sciences, 2018", "pages": [ 1 ] }, { "title": "Discrete mechanics, 'time machines' and hybrid systems", "content": "Hans-Thomas Elze 1 , a 1 DipartimentodiFisica'EnricoFermi',UniversitàdiPisa, LargoPontecorvo 3,I-56127Pisa,Italia Abstract. Modifying the discrete mechanics proposed by T.D. Lee, we construct a class of discrete classical Hamiltonian systems, in which time is one of the dynamical variables. This includes a toy model of 'time machines' which can travel forward and backward in time and which di ff er from models based on closed timelike curves (CTCs). In the continuum limit, we explore the interaction between such time reversing machines and quantum mechanical objects, employing a recent description of quantum-classical hybrids.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Time travel and time machines have been the stu ff of science fiction for a while and possibly excited human minds much earlier than that. - However, they have become a topic of active scientific enquiry since the realization that certain cosmological solutions of Einstein's equations of general relativity allow for closed timelike curves (CTCs). Here an object can travel in an unusual geometry of spacetime, such that it encounters the past and, in particular, its own past. It is obvious that with the link between quantum mechanics and general relativity still little understood - this provides an arena for producing paradoxes ( e.g. , grandfather paradox, unproved theorem paradox) and testing new ideas how to resolve them, besides availing surprising computational resources. - Discussion of background, an overview of existing literature, and the state of the art of constructing quantum mechanical time machines can be found in Refs. [1, 2]. Our aim here is threefold. - In Sect. 2, we recall T.D. Lee's proposal of time as a fundamentally discrete dynamical variable [3, 4]. Limiting the number of events or measurements in a given spacetime region, this can have surprising consequences in the continuum limit. We modify his action principle in such a way that a Hamiltonian formulation for such discrete systems becomes available. Furthermore, we show that these systems allow for a particular kind of time machines, namely time reversing machines . - In Sect. 3, we review a recent attempt to construct a theory that describes quantum-classical hybrids , consisting of quantum mechanical and classical objects that interact directly with each other [5-7]. Hybrids might exist as a fundamentally di ff erent species of composite objects \"out there\", with consequences for the range of applicability of quantum mechanics, or they may serve as approximate description for certain complex quantum systems. We employ the concept of quantum-classical hybrids, in order to explore a hypothetical direct coupling of classical time machines to quantum objects . In Sect. 4, we introduce a specific model, obtain its equations of motion, and discuss consequences for the study of time machines, followed by conclusions in Sect. 5.", "pages": [ 1 ] }, { "title": "2 Discrete Hamiltonian mechanics", "content": "Discrete dynamical systems arise in many contexts in physics or mathematics, for example, in discrete approximations or maps facilitating numerical studies of complex systems, as regularized versions of quantum field theories on spacetime lattices, or describing intrinsically discrete processes. Here, the usual preponderance of di ff erential equations over finite di ff erence equations is given up for reasons which may be more or less fundamental. In a series of articles T.D. Lee and collaborators have proposed to incorporate discreteness as a fundamental aspect of dynamics, see Refs. [3, 4] and further references therein, and have elaborated various classical and quantum models in this vein, which share desirable symmetries with the corresponding continuum theories while presenting finite degrees of freedom. For our purposes, it will be su ffi cient to consider classical discrete mechanics which derives from the basic assumption that time is a discrete dynamical variable . This naturally invokes a fundamental length or time (in natural units), l . Which can be rephrased as the assumption that in a fixed ( d + 1)dimensional spacetime volume Ω maximally N = Ω / l d + 1 measurements can be performed or this number of events take place. In Refs. [3, 4] a variational principle was presented, based on a Lagrangian formulation of the postulated action. Various forms and (dis)advantages of such an approach have subsequently been discussed, e.g. , in Refs. [8-10]. Presently, we present a Hamiltonian formulation which di ff ers from all previous ones in that it leads to particularly transparent and symmetric equations of motion. This allows us to introduce a suitable Poisson bracket and a phase space description of the dynamics. The latter is an essential ingredient when constructing quantum-classical hybrids, as we shall see in the following Sect. 3. We describe the state n of a discrete mechanical object by its positions in spacetime in terms of the real dynamical variables xn , τ n and corresponding conjugated momenta pn , P n , with n = 0 , 1 , 2 , . . . . 1 We postulate that its dynamics is governed by the stationarity of this action : under independent variations of all variables and momenta; the finite di ff erences are defined by: and the Hamiltonian function by H : = ∑ n H n , with: where V is a su ffi ciently smooth potential and K n will be specified in due course. The variations of the action amount to di ff erentiations here and lead to the equations of motion: introducing the discrete 'time derivative', ˙ On : = On + 1 -On -1 , on the left-hand and Poisson brackets (cf. below) on the right-hand sides, respectively; furthermore, En : = 1 2 ( p 2 n + p 2 n -1 ) + V ( xn ) + V ( xn -1) . Furthermore, stationarity of the action under independent variations of τ n and xn for every state n implies invariance under translations in time and space , respectively, and the conservation of energy and momentum (modulo the e ff ect of the external force deriving from V ). This holds under the further assumption that K n does not enter through its derivatives in Eqs. (4)-(5). Several remarks are in order here. - Assuming that K n depends only on the state n , it can be shown that the set of Eqs. (4)-(7) is time reversal invariant ; the state n + 1 can be calculated from knowledge of the earlier states n and n -1 and the state n -1 from the later ones n + 1 and n . Explicit solutions of the equations of motion can be easily found in the case that K n : = 0 and the potential V is constant or a linear function, i.e. , for zero or constant external force . This recovers the behaviour discussed in Refs. [3, 8]. However, in the present formulation, we also have the possibility to study more exotic models, in which the dynamics of the time variable τ n itself plays an important role when K n /nequal 0, cf. Sect. 2.2. Considering the dynamical variables and canonically conjugated momenta as canonical coordinates for the phase space spanned by { xn , τ n ; pn , P n } , we introduce the Poisson bracket of any two regular functions f and g on this space: which has been indicated already on the right-hand sides of Eqs. (4)-(5), This allows a convenient description also of ensembles of discrete mechanical objects, which individually follow the above equations of motion. Let us collectively denote variables and momenta as Qn and Pn , respectively, such that { f , g } = ∑ n ( ∂ f ∂ Qn ∂g ∂ Pn -∂ f ∂ Pn ∂g ∂ Qn ) . Then, we postulate, in analogy to continuum mechanics (see Subsect. 2.1), a continuity equation to determine the flow of the probability density ρ n ≡ ρ n ( Qn ; Pn ) of the ensemble in phase space: with ˙ On : = On + 1 -On -1 , as before. Employing the equations of motion (4)-(7), this continuity equation can be rewritten as the discrete mechanics analogue of the Liouville equation : In the following, we study the continuum limit of the equations of motion, in which time remains one of the dynamical variables.", "pages": [ 2, 3 ] }, { "title": "2.1 The continuum limit", "content": "In order to discuss the continuum limit, we let the fundamental time (or length) constant become arbitrarily small, l → 0, such that the density of events or measurements becomes correspondingly large, N →∞ . Furthermore, we introduce the external time , t : = nl , with n = 0 , 1 , 2 , . . . , and define x ( t ) : = xn , τ ( t ) : = τ n , p ( t ) : = pn , P ( t ) : = P n , i.e. , in terms of the discrete dynamical variables and conjugated momenta. Thus, for example, τ n + 1 -τ n = τ ( t + l ) -τ ( t ) = ˙ τ ( t ) l + O( l 2 ) , where ˙ τ : = d τ/ d t , etc. - In this way, we obtain the equations of motion in the continuum limit: where terms containing ∑ K n ' will be defined and evaluated shortly. It su ffi ces here to assume that all K n ' are independent of { xn , pn } . This simplifies Eqs. (11)-(12): implies d / d t [ p 2 / 2 + V ( x )] = 0 , and, consequently, simplifies also Eq. (14). In this case, ˙ τ plays the role of a given 'lapse' function for the subsystem described by x and p , which can be separately determined (cf. Sect. 2.2). ıI.e., if Eqs. (13)-(14) are integrated explicitly, the remaining Eqs. (15) follow from the time dependent e ff ective Hamiltonian: with ˙ τ as a time dependent parameter. The existence of a simple continuum Hamiltonian, such as H c , is not obvious, in general, since ∆ τ n on the right-hand side of Eq. (3) becomes proportional to ˙ τ , if one performs the continuum limit directly on the discrete dynamics Hamiltonian; the presence of this factor can spoil the Hamiltonian picture of the resulting dynamics.", "pages": [ 3, 4 ] }, { "title": "2.2 Time machines", "content": "Here we illustrate the continuum limit of the discrete mechanics that we obtained. We choose K n : = l [ P 2 n + V ( τ n )] . Then, the continuum limit applied to Eqs. (13)-(14) gives simply: with ˙ τ : = d τ/ d t , etc. We observe that for suitable potentials V ( τ ) and initial conditions the internal time τ will perform a bounded periodic motion as function of the external time t . For example, for an oscillator potential, V ( τ ) : = ω 2 τ 2 , we obtain solutions τ ( t ) = ¯ τ sin( ω t ), with amplitude ¯ τ and phase determined by the initial conditions, such that ˙ τ ( t ) = ˙ τ ( -t ) is time reversal invariant. Furthermore, the Eqs. (15) can be rewritten as a single second order equation: i.e. , as an ordinary equation of motion with respect to the internal time, which is considered as a function of the external time, to be obtained from Eqs. (17). This situation describes a toy model of time machines : the x , p -subsystem moves forward in time on a particular trajectory in phase space, as long as τ ( t ) increases; when, due to its periodicity, this function decreases, this trajectory is traced identically backwards! Thus, the behaviour in the external time t is cyclic, alternating between forward and backward evolution. We remark that this dynamical implementation of 'time travel' di ff ers from a frequently considered one, which is based on modifying the background spacetime structure. In particular, Politzer's spacetime, which allows closed timelike curves (CTCs), is obtained by identifying a certain spatial region at one time with the same region at a later time [11]; thus, an object may transit instantaneously from a final state to the corresponding (identical) initial state of its evolution. In our model, it evolves identically backwards from a final state to its initial state; it is conceivable that this can be realized in physical analogue models. In Sect. 4, we explore the coupling of such a classical time machine to a quantum object in a particular framework describing quantum-classical hybrids.", "pages": [ 4, 5 ] }, { "title": "3 Quantum-classical hybrids", "content": "The direct coupling of quantum mechanical (QM) and classical (CL) degrees of freedom 'hybrid dynamics' - departs from quantum mechanics. We summarize here briefly the theory presented in Refs. [5-7], where also additional references and discussion of related works can be found. Hybrid dynamics has been researched extensively for various reasons. - For example, the Copenhagen interpretation of quantum mechanics entails the measurement problem which, together with the fact that quantum mechanics needs interpretation, in order to be operationally well defined, may indicate that it needs amendments. Such as a theory of the dynamical coexistence of QM and CL objects. This should have impact on the measurement problem [12] as well as on the description of the interaction between quantum matter and (possibly) classical spacetime [13]. Furthermore, it is of great practical interest to better understand QM-CL hybrids appearing in QMapproximation schemes addressing many-body systems or interacting fields, which are naturally separable into QM and CL subsystems; for example, representing fast and slow degrees of freedom, mean fields and fluctuations, etc. Concerning the hypothetical emergence of quantum mechanics from some coarse-grained deterministic dynamics (see Refs. [14-16] with numerous references to related work), the quantumclassical backreaction problem might appear in new form, namely regarding the interplay of fluctuations among underlying deterministic and emergent QM degrees of freedom. Which can be rephrased succinctly as: 'Can quantum mechanics be seeded?' Thus, there is ample motivation for the numerous attempts to formulate a satisfactory hybrid dynamics. Generally, they are deficient in one or another respect. Which has led to various no-go theorems, in view of the lengthy list of desirable properties or consistency requirements that ' the ' hybrid theory should fulfil, see, for example, Refs. [17, 18]: These issues have also been discussed for the hybrid ensemble theory of Hall and Reginatto, which does conform with the first six points listed but is in conflict with the last two [21, 22]. We have proposed an alternative theory of hybrid dynamics based on notions of phase space [5]. This extends work by Heslot, demonstrating that quantum mechanics can entirely be rephrased in the language and formalism of classical analytical mechanics [23]. Introducing unified notions of states on phase space, observables, canonical transformations, and a generalized quantum-classical Poisson bracket, this has led to an intrinsically linear hybrid theory, which allows to fulfil all of the above consistency requirements. Recently Buri'c and collaborators have shown that the dynamical aspects of our proposal can indeed be derived for an all-quantum mechanical composite system by imposing constraints on fluctuations in one subsystem, followed by suitable coarse-graining [24, 25]. Besides constructing the QM-CL hybrid formalism and showing how it conforms with the above consistency requirements, we earlier discussed the possibility to have classical-environment induced decoherence, quantum-classical backreaction, a deviation from the Hall-Reginatto proposal in presence of translation symmetry, and closure of the algebra of hybrid observables [5, 7]. Questions of locality, symmetry vs. separability, incorporation of superposition, separable, and entangled QM states, and 'Free Will' were considered in Ref. [6].", "pages": [ 5, 6 ] }, { "title": "3.1 Quantum mechanics - rewritten in classical terms", "content": "We recall that evolution of a classical object can be described in relation to its 2 n -dimensional phase space, its state space . A real-valued regular function on this space defines an observable , i.e. , a di ff erentiable function on this smooth manifold. There always exist (local) systems of canonical coordinates , commonly denoted by ( xk , pk ) , k = 1 , . . . , n , such that the Poisson bracket of any pair of observables f , g assumes the standard form: This is consistent with { xk , pl } = δ kl , { xk , xl } = { pk , pl } = 0 , k , l = 1 , . . . , n , and has the properties defining a Lie bracket operation, mapping a pair of observables to an observable. General transformations G of the state space are restricted by compatibility with the Poisson bracket structure to so-called canonical transformations , which do not change physical properties of an object. They form a Lie group and it is su ffi cient to consider infinitesimal transformations. An infinitesimal transformation G is canonical , if and only if for any observable f the map f →G ( f ) is given by f → f ' = f + { f , g } δα , with some observable g , the so-called generator of G , and δα an infinitesimal real parameter. - Thus, for example, the canonical coordinates transform as follows: This illustrates the fundamental relation between observables and generators of infinitesimal canonical transformations in classical Hamiltonian mechanics. Following Heslot's work, we learn that the previous analysis can be generalized and applied to quantum mechanics; this concerns the dynamical aspects as well as the notions of states, canonical transformations, and observables [23]. The Schrödinger equation and its adjoint can be derived as Hamiltonian equations from an action principle [5]. We must add the normalization condition , C : = 〈 Ψ ( t ) | Ψ ( t ) 〉 ! = 1 , for all state vectors | Ψ 〉 , which is essential for the probability interpretation of amplitudes; state vectors that di ff er by an unphysical constant phase are to be identified. Thus, the QM state space is formed by the rays of the underlying Hilbert space.", "pages": [ 6, 7 ] }, { "title": "3.1.1 Oscillatorrepresentation", "content": "A unitary transformation describes QM evolution, | Ψ ( t ) 〉 = ˆ U ( t -t 0) | Ψ ( t 0) 〉 , with U ( t -t 0) = exp[ -i ˆ H ( t -t 0) / /planckover2pi1 ], solving the Schrödinger equation. Thus, a stationary state, characterized by ˆ H | φ i 〉 = Ei | φ i 〉 , with real energy eigenvalue Ei , performs a harmonic motion, i.e. , | ψ i ( t ) 〉 = exp[ -iEi ( t -t 0) / /planckover2pi1 ] | ψ i ( t 0) 〉 ≡ exp[ -iEi ( t -t 0) / /planckover2pi1 ] | φ i 〉 . We assume a denumerable set of such states. Following these observations, it is quite natural to introduce the following oscillator representation . We expand state vectors with respect to a complete orthonormal basis, {| Φ i 〉} : where the time dependent coe ffi cients are separated into real and imaginary parts, Xi , Pi [23]. This expansion allows to evaluate what we define as Hamiltonian function , i.e. , H : = 〈 Ψ | ˆ H | Ψ 〉 : Choosing the set of energy eigenstates, {| φ i 〉} , as basis of the expansion, we obtain: hence the name oscillator representation . - Evaluating | ˙ Ψ 〉 = ∑ i | Φ i 〉 ( ˙ Xi + i ˙ Pi ) / √ 2 /planckover2pi1 according to Hamilton's equations with H of Eq. (22) or (23), gives back the Schrödinger equation. - Furthermore, the normalization condition becomes: Thus, the vector with components given by ( Xi , Pi ) , i = 1 , . . . , N , is confined to the surface of a 2 N -dimensional sphere with radius √ 2 /planckover2pi1 , which presents a major di ff erence to CL Hamiltonian mechanics. The ( Xi , Pi ) may be considered as canonical coordinates for the state space of a QM object. Correspondingly, we introduce a Poisson bracket , cf. Eq.(19), for any two observables on the spherically compactified state space , i.e. real-valued regular functions F , G of the coordinates ( Xi , Pi ): As usual, time evolution of an observable O is generated by the Hamiltonian: d O / d t = ∂ tO + { O , H} . In particular, we find that the constraint of Eq. (24) is conserved: d C / d t = {C , H} = 0 .", "pages": [ 7, 8 ] }, { "title": "3.1.2 Canonicaltransformationsandquantumobservables", "content": "In the following, we recall briefly the compatibility of the notion of observable introduced in passing above - as in classical mechanics - with the usual QM one. This can be demonstrated rigourously by the implementation of canonical transformations and analysis of the role of observables as their generators. For details, see Refs. [5-7, 23]. The Hamiltonian function has been defined as observable in Eq. (22), which relates it directly to the corresponding QM observable, namely the expectation of the self-adjoint Hamilton operator. This is indicative of the general structure with the following most important features: From these equations, the relation between an observable G , defined in analogy to classical mechanics (as above), and a self-adjoint operator ˆ G can be inferred: i.e. , by comparison with the classical result. Hence, a real-valued regular function G of the state is an observable, if and only if there exists a self-adjoint operator ˆ G such that Eq. (27) holds . This implies that all QM observables are quadratic forms in the Xi 's and Pi 's, which are essentially fewer than in the corresponding CL case; interacting QM-CL hybrids require additional discussion, see Ref. [7]. with both sides of the equality considered as functions of the variables Xi , Pi and with the commutator defined as usual. Hence, the QM commutator is a Poisson bracket with respect to the ( X , P ) state space and relates the algebra of its observables to the algebra of self-adjoint operators. In conclusion, quantum mechanics shares with classical mechanics an even dimensional state space, a Poisson structure, and a related algebra of observables. It di ff ers essentially by a restricted set of observables and the requirements of phase invariance and normalization, which compactify the underlying Hilbert space to the complex projective space formed by its rays.", "pages": [ 8, 9 ] }, { "title": "3.2 Quantum-classical Poisson bracket, hybrid states and their evolution", "content": "The far-reaching parallel of classical and quantum mechanics, as we have seen, suggests to introduce a generalized Poisson bracket for QM-CL hybrids: of any two observables A , B defined on the Cartesian product of CL and QM state spaces. It shares the usual properties of a Poisson bracket. - Note that due to the convention introduced by Heslot [23], to which we adhered in Sect. 3.1, the QM variables Xi , Pi have dimensions of (action) 1 / 2 and, consequently, no /planckover2pi1 appears in Eqs. (29)-(30). At the expense of introducing appropriate rescalings, these variables could be made to have their usual dimensions and /planckover2pi1 to appear explicitly here. - For the remainder of this article, instead we choose units such that /planckover2pi1 ≡ 1. Let an observable 'belong' to the CL (QM) sector, if it is constant with respect to the canonical coordinates of the QM (CL) sector . Then, the { , } × -bracket has the important properties: The hybrid density ρ for a self-adjoint density operator ˆ ρ in a given state | Ψ 〉 is defined by [5] : using Eq. (21) and ρ i j ( xk , pk ) : = 〈 Φ i | ˆ ρ ( xk , pk ) | Φ j 〉 = ρ ∗ ji ( xk , pk ). It describes a QM-CL hybrid ensemble by a real-valued, positive semi-definite, normalized, and possibly time dependent regular function on the Cartesian product state space canonically coordinated by 2( n + N )-tuples ( xk , pk ; Xi , Pi ); the variables xk , pk , k = 1 , . . . , n and Xi , Pi , i = 1 , . . . , N are reserved for CL and QM sectors, respectively. It can be shown that ρ ( xk , pk ; Xi , Pi ) is the probability density to find in the hybrid ensemble the QM state | Ψ 〉 , parametrized by Xi , Pi through Eq. (21), together with the CL state given by a point in phase space, specified by coordinates xk , pk . - Further remarks, concerning superposition, pure / mixed, or separable / entangled QM states that may enter the hybrid density can be found in Ref. [6]. Furthermore, the simple form of ρ as bilinear function of QM 'phase space' variables Xi , Pi , stemming from the expectation of a density operator ˆ ρ , has to be generalized for interacting hybrids, allowing for so-called almost-classical observables ; see Sect. 5.4 of Ref. [5] and a related study [7]. We are now in the position to introduce the appropriate Liouville equation for the dynamical evolution of hybrid ensembles [5]. Based on Liouville's theorem and the generalized Poisson bracket defined in Eqs. (29)-(30), we are led to: with H Σ ≡ H Σ ( xk , pk ; Xi , Pi ) and: which defines the relevant Hamiltonian function, including a hybrid interaction; H Σ is required to be an observable , in order to have a meaningful notion of energy. Note that energy conservation follows from {H Σ , H Σ } × = 0. An important advantage of Hamiltonian flow and a general property of the Liouville equation is: · G) The normalization and positivity of the probability density ρ are conserved in presence of a hybrid interaction; hence, its interpretation remains valid.", "pages": [ 9, 10 ] }, { "title": "4 Quantum control by a classical time machine", "content": "Our aim here is to combine the results on discrete mechanics (Sect. 2), where time is one of the dynamical variables and which consequently allows to model a particular kind of time machines (Sect. 2.2), with those on QM-CL hybrids (Sect. 3). We explore in this framework, how such a classical time machine interacts with a quantum object. As a concrete example, we consider an oscillator-like time machine coupled to a q-bit . The former is represented by the Hamiltonian function: cf. Eq. (16), where Ω denotes the proper oscillator frequency, while ω is the frequency of the change of time direction, cf. Sect. 2.2, and the dimensionless constant ζ parametrizes its amplitude. For Ω /greatermuch ω , the oscillator performs many oscillations (circles in phase space), before the time direction changes; conversely, for Ω /lessmuch ω , the oscillator moves only little before beginning to trace its trajectory in phase space in the opposite direction. Qualitatively similar behaviour of the time machine is expected for other than oscillator potentials. The q-bit is described, in the oscillator representation, cf. Sect. 3.1.1, by the Hamiltonian function: with E 0 an energy scale, cf. Eq. (23). Wave function normalization, Eq. (24), is required by: The model is completed by choosing a hybrid interaction, for example: using the oscillator expansion of a generic state | Ψ 〉 ; Oij : = 〈 φ i | ˆ O | φ j 〉 denotes a matrix element of the q-bit observable ˆ O ( = ˆ O † ) in the basis of energy eigenstates corresponding to H QM and λ is a coupling constant. Naturally, other and more general interactions may be considered. Then, the following Hamilton equations are obtained in the usual way from the hybrid Hamiltonian H Σ : = H CL + H QM + I : where we set ζ ≡ 1, which can always be implemented by rescaling time, E 0, and λ . In agreement with the general result in Eqs. (27)-(28) of Ref. [5], the constraint of Eq. (36) is conserved under this Hamiltonian flow, d C / d t = {C , H Σ } × = 0, and, therefore, it is su ffi cient to impose the constraint on the initial conditions of the equations of motion (38)-(41). In order to uncover some characteristic features of this hybrid model, we introduce the internal time variable τ ( t ) : = ω -1 sin( ω t ) into Eqs. (38)-(39). The resulting second order equation (for x ( τ )) of a driven harmonic oscillator can be solved with the help of its retarded Green's function: where the first term solves the homogeneous equation, incorporating integration constants x 1 and φ , and where the inhomogeneity is given by: with X 's and P 's evaluated at t ( s ), determined (modulo π/ω ) by t = ω -1 arcsin( ω s ) ; for simplicity, the q-bit obervable has been assumed to be proportional to the spin-1 / 2 Pauli matrix σ 2, such that -O 12 = O 21 = i / 2 and O 11 = O 22 = 0. Using solution (42) and Eq. (38), we obtain: We see explicitly that the time machine travels periodically forwards and backwards in time, due to the periodicity of its internal time τ with respect to the external time t governing the chronology respecting q-bit (described by the X , P -variables). Most notably, ˙ x and p are not always aligned, i.e. , of same sign. However, since from one period of forward (or backward) evolution to the next the external time increases by 2 π/ω , generally, the value of the function ˜ O in Eqs. (42)-(44) will change accordingly. This implies that the classical time machine that interacts with the q-bit, will go backwards in ( x , p ) phase space in a di ff erent way than it came! Which can entail known paradoxes of time travel, such as the grandfather paradox or the unproved theorem paradox [1, 2]. Summarizing, the interaction of a classical time machine with a chronology respecting system, the q-bit here, introduces an aspect of 'ageing' into its dynamics: despite going forwards and backwards in time, in general, its state does evolve and depends on the external time t . Unlike solutions to such paradoxes proposed in the literature for quantum systems in the presence of closed timelike curves (CTCs) by Deutsch [26], Lloyd and collaborators in the form of post-selected teleportation (P-CTCs) [1, 2], or the consideration of open timelike curves (OTCs) by Ralph and collaborators [27], our model of a classical time machine does not provide enough freedom to eliminate paradoxical situations (Novikov principle) by imposing additional constraints on its dynamics. Apparently, quantum-classical hybrids do not work here. - Considering a suitably constrained ensemble of classical time machines might help. However, its physical relevance remains to be seen. Of course, having a QM time machine consistently interacting with a classical object is not ruled out by the present model. In fact, previously considered CTC scenarios should reduce to such a hybrid situation under suitable circumstances. We consider the e ff ect of the time machine on the q-bit next. In this case, we conveniently rewrite Eqs. (40)-(41) by undoing the oscillator expansion, cf. Eq. (21): where ˆ σ 2 , 3 are the imaginary and diagonal spin-1 / 2 Pauli matrices, respectively. This is a Schrödinger equation representing a spin-1 / 2 in a magnetic field . In particular, here its 2-component is time dependent. With the time dependence arising from x ( t ) cos( ω t ), cf. Eqs. (42)-(43), this e ff ective Schrödinger equation is nonlinear and non-Markovian . The nonlinear and non-Markovian behaviour can be neglected for su ffi ciently small coupling λ , in which case the time dependence of the e ff ective magnetic field is given by the following factor: incorporating only the first term from the right-hand side of Eq. (42). Concerning the q-bit, the corresponding instantaneous eigenvalues of the e ff ective Hamiltonian are shifted in this approximation (lowest nonvanishing order in λ ) and are simply given by: This result would, in principle, allow to constrain parameters defining the present toy model of a quantum-classical hybrid, consisting of a classical time machine interacting with a q-bit, given the manifold laboratory realizations of q-bits. Furthermore, among others, there are generalizations of the hybrid interaction, Eq. (37), which could give rise to a rotating magnetic field instead of the oscillating one in Eqs. (45)-(46). This, in turn, produces e ff ects like a Berry phase , or its generalizations (see, e.g. , the recent Ref. [28] and references therein), which could serve as well to constrain such models. However, as we have discussed, the classical time machines addressed here are likely bound to reproduce the paradoxes of time travel. If they are not directly observable, for some reason, their indirect e ff ects on quantum systems may be worth further study.", "pages": [ 10, 11, 12 ] }, { "title": "5 Conclusions", "content": "Our purpose here has been to explore the possibility that classical 'time machines' couple directly to quantum mechanical objects. We invoked the discrete mechanics proposed by T.D. Lee, in which time belongs to the set of dynamical variables [3, 4]. Suitably modifying the underlying action, we have developed a Hamiltonian theory of such discrete classical dynamical systems . Choosing the dynamics of the time variable appropriately, we are led to systems which evolve forward and backward in time, time machines or, more precisely, time reversing machines. In the continuum limit and for particular choices of the dynamics of time, the motion is periodic. Thus, such an object evolves forward in time, forming a trajectory in phase space, until it comes to a halt, then traces this trajectory backwards in time, comes to halt, evolves forward again, and so on. A clock carried on board would be seen running alternatingly forwards and backwards. - These time machines are distinct from the closed timelike curves (CTCs) on which an object travels, which have been frequently discussed in the literature, see Refs. [1, 2], for example, and works referred to there. They might be realizable in physical analogue models. In order to describe the direct coupling of such a time machine with a quantum object, e.g. a q-bit, we reviewed our recent proposal for a consistent quantum-classical hybrid dynamics, which is based on a phase space formalism for classical as well as for quantum mechanics [5-7, 23, 24]. We have defined a toy model of such a QM-CL hybrid, consisting of an oscillator like classical time machine coupled to a q-bit and discussed its Hamiltonian equations of motion. While this could lead to observe the action of a time reversing machine through its e ff ects on a quantum object, we have also pointed out that common time travel paradoxes would a ff ect the classical time machine. In retrospect, the latter is understandable, since the outcome of the evolution of a classical object is deterministic and fixed, e.g. , by initial conditions, to the extend that no additional (nonlinear) constraints can be imposed, as in the quantum case. The QM-CL hybrids that we described do not alter this circumstance. For quantum mechanical objects travelling on CTCs, however, such constraints serve to suppress the paradoxes by selecting well-behaved ones from the ensemble of all possible histories [1, 2, 26]. Which poses the question whether a quantum mechanical time reversing machine , based on the Hamiltonian discrete meachanics presented here, can similarly avoid time travel paradoxes? Acknowledgements: It is a pleasure to thank M. Crosta, M. Gramegna and M. Ruggiero for the invitation to give a talk in the inspiring atmosphere of the conference Time Machine Factory (Torino, October 2012) and to thank N. Buri'c, L. Maccone, and C. Stoica for discussions and correspondence.", "pages": [ 12, 13 ] }, { "title": "The Journal's name", "content": "[26] D. Deutsch, Quantum mechanics near closed timelike lines , Phys.Rev. D 44 , 3197 (1991)", "pages": [ 14 ] } ]

[ { "title": "Could quantum decoherence and measurement be deterministic phenomena?", "content": "∗ Jean-Marc Sparenberg , R'eda Nour and Aylin Man¸co Universit'e libre de Bruxelles (ULB), Nuclear Physics and Quantum Physics, CP 229, ' Ecole polytechnique de Bruxelles, B 1050 Brussels, Belgium August 19, 2021", "pages": [ 1 ] }, { "title": "Abstract", "content": "The apparent random outcome of a quantum measurement is conjectured to be fundamentally determined by the microscopic state of the macroscopic measurement apparatus. The apparatus state thus plays the role of a hidden variable which, in contrast with variables characterizing the measured microscopic system, is shown to lead to a violation of Bell's inequalities and to agree with standard quantum mechanics. An explicit realization of this interpretation is proposed for a primitive model of measurement apparatus inspired by Mott [1]: in the case of an α -particle spherical-wave detection in a cloud chamber, the direction of the observed linear track is conjectured to be determined by the position of the atoms of the gas filling the chamber. Using a stationary-state coupled-channel Born expansion, a reduction of the spherical wave function is shown to be necessary to compensate the flux loss due to scattering on the chamber atoms. Being highly non local, this interpretation of quantum mechanics is finally argued to open the way to faster-than-light information transfer.", "pages": [ 1 ] }, { "title": "1 Introduction: advocacy for determinism", "content": "The most disturbing feature of quantum physics is probably the fundamental randomness of the outcome of a quantum measurement: the repeated measurement of the same observable on a microscopic system prepared in the same state, with the same apparatus, can lead to very different results. Think for instance about the detection of an s -wave α -particle in a cloud chamber. Though the wave function is perfectly spherical [see equation (13) below], the observed individual tracks are linear, spread isotropically in apparently totally random directions. This seemingly unavoidable lack of determinism very deeply bothered Einstein himself, as expressed by his famous opinion, expressed in a 1926 letter to Born, that '[God] does not throw dice' [2]. Indeed, determinism is one of the keystones of modern science, as experiments are only reproducible provided the same cause always produces the same effect. Giving up this very principle is in a sense equivalent to giving up science itself! Another motivation for not giving up determinism is timeless theories. In these, time does not have a fundamental character but rather emerges as a secondary quantity (see in particular the contribution by Barbour at this conference and reference [3]). In a deterministic world, the present moment is unambiguously related to the past from which it results and to the future which it determines, hence strengthening the idea that past, present and future are one and the same. A possible way to recover determinism in quantum mechanics is to assume, in the spirit of Einstein, Podolsky and Rosen [4], that the knowledge of a microscopic system provided by quantum mechanics is incomplete. Hidden variables should exist, that determine the measurement outcome, even though they might not be accessible to the experimenter. Another fundamental advantage of hidden variables in Einstein's view is that they avoid violation of special relativity by quantum mechanics: instantaneous correlations between distant particles in an intricated state can be explained by a common variable shared by the particles when they first interacted. The very logical assumption of hidden variables was however proved incorrect. Bell [5] (also reproduced in [6]) and others [7, 8, 9] established inequalities that should be satisfied by measurement outcomes on intricated particles if such hidden variables existed, and that are violated by quantum mechanics predictions. Aspect and others [10, 11, 12] then checked experimentally that quantum predictions were correct, which ruled out hidden variables at least in their simplest form. These experiments also prove that quantum mechanics is highly non local and that the measurement performed on one particle indeed has an instantaneous impact on the measurement performed on another particle intricated with the first one, even at very large distances. The lack of determinism of quantum measurements then comes as a 'savior' of special relativity [13]: since the outcome of the measurement on the first particle is random, the intrication between both particles, though it exists and is instantaneous, cannot be used to transmit information faster than light. In the present work, we try to answer the question: what actually determines the result of a given quantum measurement? In the case of the α -particle detection, what actually determines the direction of an observed track? We explore the hypothesis that hidden variables are not variables characterizing the microscopic system under study (the α -particle in the cloud chamber experiment or the pair of intricated particles in an EPR-type experiment), but rather the macroscopic apparatus (or apparatuses) used to study them. Indeed, the mysterious properties of quantum mechanics (randomness and non locality) always occur when a microscopic system, well described by its wave function, interacts with a highly complex and unstable macroscopic system. Describing the state of such a complex system at the microscopic level is of course very difficult as its internal degrees of freedom (or more generally its 'environment') cannot be monitored and appear as totally random to our 'macroscopic eyes'. It is however tempting to assume, as also suggested by Gaspard [14], that it is precisely this microscopic state of the system which is at the origin of the randomness observed during the measurement process. Our approach is also related to attempts to describe quantum measurement processes based on decoherence [15, 16, 17]. There also, it is the interaction with a macroscopic system, e.g. a measurement apparatus, that makes a microscopic system initially in a linear superposition of states evolve into a standard statistical mixture of possible outcomes. Mathematically, decoherence is usually described by ad hoc terms in a master equation, with no attempt to explain particular outcomes of the decoherence process into one or another result. In the present approach, we go a step further, similarly to Gaspard and Nagaoka [18]: the internal state of the macroscopic system precisely determines which outcome will occur, not only the mere fact that probabilistically-acceptable outcomes do occur. The aim of the present work is to test whether this simple deterministic interpretation leads to fundamental principle impossibilities or not. In the following, we first show that it leads to violations of Bell's inequalities, in agreement with standard quantum mechanics and with experimental results. The price to pay for that success is a strong non locality, which is not saved any more by quantum randomness [19]. Next we come back to the problem of α -ray tracks in a cloud chamber. We explore the hypothesis that the positions of the atoms of the chamber actually determine the direction of the observed track, hence playing the role of apparatus hidden variables randomly distributed because of the thermal agitation inside the chamber. Possible experimental tests of this idea are then proposed. Finally, this interpretation is speculated, if correct, to open the way to faster-than-light communications.", "pages": [ 1, 2 ] }, { "title": "2 Apparatus hidden variables and Bell's inequalities", "content": "Let us first recall the principle of the Einstein-Podolsky-Rosen thought experiment [4], as formulated by Bohm-Aharonov [7]. A system of two particles, 1 and 2, both with spin 1 2 denoted as s 1 and s 2 , is created in the maximally entangled singlet (i.e. spin 0) state where ˆ u is an arbitrary unit vector. The particles are assumed to stay in that spin state while moving away from each other. The spin of particle 1 is then measured in direction ˆ a while the spin of particle 2 is measured in direction ˆ b , each measurement leading to the result + ¯ h 2 or -¯ h 2 . Following Bell [5], we define the mean correlation between spin 1 and 2 measured along directions ˆ a and ˆ b as For the singlet state (1), this quantity can be shown to take the value in standard quantum mechanics. In particular, when ˆ a = ˆ b , the correlation between both measurements is perfect and one has E Q ( ˆ a , ˆ a ) = -1, whatever the distance between both particles at the time their spins are measured. This non-local correlation, typical of entangled quantum states, chagrined Einstein, hence leading to the hypothesis that the state of the particle pair was given not only by the wave function (1) but also by some hidden variable λ . This variable is assumed to be randomly distributed; its value is supposed to determine the result of any of the measurements. It thus also determines the value of correlation (3), which we now denote as E λ ( ˆ a , ˆ b ). Such a variable avoids 'spooky actions at a distance' apparently implied by state (1). However, Bell [5] brilliantly proved that the very existence of variable λ leads to inequalities for the correlations calculated for three arbitrary directions ˆ a , ˆ b , ˆ c , namely The observation by Bell is that these inequalities are violated by the quantum prediction (3) for particular directions ˆ a , ˆ b , ˆ c (for instance differing by π 8 ), hence offering the possibility to discriminate between standard quantum mechanics and hidden variables experimentally. Such experiments were realized with photon polarizations by Aspect et al. [12] and displayed a clear disagreement with inequality (4) while confirming the quantum prediction (3). This result somehow signs the death warrant of hidden variables characterizing the microscopic system (in this case the pair of particles). Let us now show that in contrast it does not prevent the hypothesis of hidden variables characterizing the internal state of the measurement apparatus. In the present situation, we introduce variables Λ 1 and Λ 2 that correspond to the apparatus measuring particle 1 and 2 respectively. In the same spirit as Bell, we do not attempt to find the explicit physical nature of these variables; we only assume that they exist and that they determine the experimentally observed results, together with the particle state and the apparatus orientation. Let us denote by the result of the measurement of the projection of the spin s of one of the spin 1 2 particles assumed to be in state | σ 〉 , in one of the measurement apparatus oriented along direction ˆ a . This result is assumed to be determined by the value of the hidden variable Λ characterizing the internal state of the apparatus. The precise value of Λ is not known but its probability distribution p (Λ) has to satisfy ∑ Λ p (Λ) = 1, where the sum covers all the possible values of Λ. The two apparatuses are assumed to be identical; a unique function (5) hence describes both of them. This function has to satisfy the following conditions to describe the measurement process correctly: The last two conditions are compatible with each other and condition (7) also implies that Within these hypotheses, the mean correlation between the two measurements reads where | σ i 〉 is the state of particle i as measured by apparatus i . To fix ideas, we assume that the measurement in apparatus 1, oriented in direction ˆ a , is made before that in apparatus 2, oriented in direction ˆ b (but the time interval between both measurements can be made arbitrarily small). Hence we choose to write the singlet state (1) with ˆ u = ˆ a . The state of particle 1, as seen by apparatus 1, is thus | σ 1 〉 = 1 √ 2 [ | + 〉 ˆ a -|-〉 ˆ a ], which implies that the first measurement gives +1 or -1 in 50% of the cases. This allows an easy calculation of the first sum in (9). Immediately after this first measurement, particle 2 is put in either a down or an up state along direction ˆ a , depending on the result obtained for the first measurement. The second apparatus being oriented along direction ˆ b , the second sum in (9) has thus to be calculated using equations (7) and (8). One has finally The result is thus identical to the standard quantum mechanics prediction (3). Apparatus hidden variables hence also lead to a violation of Bell's inequalities (4), in agreement with experiment. While still agreeing with standard quantum mechanics, the present theory is thus deterministic, as the measurement results are determined by the actual values of Λ 1 and Λ 2 . Another advantage is that these hidden variables are local, in the sense that Λ 1 only characterizes the state of apparatus 1 and Λ 2 only characterizes the state of apparatus 2. Hence there is no need to consider generalized inequalities for non-local hidden variables [20]. The price to pay for this simplicity is in contrast a strong non-locality. The measurement performed in apparatus 1 is actually a measurement of the particle pair as a whole, not only of particle 1: it immediately affects the state of both particles by reducing their wave function to either | + 〉 ˆ a 1 |-〉 ˆ a 2 or |-〉 ˆ a 1 | + 〉 ˆ a 2 . The very structure of wave function (1) thus allows an actual 'spooky action at a distance' which implies the perfect correlations between the states of both particles. The consequences of this irreducible quantum non locality will be discussed in a relativistic perspective in the speculative conclusions. Before that, we would like to consider a model of measurement apparatus in which the physical nature of these internal degrees of freedom can be explicitly described, to test the hypothesis that they determine the measurement outcome. To do that, we consider what is probably the simplest possible measurement apparatus to model schematically, namely the cloud chamber mentioned in the introduction. There the Λ internal variables could just be the positions of the atoms in the chamber. Note that a schematic model of Stern-Gerlach apparatus could be built on the same basis, with the screen being replaced by a cloud chamber. As for photon polarization measurement apparatuses, a microscopic model should be based on internal electronic degrees of freedom; mesoscopic or cold polarisers might be interesting to consider in that respect.", "pages": [ 2, 3, 4 ] }, { "title": "3 A schematic model for deterministic quantum measurement", "content": "In the present section, we revisit an idea proposed by Mott [1] (also reproduced in [21]) and discussed in [22] (also reproduced in [6]), namely the measurement of a spherical-wave α -particle state in a Wilson cloud chamber. Mott shows that the observation of linear tracks can be explained by the interaction of the α -particle spherical wave with the atoms of the gas filling the chamber. Here, we reproduce his calculation but interpret his result differently, hence leading to the interpretation that the presence of the gas might not only explain the appearance of linear tracks but also determine which linear track is actually observed for a given microscopic configuration of the gas atoms. We thus claim that Mott's model might be considered as the first model for a deterministic decoherent process of wave-function reduction appearing in a quantum measurement. Let us first consider the interaction between an α s -wave, emitted by a typical spherical α emitter like 210 Po, and a single atom in a cloud chamber [see figure 1(a)]. We assume that this atom, which is the first obstacle seen by the α wave, is immobile at a fixed position a . This is justified as the typical thermal agitation energy of molecules in a cloud chamber (a few meV) is much lower than the typical α -particle kinetic energy E α (a few MeV). This obstacle is rather generic; its precise nature will not affect the general conclusions drawn below. To simplify calculations, we assume that it is described by an internal Hamiltonian H with internal variables r and we only consider two excitations levels: its ground state E 0 and its first excited state E 1 . The corresponding wave functions ψ 0 ( r ) and ψ 1 ( r ) are orthonormal, i.e. 〈 ψ i | ψ j 〉 = δ ij for i, j = 0 , 1; they are assumed to be localised in a region of size s , e.g. the typical size of an atom. The total hamiltonian of the system is thus ✈ ④ ✁ ④ ∑ ∑ where T α is the kinetic energy of the α particle and V ( R , r ) describes the interaction between the α particle and the obstacle. In the following, this interaction is assumed to be small, which allows the use of a Born-expansion perturbative treatment [23]. Within these hypotheses, the stationary wave function of the total α -particle + obstacle system can be approximately described as a first-order Born-expanded coupled-channel wave function, which reads In this equation, the superscript refers to the order in the Born expansion while the subscript refers to the excitation state of the obstacle. The first term corresponds to the situation where the α particle is unaffected by the presence of the obstacle; its wave function is thus an outgoing spherical wave of kinetic energy E α , This wave function is represented in figure 2(a); in this figure, the exponential phase factor is visible as coloured rings, while the 1 /R modulus, necessary for flux conservation, is visible as decreasing brightness. The second term of (12) is the first-order Born expansion of the elastic scattering; the wave function f (1) 0 ( R ) hence corresponds to the same kinetic energy E α as the zero-order term. Note that this term was not considered by Mott in [1]. The third term of (12) corresponds to the excitation of the obstacle ✈ ✐ by the α particle; hence, by energy conservation, the wave function f (1) 1 ( R ) is characterized by an energy E ' α and a wave number k ' which are defined as These first-order terms can be calculated with the help of the Green-function method and read (17) ∫ ∗ j V j 0 ( R ) ≡ d r ψ ( r ) V ( R , r ) ψ 0 ( r ) . (18) The remarkable feature of these wave functions, as already stressed by Mott [1], is that they present an asymptotic behaviour strongly peaked along direction ˆ a . To show that, we first assume that the energy excitation of the obstacle is negligible with respect to the α -particle kinetic energy, which implies k ' ≈ k . Then, by defining the unit vector and the angle θ between ˆ a and ˆ e , one gets for the asymptotic behaviour of the first-order wave functions That is a spherical wave emitted in a , modulated by an angular-dependent function where we have defined the transferred momentum which is related to θ by To establish (21) and (22), we have used the fact that matrix elements (18) only have a non negligible value in a volume of size s around the obstacle position a . Expression (22) contains the Fourier transform of matrix elements (18), a well-known result for the first-order Born approximation. These matrix elements can be calculated explicitly for various assumptions on the obstacle, in particular in the case of hydrogen-like wave functions [1, 23]. Here, as we are rather interested in general properties of the scattering wave function (12), we simply assume that the matrix elements (18) have a spherical Gaussian form factor of width s . This allows us to plot a typical behaviour of the wave function in panel (b) of figure 2. In this figure, only the spherical and peaked elastic-scattering waves are shown; the inelastic-scattering wave would have a behaviour similar to the elastic-scattering one but with a broader peak. Let us now come to the delicate point of the wave-function normalization, which is not discussed in [1]. Here, we fix this normalization by imposing that the total probability flux F across a sphere centred on the α emitter and large enough to include the obstacle should be the same whether this obstacle is present or not. This seems natural, as the activity of the radioactive source should be independent of with its surroundings, but we shall show that it has important physical consequences. Without obstacle, this probability flux takes the value which only depends on the α -particle velocity v α . This derives from the probability current corresponding to the spherical wave function (13), integrated on all angles on the large-radius sphere. With an obstacle, the calculation is more complicated as a projection on the obstacle states has to be made and the current calculation makes an interference term appear between the first two terms of (12). However, these interference terms vanish when the current is integrated on a large enough sphere. Hence, the total flux reads Since the last two terms of this expression are clearly positive, this flux is larger than (25). Hence, the coupled-channel wave function (12) has to be multiplied by a factor C , with a modulus smaller than 1, defined as for the fluxes to be equal with and without obstacle. The effect of this factor is clearly visible on figure 2: the amplitude of the spherical wave is reduced by the presence of the obstacle. One has thus which is the key result of this part of the paper. Following Mott [1], we now consider the scattering on two successive obstacles located in a and b [see figure 1(b)]. A second-order Born expansion is required and the coupled-channel wave function reads The very important result by Mott is that the second-order wave function f (2) 11 , corresponding to an excitation of both atoms, is significantly different from zero only if a and b are aligned (for atoms a and b to be excited, atom b has to lie in the narrow cone generated by the presence of atom a ). This explains the appearance of linear tracks from an initially symmetric spherical wave function. Here, we complete this result by noting that, in case of an alignment, this wave function should be multiplied by a factor of the order of | C | 4 for the probability flux to be equal to the flux without obstacles. We infer that for N successive aligned obstacles, the wave function should be multiplied by a factor | C | 2 N . This would have a spectacular consequence on the spherical-wave flux, which would read That is the spherical wave would tend to zero if there is a large enough number of atoms in the cloud chamber aligned with the α emitter. We conjecture this might be a model for the phenomenon of wave-function reduction in quantum mechanics. Now, in the case of a cloud chamber consisting of randomly-distributed atoms, this mechanism would select the direction of the atoms best-aligned with the α emitter: only for these atoms would the highorder component of the wave function not vanish. The wave-function reduction would thus only occur if some atoms are aligned in a given direction; that direction would be the measured linear track. This model thus provides a deterministic explanation to the apparent randomness of quantum measurement.", "pages": [ 4, 5, 6, 7 ] }, { "title": "4 Possible theoretical flaws and experimental tests", "content": "One should keep in mind that the above reasoning rests on simplifying hypotheses, which should be tested carefully in future works. First, the calculation is based on the Born approximation, which is well-known to violate unitarity [23]. We do not expect this to be problematic here as the high energy of the α particle probably implies that the Born approximation is very good; nevertheless, convergence tests should be carried out. Second, the calculation is based on a truncated coupled-channel approximation; there again, convergence tests including more excited states should be made. These two first possible problems could be avoided by directly solving the Schrodinger equation numerically and compare the obtained result with the approximated one. Finally, last but not least, our interpretation is based on a stationary state calculation, whereas it has a temporal content in essence: the wave function is first affected by the atom nearest to the source, then by the second nearest atom and so on. This stationary-state approach is an approximation of the full solution of the time-dependent Schrodinger equation. There again, solving this equation numerically for instance for an initial spherical wave packet would provide very useful checks. Let us now assume that the reduction of the spherical wave in the presence of an obstacle shown in figure 2 is correct and briefly explore some set-ups that might be used to test it experimentally. A first important aspect to consider is the practical implementation of a spherical-wave emission. A solid-state α source might not be a good candidate as decoherence of the spherical wave might occur in the source already. An alternative option might be a mesoscopic ensemble of radioactive atoms (or ions), trapped in an atomic trap as a low-density gas, so as to limit the interactions between the emitted α particles and the other atoms of the source. A second aspect is the nature of the obstacle leading to the spherical-wave reduction. To display a substantial effect, the interaction between the α particle and the obstacle should be strong. According to (31), a possible way could be to align a large number of atoms, which seems difficult to achieve in practice. We might instead assume that the mechanism proposed above for the cloud chamber, namely that the internal state of the apparatus determines the measurement result, is actually valid for any kind of particle detector. If this is true, replacing the obstacle by a high-efficiency detector covering a limited solid angle would lead to an observable wave-function reduction in other directions. This reduction could be observed by another detector placed at a larger distance from the α -particle source, as illustrated in figure 3. A difficulty with such an experimental set-up might be to get an absolute flux measurement, both in the presence and absence of the 'obstacle detector'. Another option might be to use photons instead of α -particles. Electric dipole photons emitted by trapped polarized atoms are also characterized by a wide (though not spherical) angular distribution. Their interaction with an obstacle (say a single atom) in one particular direction might also lead to a detectable flux reduction in other directions. The advantage of such a set-up would be that the presence or absence of the obstacle could be simulated by tuning or detuning, for instance by Zeeman effect, an atomic transition of the obstacle atom with the energy of the electric-dipole photon, hence strengthening or dimming their interaction. On the other hand, absolute flux measurement and high-efficiency detection might be more difficult to achieve with photons than with α particles.", "pages": [ 8 ] }, { "title": "5 Speculative conclusions: faster-than-light or not?", "content": "As a conclusion we propose an interpretation of quantum mechanics which is deterministic in essence. The outcome of a decoherent process like a measurement can in principle be predicted from the knowledge of the microscopic state of the environment or macroscopic apparatus. This state being inaccessible in most practical situations, the outcomes of decoherence processes generally seem random. Nevertheless this randomness now appears as much less fundamental and unavoidable than in standard quantum mechanics presentations. We have first shown, without making any attempt to describe microscopic states explicitly, that this interpretation leads to a violation of Bell's inequalities, despite the fact that this microscopic state can be seen as a hidden variable. Next we have studied explicitly a simplified model of measurement apparatus, the cloud chamber, and obtained results that support our interpretation: the wave-function collapse leading to the measurement outcome is determined by the positions of the atoms in the cloud chamber. Let us stress that this interpretation, to put it on Mott's words [1], is based on 'wave mechanics unaided': no further ingredient (pilot wave, many worlds, free will. . . ) than wave functions is required. Let us next notice that both types of quantum states considered above are highly non local. The EPR intricated Bell state (1) implies a perfect and instantaneous correlation of the measured states of particles 1 and 2, despite the fact that these states are not known in advance. Similarly the α -particle spherical state (13) implies that the detection of the particle in one direction immediately prevents the particle from being detected in other directions. These states themselves do not violate special relativity and causality as they can be explained by their unique spatio-temporal origin either at the creation of the pair or at the emission of the α -particle. When submitted to a measurement process their non locality is revealed and an action at a distance is necessary to explain the perfect correlations between space-separated events. In usual interpretations of quantum mechanics, causality is however preserved by the random character of the measurement results [13], as already mentioned in the introduction. In contrast, in the present interpretation this random character is replaced by a deterministic interpretation. For the spherical wave, the presence of the obstacle in a given direction immediately leads to a reduction of the spherical wave in all other directions, even if the distance between the emitter and the obstacle can be made very large, at least in principle. Hence by controlling the presence of the obstacle at a given position one could immediately transfer information to remote space-separated regions, as proposed on a small scale in figure 3. For the EPR pair, the state of apparatus 1 formally described by the value of Λ 1 determines the result of the measurement performed on particle 1. To control Λ 1 in practice, one could for instance think of a Stern-Gerlach apparatus with the screen replaced by a cloud chamber or of a cold or mesoscopic polariser. Particles 1 and 2 being fully intricated, the value of Λ 1 also instantaneously determines the state of particle 2, even though apparatus 1 and particle 2 can be separated by very large distances (or by very long optical fibers). The present interpretation thus suggests that quantum non locality actually allows faster-than-light information transfer. This raises of course many paradoxes of special relativity, which might be considered as a very strong 'no-go' argument for our interpretation. We rather speculate that our approach might finally settle the long-standing conflict between special relativity and quantum mechanics, Einstein's two warring daughters. This conflict was made explicit by Einstein himself in the EPR paper, then brought to a climax by Bell and his inequalities, to finally burst into a triumphant victory of quantum mechanics in Aspect's famous experiments. We believe the quantum world to be fully deterministic and non local and hope this work will help testing this very strong hypothesis thoroughly. As for special relativity paradoxes, we adopt a very 'engineering-like' approach by suggesting to first build an instantaneousinformation-transfer machine and, if it works, to deal with the paradoxes it creates afterwards!", "pages": [ 9 ] }, { "title": "Acknowledgements", "content": "JMS acknowledges very interesting discussions with several colleagues at different stages of this work, in particular with N. J. Cerf, D. Baye, J. Barbour, P. Gaspard, C. Semay and P. Capel.", "pages": [ 9 ] }, { "title": "References", "content": "[1] N. F. Mott. The wave mechanics of α -ray tracks. Proc. R. Soc. , A126:79-84, 1929.", "pages": [ 9 ] } ]

[ { "title": "ABSTRACT", "content": "EPJ Web of Conferences will be set by the publisher DOI: will be set by the publisher c © Owned by the authors, published by EDP Sciences, 2018", "pages": [ 1 ] }, { "title": "Elementary cycles of time", "content": "Donatello Dolce 1 , a 1 CoEPP,TheUniversityofMelbourne,Australia&ComerinoUniversity,Italy. Abstract. Elementary particles, i.e. the basic constituents of nature, are characterized by quantum recurrences in time. The flow of time of every physical system can be therefore decomposed in elementary cycles of time. This allows us to enforce the local nature of relativistic time, yielding interesting unified descriptions of fundamental aspects of modern physics, as shown in recent publications and reviewed in [1]. Every particle can be regarded as a reference clock with time resolution of the order of its Compton time, typically many orders of magnitude more accurate than the atomic clocks. Here we summarize basic conceptual aspects of the resulting relational interpretation of the relativistic time flow.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "In recent publications [1-4] we have proposed a formalization of relativistic Quantum Mechanics (QM) in terms of relativistic Elementary Cycles (ECs). This description can be justified, for instance, by the wave-particle duality introduced by de Broglie in terms of a 'periodic phenomenon' attributed to every elementary particle [5]. In a 'periodic phenomenon' the space-time coordinates enter as angular variables, e.g. as in waves and phasors. By assuming ECs in space-time, the recurrence of elementary particle can be implemented directly at the level of space-time geometrodynamics. Technically this is realized in terms of compact space-time dimensions with periodic boundary conditions [2-4]. The resulting description provides elegant solutions of central problems of modern physics such as a unified semiclassical geometrodynamical description: of quantum and classical mechanics [2], of gauge and gravitational interaction [3], as well as an intuitive interpretation of Maldacena's conjecture [4].", "pages": [ 1 ] }, { "title": "2 Elementary cycles", "content": "In physics every system can be described in terms of a set of elementary particles and their interaction schemes. The Standard Model predicts with astonishing precision the physical properties of the experimentally observed matter in terms of basic constituents: leptons (electron, muon, tau, and their neutrinos) and quarks (up, down, charm, strange, top and bottom), constituting proper matter particles; vector bosons (photon, W ± , Z , and gluons) which, together with gravity, describe the possible interactions of the matter particles; and recently the experimental evidences from LHC of the Higgs boson, generating the mass of the the elementary particles. These elementary particles are characterized by di ff erent properties such as mass, charges, and spin, determining their relativistic evolutions in space-time. Interactions and the consequent dynamics of a single particle are denoted by corresponding relativistic variations of its energy ¯ E ( ¯ p ) and", "pages": [ 1 ] }, { "title": "EPJ Web of Conferences", "content": "sexagesimal base), but they need regular adjustments as they mimic natural cycles (e.g. Moon and Earth rotations) which form ergodic systems (or chaotic systems if we also consider interactions). The combinations of relativistic ECs depend on the reference frame of the observer, according to relativistic simultaneity: every observer experiences a di ff erent 'present'. Hence, owing this unique combinatory description of phases associated to every instant in time, the external time axis can be in principle dropped. The time flow of the system can be in fact decomposed as modulations of ECs. Besides the many fundamental physical motivations, such a description of time flow has important philosophical and anthropological justifications (the list would be too long to be mentioned here). This picture can be better understood if we consider the role of the mediator of interactions among particles (for simplicity's sake we consider only photons). As mentioned before, massive particles (expect neutrinos) have typically extremely fast periodicities. On the other hand, the time periodicity of photons can vary from zero to infinity. In fact they have zero mass and therefore infinite proper time recurrence: we say that light has a 'frozen rest clock' T τ ≡ ∞ . In a relational description of time, the long temporal and spatial periodicities of light provide the long scale temporal structure. Photons cycles provide the reference temporal axis of the ordinary interpretation of relativity (emphatically non cyclic time coordinate), allowing a common parameterization of the elementary cyclic phenomena. That is, such long space-time scales with respect to the typical periodicities of matter fields can be used as a reference upon which the ordinary relativistic structure of space-time can be built. In other words, the 'frozen' clock of light (or of gravity) set the causal structure of the ECs of nature. The assumption of intrinsic periodicity enforces the local nature of relativistic time. The helicity (clockwise or anticlockwise) of the internal clocks is arbitrary. Since the flow of time is given by the combination of the elementary clocks 'ticks', the same physical evolution is equivalently described by inverting the helicity of all the elementary clocks of the system. Furthermore, as the external time axis can be in principle defined with reference to every single particle, the inversion of the internal clock of a particle does not imply the inversion of the arrow of time of the whole system (i.e. of the external time axis). The inversion of the helicity of a single internal clock however describes a di ff erent physical system, it in fact means to transform a particle to the corresponding antiparticle (as in Feynman's interpretation).", "pages": [ 4 ] }, { "title": "3 Relativistic time flow", "content": "An isolated system or universe composed by a single particle has a purely cyclic evolution. In fact, every elementary particle is a 'periodic phenomenon' with persistent time recurrence. There are not energy variations in this elementary system. Every period cannot be distinguished by the other. The whole physical information of such a cyclic universe is contained in a single period: 'all that happens in a given period is identical with all that happens in an arbitrary period'. The evolution is parametrized by a cyclic time variable. We can now imagine to add other persistent 'periodic phenomena' into this universe, such as isolated particles and atomic clocks. From each of them the external time axis t ∈ R can be defined by counting the number of theirs 'ticks'. Such an external time axis can be regarded as an angular variable of infinite periodicity, such as that associated to the recurrence of a massless particles with very low energy. Thus the cyclic variables describing the recursive evolutions of every elementary clock can be parametrized by a common time parameter t . The evolution of such a composite system of non interacting 'periodic phenomena' however is not cyclic. The combination of periodicities not rational each other describes an ergodic system. Its orbits pass arbitrarily close to the initial phase-space point without never passing from it again (the definition of elementary systems could depend on the resolution). Now we imagine to turn on interactions among the elementary particles of this ergodic system. The energy propagates according to the retarded relativistic potential (relativistic undulatory mechanics is in fact based upon relativistic waves). After a given time delay, this induces a retarded variation of periodic regime of the ECs involved in the interaction, dependent on the amount of energy exchanged. In this case it is possible to establish a 'before' and an 'after' with respect to these retarded events in time (in this case all that happens in a given period is not identical with all that happens in an arbitrary period). Hence we have relativistic causality and time ordering in terms of local and retarded modulations of periodicity of the internal clocks [2]. This also means that physical systems composed by interacting elementary particles, are characterized by chaotic evolutions rather than ergodic ones, according to our empirical observation of nature. In such a chaotic evolution, every value ('instant') of the external relativistic time axis t ∈ R , defined by means of the atomic or 'particle' clocks, is characterized by a unique combination of the phases of all the 'internal clocks' of the system. That is, events in time can be uniquely fixed by combining reference cycles 'ticks', as in a stopwatch or in a calendar. In everyday life we in fact fix events in time in terms as combinations of reference cycles of years, months, days, hours, minutes, seconds, etc. These are conventionally assumed to be rational each other (in particular in a", "pages": [ 3 ] }, { "title": "4 The 'missing link' of quantum mechanics", "content": "The technological possibility to replace atomic clocks with the internal clock of, say, an electron would open a new frontier in physics (similarly to what happened with other improvements in time resolutions, see Galileo example above). This could allow us to control the 'gears' of QM. The enormous rapidity of these recurrences in fact justifies a novel semi-classical interpretation of QM. If we assume that an elementary particle is constrained to have intrinsic periodicity, that particle is like a 'particle in box'. That is, the assumption of intrinsic periodicity is a quantization condition. This extends the wave-particle duality: every particle can be in fact represented as a one dimensional string vibrating with characteristic space-time periodicities determined by its kinematical state. The harmonics of such vibrating strings represent the quantum excitations of the particles. The resulting theory is the relativistic generalization of the theory of sound (allowing vibrations along the time dimension) in which elementary particles are the analogous of sound sources. Roughly speaking we give a timber to de Broglie waves [3]. It is possible to show that a statistical description of the vibrations associated to the ECs leads to a formal correspondence with ordinary QM, for both the canonical and Feynman formulations. For instance, as in a cylindric geometry, a periodic phenomenon can arrive to a given final configuration by passing through infinite classical paths characterized by di ff erent windings numbers. In this way it is possible to prove that the classical evolution of modulated ECs are actually described by the ordinary Feynman path integral. The correspondence can be interpreted in terms of 't Hooft determinism [8]: 'there is a close relationship between a particle moving very fast in a circle and a harmonics quantum oscillator' - the former can be statistically described as a fluid with unitary total density, see Born rule, the latter is the basic constituent of quantum fields, see Klein-Gordon field. The periodicity of typical pure quantum systems (complete coherence) are extremely fast with respect to our resolution in time; for electrodynamical systems this is of the order of the zitterbewegung (10 -21 s ) - though coherent electromagnetic systems (lasers, superconductors, condesates, etc) can have a slower recurrence (the thermal noise destroys the quantum recurrence). A quantum system can be therefore imagined as a die rolling too fast with respect to our resolution in time, so that the outcomes can only be described in a statistical way (with implicit Heisenberg's relation); an imaginary observed with infinite resolution in time would have no fun playing dice since the outcomes could be always predicted ('God doesn't play dice' A. Einstein). Intrinsic periodicity could represent the physical principle ('missing link' [6]) behind QM. The statistical description of ECs matches quantum dynamics without involving any local hidden variable. This suggests a possible deterministic description of physics.", "pages": [ 4, 5 ] }, { "title": "5 Gauge interaction as relativistic clock modulation", "content": "A formulation of physics in terms of space-time ECs implements undulatory mechanics (e.g. the wave particle duality) in the geometrodynamcis of the space-time dimensions. Every interaction can be equivalently described as modulations of the periodicity of the ECs, i.e. as modulations of the internal clocks of the particles. As known from general relativity, the modulations of (space-time) clocks can be equivalently encoded in corresponding deformations of the underlying space-time metric. In this way we have proved [3] that gauge interactions (e.g. electromagnetism), as gravitational interaction, can be associated to space-time geometrodynamics, similar to original Weyl's proposal. The idea is to describe the oscillatory motion of a particle interacting, say, electromagnetically in terms of corresponding local variations of reference frame, i.e. similarly to the description of a particle interacting gravitationally in general relativity. In this case (linear approximation) the transformations associated to gauge interactions are local transformations of flat reference frames. Thus, such a formulation of physics in terms of ECs points out an important relationship between gauge and gravitational interactions [3].", "pages": [ 5 ] }, { "title": "6 Conclusion", "content": "The flow of time in physical systems can be decomposed in ECs. This possibility is provided by QM, which associates to every elementary constituent of matter a recurrence in time. Free elementary particles can be regarded as reference clocks for a more accurate operational definition of the time unit. A formalization of elementary particles as ECs provides natural solutions to controversial questions of physics as well as a novel concept of relativistic time flow [1-4].", "pages": [ 5 ] } ]

[ { "title": "ABSTRACT", "content": "The Journal's name will be set by the publisher DOI: will be set by the publisher c © Owned by the authors, published by EDP Sciences, 2018", "pages": [ 1 ] }, { "title": "Is time enough in order to know where you are?", "content": "one event in the manifold, thus uniquely identifying its position. The situation is represented in two dimensions in fig. 6. Here local singularities and horizons are not taken into account. Agrid is shown built by hyperplanes spaced out by the period of the signal from each source. Any world-line can in principle be identified by the intersections of the hypersurfaces of the grid.", "pages": [ 9 ] }, { "title": "1 Introduction", "content": "The problem of positioning an event within a four-dimensional manifold, such as space-time, necessarily implies various assumptions concerning what the properties of space-time are. In particular something must be clarified concerning time and its special status with respect to the three space dimensions. For this reason I shall devote the first part of my talk to the emergence of time from the geometrical properties of an \"elastically\" deformable four-dimensional manifold. The idea of such a space-time is assumed from the Strained State theory developed in ref.s [1]-[7]. Once the origin of the light cones has found a logical framework within which to fit, we may exploit both the geometrical properties of the manifold and the fact that the \"length\" of a portion of the world-line of any observer is given by the proper time measured by any clock carried along by the traveler. If the measured proper time intervals are between the arrivals of regular pulses emitted by far away sources, whose worldlines are known, I shall show that the time sequences are su ffi cient to enable the voyager to find out its own way in space-time reconstructing piecewise its world-line and locating it in a previously defined global reference frame. The positioning system developed according to this approach is intrinsically relativistic, in the sense that it automatically includes all relativistic e ff ects with no need for by hand corrections as the ones that are introduced in the GPS or similar systems. The relativistic positioning system based on proper time measurements has been developed and tested in ref.s [9]-[13]. In the following I shall outline the essence of the method.", "pages": [ 1 ] }, { "title": "2 Space-time and its properties", "content": "According to the theory of General Relativity (GR) space-time is a four-dimensional Riemannian manifold with a Lorentzian signature. A pictorial view of a space-time can be seen in fig. 1. The figure is of course bidimensional and the representation has Euclidean signature rather than the true Lorentzian signature of actual space-time, but it gives an idea of the geometrical nature of our manifold, exaggerating its warpedness. The local curvature is a manifestation of the gravitational interaction.", "pages": [ 2 ] }, { "title": "2.1 Where does time come from?", "content": "In GR curvature is related, through Einstein's equations, to the matter / energy density. When taking all matter / energy away what is left is a flat Minkowski manifold. This simple result sounds rather obvious, because the absence of any source consistently implies a flat manifold. There is however something peculiar to a Minkowski manifold, besides the flatness: it is the presence of the light cones or, otherwise stated, the peculiar signature that singles out one dimension with respect to the other three. Suppose space-time is not a mere mathematical artifact, but it represents something real. If it was not so, what meaning would we attach to the famous sentence by John Wheeler: \"matter tells space-time how to curve, and space-time tells matter how to move\"? Space-time will have properties on its own, but any non-trivial geometrical feature will depend on matter / energy. Now, any given manifold with nothing in it should have all possible symmetries and the corresponding geometry. In practice a space-time totally devoid of matter should be flat and have Euclidean geometric properties; consequently it should be perfectly isotropic, which means that no direction could have any properties distinguishing it from the others. Where would the light cones of the Minkowskian geometry come from? But of course we know that the light cones do exist in a real space-time (according to GR, which we assume to hold); they belong to a Minkowski manifold too, as far as the latter is intended as the local tangent space of a generally curved space-time. A local tangent space preserves some symmetry from the original manifold, and in particular the peculiarity of time, i.e. the local light cone; however this is not the case for a space-time totally deprived of matter / energy. In order to try and find at least a reasonable origin for the special role of time with respect to space, we can explore the possibilities lent by the idea that space-time is indeed a real continuum. The analogy we may exploit is with ordinary three-dimensional continua; we know them to be deformable media, whose behaviour, at least at the lowest order of approximation, is described by the linear elasticity theory. Besides this aspect we also know that material continua can contain such things as structural or texture defects. A defect can be built by the ideal procedure described originally by Volterra [8]. Start from a flat four-dimensional Euclidean manifold; imagine to cut out a finite patch of the manifold. By this simple proceeding two distinct families of geodetic lines have been sorted. As far as the manifold is flat, geodesic curves on it are all possible straight lines; cutting out an area we practically distinguish all the complete geodesics (i.e. the ones which do not impinge into the missing region) from the incomplete ones which are limited, on one side, by the border of the prohibited area. This is indeed a first step towards the definition of time-like world-lines contrasted with space-like ones, but does not touch the problem of signature yet. Adopting the idea that space-time can host defects just in the same sense as ordinary physical continua do, there is one more feature on which we may draw our attention. A defect is not only identifiable with a secluded region of a manifold; we must also think of pulling the rim of the hole inward until the gap is closed and the opposite portions of the border are glued together so that the manifold no longer contains voids but rather \"scars\". From the geometrical viewpoint the envisaged process is the continuous formation of a singularity in the manifold, where the singularity may be any singular sub-manifold: a point, a line, a bidimensional surface, a hypersurface... What matters now, however, is that the presence of a defect in a material continuum induces a spontaneous strained state, where \"spontaneous\" means \"in the absence of the action of external agents (\"forces\")\". Applying these concepts to space-time amounts to assume that it contains at least one global defect representing the origin of all the geodesics we call time-like and being the cause of curvature even in the absence of matter / energy (the \"external agent\"). The above thumbnail description is the basis of the cosmic defect theory [3]. It ascribes the global Robertson-Walker (RW) symmetry of the universe to the presence, in the four-dimensional manifold, of a cosmic defect, which is also responsible, via the induced strain, of what we call the accelerated expansion. The strain is accounted for by an additional term in the empty space Einstein-Hilbert Lagrangian density. The additional term is molded on the deformation energy of elastic continua; the basic ingredient is the strain tensor proportional to the di ff erence between the metric tensors of a flat undeformed Euclidean manifold (reference manifold) and of the actual space-time (natural manifold). In terms of the line elements we may write: It is understood that the same coordinates are used to identify corresponding events on the two manifolds. The strain tensor is: According to the approach I am presenting here, the full action for the strained state of space-time is [3]: R is the scalar curvature; /epsilon1 = /epsilon1 α α is the trace of the strain tensor; λ and µ are the Lamé coe ffi cients of space-time with exactly the same role as in three-dimensional elasticity. Indices are lowered and raised by means of the full metric tensor gµν and its inverse. The integrand is the Lagrangian density L .", "pages": [ 2, 3 ] }, { "title": "2.2 A Robertson-Walker space-time", "content": "Let us apply the approach described in the previous section to a Robertson-Walker space-time, i.e. to a manifold homogeneous and isotropic in space. This is the symmetry we think we see in the universe, but here matter is not included for the moment. Equations (1) may now be written as: We recognize the typical scale factor a ( τ ) depending on the cosmic time τ expressed as a length. The b ( τ ) of the reference manifold does not change the flatness of the Euclidean manifold, but it represents a \"gauge\" function expressing the fact that there are in principle infinitely many di ff erent deformation strategies leading from the reference to the natural manifold, all preserving the global symmetry. The meaning of b is easily understood looking at fig. 2 [6]. From eq.s (4) the Robertson-Walker strain tensor immediately follows: Once we have the strain tensor we can write the explicit form of the action integral (3) and from it deduce the Euler-Lagrange equations for the unknown functions. The first step is quite simple and leads us to the gauge function b that follows from ∂ L /∂ b = 0; it is [7]: Consequently the equation for the scale factor a ( τ ) is: Using the energy condition it is possible to pass to a first order di ff erential equation: From eq. (8) it is possible to desume the square of the Hubble parameter: where the shorthand notation has been used. The next step is to consider that letting µ → 0 should bring about the traditional GR result (no strain contribution), which implies W = 0. Finally we have: The double sign in front of eq. (10) expresses two options: an expanding or a contracting spacetime. Since we know the universe is expanding, we choose the + sign. I stress the fact that this solution has been obtained even in the absence of matter under the assumption of the RW symmetry and of space-time being an \"elastic\" manifold. These assumptions imply the presence of an initial singularity in the form of a texture defect of the manifold.", "pages": [ 4, 5 ] }, { "title": "2.3 Signature flip and the emergent rigidity of the manifold", "content": "I had left open the question about the origin of the signature of our space-time. Now considering eq. (10) we can solve it for a , finding [7]: C is an integration constant. If it is smaller than 1, we see that, close to the origin of the τ coordinate, a acquires imaginary values ( a 2 < 0). The interpretation of this result is that the manifold has a defect for τ = 0; the defect is surrounded by a curved region with Euclidean signature, whose boundary is at The global RW symmetry is preserved. Below τ = τ h three out of four space dimensions are homogeneousand isotropic, but the fourth is not a time at all: there are no light cones. In the Euclidean signature region τ is just a running coordinate along the space-like incomplete geodesics, that start at the defect in τ = 0. On our side of the singular τ = τ h hypersurface we find a Lorentzian signature so that now τ , which still is a running coordinate along the incomplete geodesics stemming out of the defect, acquires a time character and indeed is read as the cosmic time. In the Euclidean signature domain the three homogeneous space dimensions shrink with increasing distance from the defect (increasing τ ); in the Lorentzian signature domain they expand. The theory that I have sketched until now has been tested against the observation of the universe at the cosmic scale and the results have been good [2] [5]. Remarkably the best fit values of the Lamé coe ffi cients are ∼ 10 -52 m -2 ; such small values imply that space-time behaves like an extremely rigid stu ff . On the other side the whole description so far is based on an analogy with the three-dimensional elasticity theory and we know that that theory emerges from the properties of a microscopic discrete structure underlying the apparently continuous aspect of solid materials. We are then led to think that this could also be the case of space-time. Following this possibility we may for instance remark that high rigidities even of per se soft materials are attained when the material has a foamy structure. Why then not consider that space-time too has, at microscopic level (Plank scale?), the topology of a four-dimensional foam? This would produce an extremely rigid behavior at macroscopic as well as cosmic scale. A typical geodetic line would then appear, at the microscopic level, as in fig. 3. At a higher scale the geodetic would practically be a straight line. I would like to stress that the idea of an underlying foamy topology is here entirely classical. I am not calling in any specific attempt to quantize gravity. The whole conceptual framework in which the strained state theory is cast is classical or, maybe, e ff ectively classical, and it is applied essentially at the cosmic or at least astronomical scale. Given the numerical values of the Lamé coe ffi cients, no relevant e ff ect is expected at the local (for example at the solar system) scale.", "pages": [ 5, 6 ] }, { "title": "3 Positioning", "content": "After having given a logical frame for the presence and relevance of time, I may try and answer to the question in the title of the present article coming to the practical problem of positioning, i.e. of finding the position of an observer within space-time. Any attempt to set up a global positioning system has a number of underlying assumptions which I am recalling here: The first two assumptions were implicit in all I have written in the previous sections. As for the global reference frame, currently it is assumed to be attached to the \"fixed stars\" intended as quasars. Quasars are indeed assumed to be at distances in the order of billions of light years so that their reciprocal positions in the sky may be treated as fixed, notwithstanding their proper motions and for times as long as the human history. Fig. 4 gives an example of the positions of some quasars in the sky. They identify a corresponding bunch of fixed directions that are the same for all observers in the solar system with accuracies better than 10 -12 . Actually the use of quasars implies one more hypothesis which, strictly speaking, is improper but may be assumed to be approximately or e ff ectively true: i.e. quasars, as point-like objects (!), are represented by straight parallel world-lines.", "pages": [ 7 ] }, { "title": "3.1 Null geodesics", "content": "Once these assumptions have been accepted, a good strategy in order to define a set of coordinates is to rely on null geodesics. Four independent families of such geodesics covering all space-time are a good means to identify any event there as the unique intersection among four geodesics each from a di ff erent family. The bidimensional sketch in fig. 5 gives the idea. Each null geodesic is locally identified by its null tangent four-vector, written as: It is χ 2 = 0. The space components of (12) are the direction cosines of the line with respect to the local axes of the reference frame. The factor in front can have any value without modifying the geometrical meaning of χ ; it can be used to host some additional information: here it contains T which is the period of the electromagnetic signal propagating along the geodesic. If the space-time is flat (12) identifies not only a specific geodesic of a given family, but the whole family everywhere. Four independent vectors like (12) form a null basis, that can be used to represent any four-vector r , pointing to any position in the surroundings of an origin. It will be: The pure numbers τ a / Ta (Latin letters from the first part of the alphabet label the families of geodesics: a = 1 , 2 , 3 , 4) are called light coordinates of the event on the tip of the vector. Acomplementary view to the one based on null tangent vectors is based on the hypersurfaces dual to them: /epsilon1 abcd is the fully antisymmetric Levi-Civita tensor. If the space-time is flat then the /pi1 abc 's represent four families of null hyperplanes, covering the whole manifold. Otherwise /pi1 abc identifies the local tangent space to one of the null hypersurfaces perpendicular to the corresponding χ . Hypersurfaces from the same family never intersect each other so that four from di ff erent families intersect at only", "pages": [ 7, 8 ] }, { "title": "3.2 Finding the light coordinates", "content": "A practical implementation of the principles stated in the previous section may be obtained using discrete electromagnetic pulses coming from (not less than) four independent sources located at infinity; the a -th source emits pulses at the rate of 1 / Ta per second. The T parameter of formula (12) is now interpreted as the repetition time of the pulses rather than the period of a monochromatic continuous wave. The grid exemplified in fig. 6 is now really discrete; we have then a sort of an egg-crate whose walls are in a sense \"thick\" because they are associated to pulses which have, though short, a duration in time: in practice the hypersurfaces on the graph correspond to \"sandwich waves\" carrying a pulse. The sides of the cells are measured by the T 's, projected along the time axis of the background global reference frame (we should remember that the bundles of hypersurfaces forming the walls of the cells are all null). The world-line of an observer necessarily crosses the walls of successive boxes of the egg crate. If we are able to label each cell of the crate assigning integer numbers to the walls, we are also able to reconstruct the position of the observer in the manifold. A typical emission diagram of one of the sources will more or less be like the one sketched in fig. 7. The shape of the pulse is not important as well as it is not the spectral content of it. What matters is its reproducibility and the stability of the repetition time. Considering natural pulses, as the ones coming from pulsars, we find repetition times ranging from several seconds down to a few milliseconds and lasting a fraction of the period. As an example of artificial pulses the highest performance is obtained with lasers: GHz frequencies are possible with pulses as short as ∼ 10 -15 s. Once pulses are used, we may label them in order, by integer numbers, as it can schematically be seen in fig. 8. The integers can be though of as rough coordinates identifying the cells of the grid. At this level the approximation would be rather poor, being of the order of the size of each cell. If the periods are milliseconds this corresponds to hundreds of kilometers. Looking at fig. 8 we may however notice that the intersections of a given world-line with the walls of the cells are labeled by a quadruple of numbers, at least one of which is an integer: these numbers are the coordinates of the intersection points. We may write the typical light coordinate of a position in the crate as The n 's are the integers, whilst the x 's are the fractional parts. If we have a means to determine the x 's the localization of an intersection event can be done with an accuracy much better than the hundreds of km I mentioned above. Considering that the intersections coincide with the arrivals of pulses from di ff erent sources, the determination of the fractional part of the coordinates is indeed a trivial task, provided the traveler carries a clock, the space-time is flat and the world-line is straight. Once one measures the proper intervals between the arrivals of successive pulses, a simple linear algorithm based on elementary four-dimensional flat geometry produces the x 's [12]. In fact the proper time interval between the arrivals of the i -th and the j -th pulses from a given source is the norm of the ri j four-vector separating the two arrival events, i.e. Simple proportions tell us that [12]: and eight successive arrival events are su ffi cient for determining all the x 's of the sequence. It is: Using moving sets of eight successive arrivals we piecewise reconstruct the whole world-line of the receiver. The accuracy of the result depends on the precision of the clock which is being used in order to measure the proper intervals between pulses and on the stability of the period of the pulses, which in turn tells us what the e ff ective \"thickness\" of the walls of the cells of our space-time crate is. Just to fix some order of magnitude, let me remark that nowadays to have a portable clock with a 10 -10 s accuracy is quite easy (much better performances can be achieved in the lab); on the other side, considering pulsars, we have some, whose period is known and stable down to 10 -15 s. With these figures the final positioning can be within a few centimeters. Of course the traveler's motion will not in general be an inertial one and space-time will not be flat, however a short enough stretch of the world-line can always be confused with the tangent straight line to it and a small enough patch of space-time can always be confused with a portion of the local tangent space. In practice we work on the local tangent space and on a linearized portion of the worldline. The acceptability of these assumptions depends on the accuracy required for the positioning and on the constraints posed by the linear algorithm in use. If δτ is the maximum proper time inaccuracy that we decide to be tolerable, the final relative accuracy of the positioning will be [12]: The index i in eq. (18) labels the order of the arrival events; τ i , i + 4 n is the proper time interval between the i -th and the ( i + 4 n )-th arrival, being n ≥ 1 an integer; n should assume the highest value compatible with the straightness hypothesis for the world-line. Of course the number of pitches that can safely be considered depends on the periods Ta of the emitting sources: the shorter are the periods, the bigger is the number of paces that can be used within the linearity assumption. A pictorial view of what we are doing is as follows. Imagine to embed the real four-dimensional manifold, together with its tangent space at the start event, in a five-dimensional flat manifold; then consider the real world-line of the traveler and project it onto the tangent space. The world-line on the tangent space is what we are piecewise reconstructing by our linear algorithm: in practice we are building a flat chart containing the projection of our space-time trajectory. The time dependence of the adimensional coordinates of the projected world-line may of course be written in the form of a power series, as: The coe ffi cients ua and α a are proportional to the four-velocity and four-acceleration of the traveler. The individual segments used for the reconstruction are short enough so that the second and further terms of (19) are negligible with respect to the linear one. Given the tolerance δτ on the time measurement, the maximum acceptable duration of an elementary sequence will be: Going on, after a number of paces, the possible presence of an extrinsic curvature of the projected world-line shows up; we know that locally it is impossible to distinguish a gravitational field from a non-gravitational acceleration so we need additional information for that purpose. In the case of a gravitational field evidenced by the reconstruction process I am describing, we get from the data the gradient of the Newtonian gravitational potential Φ . Actually it becomes visible when In order not to cumulate the distortion introduced by the projection from the real curved manifold to the tangent space at a given event, we need periodically to restart from a further event on the worldline, i.e. to pass to the tangent space at a di ff erent event. If the visible curvature of the line on the tangent space as well as the tilt of the successive tangent spaces continues for long in the same sense, the linearization process, as in all similar cases, tends to produce a growing systematic discrepancy with respect to the real world-line, so that periodically one has to have recourse to some independent position fixing method in order to reset the procedure.", "pages": [ 9, 10, 11, 12 ] }, { "title": "3.3.1 Pulsars", "content": "Possible natural sources of pulses are pulsars. This kind of neutron stars are indeed good pulse emitters because of their extreme stability and long duration. As we know, their emission is in the form of a continuous beam. The apparent periodicity is due to the fact that the emission axis (the magnetic axis) does not coincide with the spin axis of the object so that it steadily rotates, together with the whole star, about the direction of the angular momentum. The pulses arise from the periodic illumination of the earth by the rotating beam. The stability is guaranteed by the angular momentum conservation. The idea of using pulsars for positioning and navigation was put forth almost as soon as they were discovered [14] and indeed the advantages of this kind of sources are numerous. Their period is extremely stable and is sometimes known with the accuracy of 10 -15 s; it tends to decay slowly (the relevant times are at least months), but with a very well known trend, determined by the emission of gravitational radiation. Typically the fractional decay rate of the period is in the order of one part in 10 12 per year. The number of such sources is rather high, so that redundancy in the choice of the sources is not a problem: at present approximately 2000 pulsars are known and their number continues to increase year after year. Being these stars at distances of thousands of light years from the earth, they can be treated as being practically fixed in the sky; in any case their slow apparent motion in the sky is known, so corrections for the position are easily introduced. Unfortunately pulsars have also major drawbacks. One is that their distribution in the sky is uneven, since they are mostly concentrated in the galactic plane, which fact brings about the so called \"geometric dilution\" of the accuracy of the final positioning: sources located on the same side of the observer produce an amplification of the inaccuracy originating in the intrinsic uncertainties. Furthermore individual pulses di ff er in shape from one another so that some integration time is needed in order to reconstruct a fiducial series of pulses; this fact, also considering the length of the repetition time, can conflict with the linearization of the world-line of the traveler. It should also be mentioned that most pulsars are subject to sudden jumps in the frequency (glitches), caused by matter falling onto the star; these unpredictable changes can be made uno ff ensive by means of redundancy, i.e. making use of more than four sources at a time. However the most relevant inconvenience with pulsars is their extreme faintness. In the radio domain their signals can be even 50 dB below the noise at the corresponding frequencies; to overcome this problem big antennas are required and convenient integration times accompanied with \"folding\" techniques must be employed. In principle at least four di ff erent sources must be looked at simultaneously and this is not an easy task, especially with huge antennas. The weakness problem has led to consider X-ray- rather than radio-pulsars for positioning [15]. A few hundreds X-ray emitting pulsars are indeed known; their signals are weak too, and can be received only outside the atmosphere, but the background noise is far smaller than the one typical in the radio domain; as for the hardware, X-ray antennas can be much smaller than the typical radio-antennas. The principle feasibility of a pulsar based positioning system, applying the method I have described in the previous sections, has been tested by a simulated exercise named \"Eppur si muove\" [13]. Using a software employed by astrophysicists in order to forecast the arrival times of the pulses from known pulsars at any point of the earth (the name of the software is TEMPO2), we have mimicked an antenna located at the Parkes observatory in Australia and the pulses from four real pulsars there, during three days. The method has been able to reconstruct the motion of the chosen location, together with the whole earth, with respect to the fixed stars, represented in this case by the pulsars. Fig. 9 shows the result, evidencing the wiggling motion of the Parkes observatory due to the combination of the revolution of our planet around the sun with the diurnal rotation.", "pages": [ 13 ] }, { "title": "3.3.2 Artificialemitters", "content": "In principle what can be done using pulsars can as well be done by means of artificial emitters of electromagnetic pulses. Artificial emitters can have far higher intensities than pulsars; the repetition time can easily be in the range of ns or less, thus making the linearization process more reliable. The stability of the source over time is not as good as for pulsars, but this can represent no inconvenience as far as the number of sources is redundant and they are kept under control. A problem is in the sources clearly not being at infinite distance, which implies a more complicated geometry and of course the need for a good knowledge of the world-line of the emitter in the background reference frame. One could think of building a Solar System reference frame made of pulse emitters laid down on the surface of various celestial bodies whose orbits are well known and reproducible: the earth of course, the moon, Mars, maybe some of the asteroids; even some space station following a well defined, highly stable orbit around the sun or a planet. A blended solution for self-guided navigation in the solar system could combine some artificial emitters, as quoted above, together with a limited number of pulsars (the most intensely emitting ones).", "pages": [ 14 ] }, { "title": "4 Conclusion", "content": "As we have seen, the answer to the question posed in the beginning is yes : it is indeed possible to determine the position of a given event, with respect to a predefined reference frame, just measuring the time sequences of the arrivals of electromagnetic pulses from at least four emitters whose worldline is known. After clarifying the special role of time as stemming from the strain induced in spacetime by the presence of a global defect responsible for the Robertson-Walker symmetry of the universe at high enough scales, I have expounded the conditions under which and the method whereby the self positioning is possible. The approach is indeed intrinsically relativistic from the beginning since it relies on the very structure of space-time in order to reconstruct the world-line of an observer. This principle feature, together with the extreme accuracy with which technology allows for very precise measurements of proper times, makes the proposed RPS (Relativistic Positioning System) rather appealing for Global Positioning purposes and especially for navigation across the solar system, where other methods in use are either impracticable or inaccurate. We may then legitimately expect that the new relativistic method will be implemented in the next generations of global positioning systems. The process will however probably be rather slow because of the pervasive presence of more traditional systems like GPS, which, though less satisfying from the principle viewpoint, have an enormous accumulated advantage due to the huge investments made to implement them for military reasons. Notwithstanding the expected slow implementation for the most practical aims in the terrestrial environment, the RPS can also be the basis of space-time geodesy for fundamental physics objectives. Consider for instance a swarm of satellites orbiting the earth and allow them to exchange with each other electromagnetic pulses (for instance laser pulses): accurate timing of the travel times of the pulses within the swarm would permit to map the region of space-time where the satellites are located, evidencing the average curvature, i.e. the gravitational field. For long enough base lines even propagating disturbances of the curvature (i.e. gravitational waves) could be detected. So finally let me conclude that light is indeed an excellent probe of the structure of space-time especially when coupled with accurate local measurements of proper time intervals.", "pages": [ 14, 15 ] } ]

[ { "title": "Holographic dark information energy: predicted dark energy measurement.", "content": "Michael Paul Gough Department of Engineering and Design, University of Sussex, Brighton, BN1 9QT, UK E-mail: [email protected]", "pages": [ 1 ] }, { "title": "Abstract.", "content": "Several models have been proposed to explain the dark energy that is causing universe expansion to accelerate. Here the acceleration predicted by the Holographic Dark Information Energy (HDIE) model is compared to the acceleration that would be produced by a cosmological constant. While identical to a cosmological constant at low redshifts, z < 1, the HDIE model results in smaller Hubble parameter values at higher redshifts, z > 1, reaching a maximum difference of 2.6 - 0.5% around z ~1.7. The next generation of dark energy measurements, both those scheduled to be made in space (ESA's Euclid and NASA's WFIRST missions) and those to be made on the ground (BigBOSS, LSST and Dark Energy Survey), should be capable of determining whether such a difference signature exists or not. The HDIE model is therefore falsifiable. Keywords: dark energy experiments, dark energy theory, cosmological constant experiments", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The expansion of the universe is accelerating, driven by a dark energy that presently accounts for around three quarters of the total energy of the universe. This conclusion, initially obtained using type 1a supernova as reference sources [1][2], is supported by more recent supernova measurements [3] and now confirmed by a number of independent measurements [4-7]. The large dark energy component is evident in gravitational weak lensing [4], in the projected scale of baryon acoustic oscillations [5], in the Cosmic Microwave Background radiation anisotropies [6], as well as in the growth rate of large scale structure, or clustering power spectrum of galaxies [7] (see reviews [8][9] ). The many explanations proposed to explain dark energy include: Einstein's cosmological constant; some form of quintessence field; and a wide range of 'alternative'models [8][9]. Two properties are required for a dark energy model to fit the observations. Firstly, the model must be capable of quantitatively accounting for the dark energy density value, a high value, three times the energy density equivalence of the universe's total mass. Secondly, at least for the recent period, redshifts z <1, the model must provide a near constant energy density, equivalent to a total dark energy that increases as ~ a 3 where a is the universe scale size (size relative to today, a =1). This corresponds to a dark energy equation of state value w ~-1, since energy densities vary as a -3(1+w) . Ideally we wish to find a model that satisfies these two requirements without recourse to exotic or unproven physics. Foremost amongst likely explanations is the cosmological constant, or vacuum energy, that satisfies the second of our two requirements, by definition exhibiting a constant energy density, equivalent to the specific equation of state value w =-1. In contrast, quintessence is a scalar field with a dynamic equation of state that varies over space and time. Experimental measurements [10] at low redshifts, z < 1, limit the dark energy equation of state to lie within the narrow range w =-0.94-0.09 and thus generally favour the cosmological constant explanation. Accurate measurements of dark energy at higher redshifts, z >1, await the next generation of measurements. Unfortunately, satisfying the first requirement is more difficult as there are only two possible energy density values expected for a cosmological constant from quantum field theories [11]. The cosmological constant should either have a preferred value some 120 orders of magnitude higher than that required - a value impossible to reconcile with our universe, or it has the value zero. Then a different dark energy explanation would, if verified, enable the cosmological constant to take this second natural value: zero. A number of holographic dark energy models attempt to account for dark energy without invoking exotic particles, exotic fields, modifications to gravity, or interactions with dark matter [12-14]. The anti-de-Sitter/conformal field theory duality leads directly to considering that all of the information describing a system can be considered as being encoded on the system's bounding surface [15][16]. Then today's dark energy density can follow from a combination of the Planck scale with a suitable cut-off dimension, for example the infrared cut-off [17]. Holographic dark energy may also be caused by vacuum entanglement [18], effectively energy from quantum information loss via Landauer's principle [19]. Holographic dark energy models [12-14],[17][18] generally aim to provide an all pervading vacuum energy in the form of a cosmological constant. In contrast the specific model considered in this paper is the Holographic Dark Information Energy, HDIE, model [20] which takes a phenomenological approach. Here dark energy is explained as the energy equivalence of the information, or entropy, associated with stars and stellar heated gas and dust. HDIE accounts for both of the required dark energy properties and, in particular, manages to satisfy the first property using well established physics (see discussion of section 2.2). Nevertheless, HDIE is only one of the many dark energy models proposed to date and science progress requires that we eliminate all but one. With that elimination in mind, the emphasis here is on the HDIE model's predicted signature that differentiates HDIE from other models/theories.", "pages": [ 1, 2 ] }, { "title": "2 The HDIE model", "content": "The HDIE model has been described in detail before [20] and only relevant features are recounted here. Essentially, HDIE combines Landauer's principle [19] with the Holographic principle [15][16]. Landauer's principle [19][21-23] states that any 'erasure' of information, or reduction of information bearing degrees of freedom, requires a minimum of kBT ln 2 of heat per erased bit to be dissipated into the surrounding environment. This dissipated heat increases the thermodynamic entropy of the surrounding environment to compensate for the loss of degrees of freedom and comply with the 2nd law. Information is not destroyed as the 'erased' information is now effectively contained in the extra degrees of freedom created in the surrounding environment. Heat dissipation from information erasure is comparatively weak and usually insignificant in our normal day to day experience. For example, world-wide, man-kind has now accumulated some 10 22 bits of stored digital data and we have the technological capacity to process a total of around 3 × 10 19 instructions per second in general-purpose computers [24]. We can assume that the main information erasure that occurs during the process of computing is caused by the overwriting the processor's instruction register when each new instruction is read from memory. In this way man-kind erases some 10 21 bits each second. At room temperature this rate of information erasure will generate a world-wide total of only 3W! This is insignificant, ~10 -11 of the total electronic heat dissipation (ohmic and inductive heating, etc) of the world's ~10 9 computer systems that each dissipate ~10 2 W. Similarly, erasing the 10 22 bit sum total of all man made stored digital data would generate a world-wide total of just 3J. Despite this low bit equivalent energy, the information-to-energy conversion process has now been demonstrated experimentally using Brownian particles under feedback control [25] and by a one-bit memory consisting of a single colloidal particle trapped in a modulated double-well potential [26]. Moreover, Landauer's heat dissipation from information erasure is still considered the best way to reconcile Maxwell's Demon with the second law of thermodynamics (see reviews [23] and [27] and references therein). Landauer's principle provides the information energy equivalence, similar to the mc 2 energy equivalence for mass. When the same degrees of freedom are considered information entropy and thermodynamic entropy are identical. Then every component of the universe has an information equivalent energy of NkBT ln 2 that depends on the quantity of information (or entropy), N bits, associated with the component and on the component's temperature, T . In this way we can consider the energy represented by information in the cosmos without requiring, or even identifying, processes whereby information may be actually 'erased'. The Holographic principle [15][16] states that the amount of information in any region, N , scales with that region's bounding area. The Holographic Principle lead directly from the discovery that the maximum entropy of a black hole is set by its surface area [28] but the principle is considered to have universal validity [29], i.e. not just limited to the maximum entropy limit of black holes.", "pages": [ 2, 3 ] }, { "title": "2.1 Stellar heated gas and dust", "content": "Stellar heated gas and dust and black holes have been found to make the greatest contributions to the entropy, N , of the universe [30][31][32]. Furthermore, stellar heated gas was estimated [20] to have the highest NT product, making the largest information energy contribution to the universe (see Table 1 of [20]). Black holes could make the next strongest contribution at a few percent of that level but it is doubtful whether the information within a black hole, and therefore its information energy, has any effect on the universe because of the 'no hair theorem' [33]. While a black hole may exert a significant gravitational force on local objects, the only information that the universe has about it is limited to just three parameters: mass; charge; and angular momentum. From the universe's information point of view a black hole is no more than just another single fundamental particle, albeit a massive one! Since the information energy, NkBT ln2 , of the universe is primarily determined by stellar heated gas and dust, the appropriate temperature, T , will be the average temperature of baryons in the universe. Figure 1(a) plots average baryon temperature, T , data and the fraction of baryons in stars, f , deduced from a wide literature survey of integrated stellar density measurements, extending the earlier HDIE work [20]. Data symbols and measurement source references are listed here: open circle [34]; open squares [35]; filled rectangles [36]; diamonds [37]; upside down triangles [38]; normal triangles [39]; crosses [40]; circles with dot [41]; filled circles [42] and blue line [43]. Figure 1(a) shows that the average temperature of baryons today is T ~2×10 6 K, which, together with the estimate of N ~10 86 from surveys [30][31][20], provides a quantitative estimate of the present HDIE energy value within an order of magnitude of the observed dark energy, satisfying dark energy requirement 1. Note that this is dependant on our estimate of N for stellar heated gas and dust, only accurate to a couple of orders of magnitude. We find that, despite the very low bit equivalent energy, information energy can provide a significant contribution on cosmic scales, primarily because the universe's mass has remained constant while both the quantity of information (entropy), N , increased continually and the average baryon temperature, T , also increased with increasing star formation. (b) Middle Panel: Log plot of energy density contributions: red continuous line, HDIE energy density corresponding to the red line fit in the upper panel; dashed red line, cosmological constant; blue line, mass; solid black line, total for HDIE case; dashed black line, total for the case of a cosmological constant; grey dashed line, to illustrate problem of two parameter dynamic w model approximation (see text); grey continuous line, the gedanken experiment considered in section 3.2. Figure 1(a) shows a distinct change in power law around z ~1 and data either side of z = 1 are therefore considered separately. Applying linear least squares curve fitting to logarithmic values of the data in the redshift range z < 1 we observe a temperature gradient of a +0.98-0.1 . Then, assuming the baryon information bit number, N , scales as a +2 from the holographic principle, total HDIE energy for z < 1 scales as a +2.98-0.1 . Since universe volume increases as a 3 , there is a nearly constant HDIE energy density at z < 1. The HDIE equation of state then lies in the narrow range -0.96 > wHDIE > -1.03 which includes the specific value, wDE = -1 and thus satisfies dark energy requirement 2.", "pages": [ 3, 4, 5 ] }, { "title": "2.2 Dependence on the Holographic principle", "content": "HDIE can account for today's high dark energy value (requirement 1) solely by applying proven physics i.e. without requiring the holographic principle. Measurements of the present average baryon temperature (figure 1(a) right-hand axis intercept [34-43]), are combined with estimates of the information (entropy) associated with stellar gas and dust [30][31], and experimentally proven [25][26] Landauer's principle. However, for HDIE to account for the constant dark energy density z < 1, (requirement 2), the measured average baryon temperature relation ( T /g302 a +0.98-0.1 at z < 1, figure 1 (a)) has to be combined with the, as yet unproven, holographic principle relating information content to bounding area, N /g302 a +2 . Now the Bekenstein-Hawking description [28] of a black hole, with entropy proportional to surface area, is widely accepted physics. But black holes exist at the maximum entropy holographic bound while the universe is some 30 orders of magnitude below the holographic bound. The holographic principle, whereby all 3-D space can be translated into a 2-D representation [15][16][44], is directly related to string theory and M-theory. Strong support for the holographic principle has been provided by a specific quantum theoretical example from string theory which allows for a holographic translation between one particular multidimensional space with gravity and another space with one less dimension but without gravity - the 'Maldacena duality' or 'anti-de-Sitter/conformal field theory' (AdS/CFT) correspondence [45]. Another theoretical work [46] effectively combines the holographic principle with Landauer's principle as in this present work, and suggests that gravity may emerge as an 'entropic force'. Note that our present work may be thought of as considering 'entropic energy'. Relevant theory in support of the holographic principle is still being developed and, as yet, there is no experimental proof of the principle. Attempts to directly verify the holographic principle by experiment are difficult and sometimes controversial [47]. Therefore the holographic principle, naturally extending the N /g302 a +2 relation to all objects including those well below their maximum entropy, remains an attractive but unproven hypothesis, and thus accounts for the main speculative aspect of the HDIE model. Nevertheless, the measurements plotted in figure 1(a), showing T /g302 a +0.98-0.1 at z < 1, closely centered round the T /g302 a +1 relation required for the HDIE explanation, provide significant support for the HDIE model. Accordingly, we continue our work below by considering HDIE primarily from a phenomenological point of view, and limit ourselves to only employing the main proposition of the holographic principle: i.e. that N /g302 a +2 .", "pages": [ 5 ] }, { "title": "2.3 Dark energy predictions for z >1", "content": "Figure 1(a) shows that the temperature gradient was much steeper at redshifts z > 1, with a wider spread in measured data points and a best power law fit of a +2.8-0.3 . Clearly HDIE energy density was increasing in this earlier period up to the point around z ~ 1 where HDIE leveled out at a near constant value that we showed above can account for both dark energy requirements from that time onwards. In the following analysis we therefore assume that the level HDIE energy density z < 1 indeed accounts for all dark energy, and thus is located at a value three times the present mass energy density equivalent. Then figure 1(b) shows the mass energy density falling as a -3 (blue line), the resulting HDIE energy density contribution with the above assumption (red line), and a cosmological constant for comparison (red dashed line). The a +2.8-0.3 temperature dependance, z > 1, corresponds to an HDIE energy density gradient of a +1.8-0.3 when information bit quantity, N , is again assumed to scale as a 2 from the Holographic principle. Then the mean a +1.8 HDIE energy density variation corresponds to an equation of state w HDIE = -1.6 for z > 1. In figure 1(b) we also compare the total energy density from HDIE plus mass (black continuous line) with the total energy density of a cosmological constant plus mass (black dashed line). At first sight the two total energy curves lie very close with little apparent difference because they are plotted on a multi-decade log versus log plot. Accordingly, in figure 1(c) the relative difference in total energy density, /g507 E/E , between HDIE plus mass and a cosmological constant plus mass is shown on a linear versus log plot. The lower, average and upper limits of the HDIE energy density gradient, a +1.8-0.3 , correspond to relative differences in total energy, /g507 E/E , in figure 1(c) that peak at -4.2%, -5.2%, and -6.2% respectively near z ~1.6 - 1.7. Although there is a clear change in gradient around z ~ 1 evident in the data points of figure 1(a), our fitting to gradients that change precisely at z = 1 may provide an overemphasized sharp transition in /g507 E/E at z = 1. However, this transition should not significantly affect the size or the location of the predicted negative peak in /g507 E/E at z ~1.7. At earlier times, z > 4, the higher mass density swamps any difference between HDIE and a cosmological constant. Later, as the massdensity falls /g507 E/E begins to reflect the difference in the energy densities of the two dark energy components, peaking at z ~1.7 as HDIE energy density rapidly increases as a +1.8-0.3 towards z ~1, after which time there is no difference between models. While the Hubble constant, H0 , is the fundamental relation between the recessional velocities of objects in the universe and their distance from us today, H0 is just the present value of the more general Hubble parameter, H . The Hubble parameter, H , varies with changes in universe expansion rate over time and is therefore a function of universe scale factor, a . Since total energy density, E , is proportional to H 2 , (from the Friedmann equation, [48]) these three curves then correspond to relative differences in Hubble parameter, /g507 H/H , that peak at -2.1%, -2.6% and -3.1% respectively. The HDIE model thus predicts that the Hubble parameter around z ~ 1.7 should be 2.6 - 0.5% less than that expected for a cosmological constant explanation for dark energy.", "pages": [ 5, 6 ] }, { "title": "2.4 Measurement capabilities of next generation instruments", "content": "The HDIE predicted ~ 2.6% difference in the Hubble parameter can not be resolved by today's instruments which still have typical resolutions > 5% (See, for example, recent BOSS results, figure 21 of [49]). Fortunately the next generation of space and ground based dark energy instruments should be capable of making such a measurement. The future European Space Agency Euclid spacecraft [50], and the planned NASA WFIRST spacecraft [51] will both cover the redshift range 0.7 < z < 2.0, while the ground based BigBOSS [52], LSST [53] and Dark Energy Survey [54] measurement campaigns will measure z < 1.7, z < 5 and z < 2, respectively. These experiments employ a combination of techniques: weak gravitational lensing to measure the growth of structure; supernova distances at low z; and baryon acoustic oscillations at higher z. The resolving limits of three of these dark energy experiments are shown in figure 1(c) (green lines) for comparison. The 2 - 3% difference in Hubble parameter around z ~ 1.7 should be resolved by all three experiments. At the time of writing these next generation measurement development timescales are: Dark Energy Survey starting a five year survey in 2012, BigBOSS first light 2016 with full science starting 2017; ESA Euclid launch 2019; LSST first light 2020 with full science starting 2022; and NASA WFIRST launch 2022. Note that HDIE effectively provides a dynamic equation of state but, rather than a smooth variation, there are two distinct regimes: w HDIE = -1 for z < 1; and w HDIE ~-1.6 for z < 1. Now it is usual when designing these experiments to characterise any dynamic equation of state, w(a) , by a smoothly varying two parameter model, typically given as : w(a) = wp + wa(1 - a) , where wp is the present value, the early value was wp + wa , and the mid-point transition occurs at a = 0.5, or z =1. The experimental figure of merit is then determined by how small the error ellipse is in the wp - wa plane. For example, the ESA Euclid measurement accuracies equivalent to 1 sigma error are expected to be 0.02 in wp , and 0.1 in wa up to z ~ 2 [50]. These accuracies are clearly sufficient to falsify HDIE where the nearest equivalent parameter values are wp =-1 and wa =-0.6, as compared to the cosmological constant values of wp = -1 and wa = 0. Note that this form of wp-wa data analysis is not the ideal for HDIE because of the difference between such a smooth variation (grey dashed line in figure 1(b) fitted to the high gradient limit, a +1.8+0.3 , for emphasis) and the more distinct transition expected at z ~ 1 from HDIE. Rather than a single cluster of measurement data points on the wp-wa plane HDIE predicts two clusters: one at wp =-1 and wa =0 independent of scale size, a , (i.e. wa = 0) over that range z > 1. Thus the more appropriate mode of data analysis to identify any signature of HDIE is by a determination of the Hubble parameter, H , as a function of scale factor, a , or redshift, z ( as in figure 21 of [49]). The HDIE model is therefore falsifiable [55] since a failure to observe its predicted specific signature would clearly exclude this model. Note that there is an inherent lack of symmetry in falsification arguments. Although a positive observation of this signature would exclude a cosmological constant, it would not necessarily exclude all other models. For example, some form of quintessence field might produce a similar signature to that described below for HDIE, but then that model would also need to be equally capable of explaining the specific form of that observed signature.", "pages": [ 6, 7 ] }, { "title": "2.5 Characteristic energy", "content": "The characteristic energy of HDIE, the energy equivalence of a bit of information, kBT ln 2 , depends solely on temperature, T . Today, some 10% of the baryons are located in stars at temperatures ~2 × 10 7 K with characteristic energies ~10 3 eV. As the remaining 90% of baryons exist at very much lower temperatures the average baryon temperature of baryons is , ~ 2 × 10 6 K, one tenth of the stellar temperature, corresponding to an average characteristic energy ~10 2 eV. We wish here to consider the characteristic bit energy of the 90% of baryons not involved in star formation and must first associate a representative temperature. We might consider the radiation temperature, T' , that would have the same energy density as matter: /g545 c 2 = /g305 T' 4 , where /g545 is the universe total mass density (including dark matter), and /g305 the radiation constant. Substituting the radiation constant by its definition in terms of fundamental constants, we obtain the characteristic bit energy, Echar = kBT'ln2 = (15 /g545/g644 3 c 5 / /g652 2 ) 1/4 ln 2. This definition was previously identified [56][57] as being identical to the characteristic energy of a cosmological constant (taking ln 2 ~ 1, identical to equation 17:14 of [58]). In this way we obtain a value T' ~ 35K, corresponding to a characteristic energy of ~ 3 × 10 -3 eV. Note that we expect a temperature around 10 times the temperature of the cosmic microwave background, CMB, since present matter energy density is ~ 10 4 times the CMB energy density. Thus the, otherwise difficult to account for, low characteristic milli-eV energy usually associated with the cosmological constant [58] may be finally explained as an information bit equivalent energy. While the characteristic energy of HDIE corresponds to information concerning stars and star formation, the characteristic energy of the cosmological constant corresponds to information concerning those parts of the universe not involved in star formation. HDIE characteristic bit energy increases as a +1 with increasing star formation while the cosmological constant characteristic energy falls as /g545 +1/4 , or a -3/4 , with the cooling universe majority. Then it is difficult to see how the cosmological constant, with a characteristic energy falling as a -3/4 , can produce a total energy that increases as a +3 as required for a constant energy density. In contrast, HDIE characteristic bit energy increases as a +1 and total bit number, N , increases as a +2 by the holographic principle to provide the required a +3 total energy variation.", "pages": [ 7 ] }, { "title": "3 Gedanken experiment", "content": "We have used Landauer's principle above to show that the HDIE model may be considered a serious contender to explain dark energy. Landauer's principle can also explain Maxwell's Demon, the famous gedanken (thought) experiment of physics [23][27]. We continue here with what might be considered a less serious thought experiment, but one that nevertheless provides an interesting, information related, aspect to the universe's accelerating expansion.", "pages": [ 8 ] }, { "title": "3.1 Hypothetical computer simulation", "content": "Consider the amount of information that a hypothetical super computer located outside the universe would need to fully simulate the universe's baryons. We conveniently ignore how such a computer can be located outside the universe, how this information would be gathered, and any measurement limitations imposed by the uncertainty principle. For a full physics simulation we require that each spatial parameter of every baryon be registered to the maximum physically meaningful accuracy, i.e. at the resolution of the Planck length, lp =1.6×10 -35 m. Intergalactic baryons in the present universe, size lu ~10 27 m, will then require an accuracy of one part in 6×10 61 (~ 2 205 ) and hence require 205 bits per spatial parameter. Similarly baryons located in giant molecular clouds, size lgmc ~10 18 m, and baryons located in typical stars, e.g. the sun size ls ~10 9 m, require ~175 bits and ~145 bits respectively per spatial parameter. Rather than consider the total information required for our simulation it is more convenient to limit ourselves to estimating the average number of bits per spatial parameter per baryon. We assume that the 10% of baryons presently located in stars that now require ~145 bits per parameter were, at earlier times, located in giant molecular clouds that required ~30 bits more at ~175 bits per parameter. We model this change from giant molecular clouds to stars from the variation of the fraction, f(a) , of baryons in stars as a function of scale size using the power law fits to the measurements plotted in figure 1(a). Meanwhile the remaining 90% of baryons, intergalactic baryons not involved in star formation, increased as log2( a ) up to their present 205 bits per parameter. The average number of bits per parameter per baryon, nav , is a function of scale size, a , given by: nav = (1f(1) ) log2( a lu / lp ) + ( f(1) -f(a) ) log2( lgmc / lp ) + f(a) log2( ls / lp ). Inserting values of f(a) from figure 1(a) we find that nav increased with intergalactic baryons but reached a peak value of 200.03 bits at a ~0.32, but then decreased due to increasing star formation to today's value of 199.02 bits, almost exactly one bit below the peak value. This one bit loss can be explained by the 10% of baryons that formed stars lost 30 bits per spatial parameter, contributing a loss of 3 bits to nav , while the 90% intergalactic baryons only added 2 bits to nav between a = 1/4 and the present, a = 1. The amount of information required to simulate an independant system should never decrease, otherwise it must imply a decrease in that system's number of states or information, contrary to the 2nd law. Now, if the universe expanded faster to double its expected size over the recent period, this would increase the contribution of the 90% intergalactic baryons to nav by a further 1 bit. Then we could effectively compensate for our loss of one bit in nav and satisfy the 2nd law again. Interestingly, dark energy has indeed doubled the size of the universe, exactly as we require, since it has increased the energy density by a factor of four, corresponding to a doubling of the Hubble parameter. Figure 1(b) grey continuous line, uses the above relation and assumptions to show the minimum required variation in total energy density that ensures there is no decrease in the amount of information required as input to our simple computer simulation during this period. This variation can be seen to lie close to that deduced from the effects of dark energy (whether due to HDIE or a cosmological constant). It is a surprise to find that the accelerating expansion was necessary for the universe to comply with the 2nd law and ensure that there was no decrease in the amount of information required as input to our simple thought experiment! Note that while the approach here is based on just a few simple assumptions they are all none the less quite reasonable. For example, many of the 90%, intergalactic baryons, not involved in star formation, do not move freely throughout the whole universe but are probably constrained to intergalactic filaments whose dimensions presumably stretch with the increasing space between galaxies and hence still have dimensions that scale with a . We have ignored the information represented by CMB as it has remained near constant since decoupling with CMB wavelength increasing in proportion to universe size. We have also ignored those baryons still in giant molecular clouds, yet to take part in star formation, but these will just add a constant amount to nav . If we had used a different minimum resolution, for example the Fermi length, 10 -15 m, the above bit numbers would be 66 bits less but, without the doubling of universe size from dark energy, there would still have been the same reduction of 1 bit in nav . So, although there is considerable uncertainty in absolute quantity of information required for our simulation, we can reasonably say that the doubling in universe size due to dark energy was just what was required to ensure that the amount of information needed as input to our computer simulation did not decrease. Of course it is still possible that our requirement for about one bit is just because we chose giant molecular clouds (size ~10 18 m) as the starting points for star formation. For comparison, at the two extremes of starting point either side, we would have obtained a value close to two bits drop in nav if we had considered star formation as starting all the way from the parent galaxies (size ~10 21 m) or a value close to zero change in nav if we had considered that star formation only started much later, at the final pre-stellar stage of proto-stellar nebula (size ~10 15 m). However, given the typical star formation sequence and the timescale considered in figure 1, it seems most reasonable to consider pre-existing giant molecular clouds as the effective beginning points for star formation.", "pages": [ 8, 9 ] }, { "title": "3.2 Algorithmic information content", "content": "The relation between this simulation information and the actual information intrinsic to the universe is analogous to the relation between the algorithmic information content (algorithmic entropy or Kolmogorov complexity), the size of the smallest algorithm that can generate a dataset and the actual amount of information contained within that dataset [59,60]. For non-random datasets algorithmic information is always less than the information in the dataset. For a truly random dataset, i.e. one that can not be calculated by an algorithm, the algorithmic information is always just slightly greater than the amount of information in the dataset. At a minimum it is greater by the size of the small program required to access the random dataset that must then be completely included as data, constituting the bulk of program code. At ~200 bits per spatial parameter each of the ~10 80 baryons in the universe requires ~10 3 bits, giving a total baryon simulation requirement ~10 83 bits. Note that this value is not very dependent on simulation resolution, whether say at Planck or Fermi lengths. Then, by analogy to algorithmic information content, we see that this simulation requirement is, as expected, less than the above N ~10 86 bits of HDIE because significant structure, or non-randomness, exists in the form of galaxies, stars etc. We can not deduce much from the actual size of this difference because of both the uncertainties in entropy estimation and the simplicity of our thought experiment argument. However, in future, the maximum rate of increase of simulation information is limited to the slow log2 a rate of simulated intergalactic baryons while holographic information should continue to increase at the much faster rate of a 2 . Then we should expect a growing significant difference that must further reflect the evolving level of non-randomness in the universe caused by increased structure formation.", "pages": [ 9 ] }, { "title": "4 Implications for the cosmos", "content": "The information based approach followed in this work leads directly to several implications for the cosmos, especially if the predicted HDIE model signature is observed, and HDIE thus found to be the correct explanation for dark energy. The first implication concerns the reason why the temperature variation a +0.98-0.1 so closely follows a +1 since z ~1 to provide the near constant HDIE energy density, -0.96 < w HDIE<-1.03. If star formation had continued to proceed at the earlier faster rate, then it would have continued the steep a +2.8-0.3 average baryon temperature increase after z ~1. This would have increased HDIE dark energy well above its present value, lead to much greater acceleration and greater expansion, but in turn, would have resulted in much less star formation. It would appear that since z ~1 there has been a balance, or feedback, between expansion acceleration and star formation that has naturally maintained the star formation rate close to a +1 for a constant dark energy density. Note that the reduced rate of star and structure formation starting at z ~1 was previously attributed to the onset of acceleration [61]. Thus HDIE provides a natural explanation for the reason why wDE =-1 since z ~1. The second implication concerns the cosmic coincidence problem. Our existence just now in the era dominated by dark energy is considered an unlikely coincidence. However, HDIE dark energy density increased with increasing entropy and increasing baryon temperature while mass density decreased with increasing universe scale size. There had to be a time when HDIE energy density reached a level comparable to mass energy density to initiate acceleration (provided that time was reached before f(a) =1). Similarly, the likelihood of our existence also increased as overall star formation increased, and thus more likely to occur after HDIE started to make a significant contribution to the universe energy budget, effectively removing the cosmic coincidence problem. The third implication concerns how long the present period of accelerating expansion will last. Acceleration will continue provided that the overall universe equation of state, w <-1/3 [9]. This threshold corresponds to HDIE energy density falling off as a -2 , and, assuming the total information, N , continues to follow the Holographic principle as a +2 , provides a limiting average baryon temperature, T , variation of a -1 . Thus, acceleration due to HDIE will continue providing T does not fall off more steeply than a -1 . Computer simulations of future average baryon temperatures, T , up to a =200 [62], predict a leveling off of T since f(a) is limited by definition to f(a) < 1, with a slow eventual fall as star formation ceases, but falling less steeply than the threshold gradient of a -1 . Thus acceleration should continue, until at least the universe has increased in size by a factor of 200. Clearly the fourth implication is that, should the predicted signature of HDIE be observed, it would provide very strong support for the holographic principle (see section 2.2). The final implication concerns how the universe as a whole still manages to satisfy the 2nd law when degrees of freedom are lost as matter becomes denser when stars are formed. It has been suggested [63] that the loss of thermodynamic entropy due to structure and star formation is counteracted by a gain in gravitational entropy. However, our simple gedanken experiment above implies that the extra expansion from dark energy acceleration provides enough of an increase in inter-galactic states to compensate for those states lost during star formation. Interestingly, with the HDIE explanation for dark energy, the extra expansion is itself a direct result of star formation.", "pages": [ 9, 10 ] }, { "title": "5 Summary", "content": "Computer scientist Landauer [64] emphasized that 'Information is Physical' and astrophysicist Wheeler [65] went further, declaring with his famous slogan 'It from Bit', that information may be more fundamental than matter. All of the arguments put forward in this paper for the HDIE dark energy explanation, as well as those used in the above thought experiment, also combine to point to the importance of considering information as one of the fundamental properties of the universe. Most importantly, we have shown that HDIE can account for dark energy both qualitatively and quantitatively, accounting for both key dark energy properties in the redshift range z < 1: the constant dark energy density and that energy density value. Furthermore, with the HDIE explanation for dark energy we no longer have the coincidence problem. At higher redshifts, HDIE should produce a clear signature, predicting that at z ~1.7 the Hubble parameter will have a value 2.6-0.5% less than that expected for a cosmological constant. Then the HDIE model is falsifiable as the size and location of this predicted signature lies within the resolvable ranges of the next generation of dark energy measurements.", "pages": [ 10, 11 ] } ]

[ { "title": "ABSTRACT", "content": "entropy ISSN 1099-4300 www.mdpi.com/journal/entropy Article", "pages": [ 1 ] }, { "title": "Optimization of Curvi-Linear Tracing Applied to Solar Physics and Biophysics", "content": "Markus J. Aschwanden 1 ∗ , Bart De Pontieu 1 , and Eugene A. Katrukha 2 Version November 10, 2021 submitted to Entropy . Typeset by L A T E X using class file mdpi.cls Abstract: We developed an automated pattern recognition code that is particularly well suited to extract one-dimensional curvi-linear features from two-dimensional digital images. A former version of this Oriented Coronal CUrved Loop Tracing (OCCULT) code was applied to spacecraft images of magnetic loops in the solar corona, recorded with the NASAspacecraft TransitionRegionAndCoronalExplorer(TRACE)in extreme ultra-violet wavelengths. Here we apply an advanced version of this code (OCCULT-2) also to similar images from the SolarDynamicsObservatory(SDO), to chromospheric Hα images obtained with the SwedishSolarTelescope(SST), and to microscopy images of microtubule filaments in live cells in biophysics. We provide a full analytical description of the code, optimize the control parameters, and compare the automated tracing with visual/manual methods. The traced structures differ by up to 16 orders of magnitude in size, which demonstrates the universality of the tracing algorithm.", "pages": [ 1 ] }, { "title": "1. Introduction 15", "content": "23 features of interest, using a priori information from the data. In this study we optimize an automated", "pages": [ 2 ] }, { "title": "2. Description of the Automated Tracing Code 38", "content": "65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 The original automated loop tracing code (OCCULT) employed the following control parameters: 91 the highpass filter boxcar n sm 2 , the noise threshold level N σ , the minimum curvature radius r min , the 92 moving segment length n s , the directional angle range ∆ α , and a filling factor q fill for the guiding 93 criteron. In the advanced code (OCCULT-2) we use only two free parameters: the lowpass filter n sm 1 , 94 and the minimum curvature radius r min . In addition, noise treatment is handled by selecting a typical 95 noise area in the image, as well as the control factor q med that ignores faint structures below the base level 96 z min = z med × q med . Thus, the new version of the code offers a simpler choice of control parameters for 97 automated detection of structures. 98", "pages": [ 3, 4 ] }, { "title": "3. Application to Solar Physics 99", "content": "In the following three subsections we process three different types of solar images, one from the 100 TRACE spacecraft (Section 3.1), one from the SDO spacecraft (Section 3.2), and one from a ground101 based solar telescope (Section 3.3). Some parameters of the analyzed images are listed in Table 1. 102", "pages": [ 4 ] }, { "title": "3.1. TRACE Data 103", "content": "The first image we are processing has been used as a standard in a number of previous publications 104 [3,4,18,19], which shows coronal loops in a dipolar active region that was recorded by the TRACE 105 spacecraft on 1998 May 5, 22:21 UT, in the 171 ˚ A wavelength (Fig. 1, center). The original image 106 has a size of 1024 × 1024 pixels with a pixelsize of 0.5 '' ( ≈ 360 km on the solar surface). The EUV 107 brightness or intensity in every pixel is quantified by a data number (DN), which has a minimum of 56 108 DN and a maximum of 2606 DN in this image. We show the bandpass-filtered image (with a lowpass 109 boxcar of n sm 1 = 5 pixels and a highpass boxcar of n sm 2 = 7 pixels) in Fig. 2. The image shows at 110 least four different textures (Fig. 1, side panels): coronal loops (L: curvi-linear features), so-called moss 111 regions (M: high-constrast reticulated or spongy features in the center of the image, which represent 112 the footpoints of hot coronal loops), transition region emission (T: low-contrast irregular features in the 113 background), and faint emission areas. The faint image areas show in addition a ripple with diagonal 114 Figure 1. A solar EUV image of an active region, recorded with the TRACE spacecraft on 1998 May 15, is shown with a colorscale that has the highest brightness in the white regions (center). In addition we show ( 100 × 100 pixel) enlargements of four subregions with different textures, which contain coronal loops (top left panel), electronic ripple (bottom left panel), chromospheric and transition region emission (top right panel), and moss regions with footpoints of hot coronal loops (bottom right panel). stripes at a pedestal level of ≈ 57 ± 1 DN(Fig. 2 bottom left panel R) that results from some interference 115 in the electronic readout, which can produce unwanted non-solar structures in the automated detection 116 of curvi-linear features. 117 The automated detection of curvi-linear features with the OCCULT-2 code yields a total of 437 118 loop structures with lengths of L ≥ 30 pixels (Fig. 2), whereof the longer loops (with lengths of 119 L > ∼ 50 ) coincide well with the 210 visually/manually traced loops (see Fig.7 in [4]). The good 120 agreement between the automated and visually detected loops can also be seen from the cumulative 121 size distributions of loop lengths obtained with both methods (Fig. 1, bottom right panel). The two 122 methods detect N = 154 and N = 134 loops with a length of L ≥ 70 pixels, and both distributions have 123 a powerlaw slope of α L ≈ 2 . 9 ± 0 . 1 . Challenges of coronal loop detection in this image are confusion 124 and interference from all three types of non-loop structures (Fig. 1): chains of 'dotted moss structures', 125 filamentary and spicular transition region emission, and the diagonal stripes of electronic ripple in the 126 background, which become all comparable with the signal of loop structures once the image contrast is 127 enhanced with a highpass filter. 128 The successful or false detection in an image thus depends most strongly on the chosen low and 129 highpass filter constant n sm 2 and the background base level z min = z med × q med , and to a lesser degree 130 on the other control parameters. A good indicator of the completeness and efficiency of automated 131", "pages": [ 4, 5 ] }, { "title": "Code = OCCULT-2", "content": "Image flux min/max Image threshold, FOV noise Loop lowpass/highpass Minimum curvature radius Loop width, min length Number of angles, curvatures Number of loops Manual: L m = 463 OCCULT-2: L = 387 m sm2 min loop detection is the number of coherently detected long loops, say with a length above 70 pixels here, 132 N det ( L > 70) (which is marked with a dotted line in the size distribution in Fig. 2, bottom right panel). In 133 Fig. 3 (top left panel) we show how this detection efficiency N det ( n sm 2 ) varies as a function of the chosen 134 control parameter n sm 2 . The detected number has a maximum of N det = 134 at n smi 1 = 5 and n sm 2 = 7 , 135 which corresponds to a bandpass filter in the range of 5-7 pixels ( 2 . 5 '' -3 . 5 '' or 1800-2500 km on the solar 136 surface). It appears that this is the most typical cross-section width (FWHM) of coronal loops. Other 137 statistical studies of coronal loops observed with TRACE yield similar values (e.g., FWHM= 1420 ± 340 138 km; [2]). Thus, if an image contains structures with a preferential cross-section width, the relevant 139 cross-section range can be bracketed with a bandpass filter ( n sm 1 , n sm 2 ), providing a useful a priori 140 information for automated detection of curvi-linear features. We conducted tests with all possible filter 141 widths n sm 1 = 1 , 3 , ..., 21 and n sm 2 > n sm 1 and found that the largest number of detected structures 142 virtually always occurs at n sm 2 = n sm 1 +2 , which can be explained also by the theoretical argument that 143 the best signal-to-noise ratio is obtained for maximum smoothing of the highpass-filtered (unsharp mask) 144 image. We vary also the minimum curvature radius r min and find a maximum of detected structures at 145 r min ≈ 30 pixels (Fig. 3 top right panel). 146", "pages": [ 6, 7, 8 ] }, { "title": "3.2. SDO/AIA Data 147", "content": "The next image to which we apply our automated loop tracing code is from the AtmosphericImager 148 Assembly(AIA)onboard the SolarDynamicsObservatory(SDO), which replaced the TRACE mission 149 and is operating since 2010 [20]. AIA has a similar spatial resolution (pixel size 0.6 '' ) as TRACE (pixel 150 size 0.5 '' ), but covers the full Sun disk, with an image size of 4096 × 4096 pixels. Fig. 4 shows a subimage 151 with a size of 1450 × 650 pixels, which contains a complex of two magnetically coupled active regions, 152 observed on 2011 Aug 03, 01 UT, in the 171 ˚ A wavelength. This image is currently subject of nonlinear 153 force-free magnetic modeling (Mark DeRosa, private communication 2012), and thus requires automated 154 loop tracing to constrain the coronal part of the magnetic field configuration [5]. Differences to the 155 TRACE image are the higher sensitivity of the AIA telescopes, different exposure times, the availability 156 of simultaneous images in 8 other wavelengths, different image compression, and no apparent electronic 157 ripple in the CCD readout, which all affect the automated detection of faint structures. Synthesized loop 158 tracings from multiple wavelength filters has been proven to provide a more robust and representative 159 subset of loop structures for magnetic modeling than loop tracings from a single filter image ([6]). 160 We vary the lowpass filter constant in the range of n sm 1 = 1 , ..., 21 , set the highpass filter constant 161 to n sm 2 = n sm 1 + 2 , and find a maximum detection rate of N det ( L > 70) = 121 loop structures for 162 n sm 1 = 9 and n sm 2 = 11 (Fig. 3, second row left panel). We vary also the minimum curvature radius 163 in the range of r min = 10 , ..., 100 pixels and find a maximum detection rate of N det ( L > 70) = 121 at 164 r min = 30 pixels (Fig. 3, second row right panel). 165", "pages": [ 8 ] }, { "title": "3.3. SST Data 166", "content": "Now we apply automated loop tracing to a solar image in a completely different wavelength, namely 167 in the H α 6563 ˚ A line, the first line of the Balmer series of hydrogen. Fig. 5 shows such an image of 168 the solar upper chromosphere, which displays chromospheric spciules in the right side of the picture and 169 Figure 4. Bandpass-filtered image of an active region complex observed with AIA/SDO on 2011 Aug 3, 01 UT, 171 ˚ A , shown as intensity image (top panel), as bandpass-filtered version with n sm 1 = 9 and n sm 2 = 11 (middle panel), and overlaid with automatically traced loop structures (bottom panel), where the low-intensity values below the median of f = 75 DN are blocked out (grey areas). 800 600 400 200 0 0 200 400 1000 filamentary structures in the upper chromosphere and transition region (in altitudes ≈ 2000 -5000 km 170 above the solar surface) [10,11]. The image was taken with the Swedish 1-m Solar Telescope (SST) 171 on La Palma, Spain, using a tunable filter, tuned to the blue-shifted line wing of the H α 6563 ˚ A line. 172 The spicules are jets of moving gas at a lower temperature than the million degree hot corona and flow 173 upward from the chromosphere to the transition region with a speed of ≈ 15 km s -1 [10]. The image has 174 a size of 1507 × 999 pixels ( 62 × 41 Mmon the solar surface), with a pixel size of 0 . 041 '' (or 30 km on 175 the solar surface). This picture is particularly intriguing for automated tracing of curvi-linear structures 176 because of the ubiquity and complexity of fine structure. 177 This SST image has such a high contrast so that there is no significant noise that affects curvi-linear 178 tracing. Thus, we set the background level to zero ( q med = 0 . 0 ) in the OCCULT-2 code. We vary the 179 lowpass filter constant in the range of n sm 1 = 1 , ..., 21 , set the highpass filter constant to n sm 2 = n sm 1 +2 , 180 and vary the minimum curvature radius in the range of r min = 10 , ..., 100 pixels. We find a maximum 181 number of detected structures of N det ( L > 70) = 376 (with a length above 70 pixels) at n sm 1 = 3 182 and N sm 2 = 5 (Fig. 3, middle row left panel), and r min = 40 pixels (Fig. 3, middle row right panel). 183 Extending to shorter segments with lengths of L > 30 pixels, the automated tracing code identifies a total 184 of 1757 curvi-linear segments, which outline the patterns of the flow field in the upper chromosphere 185 (Fig. 5, bottom panel). 186", "pages": [ 8, 9, 10, 11 ] }, { "title": "4. Applications to Biophysics 187", "content": "Finally we apply our automated tracing code to images obtained in cellular biophysics, in order to test 188 the versatility and universality of the OCCULT-2 code. As an example, we chose microscopy images of 189 microtubules, but the same approach could be applied to any other filament-like structures: intermediate 190 or actin filaments, fibrin, etc. Microtubules are long stiff polymers that are part of cytoskeleton (internal 191 cellular scaffolding). They are important for cell division, motility and organization ([16], [9]). The 192 entangled network of microtubules serves as a 'railroad' system for delivery of cargos by molecular 193 motors and also as a stiff carcass controlling cellular mechanics ([9], [26]). It is a dynamic network 194 adapting and changing in time in response to external cues. The automated extraction of the microtubules 195 network's configuration from microscopy images and movies can provide insights on the mechanisms of 196 these changes. 197 Fig. 6 shows two images of cells from two different cell lines with microtubules networks of different 198 density. Cells belonging to CHO cell line (Fig. 6 top, image Cell-HC) are small and have sparse 199 microtubule network. Cells from U2OS line (Fig. 6 bottom, image Cell-LC) are larger and contain 200 more dense radial network. Since microtubules are transparent to the visible light, a fluorescent tag is 201 used to observe them in the living cells. In the analyzed images (Fig. 6) microtubules were labeled 202 with a green fluorescent protein (GFP) ([28]) that adsorbs light at 490 nm and has an emission peak 203 at 510 nm wavelength. The images were acquired with a spinning disk confocal microscope using the 204 corresponding GFP's emission filter. The magnified image was projected on the 16-bit chip of camera 205 with dimensions of 512 × 512 pixels resulting in the final image pixel size of 66 nm. The width of 206 microtubules filaments measured in the electron microscopy studies is about 25 nm ([29]). Due to the 207 diffraction limit the effective width of microtubules in the image is much larger and is defined by the 208 microscopy setup and the wavelength used ([1]). In our case it is approximately equal to a half of the 209 emission wavelength (510 nm) that is close to the measured microtubule width of 4 . 7 ± 2 . 0 pixels. The 210 two pictures shown in Fig. 6 (left) show cases of opposite contrast, one with high contrast (Fig. 6 top, 211 image Cell-HC), and one with low contrast (Fig. 6 bottom, image Cell-LC). The difference in contrast 212 is explained by a different amount of GFP fluorescent tag associated with microtubules. The automated 213 tracing of the cell filaments provides their locations inside cell and curvature radii, from which flexural 214 rigidity and mechanical stress can be inferred. 215 216 217 218 219 We optimize the automated filament tracing by varying the bipass filter constants and find a maximum number of detected filaments (with lengths L > 70 pixels) with n high-contrast cell image, and n sm 1 = 7 , n sm 2 = 9 , r min = 40 for the low-contrast cell image (Fig. 3). The low-contrast image has a higher degree of noise, and thus the filter constants have to be adjusted to", "pages": [ 11, 13 ] }, { "title": "5.1. Optimization of Automated Curvi-Linear Tracing 222", "content": "The efficiency and accuracy of automated curvi-linear tracing can be controlled by a number of tuning 223 or control parameters. For the present OCCULT-2 code we have three independent control parameters 224 ( n sm 1 , r min , q med ). A prerequisite for the application of curvi-linear tracing codes is the assumption that 225 the structures of interest have a much smaller width w than their length l , so that they can be represented 226 by a 1D path, which is generally curved, possibly limited by a minimum curvature radius r min . The larger 227 the minimum curvature radius is, the less ambiguity there is for tracing of crossing 1D structures. In this 228 study we explored the parameter space of the control parameters n sm 1 and r min in order to optimize the 229 performance of the automated tracing code. 230 The lowpass filter ( n sm 1 ) and highpass filter ( n sm 2 ) represent the brackets or scale range of a bandpass 231 filter, bracketing the range of cross-sections of detected structures. We expect to find structures with the 232 smallest width preferentially in images with a high signal-to-noise ratio, while noisy images require 233 more smoothing to enhance fine structure, and thus tend to have larger widths due to the smearing effect 234 of the smoothing. We found the narrowest structures indeed in the two images with the highest contrast 235 (i.e., the SST and Cell-HC image), where highpass filters with boxcars of n sm 2 = 5 pixels were used. 236 In images with lower contrast, optimum performance occurred for highpass filters of n sm 1 = 7 , ..., 11 237 pixels. In addition, we found that the smallest bandpass filters produce the sharpest structures and thus 238 the highest detection rate of structures. Since a symmetric boxcar requires odd numbers n sm = 1 , 3 , 5 , ... , 239 the smallest difference between a lowpass and a highpass filter is 2, and thus the optimum combination 240 is expected to be n sm 2 = n sm 1 +2 , which we indeed confirmed also experimentlly. 241 For the minimum curvature radius we found optimum performance for a typical range of r min ≈ 242 30 , .., 50 pixels (Fig. 3), which seems not to depend on the contrast of the image. 243 The control parameter q med = 1 suppresses faint structures in an image that are below the median 244 value of the image brightness. A meaningful value for this parameter depends very much what fraction 245 of the image contains bright structures of interest, which has to be decided depending on the area ratio 246 sm 1 = 3 , n sm 2 = 5 , r min = 50 for the of structures of interest to non-relevant background area. If the background has comparable brightness 247 to the structures of interest, a thresholded separation may be impossible, but may still succeed if the 248 background has a different texture than the curvi-linear features (see examples in Fig. 1, where moss and 249 transition region emission have a different texture than coronal loops, while background ripples have 250 the same texture as straight loops and can only be separated by a threshold control parameter). Future 251 efforts aim to synthesize the loop tracings from multi-wavelength image sets, which are more robust and 252 representative for magnetic modeling than single-wavelength images ([6]). 253 In conclusion, we recommend the following procedure to achieve optimum performance of the curvi254 linear tracing code (OCCULT-2): (1) Start with the following recommended control settings: n sm 1 = 1 , 255 n sm 2 = 3 , r min = 30 , and q med = 1 . 0 ; (2) Vary the filter combination in the range of n sm 1 = 1 , 3 , ..., 15 , 256 while setting n sm 2 = nsm 1 + 2 , to find the maximum detection rate for a given loop length (e.g., here 257 we used L > 70 pixels); (3) Low-contrast images are likely to require higher highpass filter values 258 n sm 2 than high-contrast images. (4) Vary the minimum curvature radius r min within some range to find 259 the maximum detection rate of structures; (5) If the code yields a lot of random structures in obvious 260 background areas, increase the base level factor q med > 1 . 261 The software of the numerical code OCCULT-2 is publicly accessible in the InteractiveDataLanguage 262 (IDL) in the Solar Software (SSW) package. A tutorial and example is accessible at the authors 263 homepage http://lmsal.com/ ∼ aschwand/software/. 264", "pages": [ 13, 14 ] }, { "title": "5.2. Solar Applications 265", "content": "The automated tracing of curvi-linear structures in solar physics has mostly been applied to 266 extreme-ultraviolet or soft X-ray images, which show magnetized coronal loops that outline the coronal 267 magnetic field, due to the low plasmaβ parameter in the solar corona (which is defined as the ratio of 268 the thermal to the magnetic pressure). Thus, coronal loops represent the perfect tracers of the otherwise 269 invisible coronal magnetic field. Standard magnetic field models of the solar corona or parts of it, such 270 as sunspot regions and active regions, have been modeled by extrapolating a photospheric magnetogram, 271 either with a potential field solution or a force-free solution of Maxwell's equations. However, since 272 the chromosphere was found not to satisfy the force-free condition [23], extrapolations of photospheric 273 magnetograms do not exactly render the coronal magnetic field, while coronal loops outline the true 274 coronal magnetic field. It is therefore desirable to trace such coronal loops in EUV and soft X-ray images 275 and to use them to constrain a magnetic field solution. Such attempts have been performed with single 276 EUV images as well as with stereoscopic EUV image pairs [5]. Forward-fitting of theoretical magnetic 277 field models to traced coronal loops is able to discriminate between potential field and force-free field 278 models, as well as to quantify the free magnetic energy (that is released in solar flares) and Lorentz 279 forces, e.g., [13]. Curvi-linear tracing of filamentary structures in the chromosphere and transition region 280 (Fig. 5) may also help to constrain the horizontal magnetic field components in the non-forcefree regions, 281 which has been used in pre-processing of solar force-free magnetic field extrapolations [12]. 282 One fundamental limitation of automated coronal loop tracing is the confusion by background 283 structures resulting from EUV emission from the transition region, which has generally a cooler 284 temperature than the coronal loops. Future efforts may use multi-wavelength image data sets to 285 discriminate EUV emission from the chromosphere, transition region, and the corona by its temperature, 286 using a a deconvolution of the multi-wavelength temperature filter response functions in terms of a 287 differential emission measure (DEM) method. 288", "pages": [ 14, 15 ] }, { "title": "5.3. Biological Applications 289", "content": "The method of curvi-linear tracing is increasingly used in the analysis of biological and medical 290 images, such as to characterize blood vessel tracking in retinal images [15,21,27], neurons [30], dendritic 291 spines [31], or microtubule tracing in fluorescent and phase-contrast microscopy [8,17,25], as shown in 292 Fig. 6. In our experiment with a high-contrast (Fig. 6, top panel) and low-contrast image (Fig. 6, bottom 293 panel), we demonstrated that a bipass or bandpass filter with a bandpass factor of n sm 2 = n sm 1 + 2 294 enhances the structures to an optimum contrast for automated tracing. Moreover we found that the 295 bandpass filter of low-contrast images requires a larger width ( n sm 2 = 9 pixels in Fig. 6, bottom panel) 296 than in a high-contrast image ( n sm 2 = 5 pixels in Fig. 6, top panel). 297 In our optimization exercise we concentrated mostly on the completeness of detected long curvi-linear 298 segments, but future efforts may also consider the optimization of linking multiple segments that are 299 interrupted with gaps or subject to crossings. The efficiency and reliability of automated curvi-linear 300 algorithms became more important with the massive increase of imaging data over the last decade, 301 which exceeds our limited capabilities of visual inspection. Note that the algorithm used here tracks 302 curvi-linear structures that differ by 16 orders of magnitude in size, from 66 nm to 360 km (pixel size in 303 images). 304", "pages": [ 15 ] }, { "title": "6. Appendix A: Analytical Description of the OCCULT-2 Code 305", "content": "The Oriented Coronal CUrved Loop Tracing (OCCULT-2) code version 2 is an improved version 306 of the original OCCULT code described in Aschwanden (2010). The improvements include: (1) a 307 'curved' guiding segment that is adjusted to the local curvature radius of a traced loop (representing 308 the second-order term of a polynomial), rather than the linear (first-order polynomial) guiding segment 309 used in the original version, (2) suppression of faint structures, (3) bypass filtering instead of highpass 310 filtering, and (4) simplification of selectable free parameters. 311 The input is a simple 2-dimensional (2D) image z ij , with pixel numbers i = 0 , ..., n x -1 on the x-axis, and j = 0 , ..., n y -1 on the y-axis, respectively. The output is a number of curvi-linear structures (also called 'loops' for short), which are parameterized in terms of x and y-coordinates, [ x ( s k ) , y ( s k )] , where the loop length coordinate s k = 0 , ..., n s is given in steps of ∆ s = 1 pixel, so that for all k = 0 , ..., n s -1 , with n s the number of loop points for each loop. 312 The goal of the algorithm OCCULT-2 is to retrieve as many curvi-linear structures as possible, without picking up false signals of non-existing structures in the noise of the image, which has some probability to form chains of random points in a curved array configuration. The challenges are therefore to evaluate an optimum threshold level that separates existing loops from noise structures, and to retrieve the real curvi-linear structures as completely as possible, without subdividing them into partial loop segments. Our strategy to obtain a fast numeric code is to retrieve the loops in a one-dimensional search algorithm, because any two-dimensional concept has a computation time that grows with the square of the image size. The one-dimensional parameter space is essentially the loop length coordinate s k , k = 0 , ..., n s -1 . In addition we define two other independent parameters in each loop point, which can be considered as the first-order and second-order term of a polynomial, namely the local direction angle α l , l = 0 , ..., n α , and the curvature radius r m , m = 0 , ..., n r . The algorithm selects iteratively the brightest position ( x i , y j ) in the image and starts a bi-directional loop tracing, determining first the local direction α l ( x i , y j ) and curvature radius r m ( x i , y j ) at the starting point, and continues tracing the loop within a small (guided) range of the local curvature radius, which is the principle of 'orientation-guided tracing'. So we are dealing essentially with 1D-tracing in a 5D-parameter space ( x i , y j , s k , α l , r m ) , which we parameterize with the 5 indices ( i, j, k, l, m ) that have the index ranges i = 0 , ..., n x -1 , j = 0 , ..., n y -1 , k = 0 , ..., n s , l = 0 , ..., n α , m = 0 , ..., n r . Specifically, we define the arrays, for a symmetric bi-directional array, used in the search of the local direction α l , and for a uni-directional array, used in the search for the curvature radius in the forward-direction of a traced structure. For the directional angle α l , which is only determined at the starting point of the loop, we use a fixed array with a resolution of one degree (or π/ 180 radian), For the curvature radii r m we use a reciprocal scaling in order to obtain a uniform distribution of directional angles at the end of a curvature segment, This parameterization covers positive and negative curvature radii in the ranges of [ -∞ , -r min ] and 313 [ r min , + ∞ ] . The choice of an even number n r prevents the singularity r m = ±∞ . 314 Now we describe the consecutive steps of the algorithm one by one, which follow more or less the 315 same flow chart as depicted in Figure 1 of Aschwanden (2010). 316 (1) Image base level ( q base ): If the image consists of high-contrast structures without unwanted secondary structures in the background, we do not have to worry about the image base level and can set it to the lowest value (which should be zero or positive in astrophysical images that record an intensity or brightness). However, if there are unwanted structures at a faint brightness level, we can just set the image base level z base above the brightness level of unwanted structures, which we parameterize with the factor q med in units of the median brightness level z med = median [ z i,j ] , so that the corrected brightness z ' i,j in each pixel fulfills the condition For q med = 0 the image is unchanged, while the image z ' i,j appears to be flat in the fainter half area of the 317 image, if set q med = 1 . 0 . So, the value q med can be adjusted depending on the estimated fraction of the 318 image that is covered with structures of interest. For solar images, this feature offers a convenient way 319 to filter out coronal loops in active regions (which are bright) from unwanted structures in the Quiet Sun 320 (which are faint). 321 (2) Bandpass Filtering ( n sm 1 , n sm 2 ): The tracing of curvi-linear structures is considerably eased by enhancing of fine structures within a chosen bandpass that corresponds to the typical width n w of structures of interest, which is typically a few pixels, and assuming that the curvi-linear structures has a much longer length n s than width, i.e., n s /greatermuch n w . We accomplish the enhancement with a bandpass filter, which consists of a lowpass filter with a boxcar n sm 1 and a highpass filter with a boxcar n sm 2 , which filters out broad structures with widths n w > ∼ n sm 2 (highpass filter), but smoothes out fine structure with a boxcar of n sm 1 < n sm 2 . We experimented with a large number of combinations ( n sm 1 , n sm 2 ) and found that the optimum choice is, which follows the principle of maximum possible smoothing of fine structures with a given width n sm 2 . 322 The values for the smoothing with a symmetric boxcar has to be an odd integer, i.e., n sm 1 = 1 , 3 , 5 , ... ) , 323 which implies n sm 2 = 3 , 5 , 7 , ... , where the lowest value n sm 1 = 1 corresponds to the original image 324 without any smoothing. This experimentally tested relationship for the optimum choice of bandpass 325 filters ( n sm 1 , n sm 2 ) reduces also the possible parameter space by one dimension, and thus we have search 326 only for n sm 1 = 1 , 3 , 5 , ... , while using n sm 2 = n sm 1 +2 . 327 (3) Noise Threshold: Our algorithm starts with the brightest curvi-linear structure and proceeds to fainter structures, and thus we have to find a stop criterion that halts the procedure when it reaches the level of data noise. Such a noise threshold has to be determined empirically for every image, since there are many sources of possible data noise. Testing many images with completely different data types, we found that a most reasonable threshold level can be determined by an interactive choice of a noise area in the image that contains typical data noise but little structures of interest. Such an image area can be characterized by the pixel ranges ( i n 1 : i n 2 , j n 1 : j n 2 ) . In this noise area we determine the median brightness z noise med and define a noise threshold at the doubled value, using only the pixels with positive values in the noise area (in order to prevent a too low value of z thresh = 328 0 in the case when more than 50% of the noisy pixels are below the previously chosen base value z base ). 329 The rationale for the factor 2 in the threshold level comes from the fact that the median separates out 330 only half of the noisy pixels, while the double value would separate out all noisy pixels if the distribution 331 of noisy pixel values follows a linear relationship. 332 (4) Start of Curvi-Linear Structures: We are ready now to trace the first curvi-linear structure. We determine the location ( i 0 , j 0 ) of the absolute brightness maximum z 0 in the bandpass-filtered image z filter i,j , The rationale for the choice of this starting point is the expectation to trace first the most significant 333 structure in the bandpass-filtered image, which can then be continued by going to the next-significant 334 structure, once the tracing of the first structure has been successfully completed and the corresponding 335 loop area is eliminated in a residual image. Consequently, the maximum of the brightness in the residual 336 image marks the second-brightest structure and we can repeat the same procedure by tracing the next 337 loop. 338 (5) Loop Direction at Starting Point: The next element of the structure to be traced is the first-order term of a polynomial, the direction angle α l , which is also the direction of a possible ridge that outlines the local segment of the structure. We determine this directional angle simply by measuring the flux averaged over a straight loop segment symmetrically placed over the starting point ( i 0 , j 0 ) and rotated over a full range of possible angles α l from 0 · to 180 · degrees. The x,y-coordinates of this linear segment are, with the array s bi k defined by Eq. (5), where the index k runs along the length s k of the segment, and the index l denotes a particular angle α l . Among the set of angular values α l we determine the maximum of the summed flux in each rotated segment, which yields the local direction α max = α l ( l = l max ) . In the ideal case of a straight ridge with a constant 339 value z 0 along the ridge segment with length n s pixels and zero-values outside, a value of z ‖ = z 0 is 340 found, while the value for a segment in perpendicular direction to the ridge is much smaller, namely 341 z ⊥ = 1 /n s . This ridge criterion works even for a close succession of parallel ridges, in which case it is 342 still a factor of 2 smaller than in parallel direction, i.e., z ⊥ = 1 / 2 . 343 (6) Local curvature radius: Next we determine the second-order term of a polynomial that follows the loop to be traced. We define a directional angle β 0 that is in perpendicular direction to the loop and defines the direction where all centers of possible curvature radii are located (that intersect the structure at location ( x 0 , y 0 ) ), see geometric definition of the angles α and β in Fig. 7, The location ( x c , y c ) of the curvature center with a minimum curvature radius r min is then found at, The loci of all curvature centers ( x m , y m ) of a set of curvature radii r m = r min / [ -1 + 2 m/ ( n r -1)] (Eq. 6) is then found at, Since we want to follow a loop along a curved segment for every possible curvature radius r m , we determine the coordinates for each segment point s k . It is useful to define the angle β m of the line that connects a curvature center ( x m , y m ) with a curved segment point s k , where σ dir = ± 1 has two opposite signs, depending on the forward or backward tracing of a loop. The x,y-coordinates ( x km , y km ) of the loop segment s k is then, In order to determine the curvature radius r m that fits the local loop segment best, we search for the segment with the maximum flux along the curve with radius r m , Since we know now the optimum curvature radius r m , we can trace the loop incrementally by a step ∆ s and extrapolate the position ( x k +1 , y k +1 ) and angle α k +1 by (7) Bidirectional tracing: The step (6) describes the extrapolation or tracing of the loop segment from 344 position s k = ( x k , y k ) to s k +1 ( x k +1 , y k +1 ) by an incremental length step ∆ s . This step is repeated, 345 starting from some arbitrary starting point s 0 inside the segment until the first endpoint s n s 1 , where the 346 traced curvi-linear structure seems to end. The first endpoint generally is demarcated at a location where 347 the bandpass-filtered image has zero or negative values, so one could just detect the first image pixel 348 along the guiding segment s k that is non-positive. However, in order to allow for some minor gaps in 349 noisy structures, it turned out to be more reliable to define a stop criterion when a least a few pixels 350 have a nonpositive value, say n gap = 3 pixels. An example of a traced loop is shown in Fig. 8, which 351 illustrates that a loop can reliably be traced even in the presence of secondary loops that intersect at small 352 angles. 353 After completing the first half segment ( σ dir = +1 ), we repeat the same procedure from the midpoint 354 ( x 0 , y 0 ) in opposite direction ( σ dir = -1 ), until we stop at the second endpoint at s n s 2 . We combine than 355 the two segments [ s 0 , s n 1 ] and [ s 0 , s n 2 ] by reversing one segment in order to maintain the same direction, 356 and concatenating the two equal-directed segments into a single loop structure with indices [ s 0 , ..., s n s ] , 357 where the length is s n = ( s n 1 + s n 2 ) . 358 Structure # 28 Loop # 19 680 550", "pages": [ 15, 16, 17, 18, 20, 21, 22 ] }, { "title": "Acknowledgements 373", "content": "377 Fatemeh Amirkhanlou, Hossein Safari, and participants of the 3rd Solar Image Processing Workshop", "pages": [ 23 ] }, { "title": "References 383", "content": "394 6. Aschwanden, M.J. and Sun. Y. The Active Region 11158 during the 2011 February 15 X-Class Flare: 409 410 411 412 415 416 417 418 419 420 421", "pages": [ 23, 24 ] } ]

[ { "title": "Might quantum-induced deviations from the Einstein equations detectably affect gravitational wave propagation?", "content": "Adrian Kent 1, 2, ∗ 1 Centre for Quantum Information and Foundations, DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, U.K. 2 Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON N2L 2Y5, Canada. (Dated: October 2012 (revised March 2013)) A quantum measurement-like event can produce any of a number of macroscopically distinct results, with corresponding macroscopically distinct gravitational fields, from the same initial state. Hence the probabilistically evolving large-scale structure of space-time is not precisely or even always approximately described by the deterministic Einstein equations. Since the standard treatment of gravitational wave propagation assumes the validity of the Einstein equations, it is questionable whether we should expect all its predictions to be empirically verified. 'In particular, one might expect the stochasticity of amplified quantum indeterminacy to cause coherent gravitational wave signals to decay faster than standard predictions suggest. This need not imply that the radiated energy flux from gravitational wave sources differs from standard theoretical predictions. An underappreciated bonus of gravitational wave astronomy is that either detecting or failing to detect predicted gravitational wave signals would constrain the form of the semi-classical theory of gravity that we presently lack.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "One difficulty theoretical physicists currently face is that, as the subject has grown larger and speculative attempts to address fundamental problems have multiplied, our collective knowledge has become increasingly fragmented. Questions which are at the forefront of the attention of one group of people can be pretty much neglected, or not even recognised, by others. Indeed, even individuals may display a version of this: because our attention is selective and trained, we can end up functioning according to a sort of doublethink, in which we note a problem in one context and neglect it in others. I suspect this is actually much more common than we generally realise. Here, following a broader programme ([1, 11-14]; see also [8-10] for earlier distinct but related ideas in this area) of trying to test underexplored foundational questions relating quantum theory and gravity, I suggest one area - gravitational wave physics - where this phenomenon may be at work. The problem is this. On the one hand, the standard theory of propagating gravitational waves treats them as perturbations of the Einstein gravitational field equations. On the other hand, there is no way of incorporating unpredictable quantum events into a classical theory of gravity described by the Einstein equations. This is not only because of small corrections arising from an as yet unknown quantum theory of gravity. If this were the only issue, it might be easier to defend the case that the standard perturbative treatment of gravitational waves is likely to be essentially unaltered by quantum corrections, at least in regions where the grav- itational field is not too strong. The more immediate problem is that, whenever a quantum measurement type interaction takes place - whether a deliberate measurement by a human observer or a naturally occurring event -it can produce any of a number of macroscopically distinct results from the same initial state. According to the standard understanding of quantum theory, these measurement outcomes are intrinsically probabilistic. Not only can they not be predicted in advance by quantum theory, but there are very strong reasons [2-5] to believe that no underlying deterministic theory allows us to predict them. These macroscopically distinct outcomes lead to distinct space-times and matter distributions. Each of several possible space-times and matter distributions could thus emerge from the same initial state. If the measurement event is suitably amplified, their differences can be arbitrarily large. Since general relativity is deterministic, it follows that the Einstein equations cannot even approximately describe the large-scale structure of spacetime around measurement events. If we try to describe the space-time physics in the vicinity of the measurement event by some classical stressenergy tensor and metric, it seems that we need to introduce some stochastic source which creates an unpredictable macroscopic perturbation of the metric and tensor, at some point - or, probably more accurately, in some region - in the vicinity of the event. However, not only are the Einstein equations inconsistent with these perturbations, but moreover we know of no generally valid way of incorporating them into some semi-classical theory coupling the metric to a classical quantity derived from the quantum stress-energy tensor, or by some stochastic modification of the Einstein equations. Looking at current knowledge without a predefined theoretical agenda, then, all we can really say for sure is that general relativity and quantum theory both work well in their respective domains. However, even charac- terizing precisely those domains of confirmed validity is subtler and harder than it first seems, as the comments above illustrate. It is even harder to justify confidence that we now understand all the principles underlying a unified theory. It's not evident that mainstream ideas work, and while it's certainly not evident that comparatively undeveloped alternatives will work either, they do exist. For example, it is certainly possible to imagine unifications in which both general relativity and quantum theory work as good approximations in their respective domains, and in which gravity is quantized, but nonetheless the structure of space-time is also constrained by additional laws that modify the probability of each spacetime and are defined by intrinsically geometric rules that do not follow from any quantum theory [1]. It's also possible to imagine theories in which gravity is not quantized at all, and the laws of nature define some probability distribution on classical space-times with quantum matter distributions [1]. The general hypothesis that gravity and quantum state collapse could be linked [8-10], via fundamentally nonunitary dynamical laws is also intriguing, even though it too is hard to make into a precise theory. Also, even if one of the more currently popular approaches to quantum gravity is correct, it is very unclear how to derive from it a phenomenological higher-level theory that couples microscopic quantum matter with macroscopic events in space-time, or precisely what features such a theory would have. To reiterate, the point here is not to advocate these comparatively undeveloped alternative ideas, but simply to underline that unifying gravity and quantum theory is an open subject and there are many un(der)explored possibilities. Even in macroscopic regimes with weak gravitational fields, we simply do not have a theory of matter and space-time good enough to fit all observable data. This problem occupies a peculiar status in modern physics: it cannot be denied that the problem is there, but yet most discussions of quantum theory and gravity ignore it. The gap in the literature is so glaring that one almost gets the impression that it is somehow seen as scientifically unsophisticated to look for a theory - even a provisional phenomenological theory - that actually fits the available empirical data. To sum up: the Einstein equations do not actually correctly describe the large-scale structure of space-time and finding equations that do is an open question . It thus doesn't seem so obvious that the long-range propagation of gravitational waves is necessarily correctly described by considering them as perturbations of the Einstein equations. The rest of this paper attempts to flesh out this point mostly by conceptual, rather than mathematical, argument. In mitigation, I would stress again that we know nothing for certain about unifying quantum theory and gravity, and the world possibly already has more than enough mathematically rigorous, but conceptually prob- lematic, and quite likely ultimately irrelevant, calculations based on speculative mathematical ideas about how to solve the problem. Our ultimate aim, of course, is to test precise mathematical theories against quantitative experimental data, but just at the moment we need new ideas about where to look. It seems to me there are strong reasons to try a different style of scientific reasoning: namely, to look at interesting experiments and observations where we don't yet know for sure what theory predicts but can - now, or soon - get an empirical answer, and to ask whether there is any semi-plausible phenomenological model or intuition that might contradict the standard expectation (if there is one). Either we verify with certainty interesting features of gravitational physics that were previously either ignored completely or assumed without compelling evidence, or (even better) we learn something new and surprising.", "pages": [ 1, 2 ] }, { "title": "A. The Page-Geilker Experiment", "content": "The fundamental problem in constructing semiclassical gravity theories was illustrated by a very simple experiment carried out some time ago by Page and Geilker [6]. Page and Geilker's aim was to refute conclusively the hypothesis that a classical gravitational field couples to quantum matter via the semiclassical Einstein equations where G µν is the Einstein tensor and T µν the quantum stress-energy operator. Their hypothesis presupposes an Everettian interpretation of quantum theory, so that the matter field quantum state, | ψ 〉 , is supposed to evolve unitarily without collapse. As Page and Geilker noted, this hypothesis already seemed intrinsically unlikely (perhaps even incredible) before they carried out their experiment, since it is hard to imagine how it could lead to a cosmological theory which accounts for our observation of gravitational fields generally consistent with those predicted by the Einstein equations from the observed positions of astronomical bodies. Page and Geilker's motivation for their experiment is thus seriously questionable. Perhaps, though, one could imagine some form of theory in which a classical gravitational field couples to the expectation value of quantum matter, while the quantum matter state over time collapses towards definite values for T µν . In any case, it still seems good to have conclusive experimental confirmation even of very solidly based theoretical expectations when, as here, we have no complete theory. The experiment proceeded by counting the number of decays detected from a radioactive source in a given time interval, and then manually placing large ( ≈ 1 . 5 kg) lead balls in one of two configurations, with the choice of configuration depending on the decay count. A Cavendish torsion balance, sensitive enough to distinguish between the two configurations, was used to measure the local gravitational field. To good approximation, Eq. (1) predicts that in each run the experimenter should (whichever of the two configurations they place the masses in) observe a gravitational field defined by the weighted average of the fields corresponding to the two possible configurations. As most expected, the results were consistent with the hypothesis that the gravitational field is determined by the configuration of the masses chosen in any given run of the experiment, and inconsistent with Eq. (1). We can flesh out the implications of the experiment or, to capture the historical flow of ideas better, perhaps one should say the implications of the generally held prior intuition that its results would be those which were in fact observed - by looking at three possible solutions to the Einstein equations. First, consider the classical metric and matter fields, which we denote by ( g 0 µν , φ 0 ), in the neighbourhood of some spacelike hypersurface S before the point at which a Page-Geilker experiment is carried out. By 'the classical metric and matter fields', we mean here the fields that would ordinarily be defined by someone trying to model the local space-time neighbourhood using general relativity - for instance, an engineer, trying to predict how small lumps of matter will evolve, and doing so as precisely as is possible without taking quantum theory into account. Let us suppose there is some way of fitting these data to a solution of the Einstein equations, using some well-defined and natural recipe (not necessarily Eq. (1)) to obtain a classical stress-energy tensor from the quantum matter field, and denote the corresponding spacetime by Σ 0 . Now consider the classical metric and matter fields, ( g 1 µν , φ 1 ) and ( g 2 µν , φ 2 ), in the neighbourhood of a spacelike hypersurface S ' after the point at which a PageGeilker experiment is carried out. Let us suppose these data can also be fitted to solutions of the Einstein equations, using the same recipe for a classical stress-energy tensor as before, and denote the corresponding spacetimes by Σ 1 and Σ 2 . Since Σ 1 and Σ 2 describe macroscopically distinct matter configurations on S ' , they are not identical, and so cannot both be identical to Σ 0 . In fact, since neither of them is preferred in any way, one would not generally expect either of them to be identical to Σ 0 , assuming that the recipe used to obtain the stress-energy tensors depends in any natural way on the relevant fields. (Obviously, if completely arbitrary recipes are allowed, one could contrive things so that one of them equals Σ 0 . For instance, one could define the recipe for obtaining Σ 0 to involve first studying the possible experimental outcomes, then constructing Σ 1 and Σ 2 , and then simply setting Σ 0 to be equal to one of them.) To sum up, then, if we can find a way of describing the data before and after the experiment by piecewise continuous solutions of the classical Einstein equations, they will generally be solutions that form part of different spacetimes. In this description (if there is indeed such a description), it is as though some sort of stochastic jump takes place, starting from one solution, and arriving at one of two alternative solutions, both distinct from the original. Is there such a description? Can the spacetime we actually observe be described by piecewise continuous solutions of the Einstein equations? I don't think we know for sure: some sort of smoothing could take place in the vicinity of Page-Geilker experiments, for instance. But it seems to be generally tacitly assumed - and it seems to be necessary to assume, in order to explain experimental data - that this is at least approximately the case. For, on the one hand, if it were not the case that large regions of spacetime are well described by a solution to the Einstein equations, we would not be able to account theoretically for any of the confirmed predictions of general relativity. On the other hand, as we have just argued, the sort of macroscopic indeterminacy exhibited by the Page-Geilker experiment implies that spacetime cannot be described globally by a single solution of the Einstein equations.", "pages": [ 2, 3 ] }, { "title": "B. Probing gravitational non-locality and the Salart et al. experiment", "content": "A more recent proposal [12] with some related motivations was to look for direct evidence of violations of Bell's local causality in the gravitational field. Recall that Bell experiments (modulo loopholes) show that any hidden variable theory underlying quantum theory must violate Bell's condition of local causality. Since non-local hidden variable theories are theoretically uncompelling and difficult to reconcile with relativity, this gives much stronger evidence that the outcomes of quantum experiments are indeed inherently unpredictable. The evidence that the classical gravitational field evolves probabilistically, though, is less direct. While the Page-Geilker experiment appears to confirm that the gravitational field evolves probabilistically, one could still imagine an underlying deterministic semi-classical gravity theory, sensitive to microscopic variables, that predicts each observed evolution, in the same way that de Broglie-Bohm theory and other deterministic hidden variable theories reproduce the predictions of quantum theory. We would like to be able to argue directly that any deterministic phenomenological theory of gravity must be non-local. Since a non-local gravity theory would be very hard to reconcile with either special or general relativity, this would be a compelling reason for abandoning all hope for deterministic semi-classical theories. Another strong motivation for verifying this point directly is that gravitational collapse models highlight another possible loophole in the interpretation of all Bell experiments to date. This 'collapse locality loophole' [11] arises because collapse models suggest that a definite measurement outcome occurs only after macroscopic amplification to a particular scale (which depends on the parameters of the collapse model). If this is correct, to exclude locally causal explanations we need Bell experiments that ensure that collapses, and thus definite measurement outcomes, take place in spacelike separated regions in the two wings. No Bell experiment to date ensures this for the full range of collapse model parameters consistent with known experiment [11]. This motivation is further reinforced by the observation [11] that there are ways in which a consistent theory combining quantum theory and gravity could conceivably produce the outcomes observed in all Bell experiments to date and nonetheless allow only locally causal evolutions of the metric. Of course, models incorporating these ideas have highly non-standard properties, and may be theoretically problematic as well as ad hoc. Still, as with the Page-Geilker experiment, clear experimental data would be much preferable to strong theoretical intuitions and arguments. A beautiful experiment investigating this possibility was carried out by Salart et al. [13], showing that the local causality loophole can indeed be closed at least for gravitational collapse models whose collapse times and scales agree with theoretically motivated estimates proposed by Penrose [8] and Diosi [9]. Further more conclusive experiments have been proposed [14], with the ultimate aim of directly measuring non-locally correlated gravitational fields in such a way that these measurements are themselves completed in space-like separated regions.", "pages": [ 3, 4 ] }, { "title": "III. NATURAL MEASUREMENT-LIKE EVENTS", "content": "Of course, quantum measurement-like events with macroscopically distinct consequences take place without any artificial help. On Earth, fissioned particles and cosmic rays leave tracks in mica; frogs can respond to the stimulus of a single photon; a single gamma ray or charged particle can trigger a cancer; the bouncing of photons and cosmic rays off dust particles must from time to time determine the formation or otherwise of a particular macroscopic clumping. Each of these outcomes is effectively a quantum measurement of the position of a particle whose wave function was previously delocalised. On the cosmological scale, quantum fluctuations are believed to have seeded the instabilities that led to galaxy formation. It would be very interesting indeed to try to quantify the degree to which quantum noise, bubbling up from the microscopic realm, affects the predictability of the macroscopic world in general, and in particular to characterise the degree and type of the resulting deviations from Einstein's equations. This project is beyond our present scope, though. For the purposes of the present discussion, we need only take the point that these deviations occur naturally, and presumably have been doing so since very early cosmological times. This is why the outcome of the PageGeilker experiment was generally (perhaps even universally) anticipated. In other words, while the Page-Geilker experiment is a good illustration of the point that our observed space-time deviated from Einstein's equations, we do not actually need to appeal to it to make that point.", "pages": [ 4 ] }, { "title": "IV. QUANTUM GRAVITY: RESOLUTION OR DISTRACTION?", "content": "According to one school of thought, at this point in the discussion one should throw up one's hands, regret the fact that we don't yet have a quantum theory of gravity, and accept that we can't productively advance the discussion further without one. It seems to me far from evident that we should heed this counsel of despair. I can see two reasons for optimism. First, we might not actually need a quantum theory of gravity at all. Second, even if we do, it ought to imply some effective phenomenological theory of classical gravity which incorporates stochastic fluctuations into general relativity. In the first case, we might hope for some fundamental theory which incorporates stochastic fluctuations into general relativity; alternatively, we might hope for a theory based on different principles, which again should imply an effective phenomenological theory of classical gravity of the type just mentioned. In all these cases, it is reasonable to try to explore how a classical gravity theory with stochastic fluctuations might be probed experimentally. But we don't have such a theory. Perhaps the best hope, then, is that experiment might guide us to the right theory, if we can at least identify what to look for experimentally. The following two sub-sections flesh out these arguments in more detail.", "pages": [ 4 ] }, { "title": "A. Do we need quantum gravity to explain the Page-Geilker experiment?", "content": "Embarrassingly, our best theory of gravity, general relativity, has no way of consistently incorporating the results of macroscopically amplified quantum measurement events, whether they occur naturally or are created artificially as in a Page-Geilker experiment. The current conventional wisdom suggests that this embarrassment stems from our failure to devise a consistent quantum theory of gravity. There is - the standard argument runs - no fully consistent way to couple a classical gravitational field to quantum matter fields: the gravitational field also needs to be quantised. We would expect - the argument proceeds to suggest - that in a full quantum theory of gravity, the gravitational field would evolve so as to be (to very good approximation) correlated with the matter fields in any given branch of the universal wave function. In particular, we would expect a full quantum theory of gravity to predict the observed outcome of the Page-Geilker experiment: this is why Page and Geilker provocatively titled their paper 'Indirect Evidence for Quantum Gravity'. More generally, we would expect a full quantum theory of gravity to predict that the gravitational field should be correlated with the positions of astronomical bodies in the way we observe. Of course, this argument begs several key questions. Aside from the problems of principle in unifying quantum theory and gravity, discussed above, there is the problem of making scientific sense of purely unitary quantum theory. We do not have, despite nearly fifty years' of effort, any clearly consistent and logically compelling account of how Everett's original intuition might be fleshed out into a clearly and carefully justified interpretation of a unitarily evolving universal wave function. (State-of-the-art reports and assessments of recent attempts can be found in Ref. [7].)", "pages": [ 4, 5 ] }, { "title": "B. Probing an effective theory derivable from quantum gravity", "content": "What if some version of quantum gravity is correct, though? Suppose, for example, we find some rigorously defined way of carrying out path integrals over gravitational and matter field configurations, and find some evidence that it gives correct answers. In order to understand large-scale gravitational physics, we would still need some (presumably) phenomenological effective theory, derived from our fundamental quantum gravity theory, which characterises the quasiclassical behaviour of matter and gravity that we actually observe. (In Gell-Mann and Hartle's terminology [15], we would need some way of characterising our own quasiclassical domain within this hypothetical quantum gravitational or quantum cosmological theory.) In particular, this higher-order theory would need to be consistent with the Page-Geilker experiment and with the observed correlations of gravitational fields and astronomical bodies. It thus seems a reasonable conjecture - suggested by observational evidence, and contradicted by nothing we know about quantum gravity that we would end up with some sort of stochastically modified version of general relativity, albeit in this case as a derived effective theory rather than a fundamental theory. If so, one might make the further reasonablelooking guess that the propagation of gravitational waves is approximately described by considering them as perturbations of the gravitational field within this higherorder quasiclassical theory", "pages": [ 5 ] }, { "title": "V. WHAT HAPPENS TO GRAVITATIONAL WAVES IN A STOCHASTIC MODIFICATION OF GENERAL RELATIVITY?", "content": "Without knowing the details, one can only guess. So, without further ado, I shall. A plausible guess, it seems to me, is that stochastic fluctuations break up the coherence of propagating waves. It is difficult to hear someone shouting in a high wind, not only because the noise of the wind drowns out the propagating sound wave, but also because the turbulence causes its amplitude to decay faster than in still air. If the level of stochastic fluctuations is constant throughout a region in which a wave propagates, the simplest guess would be that the wave amplitude decays by a factor exponential in the region length, in addition to the normal approximately inverse square law decay. Without knowing the theory, one can't estimate the value of the exponential constant - but if this guess is right, and if gravitational wave astronomy turns out nonetheless to be viable, one might be able to estimate it from observational data, and thereby get quantitative data characterising an important feature of the relation between quantum theory and gravity. This raises the possibility that the stochastically induced decay of gravitational waves could conceivably prevent gravitational wave astronomy from being viable with presently envisaged gravitational wave detectors. If so, of course, gravitational wave astronomy's loss would be gravitational theory's gain.", "pages": [ 5 ] }, { "title": "VI. WHAT ABOUT THE BINARY PULSAR OBSERVATIONS?", "content": "If one suggests the possibility that the standard account of gravitational wave physics might not be correct, one has to deal with the counter-argument that observations of binary pulsars [16] have already confirmed the standard account to a very impressive degree of precision. This counter-argument has no force against the speculations considered here, though. The suggestion is not that binary pulsars do not emit gravitational waves, and thereby lose energy, as standard theory predicts. The suggestion is, rather, that the gravitational waves lose coherence, and thus decay faster than expected, as they propagate through space, and hence in particular that gravitational wave signals reaching Earth might be weaker than anticipated. Careful observation of a drum vibrating in the distance would reveal that it is losing energy by radiating sound waves; nonetheless, the sound of the drum will not propagate as far in a strong wind. There is no evident inconsistency here.", "pages": [ 5 ] }, { "title": "VII. COMPARING QUASICLASSICAL GRAVITY AND QUASICLASSICAL ELECTRODYNAMICS", "content": "To what extent are the problems we raise about our understanding of quasiclassical physics specific to gravity? In particular, are there any reasons to think that classical gravitational waves might behave differently from classical electromagnetic waves? In considering these questions, it's helpful first to consider quasiclassical electrodynamics in Minkowski space. Clearly, some of the points made above apply. In particular, we can carry out Bell experiments and ensure that, on each wing, a source of electromagnetic waves behaves differently depending on the measurement setting and outcome on that wing, and that the measurement settings themselves are locally determined by random quantum events. For example, a charged sphere could be move in any of four different ways, depending on the two measurement choices and two possible outcomes, and the measurement choices could be determined, just before the measurements are made, using bits produced by quantum random number generators. Since the outcomes of Bell experiments are nonlocally correlated, we expect this to produce nonlocal correlations in the electromagnetic fields propagating from the regions of the two measurements. Now, this probably has not been directly tested in experiments to date, and I am not sure we can in principle rigorously exclude models (with very counterintuitive features) that agree with experiments to date but predict that nonlocal correlations of classical electromagnetic fields cannot in fact be observed. Of course, this would be a very surprising outcome indeed. We ignore the possibility here, since our aim is to understand whether one might have possible reasons to look for unexpected behaviour in quasiclassical gravitational physics even if there are no analogous surprises in quasiclassical electrodynamics. Assuming, then, that nonlocal correlations can be created in macroscopic electromagnetic fields, it follows that quasiclassical electrodynamics in the real world cannot be described by an underlying local deterministic model. Note, though, that the nonlocalities we introduced arise entirely from nonlocal correlations in the motion of sources. Given a description of the motion of each source, we can calculate the subsequent behaviour of the electromagnetic fields it generates. Since electrodynamics is a linear theory, we can obtain a complete solution by superposing the contributions from the various sources. This gives a strategy for building a phenomenological model of quasiclassical electrodynamics in the presence of quantum unpredictability and nonlocality: first apply the predictions of quantum theory to give a model of additional stochastic (and nonlocally correlated) forces acting on the sources, and then solve to obtain the fields. Adding forces that alter the motion of the sources does not affect charge conservation, so in such a model we still have ∂ µ J µ = 0. It would be wrong to suggest this gives a rigorous understanding of the relationship between quantum and quasiclassical electrodynamics in Minkowski space. We do not even have a completely rigorous definition of quantum electrodynamics as a non-trivial theory. Nor do we have a precise general prescription for how to obtain quasiclassical equations of motion from quantum theory, either for electrodynamics or for any other physically relevant theory. However, we do at least have an ansatz for dealing with the quasiclassical consequences of quantum experiments with unpredictable and nonlocally correlated outcomes, and this ansatz does not violate the conservation laws necessary for a consistent solution of the electrodynamic equations. Now compare the situation when we try to model an analogous experiment in which the measurement choices and outcomes of Bell experiments determine the motion of massive objects on each wing, with the measurement choices again locally determined by quantum random number generators. As noted earlier, we cannot model the quasiclassical physics by extrapolating the predictions of general relativity from data on a spacelike hypersurface before the Bell experiment, since general relativity is deterministic and the Bell data are not. So far the analogy with electrodynamics holds, since electrodynamics is also deterministic. However, we run into further problems in this case. To define any consistent solution of the Einstein equations, we need the local conservation of stress-energy, D ν T µν = 0. We know of no generally covariant quasiclassical model of the possible outcomes of quantum measurement-like processes that preserves stress-energy and is consistent with general relativity where quantum effects are negligible. (Indeed, even non-relativistic dynamical collapse models [17, 18], which might be the best guesses at phenomenological descriptions of the quasiclassical physics emerging from measurement interactions, violate conservation of energy.) Without such a model, it seems our best description of quasiclassical gravitational physics would be by models which generally obey the Einstein equations but have singularities or discontinuities at or in the vicinity of quantum measurement events. And if that were the best possible description, the standard classical derivation of gravitational wave propagation would break down in these regions. To be sure, there are further uncertainties here. If these discontinuities are physically real, should we expect them to affect the propagation of electromagnetic radiation in the same way as they affect the propagation of gravitational waves? If so, of course, any effect is likelier to be evident in standard (electromagnetic wave observation) astronomy than in gravitational wave astronomy, and the absence of any observed effect to date is a strong constraint. On the other hand, we have a quantum theory of electromagnetism and no quantum theory of gravity. And, if there is a quantum theory of gravity from which quasiclassical solutions obeying Ein- stein's equations with discontinuities emerges, we have no clear reason to think that coherent beams of gravitons and photons should scatter similarly from the discontinuities - indeed one might guess that gravitons are more directly affected than photons by a discontinuity in the classical field generated by gravitons. Some may nonetheless hold the intuition that we should expect exactly the same effects in quasiclassical gravity and quasiclassical electrodynamics. The points made here do not refute this possibility, but they do give significant reasons to query it.", "pages": [ 6, 7 ] }, { "title": "VIII. SUMMARY", "content": "In this paper, we raised a possibility that does not seem to have been considered: that stochastic corrections to the Einstein equations dissipate gravitational waves. Such stochastic corrections could either arise directly from a fundamental theory or as a phenomenological effect resulting from quantum gravity (or some other presently unknown type of theory). Either way, our guess at their effect on gravitational wave propaga- tion is not provable given the present state of theoretical understanding. But is it obviously wrong, or totally implausible? If, as we suggest, not, it seems a possibility to be kept in mind if and when gravitational wave astronomy produces data, null or otherwise. We hope too that the questions raised here may encourage more attention to be focussed on the problem of finding realistic quasiclassical descriptions of gravitational physics in the presence of quantum measurements, through Bell experiments and otherwise.", "pages": [ 7 ] }, { "title": "IX. ACKNOWLEDGEMENTS", "content": "This work was partially supported by a Leverhulme Research Fellowship, a grant from the John Templeton Foundation, and by Perimeter Institute for Theoretical Physics. 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. 100(22):220404, 2008.", "pages": [ 7 ] } ]

[ { "title": "Non-Inertial Frames", "content": "Timothy H. Boyer Department of Physics, City College of the City University of New York, New York, New York 10031", "pages": [ 1 ] }, { "title": "Abstract", "content": "Classical electron theory with classical electromagnetic zero-point radiation (stochastic electrodynamics) is the classical theory which most closely approximates quantum electrodynamics. Indeed, in inertial frames, there is a general connection between classical field theories with classical zero-point radiation and quantum field theories. However, this connection does not extend to noninertial frames where the time parameter is not a geodesic coordinate. Quantum field theory applies the canonical quantization procedure (depending on the local time coordinate) to a mirror-walled box, and, in general, each non-inertial coordinate frame has its own vacuum state. In particular, there is a distinction between the 'Minkowski vacuum' for a box at rest in an inertial frame and a 'Rindler vacuum' for an accelerating box which has fixed spatial coordinates in an (accelerating) Rindler frame. In complete contrast, the spectrum of random classical zero-point radiation is based upon symmetry principles of relativistic spacetime; in empty space, the correlation functions depend upon only the geodesic separations (and their coordinate derivatives) between the spacetime points. The behavior of classical zero-point radiation in a noninertial frame is found by tensor transformations and still depends only upon the geodesic separations, now expressed in the non-inertial coordinates. It makes no difference whether a box of classical zero-point radiation is gradually or suddenly set into uniform acceleration; the radiation in the interior retains the same correlation function except for small end-point (Casimir) corrections. Thus in classical theory where zero-point radiation is defined in terms of geodesic separations, there is nothing physically comparable to the quantum distinction between the Minkowski and Rindler vacuum states. It is also noted that relativistic classical systems with internal potential energy must be spatially extended and can not be point systems. Based upon the classical analysis, it is suggested that the claimed heating effects of acceleration through the vacuum may not exist in nature.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Classical electron theory with classical electromagnetic zero-point radiation (stochastic electrodynamics) is the classical theory which comes closest to quantum electrodynamics.[1] However, there seems to be little interest in the physical interpretations provided by this classical theory. This lack of interest in the related classical theory holds even when quantum theory ventures into untested areas involving noninertial coordinate frames such as appear in connection with black holes and acceleration through the vacuum. In this article, we illustrate the contrasting classical and quantum interpretations surrounding vacuum behavior in an inertial and in a noninertial (Rindler) frame. Although the ideas are believed to have much wider implications, the illustrations here focus on a massless relativistic scalar field in two spacetime dimensions in flat spacetime. There is a general connection between the classical and quantum field theories in an inertial frame.[2] However, this connection does not extend to noninertial frames where the time parameter is not a geodesic coordinate. Irrespective of the spacetime metric, quantum field theory regards one box as good as another when applying the canonical quantization procedure to a mirror-walled box. In general, each non-inertial coordinate frame has its own vacuum state. In particular, there is a distinction between the 'Minkowski vacuum' for a box at rest in an inertial frame and a 'Rindler vacuum' for an accelerating box which has fixed spatial coordinates in an (accelerating) Rindler frame. It has been claimed[3] that the radiation in a box in the Minkowski vacuum which is very gradually speeded up to become a box in uniform acceleration, will end up in the Rindler vacuum state; on the other hand, if the box in the Minkowski vacuum is suddenly accelerated, then the box will contain Rindler quanta. This quantum situation is completely different from that found in classical physics. In the first place, the spectrum of random classical zero-point radiation is based upon symmetry principles of relativistic spacetime; the spectrum is such as to give correlation functions which depend only upon the geodesic separations (and their coordinate derivatives) between the spacetime points. In an inertial frame, the zero-point radiation spectrum is Lorentz invariant,[4] scale invariant, and conformal invariant.[5] The behavior of zero-point radiation in a noninertial frame is found by tensor transformations to the non-inertial coordinates. In particular, we can calculate the spectrum of classical zeropoint radiation in an accelerating box, and we find that, except for small endpoint (Casimir) effects, the spectrum and correlation functions are the same as observed by a Rindler observer accelerating through zero-point radiation. It makes no difference whether or not the box of classical zero-point radiation is gradually or suddenly set into uniform acceleration; the radiation in the interior retains the same zero-point spectrum. In classical theory where zero-point radiation is defined in terms of geodesic separations, there is nothing physically comparable to the quantum distinction between the Minkowski and Rindler vacuum states. The work presented here involves only the free scalar field in a box with Dirichlet boundary conditions in one spatial dimension. Also, we will be interested only in the large-box approximation and will not treat the Casimir effects associated with a the discrete normal mode structure of the box. We start out in an inertial frame. We review the determination of the classical zero-point spectrum in the box and also the canonical quantization procedure for the corresponding quantum scalar field in the same box. Then we turn to the situation of thermal equilibrium in the box and note the contrasting classical and quantum points of view for thermal radiation. All of this work confirms the general connection between classical and quantum free fields in an inertial frame in two spacetime dimensions. This connection was treated earlier in four spacetime dimensions for electromagnetic fields[2] and for scalar fields.[6] Next we turn to the situation for a coordinate frame undergoing uniform proper acceleration through Minkowski spacetime (a Rindler frame). Quantum field theory introduces a canonical quantization in a box at rest in a Rindler frame which parallels that in an inertial frame, without making any adjustment because of the nongeodesic time coordinate involved in the quantization. In complete contrast, classical theory takes the correlation function for zero-point radiation as dependent only upon the geodesic separations of the field points, with tensor coordinate transformations between various coordinate frames. In the limit of a large Rindler-frame box, the classical radiation inside the box is shown to agree exactly with the empty-space zero-point radiation of an inertial frame. However, in the limit of a large Rindler-frame box, the quantum vacuum remains distinct from the quantum empty-space inertial vacuum. It is also emphasized that relativistic classical systems with internal potential energy must be spatially extended and can not be point systems. In contrast, systems used within quantum theory are often described as small (point) systems.[7] Based upon the classical analysis, it is suggested that the claimed", "pages": [ 2, 3 ] }, { "title": "A. Scalar Field in Two Spacetime Dimensions", "content": "We will consider a relativistic massless scalar field φ which is a function of ( ct, x ) in an inertial frame with spacetime metric ds 2 = g µν dx µ dx ν , where the indices µ and ν run over 0 and 1, x 0 = ct , x 1 = x , and The behavior of the field φ follows from the Lagrangian density L = (1 / 8 π ) ∂ µ φ∂ µ φ corresponding to[8] The wave equation ∂ µ [ ∂ L /∂ ( ∂ µ φ )] = 0 for the field is The associated stress-energy-momentum tensor density T µν = [ ∂ L /∂ ( ∂ µ φ )] ∂ ν φ -g µν L gives the energy density u as and the momentum density as The energy U in the field in a one-dimensional box extending from x = a to x = b is", "pages": [ 4 ] }, { "title": "B. Radiation Spectrum in a Box", "content": "Both classical and quantum field theories start with the normal mode structure of the radiation field in a box. We consider standing wave solutions which vanish at the walls x = a and x = b of the box (Dirichlet boundary conditions) so that a normalized normal mode can be written as where f n is the amplitude of the normal mode. The radiation field in the box can be written as a sum over all the normal modes where θ n is an appropriate phase. From Eq. (4) we find that each mode φ n ( ct, x ) has the time-average spatial energy density which is uniform in space. The total mode energy U n found by integrating over the length of the box is given by where the wave amplitude f n must be determined by some additional physical considerations.", "pages": [ 4, 5 ] }, { "title": "C. Canonical Quantization of the Quantum Scalar Field", "content": "Classical and quantum theories take different points of view regarding the vacuum radiation field. Quantum field theory follows the canonical quantization procedure which rewrites the cosine time dependence in terms of complex exponentials (the positive and negative frequency aspects) and introduces annnihilation and creation operators a n , a + n for each normal mode n so that the field becomes an operator field φ ( ct, x ) with a vacuum energy per normal mode. Thus from Eqs. (8), (10), and (11) the quantum field is Here the operator a n annihilates the vacuum, a n | 0 > = 0 , and the operator commutation relations are [ a n , a n ] = [ a + n , a + n ] = 0 , [ a n , a + n ] = 1 . In the quantum vacuum state | 0 > in the inertial frame, the two-point vacuum expectation value which is symmetrized in operator order is easily calculated and takes the form", "pages": [ 5, 6 ] }, { "title": "D. Zero-Point Radiation for the Classical Scalar Field", "content": "The vacuum state for the classical scalar field involves random classical zero-point radiation which is featureless, so that its correlation functions depend only on the geodesic separations (and coordinate derivatives) between the field points. In an inertial frame, the zeropoint radiation is Lorentz invariant,[4] scale invariant, and indeed conformal invariant.[5] Random classical radiation can be written in the form given by Eq. (8) with the phases θ n randomly distributed in the interval [0 , 2 π ) and independently distributed for each n . In an inertial frame, the invariance properties of the spectrum can be shown to lead to a spectral form corresponding to an energy per normal mode which is a multiple of the frequency with an undetermined multiplicative constant, U n = const × ω n .[5] In order to give a close connection between the classical and quantum theories, we choose the energy per normal mode to agree with that used in the quantum theory as given in Eq. (11). In order to make the classical and quantum field expressions look as similar as possible, we rewrite Eq. (8) in the form parallel to Eq. (12), (note the change from 8 π over to 4 π ) , It is convenient to characterize random classical radiation by the two-point correlation function 〈 φ ( ct, x ) φ ( ct ' , x ' ) 〉 obtained by averaging over the random phases as 〈 cos θ n sin θ n ' 〉 = 0 , 〈 cos θ n cos θ n ' 〉 = 〈 sin θ n sin θ n ' 〉 = (1 / 2) δ n,n ' , or as 〈 exp[ θ n ] exp[ θ n ' ] 〉 = 〈 exp[ -θ n ] exp[ -θ n ' ] 〉 = 0 , 〈 exp[ θ n ] exp[ -θ n ' ] 〉 = δ n,n ' From these relations, we can easily show, for example, that 〈 cos( A + θ n ) cos( B + θ n ' ) 〉 = cos( A -B )(1 / 2) δ nn ' . The two-point correlation function for a general distribution of random classical scalar waves is found by averaging over the random phases θ n We notice that the classical correlation function (15) and vacuum expectation value of (symmetrized) quantum operators (13) agree exactly. Indeed it has been shown that in an inertial frame, there is a general connection[2] between the correlation functions of the classical zero-point radiation field and the vacuum expectation values of the corresponding symmetrized operator products for all the correlation functions including the correlation functions of arbitrarily high order. If we take the limit b → ∞ , corresponding to the presence of a reflecting mirror at the left-hand end x = a of the box but infinite extent on the right, then we obtain the correlation function as an integral where the wave numbers k n = nπ/ ( b -a ) become continuous, This integral is convergent. It can be rewritten as a sum of terms of the form ∫ ( dk/k ) cos( ka ) and evaluated as an indefinite integral. Thus we find ∣ ∣ The correlation function for empty space can be found by moving the mirror at the lefthand edge x = a of the box out to spatial infinity, a → -∞ . However, this procedure introduces a divergence going as /planckover2pi1 c ln | (2 a ) 2 | . One way to eliminate this divergence is to take the spatial derivatives of the correlation function. Indeed, we can go back to the integral of Eq. (16) and use the identity 2 sin A sin B = cos( A -B ) -cos( A + B ) to rewrite the correlation function as Both integrals in Eq. (18) are divergent at k → 0. In the limit a →-∞ , corresponding to moving the left-hand reflecting mirror at x = a out to spatial minus infinity, we can drop the second line in Eq. (19) as a very rapidly oscillating cosine function. Thus for a box extending infinitely far in both directions, we find the free-space correlation function where in the second line and third lines we have used the identity 2 cos A cos B = cos( A + B ) + cos( A -B ) and in the last line have incorporated both the sum and difference cosine terms by extending the integral over negative values of k . The integrals in Eqs. (18) and (19) are divergent as k → 0 . This divergence can be removed by considering the coordinate derivatives of the correlation functions. Thus in free space, we consider 〈 φ 0 ( ct, x ) ∂ ct ' φ 0 ( ct ' , x ' ) 〉 and 〈 φ 0 ( ct, x ) ∂ x ' φ 0 ( ct ' , x ' ) 〉 . The resulting expressions are convergent as k → 0 but now divergent as k →∞ . However, the divergence at large values of k involves oscillating sine functions. Thus we may introduce a convergence factor such as exp[ -Λ k ] into the integrand, carry out the integrals in terms of exponentials, and then take the no-cutoff limit Λ → 0 to obtain the singular Fourier sine transforms of the form[9] In this fashion we obtain the closed-form expression and similarly obtain both of which agree with the limit a → -∞ in Eq. (17). We note that in empty space there is no length or time parameter which is singled out by the zero-point radiation in an inertial frame. The zero-point correlation functions depend upon the geodesic separation c 2 ( t -t ' ) 2 -( x -x ' ) 2 between the field points ( ct, x ) and ( ct ' , x ' ). For later comparisons, it is useful to have the closed-form expressions for the zero-point correlation functions in empty space as a function of time at a single spatial coordinate x = x ' and as a function of space at a single time t = t ' . Thus we have for the nonvanishing correlations from Eqs. (21) and (22) and The spatial derivatives of the correlation function for a mirror at x = a at the left-hand end of the spatial region can be written explicitly as", "pages": [ 6, 7, 8, 9, 10 ] }, { "title": "E. Thermal Scalar Radiation", "content": "Within classical theory with classical zero-point radiation, zero-point radiation represents real radiation which is always present, and thermal radiation is additional random radiation above the zero-point value. Thus if U ( ω, T ) is the energy per normal mode at frequency ω and temperature T , the thermal energy contribution U T ( ω, T ) is found by subtracting off the zero-point energy, U T ( ω, T ) = U ( ω, T ) -U ( ω, 0) . The additional thermal energy is distributed across the low-frequency modes of the radiation field. The (finite) total thermal energy U T ( T ) in a box is found by summing the thermal energy per normal mode U T ( ω, T ) over all the normal modes at temperature T in a box of finite size. The spatial density of thermal energy is given by u ( T ) = U T ( T ) / ( b -a ) = a Ss T 2 where a Ss is the constant for one-spatial-dimension scalar radiation corresponding to Stefan's constant for electromagnetic radiation.[9] Classical thermal radiation is described in exactly the same random-phase fashion as the zero-point radiation except that the spectrum is changed. The thermal radiation spectrum for massless scalar radiation can be derived from classical theory involving zero-point radiation and the structure of spacetime.[9][10][11] One finds for the energy per normal mode at frequency ω and temperature T The calculation for the classical two-point field correlation function at finite temperature accordingly takes exactly the same form as given above in Eqs. (15), except that the spectrum is changed so that now The quantum point of view regarding thermal radiation is strikingly different from the classical viewpoint. The vacuum of the quantum scalar field is said to involve fluctuations but no quanta, no elementary excitations, no scalar photons, whereas the thermal radiation field involves a distinct pattern of scalar photons. If the index m is used to label the normal modes in a one-dimensional box, the quantum expectation values correspond to an incoherent sum over the expectation values for the fields for all numbers n m of photons of frequency ω m = mπc/ ( b -a ) with a weighting given by the Boltzmann factor exp[ -n m /planckover2pi1 ω m / ( k B T )] . Thus the quantum two-point field correlation function for our example involving a box in one spatial dimension is given by[2] where we have noted that and have defined Thus for symmetrized products of quantum fields, the quantum expectation value in Eq. (29) is in exact agreement with the corresponding classical average value found in Eq. (28). Again the agreement holds for higher order correlation functions provided the quantum operator order is completely symmetrized.[2] The agreement between the classical and quantum correlation functions remains in the limits of a large box b →∞ analogous to the transition from Eq. (15) over to Eq. (16) and in the removal of the left-hand mirror to negative spatial infinity as in the transition from Eq. (16) over to Eq. (19). It should be emphasized that although there is complete agreement between the correlation functions arising in classical theory and the symmetrized expectation values in quantum theory, the interpretations in terms of fluctuations arising from classical wave interference or in terms of fluctuations arising from the presence of photons are completely different between the theories.[12] The contrast in interpretations and indeed in predictions becomes even more striking when an accelerating coordinate frame is involved.", "pages": [ 10, 11, 12 ] }, { "title": "A. Rindler Coordinate Frame", "content": "Although there is close agreement between classical and quantum field theories in an inertial frame, the two theories part company in noninertial frames. The noninertial frame which we will consider in this article is a Rindler coordinate frame accelerating through Minkowski spacetime in two spacetime dimensions.[13][14] If the coordinates of a spacetime point in an inertial frame are given by ( ct, x ), then the coordinates ( η, ξ ) of the spacetime point in the Rindler frame which is at rest with respect to the inertial frame at time t = 0 = η are given by with -∞ < η < ∞ , and 0 < ξ . Using the relation cosh 2 η -sinh 2 η = 1, it follows that a point with fixed spatial coordinate ξ in the Rindler frame has coordinates x ξ ( t ) in the inertial frame given by and so moves with acceleration a ξ = d 2 x/dt 2 = c 2 /ξ at time t = 0 , and indeed in the Rindler frame has constant proper acceleration at all times. Thus for large coordinates ξ, the acceleration a ξ becomes small whereas for small ξ , the proper acceleration diverges. The point ξ = 0 is termed the 'event horizon' for the Rindler coordinate frame. The metric in the Rindler frame can be obtained from Eqs. (32) and (33) as It is clear from this expression that the time coordinate η in the Rindler frame is not a geodesic coordinate. Indeed, the geodesic separation between two spacetime points which takes the form c 2 ( t -t ' ) 2 -( x -x ' ) 2 in the geodesic coordinates of the inertial frame becomes in Rindler coordinates", "pages": [ 12, 13 ] }, { "title": "B. Normal Modes in a Box in a Rindler Frame", "content": "We now consider the spectrum of random radiation as observed in the Rindler frame. First we obtain the radiation normal modes. The wave equation (3) in an inertial frame can be transformed to the wave equation in the Rindler frame by using the transformations (32) and (33) together with the scalar behavior of the field φ under a coordinate transformation. The scalar field takes the same value in any coordinate frame. Thus the field ϕ ( η, ξ ) in the Rindler frame is equal to the field φ ( ct, x ) in the inertial frame at the same spacetime point, If we use the usual rules for partial derivatives, we find that Eq. (3) becomes in the Rindler frame The solutions of Eq. (39) take the form H (ln ξ ± η ) where H is an arbitrary function. Thus, whereas the general solution of the scalar wave equation (3) in an inertial frame is φ ( ct, x ) = h + ( x -ct ) + h -( x + ct ) where h + and h -are arbitrary functions, the general solution in a Rindler frame is ϕ ( η, ξ ) = H + (ln ξ -η ) + H -(ln ξ + η ) where H + and H -are arbitrary functions. The normal mode solutions of the wave equation in the Rindler frame for a box extending from 0 < ξ = a to ξ = b with Dirichlet boundary conditions can be obtained by separation of variables and expressed as a time-Fourier series where F n is the amplitude of the normal mode and the spatial functions arise from a Sturm-Liouville system[15] and form a complete orthonormal set with weight 1 /ξ on the interval a < ξ < b . Thus we find where we have used the substitution v = [ π ln( ξ/a )] / ln( b/a ) in evaluating the integral. For a radiation normal mode, the Rindler time parameter η agrees with all local clocks when adjusted by ξ , and thus the time τ = ξη gives the proper time of a clock located at fixed Rindler spatial coordinate ξ . For time-stationary random radiation in the Rindler frame with an unknown time-spectral amplitude F n , the field ϕ ( η, ξ ) can be written as a sum over the normal modes ϕ n ( η, ξ ) in Eq. (40) with random phases θ n distributed randomly over the interval [0 , 2 π ) and distributed independently for each value of n Then the two-field correlation function is obtained in analogy with Eqs. (14)-(15) For a large box b →∞ , The normal mode frequencies κ n = nπ/ ln( b/a ) become continuous and the sum in Eq. (44) becomes the integral for the correlation function for a mirror at the left-hand edge ξ = a of the box The expression (45) can be rewritten in the form In the limit a → 0 in which the mirror at ξ = a is moved to the event horizon, the last integral in Eq. (46) involves a rapidly oscillating cosine function; it can be taken to vanish when considering the time derivative at ξ = ξ ' . Thus we find the free-space expression where the spectral amplitude F ( κ ) of the random radiation is still unspecified.", "pages": [ 13, 14, 15 ] }, { "title": "C. Classical Zero-Point Radiation in the Rindler-Frame Box", "content": "It was noted earlier that the spectrum of classical zero-point radiation follows from the assumed symmetry properties of the vacuum. Thus the spectrum of random classical radiation in empty space is assumed to be featureless; the two-point correlation function can depend upon only the geodesic separation (and its coordinate derivatives) between the spacetime points. This dependence upon the geodesic separation has been exhibited in earlier articles for the relativistic scalar and electromagnetic fields in four spacetime dimensions.[5][6] For the example of two spacetime dimensions used in the present article, the derivative correlation functions (21) and (22) involve the partial derivatives of the logarithm of the spacetime separation | c 2 ( t -t ' ) 2 -( x -x ' ) 2 | between the spacetime points ( ct, x ) and ( ct ' , x ' ) . In classical theory, the zero-point radiation is physically present. There is no notion of 'virtual' photons which may come into and then out of existence. Thus in empty space, the spectrum of radiation which is found in the Rindler frame follows directly by tensor transformation from the radiation found in the inertial frame. We find for a scalar field that the correlation function is the same in the inertial frame and the Rindler frame for the same spacetime points However, it is clear from this equation (48) that the functional dependence of the correlation function upon ξ, ξ ' , η, η ' will in general be quite different from the dependence upon x, x ' , t, t ' since from Eq. (37), the geodesic separation takes the form c 2 ( t -t ' ) 2 -( x -x ' ) 2 = 2 ξξ ' cosh( η -η ' ) -ξ 2 -ξ ' 2 , and the Rindler frame time parameter η is not a geodesic coordinate. In empty space, the closed form expressions for the spatial derivatives of the correlation function in the Rindler frame follow from Eqs. (21), (22), (37)and (48) as ∣ ∣ The time-spectrum found in the Rindler frame may be obtained by taking the singular Fourier sine transform of the time correlation at a single spatial coordinate ξ = ξ ' . Thus from Eq. (47) and (49), we find for the spectral function corresponding to classical zero-point radiation[16] In a Rindler frame box of finite length, this spectral function (51) is restricted to the allowed normal modes κ n = nπ/ ln( b/a ) , so that and the two-point correlation function in the box is given by In the limit as b →∞ , corresponding to the right-hand edge of the box going to positive spatial infinity, the normal mode frequencies κ n = nπ/ ln( b/a ) become continuous, and the correlation function (53) becomes that for a mirror at the left-hand edge ξ = a of the box , This is a convergent integral which can be evaluated as[16] ∣ ∣ In the limit where the mirror at ξ = a is moved to the event horizon, a → 0, the correlation function in Eq. (55) diverges as /planckover2pi1 c ln | ξξ ' / ( a ) 2 | = /planckover2pi1 c ln | ξξ ' | -/planckover2pi1 c ln | ( a ) 2 | , which appears similar to the divergence in Eq. (17), except that in previous case a →-∞ whereas here a → 0 . Just as was done earlier, the divergence can be eliminated by taking coordinate derivatives. In this limit, the correlation function (55) should correspond to that for empty space since as the mirror goes to the event horizon of the Rindler frame, the phases of waves change very rapidly with distance, and we expect that the phases of the incident and reflected waves should become uncoupled. In the limit a → 0 , the correlation function for the mirror (55) becomes (with divergence-eliminating coordinate derivatives) This expression indeed agrees with the correlation functions for empty space given in (49) and (50) since Thus a box with classical zero-point radiation takes on the empty-space zero-point correlation function when the box is expanded to cover the entire Rindler spacetime region (the Rindler wedge). The presence of any reflecting walls on the Rindler box becomes ever less important as the walls recede to the limits of the Rindler region. If we consider the zero-point correlation function in free space as a function of space for a single time η = η ' or as a function of time for a single coordinate ξ = ξ ' in the Rindler frame, then we find the non-vanishing two-point correlations in free space from Eqs. (49) and (50), and", "pages": [ 15, 16, 17, 18 ] }, { "title": "D. Canonical Quantization in a Rindler-Frame Box", "content": "Quantum theory regards canonical quantization as a fundamental procedure which can be followed in any box, no matter whether the box is at rest in an inertial frame or is at rest in a noninertial coordinate frame. Thus for a box in a Rindler frame, the quantum field can be expressed in a form parallel to Eq. (12) as where b n and b + n are the annihilation and creation operators for particles in the Rindlerframe box. Notice that the amplitude appearing in the sum is the same factor involving the square root of 8 π /planckover2pi1 c times the wave number, just as in Eq. (12) in the inertial frame in empty space. In contrast, the classical theory involves the amplitude factor F 0 ( κ ) given in Eq. (51) in order to compensate for the fact that the time coordinate for the normal modes is not a geodesic coordinate. In quantum theory, there is a Rindler-frame vacuum state | 0 R > which is annihilated by the Rindler operator b n . The two-point Rindler-vacuum expectation value for the symmetrized product of the field operators gives the result parallel to Eq. (13) as In the limit as b → ∞ , this expression becomes the Rindler-vacuum expectation value for the situation of continuous normal mode frequencies κ n = nπ/ ln( b/a ) and a mirror at ξ = a, analogous to Eqs. (16), and (17), ∣ ∣ In the limit a → 0 that the mirror is moved to the event horizon, the expectation value for the quantum fields in the Rindler vacuum becomes divergent as 2 /planckover2pi1 c ln[2 ln( a )] . Again the divergence can be eliminated by taking coordinate derivatives Thus we obtain and If we consider the spatial dependence at a single time and the time dependence at a single spatial point, we find for the symmetrized expectation value for the Rindler vacuum that the non-vanishing values from Eqs. (65) and (66) are and The Rindler vacuum expectation value in (67) with its dependence upon the inverse time separation is analogous to the free-space inertial frame vacuum expectation value (23) in an inertial frame. However, the Rindler vacuum expectation value (68) with its logarithmic dependence on ξ and ξ ' has no analogue in an inertial frame. The 'Rindler vacuum' is different from the 'Minkowski vacuum' under canonical quantization.", "pages": [ 18, 19, 20 ] }, { "title": "E. Contrasting Classical-Quantum Viewpoints in a Rindler Frame", "content": "Although the classical zero-point correlation functions and the quantum symmetrized vacuum expectation values agree in inertial frames, they are no longer in agreement in noninertial frames. The vacuum states arise from very different concepts in the classical and the quantum theories. The essential feature of classical zero-point radiation is that the spectrum of random radiation is featureless. Therefore in empty space, classical zero-point radiation depends only upon the geodesic separation of the field points. The spectrum obtained from the continuous frequencies of empty space is then restricted to the allowed normal mode frequencies in a box of finite size. In the limit where the sides of the box are moved to the limits of the spacetime, the spectrum in the box becomes that of empty space. Thus a box with walls at rest in an inertial frame and a box at rest with respect to the coordinates of a Rindler frame have very different normal modes, and the spectral amplitudes are readjusted to reflect the change from a geodesic to non-geodesic time coordinate. In terms of a geodesic time coordinate such as appears in an inertial frame, the spectrum of zero-point radiation is given by f 2 0 ( k ) = 4 π /planckover2pi1 c/ | k | where the constant is chosen to give an energy (1 / 2) /planckover2pi1 c | k | per normal mode. In terms of the non-geodesic time coordinate η appearing in a Rindler frame, the spectrum of zero-point radiation is given by F 2 0 ( κ ) = (4 π /planckover2pi1 c/κ ) coth( πκ ) . If the walls of the box are moved to the limits of the Rindler wedge, the random radiation in the Rindler space is exactly that of the inertial space. The classical vacuum is unique. In complete contrast, the vacuum of quantum field theory arises from a prescriptive process which takes no account of the spacetime metric. In any box, the amplitude for the normal modes is fixed, and annihilation and creation operators are introduced for the positive and negative time aspects. Thus a box with walls at rest in an inertial frame and a box at rest with respect to the coordinates of a Rindler frame have very different normal modes but the same spectral amplitude, and accordingly have very different vacuum states. If the walls of the Rindler box are moved out to the limits of the Rindler spacetime wedge, the quantum fluctuations associated with the Rindler vacuum state remain quite different from the quantum fluctuations associated with the inertial vacuum state. The 'Rindler vacuum' is different from the 'Minkowski vacuum' even for a large box. There is a non-uniqueness for the quantum vacuum in non-inertial frames. Of course, one can apply tensor transformations to the vacuum expectation values of the symmetrized quantum operators which were found in an inertial frame. Since the symmetrized quantum expectation values agree exactly with the corresponding classical correlation functions in an inertial frame, we obtain exactly the same expressions (53)(60) as found for the classical correlation functions in the Rindler frame. The spatial dependence on the geodesic coordinate ξ found in Eq. (59) for the correlation function at a single time η = η ' agrees exactly with that found in the corresponding expression (24) in an inertial frame (for x = ξ, x ' = ξ ' ), as we indeed expect since a fixed time η = η ' corresponds to a single time t = t ' in the momentarily comoving inertial reference frame, and all inertial frames have the same correlation functions for zero-point radiation. The absence of any spatial correlation length in Eq. (59) corresponds to zero-temperature T = 0 . However, the time dependence in Eq. (58) for the correlation function at a single spatial coordinate ξ = ξ ' is quite different from the time dependence (23) found in an inertial frame. Indeed, The appearance of the hyperbolic cotangent function for the timeFourier spectrum in Eq. (51) has led some physicists to speak of the 'thermal effects of acceleration through the vacuum'[17][18][19][20][21] with temperature T = /planckover2pi1 a/ (2 πck B ) . After all, the hyperbolic cotangent function appeared in Eq. (27) for the spectrum of thermal radiation in an inertial frame. Thus the spectra in the Rindler frame can be used to suggest either finite temperature T = /planckover2pi1 a/ (2 πck B ) or zero-temperature T = 0 depending upon one's point of view. This ambiguity arises precisely because the Rindler frame is not an inertial frame and the Rindler time parameter η is not a geodesic coordinate. Indeed one may inquire as to just what spectrum corresponds to thermal radiation in a noninertial frame. Within classical physics, this question has been discussed in connection with time-dilating conformal transformations which allow us to derive the Planck spectrum from the structure of relativistic spacetime.[9][10][11] Despite the classical-quantum agreement of the tensor-transformed inertial expectation values, the quantum viewpoint is more complicated since quantum theory introduces a new vacuum state associated with canonical quantization in the Rindler frame. Canonical quantization within a box in a Rindler frame leads to field fluctuations which are quite different from those found from quantization in an inertial frame. Thus the time dependence of the symmetrized Rindler vacuum expectation value at a single spatial coordinate in (67) (with its inverse time dependence) is indeed analogous to the inverse time dependence found in (23) for the inertial frame. However, the logarithmic spatial dependence of the Rindler vacuum expectation value at a single time in (68) is quite different from that given in (24) for the inertial frame. Thus the quantum vacuum in a Rindler frame has quite different properties from the quantum vacuum in an inertial frame. Indeed, over 30 years ago, Fulling called attention to this 'Nonuniqueness of Canonical Quantization in Riemannian SpaceTime.'[17] And what is the physical meaning of the 'Rindler vacuum state' which is different from the familiar 'Minkowski vacuum state'? According to some quantum field theorists,[3] the vacuum is established by the walls of the box which confine the radiation. If the walls of the box are established at temperature T = 0 in the inertial frame vacuum and then the box has its acceleration slowly increased to the final acceleration, its interior will be in the Rindler vacuum. On the other hand, if the box at temperature T = 0 in the inertial frame is suddenly accelerated, the box will contain Rindler excitations corresponding to the Fulling-Davies-Unruh temperature T = /planckover2pi1 a/ (2 πck B ) as measured in the Rindler frame where the Rindler vacuum is the lowest energy state.[3] The classical theory with zero-point radiation lends no support to this quantum interpretation. The classical vacuum state involving classical zero-point radiation is unique; its description between any two coordinate frames is found by tensor transformations. In particular, classical physics has nothing like the scenario described above for a box of zeropoint radiation which is moved from an inertial to a Rindler frame. According to classical theory, (except for small Casimir effects) it matters not how the box of (featureless) classical zero-point radiation is moved from the inertial frame into the accelerating Rindler frame; the box of radiation will always correspond to zero-point radiation as described by tensor transformation from an inertial frame. This statement seems to come as a surprise to many physicists who are misled by their experience with spectra involving finite total energy. The invariant result for a box of zero-point radiation follows from the very special character of the zero-point spectrum which has no structure other than that which is given to it by the coordinates associated with the metric of the spacetime. In an inertial frame in empty space, the zero-point radiation spectrum is Lorentz invariant and scale invariant; it depends only upon the separation (and coordinate derivatives) between the two spacetime points measured along a geodesic between the points.[5][6] Perhaps the reader can obtain a better sense of the special character of zero-point radiation from the following considerations. We saw in Eqs. (21) and (22) that the spectrum of random classical zero-point radiation for the scalar field in an inertial frame depends upon the logarithm of the invariant separation between the two spacetime points. Since we are dealing with a scalar field, the correlation function takes the same value in the Rindler frame. If we transform the Minkowski coordinates to Rindler coordinates, as given in Eqs. (49) and (50), we find that the correlation function is time stationary; it depends upon only the time difference ( η -η ' ) and not on any initial time. There is no spectrum of finite energy density which has such behavior; time-translation invariance both in all inertial frames and in all Rindler frames is a property unique to the zero-point radiation spectrum. The solutions for the wave equations (3) and (39) are unique for boundary conditions which specify both the function and its first time derivative at a single time coordinate. We can imagine a box of zero-point radiation which is at rest in an inertial frame and then is suddenly accelerated so as to remain at the fixed coordinates of a Rindler frame. If we have a box at rest with respect to the coordinates of a Rindler frame, it will be instantaneously at rest with respect to some inertial frame. Within the classical theory, the zero-point radiation within the box differs from the zero-point radiation in the inertial frame by simply the fact that the box modes are restricted to the normal modes of the box rather than being the continuous modes of empty space. As was proved in our analysis above, the zero-point radiation in a Rindler box whose walls are moved to the limits of the Rindler wedge is in complete agreement with the radiation in the empty-space Rindler frame and the radiation in the empty-space inertial frame. Thus the only difference between the radiation inside the box and the radiation of the empty-space inertial frame outside the box are the Casimir aspects associated with the discreteness of the normal mode spectrum of a finite box. For a large box, the zero-point radiation can be accelerated without changing its spectrum. In work published earlier,[9][10] it has been pointed out that the Planck spectrum for classical thermal radiation arises naturally by considering the time-dilation symmetry of thermal radiation in a Rindler frame. Thus in an inertial frame, a time-dilating conformal transformation carries thermal radiation at temperature T into thermal radiation at temperature σT where σ is a positive real number. Under such a transformation, zero-point radiation in an inertial frame remains zero-point radiation. However, in a Rindler frame, a time-dilating conformal transformation carries zero-point radiation into thermal radiation at a non-zero- temperature.[11] The perspective from classical physics suggests that the canonical quantization procedure in a non-inertial frame may be predicting results which have no realization in nature.", "pages": [ 20, 21, 22, 23, 24 ] }, { "title": "F. Detectors Accelerating through Classical Zero-Point Radiation", "content": "Although during the 1970's there were discussions as to whether or not acceleration through the quantum Minkowski vacuum turned virtual photons into real photons, today quantum field theory claims merely that 'detectors' accelerating through the quantum vacuum behave as through they were in a thermal bath.[21] Indeed one quantum theorist has asserted that on acceleration through the vacuum, 'Steaks will cook, eggs will fry.'[22] Of course, there is no experimental basis for such an assertion. And our suggestion is that such an assertion may be wrong. Quantum theorist often speak of using a very small system which would not be affected by gravity in order to examine the thermal bath behavior of mechanical systems.[7] Indeed, within classical physics, there are calculations for point harmonic oscillators[23][24] and point magnetic dipole rotators[25] accelerated through classical zero-point radiation; these systems indeed take on values for the average energy as through they were located in an inertial frame in a thermal bath with temperature T = /planckover2pi1 a/ (2 πck B ) . Point systems respond simply to the time correlation function and so do not sample anything regarding spatial extent. Indeed, by using time-dilating conformal transformations it can be shown that if we consider only the correlations in time at a fixed spatial coordinate without measuring anything involving spatial extent, then we can not separate out the effects of acceleration from those of non-zero temperature.[26] However, are point mechanical systems reliable indicators of thermal behavior? We suggest that point systems with internal structure are not relativistic systems and can not be expected to illustrate accurately the ideas of a relativistic field theory. Point systems do not exist as relativistic systems except for point masses. Point systems (such as a harmonic oscillator of vanishing spatial extent) which contain potential energy have no mechanism to show the dependence of the supporting force on the internal potential energy of the system when the system is located in a gravitational field or in an accelerating coordinate frame. This situation is in complete contrast with electromagnetic systems of charged particles; such systems must have finite spatial extent and will be affected by gravity. When the mechanical system contains electromagnetic energy, then the mechanism for the connecting the supporting force to the system potential energy in a gravitational field (or in an accelerating coordinate frame) involves the droop of the electromagnetic field lines.[27] However, a mechanical system with electromagnetic potential energy, such as a classical hydrogen atom, must have finite spatial extent, and therefore responds to both the temporal and spatial correlation aspects of the fluctuating field. Indeed, this question of finite spatial extent has direct relevance to the arguments given previously regarding 'sudden' versus 'adiabatic' acceleration of boxes of radiation. We can imagine a mechanical system located at a fixed position in the interior of a box of radiation which is moved from an inertial frame over to a Rindler frame. Within classical theory, this mechanical system takes on the same value whether alone and accelerated through the zero-point radiation of a Minkowski frame or whether at rest inside a (large) box in an accelerating Rindler frame because the spectrum of classical zero-point radiation is the same inside or outside the (large) box. However, quantum theory might suggest different behavior for the mechanical system in these two cases; in the first case the system is responding to the tensor transformations of the fluctuations of the Minkowski vacuum and in the second case the system is (presumably) responding to the fluctuations of the Rindler vacuum. Indeed a point system will simply respond to the local time-fluctuations of the radiation inside the box. This is not true for a hydrogen atom or any spatially extended relativistic system. The field lines of a Coulomb potential 'droop' in a gravitational field and the extent of the 'droop' is a measure of the strength of the gravitational (or acceleration) field. The final droop of the field lines of a Coulomb potential in a Rindler frame has nothing to do with the way in which the potential may have been moved from an inertial to the Rindler frame. When a point harmonic oscillator is moved up and down in thermal radiation in a gravitational (or acceleration) field, it can be used to violated fundamental laws of thermodynamics precisely because it does not readjust to the gravitational field. A hydrogen atom, which is truly a relativistic system, will readjust to the gravitational field by the droop of the field lines as it is moved up or down in a Rindler frame. Only relativistic systems should be considered seriously when dealing with relativistic situations. It seems possible that all the claims that acceleration through the vacuum provides a thermal bath may be in error.", "pages": [ 24, 25, 26 ] }, { "title": "IV. CLOSING SUMMARY", "content": "Although quantum field theory and classical field theory with classical zero-point radiation have related vacuum states in inertial frames, the theories part company in non-inertial frames. The vacuum correlation functions of the classical theory depend upon geodesic separations in the spacetime whereas the expectation values of the quantum theory depend upon a canonical quantization procedure which makes no distinction between geodesic and non-geodesic coordinates. The classical vacuum is unique. The non-uniqueness of the quantum vacuum was noted by Fulling over thirty years ago. This contrast invites deeper exploration.[28] Introduction to Stochastic Electrodynamics (Kluwer, Boston 1996). See also, T. H. Boyer, 'Random electrodynamics: The theory of classical electrodynamics with classical electromagnetic zero-point radiation,' Phys. Rev. 11 , 790-808 (1975). namely, that under uniform acceleration a small thermometer will register a temperature. Accelerated enough steaks in Minkowski vacuum will cook, and eggs will fry.'", "pages": [ 26, 27, 29 ] } ]

[ { "title": "Distinguishing initial state-vectors from each other in histories formulations and the PBR argument", "content": "Petros Wallden ∗ September 12, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "Following the argument of Pusey, Barrett and Rudolph [1], new interest has been raised on whether one can interpret state-vectors (pure states) in a statistical way ( ψ -epistemic theories), or if each one of them corresponds to a different ontological entity. Each interpretation of quantum theory assumes different ontology and one could ask if the PBR argument carries over. Here we examine this question for histories formulations in general with particular attention to the co-event formulation. State-vectors appear as the initial state that enters into the quantum measure. While the PBR argument goes through up to a point, the failure to meet some of the assumptions they made does not allow one to reach their conclusion. However, the author believes that the 'statistical interpretation' is still impossible for co-events even if this is not proven by the PBR argument.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "In quantum theory the state of a system is represented by the wavefunction 1 . While there is general agreement on how to use this state in order to extract predictions in the form of relative frequencies of outcomes of multiple copies, there is strong debate on the meaning of the wavefunction for single systems and of the interpretation that is given to it. On the one hand, one can claim that the state represents some real properties of the (single) system and thus attain an ontological status. On the other hand, one can claim that the state reflects the experimenters information about properties of the system. In the latter view, the state is understood as a statistical distribution of different potential realities (if one assumes that it makes sense talking about the real properties of the system). Here we should stress, that the above considerations, concern pure states, or in other words state-vectors, and in the following text when we use the term 'state' with no further specification, it should be understood as a state-vector.", "pages": [ 1, 2 ] }, { "title": "1.1 Statistical distribution or distinct ontological entities?", "content": "If one was to adopt the statistical view of the state, one would be lead to conclude the following: There is no way with a single experiment to be able to deduce with 100% certainty which of two non-orthogonal states was your initial state. This is in analogy with classical statistical physics, in the following sense. If the experimenter has one of two possible distributions as initial information (corresponding to two different initial states), and the distributions are overlapping, there is no single experiment that can determine with certainty, which of the two initial distributions was correct. This is due to the fact, that reality corresponds to a single point, and since the distributions are overlapping, there are some potential realities that is consistent with either initial distributions/states. If on the other hand, one takes the view that the state reflects some ontological property of the system, then one should be able in principle to construct a series of measurements (for the single system considered) that would be able to distinguish between any two non-orthogonal states. This statement concerns the wavefunction of single system and not about some statistical distribution of the relative frequencies of outcomes of many identically prepared copies. Here we should note, that in principle one can maintain the opinion that different states corresponds to different realities, even if we cannot possibly distinguish them experimentally. However, what is certain is that if one can distinguish between any two states, then it is difficult to maintain the view that the state corresponds to a statistical distribution. The argument of Pusey, Barrett and Rudolph (from now on referred to as PBR) [1], provides an algorithm distinguishing between non-orthogonal states and thus (according to the authors) ruling out a statistical interpretation or what they call ' ψ -epistemic theories '. In [2] the argument has been tested experimentally, and following the assumptions that they made and re-stated, quantum theory is confirmed and ψ -epistemic theories ruled out. The argument along with its assumptions, will be presented in detail later. The specifics of the PBR argument was claimed to be independent of the actual ontology of quantum theory and thus potentially independent of the interpretation one chooses. However it would be interesting to examine the arguments details for particular interpretations and how the assumptions made carry over. Particular interest, would be to do so for alternative formulations of quantum theory (where the argument does not follow trivially) such as the histories formulation, and the relation this has with the ability to retrodict properties of the initial state of the universe. This precisely is the topic of this paper.", "pages": [ 2 ] }, { "title": "1.2 Histories and initial state", "content": "In order to do this comparison, one first needs to understand what is the meaning and role both conceptually and mathematically, of state-vectors in histories formulations. Here, we should specify what we mean as histories formulation. They are formulations of quantum theory, that assign (or use) a quantum amplitude or a quantum measure to histories of the system. In other words, formulations based on the Feynman path integral 2 . Examples of full interpretations based on the path integral, are the decoherent (or consistent) histories approach (e.g. [3]) and the co-event formulation (e.g. [4, 5]) which we will introduce in section 3. In these formulations, the concept of the state of the system at a (random) moment of time does not make sense. The only place that the state enters the picture is the initial state of the universe strictly speaking or more practically, the initial state of a subsystem we consider 3 . In the latter case, the initial state, represents the complete summary of the past of our system. Mathematically it enters by modifying the amplitudes of histories and thus the decoherence functional and the quantum measure (see below). In histories formulations, depending on the particular interpretation, one gives ontological status to either a single history or a subset of histories (or to something else such as a co-event which will be defined properly in section 3.3) but not to the state of the system itself. However, one is still able to ascribe ontological status to the initial state in the following sense. Since we are not doing deterministic physics, starting from some initial state gives rise normally to several potential realities. If one can show that the set of potential realities that arise, if we start with one initial state is completely disjoint with the set of potential realities of any other distinct initial state, then by determining the actual realised reality, we can retrodict uniquely the state we started from. One can speak about properties of the initial state which, in this sense, attains ontological status. In other words, the universe where the initial state is | Ψ 1 〉 is a different one from one that has a distinct initial state | Ψ 2 〉 . If on the other hand, one wishes to interpret the initial state as some short of statistical distribution, then it is necessary that for a given reality there are more than one initial states that are compatible with. In this case, it would be impossible, no matter how fine-grained description one has, to have completely disjoint sets of potential realities corresponding to (non-orthogonal) distinct state-vectors. The procedure to disprove the latter view, is to consider some sufficient fine description, i.e. a sequence of measurements 4 at suitable moments of time that would give rise to sets of potential realties that are disjoint for distinct state-vectors. This will be done in section 4 and in the Appendix. Here, we should stress, that to draw conclusions for the histories formulations, one needs to have a specific interpretation (what are those potential realities) and is not possible to do so fully, only by looking at the quantum measure. If one is able to distinguish between different state-vectors, that would have an important consequence for histories. We would be able, at least in principle, to retrodict the initial state (if pure), uniquely. This is of great importance, e.g. for cosmological considerations, where the histories formulations have been applied extensively.", "pages": [ 3, 4 ] }, { "title": "1.3 This paper", "content": "In this paper, in section 2 we will review the PBR argument illustrated with the simplest example of the | Ψ 1 〉 = | 0 〉 , | Ψ 2 〉 = | + 〉 states, and point out the assumptions made. In section 3 we will present the histories formulation, by introducing the decoherence functional and the quantum measure and briefly mentioning the co-event and the decoherent histories formulations. In section 4 we analyse the PBR argument for histories, looking the quantum measure and examining the potential co-events in detail for two versions of the example considered in section 2. In section 5 we will see how the assumptions of PBR argument affect our conclusion for the co-events formulation and in section 6 we will summarise and conclude.", "pages": [ 4 ] }, { "title": "2 The PBR argument", "content": "In this part, following [1], we provide an algorithm that one could follow to show that two non-orthogonal state-vectors correspond to distinct realities and could not possibly be confused as a statistical interpretation would imply. /negationslash Assume that two states 5 Ψ 1 , Ψ 2 correspond to a statistical distribution of some underlying true properties. This would imply that with some probability p = 0 the true properties of the system are such that they are compatible with both the system being in state Ψ 1 and Ψ 2 . Now imagine we consider a pair of identical systems such that each of them can be either in state Ψ 1 or in state Ψ 2 . This can be realised by considering two boxes that each of them generates either Ψ 1 or Ψ 2 with some probability p . Then, provided that the two systems are independent, with probability p 2 the true underlying properties of the composite system would be compatible with the system being in any of the four states 6 | Ψ 1 , Ψ 1 〉 , | Ψ 1 , Ψ 2 〉 , | Ψ 2 , Ψ 1 〉 , | Ψ 2 , Ψ 2 〉 . Then by choosing to make a measurement in a suitable basis for the composite system, one can show that no outcome of that measurement is compatible with all of those potential initial states. Let us see it here with a simple example, of two particular non-orthogonal states of a qubit, the | Ψ 1 〉 = | 0 〉 and the | Ψ 2 〉 = | + 〉 = 1 / √ 2( | 0 〉 + | 1 〉 ) states. /negationslash If | Ψ 1 〉 and | Ψ 2 〉 were not distinct entities but corresponded to statistical distribution of some underlying true properties, then there must be some possible realities, where one cannot distinguish between the four states | 00 〉 , | 0+ 〉 , | + 0 〉 , | + + 〉 . In particular the chance of being in one such case is p 2 where p = |〈 0 | + 〉| 2 = 1 / 2. For those cases, no measurement should be able to distinguish between the four above states. In other words, it should not be possible to measure the composite system and get with certainty (probability 1 or 0) that any one of the four above states is not possible, since that would not be compatible with the cases that appear with probability p 2 = 0 that those four states are indistinguishable. Assume now that we measure the composite system in the following basis 7 : We are lead to paradox because, there is no outcome of this measurement that is compatible with the system being all four initial states. In particular, if the initial state was | 00 〉 then | ξ 1 〉 never occurs, if | 0+ 〉 then | ξ 2 〉 never occurs, if | + 0 〉 then | ξ 3 〉 never occurs and if | + + 〉 then | ξ 4 〉 never occurs. But since {| ξ i 〉} forms a complete basis, one outcome always occurs, and thus with certainty we can conclude that the system never is compatible with all four states | Ψ i , Ψ j 〉 . Depending on the outcome, every time we rule out one of the four initial states. From this observation, the authors conclude that the single system is not compatible with both | Ψ 1 〉 and | Ψ 2 〉 and thus a statistical interpretation is not possible. The argument can be extended for any two non-orthogonal states provided one considers a suitable number n of identical (and non-interacting) copies and performs a measurement in a particular basis on the total tensor product Hilbert space. The reader is referred to the original papers [1],[2] for details. At this point, we should stress what assumptions were made (and stated) by the authors in order to reach their conclusion. using the same apparatus at much later times. Note, that due to this point we are allowed to deduce that if with probability p something occurs in the single system then we get with probability p 2 this thing occurring at both copies of the composite system. The experimental test of the argument in [2] also assumes the above assumptions, and in particular failure to satisfy them (as we will see in section 5), allows for reaching different conclusion. Further discussion of the assumptions will follow at section 5, and in relation with histories formulations.", "pages": [ 4, 5, 6 ] }, { "title": "3 Histories and co-events", "content": "In histories formulations the central mathematical structure of interest, is the history space Ω, the space of all finest grained descriptions 8 . It is the set of all possible histories, and each element of it h i ∈ 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 nonrelativistic particle, Ω would be the space of all trajectories in the physical space. Subsets of Ω are called events and correspond to all the physical questions one can ask. 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 ∆.", "pages": [ 6 ] }, { "title": "3.1 Amplitudes, decoherence functional and quantum measure", "content": "One can assign an amplitude (complex number) to each history following the Feynman path integral approach. This amplitude, depends on the initial state and on the dynamics of the system encoded in the action S : Using this amplitude one can recover the transition amplitudes from ( x 1 , t 1 ) to ( x 2 , t 2 ) by summing through all the paths P obeying the initial and final condition: The mod square of this amplitude is the transition probability. Using the Feynman amplitudes Eq. (2) one can define the decoherence functional: Where A and B are any subsets of Ω, t f is the final while t 0 the initial moment of time considered and ρ the initial state. The decoherence functional obeys the following conditions: The decoherence functional can also be defined using time ordered strings of projection operators. In particular Where C A and C B are the class operators, which are strings of time ordered projection operators corresponding to the histories A and B respectively that we will specify below and is defined in the following way. Where the 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 . The history A is the subset of Ω that contains all the histories that the system lies in the subspace that P A 1 projects to, at time t 1 and in the subspace that P A 2 projects to, at time t 2 , etc. The projection operators in general can be at any subspace of the Hilbert space. Note that the expression for the class operator, is precisely the one used in ordinary quantum mechanics to obtain the amplitude, if some external observer carried out those measurements at the given times. By the linearity property of the decoherence functional the Eq. (5) can be extended to subsets of Ω that are not just strings of projection operators (called inhomogeneous histories). Here we are not going into deeper discussion of the differences of those two definitions and their interpretational consequences. We will simply use the operator expression for finite moments of time histories, since it is more easy to deal with. However, one can see that the two definitions are essentially equivalent for the examples considered in this paper. From the positivity condition, we can see that the diagonal elements of the decoherence functional are non-negative. Those terms are also referred to, as quantum measure ([6]) and are labelled as µ ( A ) := D ( A,A ). It is important to note, that only the real part of the decoherence functional affects the quantum measure. This property, in relation with extending the quantum measure for composite systems, will be discussed at section 5. One could be tempted to interpret the quantum measure as probability. However this is not possible, due to interference. The additivity condition of probabilities, is not satisfied: /negationslash However a weaker condition holds that shows that there is no three-paths interference: Interpreting the quantum measure, is the issue of histories formulations, and in this paper we will focus on the decoherent histories and the co-event formulations 11 . Provided the initial state is pure, we can define the quantum measure in terms of amplitudes for histories It is important to note the delta function that guarantees that histories that do not end (final time t f ) at the same point, do not interfere. In other words, the quantum measure becomes additive if one considers alternatives that differ at the final moment of time. Another thing to note, when one considers the operator definition of the decoherence functional, is what happens if one introduces more moments of time. This is simply a fine graining of the previous histories as it is easily seen from the definition using the paths/trajectories. A final issue to discuss, before introducing interpretations of the quantum measure, is the sets of histories that have quantum measure zero. Those sets of histories will be referred to as precluded sets. In general, there are two kind of quantum measure zero sets. The trivial ones, that all their subsets have also quantum measure zero (similarly with classical measure zero sets), and the nontrivial, that have subsets that are non-zero. The latter are due to interference and are the source of any counter-intuitive property. If a set A has quantum measure zero then it decoheres with its compliment 12 µ ( A ) + µ ( ¬ A ) = 1 = µ ( A ∪ ¬ A ). This along with other considerations lead us to the conclusion that quantum measure zero sets, do not occur in nature. We further need to assume that any set B ⊆ A also does not occur if A does not occur, if one wishes to maintain classical deductive reasoning such as the Modus Ponens (see [8]). However, this could lead us to trouble, because it is known that generally, one can cover the full history space Ω with zero quantum measure sets [9].", "pages": [ 6, 7, 8, 9 ] }, { "title": "3.2 Decoherent histories approach to quantum theory", "content": "Decoherent histories (also known as consistent histories) is an approach developed, initially, mainly by Griffiths, Omnes and Gell-Mann and Hartle (eg. [3]). The decoherence functional, first appeared in relation with this approach. The mathematical aim of the approach is to tell when it is possible to assign probabilities to (coarse-grained) histories of a closed quantum system, or in the language we developed above, when is it possible to assign the quantum measure of a set of histories A as the probability of this set A actually occurring. Different people have given different motivations for the approach, but the general aim is to be able to reason about a closed system with no reference to observer or an a-priori distinction of microscopic and macroscopic degrees of freedom, or distinction of quantum and classical systems. The field of quantum cosmology which is the ultimate closed quantum system, was one of the motivation for this approach, while the way that classicality emerges and its connection with decoherence, was another. In order to make the quantum measure into a proper classical measure, one needs to restrict attention to some particular collection of subsets of Ω rather than the full collection of all possible subsets of Ω. The failure to satisfy the additivity condition can be traced at the off-diagonal terms of the decoherence functional as one can see from the very definition 13 . Let us take a partition of Ω which is defined to be a collection of subsets P 1 = { A 1 1 , A 1 2 , · · · , A 1 n } where A 1 i ∩ A 1 j = ∅ and ∪ i A 1 i = Ω. The superscript labels the partition considered, while the subscript labels the different cells of one partition. If for any pair of cells of one partition A 1 i , A 1 j it holds that /negationslash then the partition is called a consistent set . For this partition and any further coarse-graining, the standard rules of probability theory hold. The quantum measure, when restricted to those questions, becomes a classical measure. One would be tempted to assign these probabilities to the coarse-grained histories of the partition. However, one can consider other partitions, say P 2 = { A 2 1 , A 2 2 , · · · , A 2 n } . It is possible that this partition also forms a consistent set obeying Eq. (10). Important thing to note, is that there does not exist, one finest-grained consistent set, that all other consistent sets are simply coarse-graining of that. One is not allowed to make propositions involving sets that belong to separate consistent sets, and thus cannot properly assign probabilities to histories once and for all, but it is dependent (contextual) to the consistent set one considers. Counterintuitive consequences have been analysed (eg. by Dowker and Kent [10]) and at this point we can only say that one cannot assign probabilities to histories in a classical sense. The minimalist view, is that one could use present records corresponding to one consistent set, to deduce things about other present records related with the same consistent set. In this language, being able to deduce things about the initial state of the universe from present records, would imply the ability to make further present time predictions. Being able to distinguish between different initial state-vectors, which is the discussion of the present paper, lie within this scope.", "pages": [ 9, 10 ] }, { "title": "3.3 The co-events formulation of quantum theory", "content": "The co-event formulation was developed mainly by Sorkin [4, 11] and collaborators (e.g. [9, 12, 13, 14]). A review can be found here [5]. It is a more recent attempt, to maintain a realistic picture of closed systems quantum theory and being able to speak about properties of the closed system being possessed objectively. In classical physics, there are three structures if one wishes to use the histories language. First, the histories space Ω and the collection of all subsets of Ω which form a boolean algebra U , second the space of truth values T = { 1 , 0 } also forms a boolean algebra and finally third the valuation maps φ which assign a truth value 1 , 0 to all questions/subsets of Ω. These maps we call them co-events and in classical physics need to respect the boolean structure of U and T and be a homomorphism One can show that there is a one-to-one correspondence between single histories (points at Ω) and homomorphic co-events, in this way i.e. when the co-event φ h is a characteristic map of h . We usually assume that reality is one element (say h ) of Ω, the one that is actually realised. We can see here, due to the above correspondence, that we could have a dual view, and say that the co-event/characteristic map φ h is what is truly realised. The potential realities, thus are all the co-events that correspond to some history h that does not have zero (classical) measure. In quantum theory the above picture cannot be maintained, due to the fact that we have a quantum measure on Ω. One could weaken the requirement that the maps are homomorphisms and allow them to be non-additive. In particular, we could have multiplicative co-events that /negationslash Then one can show that all the multiplicative co-events are to one-to-one correspondence not with single histories but to subsets of histories, in other words they are characteristic maps of non-trivial subsets A . One can see that if A is neither subset of B nor of ¬ B , then both B and ¬ B are false. This precisely is where the strangeness of quantum theory is encoded. However, if the possible A are small enough, one would expect that all classical questions would be too coarse-grained to intersect non-trivially A , and no paradox would appear. One should note, that if a particular co-event corresponds to a characteristic function of a set with a single history, it gives rise to completely classical logic (homomorphism) and in this sense we will refer to it as a 'classical co-event' even if the system it arises may allow other co-events that are not of this type. Other than the requirement to be multiplicative, the allowed co-events must (a) be preclusive, i.e. respect that µ ( A ) = 0 = ⇒ φ ( A ) = 0 and (b) be minimal (called primitive), i.e. be as small as possible, providing they obey multiplicativity and preclusivity (e.g. [14] for greater detail). The first requirement, uses the quantum measure and it is at this point where the dynamics (Hamiltonian) and initial state of the particular system enters the picture. In order to recover the full probabilistic predictions, one apparently needs to use the quantum measure and resort to the Cournot principle. Cournot's principle, is the following: 'In a repeated trial, an event A singled out in advance, of small measure ( ≤ /epsilon1 ), rarely occurs'. The details on how this can be used to recover the probabilistic predictions of quantum theory (e.g. double slit pattern) can be found in [12], and a shorter version in section 4.2 of reference [5]. We should stress however, that ontological status of the state-vector that we examine in this paper, is not directly related with the way of recovering the probabilistic predictions. In PBR argument, for example, the particular values of probabilities, play no role. To summarise, the potential realities for the co-event formulation, are the set of all primitive, preclusive and multiplicative co-events. From now on, when we mention co-events or potential realities, we will refer precisely to these. Two important features of the co-event formulation, are the following. First, that the deductive logical inferences are possible, namely the Modus Ponens rule of inference holds for multiplicative co-events (and only for those!) [8]. Second, that one has a unique finest grained classical partition. In other words, there exist a finest grained description, such that, no matter which co-event is realised, the resulting (coarse-grained) logic is classical (see appendix of [12]). This is different than in decoherent histories, where one has incompatible consistent sets. As a final point at this section, we should stress that in general there are many potential co-events, given a quantum measure. This is in analogy with classical stochastic physics and we could claim that quantum theory constitutes generalisation of stochastic physics rather than deterministic. We thus have a set of possible realities C . Measurements made, narrow down this set of potential co-events and we may or may not be able to narrow it down to the single co-event that is actually realised. If unsure of which was the initial state, we have many possible sets C i of co-events, each set corresponding to one candidate initial state ψ i . If we carry out a measurement and get a result that is not possible in any of the co-events of one particular set (say for example C 3 ), we can safely deduce that the initial state was not the one that has as possible co-events this set ( ψ 3 in this example). In other words we would be able to deduce things about the initial state of the system (or more ambitiously stated, of the universe). The reader is referred to the original references for a more complete and detailed presentation. Here we only introduced the necessary material and stressed few things that are important for this paper.", "pages": [ 10, 11, 12 ] }, { "title": "4 PBR for co-events", "content": "In order to see the PBR argument for co-events one has to first construct the above experiment in a histories version, then compute the quantum measure and finally find the potential realities, i.e. the potential co-events. The first two steps are common in all histories formulations, however the conclusion of the argument is possible only after one considers the potential realities. Different initial states give rise to different quantum measures and thus different set of possible realities/co-events. The PBR argument, needs to show that there exists no common co-event, between the different sets of possible coevents corresponding to different initial states. If this is shown we will be able to conclude that there is no possible reality/co-event compatible with the state being both | Ψ 1 〉 and | Ψ 2 〉 . Here we will consider the simple example presented in section 2 with the qubit starting in either | Ψ 1 〉 = | 0 〉 or | Ψ 2 〉 = | + 〉 . The way to realise in histories language the PBR argument is not unique. We will first see the simpler version, where we have a single moment of time and it is essentially the PBR argument casted into co-events language. Then we will consider an apparatus with two moments of time, that gives a better picture of the histories formulations, having given preferred status in the {| 0 〉 , | 1 〉} basis in analogy with the preferred fine grained set of histories being the configuration space basis. The latter version, involves more of the features of the co-event formulation, since both non-trivial quantum measure zero sets exist and non-classical co-events.", "pages": [ 12, 13 ] }, { "title": "4.1 Version 1", "content": "We consider two copies (uncorrelated and with no interaction) of a qubit, that can be in either the state | 0 〉 or in the state | + 〉 . In other words the initial state is one of the following: After preparing the initial state in one of those four states, we measure it in the {| ξ i 〉} basis given in Eq.(1). We label history h i the one that the system is found in state | ξ i 〉 . In other words, for example, h 1 is the history that the system starts from one the four states of Eq. (15) and then is found being in the | ξ 1 〉 state. The quantum measure, and thus the possible co-events are different for each possible initial state. For initial state | Ψ 1 , Ψ 1 〉 = | 00 〉 , the quantum measure of different fine grained histories is the following: Where the subscript at the quantum measure, denotes that it corresponds to the initial state | Ψ 1 , Ψ 1 〉 . Note that this is a trivial histories space, since it has only a single moment of time. Alternative histories, decohere, since they lie at the final moment of time, and thus the quantum measure is additive and fully given by the quantum measure on the fine grained histories. If there were more moments of time, in order to fully characterise the quantum measure, one could give the amplitudes for different histories as we will do at version 2. There is only one quantum measure zero set, which is the single history { h 1 } and thus the potential co-events are all classical. The set of possible co-events for the initial state | 00 〉 are For initial state | Ψ 1 , Ψ 2 〉 = | 0+ 〉 , the quantum measure of different fine grained histories is the following: And the set of possible co-events for the initial state | 0+ 〉 are For initial state | Ψ 2 , Ψ 1 〉 = | +0 〉 , the quantum measure of different fine grained histories is the following: And the set of possible co-events for the initial state | +0 〉 are For initial state | Ψ 2 , Ψ 2 〉 = | ++ 〉 , the quantum measure of different fine grained histories is the following: And the set of possible co-events for the initial state | ++ 〉 are The important thing to notice here, is that there is no common co-event for these four different initial states, namely: This implies that there is no potential reality, that is compatible with all four above initial states. Carrying out the measurement suggested in PBR argument, leads us to exclude one of the four initial states as the initial state of the system. Which one is excluded, depends on the outcome of the measurement. This is close, but not quite the same, as saying that we can definitely distinguish between the state | Ψ 1 〉 and | Ψ 2 〉 . We will come back to this point in section 5 in the discussion.", "pages": [ 13, 14 ] }, { "title": "4.2 Version 2", "content": "Since the quantum measure is affected from the particulars of the measurements/histories considered, one could expect that realising the above set up in a different way could affect the conclusion in relation with the possible coevents. Since in histories formulation, preferred status is given to the 'actual' fine grained histories that correspond to paths at the (extended) configuration space, one could see that the analogous thing to do for a qubit, is to measure it in the {| 0 〉 , | 1 〉} basis solely. So while we start with the same four possible initial states given by Eq. (15) as in version 1 we consider different histories which in other words corresponds to a more fine grained description. We have fine grained histories, starting at one of the four initial states, then both particles are 'measured' in the {| 0 〉 , | 1 〉} basis, and finally they are measured in the {| ξ i 〉} basis. The possible histories are labeled as h 00 ξ i if the system (no matter which initial state we consider) is found in | 00 〉 and then in | ξ i 〉 , h 10 ξ i if it is in | 10 〉 and then in | ξ i 〉 etc. In order to find the co-events, one needs for a given initial state to compute the quantum measure and find the quantum measure zero sets. Since histories in this version do not trivially decohere (unless they end at different final time) the most convenient way to write down the quantum measure, is in terms of the amplitudes of different fine grained histories. If a set of histories ends at the same outcome at the final time, then the quantum measure is simply the mod square of the sum of the amplitudes of fine grained histories. For the fine grained histories the quantum measure is given by µ ( h i ) = | α ( h i ) | 2 . For initial state | Ψ 1 , Ψ 1 〉 = | 00 〉 we have these quantum amplitudes: and zero amplitude for all the other histories. Here, too, there are no non-trivial quantum measure zero sets. We therefore have only three classical co-events Important to note here that there is no co-event ending at ξ 1 . For initial state | Ψ 1 , Ψ 2 〉 = | 0+ 〉 we have these quantum amplitudes: In this case, we see that there is one non-trivial quantum measure zero set. The set { h 00 ξ 2 , h 01 ξ 2 } has quantum measure zero, while the fine grained histories have both quantum measure 1 / 8. Here again we have the following allowed co-events: And we should note, that none of these co-events, ends at ξ 2 . we have these quantum amplitudes: For initial state | Ψ 2 , Ψ 1 〉 = | +0 〉 we see that here too there is one non-trivial quantum measure zero set. The set { h 00 ξ 3 , h 10 ξ 3 } has quantum measure zero, while the fine grained histories have both quantum measure 1 / 8. We have the following allowed co-events: And we should note, that none of these co-events, ends at ξ 3 . Finally, for initial state | Ψ 2 , Ψ 2 〉 = | ++ 〉 we have these quantum amplitudes: We have several non-trivial quantum measure zero sets, namely: { h 00 ξ 4 , h 11 ξ 4 } , { h 00 ξ 2 , h 01 ξ 2 } , { h 01 ξ 2 , h 10 ξ 2 } , { h 01 ξ 2 , h 11 ξ 2 } , { h 00 ξ 3 , h 10 ξ 3 } , { h 01 ξ 3 , h 10 ξ 3 } and { h 10 ξ 3 , h 11 ξ 3 } . The allowed co-events are more complicated in this case and there exist some non-classical co-events (corresponding to pairs of histories). There are two classical co-events ending at ξ 1 , three co-events consisting of pairs of histories, ending at ξ 2 and the same at ξ 3 while there is no co-event ending at ξ 4 . In particular, the co-events are: The important thing to conclude from this analysis is that there is no common co-event for these different initial states either: Thus we can see that if the ontology of quantum theory is that of a co-event, we deduce that there is no conceivable reality that is compatible with the state being all four states | 00 〉 , | 0+ 〉 , | +0 〉 , | ++ 〉 . This result, that was also present in version 1, is therefore maintained even if we further fine-grain the system. We should note that in this version of the example, we had more than one moments of times and there were non-trivial quantum measure zero sets that also lead to having some possible non-classical co-events. All this complication, did not affect the above conclusion. However, to make the final step, and conclude that we can always distinguish between distinct state-vectors, as the PBR arguments claims, we need to review and examine the assumptions made 14 .", "pages": [ 14, 15, 16, 17 ] }, { "title": "5 Assumptions and discussion", "content": "In the end of Section 2, the assumptions of this theorem were briefly stated. The first assumption was that one can prepare a state of a system in isolation from the rest of the universe and its individual well defined physical properties depend only on this state and not in any sense from the rest of the universe, and the second that constructing multiple unrelated copies of the system is possible. The third was that the outcomes of the measuring deices respond solely to physical properties of the measured systems. The first two assumptions are not (at least obviously) valid in general if one uses the Feynman path integral and the quantum measure. For co-events, for example, we know it is not true. The property that forbids one to take these two assumptions, is how the quantum measure of a composite system relates to the quantum measure of individual subsystems. The quantum measure of a composite system is not in general the product of the quantum measures of the individual subsystems, even if there is no interaction between the subsystems. It is not even clear, how one would define the composite quantum measure if he was simply given the quantum measure of the subsystems. One of the reasons for this failure, is related to the fact that to each quantum measure there correspond infinite different decoherence functionals. In particular, any two decoherence functionals that differ only in their complex part, give rise to the same quantum measure. While this does not affect the predictions for a single system, no matter the approach one takes (decoherent histories or co-events), it does affect considerations of composite systems. If one is given the decoherence functional of a single system, there is a preferred way to construct the composite (two copies of) system decoherence functional, by simply taking the product of the two individual decoherence functionals. However, this also leads to some paradoxes. In particular, we could have two uncorrelated and non-interacting systems, that all the individual histories are possible, i.e. no history has zero amplitude and moreover no coarse-grained history of the individual subsystems has quantum measure zero. By considering the quantum measure that arises from taking the product decoherence functional, we now have some coarse grained histories having quantum measure zero, and therefore not all combinations of individual outcomes are allowed. Moreover, this property arises from simply considering the two uncorrelated, non-interacting systems together. The reason that this problem that appears for sets with quantum measure zero is important, is twofold. First, because sets with quantum measure zero, decohere with their negation, as we show earlier, and thus necessarily belong to one decoherent set. The second, is because those sets are used in the co-event formulation in order to find the potential co-event. A simple example of what could go wrong is given here. Assume we have a decoherence functional given by: In this example, no history is precluded, since there exist no quantum measure zero set. History h 1 corresponds to the (1 , 1) entry of the matrix and h 2 to the (2 , 2). One can also note, that for this system there is no non-trivial decoherent set since we adopt the medium decoherence condition that requires both the real and imaginary part of the off-diagonal parts of the decoherence functional to vanish 15 . Consider two identical systems and take their deoherence functional be their product: We can now see that there is a nontrivial quantum measure zero set. We can conclude from that that the coarse-grained history { h 11 , h 22 } is not possible. A subset of this history, and thus also precluded, is the history where both subsystems are at h 1 . Similarly the history that both subsystems are at h 2 is subset of { h 11 , h 22 } and thus precluded. However, no such preclusion was possible if we were simply looking the single systems decoherence functionals. One can claim that there is an (anti)correlation of the two systems, even though there is no entanglement and interaction. Another aspect of this property can be seen, if we note that for the composite system, there is a non-trivial decoherent set, that follows from the existence of the zero set. The partition of Ω = { A,B } where A = { h 11 , h 22 } and B = { h 12 , h 21 } is a decoherent set. For the individual systems however, there was no such set. Note that, while it is true that the product of two non-interacting (medium) decoherent sets of histories give rise to a decoherent set of histories, the converse is not true. We can have a decoherent set of histories on the product system of two non-interacting subsystems, with no analogue for the individual subsystems. This observation, highlights the failure of the assumptions of the PBR argument for histories. A side-note, here is that it was already known that there are issues with composite systems if one considers the decoherence functional. For example in the consistent histories framework, Diosi in [15] showed that if one requires weak decoherence, i.e. that only the real part of the decoherence functional needs to vanish, then one is lead to contradictions by considering non-interacting composite systems. In particular, the weak decoherence condition might hold for subsystems but not for the total composite system 16 . This was probably the strongest reason of adopting the medium (or stronger) decoherence condition for the decoherent histories approach. From the latter observation, one could be tempted to conclude, that issues with composite systems, arise from the complex part of the decoherence functional and that a possible restriction to real decoherence functional could resolve them 17 . However, this is also not true. One can construct a purely real decoherence functional and when considering a composite system, still getting new quantum measure zero sets and their interpretational consequences 18 . The analysis of this property is the subject of an ongoing work [17]. To this end, we must stress that there is a special type of correlation between systems in histories formulations that is not related to entanglement. This precise property, forbids one to make the final step at the PBR argument. To recall the full argument, it was first shown that if one measures in the {| ξ i 〉} basis, any possible outcome is consistent with the system having started with only three of the total four possible initial states of the composite system | Ψ 1 , Ψ 1 〉 , | Ψ 1 , Ψ 2 〉 , | Ψ 2 , Ψ 1 〉 and | Ψ 2 , Ψ 2 〉 . e.g. if ξ 1 is the outcome of the measurement, we know definitely the system did not start at | Ψ 1 , Ψ 1 〉 , but could have started form any of the other three composite states. The second part of the argument, was to relate this with properties of the single system. In particular, they concluded that there is no outcome that can be consistent with both systems being both | Ψ 1 〉 and | Ψ 2 〉 . If | Ψ 1 〉 and | Ψ 2 〉 were merely overlapping distributions of some underlying potential realities, then with some non-zero probability the reality that is actually realised, would lie in the overlapping part and thus, in these cases, any possible measurement would not be able to tell which was the initial state. The initial state would no longer have any ontological status. We need to stress here, that the second part of the argument, lies on the assumption, that the joint distribution of the two single systems is merely the product of the distributions of the individual systems (for unrelated, disentangled, non-interacting systems). However, as we analysed above, this is not true for the histories and the quantum measure of composite systems, and it is not at all clear if and how can this argument be completed. The conclusion of the PBR argument, namely that state-vectors cannot be interpreted statistically, could still hold for co-events (in the sense that the set of allowed co-events for different initial states are completely disjoint), but it is not proven with this thought experiment. For example, it would be interesting to explore other possibilities such as extending alternative proposals to the PBR argument (one very recent one is [18]) for the co-events formulation. In our case, for histories formulations and the co-events in particular, we actually expect it to be true. By extending the histories for sufficient time and sufficiently fine-grained, we conjecture (and give further evidence for it) that the set of possible co-events for any separate state-vectors would be distinct. This would suggest, that if we observe with sufficient detail the system for long enough, we will be able to deduce uniquely which was the initial state, provided that it was a pure state. Evidence that this conjecture holds, is given in the appendix, where we show that indeed for essentially any two distinct initial state-vectors | Φ 1 〉 and | Φ 2 〉 of a qubit (2-dim Hilbert space), we can explicitly construct sufficiently fine grained histories, such that the set of co-events C 1 and C 2 have no common elements ( C 1 ∩ C 2 = ∅ ). Distinguishing between two states is something that always concerns the 2-dim subspace of the Hilbert space that is spanned by those states (as was stressed in [1] and explained in detail in the appendix). Therefore the evidence given in the appendix, that the conjecture holds, is very strong, failing to form a full proof due to some special cases and some extra attention needed for the infinite dimensional case.", "pages": [ 17, 18, 19, 20 ] }, { "title": "6 Summary and conclusion", "content": "We first reviewed the PBR argument for why a statistical interpretation of the quantum state is not possible. We introduced the histories formulations and in particular the decoherent histories approach and the co-event formulation. The reason to examine the PBR argument for histories is twofold. First, in order to make contact with standard one-moment-of-time quantum theory and compare the role of the state. Second reason, is related with retrodiction. While it is not clear that one can determine uniquely to arbitrary precision which one of the potential realities is eventually realised, it is clear that if two different initial state-vectors can give rise even to one common reality, there is no hope of one being able to distinguish between the two situations with certainty. Even speaking about which was the initial state-vector, in this case, is meaningless. Note that histories formulations, are frequently applied to the field of quantum cosmology where the initial state and the ability to retrodict are very important. We then showed that the PBR argument applies to histories formulations looking in section 4, at two versions of the specific example considered in section 2. One is always able to distinguish between the four (at the example) statevectors | Ψ 1 , Ψ 1 〉 , | Ψ 1 , Ψ 2 〉 , | Ψ 2 , Ψ 1 〉 and | Ψ 2 , Ψ 2 〉 . However one cannot make the further step and conclude that we can distinguish between single system statevectors | Ψ 1 〉 and | Ψ 2 〉 , as we saw in section 5. The reason is that some of the assumptions of PBR, do not hold in histories formulations. In particular, there is a strange correlation, not related with entanglement, between sub-systems that the decoherence functional of composite systems has. This property needs to be further examined and understood [17]. The author however conjectures, that it would still be possible to distinguish between different initial states, if one extends the histories suitably, for the co-event formulation. Evidence for the validity of the conjecture is given in the appendix, where it is proven explicitly that this is the case for 2-dim Hilbert space except some very special case, and that it can be generalised to higher (finite) dimensions. To sum up, the conclusion of the PBR argument is expected to be valid for histories as well, but it cannot be proven with their gedanken experiment. Acknowledgments: The author is very grateful to Rafael Sorkin, for bringing the problem to his attention, many discussions and reading and commenting an earlier draft. He acknowledges the COST Action MP1006 'Fundamental Problems in Quantum Physics' and also the Perimeter Institute for Theoretical Physics, Waterloo, Canada, for hospitality while carrying out part of this work.", "pages": [ 20, 21 ] }, { "title": "Appendix", "content": "In this appendix, we will attempt to prove the conjecture made in the text, that the allowed co-events for two different state-vectors, are disjoint. We will use the example of a qubit. However, the importance of this example, is much greater. When comparing two state-vectors 19 , we can do so, by restricting attention to the 2-dimensional subspace that is spanned by the two state-vectors. While it is true, that many results hold in 2-dimensions and not for higher dimensions, the comparison of two pure states, is not one of them. The reason this is the case is because if one has two state-vectors of higher dimensions | Φ 1 〉 , | Φ 2 〉 ∈ H , he can find the two-dimensional subspace that is spanned by those two vectors H Φ 1 , Φ 2 = span( {| Φ 1 〉 , | Φ 2 〉} ). One can then choose | Φ 1 〉 = | 0 〉 and define | 1 〉 ∈ H Φ 1 , Φ 2 such that it is orthogonal to | 0 〉 , i.e. 〈 0 | 1 〉 = 0. Then, we can express | Φ 2 〉 = cos θ | 0 〉 +sin θ | 1 〉 for some angle θ , and without loss of generality, we can proceed as if our initial states | Φ 1 〉 , | Φ 2 〉 were 2-dimensional. Our attempt to prove the conjecture, consist of the explicit construction of the sets of co-events corresponding to two different initial states, that are non-orthogonal. By choosing finer or coarser grained histories (i.e. by having more moments-of-time) we can construct finer or coarser sets of co-events. The direction of proof we will follow, is to exploit some property that certain coarsegraining of histories have, namely the existence of zero covers [5]. This simplifies considerably the analysis and allows us to find a suitable coarse-graining that distinguishes the states with only 3-moments-of-time, and therefore 2 3 different histories for almost all cases. Unfortunately, as we will see below, for the very special case that the angle between the two states we want to distinguish is tan θ = ± 1 / 3, this strategy is not successful. To fully prove the conjecture, one needs to consider different coarse-grainings with more moments-of-time (and therefore exponentially more possible histories). With the current development of the co-event formulation, its extremely difficult to compute the possible coevents, as the moments of time increase. Moreover, the technical trick used to prove the conjecture for all the other cases (the use of that particular zero cover), cannot be used here. However, it seems very implausible, that one can distinguish between any two state-vectors in this formalism, unless their angle is tan -1 ( ± 1 / 3). If that was the case, it would certainly be a strange property that would require further study. It is most likely that some other finer-grained description would complete the proof, but until the relevant technical methods to efficiently compute co-events for many moments-of-time appears, our claim will remain a conjecture. We now return to prove the general case. The histories we will consider, is essentially 3-moments-of-time. We start with some initial state | Φ 〉 and then measure it three times in the basis we will give below. We assume trivial evolution (the identity), but we could easily have any Hamiltonian, and then have to choose the basis measured suitably. Given a particular Hamiltonian (non-trivial this time), one can also reproduce the result we will give, considering measurement done only in the {| 0 〉 , | 1 〉} basis, by suitably choosing the time t 1 , t 2 , t 3 that the measurements take place as we will see in the end of the appendix. The initial state will be either | Φ 1 〉 = | 0 〉 or any other state | Φ 2 〉 = cos θ | 0 〉 + sin θ | 1 〉 . We consider the following two orthogonal bases 20 : and The histories considered will be: They start with the initial state | Φ i 〉 , and then are measured in the {| Ψ + 〉 , | Ψ -〉} basis then in the {| Ψ 0 〉 , | Ψ 1 〉} basis and then again in the {| Ψ + 〉 , | Ψ -〉} . We will label the histories depending on the outcome of each measurement in the following way (measurements are from right to left): Histories h 1 , h 2 , h 3 and h 4 end at final time in the | Ψ + 〉 while h 5 , h 6 , h 7 and h 8 end in | Ψ -〉 . We compute the amplitudes of histories for | Φ 1 〉 = | 0 〉 (the subscript at the amplitudes α 1 signifies that it correspond to initial state | Φ 1 〉 ): The only zero quantum measure sets are the { h 3 , h 4 } and { h 5 , h 6 } for a general angle θ . Here we should note that there are other zero quantum measure sets only in the cases where θ = 0, that reduces to | 0 〉 which we will see below, and for tan θ = ± 1 / 3 which is the exceptional case mentioned earlier that prevents us from providing a full proof of the conjecture. For these very special cases, the fine grained description we used here is not sufficient to prove the conjecture, and further fine graining (measurements) are required. Returning to the general θ case, we have 4 classical co-events and the set of allowed co-events are: For | Φ 2 〉 = cos θ | 0 〉 +sin θ | 1 〉 the amplitudes are (note that they are independent of θ ): The subscript α 2 signifies that that the initial state is | Φ 2 〉 . The sets with quantum measure zero are: We see that all fine grained histories are contained in one quantum measure zero set and thus there are no classical co-events. Only pairs of histories are allowed, and we have 6 potential co-events: It is easy to see that C 1 ∩C 2 = ∅ . This is general for an arbitrary θ (other than a very special case mentioned above), and thus any two state-vectors of a qubit give rise to completely disjoint set of potential co-events and thus correspond to different ontology, in the sense discussed in the main text. We now return at the earlier remark, that all of the above can be re-expressed in terms of measurements in the {| 0 〉 , | 1 〉} basis given a Hamiltonian, if we suitably choose the times that the measurements take place. If for example we this Hamiltonian It gives rise to the following unitary evolution We choose to measure at t 1 = ( θ -π/ 4) and at t 2 = θ and finally at t 3 = ( θ + 7 π/ 4), always in the {| 0 〉 , | 1 〉} basis. It is easy to calculate that the amplitudes we get for these histories for any of the two initial states, are exactly the same as the ones we calculated earlier in the appendix in Eqs. (39 , 41), with the following adjustments. (a) In the labeling we replace Ψ + and Ψ 1 with 1 while we replace Ψ -and Ψ 0 with 0 (i.e. the history h 3 for example, that was (Ψ + Ψ 0 Ψ -) is now (100)) and (b) there is an overall factor of exp( -i ( θ -π/ 4)) in the amplitudes of all histories, which however, does not affect the quantum measure. Since the quantum measure is the same, it follows that set of allowed co-events is also the same.", "pages": [ 21, 22, 23, 24 ] } ]

[ { "title": "CONFORMALLY FLAT SOURCES FOR THE LINET-TIAN SPACETIME", "content": "Irene Brito 1 ∗ , M. F. A. da Silva 2 † , Filipe C. Mena 1 ‡ , and N. O. Santos 3 § 1 Centro de Matem'atica, Universidade do Minho, 4710-057 Braga, Portugal. 2 Departamento de F'ısica Te'orica, Instituto de F'ısica, Universidade do Estado do Rio de Janeiro, Rua S˜ao Francisco Xavier 524, Maracan˜a, 20550-900, Rio de Janeiro, Brazil. 3 School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS, U.K. 3 Observatoire de Paris, Universit'e Pierre et Marie Curie, LERMA(ERGA) CNRS - UMR 8112, 94200 Ivry sur Seine, France. June 16, 2021", "pages": [ 1 ] }, { "title": "Abstract", "content": "We investigate the matching, across cylindrical surfaces, of static cylindrically symmetric conformally flat spacetimes with a cosmological constant Λ, satisfying regularity conditions at the axis, to an exterior Linet-Tian spacetime. We prove that for Λ ≤ 0 such matching is impossible. On the other hand, we show through simple examples that the matching is possible for Λ > 0. We suggest a physical argument that might explain these results.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The Levi-Civita spacetime [1] describes the vacuum field exterior to an infinite cylinder of matter. In its general form, it contains two independent parameters [2, 3, 4], one, usually denoted by σ , describing the Newtonian energy per unit length, and another related to the angle defects. At first sight, global considerations in General Relativity seem to make cylindrical solutions to the Einstein field equations not so physically relevant: fields with cylindrical symmetry impose infinitely long sources, suggesting a peculiar physical situation. Nonetheless its importance cannot be underestimated, and under controlled circumstances, they provide a very good description of systems of physical interest (see e.g. [5]). Furthermore, Newtonian cylindrical models correspond well to observations [6, 7, 8]. In General Relativity, cylindrical solutions have been used to study various fields like cosmic strings [9, 10], exact models of rotation matched to different sources [11], and models for extragalactic jets [12, 13, 14] and gravitational radiation [15]. The generalization of the Levi-Civita spacetime to include a nonzero cosmological constant Λ was obtained by Linet [16] and Tian [17] and it is shown by da Silva et al. [18] and Griffiths and Podolsky [19] that it changes the spacetime properties dramatically. The Linet-Tian (LT) solution has also been used to describe cosmic strings [17, 20, 21] and, in [22], static cylindrical shell sources have been found for the LT spacetime with negative cosmological constant. Considering this extensive interest in cylindrically symmetric solutions we assume worthwhile to analyze some further properties of LT spacetime. In [23], while being studied conformally flat sources, it is proved a seemingly unexpected property that static cylindrical sources matched smoothly to the Levi-Civita spacetime exteriors do not admit conformally flat solutions. For spherical symmetry, there is the well known interior isotropic pressure and incompressible Schwarzschild solution, which is conformally flat [24], matched to the Schwarzschild vacuum exterior spacetime. Senovilla and Vera [25] obtained another disturbing result being the impossibility of the cylindrically symmetric Einstein-Straus model. In order to prove this impossibility they show that a Robertson-Walker spacetime, which is conformally flat, cannot be matched to any cylindrically symmetric static metric across a nonspacelike hypersurface preserving the symmetry. This result was subsequently generalised in [26, 27, 28]. Another result that might be linked to this trend was obtained by Di Prisco et al. [15] and is the following: A cylindrically symmetric shear free collapsing anisotropic fluid can be matched to Einstein-Rosen spacetime as obtained in [15], however, by considering that the exterior spacetime reduces to the static Levi-Civita spacetime, it imposes through its matching conditions, that the cylindrical source must be static 1 . We recall that a collapsing cylindrical shear free fluid, if it is isotropic, reduces to the conformally flat Robertson-Walker spacetime. Here, we study static conformally flat solutions to an anisotropic fluid distribution bearing a non-zero cosmological constant and the possibility of matching them to the exterior LT spacetime. The plan of the paper is as follows. In Section 2, we present the field equations for static anisotropic sources with non zero cosmological constant. In Section 3, the matching conditions for the interior static anisotropic fluid to the LT exterior spacetime are given. Section 4, is devoted to conformally flat solutions. The matching conditions when the interior spacetime is conformally flat and their consequences are analyzed in Section 5. We finish the paper with a conclusion suggesting a physical justification to our matching results. We use latin indices a, b, ... = 0 , 1 , 2 , 3 and use units such that the speed of light c = 1.", "pages": [ 2, 3 ] }, { "title": "2 Static cylindrically symmetric anisotropic sources with Λ = 0", "content": "/negationslash We consider a static cylindrically symmetric anisotropic fluid bounded by a cylindrical surface S and with energy momentum tensor given by where µ is the energy density, P r , P z and P φ are the principal stresses and V a , S a and K a satisfy We assume for the interior to S the general static cylindrically symmetric metric which can be written where A , B and C are C 2 -functions of r . To represent cylindrical symmetry, we impose the following ranges on the coordinates and φ = 2 π is identified with φ = 0. We number the coordinates x 0 = t , x 1 = r , x 2 = z and x 3 = φ and we choose the fluid being at rest in this coordinate system, hence from (2) and (3) we have For the Einstein field equations, G ab = κT ab -Λ g ab , where Λ is the cosmological constant, with (1), (3) and (5) we have the non zero components, where ¯ µ = µ +Λ /κ , ¯ P r = P r -Λ /κ , ¯ P z = P z -Λ /κ , ¯ P φ = P φ -Λ /κ and the primes stand for differentiation with respect to r . The extension of the expression for the mass of an isolated system proposed by Tolman [30] and Whittaker [31] to a non isolated system, bearing cylindrical symmetry, has been obtained by Israel [32]. Other proposals for the mass per unit length exist, like by Marder [33] and Vishveshwara and Winicour [34], but they proved to do not reproduce the expected Newtonian limit [4], while Israel's does. For this reason we use here the expression for mass per unit length obtained by Israel, which is where g is the determinant of the metric. Substituting (3) and (6)-(9) into (10) one obtains and by considering the following regularity conditions on the axis [35] equation (11), at r = r S , becomes where S = denotes equality on S . Since we are concerned with conformally flat sources for the LT spacetime, we need in the sequel the square of the magnitude of the Weyl tensor C 2 = C abcd C abcd , which can be written with the aid of the field equations (6)-(9) as where", "pages": [ 3, 4, 5 ] }, { "title": "3 LT spacetime and matching conditions", "content": "In this section, we match the interior spacetime, bounded by the surface S and given by the metric (3), to an exterior described by the LT spacetime containing the cosmological constant. The generalized static cylindrically symmetric Levi-Civita metric with non zero Λ, given in its usual form by the LT metric [16, 17] is where Σ = 1 -2 σ +4 σ 2 , and for Λ < 0, with and a , b , c and σ ≥ 0 are real constants. The case Λ > 0 is obtained by replacing the hyperbolic functions by trigonometric ones [16, 17]. The coordinates t , z and φ in (16) can be taken the same as in (3) and with the same ranges (4). The radial coordinates r and ρ are not necessarily continuous on S as we see below by applying the junction conditions. The constants a and b can be removed by scale transformations (although we don't do this ahead in order to use these constants as free parameters for the matching), while c cannot be transformed away if we want to preserve the range of φ . The constant σ represents the Newtonian mass per unit length. Following Darmois junction conditions [36] we impose that, on the surface S , the first and second fundamental forms which S inherits from the interior metric (3) and from the exterior metric (16) are equal, hence we obtain the following two sets of equations on S , By replacing the matching conditions (19)-(24) in (7) we get, for both cases Λ < 0 and Λ > 0, that and 2 as expected. The mass per unit length (13) with (19)-(22) and considering the gravitational coupling constant G = 1, then κ = 8 π , can be written as where the mass per unit length for the Levi-Civita metric, with Λ = 0, is thus showing that the presence of Λ < 0 increases the mass per unit length. However, for Λ > 0 we obtain producing an opposite effect, diminishing the mass per unit length. In the Conclusion we consider these results as a possible justification for the possibility, or impossibility, of matching conformally flat interior spacetimes to LT exteriors.", "pages": [ 5, 6, 7 ] }, { "title": "4 Conformally flat interior sources", "content": "The conformally flat spacetime solution, where all Weyl tensor components vanish, C abcd = 0, for (3) with the regularity conditions (12) satisfied produces [23] /negationslash /negationslash where a 1 = 0 and a 2 = 0 are integration constants, and by rescaling t we can assume a 1 = 1. The interpretation of a 2 can be given in the following way. From (29) and (30) we can write and with (14) and (15) it follows, producing At the centre of the source, r = 0, considering the regularity conditions (12) we have from (35)", "pages": [ 7, 8 ] }, { "title": "5 Interior static conformally spacetime matched to exterior LT spacetime", "content": "We start by considering the matching on S for Λ < 0. Then, (32) with (23) and (24) becomes From the equality of the interior and exterior first fundamental forms on S we have B 2 dr 2 S = dρ 2 which, using (20), leads to the relation with b > 0. Then, using the equality at the boundary S, (37) becomes Since the left hand side of (40) is always bigger than 1 this condition can never be satisfied. When Λ = 0, equation (40) reduces to obtained in [23] for the case of a Levi-Civita exterior, which again shows the impossibility of matching a cylindrical conformally flat interior spacetime to a Levi-Civita exterior. Then we can state the following: It is impossible to match any conformally flat static cylindrically symmetric interior spacetime (29) and (30) satisfying the regularity conditions (12) to an exterior LT spacetime, with Λ < 0 , or to an exterior Levi-Civita spacetime, with Λ = 0 , across a timelike cylindrical hypersurface S . For Λ > 0, the corresponding equation to (40) becomes and since the left hand side is less than 1 it does not discard, a priori, conformally flat sources matched to the LT spacetime with Λ > 0. Now, we give simple examples 3 for which a conformally flat interior source can be matched to an exterior LT spacetime with Λ > 0. In this case, the solution of (6)-(9) with (29) and (30) can be easily demonstrated to be and By matching this solution on S to the exterior LT spacetime we have from (25), reducing the interior solution to the Einstein static universe. The junction conditions (19)-(24) for Λ > 0 and (43) become From (49)-(51) we have a 2 2 = Λ, as can be obtained too from (36), and where 0 ≤ sin R S ≤ 1 and 0 ≤ sinh( √ Λ r S ) < ∞ are satisfied by 0 < σ < 1 / 4. While (46)-(48) with (52) define the exterior parameters a , b and c in terms of Λ and σ . Hence, it is possible to match a conformally flat static cylindrically symmetric interior spacetime (29) and (30), satisfying regularity conditions (12), to an exterior LT spacetime with Λ > 0, across a timelike cylindrical hypersurface S . We call attention to the fact that the LT spacetime with Λ > 0 has, besides the singularity at ρ = 0 where we placed the source, another singularity at ρ = π/ √ 3Λ where another source has to be placed. In that case, the matching is still possible by substituting the cylindrical region by a toroidal one following the methods of [19].", "pages": [ 8, 9, 10 ] }, { "title": "5.2 ¯ P r = ¯ P φ", "content": "In this case, the solution of (6)-(9) with (29) and (30) is and and /negationslash where a 4 = 0 is a constant. We note that if a 4 = 1, the function B corresponds to the solution (43). The density and pressures have the following form In this case, the matching conditions (19)-(24) read From (60)-(62) we obtain and as well as (which also follows from (25)) The inequality 0 ≤ sin 2 R S ≤ 1 in (63) and the positivity of the right hand side of (65), for any 0 < σ < 1 / 4, are satisfied if 1 / 2 ≤ a 4 ≤ 1. We conclude that, in this case, the matching is possible in the following sense: For any 1 / 2 ≤ a 4 ≤ 1, 0 < σ < 1 / 4 and Λ > 0, the parameter a 2 is fixed by (65) while ρ S and r S are determined from (63) and (64). In turn, (57)-(59) fix the exterior parameters a, b and c . If a 4 = 1, this solution reduces to the example of the previous section.", "pages": [ 10, 11, 12 ] }, { "title": "6 Conclusion", "content": "The main result obtained here is that it is not possible to match a static interior cylindrically symmetric conformally flat spacetime smoothly across a cylindrical surface to an exterior given by the LT spacetime, when Λ < 0, or, by the Levi-Civita spacetime when Λ = 0. For Λ > 0, it is possible to perform such matching as we showed with two examples. We also showed, that the mass per unit length is increased by the presence of Λ < 0, while it is diminished by Λ > 0. The Levi-Civita spacetime does not possess any horizons, which may seem to indicate, according to our understanding of black hole formation, that there is an upper limit allowed by the mass per unit length. This limit is always below the critical linear mass above which horizons may be formed [4, 37]. The fact that conformally flat spacetimes cannot be matched to Levi-Civita might be physically justified from the fact that sources producing these spacetimes have linear masses higher than this limit. If this is the case, then the inclusion of Λ < 0 would further unbalance this limit since, from (26), the mass per umit length would be further increased. On the other hand, for Λ > 0, we see that the matching is possible and, from (28), the corresponding mass per unit length is diminished as compared to the LeviCivita linear mass. This fact might suggest that, in this case, the linear mass is sufficiently diminished as compared to the critical mass limit.", "pages": [ 12 ] }, { "title": "Acknowledgments", "content": "We thank the referees for useful criticisms. IB and FM thank CMAT, Univ. Minho, for support through the FEDER Funds - 'Programa Op- eracional Factores de Competitividade COMPETE' and FCT Project EstC/MAT/UI0013/2011. FM is supported by FCT projects PTDC/MAT/108921/2008 and CERN/FP/116377/2010. MFAdaSilva acknowledges the financial support from FAPERJ (no. E-26/171.754/2000, E-26/171.533.2002, E-26/170.951/2006, E-26/110.432/2009 and E-26/111.714/2010), Conselho Nacional de Desenvolvimento Cient'ıfico e Tecnol'ogico - CNPq - Brazil (no. 450572/2009-9, 301973/2009-1 and 477268/2010-2) and Financiadora de Estudos e Projetos - FINEP - Brazil.", "pages": [ 12, 13 ] } ]

[ { "title": "ON UNITARY SUBSECTORS OF POLYCRITICAL GRAVITIES", "content": "Axel Kleinschmidt ∗/diamondmath , Teake Nutma ∗ , Amitabh Virmani ∗ ∗ Max-Planck-Institut fur Gravitationsphysik (Albert Einstein Institut) Am Muhlenberg 1, 14476 Golm, Germany /diamondmath International Solvay Institutes Campus Plaine C.P. 231, Boulevard du Triomphe, 1050 Bruxelles, Belgium { axel.kleinschmidt, teake.nutma, amitabh.virmani } @aei.mpg.de", "pages": [ 1 ] }, { "title": "Abstract", "content": "We study higher-derivative gravity theories in arbitrary space-time dimension d with a cosmological constant at their maximally critical points where the masses of all linearized perturbations vanish. These theories have been conjectured to be dual to logarithmic conformal field theories in the ( d -1)-dimensional boundary of an AdS solution. We determine the structure of the linearized perturbations and their boundary fall-off behaviour. The linearized modes exhibit the expected Jordan block structure and their inner products are shown to be those of a nonunitary theory. We demonstrate the existence of consistent unitary truncations of the polycritical gravity theory at the linearized level for odd rank.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The perturbative properties of ordinary general relativity in d = 4 space-time dimensions can be improved by adding higher derivative terms to the action. The price one has to pay for rendering the theory renormalizable in this way is typically the loss of unitarity [1, 2]. Recently, specific models in d ≥ 3 with special choices of higher derivative terms have attracted renewed attention for several reasons. One is that in d = 3 they can provide consistent ghostfree theories of massive gravitons. This was first observed in the parity violating 'topologically massive theory of gravity' (TMG) [3, 4] with three derivatives and more recently for the parity preserving 'new massive gravity' (NMG) [5] with four derivatives. A crucial feature in the construction of NMG is the choice of coefficients in the four-derivative Lagrangian such that the problematic scalar mode of the massive graviton becomes pure gauge. Furthermore, there is a critical point where the mass of the massive graviton vanishes and degenerates with that of the massless graviton. Both features were later extended to higher dimensions by the discovery of 'critical gravity' theories with four derivatives [6-11]. At the critical points one typically encounters logarithmic graviton modes that emerge as the replacement for the massive modes. These theories importantly have a non-vanishing cosmological constant. Similar parity preserving theories now also exist in arbitrary dimension and with an arbitrary (even) number of space-time derivatives and critical points, the so-called polycritical gravities [12]. (For other work on massive gravity see [13-19].) Another reason for studying polycritical models is provided by the AdS/CFT correspondence where one would expect a non-unitary logarithmic CFT as the dual of a polycritical gravity theory [8, 20, 21] (see also [22, 23]). The non-unitarity of the logarithmic CFT is related to the fact that the Hamiltonian cannot be diagonalized on the fields; there is a Jordan structure [24, 25]. However, the precise structure of the two-point correlation functions suggests the existence of unitary truncations, and by AdS/CFT also in the gravity theory [21]. The example of six-derivative gravity in d = 3 was treated recently in [26] whereas fourderivative critical gravity in d = 4 appeared in [27]. To explore this question further, the present paper analyzes the structure of the various gravitational modes in polycritical gravity in space-time dimensions d ≥ 3 at the linear level. We find that an inner product can be defined that reproduces the structure expected from logarithmic CFTs. The linearized graviton excitations around an AdS background can be organized into a hierarchy of higher and higher logarithmic dependence near the boundary of AdS. The lowest mode is the usual Einstein mode, the next one has an additional logarithmic dependence on the AdS radius, the next one contains log 2 terms and so on. This allows us to truncate the linearized theory by imposing appropriate boundary conditions on the graviton fall-off behaviour. A suitable truncation then renders the inner product matrix between the various modes positive semi-definite. The null states can also be factored out, but the resulting theory is quite different depending on the rank of the polycritical gravity theory. The rank is defined as half the maximum number of space-time derivatives. When the rank is odd, one arrives at a unitary model of a single graviton mode. By contrast, the theory becomes trivial for even rank; the surviving mode has zero energy. This confirms a conjecture of [21]. An alternative description of this truncation can be given by defining a hierarchy of (conserved) charges and then restricting to a superselection sector in this charge hierarchy. While this paper was being completed, the preprint [28] appeared that discusses the specific case of non-linear critical gravity of rank 3 in d = 3 and d = 4 with the result that truncations that appear to be unitary at the linearized level may be inconsistent at the non-linear level, i.e., the truncation is flawed by a linearization instability. The argument given there seems to extend to the general case independently of how the linearized theory is completed and this would suggest that our unitary subsectors exist only in the linearized approximation. Our paper is structured as follows. In section 2, we give the Lagrangian of the polycritical theory around AdS space whose various modes will be obtained in section 3. Then in section 4 we define and compute the inner product for these modes. Using either the hierarchy of charges established in section 5 or appropriate boundary conditions, we will be able to define a unitary truncation of the polycritical model in section 6. An appendix shows that our inner product is equivalent to one derived canonically from a two-derivative master action.", "pages": [ 2, 3 ] }, { "title": "2 Quadratic Lagrangian", "content": "In this section we briefly review the quadratic Lagrangian around AdS space of polycritical models of arbitrary rank. But before doing so, it is useful to first go over the rank one (i.e. two derivative) case: Einstein gravity with a cosmological constant.", "pages": [ 3 ] }, { "title": "2.1 Rank one: Einstein gravity", "content": "Recall that for Einstein gravity with a cosmological constant, we have the Lagrangian The equations of motion state that the cosmological Einstein tensor (that is, the Einstein tensor plus a term proportional to the cosmological constant) vanishes, We will perform perturbations around solutions of the equations of motion as follows, The bar indicates the background solution, and the superscript L the linear perturbations around it. Thus the linear perturbation of the metric is given by h µν . We take the background solution to be an AdS space, which means that the curvature tensors satisfy with d being the number of space-time dimensions and Λ < 0. Instead of the cosmological constant, we can also use the AdS length /lscript as a measure for the background curvature. The two are related via Note that (4d) indeed solves the equations of motion (2). On this background, the linear equations of motion become with Taking the trace of the linear equation of motion (6) is the same as linearizing the trace of the non-linear equation of motion (2), because the cosmological Einstein tensor vanishes by construction on the background. Either way, we find Furthermore, the linear equations of motion (6) have a gauge invariance that stems from the diffeomorphism invariance of the non-linear theory. To be precise, they are invariant under the gauge transformation for any vector v µ . This gauge invariance, combined with the on-shell vanishing of the linearized Ricci scalar R L , implies [29] that we can go to the so-called 'transverse traceless' gauge, This gauge eliminates the scalar mode (that would otherwise be a ghost) of h µν , making it a proper spin-2 field. 1 In the transverse traceless gauge, the linearized equation of motion (6) simplifies considerably to ( ) Lastly, the linear equations of motion (6) can also be obtained from the quadratic perturbation of the Lagrangian (1), which, after partial integration, reads The term 2 /lscript -2 may look like a mass term, but it is not. Mass terms in general break gauge invariance, but the linearized equations of motion were in fact gauge invariant. Instead, if one were to introduce a mass for the spin-2 field, its equation of motion would read ¯ +2 /lscript -2 -m 2 h µν = 0, with m being the proper mass parameter. Indeed, upon varying this quadratic action with respect to h µν we recover (6).", "pages": [ 4, 5 ] }, { "title": "2.2 Einstein and Schouten operators", "content": "The fact that the Lagrangian (13) is quadratic in h µν is obscured as the linear Einstein tensor ( G Λ µν ) L also contains h µν . We can make the quadratic dependence a bit more transparent by introducing the so-called Einstein operator G , upon which the Lagrangian reads The (cosmological) Einstein operator G is defined as Here and in the following we have suppressed the indices on G . But it is in fact a tensorial operator, so when we write G h µν we implicitly mean G µν ρσ h ρσ . Reading off from equation (6), the explicit form of the Einstein operator is The Einstein operator has a number of nice properties: Here the symmetric A µν , B µν , and v µ are completely arbitrary. In the following we will also need another operator, the so-called (cosmological) Schouten operator S [12]. It is defined similarly as the Einstein operator, the difference being that it yields the linearized cosmological Schouten tensor when applied to h µν : In turn, the cosmological Schouten tensor is the usual Schouten tensor 2 plus a term proportional to the cosmological constant: The extra term proportional to the cosmological constant is chosen such that the cosmological Schouten tensor vanishes on AdS backgrounds, The linearized cosmological Schouten tensor reads Note that it differs from the linearized cosmological Einstein tensor (6) by a factor of R L : The Schouten operator on its own does not have striking properties: it is not self-adjoint, nor is it conserved. However, in combination with the Einstein operator, things become more interesting: And because G on its own is also self-adjoint, GS k (the k -fold application of S followed by G ) is so too. 2. S can be traded for a cosmological constant when taking the trace: 3. S is gauge invariant: 4. For a symmetric, transverse and traceless tensor (say C µν ), G and S are the same: The first two properties are crucial for constructing a quadratic theory of general rank, which we will do now.", "pages": [ 5, 6, 7 ] }, { "title": "2.3 General rank", "content": "The rank r polycritical Lagrangian around an AdS background is given by with τ = ∏ r -1 i =1 ( m 2 i + d -2 /lscript 2 ) . For rank one, it reduces to the quadratic Einstein Lagrangian (14), as required. The non-linear completion for rank one is unique [30]; it is simply the Einstein-Hilbert Lagrangian (1). For rank two in d = 3 [5] or d = 4 [7] the non-linear completion is also unique, because the number of independent curvature invariants is sufficiently small in those cases. However, for higher rank the quadratic theory no longer uniquely fixes the non-linear theory, due to the growth of curvature invariants. One can still find some non-linear Lagrangian that reproduces the above theory (29) for quadratic perturbations. For d ≥ 4 and arbitrary rank this was done in [12], while [26] has a non-linear action for r = 3, d = 3. However, finding a unitary interacting theory is not so easy [1, 2]. We will content ourselves with knowing one can always write down a non-linear completion. Since GS k is self-adjoint, the equations of motion that follow from (29) are simply Upon taking the trace of this, we find with the help of (26), Note that the use of Schouten operators is crucial in order for the trace to reduce to the linear Ricci scalar. If one were to use only Einstein operators in the action (29), the trace of the equations of motion would not be equal to the linear Ricci scalar. Similarly as in the rank one case, the on-shell vanishing of the linear Ricci scalar allows us to go to the transverse and traceless gauge. The equations of motion then become with m 0 = 0. The theory thus contains r propagating gravitons h [ i ] µν , one of which is always massless while the others can be massive. Such a graviton mode is a solution to a different equation of motion than (32). Instead it is annihilated by a single factor of the product of (32c), h [ i ] µν : ( ¯ +2 /lscript -2 -m 2 i ) h [ i ] µν = 0 , (33) while of course still being transverse and traceless. Their on-shell quasilocal energies can be computed by taking an appropriate integral of the effective stress-energy tensor, as we will explain in more detail in subsection 4.1. For the modes h [ i ] defined above the result is [12] /negationslash Here E 0 is the energy of the massless graviton for rank one, that is, the usual graviton energy in Einstein gravity. If we arrange the masses by size, the sign of the energies alternates: So unfortunately, when the masses are not degenerate some of the gravitons will always be ghosts, no matter how one chooses the overall sign of the action. A notable exception is the d = 3, r = 2 case, NMG [5]. In three dimensions the massless graviton does not propagate, but the massive one does. One can then choose the overall sign of the action such that massive graviton has positive energy and is not a ghost. However, for generic dimensions and rank, such a thing is not possible. In an attempt to ameliorate the situation, one can send all the masses to zero, thereby reaching the polycritical point .", "pages": [ 7, 8 ] }, { "title": "2.4 Polycritical point", "content": "We define the maximally polycritical point to be the point in parameter space where all the masses are zero. The Lagrangian (29) then reads and the equations of motion that follow from it are We know from before that this allows us to let h µν be transverse and traceless. In this gauge, the Schouten and Einstein operator become the same, and the remaining equations of motion read It is worth stressing that the above is not the correct complete equation of motion; one must take into account that h µν is already transverse and traceless. If this was not the case we would have to go back to (38). At the polycritical point the r -1 massive modes degenerate with the massless mode into a single mode. In addition r -1 new modes appear, the so-called log modes h ( I ) µν (with I = 1 , . . . , r -1). These log modes satisfy different equations of motion than the massless mode; they are annihilated by two or more Einstein operators: /negationslash with I = 0 , 1 , . . . , r -1. Note that h (0) µν is the usual massless graviton. The action of a single Einstein operator on a log mode gives a 'lower' log mode, If we use the convention h ( -1) µν = 0 then (40) reproduces the equations of motion. The main aim of this paper is to analyse the inner product of the log modes and study the unitarity of the linearized theory. Next, we will examine explicit solutions to the equations of motion of the log modes.", "pages": [ 8, 9 ] }, { "title": "3 Linearized log modes", "content": "In this section we explicitly present modes of the linearized equations of motion of the polycritical gravities at their maximally polycritical point. We first recall certain basic facts about constructing such modes from [31, 32] and then give some explicit expressions. A similar analysis has been recently performed by other authors [28, 32, 33]. To construct log modes, one first constructs massive as well as massless transverse traceless spin-2 modes in terms of the highest weight representations of the symmetry group SO(2 , d -1) of AdS d spacetime. From the highest weight states all other states are obtained by acting with the negative root generators of the algebra. Recall that the mass parameter for spin-2 modes is defined as in (33), therefore a general transverse traceless massive spin-2 mode ψ µν satisfies In higher derivative theories at the maximally polycritical point the equations of motion read (39) in the transverse traceless gauge. At the critical points log modes emerge that satisfy different equations of motion (40). It has been previously observed [22, 32, 33] that the highest weight log modes are related to the corresponding massless mode. The relation is through an overall factor. For the log mode h ( I ) µν of index I , the factor is a polynomial of order I in a function f . To introduce the function f , we first introduce global coordinates on AdS d in which the metric of AdS d takes the form ds 2 = /lscript 2 ( -cosh 2 ρdτ 2 + dρ 2 +sinh 2 ρd Ω 2 d -2 ) , (43) where d Ω 2 d -2 is the unit metric on the round d -2 sphere. In terms of the coordinates τ and ρ the function f takes the form The highest weight massless spin-2 mode ψ µν in general dimension d is constructed in [33]. We will present explicit expressions for ψ µν in four-dimensions in subsection 6.2. The massless spin-2 mode ψ µν is exactly the mode h (0) µν of the preceding section. We observe the following properties of the function f and of the highest weight massless spin-2 mode ψ µν in general dimension d , From these equations it follows that where I = 0 , . . . , r -1. As a result, Using the above equations one can easily find log modes in the basis (41). The first few modes are To find a general expression for the index I mode, one needs to solve a combinatorial equation. This equation can perhaps be solved explicitly, but the resulting expressions will not be illuminating. Instead one can develop a recursive algorithm to find log modes that solves (41) to whatever order one wants. This is how we have obtained (48).", "pages": [ 9, 10 ] }, { "title": "4 Quasilocal energies and the inner product", "content": "In this section we define an inner product on the (log) solutions. The first step is to define a bilinear norm 〈·|·〉 by means of the quasilocal energy of a solution. Once we have such a norm, the inner product between two states follows readily. It turns out that the formulas for the generic case involve the inner product of Einstein-Hilbert gravity. So, as a starting point, let us see how things work for rank one.", "pages": [ 11 ] }, { "title": "4.1 Rank one", "content": "The on-shell energy of a solution of the equations of motion can be computed by where ¯ ξ ν is a time-like Killing vector, n µ is the normal to the Cauchy surface that is being integrated over. The effective stress-energy tensor T µν is given by varying the action w.r.t. the background metric as if it were dynamical [34], In computing the on-shell stress-energy tensor, one has to be a bit careful in first taking the variation and then going on-shell, because the on-shell Lagrangian vanishes. For the Einstein-Hilbert case (14), the on-shell stress-energy tensor can be written as where we have used the short-hand notation δ G δ ¯ g µν h ρσ = δ δ ¯ g µν ( G h ρσ ) . There would have been other contributions, such as the variation w.r.t. the background metrics that contract h ρσ and G h ρσ , but since they are proportional to the equations of motion they vanish on-shell. The complete form of the energy of a mode h µν is thus For physical excitations on AdS spaces this yields a real number. We can use the above expression to define a norm: We have dropped the indices on the fields in the norm 〈·|·〉 in order to simplify notation, but it is to be understood that we mean the 'full' tensor h µν in the above, not its trace. The norm defines an inner product on the space of solutions as follows: Because the expression h ρσ δ G δ ¯ g µν k ρσ is not obviously symmetric in h and k , we have kept the symmetrization in the above formula for clarity.", "pages": [ 11 ] }, { "title": "4.2 General rank", "content": "For general rank one can define an inner product on the space of solutions in a similar manner as for Einstein gravity. At the polycritical point, the effective stress-energy tensor is Here we have used the fact that when acting on transverse and traceless tensors, the Einstein and Schouten operator are the same. Furthermore we made use of their variation w.r.t. the background metric being identical after going on-shell: ∣ The norm for rank r , 〈·|·〉 r , then becomes where 〈·|·〉 1 is the norm of the rank one (Einstein-Hilbert) case, as calculated in the last section. This leads to the following inner product on the log modes: Since h ( -1) µν = 0, the inner product between log modes h ( I ) µν and h ( J ) µν vanishes if I + J < r -1. Furthermore, we can relate the inner product for rank r to the inner product at lower rank if I > J : ∣ In the first line we used the fact that h ( J -r +1) = 0 for J < I ≤ r -1, and in the second we relabeled the summation index. This allows us to inductively compute the complete inner product matrix for generic rank starting from rank 1. The only new bit of information at every step is 〈 h ( r -1) ∣ ∣ ∣ h ( r -1) 〉 r , which is the energy of the maximal log mode at the given rank. We denote this quantity by E r -1 : The inner product matrix for generic rank then reads In Appendix A we show that the structure of this inner product is same as the one derived canonically from a two-derivative master action. Since E 0 is the energy of the massless graviton in Einstein gravity, it is a positive number. In four dimensions E 1 was also found to be positive [7]. Based on the explicit solutions of the log modes (48) and the form of the inner product, we expect all norms to be non-zero. The inner product matrix is indefinite. Regardless of the exact values of the energies E I , one can always find linear combinations whose energies have opposite sign. One example is ∣ ∣ Thus one of the above modes is a ghost, implying that the untruncated linear theory is not unitary. It is possible, however, to truncate some of the log modes such that the resulting submatrix of (61) is semi-positive definite. After we have developed some of the necessary machinery in the next section, we will demonstrate this in section 6.", "pages": [ 12, 13 ] }, { "title": "5 Conserved charges", "content": "We now turn to the construction of conserved charges in these models following the method of Abbott-Deser for asymptotically AdS spaces [35-37]. It will turn out that one can define an extended hierarchy of charges that makes reference to the hierarchy of graviton modes defined above.", "pages": [ 13 ] }, { "title": "5.1 Abbott-Deser charge", "content": "The method of [35-37] was applied to polycritical theories in [12] and makes use of the split of the metric into background and perturbation g µν = ¯ g µν + h µν , where h µν does not need to be small. In the formalism, one splits the equations of motion for h µν into linear terms and nonlinear terms. The non-linear terms are then interpreted as an effective energy-momentum tensor T µν that, by the equations of motion, can be equivalently expressed linearly in h µν on-shell. T µν is then conserved by the linearised equations of motion. From T µν one can define a conserved current by in terms of a time-like Killing vector ¯ ξ ν . Writing this current in terms of a divergence is possible by virtue of ¯ ∇ µ J µ = 0 and leads to where F µν = F [ µν ] . In our specific case we have [12] where F ¯ ξ is an operation that creates an antisymmetric tensor out of a symmetric one. It is constructed in such a way that its derivative relates to the Einstein operator by when acting on any symmetric tensor A µν . This ensures that which vanishes by the linearized equations of motion (39). The conserved Abbott-Deser (AD) charge is then given (up to normalization) by the integral at infinity via", "pages": [ 13, 14 ] }, { "title": "5.2 Hierarchy of charges", "content": "We can define a more refined object by considering the following generalization of (65) A generalized current can be defined as the divergence of this antisymmetric tensor via For I = r this gives the conserved Abbott-Deser current (64): J µ ( r ) ≡ J µ . For I < r , the divergence of this current does not vanish in general and hence the current is not conserved on the full space of solutions of the linearized theory. But we see from the definition of J µ ( I ) that it is proportional to the equations of motion for a lower log graviton mode. Explicitly, for a transverse traceless mode since the Einstein and Schouten operators then coincide. Hence, by virtue of the definition of the various graviton modes in (40), we find that J µ ( I ) = 0 when evaluated for modes h ( K ) µν with K < I . So, if one restricts to the part of the space of solutions spanned by modes h ( K ) µν with K < I , the current J µ ( I ) is conserved and can be used to define a conserved charge Q ( I ) on that subspace. Hence, the AD charge Q of (69) is equal to Q ( r ) . Whether a given 'charge' Q ( I ) vanishes or not can be calculated explicitly for the various modes h ( I ) µν . We find the distribution of charges displayed in Table 1. In that table we use 'n.d.' to indicate when a certain charge is not well-defined for a given mode. It is easy to see that the only mode that has a non-vanishing charge Q ( I -1) is the mode h ( I ) , all others have vanishing charge (if it is defined). This means that the only mode that has non-vanishing AD charge Q ( r ) is the highest log-mode h ( r -1) µν . · · ·", "pages": [ 14, 15 ] }, { "title": "6 Unitary subsectors", "content": "In this section, we discuss possible unitary truncations of the polycritical theory at the linearized level. As anticipated in [21] we will find a difference between odd and even rank.", "pages": [ 15 ] }, { "title": "6.1 Truncation by superselection", "content": "Looking at Table 1 one sees that one can consistently truncate to superselection sectors by demanding that certain charges vanish. Starting with the AD-charge Q ( r ) = 0 we truncate out the mode h ( r -1) ∼ log r -1 . In the linearized approximation this truncated sector is dynamically closed. Furthermore, the 'charge' Q ( r -1) becomes a perfectly well-defined conserved charge in this truncated model. We can repeat the superselection now to further truncate consistently to Q ( r -1) = 0. One would like to continue the truncation until one obtains a standard positive semidefinite inner product matrix from (61). Every new step in the truncation corresponds to removing the last row and column of the inner product matrix. For even rank this leads to removing all the modes h ( r/ 2) , . . . , h ( r -1) ; the resulting inner product matrix is identically zero and the theory has become trivial. This was already observed in the rank r = 2 case (four derivatives) in [7, 9]. For odd rank, a truncation to a theory with positive semi-definite two-point functions can be achieved by restricting to the sector Q ( r ) = . . . = Q ( r +1 2 +1) = 0. In this case the inner product matrix becomes almost identically zero except for one standard correlator in the lower right-hand corner. This is the structure of a theory with many null states that need to be quotiented out. The resulting theory then is that of a single mode h ( r -1 2 ) (defined up to the definition of lower log-modes). This mode has the standard correlator and positive energy. It appears to be the same model as standard Einstein gravity in the linearized approximation. For low rank, the truncated models (before removing unphysical null states) have the following inner product matrices: Factoring out the null states (that decouple) one is then left with a standard CFT for odd rank, identical to that of Einstein-Hilbert gravity, but this time for the log ( r -1) / 2 -mode. For even rank, the theory trivializes completely.", "pages": [ 15, 16 ] }, { "title": "6.2 Truncation by boundary condition", "content": "In this subsection we discuss fall-offs of various modes, and truncation to a unitary subsector from the point of view of boundary conditions at spatial infinity. For concreteness we work with explicit coordinates and we write expression only in four dimensions. We expect our considerations to apply more generally. Let us introduce global coordinates in AdS 4 in which the metric of AdS 4 takes the form In these coordinates explicit expressions for the various components of the massless spin-2 highest weight mode are [32] Physical excitations correspond to real or imaginary part of these modes. Let us now introduce another set of global coordinates r and t as In these coordinates the AdS metric takes the form The boundary of AdS space lies at r →∞ , or equivalently, ρ →∞ . In these new coordinates it is fairly easy to check that the mode (75) satisfies the Henneaux-Teitelboim boundary conditions [38]. In particular, we have In fact, except for the rr component these fall-offs saturate the Henneaux-Teitelboim boundary conditions. This is expected: since we are working in the transverse traceless gauge we do not expect to reproduce all fall offs of [38], but we do expect that the fall-offs to be strong enough for the linearized mode (75) to be contained in the phase space defined by those boundary conditions. We schematically denote a mode saturating the Henneaux-Teitelboim boundary conditions as ψ HT . The index I log mode h ( I ) µν behaves asymptotically as This is simply because the function f behaves asymptotically as f ∼ -log r . Thus, if for a rank r polycritical theory one imposes boundary conditions such that, then one clearly truncates away the highest logarithmic mode, that is, log r -1 rψ . One can also choose to impose a stronger boundary condition to truncate way more logarithmic modes. In this way one can continue truncating away higher index logarithmic modes until one arrives at the boundary conditions where one obtains a standard positive semi-definite inner product matrix (61). This happens for a rank r theory when one removes the highest /floorleft r 2 /floorright log modes. This can be done by imposing boundary conditions At this stage it becomes quite clear that the discussion about the choice of boundary conditions exactly parallels the discussion of the previous subsection based on the superselection sector. Note that in comparison to the corresponding three-dimensional discussion [26], our fourdimensional discussion of boundary conditions is rather schematic. In three-dimensions there are other independent studies checking the consistency of log- [23, 39, 40] and log 2 - [41] boundary conditions. Similar studies do not yet exist for the four- and higher-dimensional settings.", "pages": [ 16, 17 ] }, { "title": "7 Discussion and conclusions", "content": "In this paper we have analyzed the structure of polycritical gravity of rank r in d spacetime dimensions at the linear level. We found that the r different graviton modes on an AdS background satisfy a hierarchical structure h ( I ) ∼ log I r in terms of the AdS radius r . Their inner products can be calculated by using quasilocal energies and the inner product matrix was found to exhibit a specific triangular structure as expected from a putative dual logarithmic CFT description. In particular, the inner product matrix is indefinite, reflecting the non-unitary structure of the theory. Following the method of Abbott and Deser one can define a conserved charge in these models and only the highest logarithmic mode h ( r -1) carries a non-vanishing charge. Restricting to a superselection sector where this charge vanishes -or equivalently modes that have faster fall-off near the boundary- one can truncate the model. As we showed, this process can be iterated until one ends up in a truncated model with positive semi-definite inner product matrix. After modding out the null states one is then left with a unitary model of a single graviton mode in the odd rank case. For even rank, the truncated model trivializes completely. In either case, the truncated model is unitary at the linear level. The correlator of the single remaining mode in the odd rank case is non-trivial and identical to that of the Einstein mode in usual two-derivative general relativity theory. This raises the question whether our truncated model is nothing but a reformulation of standard general relativity, albeit a rather complicated one. Since one has to impose appropriate boundary conditions on the graviton modes to implement the truncation, this idea is reminiscent of the proposal of [42] to obtain Einstein gravity from a conformal higher-derivative gravity theory with appropriate boundary conditions. However, since our truncation will probably not remain unitary and consistent when embedded in a non-linear theory [28], it appears to be impossible that our model is equivalent with general relativity.", "pages": [ 18 ] }, { "title": "Acknowledgements", "content": "We would like to thank E. A. Bergshoeff, G. Comp'ere, S. Fredenhagen, S. de Haan, M. Kohn and I. Melnikov for useful discussions.", "pages": [ 18 ] }, { "title": "A Master action and the other inner product", "content": "In this appendix we relate our inner product to the symplectic inner product derived from a two-derivative action. To this end we first observe that the equations of motion (38) can also be obtained from an auxiliary field action, which we call the 'master action.' It only contains two derivatives but it has r -1 auxiliary fields k ( I ) µν . It takes the form with I, J = 0 , . . . , r -1 and the symmetric matrices A and B are given by The equations of motion for the field k ( r -1 -I ) µν read from which we have, after taking the trace, The complete set of equations of motion become /negationslash Upon eliminating the r -1 auxiliary fields k ( I = r -1) µν , and calling k ( r -1) µν to be h µν , we recover the original equations of motion (38). For the rest of the discussion we take the k ( I ) s to be transverse and traceless, which follows from the equations of motion. This allows us to freely replace Schouten operators on k ( I ) with Einstein operators. For the two derivative master action (82) the symplectic inner product can be simply computed following the formalism reviewed in [11]. Up to an over-all normalization, the symplectic inner product for a rank r theory takes the form where ψ ( I ) and φ ( I ) are the auxiliary field configurations associated with the configurations ψ and φ respectively. The integration is done over a constant τ Cauchy surface Σ and the index 0 denotes the time components of the various tensors in the coordinates (43). Furthermore, to distinguish this inner product from that of section 4 we use the double line notation 〈 ψ || φ 〉 r . The subscript r denotes the rank of the theory. From expression (88) and from the form of the matrix A IJ it follows that where in going from the first to the second line we have used (87), in going from the second to the third line we have used the notation 〈 ψ || φ 〉 1 which denotes the (appropriately normalized) symplectic inner product for the rank one theory, and finally in going from the third to the fourth line we have renamed the dummy variable I to j = I +1. From this last equality and equation (57) we immediately see that the two inner products give rise to the identical matrix structure over log modes.", "pages": [ 18, 19, 20 ] } ]

[ { "title": "Wahlquist's metric versus an approximate solution with the same equation of state", "content": "J. E. Cuch'ı, 1 J. Mart'ın, 1 A. Molina, 2 and E. Ruiz 1 1 Dpto. F'ısica Fundamental, Universidad de Salamanca ∗ 2 Dpt. F'ısica Fonamental, Institut de Ci'encies del Cosmos, Universitat de Barcelona † We compare an approximation of the singularity-free Wahlquist exact solution with a stationary and axisymmetric metric for a rigidly rotating perfect fluid with the equation of state µ +3 p = µ 0 , a sub-case of a global approximate metric obtained recently by some of us. We see that to have a fluid with vanishing twist vector everywhere in Wahlquist's metric the only option is to let its parameter r 0 → 0 and using this in the comparison allows us in particular to determine the approximate relation between the angular velocity of the fluid in a set of harmonic coordinates and r 0 . Through some coordinate changes we manage to make every component of both approximate metrics equal. In this situation, the free constants of our metric take values that happen to be those needed for it to be of Petrov type D, the last condition that this fluid must verify to give rise to the Wahlquist solution. PACS numbers: 04.25.Nx, 04.40.Dg Keywords: Wahlquist, approximate, post-Mikowskian, CMMR, rotating stars, Petrov type, stellar models", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "There are a few exact solutions of the Einstein equations describing the gravitational field inside a stationary and axisymmetric rotating perfect fluid, the basic candidates to form a stellar model in General Relativity [1, 2]. Among them, only one is known to admit a spheroidal closed surface of zero pressure, the key component to build a stellar model matching the interior (source) spacetime with a suitable asymptotically flat exterior. It is the Wahlquist metric, that describes a rigidly rotating perfect fluid, possesses the energy densitypressure equation of state (EOS) µ +3 p = µ 0 and has Petrov type D [3]. Nevertheless, it has been shown in several different ways that it can not correspond to an isolated object nor be matched with an asymptotically flat exterior [3-5]. Accordingly, General Relativity still lacks any exact solution that can describe the interior of such stellar model. To find these global models, numerical methods and analytic approximations are therefore the pragmatical way to go. A very influential work for both paths is due to Hartle and Thorne [6, 7]. They show how to build and match an asymptotically flat vacuum exterior to an interior corresponding to a barotropic and uniformly rotating perfect fluid in slow rotation. The scheme perturbs analytically the non-rotating initial configuration obtaining results up to second order in the slow-rotation parameter. Nevertheless, it usually relies in numerical integration to get them and the matching is not as general as it could. Numerical approximations have been very successful in this field, although some of its most modern and precise codes RNS [8], rotstar [9, 10], AKM [11, 12], rotstar-dirac [13]- are inspired by the work of Ostriker and Mark [14]. Fully analytic stellar models on the contrary are quite hard to find; this led some of us to introduce a new approximation scheme in [15, 16] focused in this kind of problem. It is a double approximation. The first one is post-Minkowskian with associated parameter λ , which is related with the strength of the gravitational field and the second one is a slow rotation approximation with parameter Ω, measuring the deformation of the matching surface due to the rotation of the fluid. We have applied this scheme to find an approximate global solution for a fluid with simple barotropic EOS up to order λ 5 / 2 and Ω 3 in the following cases. We have found solutions for constant density [16] and for a polytropic fluid [17] with Lichnerowicz matching conditions [18] and more recently, also for the linear equation of state with both Darmois-Israel [19, 20] and Lichnerowicz matching conditions [21] (hereinafter CGMR) 1 . In particular -and contrarily to what happens for anisotropic fluids [25]- the exact perfect fluid solution in closed form for this EOS is unknown even in the non-rotating case. When n = -2, this EOS becomes the one of the Wahlquist solution. It is worth noting that despite the inherent interest of Wahlquist's metric as an exact solution, our metric is in some sense more general even after fixing n = -2. This comes from the fact that although Wahlquist's is the most general Petrov type D solution for this EOS, symmetries and motion of the fluid [26], our metric can also be of type I. More interestingly, for special values of its constants, our approximate solution, unlike Wahlquist's, can be matched to an asymptotically flat exterior and thus can describe a compact isolated object. Another problem of Wahlquist's solution is its rotation parameter. It is written in corotating coordinates and lacks of any parameter that can be directly related with the angular velocity of the fluid. All that is known is that letting the Wahlquist's r 0 parameter go to zero in a limiting procedure (that involves a coordinate change that is singular when r 0 → 0) we get the static spherically symmetric Whittaker solution [2, 3]. Despite the singular character of this limit, the relation it shows between the Whittaker and Wahlquist metrics seems to be a sound one since in the slow rotation formalism of [27], which is first order in the rotational parameter, the Whittaker-like and Wahlquist-like slowly rotating metrics coincide [28, 29]. The question we try to answer in this paper is whether or not we can include an appropriate approximation of the Wahlquist solution in our family of approximate solutions. There are two ways to answer. One of them is asking our n = -2 solution to be of Petrov type D, since a metric with its characteristics and this Petrov type must belong to the Wahlquist family [26]. The other way is finding a coordinate change to make them to coincide. Regarding the first one, we have already verified that our solution can take Petrov type D in [21]. Some of its free constants are then fixed and we have found their values do not coincide with the ones they are forced to take when we matched our interior solution with an asymptotically flat exterior solution, as one expects. The coordinate change way involves writing our solution in a co-rotating frame, and then making a series expansion of Wahlquist's solution with µ 0 as post-Minkowskian parameter. When the parameter µ 0 tends to zero the Wahlquist solution becomes Minkowski's metric, what shows that µ 0 plays an equivalent role to the one of the parameter λ in our scheme. This way is more meaningful since, in spite of any result we can get from our approximate metric alone, there is always the question of whether our solution really corresponds to a parametric expansion of an exact metric. Working this way we verify explicitly this correspondence. In Section 2 we give some notation and definitions used along the paper and we write the CGMR interior metric for n = -2 and perform the rotation to write our metric in a co-rotating coordinate system. In Section 3 we give the Wahlquist metric, write it in spheroidal-like coordinates and then write the approximate post-Minkowskian Wahlquist metric. Finally, in Section 4 we compare both solutions and determine the value of our constants and the relation between r 0 and the rotation parameter.", "pages": [ 1, 2, 3 ] }, { "title": "2. The approximate interior metric", "content": "We work within the analytical approximation scheme developed in [15, 16] and [21]. It allows to build an approximate stationary and axisymmetric solution of the Einstein's equations for a source spacetime and an asymptotically flat vacuum region around it, although in this paper we put the focus on the interior, source spacetime. This section is devoted to a brief review of its main points in the general formulation. With ξ the time-like Killing vector and ζ the space-like closed-orbits Killing vector associated to the stationarity and axisymmetry, let us choose t and ϕ to be coordinates adapted to ξ and ζ , respectively. Our interior is filled with a perfect fluid without convective motion so we can write its 4-velocity as Here, ψ is a normalization factor and ω the angular velocity of the fluid in this coordinates. The EOS of the fluid is the n = -2 sub-case of the linear one µ + (1 -n ) p = µ 0 already studied in [21], i. e. Integrating the Euler equations ∇ α T α β = 0 with this EOS we get the explicit expressions for the mass density µ and the pressure p in terms of ψ and its value on the p = 0 surface, ψ Σ With this kind of interior, we can choose coordinates { r, θ } spanning the 2-surfaces orthogonal to the ones containing ξ and ζ [30, 31], then we can write the interior -and exteriormetric with the structure in the associated cobasis ω t = dt , ω r = dr , ω θ = r dθ , ω ϕ = r sin θ dϕ . We will require these coordinates to be spherical-like in the sense that they are associated through the usual relations to a set x α = { t, x, y, z } of harmonic coordinates ( glyph[square] x α = 0), a particularly relevant gauge choice [32, 33]. In CGMR we solved the Einstein equations with this coordinate condition using a postMinkowskian expansion for the metric so that g αβ = η αβ + λh (1) αβ + λ 2 h (2) αβ + · · · , with η αβ the flat metric and the approximation parameter (note that here we work in units where 8 πG = 1 so this definition is different from the one in CGMR; r s is the coordinate radius of the surface in the static limit), using a tensor spherical harmonic expansion truncated using a secondary slow rotation approximation with parameter that gives a measure of the deformation of the source. For n = -2, the CGMR interior metric is, up to O ( λ 2 , Ω 3 ) 2 where P l n stands for the associated Legendre polynomials P l n (cos θ ). With the EOS fixed, the interior in CGMR depends on nine free constants. Two of them, r s and ω , are part of the approximation parameters λ, Ω. The other seven are ( m 0 , m 2 , j 1 , j 3 , a 0 , a 2 , b 2 ). These arise from the harmonic expansion we use to solve the homogeneous part of the Einstein equations at each order. Accordingly, they are also series expansions in positive powers of ( λ, Ω). The first four of them are the ones that a Darmois matching fixes and choosing values for them amounts to choosing a 'particular metric' from the CGMR family -although in a strict sense, such particular metric would still be a family of metrics because of the free values of ( λ, Ω)-. The last three parametrize changes between the harmonic coordinates used. Here, to simplify we have taken these purely gauge constants a 0 = 0, a 2 = 0, b 2 = 0 without losing generality because they are not needed hereafter. The static limit (Ω = 0) of CGMR for a certain EOS is characterised with only ( r s , m 0 ) (see Table I). The constant S is defined as This value ψ Σ comes from the value of ψ on the zero pressure surface where We have not replaced S in the expressions for both brevity and to check the behaviour of ψ Σ when we compare with the parameters in the Wahlquist solution. These expressions for ψ and ψ Σ lead to the following one for the pressure from where µ ( r, θ ) can be directly obtained using the EOS (3). Here we see that, as already happens with Newtonian results for spherical sources [34], writing µ 0 in terms of λ with (7) their lowest orders go as µ ∼ λ , p ∼ λ 2 . The range of applicability of CGMR is given by the set of values ( r s , µ 0 , ω ). The parameter λ will be small whenever r s or µ 0 are small enough. For Ω, small values ω are in principle required, but the greater λ is, the higher ω can be. This comes from the fact that a strongly gravitationally bounded source deforms much less with rotation than a lightly bounded one. This solution is apparently less interesting than the Wahlquist exact solution for the same kind of source because it is an approximation. Nevertheless, it is more general in a sense because it is a Petrov type I solution unless in which case it becomes a Petrov type D solution. It is worth noticing though that when finding the Petrov type of a metric, the more special the algebraic type is, the bigger is the number of conditions to verify. Then, while an approximate metric can satisfy these constraints up to a certain order, it is possible that its higher orders do not. Accordingly, the Petrov type of an approximate metric must be regarded generally as an upper bound to the algebraic speciality of its Weyl tensor (see [21]). Another feature of the CGMR interior is that imposing Darmois-Israel matching conditions [19, 20] shows that when the interior can be matched with an asymptotically flat vacuum exterior [21]. Additionally, in our solution the parameter ω = u ϕ /u t is the angular velocity of the fluid with respect to our harmonic coordinate frame and its vanishing leads to a static solution (i. e. γ tϕ = 0). There is no parameter in Wahlquist's metric with these two features. If we want to compare this approximate solution with Wahlquist's metric we have to start finding their expressions in the same coordinates. The first problem is that the Wahlquist metric is written in a co-rotating coordinate system and CGMR is not, so first we must choose between the two kinds of coordinates. Changing the CGMR interior to a co-rotating system is straightforward doing and then in the co-rotating system the metric components are: and the other components remain unchanged. Let us remark that the γ tϕ component is now of order λ 1 / 2 instead of the order λ 3 / 2 it was in the original coordinates (see [21] for some comments).", "pages": [ 3, 4, 5, 6, 7 ] }, { "title": "3. The Wahlquist metric", "content": "The next steps in the comparison are, using the singularity free Wahlquist metric, first expand it in the appropriate approximation parameters and then make coordinate changes to reduce it to a particular case of the CGMR interior. The singularity free Wahlquist metric reads [2, 3] 3 where and Here µ 0 , b, r 0 are free constants and η 0 and c are related with the behaviour of the solution on the axis. The symmetry axis is located at η = η 0 where and to satisfy the regularity condition of axisymmetry, c must be Therefore η 0 and c become functions of the constants µ 0 , r 0 and b , which thus characterise completely the singularity free Wahlquist's solution. It is generated by a perfect fluid with 4-velocity and with energy density and pressure are given by where we can see now more clearly that the constants b and µ 0 are the values of the normalization factor f -1 / 2 and the energy density on the matching surface of zero pressure (see also (4)). Regarding rotation in Wahlquist's solution, the full expression of the module of its twist vector glyph[pi1] W ( η, ξ ) can be found in [3] and its value at ( η = 0 , ξ = 0) is We can also get a static limit for it -Whittaker's metric [35]- making the change and letting r 0 go to zero [3] although it must be noted that this coordinate change is singular when r 0 = 0. Expression (38) and the limiting procedure suggest a relation between r 0 and the rotation of the fluid. It is actually the case since everywhere so r 0 → 0 implies vanishing rotation and should lead to a static spacetime. Nevertheless, it must be done through the limiting procedure (39). It is also worth noticing that the only other parameter choice capable of giving glyph[pi1] W = 0 everywhere is µ 0 = 0 but it gives an empty interior. The parameters { r 0 , µ 0 } will be the natural choice for us to make the formal expansions -they do not need to be small at all- of the Wahlquist metric if we want to compare with the post-Minkowskian and slow rotation expansions of CGMR, but first we must find the change to spherical-like coordinates.", "pages": [ 7, 8 ] }, { "title": "3.1. The Wahlquist metric written in spherical-like coordinates", "content": "Our approximate metric (9) to (12), (26) and (27) is written in 'standard' spherical coordinates -in the sense that when λ = 0 the metric becomes the Minkowski metric in standard spherical coordinates- so we need to find a consistent way to write the Wahlquist metric in a set of coordinates as close to ours as possible to begin with. In this regard we note first that if we put µ 0 = 0 in the Wahlquist metric (28) we obtain the Minkowski metric in oblate spheroidal coordinates { ξ, η } , whose coordinate lines are oblate confocal ellipses and confocal orthogonal hyperbolas. From these coordinates it is easy to go to Kepler coordinates { R,χ } changing ξ = R/r 0 , η = cos χ , where R represents the semi-minor axis of the ellipses, χ the Kepler eccentric polar angle and r 0 the focal length. Finally we get standard spherical coordinates { r, θ } by changing Moreover, the limiting procedure (39) from Wahlquist's solution to its static limit (the Whittaker metric) has a similar form, in this case leading to Kepler-like coordinates. These considerations suggest to look for a change of coordinates in the Wahlquist metric (prior to any limit) so that the new coordinates 'directly represent' spheroidal-like coordinates. We use a heuristic approach here and start plotting the graphs of h 1 ( ξ ) and h 2 ( η ) (Fig. 1.) We can see that these curves have the appearance of a hyperbolic cosine and a squared sine for some values of b and k , respectively. Taking this into account we write as an educated guess where we introduce R 1 , R 2 just to simplify calculations later. Let us write now the two dimensional metric spanned by { ξ, η } in terms of { R 1 , R 2 } . Since taking h 1 , h 2 as functions of R 1 and R 2 we get where Now let us do another coordinate change to a kind of spherical coordinates { r , θ } using the previous relations Eqs. (41) and (42). Then, R 1 and R 2 are the following functions of r and θ and if we define the function we have that the metric (43) in { r, θ } coordinates has the coefficients Only the terms m 11 + m 22 and m 11 -m 22 depend on both { µ 0 , r 0 } ; the remaining terms depend on r 0 alone. Now we are going to write the full metric in terms of { r, θ } ; notice that the inversion can only be approximately done (in a series of µ 0 ). First, we determine η 0 up to order µ 2 0 , and then c to the same order. This last series depend on b 2 , so before that must determine how b depends on µ 0 . We recall that in the µ 0 = 0 limit the Wahlquist metric becomes Minkowski's metric written in oblate spheroidal coordinates so and hence, since b = f -1 / 2 ∣ ∣ p =0 , its series expansion must begin as b 2 = 1 + O ( µ 0 ). Besides, since Eq. (37) takes the form the expansion of b makes the pressure start with p ∼ µ 2 0 , behaving like in CGMR. Accordingly, we are going to use where σ 1 and σ 2 are two new constants introduced merely for calculation convenience. Inserting it into Eqs. (32) to (34), we obtain for the constants η 0 and c up to O ( µ 3 0 ) Next, we invert the change of coordinates Eq. (42), which gives And finally, by doing the coordinate change { R 1 , R 2 } → { r, θ } we obtain the metric coefficients up to O ( µ 3 0 ) in the spherical-like coordinates desired", "pages": [ 8, 9, 10, 11 ] }, { "title": "4. Comparing the approximate Wahlquist solution with the CGMR solution in co-rotating coordinates", "content": "Now we face the problem of identification of the parameters and to perform the final adjustments of coordinates needed to make every term in Wahlquist's metric and the CGMR interior equal. To get an idea of the problems arising, we analyze first the static limit. Using Eq. (7) and making r 0 = 0 in Eq. (59) we obtain the expression for the γ rr coefficient of the static metric and upon comparison with the corresponding static limit of CGMR we can see that there are discrepancies among r 4 terms in the sense that they can not be made equal adjusting parameters. To some extent this was to be expected since CGMR was written in coordinates associated to harmonic ones and no such a condition has been imposed on the Wahlquist metric. In this particular case, the two metrics can be rendered exactly equal with a change of radial coordinate in γ W αβ ( r 0 = 0) and making m 0 = 0 , σ 1 = r 2 s S .", "pages": [ 12 ] }, { "title": "4.1. Adjusting parameters", "content": "We go back now to the non-static case. If we compare the lowest order term in g tϕ and g tt of both solutions (Eqs. (26), (27), (63) and (64)) we can see that he relation between λ and µ 0 is (7) as expected and the constant Ω of the CGMR solution must be related with the r 0 constant of the Wahlquist metric as follows with κ a factor to be determined later on. If we perform this identification we get to a new difficulty because Wahlquist's solution has λ -free terms with Ω dependence. These terms appear associated with powers of µ 0 r 2 0 [ or κ 2 Ω 2 using Eq. (68) ] . This is not possible in our self-gravitating solution building scheme. This issue can be solved using the remaining freedom in time scale and { r, θ } coordinates. The changes we can do are 4 with F i constants depending on the parameters and G i , H i undetermined functions. Imposing vanishing of these unwanted terms, we get the time scale change and the { r, θ } changes Note that the symmetry axis for the old coordinates is located at θ = 0 , π and due to the presence of the sin Θ it remains at Θ = 0 , π . We will maintain this condition for all the coordinate changes of the θ coordinate. Now, we introduce these changes in our last expression of Wahlquist metric obtaining up to O ( µ 3 0 ) γ W tϕ = - µ 0 6 r 0 R sin Θ { 1 + µ 0 30 ( R + r 2 0 2 After dealing with µ 0 and r 0 , we have to find expressions for b and κ . Recalling Eq. (54), we wrote b 2 as a series in µ 0 with coefficients σ 1 , σ 2 . To help with its determination, we can give more details about σ 1 and σ 2 . When b 2 is written in terms of λ and Ω, its O ( λ 0 ) terms will in general contain order Ω 2 terms. These arise from µ 0 r 2 0 factors and, using dimensional arguments, we can redefine to make this possibility more explicit during calculations. Now we can write the approximate Wahlquist metric in terms of our parameters λ and Ω using (68). Comparing the lower terms in λ for g tϕ of the CGMR co-rotating interior solution and the approximate Wahlquist metric just built, we can determine the proportionality constant κ to be a series in our rotation parameter Ω", "pages": [ 12, 13, 14 ] }, { "title": "4.2. Adjusting terms", "content": "Once the relations between the approximation parameters of both metrics are determined we can obtain the expression of the approximate Wahlquist metric written in the same parameters we have used for the CGMR co-rotating interior. With the coordinate change (71) to (73) we eliminated terms that can not be present in CGMR. Now, to make both solutions coincide we can use changes of coordinates in the Wahlquist metric as long as they do not reintroduce undesired terms; also, we have freedom to adjust the ( m 0 , m 2 , j 1 , j 3 ) constants of CGMR. Regarding the first, the remaining freedom is a change in the { r, θ } coordinates of the type displayed in Eq. (70). If we make this change in the Wahlquist metric ) 2 we get that, for the two metrics to be exactly equal up to O ( λ 2 , Ω 3 ) the free constants (apart from λ and Ω) of the CGMR co-rotating interior must be and the free constants of the approximate Wahlquist metric must be This gives b 2 as thus coinciding with the expansion of ψ 2 Σ from (15) if we take into account that the term (1 + Ω 2 ) comes from the change of the normalization factor over the transformation of the temporal coordinate Eq. (71) we have done. This gives a first check of the consistency of the comparison since b is the Wahlquist counterpart of ψ Σ . The final expressions for the metric components of either Wahlquist's solution or the CGMR interior in the orthonormal basis are, up to O ( λ 2 , Ω 2 ) -and O ( λ 3 / 2 , Ω 3 ) in γ tϕ -, To give another check of the whole procedure we can compare now with the conditions necessary for our n = -2 approximate metric to be of type Petrov D [21, 23, 36], i.e., Eq. (20). They are compatible with the values of the constants m 2 and j 3 we have just found in (85) and (86), as wished. Also, when matched with an asymptotically flat vacuum exterior, m 2 , j 3 and the rest of the metric free constants can only have the expressions we found in [21]. Since the n = -2 fluid for a type D interior does not satisfy the matched expressions, we concluded then that it can not be the source of such exterior in accordance with previous works [3-5]. Nevertheless, it is worth noting here that CGMR contains a n = -2 sub-case that lacks this problem and can indeed be matched that way. It has then all the characteristics of Wahlquist's fluid but Petrov type I instead of D. Note, finally, that the Cartesian coordinates associated to the spherical-like coordinates used above are not harmonic. Nevertheless, since Eqs. (89) to (94) correspond as well to the co-rotating n = -2 CGMR interior with particular values of the free constants, undoing the change (25) they become harmonic again.", "pages": [ 14, 15, 16 ] }, { "title": "5. Remarks", "content": "In this work we have taken the singularity free Wahlquist metric and managed to transform it into the form the CGMR interior metric takes when written in a co-rotating coordinate system. We have started from a formal expansion of Wahlquist's solution in ( µ 0 , r 0 ) and found its expression in terms of the parameters of CGMR, so it possesses the range of applicability already discussed for CGMR. We have identified Wahlquist's parameters corresponding to λ and Ω of [15]. Doing this, we have found an expansion of the parameter r 0 of Wahlquist's metric in terms of our Ω. Accordingly, now we have an approximate expression of r 0 in terms of the better characterised quantities ω and µ 0 To the best of our knowledge its qualitative relation with the angular velocity was previously only guessed through the singular limiting procedure that takes the Wahlquist solution and leads to Whittaker's metric but no parametrization of it in terms of well defined quantities had been given. In the context of fixed EOS, this last equation, together with Eq. (88), completes the map from the free parameters of Wahlquist's solution ( b, r 0 ) to the free parameters of a particular CGMR metric ( r s , ω ). Curiously, we have gained insight in both sets. The role of r 0 as key to a vanishing twist vector and its good behaviour in the comparison with Ω shows far more clearly than the limiting procedure Eq. (39) its relation with the rotation in the Wahlquist metric. But also, the role of b as fundamental parameter in Wahlquist's solution hints towards the possibility of trying to build our post-Minkowskian approximation with a stronger emphasis on ψ Σ instead of the coordinate dependent r s . Last, notice that the usual interpretation of ω = u ϕ /u t as angular velocity of the fluid as seen from the infinite lacks sense if we deal with a metric that is not matched with an asymptotically flat exterior. In our interior though, it is still singled out by the harmonic coordinate condition. Besides, the definition of stationarity and axisymmetry allows a change of coordinates { t = t ' , ϕ = ϕ ' + at ' } that can modify the value of ω to ω ' = u ' ϕ /u ' t = ω -a or make it zero (the case of co-rotating frames). Nevertheless, when dealing with a family of metrics explicitly dependent on ω , its value can be important. In the case of, e. g., CGMR, we see that written in co-rotating coordinates u t ' /u ϕ ' = 0 but ω is part of the metric functions and actually, ω → 0 still leads to a static metric. It is actually the only way for the module of the CGMR twist vector to vanish (its O ( λ 3 / 2 , Ω 3 ) terms are proportional to ω as well). In this sense, the characterization of r 0 (95) is meaningful.", "pages": [ 16, 17 ] }, { "title": "Acknowledgments", "content": "We are very grateful to our reviewers for some important improvements on the original manuscript. This work was supported by the Spanish government grants FIS2006-05319, FIS2007-63034, FIS2009-07238 and FIS2012-30926. JEC thanks Junta de Castilla y Le'on for PhD grant EDU/1165/2007. on theory solution,' in Journal of Physics: Conference Series , Vol. 229 (IOP Publishing, Bristol, 2010) p. 012032. [37] Oscoz, A., Mediavilla, E., and Serra-Ricart, M., eds., Spanish Relativity Meeting 2007. Relativistic Astrophysics and Cosmology , Vol. 30 (EAS Publications Series, 2008).", "pages": [ 17, 19 ] } ]

[ { "title": "Rigid motions and generalized Newtonian gravitation.", "content": "Lost in Translation", "pages": [ 1 ] }, { "title": "Xavier Ja'en", "content": "Departament de F'ısica i Enginyeria Nuclear, Universitat Polit'ecnica de Catalunya, Spain ∗", "pages": [ 1 ] }, { "title": "Alfred Molina", "content": "Departament de F'ısica Fonamental, Universitat de Barcelona, Spain † (Dated: 20 de juny de 2018) We try to lay down the foundations of a Newtonian theory where inertia and gravitational fields appear in a unified way aiming to reach a better understanding of the general relativistic theory. We also formulate a kind of equivalence principle for this generalized Newtonian theory. Finally we find the non-relativistic limit of the Einstein's equations for the space-time metric derived from the Newtonian theory. PACS numbers: 04.20.Cv, 02.40.Yy, 02.40.Ky, 45.20.D-, 03.50.De, 45.20.Jj", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Since Charles-Augustin de Coulomb in 1785 introduced the law of force for rest electrically charged particles till 1905 when Albert Einstein introduced the theory of special relativity, first electrostatics and later electromagnetism never stoped to evolve. This fact contrasts with the laws of gravity introduced by Newton in 1687 that remained unmodified untill 'impelled' by the birth of special relativity, lead to the theory of general relativity in 1916. In this sudden evolution of Newtonian gravitation something was lost along the way. The concept of rigid motion in non-relativistic mechanics disappears swallowed by the Principle of general covariance[1]. Furthermore the principle of equivalence doesn't establish clearly the kinship between inertial and gravitational fields. Also the principle of covariance has no physical meaning at all [2] and also conceals the fact that the General Theory of Relativity has no dynamic invariance group. In section II we describe the notions of rigid motion and non-inertial reference frame, and write some field equations for the velocity field of the rigid motion. In section III we write the usual equations for the electromagnetic field and the Lorentz force using a suitable unit system and a gauge where only the vector potential is needed. We also find a non-relativistic limit for the electromagnetic field equations and compare them with the usual Newtonian gravitational field in an inertial frame. In the section IV we propose a non-relativistic equivalence principle and introduce two generalizations of the Newtonian gravitation theory where, instead of a scalar potential, we have a velocity field which plays the same role as the velocity field used to describe a rigid motion. The second of these generalizations includes non-inertial reference frames and gravitational fields in the same formulation. In section V we obtain the Lagrangian formulation of the theory where the role of the new principle of equivalence can be clearly expressed. Finally we write the space-time metric whose nonrelativistic limit leads to the Newtonian gravitation. Of course this is not the first attempt to generalize the Newtonian theory [3-7]. This paper is strongly inspired by the first part of the article Rigid motion invariance of Newtonian and Einstein's theories of General Relativity by Ll. Bel[8].", "pages": [ 1 ] }, { "title": "II. INERTIAL OBSERVERS, NON-INERTIAL OBSERVERS AND RIGID MOTION", "content": "In a Galilean frame of reference, according to the law of inertia, the equation of motion for a free particle is: As it is well known, rigid motions are defined as those motions in which the Euclidean distances between space points remain constant. A non-inertial observer is defined by a clock (absolute time) and a rigid motion (three orthogonal axis moving rigidly). According to Chasles theorem, the most general motion of this rigid non-inertial frame can be decomposed in a rotation and a translation with respect to an inertial observer. The same free particle described by (1) in an inertial frame can be described in a non-inertial frame with the origin at /vector X ( t ) and with a set of orthogonal axis rotating with an angular velocity /vector Ω( t ) with respect to the inertial frame. The space points /vectorx can be written in the noninertial system as /vectorx = /vector X + /vectory where /vectory is a vector with origin at /vector X . The acceleration in the non-inertial frame is (Annex I)(42) where /vector w is the particle velocity in the non-inertial frame. The particle behaves as if it where subjected to some 'inertial force fields', /vectorg I and /vector β I defined by Using these fields the equation of motion of a free particle in a non-inertial reference frame can be written as That is, the equation of motion of a free particle can be written as (1) in an inertial frame and as the equation (5) in an non-inertial reference system. Alternatively we can write (5) as where η i jk is the unit antisymmetric tensor of rank three. From (6) we can interpret the inertial fields as a Newtonian affine connection whose connection symbols vanish except Γ i 00 = -g i I and Γ i j 0 = -η i jk Ω k , and therefore where y 0 ≡ t . It can be easily seen that the class of these symmetric connections with these properties and Γ k j 0 δ ki + Γ k i 0 δ kj = 0 are invariant by the rigid motions group. The velocity field /vectorv 0 ( /vectorx, t ) for the rigid motions can be written as where /vector X ( t ) and /vector Ω( t ) are two vector fields, which are arbitrary functions of time. This velocity field (8) is usually found in the literature, but to our knowledge nobody has used it as a 'vector potential' from which the 'inertial force fields', /vectorg I and /vector β I can be derived as the 'inertial force fields' verify the following field equations The first of these equations is formally identical to the electromagnetic Faraday law, the second equation coincides with the Amp'ere's law (no magnetic poles) and the third one is a nonlinear modification of the Coulomb's law without electrical charges. In General Relativistic mechanics we have no way to characterize a class of physically relevant observers. This is why General Relativistic mechanics lacks of a dynamical symmetry group, contrarily to what happens in Newtonian mechanics. And in connection with this, a satisfactory way of inplementing the notion of kinematic rigidity is not known [9], [10].", "pages": [ 1, 2 ] }, { "title": "III. ELECTROMAGNETIC INTERACTION AND NON-RELATIVISTIC GRAVITATIONAL INTERACTION", "content": "The purpose of this section is to analize the similarities and differences between the electromagnetic interaction and its limit c → ∞ on one hand and the Newtonian gravitational interaction on the other. We will compare the corresponding electromagnetic field equations with those fullfilled for the non-inertial fields, equations (9) and (10). The most common version of the basic equations for the electromagnetic interaction uses the rationalized MKS system of units, in which, besides the usual mechanical units, a new unit is added, the ampere. Here we will write the electromagnetic equations in a general unspecified system of units. It is known that in the most general form of writing the electromagnetic equations we can introduce four constants k i , i = 1 . . . 4. [11][12] 1)The Lorentz force : the trajectory /vectorx of a particle of charge q and mass m inside an electromagnetic field, /vector E and /vector B , (provided that the particle moves with a small velocity compared to the speed of light in order to avoid the relativistic linear momentum in the left hand side), is a solution of the equation of motion: Given the fields /vector E and /vector B , equation (11) belongs to the non-relativistic mechanics. It is so as it is usually used. But, for (11) to be a genuinely non-relativistic equation, the fields /vector E and /vector B , should also to be derived from non-relativistic equations and to transform in accordance with the Galileo transformations. 2) The Maxwell equations : the electromagnetic fields /vector E and /vector B are a solution of The constants k i can be freely chosen with the following dimensional restrictions and where c is the vacuum propagation speed for electromagnetic waves and ρ and /vector J are respectively the charge and current densities. Depending on the application different unit systems have been chosen. These are described by different values k i , as shown in the table below: [11][12] where UES refers to the electrostatic cgs units system, UEM to the electromagnetic cgs system, H-L to the Heaviside-Lorentz system and the last line to the rationalized MKS system. The first one is more appropriate to our intentions but, as we want to be closer to the Newtonian mechanics definitions, we will use a units system in which electric charge is measured in units of mass, then only remain the mechanical units, the meter, the kilogram and the second. In fact the proposed system of units is simply MKS without adding any additional unit defined from the electromagnetic equations. To emphasize this fact we call this system, specially when is used in electromagnetism, pure MKS system of units: where G is the gravitational constant. In this pure MKS system the Maxwell-Lorentz equations are We have changed the charge symbol q to the more convenient m e to emphasize the fact that in this units system masses and charges are measured in the same mass units. As it is well know, the equation where the sources are not involved, /vector ∇· /vector B = 0, guarantees the existence of the vector potential /vector A through /vector B = /vector ∇× /vector A . Now the dimensions of /vector B are T -1 so /vector A has the dimensions of a velocity. The remaining equation without sources lead to which tells us that /vector E + ∂ /vector A/∂t is the gradient of a potential φ whose dimensions are the square of a velocity. We can also see that in the limit c →∞ we get the same Lorentz equation and whereas Maxwell equations become  Notice the similarity of these Maxwell and Lorentz equations when there is no charge density, ρ e = 0, with the non-inertial fields equations (10). It can be even closer if for the electromagnetic fields we use the gauge φ = ( /vector A ) 2 / 2, then we have (16) and which are almost the same as (9) where the vector potential plays the role of the velocity field in rigid motions. But the Gauss equation for electromagnetic field, namely that with ∇· /vector E , doesn't contain a term /vector B 2 / 2 as it is contained in the counterpart equation for the velocity field (10) for rigid motions. Now, in the usual conditions of null fields at infinity, equations (15) and (16) become The magnetic field does not appear due to the fact that the equations for the magnetic field are /vector ∇· /vector B = 0 , and /vector ∇× /vector B = 0 and the only solution vanishing at infinity is /vector B = 0. A remarkable fact is that according to the limit c →∞ of the Maxwell equations, written in the system of units that we propose, the magnetic field is entirely a relativistic effect . In this way the likeness between electromagnetism and gravitation increases. The equations for the gravitational (non-relativistic) interaction are To stress the differences between the electromagnetic non-relativistic equations and the gravitational Newtonian, notice that /negationslash", "pages": [ 2, 3, 4 ] }, { "title": "IV. GENERALIZED NON-RELATIVISTIC GRAVITATION AND EQUIVALENCE PRINCIPLE", "content": "So far, except for equation (9), we have only obtained expressions that were already known and that perhaps we have written in a unusual way. We shall now attempt a revision of Newtonian theory of gravity, aiming to get a better understanding of General Relativity theory. To this purpose we start analyzing the weak relativistic equivalence principle:", "pages": [ 4 ] }, { "title": "Relativistic equivalence principle", "content": "At every space-time point in an arbitrary gravitational field it is possible to choose a 'locally inertial system of coordinates' such that, within a sufficiently small region around this point the laws of mechanics are the same as in an inertial Cartesian coordinate system in the absence of gravitation. Notice that this principle says nothing about how to built the 'locally inertial coordinate system'. It only states its existence. In section III when studying the forces of inertia for rigid motions, we have seen that a velocity field opened the possibility to define a local frame, and how the velocity field of the rigid motion /vectorv 0 can be used as a vector potential to define the acceleration and the rotation fields. Also in section III we used a gauge where only a vector potential, with dimensions of velocity, is needed to build the electromagnetic field. In a similar way as we did with the Newtonian rigid velocity field, we shall introduce a generalized gravitational vector potential from which the gravitational field can be derived. Recall that from the viewpoint of the inertial frame there is no force field for a free particle but acording to a non-inertial frame -which follows a Newtonian rigid motion- two inertial force fields arise which can be associated to a vector potential, /vectorv 0 ( /vectorx, t ), (the rigid velocity field). Assume now that we have a Newtonian gravitational field in an inertial frame. We want to introduce a velocity field /vectorv g ( /vectorx, t ) such that a particle in the local non-inertial frames associated to it does not feel any field of force. At each point in space and for every time we define the local system as a reference frame whose origin moves with the velocity /vectorv g ( /vectorx, t ), i.e. the interaction vector potential, and with a triad of orthogonal axes (with respect to the Euclidean metric) that 'rotates rigidly' with a local angular velocity /vector Ω g = 1 2 ∇× /vectorv g ( /vectorx, t ), that is the vorticity of the velocity field. Now our non-relativistic equivalence principle states that The trajectory of a particle due to gravitational interaction /vectorv g ( /vectorx, t ) , is such that with respect to the origin of the local system has no acceleration. Notice that unlike the weak relativistic equivalence principle in this non-relativistic equivalence principle we know the motion of the local frame. We know that for a velocity field /vectorv g ( /vectorx, t ) the acceleration field is given by and this will be the acceleration of our local frame. The relation between the accelerations referred to the inertial system and to the local one is (see Annex I)(42): Consider now a particle at /vectorx (in the inertial frame) in a gravitational field. As seen from the local reference it is at the origin, /vectory = 0, without acceleration and with velocity /vector w . By the non-relativistic equivalence principle, its acceleration referred to the inertial frame is: where we have included that /vectorv g + /vector w is the particle velocity in the inertial system /vectorv . The acceleration produced by our generalized Newtonian gravitational fields has two components, one depending on the particle velocity (as the Coriolis component of non-inertial fields) and the other that does not depend on the particle velocity This equation is very similar to the Lorentz equation, the only difference is that the gravitational charge coincides with the inertial mass, a feature which is on the basis of the non-relativistic equivalence principle. We have introduced the acceleration and rotation gravitational fields as follow: Notice that with this definition /vector β = -2 /vector Ω g . Furthermore we have the field equations which are equivalent to the first pair of Maxwell equations. Now we can add the source equation for nonvanishing mass density and a non-relativistic version of the Ampere law so we have a system of gravitomagnetic equations like the non-relativistic Maxwell equations (17). The condition that the fields vanish at infinity implies /vector β = 0 and we obtain the usual results, the equation of motion is /vectora = /vectorg ; and the source equation (28). We can go further and propose a gravitational interaction that includes the inertial fields. We keep the equation of motion (25) and propose the following field equations  Equations (30) are the equations for the gravitational field as seen by any observer, inertial or not. If we don't have mass density, the velocity field is the non-inertial one. If we don't have fields at infinity the solutions are the usual ones in Newtonian gravitation. But if there is a mass density and fields do not vanish at infinity we have something new. An interesting fact is that the 'magnetic gravitational field' /vector β is a non-relativistic effect. This is truly remarkable because in section III we have arrived to the conclusion that the magnetic field /vector B in electrodynamics is a purely relativistic effect.", "pages": [ 4, 5 ] }, { "title": "V. LAGRANGIAN AND HAMILTONIAN FOR THE NON-RELATIVISTIC GENERALIZED GRAVITATIONAL INTERACTION", "content": "Given the field of gravitational interaction /vectorv g ( /vectorx, t ), from the equivalence principle in the non-inertial reference system the particle is free. So in the Lagrangian only the kinetic term appears L = mw 2 / 2 which can be written in terms of the velocity in the inertial system The Euler-Lagrange equations are: Now using the identity and taking into account that ˙ /vectorx and /vectorx are independent variables (in phase space), we have that which substituted in (32) leads to the equations of motion (24), (25). We can also also set up the Hamiltonian formulation. First performing the Lagrange transformation then the energy function is and the Hamiltonian", "pages": [ 5 ] }, { "title": "VI. NON-RELATIVISTIC LIMIT OF EINSTEIN'S EQUATIONS", "content": "The free particle relativistic Lagrangian related to (31) is and the action functional this can be associated to a metric which has the following properties: 1) This is a so-called Newtonian metric [8] invariant under the rigid motion group. The transformation of the metric tensor under a change of coordinates associated to the rigid motion x i = X ( t ) i + R i j y j and t = t ' , is where That is, the metric given by the expression (36) is of this type in every non-inertial frame. More specifically, the metric is invariant provided that the velocity field transforms as /vectorv g → /vectorv ' g = /vectorv g -( /vector Ω × /vectory + ˙ /vector X ) 2) In the limit c → ∞ the metric connection for (36) leads to the Newtonian affine connection, because the inverse metric in the c →∞ limit is g 0 µ = 0 g ij = δ ij and neglecting the small terms when c → ∞ the Christoffel symbols are 3) The Ricci tensor of the metric (36) when c →∞ is: This means that, using the metric (36) the Einstein's equations for vacuum, R µν = 0 in the limit c →∞ yield the Newtonian equations (30) for ρ = 0. It is very interesting to note that, without any other consideration, the limit c →∞ yields the right Newtonian limit. 4) Another interesting result, showing the strength of this way to obtain a relativistic mechanics from this Newtonian generalization, is that the velocity field needed to obtain the spherical symmetric solutions for vacuum in the Newtonian and in the relativistic case are the same. If we take where ˆ r is the unit radial unitary vector, then the vacuum solution of the metric (36) is the Schwarzschild solution and v g ( r ) function must be but what is not usually done is to write it in this form [13]. The same function (39) using (26) gives the solution for the Newtonian equations (30) with ρ = 0.", "pages": [ 5, 6 ] }, { "title": "VII. CONCLUSIONS", "content": "We study the velocity field for the classical rigid motion and derive the field equations for its acceleration and rotation fields, the inertial fields. We also write the Maxwell-Lorentz equations for the electromagnetic field using a suitable system of units and a gauge where only the potential vector, which has the dimensions of a velocity, is necessary. For them we find the limit for c →∞ . Taking into account the null limit condition at infinity of the electromagnetic field we get the same equations, with some change in the constants, as for the Newtonian gravitational problem. We build a generalized Newtonian gravitational theory where the potential is a velocity field as in the two previous examples and that includes in an unified way the inertial forces fields and the gravitational Newtonian field. If the fields are nulls at infinity only the gravitational field remain, and if the mass density is zero we obtain the equations for inertial fields. We introduce a non-relativistic equivalence principle which is very useful to construct a Lagrangian theory for test particles moving in this generalized Newtonian gravitational field. Finally we use this Lagrangian to build a relativistic theory where we have a Newtonian metric invariant for the rigid motion group. Some interesting features of this metric are that the limit metric connection when c → ∞ is the Newtonian affine connection, and the Einstein's equations in the same limit lead to the Newtonian field equations wihtout using any kind of weak field aproximation. Another interesting fact is that the same spherical symmetric velocity field √ ( k/r )ˆ r gives us the Schwarzschild solution when we put it in the metric and the Newtonian known result when we use the Newtonian field equations.", "pages": [ 6 ] }, { "title": "ANNEX I", "content": "We give here a detailed derivation of the transformation of position, velocity and acceleration between two orthonormal reference frames. We first change to a noninertial reference system moving rigidly. From the Chasles theorem the most general motion of this frame is such that its origin moves arbitrarily with respect to the origin of an inertial frame and its axis rotates with angular velocity /vector Ω( t ). We have an inertial frame with origin in O (0 , 0 , 0) and a system of orthonormal axis /vectorε i , i = 1 . . . 3 and one non-inertial frame with origin Q ( X 1 ( t ) , X 2 ( t ) , X 3 ( t )) and three orthonormal axis /vectore i ( t ) , i = 1 . . . 3 where R i j ( t ) is a rotation matrix. A point P can be referred to both frames, as /vectorx in the inertial frame and as /vectory in the non-inertial frame The velocity transformation is: where So it can be also written as or alternatively Note The acceleration transformation is and That can be also written or Let's remark", "pages": [ 6, 7 ] }, { "title": "Acknowledgments", "content": "Thanks to three good friends, to Jes'us Mart'ın of who we have copied the Annex, to Josep Llosa for doing a carefully reading and an useful criticism that lead us to improve this paper and finally to Llu'ıs Bel that without his inspiration and insistence almost nothing of this article had occurred to us. guer, Garriga, Cespedes Worl Scientific (1990).", "pages": [ 7 ] } ]

[ { "title": "An Alternative f (R, T ) Gravity Theory and the Dark Energy Problem", "content": "Subenoy Chakraborty ∗ Department of Mathematics, Jadavpur University, Kolkata-700 032, India. Recently, a generalized gravity theory was proposed by Harko etal where the Lagrangian density is an arbitrary function of the Ricci scalar R and the trace of the stress-energy tensor T, known as F(R,T) gravity. In their derivation of the field equations, they have not considered conservation of the stress-energy tensor. In the present work, we have shown that a part of the arbitrary function f(R,T) can be determined if we take into account of the conservation of stress-energy tensor, although the form of the field equations remain similar. For homogeneous and isotropic model of the universe the field equations are solved and corresponding cosmological aspects has been discussed. Finally, we have studied the energy conditions in this modified gravity theory both generally and a particular case of perfect fluid with constant equation of state. Keywords: f(R,T) gravity theory; conservation of stress-energy tensor; dark energy.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Recent observational predictions [1] that our universe is going through a phase of accelerated expansion put new avenues in modern cosmology. A class of people are making attempts to accomodate this observational fact by choosing some exotic matter(known as dark energy) in the framework of general relativity.There are several choices for this exotic matter namely a) the quintessence scalar field models[2], the phantom field[3], K-essence [4], tachyon field [5], quintom [6] etc., b) the dark energy models including Chaplygin gas[7] and so on. On the other hand, there are attempts to modify the gravity theory itself to accomodate the present accelerated phase. A natural generalization is to choose a more general action in which the standard Einstein-Hilbert action is replaced by an arbitrary function of the Ricci scalar R [8] (i.e, f(R)) and is known as f(R) -gravity. This modified theory may explain this late time cosmic acceleration [9]. These f ( R ) models can satisfy local tests and unify inflation with dark energy[10]. Also it is possible to explain the galactic dynamics of massive test particles in this modified gravity theory without any dark matter [11-15]. For detailed review of f(R)- gravity one may refer to [8, 16]. Recently, a further generalization of f ( R ) - gravity theory has been done by Harko etal [17]. They choose the Lagrangian density as arbitrary function f ( R,T ) where as usual R is the Ricci scalar and T is the trace of the energy -momentum tensor. The justification of choosing T as an argument for the Lagrangian is from exotic imperfect fluids or quantum effects (conformal anomaly). They have argued that due to the coupling of the matter and geometry, this gravity model depends on a source term, which is nothing but the variation of the matter stress-energy tensor. As a result, the motion of test particles is not along geodesic path due to the presence of an extra force perpendicular to the four velocity. The cosmic acceleration in this modified f ( R,T ) theory results not only from geometrical contribution but also from the matter content. Subsequently, Houndjo [18] has chosen f ( R,T ) as f 1 ( R ) + f 2 ( T ) and discussed transition of matter dominated era to an accelerated phase. Very recently, Sharif etal [19] have studied thermodynamics in this f ( R,T ) theory and Azizi [20] have examined the possibility of wormhole geometry in f ( R,T ) gravity. In the present article, we have formulated the f ( R,T ) gravity theory in an unorthodox manner. Though the action is a coupling of geometry and matter, but still we restrict ourselves to the special cases where test particles move in a geodesics. As a result, the Lagrangian has some restricted form, keeping the field equations same. The alternative derivation of f ( R,T ) gravity and some specific choice for f ( R,T ) has been presented in section II. Also admissibility of some known matter fields has been examined in this section. Section III deals with cosmological solutions for homogeneous and isotropic model of the universe with some physical interpretations. Energy conditions in this modified gravity theory has been examined both in a general way as well as for perfect fluid in section IV. Finally, at the end there is a brief summary of the entire work in section V.", "pages": [ 1, 2 ] }, { "title": "II. f ( R,T ) GRAVITY THEORY: A MODIFICATION", "content": "In this gravity theory [17], the gravitational Lagrangian density is given by an arbitrary function f ( R,T ) of two variables: One is the Ricci scalar R and the other is the trace of the energy-momentum tensor T (= T µν g µν ). So the complete action of this theory is written as [17] where the stress-energy tensor of the matter ( T µν ) can be obtained from the matter Lagrangian density L m as [21] This can be simplified further assuming L m depends only on g µν but not on its derivatives as Using the standard text book result namely and the shortcut notations: the variation of the above action can be written as Now performing by parts integration to the second and third terms in the r.h.s. of equation (6), one obtains the field equations in f ( R,T ) gravity theory as [17], with It is to be noted that if f ( R,T ) = f ( R ) then we get back to the field equations for f(R) gravity. Now we can proceed further, with the field equations (7) in the following three cases: a) f ( R,T ) = R + h ( T ) b) f ( R,T ) = R.h ( T ) c) f ( R,T ) is arbitrary · Case-(a) : f ( R,T ) = R + h ( T ) For this choice of f ( R,T ) the field equations (7) now simplify to Now taking divergence of both sides of the above field equations(9) and assuming conservation of energy -momentum tensor (i.e, ∇ µ T µν = 0) we obtain This shows that the form of h(T) is not arbitrary, it depends on the choice of the matter field. We consider now some known matter fields as examples: Example-I: Electromagnetic Field. The matter Lagrangian has the form with F µν , the electromagnetic field tensor. So from equation (8) we have As a result equation (10) simplifies to ∂h ( T ) ∂x µ = 0 i.e, h ( T ) turns out to be a constant. Thus for this choice of f ( R,T ) electromagnetic field is not possible. Example-II: Perfect fluid. In case of perfect fluid, the stress-energy tensor has the usual form and the matter Lagrangian can be taken as L m = -p . Here ρ and p are the usual energy density and thermodynamic pressure and the four velocity u µ satisfies i) u µ u µ = 1 and ii) u µ ∇ ν u µ = 0. In this case Θ µν has the explicit form Now substituting equation (14) for Θ µν into equation (10) we obtain Further, if the perfect fluid has barotropic equation of state, i.e, p = ωρ , ω , a constant then for homogeneous and isotropic flat FRW model h ( T ) has an explicit form as ( ω = -1 , ± 1 3 ) /negationslash where α = 1+3 ω 2(1+ ω ) , h o is an integration constant. For the choice f ( R,T ) = Rh ( T ), the field equations become After taking divergence of both sides of equation (17) and considering energy conservation relation,one obtains the differential equation for h ( T ) as where Ricci scalar R is related to T and h by the relation Note that here also for electromagnetic field h(T) is restricted by the relation ∂h ( T ) ∂x µ = 0 i.e, h ( T ) is a constant. Here f ( R,T ) is totally arbitrary except the choices in the previous two cases. We start with the geometric identity namely [22] i.e. Now taking covariant divergence of equation (7) and using this identity we have from conservation of matter field Note that equation (21) is not identical in form to that of equation (10), there is one extra term in equation (10). Now proceeding as before we obtain the following results: where A 0 and A 1 are arbitrary functions of arguments. where A ( R ) is an arbitrary function of R (except A ( R ) = R ) and B ( T ) has the form with B 0 , an integration constant. If the fluid is in barotropic nature with constant equation of state then we have /negationslash Thus the choice of the function f ( R,T ) depends to a great extend on the matter field taken into account.", "pages": [ 2, 3, 4, 5 ] }, { "title": "III. COSMOLOGICAL SOLUTIONS AND CONSEQUENCES", "content": "We now try to find cosmological solutions for the first choice of f ( R,T ) for perfect fluid in the background of flat FRW model. The Einstein field equations are and where we have used the solution (16) for h ( T ) . In Einstein gravity, the above field equations correspond to a non-interacting two-fluid system of which one is the usual perfect fluid (Fluid -1) that we have considered in f ( R,T ) -gravity theory while the other fluid system (Fluid-2) is also a perfect fluid having energy density and pressure The equation of state of the additional fluid (i.e. Fluid-2) is with and The nature of the two non-interacting fluids in different stages of the evolution of the universe are shown in table I: Thus, although both fluids start simultaneously at the ultra-relativistic equation of state (stiff fluid) but fluid-2 advances more rapidly so that it reaches the quintessence equation of state when the actual fluid (i.e, Fluid-1) has still positive pressure and finally fluid-2 reaches the phantom era when the fluid-1 is in quintessence era. Further, it is to be noted that although both the fluid components have constant equation of state but the effective one fluid system has always variable equation of state. As for the normal fluid (i.e.Fluid-1) we have p = ωρ so from the conservation of energy - momentum tensor, i.e, and Note that on the phantom barrier ( i.e. ω = -1) we have a big rip singularity at finite time t = t 0 + 4 √ d 2 3 d 1 . The other two solutions are the usual expanding solutions starting from the big-bang singularity at finite past.", "pages": [ 5, 6 ] }, { "title": "IV. ENERGY CONDITIONS IN f ( R,T ) GRAVITY", "content": "The fundamental features to the singularity theorems as well as to those related to classical black hole thermodynamics [23] are nothing but the energy conditions which are consequences of the Raychaudhuri equation for expansion, namely, Here θ , σ µν and ω µν are respectively the expansion, shear, and rotation associated to the congruence defined by the null vector field κ µ and R µν is the usual Ricci tensor. Though the Raychaudhuri equation is not related to any gravity theory (it is purely a geometric statement) but it has some special reference to Einstein gravity. As the attractive character of gravity is reflected through the we have on integration, where ρ 0 is an integration constant. Now substituting this value of ρ into the Friedmann equation (26) we obtain an integral equation for the scale factor 'a' as with t 0 an integration constant and d 1 = 8Π ρ 0 3 , d 2 = h 0 ρ α 0 (1 -3 ω ) ( α -1) 3 . In the following we have explicit solution for 'a' with ω = 0 , ± 1 as positivity condition, i.e, R µν κ µ κ ν ≥ 0 (which implies that the geodesic congruences focus within a finite value of the parameter labeling points on the geodesics [24] ), so in Einstein gravity the above condition becomes T µν κ µ κ ν ≥ 0, which is the null energy condition (NEC). The weak energy condition (WEC), i.e, T µν v µ v ν ≥ 0 ( v µ , a time-like vector) assumes the positivity of the local energy density and by continuity, WEC ⇒ NEC . Similarly, we have two other energy conditions namely the strong energy condition (SEC): ( R µν -1 2 Rg µν ) v µ v ν ≥ 0 which by continuity implies NEC but not the WEC in general and the dominant energy condition (DEC): T µν v µ v ν ≥ 0 and T µν v ν is not space-like imply locally measured energy density to be always positive and the energy flux is time-like or null. Also DEC ⇒ WEC (and hence the NEC) but not necessarily the SEC ( For details of energy conditions see [25]). For perfect fluid the above energy conditions have the explicit form: But difficulty arises in other gravity theories, particularly where R µν may not be evaluated using the corresponding field equations. However, in a gravity theory if the Lagrangian density still have an Einstein-Hilbert term then it is possible to determine R µν κ µ κ ν . In the present f ( R,T ) gravity theory, for the first two choices of f ( R,T ) (i.e. f ( R,T ) = R + h ( T ) or Rh ( T )) the field equations (9) or (17) can be written as, and the energy conditions read as, However, explicitly if we consider the perfect fluid case of the previous section then from the field equations (26) and (27) we have and with α = (1+3 ω ) 2(1+ ω ) . Then the above energy conditions can be written as with Ω m = ρ/ 3 H 2 , the density parameter for the matter considered in the Einstein gravity .Now the deceleration parameter q (= -(1 + ˙ H H 2 )), is related to ω and Ω m by the relation Thus NEC is satisfied until fluid-1 is in the quintessence era and Ω m is restricted by the given inequality. Note that the above inequality holds for all Ω m as long as the normal fluid ( i.e. fluid-1)is not exotic (i.e, satisfies the strong energy condition). The same is true for WEC as well as DEC. However, to satisfy the SEC Ω m has a lower bound given by Ω m ≥ (1 -3 ω ) 3(1+ ω ) . The variation of q has been plotted in figure 1 and the inequalities for NEC and SEC are presented in figures 2 and 3 respectively.", "pages": [ 6, 7, 8, 9 ] }, { "title": "V. SUMMARY", "content": "The paper deals with recently introduced f ( R,T ) gravity theory with the restriction of conservation of matter. As a result, although the form of the field equations remain same but now the test particles move in a geodesics and the choice of the Lagrangian function is not totally arbitrary. We have analyzed three possible choices for f ( R,T ) and examined whether two familiar matter fields namely electromagnetic field and perfect fluid are permissible or not in this modified gravity theory. It is found that electromagnetic field is not allowed in all the cases. For homogeneous and isotropic model of the universe, the explicit field equations are written for the modified gravity theory with f ( R,T ) = R + h ( T ) and it is found that the field equations are equivalent to Einstein gravity with a non-interacting 2-fluid system of which one is the usual perfect fluid in the modified theory while the second fluid (i.e, Fluid-2) is also a barotropic fluid with constant equation of state and will become exotic when the usual fluid (i.e, Fluid-1) is still a normal fluid. For some specific choice of the equation of state parameter of the usual fluid there are possible cosmological solutions of which one corresponds to big rip singularity. The graph (see figure 1) of q for the variation of Ω m and ω shows that there is a natural transition from deceleration to acceleration although we have considered normal fluid (non-exotic), i.e, fluid-1 as the matter source in this modified gravity theory. In particular, if we consider only the baryonic matter (with Ω m = 0 . 04) as the source of matter field then transition from deceleration to acceleration occurs when ω < 0 . 099. Thus with the normal fluid model in f ( R,T ) gravity theory, there is a natural transition from deceleration to acceleration as predicted by recent observations. Also we have analyzed the energy conditions for the modified gravity theory in a general way. For the perfect fluid model of section-III we have shown the validity of the energy conditions both analytically as well as graphically. However, it should be noted that although for some simple choice of f(R, T) we have obtained a possible solution for DE but it is natural to identify the correct class of f(R, T) which are compatible to modern observations [26] as it has been done in f(R) gravity. This is termed as cosmography of f(R, T). In the background of flat FRW model cosmography is related to the taylor series expansion of the scalar factor around the present time t 0 and the first six coefficients in the expansion are [27-29] H = ˙ a a , q = -1 aH 2 d 2 a dt 2 , j = 1 aH 3 d 3 a dt 3 , s = 1 aH 4 d 4 a dt 4 , l = 1 aH 5 d 5 a dt 6 and m = 1 aH 6 d 6 a dt 6 . They are respectively known as the Hubble parameter, the deceleration parameter, the jerk parameter, the snap parameter, the lerk parameter and the m parameter. These parameters are model independent quantities and are termed as cosmographic set. As a future work, one can analyze the cosmography of f(R, T) gravity to identify the appropriate choices of f(R, T). Moreover, it is interesting to consider hybrid gravity theory related to f(R, T) gravity. In this theory both metric and Palatini formalisms are incorporated in the action [30, 31] and the dynamical scalar corresponding to scalar-tensor representation need to be massive so that it does not care about laboratory and solar system tests and can play an active role in cosmology [30]. Similar to f(R) gravity the action may be chosen as S = 1 2 κ ∫ d 4 x √ -g [ R + f ( R,T )] + S m , where R is the Palatini curvature obtained from an independent Palatini connection ˆ Γ α µν . This issue may also be considered for future. Finally, f(R, T) gravity theory can be motivated at fundamental level, i.e, at small scales and high energies provided one should take care of quantum field theory formulated on a curved space [32, 33]. Since, at scales comparable to the compton wave length, particles, matter should be quantized so one should employ a semi classical description of gravity and equation (7) is modified as where in the arguement of f R and f T , T should be replaced by 〈 T 〉 . The expectation value of a quantum stress-energy tensor is defined as [32, 33] where | Ψ 〉 is a quantum state describing the early universe and ˆ T µν is the quantum operator associated with the classical energy-momentum tensor of the matter field. In general, a quantized matter field is subject to self interactions as well as it interacts with other fields and with the gravitational background and as a result there are infinities from 〈 T µν 〉 . So to obtain a renormalizable theory, one has to introduce infinitely many counterterms in the Lagrangian density [32] of gravity. However, one can construct a truncated quantum theory of gravity by expansion in loops. In this context it should be noted that trace anomaly [33] takes a vital role to deal with infinities in regularization procedures. This is an important issue to deal with for future studies. Acknowledgements: The work is done during a visit to IUCAA, Pune (India) under associateship programme. The author is thankful to IUCAA for warm hospitality and facilities at the library. The author also acknowledges the DRS programme of UGC, Govt. of India, in the department of Mathematics, Jadavpur University.", "pages": [ 9, 10 ] }, { "title": "VI. REFERENCES", "content": "[33] N. D. Birrel and P. C. W. Davies, Quantum Fields in Curved Spacetime (Camb. Univ. Press, Camb. U. K. 1982). [34] M. Alves and J. B. Neto, Braz. J. Phys 34 , 531 (2004).", "pages": [ 12 ] } ]

[ { "title": "Thermodynamics of black plane solution", "content": "Manuel E. Rodrigues ( a,f,g )1 , Deborah F. Jardim ( b )2 , St'ephane J. M. Houndjo ( c,d )3 and Ratbay Myrzakulov ( e )4 Centro de Ciˆencias Exatas - Departamento de F'ısica Av. Fernando Ferrari s/n - Campus de Goiabeiras CEP29075-910 - Vit'oria/ES, Brazil Rua do Cruzeiro, 01, Jardim S˜ao Paulo CEP39803-371 - Teofilo Otoni, MG - Brazil CEP 29933-415 - S˜ao Mateus/ ES, Brazil Weobtain a new phantom black plane solution in 4D of the EinsteinMaxwell theory coupled with a cosmological constant. We analyse their basic properties, as well as its causal structure, and obtain the extensive and intensive thermodynamic variables, as well as the specific heat and the first law. Through the specific heat and the so-called geometric methods, we analyse in detail their thermodynamic properties, the extreme and phase transition limits, as well as the local and global stabilities of the system. The normal case is shown with an extreme limit and the phantom one with a phase transition only for null mass, which is physically inaccessible. The systems present local and global stabilities for certain values of the entropy density with respect to the electric charge, for the canonical and grand canonical ensembles. Pacs numbers: 04.70.-s; 04.20.Jb; 04.70.Dy.", "pages": [ 1, 2 ] }, { "title": "1 Introduction", "content": "It is well known that a black hole can radiate a black-body radiation when one takes into account the effects of classical gravitational field on quantized matter fields, i.e, a semi-classical analysis of the gravity [1]. So, we can make a study of the thermodynamic system of each new black hole solution. The most common method in the literature is the analysis made through the specific heat of the black hole [2], which informs us if the system is thermodynamically interacting, if there exists any case in which the black hole is extreme or it passes across a second order phase transition. Recently, attention is attached to the methods for analysing the thermodynamic system through the geometry of the so-called thermodynamic space of the equilibrium states. The most common are the methods of Weinhold [3], Ruppeiner [4], geometrothermodynamics [5] and that of Liu-Lu-Luo-Shao [6]. These methods also notify if the system possesses thermodynamic interaction and if it undergoes a second order phase transition, in addition to the properties about the stability. In this work, we desire to make a detailed analysis of the thermodynamic system of a well known class of solutions, with a particularly interesting symmetry, the planar. This class of solutions has been previously obtained for the case of planar and static symmetry in 4 D , by Cai and Zhang [7]. This symmetry was then applied to traversable wormholes [8], and later, generalized to topological black holes in [9], and its various applications. We focus our attention to a class of solutions, called phantom [10], but now with a planar symmetry. Before beginning the analysis of this new class of phantom black holes, we will present briefly our interest in obtaining and studying such exotic solutions. With the discovery of the acceleration of the universe, various observational programs of studying the evolution of our universe were deployed, including the relationship of the magnitude-versus-redshift type supernovae Ia and the spectrum of the anisotropy of the cosmic microwave background. These programs promote an accelerated expansion of our universe, which should be dominated by an exotic fluid and should have a negative pressure. Moreover, these observations show that this fluid can be phantom, i.e, with the contribution of the energy density of dark energy [11]. As the interest in obtaining these classes has increased, we also found ourselves wanting to analyse a specific phantom model. We can mention here some recent results in the literature, such as the wormhole solutions and conformal continuation [12], the black hole solutions of Einstein-MaxwellDilaton theory, [13], the higher-dimensional black holes by Gao and Zhang [14], and the higher-dimensional black branes by Grojean et al [15]. Analysis were also made in algebraic structures of this type of phantom system, as the case of the algebra generated by metrics depending on two temporal coordinates, with D ≥ 5, which provides phantom fields in 4 D , fulfilled by Hull [16], and Sigma models by Cl'ement et al [17]. Here, we will obtain and study the thermodynamic properties of a solution arising from the coupling of Einstein-Hilbert action with a field of spin 1, which can be Maxwell or anti-Maxwell (phantom), and a cosmological constant, where the spacetime possesses planar symmetry. The idea of using the ruse of negative electric energy density is quit old, Einstein and Rosen being the first to use it [28]. Recently, through the work of Babichev et al [29] and Bronnikov et al [30], we have seen a keen interest in phantom solutions [31]. The paper is organized as follows. In Section 2, we present a new phantom black plane solution. The causal structure of the solutions are studied and the thermodynamic variables are obtained. The first law of thermodynamics is established and the specific heat is calculated. In Section 3, we minutely study the thermodynamics of normal and phantom solutions, using the analysis through the specific heat, subsection 3.1, and through the geometric methods of Weinhold, subsection 3.2, the geometrothermodynamics, subsection 3.3, and that of Liu-Lu-Luo-Shao, subsection 3.4. We finish the section with the study of local and global stabilities in subsection 3.5. The conclusion is presented in Section 4.", "pages": [ 2, 3, 4 ] }, { "title": "2 The field equations and the black holes solutions", "content": "The action of the theory is given by: where the first term is that of Einstein-Hilbert, the second is the coupling of (anti)Maxwell field F µν = ∂ µ A µ -∂ ν A µ with the gravitation, and the third is the cosmological constant. Making the functional variation of the action (2.1) with respect to the field A µ and the inverse of the metric, g µν , using R = -4Λ, we get the following equations of motion Let us write the static and plane symmetric line element as with r = | z | . We will also assume that the Maxwell field is purely electric and only depends on r . With (2.4), one can integrate (2.2) and obtain with q a real integration constant. Substituting (2.5) into the equations of motion (2.3), we obtain the equations where the 'prime' denotes the derivative with respect to r . Choosing the coordinates such that with Λ = -3 α 2 , the solution of the equations of motion (2.6)-(2.8) is given by where m is the mass and q the electric charge of the (phantom) black plane. This is the same solution as that of [7], for η = 1, and phantom black plane solution for η = -1, obtained for the first time here. We can rewrite the solution in terms of the densities of mass M and electric charge Q , as calculated in [7], yielding One can calculate the horizon of this solution, vanishing A ( r ), obtaining This solution possesses two complex and two real roots. The real roots are given by For the normal solution, η = 1, one has 0 < r -< r + , and for η = -1, the corresponding is r -< 0 < r + , with r + > | r -| . We observe that in the phantom solution, r -is in the negative part, but here something happens that we do not have in the spherical symmetry, because as r ± = | z 1 , 2 | , one gets z 1( ± ) = ± r + and z 2( ± ) = ± r -. As r -< 0, one gets z 1( -) < z 2(+) < 0 < z 2( -) < z 1(+) . Then, the singular plan z = r s = z s = 0 is covered by the plans z = z 1( -) , z = z 2(+) , z = z 2( -) and z = z 1(+) (see Figure 1). In the case of spherical symmetry, the internal horizon r -could not be achieved, for a solution of non-degenerate horizon. Hence, here we have a drastic change in the causal structure of the phantom black plane solution, whose singularity is covered by two horizons in the positive part of z . This could not occur in the phantom solutions with spherical symmetry, where just one horizon covered the singularity. However, another unusual event happens, where we get two horizons but with the property of non existence of extreme case, i.e, these horizons can never be equal, when we consider only real values. The curvature scalar of the metric (2.4) is given by The scalar of Kretschmann is given by By substituting A ( r ) = B -1 ( r ) and C ( r ) in (2.11), the curvature scalar ( R = 12 α 2 ) and that of Kretschmann are finite throughout the space-time, except in the singular plane r s = z = 0. In order to construct the Penrose diagram of this solution, we define several new coordinates for getting a description (non-singular on the horizons) of this space-time of type Kruskal. So, the Eddington-Finkelstein coordinates are gives by where the tortoise coordinate is give by With these coordinates, we can rewrite the line element (2.11) as Also defining the coordinates of type Kruskal we can rewrite (2.19) as With the use of these coordinates we can construct the causal structure of this solution, which is very similar to the Reissner-Nordstrom-AdS one (see Figure 2). We can see in Figure 2 that if we think to follow the decreasing z , starting from positive infinity, we have the region Z 1 ( z 1(+) < z < + ∞ ), passing by the first horizon at z = z 1(+) , for the second region Z 2 ( z 2( -) < z < z 1(+) ). After we passed the second horizon at z = z 2( -) , for the third region Z 3 (0 ≤ z < z 2( -) ). Now, we are interested in the geometrical analysis representing semiclassical gravitational effects of the black hole solutions as mentioned before. After arriving at the singular plane at z = 0 5 . These regions z ≥ 0 are causally disconnected from those for which z ≤ 0. Regions from Z 4 to Z 6 are the exact reflection (symmetrical values of positive z ) for positive values of z . So, we can think alike to follow a direction of creasing values of z , beginning at negative infinity. Thus, we perform the reflected route, and spent from Z 6 ( -∞ < z < z 1( -) ) to Z 5 ( z 1( -) < z < z 2(+) ), and then, to the region Z 4 ( z 2(+) < z ≤ 0), reaching the singular plane at z = 0. By semi-classical we mean quantize the called matter fields, while the background gravitational field is treated classically. Therefore, we will work with the semi-classic thermodynamics of black holes, studied first by Hawking [1], and further developed by many other authors [18]. There are several techniques to derive the Hawking temperature law. For example we can mention the Bogoliubov coefficients [19] and the energymomentum tensor methods [2, 18], the euclidianization of the metric [20], the transmission and reflection coefficients [21, 22], the analysis of the anomaly term [23], and the black hole superficial gravity [24]. Since all these methods have been proved to be equivalent [25], then we opt, without loss of generality, to calculate the Hawking temperature by the superficial gravity method. The surface gravity of a black plane is given by [7]: where r + is the event horizon radius, and the Hawking temperature is related with the surface gravity through the relationship [1, 24] Then, for the black plane solution (2.11), we get the surface gravity (2.25) as and the Hawking temperature (2.26) in this case is : We define the entropy per unit of area of the black plane as two times the quarter of the horizon area where the factor 2 is due to the contribution of two planes z = ± r + . From (2.11), we can calculate the electric potential scalar at the horizon Let us check the first law for the solution (2.11). Taking the differential of the mass, isolated from (2.12), of the electric charge and of the entropy (2.29), we get which satisfies the first law of thermodynamics Note that we introduced a compensating sign η in (2.32) due to the contribution of the negative energy density, in the phantom case, the field of spin 1, F µν , which provides a work with an inverted sign in the first law. As we need to study the thermodynamic system through the geometric methods, we must first write the mass in terms of the entropy and the electric charge. We can do this by isolating the mass in (2.12) and then replace r + in terms of the entropy 6 , with the use of (2.29), which yields where we have the conditions Q 2 ≤ (3 α 6 / 4 π 2 )( πM/α 4 ) 4 / 3 for η = 1 [7] (real horizon in (2.13)) and Q 2 ≤ ( α 2 S 2 /π 2 ) for η = -1. We also write the temperature and the electric potential in terms of the entropy and the electric charge. Taking (2.28) and (2.30), for r + in terms of the entropy, we get We can then calculate the specific heat by the expression We now have in hand the basic requirements to begin our analysis of the thermodynamic system of these solutions. In the next section we will study the specific heat (2.35) and through the four geometric methods, the thermodynamic properties of these planar solutions.", "pages": [ 4, 5, 6, 7, 8, 9, 10, 11 ] }, { "title": "3 Thermodynamics of black plane", "content": "In this section we will study in detail the thermodynamic properties of the planar solutions (2.11), both for normal and phantom cases. Through the specific heat and the curvature scalar of the thermodynamic spaces of the equilibrium states, we will examine whether there is an extreme case (only by the usual method), phase transition and finally, the local and global stabilities of the thermodynamic system.", "pages": [ 11 ] }, { "title": "3.1 Analysis of specific heat", "content": "Historically, the study of specific heat for revealing the thermodynamic properties was the first to be used [2] and has been called of usual method. Here, we have the expression of the specific heat (2.35), which, equating to zero, reveals the value of the entropy for which the solution is extreme, i.e, for S = S e = πQ √ η/α √ 3, which is real only for η = 1. Therefore, there does not exist an extreme case for the phantom solution with η = -1, as we had seen in its causal structure. Similarly, we can find the value of the entropy for which the system undergoes a second order phase transition, i.e, when the specific heat diverges. In this case the specific heat (2.35) diverges for S = S t = -i √ ηπQ/α , which shows that the normal case η = 1 has no phase transition, while the phantom case possesses a phase transition in S = S t . Note that this case is the specific value where the mass (2.33) vanishes. So, here, we have a mathematical chance of the system going from a locally stable phase ( C Q > 0 and positive mass), for an unstable phase, with C Q < 0 and negative mass (2.33). The phase transition of second order is not physically possible because the energy of the phantom black plane should be reduced continuously such that it passes from the positive values to zero, and even reaching negative values. This will be well examined in the stability study of the system. We plot the evolution of the specific heat (2.35) for a specific choice of the parameters, as shown in Figure 3. We will take the results of the study of specific heat as the basis for comparing with a geometric analysis of the thermodynamic system, through the four most popular methods in the literature. All these methods have in common the definition of a metric for the thermodynamic space of the equilibrium states, where the calculation of the curvature scalar of this metric reveals the existence or not of thermodynamic interaction, phase transition points, among other thermodynamic properties. Let us calculate this object with the aid of a mathematical software. In the next subsection we will analyse the thermodynamic system through the method of Weinhold.", "pages": [ 11, 12 ] }, { "title": "3.2 The Weinhold method", "content": "Historically, Weinhold was one of the first to formulate a geometric description applicable to a thermodynamic system. The method of Weinhold [3], as it is known, aims to define a metric for the thermodynamic space of the equilibrium states, through the mass (2.33) as thermodynamic potential. The metric constructed in this way provides a curvature scalar R W , which, for this method can be interpreted as a function of extensive variables that shows the points of phase transition, when there exists, where the thermodynamic system goes by. Then, we define the metric of Weinhold as being Here we see that the curvature scalar R W of this metric is identically zero, which prevents us of doing an analysis of the phase transition of the thermodynamic system. This result does not agree with the study of the specific heat. In the next subsection we will study the thermodynamics through the method of geometrothermodynamics.", "pages": [ 13 ] }, { "title": "3.3 The Geometrothermodynamics method", "content": "/negationslash The Geometrothermodynamics (GTD) [5] makes use of differential geometry as a tool to represent the thermodynamics of physical systems. Let us consider the (2 n + 1)-dimensional space T , whose coordinates are represented by the thermodynamic potential Φ, the extensive variable E a and the intensive variables I a , where a = 1 , ..., n . If the space T has a non degenerate metric G AB ( Z C ), where Z C = { Φ , E a , I a } , and the so called Gibbs 1-form Θ = d Φ -δ ab I a dE b , with δ ab the delta Kronecker; then, the structure ( T , Θ , G ) is said to be a contact riemannian manifold if Θ ∧ ( d Θ) n = 0 is satisfied [26]. The space T is known as the thermodynamic phase space. We can define a n -dimensional subspace E ⊂ T , with extensive coordinates E a , by the map ϕ : E → T , with Φ ≡ Φ( E a ), such that ϕ ∗ (Θ) ≡ 0. We call the space E the thermodynamic space of the equilibrium states. We can then define the metric of the thermodynamic space of the equilibrium states E , through the derivation of the thermodynamic potential and its extensive variables as [27] which, by definition, is invariant under Legendre transformations. Through the metric (3.37), we can calculate the curvature scalar of the space E , which informs if the system passes by a phase transition, when the scalar diverges for some value of extensive coordinates. If the scalar is not zero, the system possesses thermodynamic interaction, i.e, the Hawking temperature is non null. Here, we will do the calculation of the metric of E , using the mass (2.33) as the thermodynamic potential, which provides The curvature scalar of this metric is given by We get the value for which the scalar (3.39) diverges, which is given by S t = -i √ ηπQ/α , in agreement with the value obtained through the specific heat (2.35). This result is consistent with the specific heat, where we have found that the normal case has no phase transition and in the phantom case has one point of second order phase transition in S = S t . In the next subsection we will see the analysis made by the geometric method of Liu-Lu-Luo-Shao.", "pages": [ 13, 14 ] }, { "title": "3.4 The Liu-Lu-Luo-Shao method", "content": "The geometric method of the analysis of the more recent thermodynamic system is that of Liu-Lu-Luo-Shao [6], which defines a metric in the thermodynamic space of the equilibrium states, based on the Hessian matrix of several free energy, the Helmholtz's one in our case, and which can be written as follows The curvature scalar of this metric is given by Then, the analysis by this method shows that the normal case does not possess phase transition and the phantom case possess a transition phase at S = S t = -i √ ηπQ/α , which is in agreement with the specific heat. In the next subsection we will study the local and global stabilities of the black plane solutions.", "pages": [ 14, 15 ] }, { "title": "3.5 The local and global stability", "content": "Let us now study the local and global stabilities of these solutions. Through the specific heat (2.35) 7 and the temperature (2.34), one can see that in the normal case, η = 1, the system is locally stable for 3 α 2 S 2 > π 2 Q 2 , with C q , T > 0, and unstable for the other values. In the phantom case, η = -1, the system presents a local stability for α 2 S 2 > π 2 Q 2 , with C q , T, M > 0 (see Figure 3). Defining the Gibbs's potential we get that in the normal case, in the grand canonical ensemble, the system is globally stable for any values of S and Q , with G < 0 , ∀ S, Q . But in the phantom case, the system is globally stable only if α 2 S 2 > π 2 Q 2 , which agrees with the local stability of the specific heat. Here it is clear that both the specific heat and the Gibbs potential are closely linked to the sign of the mass (2.33). We have already seen from the specific heat that the mass value, zero, is precisely the point of phase transition of the phantom case. Here, it is also clear from the Gibbs potential that, passing to the negative values of the energy (mass), the system is unstable, not only locally, but also globally. This shows that the system can not move to that physically impossible stage. The explanation is that, when the system loses its energy, approaching zero, this should be treated by a more elaborated quantization, and not a simple semi-classical analysis, as we see here. Thus, we can conclude here that the phase transition presented by the phantom case, is nothing more than a purely mathematical transition, showing a divergence in the specific heat, but which is physically inaccessible to the states of the thermodynamic system. In the canonical ensemble, we can define the Helmholtz free energy as which yields a globally stable system ( F < 0), for the normal case, when α 2 S 2 > 3 π 2 Q 2 , and for the phantom case F < 0 , ∀ S, Q .", "pages": [ 15, 16 ] }, { "title": "4 Conclusion", "content": "We obtained a new phantom black plane solutions in (2.11). We analysed their basic geometric properties, the causal structure, obtaining the thermodynamic variables, temperature (2.28), entropy density (2.29) and the electric potential (2.30). We established the first law of thermodynamics in (2.32) and calculated the specific heat (2.35). We analysed the thermodynamic system through the study of the specific heat and the geometric methods called Weinhold, the geometrothermodynamics and that of Liu-Lu-Luo-Shao. In the Weinhold's case, the space metric is not invariant under Legendre transformations, and thus cannot reconcile a good thermodynamic analysis, therefore, in general, this method cannot agree with that of specific heat. By the use of the geometrothermodynamics and the method of Liu-Luo-Shao, we obtain the same results as in the case of specific heat, which shows that these two geometric methods agree with the usual one. The summarized results are that the normal case possesses an extreme limit for S = S e = πQ √ η/α √ 3, and the phantom case presents a phase transition point in S = S t = -i √ ηπQ/α , which represents a solution with mass (2.33) identically null. The interpretation of massless solutions has been presented in [21], but without any conclusion about its thermodynamics. The normal case presents locally stable thermodynamic system, for 3 α 2 S 2 > π 2 Q 2 , and globally stable, in grand canonical ensemble, when G < 0 , ∀ S, Q , and in canonical ensemble for α 2 S 2 > 3 π 2 Q 2 . On the other hand, the phantom case is locally stable when α 2 S 2 > π 2 Q 2 , and globally stable, in grand canonical ensemble, when α 2 S 2 > π 2 Q 2 , and in canonical ensemble, when F < 0 , ∀ S, Q . We conclude with the most important result here, which is the demonstration that normal and phantom cases have no physical phase transition, and that the normal case is an extreme case but not the phantom one. Acknowledgement : M. E. Rodrigues thanks a lot UFES and PPGF of the UFPA for the hospitality during the elaboration of this work and also CNPq for financial support. S. J. M. Houndjo thanks CNPq/FAPES for financial support.", "pages": [ 16, 17 ] } ]

[ { "title": "Stationary scalar configurations around extremal charged black holes", "content": "Juan Carlos Degollado 1, ∗ and Carlos A. R. Herdeiro 1, † 1 Departamento de F´ısica da Universidade de Aveiro and I3N, Campus de Santiago, 3810-183 Aveiro, Portugal. We consider the minimally coupled Klein-Gordon equation for a charged, massive scalar field in the non-extremal Reissner-Nordstrom background. Performing a frequency domain analysis, using a continued fraction method, we compute the frequencies ω for quasi-bound states. We observe that, as the extremal limit for both the background and the field is approached, the real part of the quasi-bound states frequencies R ( ω ) tends to the mass of the field and the imaginary part I ( ω ) tends to zero, for any angular momentum quantum number /lscript . The limiting frequencies in this double extremal limit are shown to correspond to a distribution of extremal scalar particles, at stationary positions, in no-force equilibrium configurations with the background. Thus, generically, these stationary scalar configurations are regular at the event horizon. If, on the other hand, the distribution contains scalar particles at the horizon, the configuration becomes irregular therein, in agreement with no hair theorems for the corresponding Einstein-Maxwell-scalar field system. PACS numbers: 04.70.Bw; 04.30.Nk; 04.40.Nr", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Scalar test fields in black hole (BH) geometries do not admit, generically, stationary configurations with an asymptotic decay and with real frequencies, i.e bound states. This follows from the physical requirement that only ingoing waves can exist at the horizon, therefore preventing a real equilibrium configuration between the field and the BH. Consequently, the configurations allowed in BH backgrounds are quasi -bound states, for which the frequencies are complex, with the imaginary part revealing a time dependence for the states, signaling either their absorption or, in the case of superradiant instabilities, their amplification by the BH [1]. The profile and some of the physical properties of quasi-bound states diverge at the horizon. This is intimately related to the inability that BH backgrounds have to accommodate, in a regular fashion, the scalar field, as an exact stationary solution, a property established by no-hair theorems [2, 3]. But even with this caveat such quasi-bound states are informative. For instance in [4], performing numerical simulations and starting with regular initial data for a scalar field around a Schwarzschild BH, there were found damped oscillating solutions with frequency and decay rate described by the real and imaginary parts of quasi-bound state frequencies. These decay rates can be very small [5] and thus long lived scalar field configurations could exist around BHs, even though eternal and regular configurations are, in general, precluded by no-hair theorems. In this work we provide an example in which scalar field configurations around a BH can become stationary and an interpretation to justify why this is possible. We start by finding the quasi-bound states for a charged massive scalar field in a non-extremal Reissner-Nordstrom (RN) BH, and show that these states have a simple limiting behaviour as extremality for both the BH and the field is approached: the imaginary part of the frequency vanishes and the real part of the frequency tends to the test field mass (and charge). Then, the scalar field configurations obtained in this limit are understood as the electrostatic potential of a distribution of extremal scalar particles in equilibrium with the extremal BH, thus providing examples of scalar fields around extremal charged BHs which are regular on the event horizon. Such configurations do not preserve spherical symmetry around the BH and that is the way they circumvents no-hair theorems. This paper is organized as follows. In Sec. II we discuss solutions of the minimally coupled Klein-Gordon equation on the RN background. Quasi-bound state solutions of this equation are considered in more detail in Sec. III, where some explicit frequencies are computed and analyzed. In Sec. IV the extremal limit is discussed, by computing the states with frequencies equal to the limiting behaviour observed in Sec. III, and an interpretation for these states is given. We draw some concluding remarks and comment on the non-linear solution including the backreaction of the scalar field in Sec. V.", "pages": [ 1 ] }, { "title": "II. BACKGROUND AND TEST FIELD", "content": "We consider a massive, charged scalar field, Φ, with mass µ and charge q , obeying the wave equation where ˆ D ν ≡ D ν -iqA ν . This field is propagating in the background of a Reissner-Nordstrom BH with charge Q and mass M : where f ( r ) = ( r -r + )( r -r -) /r 2 , r ± ≡ M ± √ M 2 -Q 2 and A ν dx ν = -Q/rdt . Taking the standard ansatz for the scalar field which reflects the spherical symmetry and staticity of the background: where Y m /lscript are the spherical harmonics and ω the complex frequency of a wave, (1) yields the radial equation for each mode: where U = r 2 [ ( ωr -qQ ) 2 -f ( µ 2 r 2 + /lscript ( /lscript +1)) ] . Observe that the azimuthal quantum number m is irrelevant due to spherical symmetry. The solution for the field Φ is immediately obtained by solving the radial equation for each mode (4), due to the linearity of the wave equation (1). In terms of Z ( r ) = r R ( r ), and dropping the subscript /lscript for notation simplicity, (4) becomes where we have defined the effective potential Alternatively, introducing the Regge-Wheeler radial coordinate r ∗ , by dr ∗ = dr/f ( r ), this wave equation is rewritten as where r = r ( r ∗ ). The properties of the potential V eff ( r ) have been discussed in the past, see eg. [6]. In particular one can show that the height of the centrifugal barrier increases with the charge of the field and that the constant value of the potential near the outer horizon also increases with the charge of the field but only up to some maximum; then it starts decreasing. The main feature of this potential, however, is that for a given combination of the parameters it exhibits a well, that can be considered as one of the key ingredients to have quasi-bound states. In order to solve the differential equation (5) we must provide a set of suitable boundary conditions at the horizon and at spacial infinity. To see the most relevant feature of the near horizon behaviour we note that in this region equation (7) becomes to leading order: where φ + = Q/r + is the electrostatic potential of the external horizon. This equation is solved by a superposition of in and outgoing waves. Choosing the solution where ω c ≡ qφ + , corresponding to an ingoing wave for q = 0, one observes the salient feature that it becomes an outgoing wave for ω < ω c (in this electromagnetic gauge). This is the condition for superradiance . Asymptotically, keeping the terms of order 1 /r in equation (4) one gets where From (10) one observes a qualitatively distinct behaviour depending on the sign of the real part of χ , R ( χ ). In particular, for R ( χ ) < 0 we have quasi-bound states . These are characterized by a decaying behaviour at spatial infinity. For R ( χ ) > 0 we have scattering states . Hereafter we will be interested in quasi-bound states.", "pages": [ 1, 2 ] }, { "title": "III. SEMI-ANALYTIC GLOBAL SOLUTION: QUASI-BOUND STATES FREQUENCIES", "content": "To find the solution of equation (5) in the region r > r + we will use a continued-fraction procedure developed by Leaver to find the quasinormal modes for the Schwarzschild and Kerr BHs [7]. 1 This amounts to take a power series ansatz with a pre-factor adapted to the boundary conditions observed in the previous section where Substituting (12) into (5) we obtain a three term recurrence relationship for the a n of the form: with The constants c i are lengthy expressions but otherwise straightforward to obtain. For a non charged massive scalar field on a Schwarzschild background, these expressions reduce to the ones (in the same limit) given in Ref. [9]. The procedure to find the frequencies of the quasibound states consists in setting the three term relationship (14) in the form of a continued fraction algebraic equation and then solving it with a root finding procedure. Furthermore, we have checked the value of the frequencies obtained with the continued fraction method by numerically integrating the radial equation. We took as initial condition the behaviour of the function close to the horizon and integrated up to some large r compared with the horizon radius. /negationslash The observed trend for the frequency of the quasibounded states is that, as the test field becomes extremal ( µ = | q | ), the imaginary part of the frequency decreases. The imaginary part of the frequency is a measure of the rate at which the field falls into the horizon. For the charged massive scalar field, the closer the background black hole is to extremality the smaller this imaginary part is for the fundamental tone. The real part of the frequency, on the other hand, tends to increase and converge towards the field mass as the BH charge is increased to extremality. This trend is shared by modes with different angular momentum quantum number. To make clear the behaviour of both the real and imaginary parts of the quasi-bound states frequencies as the double extremality is attained ( | Q | , | q | → M,µ ), we plot them against the BH charge for q/µ = 1 in Fig. 1. Concerning the radial function as the double extremal limit is taken, the profile of the R 0 ( r ) mode is qualitatively different from the profiles for the others modes. When /lscript = 0 ( s -wave), the radial function is localized in the region between the external horizon and the maximum of the potential barrier. As | Q | /M → 1, R 0 ( r ) becomes narrower in such a way that its maximum tends to the horizon. For /lscript = 0, the functions R /lscript ( r ) tend to spread out as the double extremal limit is approached - Fig 2.", "pages": [ 2, 3 ] }, { "title": "IV. EXTREMAL BLACK HOLE", "content": "The results of the previous section describe the limiting behaviour of the quasi-bound states as the double extremal limit is attained. We shall now consider exactly this limit by focusing on the extremal ReissnerNordstrom BH, | Q | = M ( r ± = M ), with an extremal test field ( µ = | q | ). Then, using a new radial coordinate ρ ≡ r -M , for states with ω = µ = | q | and qQ > 0, the radial wave equation (4) reduces to This equation is the radial part of the Laplace equation on Euclidean 3-space E 3 whose solution is R /lscript = A /lscript ρ /lscript + B /lscript /ρ /lscript +1 . The spacial part of the scalar field Φ is therefore a linear combination of harmonic functions Each of these partial waves, with appropriate A /lscript , B /lscript , describes the double extremal limit of a quasi-bound state. To interpret the meaning of the modes (16) and understand their appearance, it is useful to rewrite the extremal RN background using the coordinate ρ ; this corresponds to isotropic coordinates . Then the fields take the form where furthermore ρ = √ δ ij x i x j ≡ | x | and H is a harmonic function on Euclidean 3-space with a simple pole localised at the origin: H = 1 + M/ | x | . In these coordinates x = 0 is the location of the extremal RN BH horizon. Taking the scalar field in the form Φ( t, x i ) = e -iµt ˜ H ( x i ), the wave equation (1) with µ = | q | in the background (17), yields the harmonic equation ∆ E 3 ˜ H = 0. One solution is the harmonic function with a single pole at x ' : ˜ H = µ/ | x -x ' | . This describes the electric potential of one particle located at this pole. Expressed r in terms of spherical coordinates on E 3 , ( ρ, θ, φ ), chosen such that x ' lies at coordinates ( ρ ' , θ ' = 0 , φ ' ), then /negationslash where /negationslash So ˜ H is indeed the spatial part of (16), with A /lscript = 0. This fact shows that the radial profiles (16) correspond to partial waves for an extremal point-like scalar source of mass and charge µ displaced from the BH horizon, i.e. at ρ = 0. If, one the other hand, the particle is at ρ = 0, it follows from (18)-(19) that only the s -wave appears. There is an agreement between the behaviour of the s -wave seen in Sec. III as extremality is approached, and the one exhibited for the extremal case in this section. For the latter, a pure s -wave corresponds to a source localised at the horizon, whereas in the former, the peak of the radial function tends to the origin, as displayed in the top panel of Fig. 2. A related observation is that the harmonic function ˜ H is regular at the horizon except if the source is localised there. A similar interpretation for the modes in (16) carries through if ˜ H represents multiple scalar sources instead of a single one. We then have a superposition of harmonic functions with localised poles at fixed positions x ' k , corresponding to spherical coordinates ( ρ ' k , θ ' k , φ ' k ), which may again be rewritten, in spherical coordinates, as the multipolar expansion in the right hand side of (18), with B /lscript ( ρ ' , θ ' , φ ' ) replaced by ∑ k B /lscript ( ρ ' k , θ ' k , φ ' k ), corresponding to replacing one particle by many particles. The existence of stationary scalar states and their interpretation can, furthermore, be generalized to a background with multiple extremal BHs instead of a single one. This is achieved replacing H by a superposition of harmonic functions with localised poles at different points, x i , H = 1+ ∑ i M i / | x -x i | , since the scalar field equation still reduces to a harmonic equation on E 3 . Such solution of the Einstein-Maxwell system is the well known Majumdar-Papapetrou multi BH solution [10, 11], corresponding to a collection of BHs with mass and charge M i = Q i , placed at arbitrary positions x i [12], held in equilibrium by a balance between gravitational and electrostatic forces. A multiple scalar particle configuration will be regular on each horizon of the MajumdarPapapetrou background as long as x i = x ' k , for all i, k . The scalar particles are in equilibrium with the BHs due to a 'no-force' condition, a balance between gravitational and electromagnetic forces, as can be easily checked by studying the orbits generated by the Lagrangian L = µ √ -g αν ˙ x α ˙ x ν + µA α ˙ x α in the background (17), where 'dot' denotes derivative with respect to proper time. This Lagrangian is adequate to describe the interaction of the scalar particles with the background (17) because, at linear level, there is no interaction mediated by the scalar field; only gravitational and electromagnetic interactions occur. The gravitational energy added to the system in equilibrium - the multi BH solution - by the massive scalar field is balanced by the electromagnetic energy carried by the field. /negationslash", "pages": [ 3, 4 ] }, { "title": "V. CONCLUSIONS", "content": "A scalar field on a BH geometry does not, generically, admit stationary configurations. In this note we have showed that an extremal scalar field in the background of a charged, extremal BH geometry does admit such configurations and we have provided a physical interpretation for them. Our first observation was that the frequencies of quasibound states of a massive, charged, minimally coupled scalar field in the RN background have a well defined behaviour when a double extremal limit, for both the test field and the background is taken: the imaginary part vanishes and the real part becomes equal to the field mass. Then we showed that in such double extremal limit, configurations with a real frequency equal to the particle's mass exist, corresponding to a distribution of extremal scalar particles, placed at arbitrary locations in the exterior of the extremal (multi-)BH solution. If none of these particles sits at the BH horizon, the configuration is regular therein. One may argue, however, that the field is irregular at the location of the sources. But this is the traditional problem in classical field theory associated to point-like sources. The stationary scalar field states we have exhibited are due to no-force configurations between scalar sources and extremal BHs, at linear level: the gravitational attraction is being balanced by the electromagnetic repulsion. At non-linear level, however, the scalar field will back react on the geometry and, since it is charged, it will source the Maxwell field wherever the scalar field is non-trivial and not just at the location of the sources, in contrast to the typical multi-centre solutions found in Supergravity/String theory (see, e.g. [13]). It would will be interesting, but also challenging, to study the configurations we have analyzed herein at non-linear level, as solutions of the corresponding Einstein-Maxwell-scalar field theory.", "pages": [ 4, 5 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "We would like to thank Jo˜ao Rosa for discussions and comments on this draft. JCD Acknowledges CONACyT-M'exico support. This work was also supported by the NRHEP-295189 FP7-PEOPLE-2011IRSES Grant, and by FCT - Portugal through the project PTDC/FIS/116625/2010.", "pages": [ 5 ] } ]

[ { "title": "Valerio Faraoni", "content": "Physics Department and STAR Research Cluster Bishop's University, 2600 College St., Sherbrooke, Qu'ebec, Canada J1M 1Z7", "pages": [ 1 ] }, { "title": "Abstract", "content": "A massive scalar field in a curved spacetime can propagate along the light cone, a causal pathology, which can, in principle, be eliminated only if the scalar couples conformally to the Ricci curvature of spacetime. This property mandates conformal coupling for the field driving inflation in the early universe. During slow-roll inflation, this coupling can cause super-acceleration and, as a signature, a blue spectrum of primordial gravitational waves. Keywords: inflation; non-minimal coupling; early universe.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "A period of inflationary expansion of the early universe has gradually become to be accepted by most cosmologists as a paradigm of the modern scientific picture of the universe's history. Although there is no direct proof that inflation actually occurred, and it is healthy to contemplate alternatives, such as bouncing models [1, 2], the ekpyrotic universe [3, 4, 5] or string gas cosmology [6, 7, 8], the temperature anisotropies discovered by the COBE satellite and further studied by the WMAP and PLANCK missions have a spectrum close to the Harrison-Zel'dovich one predicted by inflation, which certainly is some support for the view of an inflationary early universe. Assuming that inflation occurred early on and that it was driven by some scalar field, φ (arguably the simplest, although not mandatory, class of inflationary scenarios), research has for long focused on identifying specific scenarios of inflation corresponding to particular choices of the scalar field potential, V ( φ ), motivated by particle physics. Here, we argue that the scalar field, φ , driving inflation, should be non-minimally (in fact, conformally) coupled to the Ricci curvature of spacetime, R , in order to avoid causal pathologies. Conformal (or, in general, non-minimal) coupling was originally introduced in radiation problems [9] or in the renormalization of scalar fields in curved backgrounds [10, 11, 12, 13, 14, 15, 16]. Therefore, it certainly is not obvious that a conventional minimally coupled scalar (with timelike or null gradient) can suffer from light cone pathologies, but this is indeed the case, as was pointed out long ago for test fields [17]. Let us revisit the argument and its consequences for inflation.", "pages": [ 2 ] }, { "title": "2 Non-Minimal Coupling", "content": "A scalar field, φ , with mass, m , propagating in curved spacetime satisfies the Klein-Gordon equation: where the dimensionless non-minimal coupling constant, ξ , between the scalar and the Ricci curvature is here allowed for generality (we will see that minimal coupling, corresponding to ξ = 0, is, in fact, ruled out). Here, /square = g µν ∇ µ ∇ ν , where g µν is the spacetime metric and ∇ µ is its covariant derivative operator. Consider the solution of Eq. (2.1) corresponding to a delta-like source, which is nothing but the Green function, G R ( x ' , x ), of this equation: where δ ( x ' , x ) is the spacetime delta. By imposing the usual boundary conditions, we are restricted to the retarded Green function. It is then well known [18, 19] that the retarded Green function, G R , can be split as: where Γ( x ' , x ) is the square of the proper distance between x and x ' calculated along the geodesic connecting these two spacetime points (which is unique in a normal domain), δ R (Γ) is the usual Dirac delta and Θ( -Γ) is the Heaviside step function with support in the past of x . The first term on the right hand side of Eq. (2.3) describes a contribution to φ ( x ) coming from the past light cone of x , while the second term describes a contribution from the interior of this light cone. The functions, Σ and W , are coefficients. If the curved spacetime manifold is to be approximated by its tangent space (which, loosely speaking, is the spirit of the Equivalence Principle of relativity), in the limit, x ' → x , in which the two points coincide, the Green function must reduce to the one of Minkowski space [17], i.e. , it must be: as x ' → x . It is rather straightforward to expand all these functions in this limit, obtaining [18, 20, 21, 17]: /negationslash where O( x ' , x ) generically denotes terms, which vanish as x ' → x . Backscattering of the scalar, φ , can be due to both a non-vanishing mass, m , or to the background curvature appearing in the term, -( ξ -1 6 ) R ( x ) 8 π in Eq. (2.6). If m = 0, at spacetime points where a massive scalar, φ , will propagate strictly along the light cone , which is clearly a causal pathology. It is even possible to concoct a space of constant curvature, R , such that the backscattering tail, due to the curvature, [ -( ξ -1 6 ) R ( x ) 8 π ] , exactly compensates the tail, [ -m 2 8 π ] , due to the mass, m . This pathology is possible for ξ = 0. Indeed, the only way to eliminate this disturbing possibility is to have ξ = 1 / 6 (conformal coupling); then, the propagation of a massive φ is forced to be inside the light cone. Note that conformal invariance has not been imposed or implied in any way. It is obtained simply to avoid causal pathologies. The physical interpretation of the result is the following: because only propagation along the light cone is involved in the argument, there must be no scale in the physics of the scalar field, which implies conformal invariance. If the argument above applies to a free test field, it will also apply to a scalar field in a generic potential, V ( φ ), and to a gravitating scalar field, which always has the previous case as a limit. Let us review briefly the various formulations of the Equivalence Principle. The Weak Equivalence Principle (WEP) states that if an uncharged test body is at an initial spacetime point with an initial four-velocity, its subsequent trajectory will not depend on its internal structure and composition. The Einstein Equivalence Principle (EEP) states that (a) WEP holds; (b) the outcome of any local non-gravitational test experiment is independent of the velocity of the freely falling apparatus (Local Lorentz Invariance, LLI); and (c) the outcome of any local non-gravitational test experiment is independent of where and when in the universe it is performed (Local Position Invariance, LPI). The Strong Equivalence Principle (SEP) consists of: (a) WEP holds for self-gravitating bodies, as well as for test bodies; (b) the outcome of any local test experiment is independent of the four-velocity of the freely falling apparatus (Local Lorentz Invariance, LLI); and (c) the outcome of any local test experiment is independent of where and when in the universe it is performed (Local Position Invariance, LPI). The WEP is a statement about mechanics: it requires only the existence of preferred trajectories, the free fall trajectories followed by test particles, and these curves are the same independently of the mass and internal composition of the particles that follow them (universality of free fall). By itself, WEP does not imply the existence of a metric or of geodesic curves (this requirement arises only through the EEP by combining the WEP with requirements (b) and (c) [22]. The EEP extends the WEP to all areas of non-gravitational physics. The SEP further extends the WEP to self-gravitating bodies and requires LLI and LPI to hold also for gravitational experiments, in contrast to the EEP. All versions of the Equivalence Principle have been subjected to experimental verification, but, thus far, stringent tests only exist for the WEP and the EEP [22]. Originally [17], the argument for ξ = 1 / 6 was presented as enforcing the EEP [22] applied to a test or a gravitating field, φ . A posteriori , however, there is no need to invoke the Equivalence Principle, and φ could be a gravitational scalar field (for example, in a scalar-tensor theory of gravity), about which the EEP has nothing to say. Although the argument supporting the value, 1 / 6, of the coupling constant, ξ (rather than the value, ξ = 0), relies only on the absence of causal pathologies in the propagation of φ -waves, it is interesting to elaborate on it in light of the recent paper [23] on theories of gravity satisfying the SEP. The author of [23] looks for ways to implement the SEP on theories of gravity and, on the basis of the analogy with the Standard Model of particle physics, concludes that the SEP is embodied by the condition on the Riemann tensor: which is analogous to the condition: for non-Abelian Yang-Mills fields of strength, F µν , which satisfy [ D µ , D ν ] = iF µν (where D µ is the covariant derivative). The Riemann tensor satisfies the analogous relation: (This characterization of the SEP, however, is different from the traditional one of, e.g., [22], presented above.) Eq. (2.9) expresses the condition that 'gravitons gravitate the same way that gluons glue' [23]. Consider general scalar-tensor theories of gravity described by the (Jordan frame) action: where the Brans-Dicke-like φ is of gravitational nature (we use units in which Newton's constant, G , and the speed of light, c , are unity and the Brans-Dicke coupling, ω ( φ ), is a function of φ ). In general, the gravitational or non-gravitational nature of a field depends on the conformal frame representation of the theory; see the discussion in [24]. In short, scalar-tensor gravity can be discussed in the Jordan frame (meaning the set of variables, ( g µν , φ )), in which the scalar field, φ , couples explicitly to the Ricci curvature and matter is minimally coupled (which has the consequence that massive test particles follow timelike geodesics). Alternatively, one can describe the theory in the Einstein conformal frame, the set of variables, ( ˜ g µν , ˜ φ ) , related to the Jordan frame by the conformal redefinition of the metric: and the non-linear field redefinition: In the Einstein frame, the scalar field has canonical kinetic energy and couples minimally to gravity ( i.e. , there is no explicit coupling between φ and R ), but it couples directly to the the matter Lagrangian in the action. As a consequence, uncharged particles in the Einstein frame do not follow geodesics of the metric, ˜ g µν , but deviate from them, due to a force proportional to the gradient of the scalar field. Massless particles, the physics of which is conformally invariant, follow null geodesics in both frames (e.g., [25]). It turns out that imposing the SEP condition (2.9) selects only two possible theories [23]. These are Nordstrom's scalar gravity (in which the metric is conformally flat and there is only a scalar degree of freedom) and the theory with: In the latter case, the field redefinition, φ → ϕ , with: recasts the action as: which is the action for a conformally coupled scalar field. In other words, insisting that the gravitational Brans-Dicke-like scalar field φ satisfies the EEP (or that the theory satisfies the SEP), leads to the requirement that it be conformally coupled. The traditional SEP amounts to imposing that the Weak Equivalence Principle of mechanics is satisfied also by gravitating bodies, plus local Lorentz invariance and local position invariance [22]. Following the definition of SEP adopted in [23], it would seem that the SEP would correspond to imposing the EEP also on gravitational fields. Now, if φ is a gravitational scalar field in a theory of gravity alternative to general relativity, there is no reason for it to satisfy the EEP. Moreover, the Brans-Dicke-like field of scalar-tensor gravity is not supposed to be the one driving inflation-even in the extended and hyperextended inflationary scenarios based on Brans-Dicke gravity and on more general scalar-tensor theories, respectively; it is a second non-gravitational scalar field that is responsible for inflation (see, e.g., the review in [25]). However, any field satisfying Eq. (2.1) should be conformally coupled, ξ = 1 / 6. Let us review the consequences of conformal coupling if φ is the scalar field driving inflation in the early universe.", "pages": [ 2, 3, 4, 5, 6 ] }, { "title": "3 Consequences for Inflation", "content": "It is well known that, if one quantizes a scalar field on a curved background, a non-minimal coupling to the Ricci scalar, R , is introduced, even if it was absent in the classical theory [10, 11, 12, 13, 14, 15, 16]. In asymptotically free grand unified theories, depending on the gauge group and the matter content, ξ is a running coupling and, generically, 1 / 6 is a stable infrared fixed point [26, 27, 28, 29, 30, 31, 32, 33]. According to the previous (classical) argument, the inflation field fueling inflation should be coupled conformally. Then, one should revisit inflation, keeping in mind that conformal coupling is not an option, but is required for consistency of the theory. Over the years, several authors have studied non-minimally coupled inflatons, usually in a rather opportunistic way, i.e. , the coupling constant, ξ , was usually considered as a free parameter to be adjusted at will in order to alleviate fine-tuning problems in the potential [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50]. Now the value of ξ is forced upon us. It has been demonstrated that viable scenarios of inflation for an unperturbed universe can occur with non-minimal coupling [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50]. A possible obstacle is the fact that the effective term, -ξRφ 2 / 2, in the Lagrangian could, in principle, spoil the flatness of an inflationary potential, V ( φ ) [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], but this difficulty is not crucial. What is more, new features of the dynamics emerge, which are not possible when ξ = 0 [51, 52]. By adopting a spatially flat Friedmann-Lemˆaitre-Robertson-Walker metric: where ρ and P are the energy density and pressure of the cosmic fluid, respectively, κ ≡ 8 πG ( G being Newton's constant) and an overdot denotes differentiation with respect to the comoving time, t . Eqs. (3.19) and (3.20) yield: and, therefore, P < -ρ (a 'phantom' equation of state) is equivalent to ˙ H > 0. A regime with ˙ H > 0, due to non-minimal coupling, called superinflation , was studied already in the 1980s [53, 54]. Minimally coupled scalar fields have ρ = ˙ φ 2 2 + V ( φ ) and P = ˙ φ 2 2 -V ( φ ); hence, the derivative ˙ H in Eq. (3.22) gives: By contrast, for a non-minimally coupled scalar field, it is: and ˙ H > 0 is a possibility. Indeed, exact solutions exhibiting explicitly this super-acceleration have been found in the context of early universe inflation [55, 56] and of present-day quintessence [57]. the field equations are: Technically speaking, the non-minimally coupled scalar field action: is a scalar-tensor action, and gauge-independent formalisms have been developed to study cosmological perturbations in this class of theories. One can fix a gauge and proceed to study perturbations in that gauge, but there is the risk that the results are unphysical, an artifact of pure gauge modes. Indeed, the gauge-dependence problem plagued the early studies of cosmological perturbations produced during inflation. An alternative is to identify gaugeinvariant variables and derive equations for these gauge-invariant quantities that assume the same form in all gauges. When this is done, a gauge-invariant formalism is obtained, which has the advantage of being completely gauge-independent and the disadvantage that the gaugeinvariant variables are not physically transparent-they can receive a physical interpretation once a gauge is fixed. The original gauge-invariant formalism, due to Bardeen [58], has been refined over the years and was designed for cosmology in the context of general relativity. Here, we adopt the Bardeen-Ellis-Bruni-Hwang formalism [58, 59, 60, 61], that is, a version of the Bardeen formalism [58], refined by Ellis, Bruni and Hwang and adapted by Hwang to a wide class of theories of gravity alternative to general relativity [62, 63, 64, 65, 66, 67, 68]. In fact, non-minimally coupled scalar field theory is a special case of scalar-tensor gravity, as can be seen by tracing in reverse the path outlined in Section 2, and it is straightforward to apply Hwang's formalism to this theory. The application of this formalism to non-minimally coupled inflation was reviewed in [69]. Slow-roll inflation with de Sitter universes as attractors in phase space is possible. There are four slow-roll parameters, as opposed to the two of minimally coupled inflation (for comparison, -/epsilon1 1 and -/epsilon1 2 coincide with the usual parameters, /epsilon1 and η , of minimally coupled inflation) [62, 63, 64, 65, 66, 67, 68, 69]: (with φ 1 , 2 constants), and /epsilon1 4 vanishes for ξ = 1 / 6. The spectral indices of scalar and tensor perturbations in the slow-roll approximation are then [62, 63, 64, 65, 66, 67, 68]: There is a possible signature of conformal coupling in the cosmic microwave background sky. In an inflationary super-acceleration regime, ˙ H > 0 (which is impossible with minimal coupling and realistic scalar field potentials), it is /epsilon1 1 > 0, and one can obtain a blue spectrum of gravitational waves, n T > 0. Blue spectra of tensor perturbations are impossible in the standard scenarios of inflation (they are, however, possible in certain non-inflationary scenarios) with ξ = 0 (for which n T = 4 ˙ H/H ≤ 0). More power is shifted to small wavelengths in comparison with minimally coupled inflation, which is interesting for the gravitational wave community, because it increases the chance of detecting cosmological gravitational waves with future space-based interferometers.", "pages": [ 6, 7, 8, 9 ] }, { "title": "4 Conclusions", "content": "Supporting the idea that the inflaton field is conformally, rather than minimally, coupled is actually a pretty conservative view. Not doing so means allowing for a possible pathology in the local propagation of the inflaton, i.e. , the possibility that this field propagates along the light cone when it is massive. This problem is even more serious during inflation because, in slow-roll, the cosmic dynamics are close to a de Sitter attractor for which R is constant, and one could even have the causal pathology mentioned above (or be very close to it) at every spacetime point. This would indeed be a radical departure from known physics, which cannot be justified. The only way out of this conundrum is if ξ = 1 / 6, and inflationary scenarios should be adapted to this constraint.", "pages": [ 9 ] }, { "title": "Acknowledgments", "content": "It is a pleasure to thank Sebastiano Sonego for his leading role in the investigation of nonminimal coupling long ago, which led to the development of this project. We thank also three referees for useful remarks. This work is supported by the Natural Sciences and Engineering Research Council of Canada.", "pages": [ 9 ] } ]

[ { "title": "ABSTRACT", "content": "Article", "pages": [ 1 ] }, { "title": "A Closer Earth and the Faint Young Sun Paradox: Modification of the Laws of Gravitation, or Sun/Earth Mass Losses?", "content": "Lorenzo Iorio 1 , * 1 Italian Ministry of Education, University and Research (M.I.U.R.)-Education, Fellow of the Royal Astronomical Society (F.R.A.S.), Viale Unit'a di Italia 68, 70125, Bari (BA), Italy. Tel. +39 3292399167 Received: xx / Accepted: xx / Published: xx Abstract: Given a solar luminosity L Ar = 0 . 75 L 0 at the beginning of the Archean 3 . 8 Ga ago, where L 0 is the present-day one, if the heliocentric distance r of the Earth was r Ar = 0 . 956 r 0 , the solar irradiance would have been as large as I Ar = 0 . 82 I 0 . It would allowed for a liquid ocean on the terrestrial surface which, otherwise, would have been frozen, contrary to the empirical evidence. By further assuming that some physical mechanism subsequently displaced the Earth towards its current distance in such a way that the irradiance stayed substantially constant over the entire Archean from 3 . 8 Ga to 2 . 5 Ga ago, a relative recession per year as large as ˙ r/r ≈ 3 . 4 × 10 -11 a -1 would have been required. Although such a figure is roughly of the same order of magnitude of the value of the Hubble parameter 3 . 8 Ga ago H Ar = 1 . 192 H 0 = 8 . 2 × 10 -11 a -1 , standard general relativity rules out cosmological explanations for the hypothesized Earth's recession rate. Instead, a class of modified theories of gravitation with nonminimal coupling between the matter and the metric naturally predicts a secular variation of the relative distance of a localized two-body system, thus yielding a potentially viable candidate to explain the putative recession of the Earth's orbit. Another competing mechanism of classical origin which could, in principle, allow for the desired effect is the mass loss which either the Sun or the Earth itself may have experienced during the Archean. On the one hand, this implies that our planet should have lost 2 % of its present mass in the form of eroded/evaporated hydrosphere. On the other hand, it is widely believed that the Sun could have lost mass at an enhanced rate due to a stronger solar wind in the past for not more than ≈ (0 . 2 to 0 . 3) Ga. Keywords: Archean period; Paleoclimatology; Solar physics; Experimental studies of gravity; Relativity and gravitation; Modified theories of gravity; Celestial mechanics Classification: PACS 91.70.hf; 92.60.Iv; 96.60.-j; 04.80.-y; 95.30.Sf; 04.50.Kd; 95.10.Ce", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The so-called 'Faint Young Sun Paradox' (FYSP) [1] consists in the fact that, according to consolidated models of the Sun's evolution history, the energy output of our star during the Archean, from 3 . 8 Ga to 2 . 5 Ga ago, would have been too low to keep liquid water on the Earth's surface. Instead, there are compelling and independent evidences that, actually, our planet was mostly covered by liquid water oceans, hosting also forms of life, during that eon. For a recent review of the FYSP, see [2] and references therein. The bolometric solar luminosity L measures the electromagnetic radiant power emitted by the Sun integrated over all the wavelengths. The solar irradiance I measured at the Earth's atmosphere is defined as the ratio of the solar luminosity to the area of a sphere centered on the Sun with radius equal to the Earth-Sun distance r ; in the following, we will assume a circular orbit for the Earth. Thus, its current value is [3] Setting the origin of the time t at the Zero-Age Main Sequence (ZAMS) epoch, i.e. when the nuclear fusion ignited in the core of the Sun, a formula which accounts for the temporal evolution of the solar luminosity reasonably well over the eons, with the possible exception of the first ≈ 0 . 2 Ga in the life of the young Sun, is [4] where t 0 = 4 . 57 Ga is the present epoch, and L 0 is the current Sun's luminosity. See, e.g., Figure 1 in [2], and Figure 1 in [5]. The formula of eq. (2) is in good agreement with recent standard solar models such as, e.g., [6]. According to eq. (2), at the beginning of the Archean era 3 . 8 Ga ago, corresponding to t Ar = 0 . 77 Ga in our ZAMS-based temporal scale, the solar luminosity was just Thus, if the heliocentric distance of the Earth was the same as today, eq. (3) implies that As extensively reviewed in [2], there is ample and compelling evidence that the Earth hosted liquid water, and even life, during the entire Archean eon spanning about 1 . 3 Ga. Thus, our planet could not be entirely frozen during such a remote eon, as, instead, it would have necessarily been if it really received only ≈ 75 % of the current solar irradiance, as it results from eq. (4). Although intense efforts by several researchers in the last decades to find a satisfactory solution to the FYSP involving multidisciplinary investigations on deep-time paleoclimatology [7], greenhouse effect [8], ancient cosmic ray flux [9], solar activity [10] and solar wind [11], it not only refuses to go away [2,12,13], but rather it becomes even more severe [14] in view of some recent studies. This is not to claim that the climatic solutions are nowadays ruled out [15,16], especially those involving a carbon-dioxide greenhouse in the early Archean and a carbon dioxide-methane greenhouse at later times [8,17,18]; simply, we feel that it is worthwhile pursuing also different lines of research. In this paper, as a preliminary working hypothesis, we consider the possibility that the early Earth was closer to the Sun just enough to keep liquid oceans on its surface during the entire Archean eon. For other, more or less detailed, investigations in the literature along this line of research, see [2,19-21]. In Section 2, we explore the consequences of such an assumption from a phenomenological point of view. After critically reviewing in Section 3 some unsatisfactorily attempts of cosmological origin to find an explanation for the required orbital recession of our planet, we offer some hints towards a possible solution both in terms of fundamental physics (Section 4) and by considering certain partially neglected classical orbital effects due to possible mass loss rates potentially experienced by the Sun and/or the Earth in the Archean (Section 5). Section 6 summarizes our findings.", "pages": [ 2, 3 ] }, { "title": "2. A working hypothesis: was the Earth closer to the Sun than now?", "content": "As a working hypothesis, let us provisionally assume that, at t Ar , the solar irradiance I Ar was approximately equal to a fraction of the present one I 0 large enough to allow for a global liquid ocean on the Earth. As noticed in [2], earlier studies [22-24] required an Archean luminosity as large as 98 % to 85 % of the present-day value to have liquid water. Some more recent models have lowered the critical luminosity threshold down to about 90 % to 86 % [25,26], with a lower limit as little as [26] Since the same heliocentric distance as the present-day one was assumed in the literature, eq. (5) is equivalent to the following condition for the irradiance required to keep liquid ocean By assuming I Ar = I oc , together with eq. (3), implies i.e. the Earth should have been closer to the Sun by about ≈ 4 . 4 % with respect to the present epoch. As a consequence, if one assumes that the FYSP could only be resolved by a closer Earth, some physical mechanism should have subsequently displaced out planet to roughly its current heliocentric distance by keeping the irradiance equal to at least I oc over the next 1 . 3 Ga until the beginning of the Proterozoic era 2 . 5 Ga ago, corresponding to t Pr = 2 . 07 Ga with respect to the ZAMS epoch, when the luminosity of the Sun was according to eq. (2). Thus, by imposing The plots of eq. (10)-eq. (11) are depicted in Figure 1. It can be noticed that a percent distance rate as large as is enough to keep the irradiance equal to about 82 % of the present one during the entire Archean by displacing the Earth towards its current location. A very important point is to search for independent evidences supporting or contradicting the hypothesis of a closer Earth at the beginning of the Archean. From the third Kepler law of classical gravitational physics, it turns out that, if the heliocentric distance of the Earth was smaller, then the duration of the year should have been shorter. As remarked in [20], in principle, a shorter terrestrial year one gets /OverDot should have left traces in certain geological records such as tidal rhythmites and banded iron formations. Actually, the available precision of such potentially interesting indicators is rather poor for pre-Cambrian epochs [27,28] to draw any meaningful conclusion. However, it cannot rule out our hypothesis. Finally, we wish to mention that a hypothesis somewhat analogous to that presented here was proposed in [29], although within a different temporal context. Indeed, in [29], by analyzing the measurements of the growth patterns on fossil corals, it was claimed that, at the beginning of the Phanerozoic eon 0 . 53 Ga ago, it was r Ph = 0 . 976 r 0", "pages": [ 3, 4, 5 ] }, { "title": "3.1. The accelerated expansion of the Universe", "content": "Given the timescales involved in such processes, it is worthwhile investigating if such a putative recessions of the Earth's orbit could be induced by some effects of cosmological nature, as preliminarily suggested in [29,30]. Such a possibility is made appealing by noticing that the rate in eq. (12) is of the same order of magnitude of the currently accepted value of the Hubble parameter [31] The Hubble parameter is defined as [32] where S ( t ) is the cosmological expansion factor; the definition of eq. (14) is valid at any time t . As a first step of our inquiry, an accurate calculation of the value of the Hubble parameter 3 . 8 Ga ago, accounting for the currently accepted knowledge of the cosmic evolution, is required. Let us briefly recall that the simplest cosmological model providing a reasonably good match to several different types of observations is the so-called Λ CDM model; in addition to the standard forms of baryonic matter and radiation, it also implies the existence of the dark energy, accounted for by a cosmological constant Λ , and of the non-baryonic cold dark matter. It relies upon the general relativity by Einstein as the correct theory of the gravitational interaction at cosmological scales. The first Friedmann equation for a Friedmann-Lemaˆıtre-Roberston-Walker (FLRW) spacetime metric describing a homogenous and isotropic non-empty Universe endowed with a cosmological constant Λ is [32] where k characterizes the curvature of the spatial hypersurfaces, S 0 is the present-day value of the expansion factor, and the dimensionless energy densities Ω i , i = R , NR , Λ , normalized to the critical energy density where G is the Newtonian constant of gravitation and c is the speed of light in vacuum, refer to their values at S = S 0 . Based on the equation of state relating the pressure p to the energy density ε of each component, Ω R refers to the relativistic matter characterized by p R = (1 / 3) ε R , Ω NR is the sum of the normalized energy densities of the ordinary baryonic matter and of the non-baryonic dark matter, both non-relativistic, while Ω Λ accounts for the dark energy modeled by the cosmological constant Λ in such a way that p Λ = -ε Λ . By keeping only Ω NR and Ω Λ in eq. (15), it is possible to integrate it, with k = 0 , to determine S ( t ) . The result is [32] For the beginning of the Archean eon, 3 . 8 Ga ago, eq. (17) yields Note that in eq. (17) t is meant to be counted from the Big-Bang singularity in such a way that the present epoch is [31] thus, has to be used in eq. (17) to yield eq. (18). Incidentally, from eq. (13) and eq. (19) it turns out As a consequence of eq. (18), the dimensionless redshift parameter at the beginning of the Archean is It is, thus, a-posteriori confirmed the validity of using eq. (17) for our purposes since, according to Type Ia supernovæ (SNe Ia) data analyses, the cosmic acceleration started at [33] corresponding to about 5 . 6 Ga to 3 . 5 Ga ago. From eq. (14) and eq. (17), it can be obtained (26) eq. (24), together with eq. (13) and eq. (20), yields Since [31] for the value of the Hubble parameter at the beginning of the Archean eon. At this point, it must be noticed that eq. (12) differs from eq. (27) at a ≈ 30 σ level. Even putting aside such a numerical argument, there are also sound theoretical reasons to discard a cosmological origin for the putative secular increase of the Sun-Earth distance at some epoch such as, e.g., the Archean or the Phanerozoic. It must be stressed that having at disposal the analytical expression of the test particle acceleration caused by a modification of the standard two-body laws of motion more or less deeply rooted in some cosmological scenarios is generally not enough. Indeed, it must explicitly be shown that such a putative cosmological acceleration is actually capable to induce a secular variation of the distance of the test particle with respect to the primary. In fact, it is not the case just for some potentially relevant accelerations of cosmological origin which, instead, have an impact on different features of the two-body orbital motion such as, e.g., the pericenter ω , etc. Standard general relativity predicts that, at the Newtonian level, no two-body acceleration of the order of H occurs [34,35]. At the Newtonian level, the first non-vanishing effects of the cosmic expansion are of the order of H 2 [34,35]; nonetheless, they do not secularly affect the mean distance of a test particle with respect to the primary since they are caused by an additional radial acceleration proportional to the two-body position vector r which only induces a secular precession of the pericenter of the orbit [36]. At the post-Newtonian level, a cosmological acceleration of the order of H and proportional to the orbital velocity v of the test particle has recently been found [37]. In principle, it is potentially interesting since it secularly affects both the semimajor axis a and the eccentricity e in such a way that r = a (1 + e 2 / 2) changes as well [38]. Nonetheless, its percent rate of change is far too small since it is proportional to H ( v/c ) 2 [38]. As far as the acceleration of the cosmic expansion, driven by the cosmological constant Λ , is concerned, it only affects the local dynamics of a test particle through a pericenter precession, leaving both a and e unaffected [39].", "pages": [ 5, 6, 7 ] }, { "title": "3.2. A time-dependent varying gravitational parameter G", "content": "The possibility that the Newtonian coupling parameter G may decrease in time in accordance with the expansion of the Universe dates back to the pioneeristic studies by Milne [40,41], Dirac [42], Jordan [43]. As a consequence, also the dynamics of a two-body system would be affected according to Nonetheless, the present-day bounds on the percent variation rate of G [44,45] ∣ ∣ ∣ inferred from the analysis of multidecadal records of observations performed with the accurate Lunar Laser Ranging (LLR) technique [46], are smaller than eq. (12) by two orders of magnitude. It could be argued that, after all, the constraints of eq. (29)-eq. (30) were obtained from data covering just relatively few years if compared with the timescales we are interested in. Actually, in view of the fundamental role played by G , its putative variations would have a decisive impact on quite different phenomena such as the evolution of the Sun itself, ages of globular clusters, solar and stellar seismology, the Cosmic Microwave Background (CMB), the Big Bang Nucleosynthesis (BBN), etc.; for a comprehensive review, see [47]. From them, independent constraints on ˙ G/G , spanning extremely wide timescales, can be inferred. As it results from Sect. 4 of [47], most of the deep-time ones are 2 -3 orders of magnitude smaller than eq. (12).", "pages": [ 7, 8 ] }, { "title": "4.1. Modified gravitational theories with nonminimal coupling", "content": "If standard general relativity does not predict notable cosmological effects able to expand the orbit of a localized two-body system, it can be done by a certain class [48] of modified gravitational theories with nonminimal coupling between the matter and the gravitational field [49]. This is not the place to delve into the technical details of such alternative theories of gravitation predicting a violation of the equivalence principle [48-50]. Suffice it to say that a class of them, recently investigated in [48], yields an extra-acceleration A nmc for a test particle orbiting a central body which, interestingly, has a long-term impact on its distance. In the usual four-dimensional spacetime language, a non-geodesic four-acceleration of a non-rotating test particle [48] occurs. We adopt the convention according to which the Greek letters are for the spacetime indices, while the Latin letters denotes the three-dimensional spatial indices; in [48] the opposite convention is followed. In eq. (31), m is the mass of the test particle as defined in multipolar schemes in the context of general relativity, δ µ ν is the Kronecker delta in four dimensions, v µ , v ν are the contravariant and covariant components of the the four-velocity of the test particle, respectively, ( v 0 = v 0 , v i = -v i , i = 1 , 2 , 3 ), ξ is an integrated quantity depending on the matter distribution of the system, K µ . = ∇ µ ln F, where ∇ µ denotes the covariant derivative, and the nonminimal function F depends arbitrarily on the spacetime metric g µν and on the Riemann curvature tensor R β µνα . From eq. (31), the test particle acceleration written in the usual three-vector notation, can be extracted. In deriving eq. (32), we assumed the slowmotion approximation in such a way that v µ ≈ { c, v } . Astraightforward but cumbersome perturbative calculation can be performed with the standard Gauss equations for the variation of the Keplerian orbital elements [51], implying the decomposition of eq. (32) along the radial, transverse and normal directions of an orthonormal trihedron comoving with the particle and their evaluation onto a Keplerian ellipse, usually adopted as unperturbed reference trajectory. Such a procedure, which has the advantage of being applicable to whatsoever perturbing acceleration, yields, to zero order in the eccentricity e of the test particle, the following percent secular variation of its semimajor axis It must be stressed that, for the quite general class of theories covered in [48], m , ξ, K 0 are, in general, not constant. As a working hypothesis, in obtaining eq. (33) we assumed that they can be considered constant over the period of the test particle. Thus, there is still room for a slow temporal dependence with characteristic time scales quite larger that the test particle's period. Such a feature is important to explain the fact that, at present, there is no evidence for any anomalous increase of the Sun-Earth distance as large as a few meters per year, as it would be required by eq. (12). Indeed, it can always be postulated that, in the last ≈ 2 Ga, m , ξ, K 0 became smaller enough to yield effects below the current threshold of detectability which, on the basis of the results in [52], was evaluated to be of the order of [38] ≈ 1 . 5 × 10 -2 ma -1 for the Earth. The rate of change of eq. (33) is an important result since it yields an effect which is rooted in a well defined theoretical framework. It also envisages the exciting possibility that a modification of the currently accepted laws of the gravitational interaction can, in principle, have an impact on the ancient history of our planet and, indirectly, even on the evolution of the life on it.", "pages": [ 8, 9 ] }, { "title": "4.2. The secular increase of the astronomical unit", "content": "At this point, the reader may wonder why, in the context of a putative increase of the radius of the Earth's orbit, no reference has been made so far to its secular increase reported by [53-55] whose rate ranges from ≈ 1 . 5 × 10 -1 m a -1 [53] to ≈ 5 × 10 -2 m a -1 [54]. Actually, if steadily projected backward in time until t Ar , the figures for its secular rate present in the literature would yield a displacement of the Earth's orbit over the last 3 . 8 Ga as little as ∆ r ≈ (2 to 6) × 10 8 m , corresponding to ≈ (1 to 4) × 10 -3 r 0 , contrary to eq. (7).", "pages": [ 9 ] }, { "title": "5. Some non-climatic, classical orbital effects", "content": "It is important to point out that, actually, there are also some standard physical phenomena which, in principle, could yield a cumulative widening of the Earth's orbit.", "pages": [ 9 ] }, { "title": "5.1. Gravitational billiard", "content": "It was recently proposed [21] that our planet would have migrating to its current distance in the Archean as a consequence of a gravitational billiard involving planet-planet scattering between the Earth itself and a rogue rocky protoplanetesimal X, with m X ≈ 0 . 75 m ⊕ , which would have impacted on Venus. However, as the author himself of [21] acknowledges, 'this may not be compelling in the face of minimal constraints'.", "pages": [ 9 ] }, { "title": "5.2. Mass losses", "content": "Another classical effect, for which independent confirmations in several astronomical scenarios exist, is the mass loss of main sequence stars [56] and/or of the surrounding planets due to the possible erosion of their hydrospheres/atmospheres [57] caused by the stellar winds [11,58]. Their gravitational effects on the dynamics of a two-body system have been worked out in a number of papers in the literature, especially as far as the mass loss of the hosting star is concerned; see, e.g., [59-61] and references therein. In regard to the orbital recession of a planet losing mass because of the stellar wind of its parent star, see [62] and references therein. Let us explore the possibility that, either partly or entirely, they can account for the phenomenology described in Section 2 within our working hypothesis of a closer Earth 3 . 8 Ga ago. For a previous analysis involving only the Sun's mass loss, see [20].", "pages": [ 9, 10 ] }, { "title": "5.2.1. Isotropic mass loss of the Sun", "content": "As far as the Sun is concerned, it is believed that, due to its stronger activity in the past [2,11] associated with faster rotation and stronger magnetic fields, its mass loss rate driven by the solar wind was higher [58] than the present-day one [63] ∣ recently measured in a model-independent way from the planetary orbital dynamics. Since from the cited literature it turns out that eq. (12) tells us that a steady solar mass loss rate as large as would be needed if it was to be considered as the sole cause for the increase of the size of the Earth's orbit hypothesized in eq. (11). It is interesting to compare our quantitative estimate in eq. (36) with the order-of-magnitude estimate in [19] pointing towards a mass loss rate of the order of See also [20]. It is worthwhile noticing that eq. (36) implies In principle, eq. (38) may contradict some of the assumptions on which the reasoning of Section 2, yielding just eq. (12) and Figure 1, is based. Indeed, the luminosity of a star powered by nuclear fusion is proportional to [64] with η = η ( M ) ; for a Sun-like star, it is η ≈ 4 . Thus, by keeping eq. (7) for r Ar , it would be However, it may be that the uncertainties in eq. (2) and, especially, in η might reduce the discrepancy between eq. (6) and eq. (41). On the other hand, we also mention the fact that a Sun's mass larger by just 4 . 4 % would not pose the problems mentioned in Section 4 of [2] concerning the evaporation of the terrestrial hydrosphere. In fact, the actual possibility that the Sun may have experienced a reduction of its mass such as the one postulated in eq. (36) should be regarded as somewhat controversial, as far as both the timescale and the magnitude itself of the solar mass loss rate are concerned [20]. Indeed, Figure 15 of [11] indicates a Sun's mass loss rate smaller than eq. (35) by about one to two orders of magnitude during the Archean, with a maximum of roughly ∣ just at the beginning of that eon. A similar figure for the early Sun's mass loss rate can be inferred from eq. (34) and the estimates in [58]. In [11] it is argued that the young Sun could not have been more than 0 . 2 % more massive at the beginning of the Archean eon. On the other hand, in [19] an upper bound of for π 01 Ursa Majoris, a 0 . 3 Ga old solar-mass star, is reported. Similar figures for other young Sun-type stars have been recently proposed in [65] as well. At the post-Newtonian level, general relativity predicts the existence of a test particle acceleration in the case of a time-dependent potential. Indeed, from Eq. (2.2.26) and Eq. (2.2.49) of [66], written for the case of the usual Newtonian monopole, it can be obtained [67] where µ . = GM . The orbital consequences of eq. (44) were worked out in [67]: a secular increase of the distance occurs. It is completely negligible, even for figures as large as eq. (36) by assuming that the change in µ is due to the mass variation. Indeed, eq. (45), calculated with eq. (36), yields a distance rate as little as ˙ r ≈ 3 × 10 -7 ma -1 . 5.3. Non-isotropic mass loss of the Earth due to a possible erosion of its hydrosphere driven by the solar wind Let us, now, examine the other potential source of the reduction of the strength of the gravitational interaction in the Sun-Earth system, i.e. the secular mass loss of the Earth itself, likely due to the erosion of its fluid component steadily hit by the solar wind. To the best of our knowledge, such a possibility has never been treated in the literature so far. Let us recall that a body acquiring or ejecting mass due to typically non-gravitational interactions with the surrounding environment experiences the following acceleration [68-73] with respect to some inertial frame K . In eq. (46), F is the sum of all the external forces, while is the velocity of the escaping mass with respect to the barycenter of the body. In eq. (47), V esc is is the velocity of the escaping particle with respect to the inertial frame K , and v is the velocity of that point of the body which instantaneously coincides with the body's center of mass; it is referred to K , and does not include the geometric shift of the center of mass caused by the mass loss. If the mass loss is isotropic with respect to the body's barycenter, then the second term in eq. (46) vanishes. In the case of a star-planet system [72], F is the usual Newtonian gravitational monopole, and the mass loss is anisotropic; moreover, V esc is radially directed from the star to the planet. According to [62], the orbital effect on the distance r is where it was assumed that the characteristic timescale of the generally time-dependent percent mass loss rate is much larger than the orbital period. It is worthwhile noticing that eq. (48) does not depend on V esc ; it is the outcome of a perturbative calculation with the Gauss equations in which no approximations concerning v and V esc were assumed [62]. The eccentricity e , the inclination I and the node Ω do not secularly change, while the pericenter ω undergoes a secular precession depending on V esc [62]. If eq. (12) was entirely due to eq. (48), then the hypothesized Earth mass loss rate would be as large as It implies that, at the beginning of the Archean, the Earth was more massive than now by ≈ 2 % . Thus, by keeping the solid part of the Earth unchanged, its fluid part should have been larger than now by the non-negligible amount For a comparison, the current mass of the fluid part of the Earth is largely dominated by the hydrosphere, which, according to http://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html, amounts to the current mass of the Earth's atmosphere is 274 times smaller than eq. (51). Also for such a postulated mechanism, it should be checked if it is realistic in view of the present-day knowledge. To this aim, it should be recalled that the fluid part of the Earth at the beginning of the Archean eon is the so-called 'second atmosphere' [74], and that, to an extent which is currently object of debate [75], it should have been influenced by the Terrestrial Late Heavy Bombardment (TLHB) [76] ≈ (4 to 3 . 8) Ga ago. In particular, in regard to the composition of the Earth's atmosphere, it is crucial to realistically asses if the extraterrestrial material deposited during the TLHB was mainly constituted of cometary matter or chondritic (i.e. asteroidal) impactors [77]. Another issue to be considered is if the spatial environment of the Earth could allow for a hydrospheric/atmospheric erosion as large as eq. (49). To this aim, it is important to remark that the terrestrial magnetic field, which acts as a shield from the eroding solar wind, was only [78] ≈ 50 % to 70 % of its current level (3 . 4 to 3 . 45) Ga ago. Moreover, as previously noted, the stronger stellar wind of the young Sun had consequences on the loss of volatiles and water from the terrestrial early atmosphere [79].", "pages": [ 10, 11, 12, 13 ] }, { "title": "6. Conclusions", "content": "In this paper, we assumed that, given a solar luminosity as little as 75 % of its current value at the beginning of the Archean 3 . 8 Ga ago, the Earth was closer to the Sun than now by 4 . 4 % in order to allow for an irradiance large enough to keep a vast liquid ocean on the terrestrial surface. As a consequence, under the assumption that non-climatic effects can solve the Faint Young Sun paradox, some physical mechanism should have subsequently moved our planet to its present-day heliocentric distance in such a way that the solar irradiance stayed substantially constant during the entire Archean eon, i.e. from 3 . 8 Ga to 2 . 5 Ga ago. Although it turns out that a relative orbital recession rate of roughly the same order of magnitude of the value of the Hubble parameter 3 . 8 Ga ago would have been required, standard general relativity rules out cosmological explanations for such a hypothesized orbit widening both at the Newtonian and the post-Newtonian level. Indeed, at the Newtonian level, the first non-vanishing cosmological acceleration is quadratic in the Hubble parameter and, in view of its analytical form, it does not cause any secular variation of the relative distance in a localized two-body system. At the post-Newtonian level, a cosmological acceleration linear in the Hubble parameter has been, in fact, recently predicted. Nonetheless, if, on the one hand, it induces the desired orbital recession, on the other hand, its magnitude, which is determined by well defined ambient parameters such as the speed of light in vacuum, the Hubble parameter and the mass of the primary, is far too small to be of any relevance. Instead, a recently investigated class of modified theories of gravitation violating the strong equivalence principle due to a nonminimal coupling between the matter and the spacetime metric is, in principle, able to explain the putative orbital recession of the Earth. Indeed, it naturally predicts, among other things, also a non-vanishing secular rate of the orbit's semimajor axis depending on a pair of free parameters whose values can be adjusted to yield just the required one. Moreover, since one of them is, in principle, time-dependent, it can always be assumed that it got smaller in the subsequent 2 Ga after the end of the Archean in such a way that the current values of the predicted orbit recessions are too small to be detected. Another physical mechanism of classical origin which, in principle, may lead to the desired orbit expansion is a steady mass loss from either the Sun or the Earth itself. However, such a potentially viable solution presents some difficulties both in terms of the magnitude of the mass loss rate(s) required, especially as far as the Earth's hydrosphere is concerned, and of the timescale itself. Indeed, the Earth should have lost about 2 % of its current mass during the Archean. Moreover, it is generally accepted that a higher mass loss rate for the Sun due to an enhanced solar wind in the past could last for just (0 . 2 to 0 . 3) Ga at most. In conclusion, it is entirely possible that the Faint Young Sun paradox can be solved by a stronger greenhouse effect on the early Earth; nonetheless, the quest for alternative explanations should definitely be supported and pursued.", "pages": [ 13 ] } ]

[ { "title": "A No-Go Theorem for Rotating Stars of a Perfect Fluid without Radial Motion in Projectable Hoˇrava-Lifshitz Gravity", "content": "Naoki Tsukamoto a and Tomohiro Harada Department of Physics, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima-ku, Tokyo 171-8501, Japan (Dated: October 18, 2018)", "pages": [ 1 ] }, { "title": "Abstract", "content": "Hoˇrava-Lifshitz gravity has covariance only under the foliation-preserving diffeomorphism. This implies that the quantities on the constant-time hypersurfaces should be regular. In the original theory, the projectability condition, which strongly restricts the lapse function, is proposed. We assume that a star is filled with a perfect fluid with no-radial motion and that it has reflection symmetry about the equatorial plane. As a result, we find a no-go theorem for stationary and axisymmetric star solutions in projectable Hoˇrava-Lifshitz gravity under the physically reasonable assumptions in the matter sector. Since we do not use the gravitational action to prove it, our result also works out in other projectable theories and applies to not only strong gravitational fields, but also weak gravitational ones.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Recently, Hoˇrava proposed a power-counting renormalizable gravitational theory [1, 2]. The theory is called Hoˇrava-Lifshitz gravity, because it exhibits the Lifshitz-type anisotropic scaling in the ultraviolet: where t , x i , b and z are the temporal coordinate, the spatial coordinates, the scaling factor and the dynamical critical exponent, respectively, and i runs over one, two and three. Since this theory is expected to be renormalizable and unitary, its phenomenological aspects [3, 4] and variants [5, 6] strenuously have been investigated, including black holes [7-16], dark matter [17, 18], dark energy [19], the solar system test [20] and so on. The field variables in this theory are the lapse function, N ( t ), the shift vector, N i ( t, x ), and the spatial metric, g ij ( t, x ). Note that the shift vector, N i , and the spatial metric, g ij , can depend on both t and x i , but that the lapse function, N , can only do so on t . Since the lapse function, N , can be interpreted as a gauge field associated with the time reparametrization, it is natural to restrict it to be space independent. This assumption, called the projectability condition, is proposed in Hoˇrava's original paper [1] from the view point of quantization. However, the pathological behaviors of the projectability condition, such as the infrared instability and the strong coupling, are found [1, 2, 21-26], and the theory has been extended to avoid the adverse situation [27, 28]. Since higher derivative terms do not contribute at large distances, the action of this theory can recover the apparent form of general relativity if we tune a coupling parameter. In this context, it seems that projectable Hoˇrava-Lifshitz gravity passes astrophysical tests. However, we will show that, actually, this is not true in this paper. In this theory, black holes have been investigated eagerly, while stars have not been studied so much [29, 30]. The comparison of the features of star solutions in Hoˇrava-Lifshitz gravity with the corresponding ones in Einstein gravity would be one of the astrophysical tests for Hoˇrava-Lifshitz gravity. It is important to investigate star solutions, gravitational collapse [31] and the formation of black holes. The first study of stars in Hoˇrava-Lifshitz gravity was done by Izumi and Mukohyama [29]. They found a no-go theorem that no spherically symmetric and static solution filled a perfect fluid without radial motion exists in this projectable theory under the assumptions that the energy density is a piecewise-continuous and non-negative function of the pressure and that the pressure at the center is positive. Their result is powerful, because it does not depend on the gravitational action. To construct star solutions, we have to change at least one of their assumptions for the matter sector, the symmetry of spacetime, the projectability and the invariance under the foliation-preserving diffeomorphism. Greenwald, Papazoglou and Wang found spherically symmetric static solutions, which are filled with a perfect fluid with radial motion and a class of an anisotropic fluid in the projectable Hoˇrava-Lifshitz gravity without the detailed balance condition [30]. It seems that static solutions are too simple to describe realistic stars, which are generally rotational. In this paper, we investigate a stationary and axisymmetric star in projectable Hoˇrava-Lifshitz gravity. We find a no-go theorem that the stationary and axisymmetric star filled with a perfect fluid without radial motion in the reflection symmetry about the equatorial plane does not exist under the physically reasonable conditions on the matter sector. Since we do not use the gravitational action to prove it, our result also works out in other projectable theories [5, 32] and applies to not only strong gravitational fields, like neutron stars, but also weak gravitational ones, like planets or moons. Our proof implies another ill behavior of the projectability condition if we follow a principle that stars should be described by stationary solutions of a low-energy effective theory. On the other hand, even if we do not follow this principle, our result would be useful to investigate rotating-star solutions in this theory and then to compare the solutions with the corresponding ones in Einstein gravity for astrophysical tests of this theory. This paper is organized as follows. In Section 2, we shall describe the definitions, the basic equations and the properties of Hoˇrava-Lifshitz gravity. In Section 3, we give the main result that there are no stationary and axisymmetric star solutions with a perfect fluid, which does not have the radial component of the four-velocity under a set of reasonable assumptions in the matter sector. In Section 4, we summarize and discuss our result. In Appendix A, we show the explicit expression for the equation of motion. In Appendix B, we show the triad components of the extrinsic curvature tensor. In this paper, we use the units in which c = 1.", "pages": [ 2, 3 ] }, { "title": "II. PROPERTIES OF HO ˇ RAVA-LIFSHITZ GRAVITY", "content": "In this section, we shall describe the definitions, the basic equations and the properties of Hoˇrava-Lifshitz gravity. Hoˇrava-Lifshitz gravity does not have general covariance, since the Lifshitz-type anisotropic scaling treats time and space differently. Instead, this theory is invariant under the foliation-preserving diffeomorphism: This means that the foliation of the spacetime manifold by constant-time hypersurfaces has a physical meaning. Thus, the quantities on the constant-time hypersurfaces, such as the extrinsic curvature tensor and the shift vector, must be regular. It is useful to describe the line element in the Arnowitt-Deser-Misner (ADM) form [33]: The action proposed by Hoˇrava [1] is given by: where I m is the matter action, R is the Ricci scalar of g ij , R ij is the Ricci tensor of g ij , D i is the covariant derivative compatible with g ij , K ij is the extrinsic curvature of a constant-time hypersurface, defined by: K = g ij K ij , C ij is the Cotton tensor, defined by: ε ikl = /epsilon1 ikl / √ g is the antisymmetric tensor, which is covariant with respect to g ij , and κ, ω, µ, λ and Λ W are constant parameters. We can rewrite the gravitational action (2.4): where parameters α, β, γ, ζ, η, ξ and σ are given by: If we take λ = 1 to recover the apparent form of general relativity and the apparent Lorentz invariance, we can compare this action to that of general relativity. Then, we obtain: where Λ is the cosmological constant and G is Newton's constant. Under the infinitesimal coordinate transformation: g ij , N i and N transform as: where L ζ is the Lie derivative along ζ i ( t, x ). L ζ g ij and L ζ N i are given by: By the variation of the action with respect to N , we get the Hamiltonian constraint: where: and: Here, n µ is defined as: Notice that due to the projectability condition N = N ( t ), the Hamiltonian constraint is global in Hoˇrava-Lifshitz gravity, while it is local in general relativity. From the variation of the action with respect to N i , we obtain the momentum constraint: where: By the variation of the action with respect to g ij , we get the equation of motion: where: The explicit expression for the equation of motion is given in Appendix A. By the invariance of the gravitational action and the matter action under the infinitesimal transformation (2.10), we get the energy conservation: and the momentum conservation: where α represents g or m . In the next section, we will only use the momentum conservation of the matter to show that no stationary and axisymmetric star solution exists. Therefore, our result does not depend on the gravitational action.", "pages": [ 4, 5, 6 ] }, { "title": "III. NO STATIONARY AND AXISYMMETRIC STAR SOLUTIONS", "content": "In this section, we show a no-go theorem for stationary and axisymmetric star solutions in projectable Hoˇrava-Lifshitz gravity. To prove it, we assume that a star is filled with a perfect fluid, which does not have the radial component of the four-velocity, that it has the reflection symmetry about the equatorial plane, that the energy density is a piecewise-continuous and non-negative function of the pressure, that the pressure is a continuous function of r and that the pressure at the center of the star is positive.", "pages": [ 7 ] }, { "title": "A. Stationary and Axisymmetric Configuration", "content": "We consider stationary and axisymmetric configurations with the timelike and spacelike Killing vectors, respectively, given by: Under the stationary configurations, the lapse function, N , does not depend on t . In the original theory, the projectability condition N = N ( t ) is proposed [1]. This condition means that the lapse function, N , does not depend on the spatial coordinates, x i , but only can do so on the temporal coordinate, t . Thus, the lapse function, N , is a constant. The timelike Killing vector, t µ , implies everywhere: The spacelike Killing vector, φ µ , implies that: is a geometrical invariant. As a part of the gauge condition, we take: Under this gauge condition, the general form for the spatial line element is described by [34]: where ψ , A , B and ξ are functions of r and θ , but neither t nor φ for stationarity and axisymmetry. Now we assume that the spacetime has a rotation axis, where sin θ = 0. This means: there [35].", "pages": [ 7, 8 ] }, { "title": "B. Triad Components of Shift Vector", "content": "We define triad basis vectors { e ( i ) } . e (1) is along the radial direction; e (3) is along the axial Killing vector and e (2) is fixed by the orthonormality and the right-hand rule. The coordinate components for the orthonormal triad are: where we have used the spatial line element (3.6). The projection of the shift vector on the triad is related to its coordinate components by:", "pages": [ 8 ] }, { "title": "C. Regularity Conditions at the Origin", "content": "Here, we give the regularity conditions of the shift vector, N i , near the origin. A tensorial quantity is regular at r = 0 if and only if all its components can be expanded in non-negative integer powers of x , y and z in locally Cartesian coordinates, defined by: The Lie derivative of the shift vector, N i , along the spacelike Killing vector vanishes, or: In locally Cartesian coordinates, the spacelike Killing vector is written as: Then, its components of Equation (3.17) are: The general regular solution of these equations is: where F 1 , F 2 and F 3 are independent and regular functions, which depend on z and ρ 2 ≡ x 2 + y 2 . Now, transforming N i back to the spherical coordinates, r, θ and φ , we get the spherical components: On the rotation axis (sin θ = 0), thus, we obtain: Using Equations (3.6), (3.8)-(3.10) and (3.25)-(3.27), the triad components are given by: Here, we additionally assume the reflection symmetry about the equatorial plane z = 0 or θ = π/ 2. Then, N x and N y must be even functions of z , and N z must be an odd function of z . This implies that F 1 , F 2 must be even functions of z , and F 3 must be an odd function of z . Since N r is an odd function of z on the rotation axis (sin θ = 0), we get: at the origin.", "pages": [ 8, 9, 10 ] }, { "title": "D. Matter Sector and Momentum Conservation", "content": "For simplicity, we assume that the matter consists of a perfect fluid. The stress-energy tensor is given by: where P and ρ represent the pressure and the energy density, respectively. We assume the four-velocity given by: where: is the normalization factor and ω is a function of r and θ . For the four-velocity, u µ , to be timelike, we shall have N 2 -N i N i -2 ωN φ -ω 2 g φφ > 0. We set α = m , and then, the momentum conservation equation (2.28) of the matter becomes: After some calculation, we obtain the r component: Now, we use the projectability condition N = N ( t ). As we mentioned above, the projectability condition means that the lapse function, N , does not depend on the spatial coordinates, x i , but only can do on the temporal coordinate, t . Thus, the r component of the momentum conservation equation (3.37) becomes: We do not use the θ and φ components to prove that no stationary and axisymmetric star exists. Here, we concentrate on the r component of the momentum conservation of the matter on the rotation axis sin θ = 0. On the rotation axis, g φφ and g φφ,r vanish from Equation (3.7). From Equation (3.13), the regularity of the triad component of the shift vector, N (3) , implies: on the rotation axis. Thus, N φ,r = 0. Thus, the r component of the momentum conservation equation (3.38) on the rotation axis becomes:", "pages": [ 10, 11 ] }, { "title": "E. Contradiction of Momentum Conservation", "content": "We assume that the star has the reflection symmetry about the equatorial plane θ = π 2 , that the energy density, ρ , is a piecewise-continuous and non-negative function of the pressure, P , that the pressure, P is a continuous function of r and that the pressure at the center of the star P c ≡ P ( r = 0) is positive. Thus, ρ + P is a piecewise-continuous function of r . We have assumed that the energy density, ρ , is non-negative everywhere and that the pressure at the center, P c , is positive; hence, ρ + P is positive at the center. We define r s as the minimal value of r for which at least one of ( ρ + P ) | r = r s , lim r → r s -0 ( ρ + P ) and lim r → r s +0 ( ρ + P ) is nonpositive. Dividing the momentum conservation equation (3.40) by 1 2 ( ρ + P ), we have: Under the assumption that the energy density is a function of the pressure, ρ = ρ ( P ), integrating the momentum conservation equation (3.41) over the interval 0 ≤ r < r s , we obtain: where P s ≡ P ( r = r s ). The definition of r s implies that at least one of ( ρ + P ) | r = r s , lim r → r s -0 ( ρ + P ) and lim r → r s +0 ( ρ + P ) is nonpositive. Since we have assumed that P ( r ) is a continuous function and that ρ is non-negative everywhere, P s = lim r → r s -0 P = lim r → r s +0 P is non-positive. Thus, we get: This implies that the right-hand side of Equation (3.42) is positive. However, the left-hand side of Equation (3.42) is nonpositive, since we have the projectability condition N = N ( t ) and we obtain from Equations (3.28), (3.32) and (3.39): at the center of the star. This contradicts that the right-hand side of Equation (3.42) is positive.", "pages": [ 11, 12 ] }, { "title": "IV. DISCUSSION AND CONCLUSIONS", "content": "Hoˇrava-Lifshitz gravity is only covariant under the foliation-preserving diffeomorphism. This means that the foliation of the spacetime manifold by the constant-time hypersurfaces has a physical meaning. As a result, the regularity condition at the center of a star is more restrictive than the one in a theory that has general covariance. Under the assumption that a star is filled with a perfect fluid that has no radial motion, that it has reflection symmetry about the equatorial plane and that the matter sector obeys the physically reasonable conditions, we have shown that the momentum conservation is incompatible with the projectability condition and the regularity condition at the center for stationary and axisymmetric configurations. Since we have not used the gravitational action to prove it, our result is also true in other projectable theories [5, 32]. Note that our result is true under not only strong-gravity circumstances, like neutron stars, but also weakgravity ones, like planets or moons. However, it is not certain that star solutions can exist in non-projectable theories. Since we have used the covariance under the foliation-preserving diffeomorphism, the projectability condition and the assumptions of the matter sector to prove the no-go theorem for stationary and axisymmetric stars, our proof will not apply if we do not assume all the above. Izumi and Mukohyama found that no spherically symmetric and static solution filled with a perfect fluid without radial motion exists in this theory under the assumption that the energy density is a piecewise-continuous and non-negative function of the pressure and that the pressure at the center is positive [29]. They concluded that a spherically symmetric star should include a time-dependent region near the center. Although we cannot deny that stars should be described by dynamical configurations, the fact that we cannot find simple stationary and axisymmetric star solutions with the four-velocity generated by the Killing vectors will be an unattractive feature of this theory. Greenwald, Papazoglou and Wang found static spherically symmetric solutions with a perfect fluid plus a heat flow along the radial direction and with a class of an anisotropic fluid under the assumption that the spatial curvature is constant in a projectable theory without the detailed balance condition [30], although it is doubtful that the constant-spatialcurvature solutions represent realistic stars. This, however, implies that rotating star solutions with a perfect fluid plus a radial heat flow and with an anisotropic fluid can also exist. We might get star solutions by introducing an exotic matter with a negative pressure, but it seems that the physical justification to introduce it is difficult. Our result does not imply the non-existence of rotation star solutions in this theory. However, it would be useful to investigate rotating-star solutions in this theory and then to compare the solutions with the corresponding ones in Einstein gravity for astrophysical tests of this theory. Furthermore, although we do not disprove the existence of rotation star solutions with radial motion, it is doubtful whether such star solutions describe realistic astrophysical stars. Recently, the property of matter in the non-projectable version of the extended HoˇravaLifshitz gravity [28] at both classical and quantum levels has been investigated by Kimpton and Padilla [36]. Although the gravity sector in Hoˇrava-Lifshitz has been investigated eagerly, the matter sector has not, relatively. It is left as future work to answer the question of whether or not the no-go theorem applies at a quantum level.", "pages": [ 12, 13 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "The authors would like to thank M. Saijo, U. Miyamoto, S. Kitamoto, N. Shibazaki, T. Kuroki, S. Mukohyama, K. Izumi, M. Nozawa, R. Nishikawa, M. Shimano, H. Nemoto, S. Kamata, S. Okuda and T. Wakabayashi for valuable comments and discussion. Naoki Tsukamoto thanks the Yukawa Institute for Theoretical Physics at Kyoto University, where this work was initiated during the YITP-W-11-08 on 'Summer School on Astronomy and Astrophysics 2011'. Tomohiro Harada was supported by the Grant-in-Aid for Scientific Research Fund of the Ministry of Education, Culture, Sports, Science and Technology, Japan [Young Scientists (B) 21740190].", "pages": [ 14 ] }, { "title": "Appendix A: Explicit Expression for Equation of Motion", "content": "After a long straightforward calculation, we obtain the explicit expression for the equation of motion: where ( i ↔ j ) means the terms, i and j , exchanged each other.", "pages": [ 14 ] }, { "title": "Appendix B: Triad Components of Extrinsic Curvature Tensor", "content": "In this theory, the triad components of the extrinsic curvature tensor also should be regular. The Lie derivative of g ij along N i is: The extrinsic curvature tensor (2.5) and (B1) yield: where: By projecting Equation (B2) onto the triad (3.8)-(3.10), we obtain the following equations [34]: where:", "pages": [ 15 ] } ]

[ { "title": "A. G. Tlatov", "content": "Kislovodsk Mountain Astronomical Station, Central (Pulkovo) Astronomical Observatory, Russian Academy of Sciences, Kislovodsk, Russia e-mail: [email protected] Relative variations in the number of sunspots and sunspot groups in activity cycles have been analyzed based on data from the Kislovodsk Mountain Astronomical Station and international indices. The following regularities have been established: (1) The relative fraction of small sunspots decreases linearly and that of large sunspots increase with increasing activity cycle amplitude. (2) The variation in the average number of sunspots in one group has a trend, and this number decreased from ~12 in cycle 19 to ~7.5 in cycle 24. (3) The ratio of the sunspot index (Ri) to the sunspot group number index (Ggr) varies with a period of about 100 years. (4) An analysis of the sunspot group number index (Ggr) from 1610 indicates that the Gnevyshev-Ohl rule reverses at the minimums of secular activity cycles . (5) The ratio of the total sunspot area to the umbra area shows a long-term variation with a period about eight cycles and minimum in cycles № 16-17. (6) It has been indicated that the magnetic field intensity and sunspot area in the current cycle are related to the amplitude of the next activity cycle.", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "The Gnevyshev-Ohl (G-O) rule, which was valid for about 150 years (beginning from cycle 10), was violated in cycles 22 and 23. This can indicate that the solar cyclicity regime will change, which possibly took place previously (Vitinsky et al., 1986). The G-O rule is violated during the decline stage of the secular cycle and can indicate that activity will decrease during a long period similar to the Maunder minimum. It is still unknown why prolonged activity cycles exist. The characteristic periods of these cycles (Hathaway, 2010) and variations in sunspot characteristics (magnetic fields, area, group properties, etc.) during secular cycles are still among the problems to be solved. The aim of this work is to trace relative variations in the properties of sunspots and sunspot groups with different areas in solar activity cycles and in the secular activity cycle.", "pages": [ 1 ] }, { "title": "2 . VARIATION IN THE RELATIVE CONTRIBUTION OF DIFFERENT SUNSPOTS TO ACTIVITY INDICES", "content": "As initial data for this analysis, we took daily observations of sunspot groups at Kislovodsk Mountain Astronomical Station (GAS) from 1954 to 2012 and other data. In addition to the coordinates and area, the number of umbrages and pores ( N sp), participating in the calculation of the Wolf number, as well as the area of the maximal sunspot in a group ( S max), are also present in the GAS data. This makes it possible to analyze different activity indices depending on the group or maximal sunspot area. An analysis of the total number of small and large groups indicates that small and large sunspots differently contribute to the Wolf number. The relative number of small sunspots decreases, depending on the activity cycle amplitude, and the fraction of large sunspots increases with increasing activity cycle amplitude. This conclusion does not confirm the conclusion drawn in (Lefe'vre and Clette, 2011) that small sun spots were rarely encountered in cycle 23. The relative contribution of large sunspot groups ( S > 500 millionths of solar hemisphere, msh) to the Wolf number increases with increasing activity cycle amplitude W max: 0,93 R , W 0,001 0,065 / max 500     tot W W ,74. 0 R , W 10 9 . 6 35 . 0 max -4       tot . The contribution of small sunspot groups ( S < 50 msh) decreases with increasing activity cycle amplitude: . Such a regularity is also valid for the total number of groups, including the maximal sunspots with areas of S / 50 W W max < 20 msh (small sunspots) and S max > 700 msh (large sunspots) (Fig. 1). The relative number of small and large sunspot groups differs, depending on the increase in the activity cycle amplitude.", "pages": [ 1, 2 ] }, { "title": "3. LONG-TERM VARIATIONS IN THE SUNSPOT NUMBER IN GROUPS", "content": "The average number of umbrages and pores in sunspot groups decreased monotonically from cycle to cycle during the last five activity cycles (Fig. 2). This is especially pronounced for medium and large sunspots ( S > 50 msh). Umbras and pores usually play the main role in the calculation of the W index for medium and large sunspots. The average area of individual umbra possibly increased during this period, and their number decreased in this case. These variations can be verified based on other activity indices. As is known, the number of groups with factor 10 and the total sunspot (umbra) number are taken into account when the Wolf number is calculated. At the same time, the index of the sunspot group number exists (Hoyt and Schatten, 1998). Figure 3a presents the variation in this ratio from 1748, found from daily data and averaged over the solar cycles. There exists a long-term variation with a period of about ten solar cycles. The number of sunspots in one group was maximal in cycles 10 and 19. Based on the sunspot group characteristics at the Greenwich observatory (RGO) (http://solarscience.msfc.nasa.gov), we can reconstruct the ratio of the total sunspot area to the umbra area (Fig. 3b). This ratio also shows a long-term variation with a slightly shorter period (about eight cycles); however, the maximum falls on cycle 13 and 20 and minimum falls on cycle 16-17. Variations also exist in the relative contribution of the areas of different groups, which is confirmed by the variations in the sunspot group area ranging from 30 to 100 msh on average over the cycle. Variations with close periods also exist for other ranges of areas. Figures 3a-3c indirectly confirm the conclusion that the number of sunspots in one group in cycle 19 is large (Fig. 2). However, the average magnetic field strength in umbrages would decrease in this case since the magnetic field of umbra increases with increasing their area (Vitinsky et al., 1986). Such a tendency is observed in Fig. 3d for the average magnetic field strength in the cycle according to the Mount Wilson (MNTW) observatory data. The magnetic field ( B ) increased from the middle of the past century to cycle 22. In this case, the magnetic field strength in even cycles was on average higher than in odd cycles. The long-term variations in the sunspot magnetic field are not random and are directly related to the activity cycle amplitude. The relationship between the magnetic field and area was found for cycles № 15-19 from the MNTW data: (Petsov et al., 2013), where a and b are the coefficients varying from cycle to cycle. Figure 4 presents the relationship between the product of the total area logarithm during a cycle into coefficient b : ) log( S b a B    ) log( S b in a given cycle and the amplitude of the ext activity cycle ( Wn + 1). The correlation between these parameters is high ( R =0.96).", "pages": [ 2, 4 ] }, { "title": "4. LO NG-TERM VARIATIONS IN THE G-O RULE", "content": "nd several observations were not onsidered. Based on additional data, Hoyt and Schatten (1998) proposed a sunspot group index, reconstructed by them for 1610-1995. The G-O empirical rule (Vitinsky et al., 1986) was formulated for a pair of successive solar cycles. There are several definitions of this rule, but the main interpretation is as follows: the amplitude of an even activity cycle is smaller than the height of the next odd cycle. The Wolf number series, which was reconstructed by R. Wolf from 1748, is usually used to verify the G-O rule. However, as was indicated in (Hoyt and Schatten, 1998), this series has a rather large noise level since it was difficult to take into consideration small sunspots a c To characterize activity cycles, we can use the daily average group number in a cycle. Figure 5 present s such a number calculated based on the sunspot group index    1 / ) ( k k T d Nd Rg k G where N  1 T d is the number of observation days in cycle k and Tk is the took from th time of the cycle k onset. The times of the cycle onset and end we e NGDC site. During cycles 12-21, the average sunspot group number ( Gd ) in even cycles was smaller than in the next odd cycles, and this ratio is 39 , 1 /  even odd . There d d G G fore, the index of the daily average sunspot group number can be used to verify the G-O rule. the cycles in the previous Figure 6 presents the even d odd d G G / ratio from 1610 to 2009. The group number according to the (http://solarscience.msfc.nasa.gov) data for cycle 23 was added here to the daily data on the sunspot group number (Hoyt and Schatten, 1998). Figure 6 indicates that the even d odd d G G / ratio corresponds to the standard definitions of the G-O rule after cycle № 10 but also shows a smooth envelop for epoch, except several individual cycles. The line where this ratio is 1 was drawn for comparison. The data of cycle № -4/-3 are not presented .", "pages": [ 5, 6 ] }, { "title": "5. CO NCLUSIONS", "content": "ig. 3a); the param An increase in activity in the middle of the 20th century was accompanied by a variation in the sunspot group properties. The sunspot number in groups increased during cycles 14-19. After cycle 19, this parameter tended to decrease (Figs. 2, 3). In the first half of the 20th century, the ratio of the umbrage area to the total group area varied (Fig. 3b) and the average magnetic field strength decreased (Fig. 3d) during the secular cycle activity growth stage. On the whole, this analysis confirms the assumption that the average sunspot number in a group varies following the secular cycle (Vitinsky et al., 1986). In this case, a decrease in the umbrage area with increasing area and sunspot number indicates that the magnetic flux is redistributed in sunspots from cycle to cycle. At the same time, different sunspot parameters have different durations of the secular cycles. The parameters related to the number of sunspots and sunspot groups have a duration of about ten cycles (F eters related to the sunspot area, about eight activity cycles. It was also noted previously (Hathaway, 2010) that the secular cycle has different periods. tal sunspot magnetic field in the current cycle and is res The relation between the sunspot area and the magnetic field changes during a secular cycle. In this case, the product of the total area logarithm into the coupling coefficient ( b ) characterizes the to ponsible for the next cycle level (Fig. 4). This fact can be the key to understanding the solar cyclicity. Studying the G-O rule can also give important information regarding the solar cyclicity nature, specifically, the possible relic field, to which this effect is usually related (Mursula et al., 2001). Some authors consider that even cycles are constantly less intense than odd ones (Mursula et al., 2001; Nagovitsyn et al., 2009) and, therefore, introduce additional activity cycles. However, the pair of cycles 22 and 23 demonstrates that this rule is violated. Therefore, this rule possibly also reversed in zone (Tlatov, 2007). If this is the case, the reversal of the G-O ule in cycles 22-23 indicates that the reversal will also be valid for the next pair of the previous centuries and varies cyclically. The usage of the average group number in a cycle makes it possible to interpret the G-O rule as a long-term variation (Fig. 6). Long-term activity variations are possibly caused by a residual slowly varying (permanent) poloidal solar magnetic field. This field can nevertheless reverse its sign (which results in a reversal in the series of 22_year cycles) and modulate secular activity cycles. Such a permanent field is caused by 'magnetic memory' below the sunspot generating r even-odd cycles.", "pages": [ 6, 7 ] }, { "title": "ACKNOWLEDGMENTS", "content": "his work was partially supported by the Russian Academy of Sciences and the ussian Foundation for Basic Research. T R", "pages": [ 7 ] } ]

[ { "title": "Frederik J. Simons 1 , 2 and Sofia C. Olhede 3", "content": "1 Department of Geosciences, Princeton University, Princeton, NJ 08544, USA 3 Department of Statistical Science, University College London, London WC1E 6BT, UK E-mail: [email protected], [email protected] 20 May 2018", "pages": [ 1 ] }, { "title": "S U M M A R Y", "content": "Topography and gravity are geophysical fields whose joint statistical structure derives from interface-loading processes modulated by the underlying mechanics of isostatic and flexural compensation in the shallow lithosphere. Under this dual statistical-mechanistic viewpoint an estimation problem can be formulated where the knowns are topography and gravity and the principal unknown the elastic flexural rigidity of the lithosphere. In the guise of an equivalent 'effective elastic thickness', this important, geographically varying, structural parameter has been the subject of many interpretative studies, but precisely how well it is known or how best it can be found from the data, abundant nonetheless, has remained contentious and unresolved throughoutthe last few decades of dedicated study. The popular methods whereby admittance or coherence, both spectral measures of the relation between gravity and topography, are inverted for the flexural rigidity, have revealed themselves to have insufficient power to independently constrain both it and the additional unknown initial-loading fraction and load-correlation factors, respectively. Solving this extremely ill-posed inversion problem leads to non-uniqueness and is further complicated by practical considerations such as the choice of regularizing data tapers to render the analysis sufficiently selective both in the spatial and spectral domains. Here, we rewrite the problem in a form amenable to maximum-likelihood estimation theory, which we show yields unbiased, minimum-variance estimates of flexural rigidity, initial-loading fraction and load correlation, each of those separably resolved with little a posteriori correlation between their estimates. We are also able to separately characterize the isotropic spectral shape of the initial-loading processes. Our procedure is well-posed and computationally tractable for the two-interface case. The resulting algorithm is validated by extensive simulations whose behavior is well matched by an analytical theory with numerous tests for its applicability to real-world data examples. Key words: flexural rigidity, lithosphere, topography, gravity, maximum-likelihood theory", "pages": [ 1 ] }, { "title": "1 I N T R O D U C T I O N A N D M O T I V A T I O N", "content": "With a remarkable series of papers, all entitled Experimental Isostasy , Dorman and Lewis heralded in an era of Fourier-based estimation in geophysics, using gravity and topography to study isostasy 'experimentally', that is, without first assuming a particular mechanistic model such as Airy or Pratt compensation (Dorman & Lewis 1970; Lewis & Dorman 1970a,b; Dorman & Lewis 1972). All three papers remain essential reading for us today. The first in the series introduced the basic point of view by which Earth is regarded as a linear time-invariant system and the unknown 'isostatic response' is the transfer function: The linear system here is the earth: The input is the topography, or more precisely, the stress due to the topography across some imaginary surface, say sea level, and the output is the gravity field due to the resulting compensation. (Dorman & Lewis 1970, p. 3360.) In keeping with classical systems identification practice, or in their words, through the fruits of linear mathematics, in particular, harmonic analysis and the convolution theorem (Dorman & Lewis 1970, p. 3358), the recovery of the impulse response practically suggested itself: If the earth is linear in its response to the crustal loading of the topography, the response of the earth's gravity field to this loading can be represented as the two-dimensional convolution of the topography with the earth's isostatic response function. [...] Through transformation into the frequency domain, the convolution becomes multiplication, and one is led directly to the result that the isostatic response function is equal to the inverse transform of the quotient of the transforms of the Bouguer gravity anomaly and the topography. (Dorman & Lewis 1970, p. 3357.)", "pages": [ 1 ] }, { "title": "2", "content": "Contingent upon establishing the validity of the linear assumption in interpreting the data, subsequently, the isostatic response function was to be 'inverted', i.e. by computing the density changes at depth that would be required to fit the experimentally determined response function (Dorman & Lewis 1970, p. 3361). However, due to various forms of measurement, geological or modeling 'noise', [t]he problems involved in computing the inverse [...] of an experimentally determined function are formidable (Dorman & Lewis 1970, p. 3361), even when strictly local compensation is assumed and the solution is, in principle, unique. The second paper (Lewis & Dorman 1970a,b) was devoted to discussing the numerous geophysical and numerical strategies by which the least-squares inversion of the experimentally derived response can be accomplished at all. Broadly speaking, these involve any or all of (a) modification of the data, e.g. by windowing prior to Fourier transformation, (b) modification of the recovered response, e.g. by averaging, smoothing, or limiting the frequency interval of interest, (c) conditioning of the unknown density profile, e.g. by series expansion or imposing hard bounds, and (d) stabilizing the inversion, e.g. by iteration, frequency weighting, or the addition of minimum /lscript 1 norm constraints on the density profile. As a result, many possible local density profiles can be found that 'explain', in the /lscript 2 sense, the observed response curves, and an appeal has to be made to independent outside information, e.g. from seismology and geodynamics, to make the final selection. Regardless of the ultimate outcome of this exercise in deciding over which depth the compensating mass anomalies occur, the modeling procedure allows for the computation of the so-called 'isostatic anomaly'. The latter is thereby defined as that portion of the variation in the observed terrestrial gravity field that cannot be explained by the difference in measurement position on or above the reference geoid (which leads to the free-air anomaly), nor of the anomalous mass contained in the topography above the reference geoid (hence the Bouguer anomaly) but, most importantly, also not by the assumption of a linear isostatic compensation mechanism, at whichever depth or however regionally this is being accommodated (Lambeck 1988; Blakely 1995; Watts 2001; Turcotte & Schubert 2002; Hofmann-Wellenhof & Moritz 2006). In their third and final paper (Dorman & Lewis 1972) the authors employed Backus & Gilbert (1970) theory to obtain and interpret the result of the inversion of isostatic response functions by way of depth-averaging kernels rather than solving for particular profiles, which had shown considerable non-uniqueness and possibly unphysical oscillations. But even admitting that only localized averages of the anomalous density structure could be considered known, the authors concluded that the available data called for the compensation of terrestrial topography by density variations down to at least 400 km depth, i.e. involving not only Earth's crust but also its mantle. If in these papers the main objective was to make isostatic anomaly maps and to recover local density variations at depth to explain the cause of isostasy where possible, to do the latter reliably arguments needed to be made that involve the strength of the crust and upper mantle (Lewis & Dorman 1970a, p. 3371). In practice, this led the authors to decide that the constitution of the earth is such that it is at least able to support mass anomalies of wavelengths equal to the depth at which they occur (Lewis & Dorman 1970a, p. 3383). This contradictio in terminis (it is no longer a strictly local point of view) was the very one that led Vening Meinesz (1931) to argue against the hypotheses of Airy and Pratt: strength implies lateral transfer of stress which is incompatible with the tenets of local isostasy (Lambeck 1988; Watts 2001). Following a similar line of reasoning in replacing local by regional compensation mechanisms, McKenzie & Bowin (1976) and Banks et al. (1977) presented a new theoretical framework by which the observed admittance, indeed the ratio of Fourier-domain gravity anomalies to topography (Karner 1982), could be interpreted in terms of a regional compensation mechanism that involves flexure of a thin (compared to the wavelength of the deformation) elastic plate (a 'lithosphere' defined in its response to long-term, as opposed to seismic stresses) overlying an inviscid mantle (an 'asthenosphere', again referring to its behavior over long time scales). No longer was the local density structure the driving objective of the inversion of the isostatic response curve, but rather the thickness over which the density anomalies could plausibly occur, assuming a certain limiting mantle density. This subversion of the question how to best explain gravity and topography data became the now dominant quest for the determination of the flexural rigidity or strength, D , of the lithosphere thus defined. The theory of plates and shells (Timoshenko & Woinowsky-Krieger 1959) could then be applied to translate D into the 'effective' elastic plate thickness, T e , upon the further assumption of a Young's modulus and Poisson's ratio. A tripartite study entitled An analysis of isostasy in the world's oceans (Watts 1978; Cochran 1979; Detrick & Watts 1979) went around the globe characterizing T e in a plate-tectonic context. Subsequent additions to the theory involved a few changes to the physics of how deformation was treated, e.g. by considering that the isostatic response may be anisotropic (Stephenson & Beaumont 1980), taking into account non-linear elasticity and finite-amplitude topography (Ribe 1982), visco-elasticity and erosional feedbacks (Stephenson 1984), and updating the force balance to include also lateral, tectonic, stresses (Stephenson & Lambeck 1985). None of these considerations changed the basic premise. With the methodology for effective elastic thickness determination firmly established, the way was paved for its rheological interpretation (e.g. McNutt & Menard 1982; McNutt 1984; Burov & Diament 1995). A first hint that not all was well in the community came when transfer function theory was applied to measure the strength of the continents. McNutt & Parker (1978) concluded from admittance analysis that, on the whole, Australia (an old continent) might not have any strength, and would thus be in complete local isostatic equilibrium. On the contrary, Zuber et al. (1989) concluded on the basis of coherence analysis that the Australian continental effective elastic thickness well exceeded 100 km. This apparent contradiction was found despite the observed admittance and coherence being merely different 'summaries' of gravity and topography: spectral ratios that both estimate the underlying isostatic transfer function. At least part of the discrepancy could be ascribed to the treatment of subsurface loads in the formulation of the forward model (Forsyth 1985). With Bechtel et al. (1990), and numerous others after them, these authors led the next decade in which a 'thick' (greater than 100 km) continental lithosphere was espoused. Then, McKenzie & Fairhead (1997) started a decade of making effective arguments for 'thin' continents (no more than 25 km), a controversial position with many ramifications (Jackson 2002; Burov & Watts 2006) that was hotly contested and remains so today (Banks et al. 2001; Swain & Kirby 2003b; McKenzie 2003, 2010). Three developments happened on the way to the current state, with sound arguments made on both sides of the debate. Inverting coherence between Bouguer gravity and topography yielded thicker lithospheres than working with the admittance between the free-air gravity and the topography. There was discussion over the treatment of 'buried loads' and how to solve for the subsurface-to-surface loading ratio. Finally, there were arguments over the best way by which to form spectral estimates of either admittance or coherence. Among others, P'erez-Gussiny'e et al. (2004), P'erez-Gussiny'e &Watts (2005) and Kirby & Swain (2009) provided some reconciliation by making estimates of effective elastic thickness that were based on both free-air admittance and Bouguer coherence, respectively. They argued the equivalence of the results when either method was applied in a 'consistent' formulation, taking into account the finite window size of any patch of available data. Still, large differences remained, experiments on synthetic data showed significant bias and large variance, and a clear consensus failed to arise. Macario et al. (1995), McKenzie (2003) and Kirby & Swain (2009) investigated the effect of the statistical correlation between surface and subsurface loads. For their part, Diament (1985), Lowry & Smith (1994), Simons et al. (2000, 2003), Ojeda & Whitman (2002), Kirby & Swain (2004, 2008a,b) and Audet & Mareschal (2007) focused on the spectral estimation of admittance and coherence via maximum-entropy, multitaper and wavelet-based methods, and identified the spectral bias, leakage and variance inherent in those. Much as the controversy involved the geological consequences of a thick versus a thin lithosphere, with only gravity and topography as the primary observations and no significant divergence in viewing the physics of the problem, that is, of elastic flexure in a multilayered system, over time the arguments evolved into a debate that was mostly about spectral analysis. Least-squares fitting of admittance and coherence functions, however determined, had become synonymous with the process of elastic-thickness determination. The appropriateness of using least squares is not something that can be taken for granted but rather needs to be carefully assessed, as was pointed out early on in this context by Dorman & Lewis (1972), Banks et al. (1977), Stephenson & Beaumont (1980) and Ribe (1982), which, however, also focused on other issues that have since received more attention. Admittance and coherence are 'statistics': functions of the data with non-Gaussian distributions even if the data themselves are Gaussian (Munk & Cartwright 1966; Carter et al. 1973; Walden 1990; Thomson & Chave 1991; Touzi & Lopes 1996; Touzi et al. 1999). Estimators for flexural rigidity based on any given method have their own distributions, though not necessarily ones with a tractable form. Without knowledge of the joint properties of admittance- and coherence-based estimators it is impossible to assess the relative merits of any method for a given data set or true parameter regime; with current state-of-the-art understanding it is not even clear if the two methods are statistically inconsistent. At this juncture this paper aims for a return to the basics, by asking the question: 'What information does the relation between gravity and topography contain about the (isotropic) strength of the elastic lithosphere?' and by formulating an answer that returns the full statistical distribution of the estimates derived from such data. As such, it should provide a framework for the interpretation of the early work on which we build: as others before us we are merely using the measurable ingredients of gravity, topography and the flexure equations. However, as we shall see, we do not need to consider this a two-step process by which first the transfer function needs to be estimated non-parametrically and then the inversion for structural parameters performed with the estimated transfer function as 'data'. This approach amounts to a loss of most of the degrees of freedom in the data, replacing them with spectral ratios estimated at a much smaller set of wavenumbers, and with much of the important information on the flexural rigidity compromised due to lack of resolution at the low wavenumbers. Rather, we can treat it as an optimization problem that uses everything we know about gravity and topography available as data to directly construct a maximum-likelihood solution for the lithospheric parameters of interest. These are returned together with comprehensive knowledge of their uncertainties and dependencies, and with a statistical apparatus to evaluate how well they explain the data; the analysis of the residuals then informing us where the modeling assumptions were likely violated. By the principle of functional invariance the maximum-likelihood solution for elastic thickness and loading ratio also returns the maximum-likelihood estimates of the coherence and admittance themselves, which can then be compared to those obtained by other methods. Admittance may be superior to coherence, or vice versa, in particular scenarios, but only maximum-likelihood, by definition, produces solutions that are preferred globally for all parameter regimes (Pawitan 2001; Severini 2001; Young & Smith 2005). Finally, we note that understanding the likelihood is also a key component of fully Bayesian solution approaches (e.g. Mosegaard & Tarantola 1995; Kaipio & Somersalo 2005).", "pages": [ 2, 3 ] }, { "title": "2 B A S I C F R A M E W O R K", "content": "Despite their singular focus on deriving density profiles to reconstruct the portion of the Bouguer gravity field that is linearly related to the topography and thereby 'explain' the isostatic compensation of surface topography to first order, even when the strength of the lithosphere had to be effectively prescribed, Dorman and Lewis' Experimental Isostasy 1, 2 and 3 contained virtually all of the elements of the analysis of gravity and topography by which the problem could be turned around to the, in the words of Lambeck (1988) 'vexing', question 'What is the flexural strength of the lithosphere'? The elements applicable to the analysis were the expressions for admittance and coherence between topography and the Bouguer, free-air, and isostatic residual gravity anomalies, the averaging or smoothing required to statistically stabilize the estimate of the transfer function that is the intermediary between the data and the model obtained by inversion for the unknown parameters (if not the density distribution, then the mechanical properties of the plates), the notion of correlated and uncorrelated noise of various descriptions: indeed all of the ingredients that will form the vernacular of our present contribution. In this section we redefine all primary quantities of interest in a manner suitable for the statistical development of the problem. We treat Earth locally as a Cartesian system. Our chosen coordinate system has x = ( x 1 , x 2 ) in the horizontal plane and defines ˆz pointing up: depths in Earth are negative. A density contrast located at interface j is found at depth z j ≤ 0 , and is denoted Two layers is the minimum required to capture the full complexity of the general problem which may, of course, contain any number of layers. In a simple two-layer system, the first interface, at z 1 = 0 , is the surface of the solid Earth, and ρ 0 is the density of the air (or water) overlying it. The density of the crust is ρ 1 , and the second interface, at z 2 ≤ 0 , separates the crust from the mantle with density ρ 2 . For now we use the term 'topography' very generally to describe any departure from flatness at any surface or subsurface interface. By 'gravity' we mean the 'anomaly' or 'disturbance'; both are differences in gravitational acceleration with respect to a certain reference model. These departures in elevation and acceleration are all small: we consider topography to be a small height perturbation of a constantdepth interface, and neglect higher-order finite-amplitude effects on the gravity. We always assume that the 'loads', the stresses exerted by the topography, occur at the density interfaces and not anywhere else. If not in the space domain, x , we will work almost exclusively in the Fourier domain, using the wave vector k or wavenumber (spatial frequency) k = ‖ k ‖ . We only distinguish between both domains when we need to, and then only by their argument. All of this corresponds to standard practice (Watts 2001). Looking ahead we draw the readers' attention to Fig. 1, which contains a graphical representation of the problem. Fig. 1 is, in fact, the result of a data simulation with realistic input parameters. Many of the details of its construction remain to be introduced and many of the symbols remain to be clarified. What is important here is that we seek to build an understanding of how, from the observations of gravity and topography, we can invert for the flexural rigidity of the lithosphere in this two-layer case. The observables (rightmost single panel) are the sum of the flexural responses (middle panels) of two initial interface-loading processes (leftmost panels) which have occurred in unknown proportions and with unknown correlations between them.", "pages": [ 3, 4 ] }, { "title": "2.1 Spatial and spectral representation, theory and observation", "content": "Writing H and G without argument we will be referring quite generically to the random processes 'topography' and 'gravity' respectively, though when we consider either physical quantity explicitly in the spatial or spectral domain we will distinguish them accordingly as where they depend on spatial position x or on wave vector k , respectively. In doing so we use to the Cram'er (1942) spectral representation under which d H ( k ) and d G ( k ) are well-defined orthogonal increment processes (Brillinger 1975; Percival & Walden 1993), in the sense that at any point in space we may write Wemake the assumption of stationarity such that for every point x under consideration all equations of the type (3) are statistically equivalent. We further assume that both processes will be either strictly bandlimited or else decaying very fast with increasing wavenumber k = ‖ k ‖ such that we may restrict all integrations over spectral space to the Nyquist plane k ∈ [ -π, π ] × [ -π, π ] . While this is certainly a geologically reasonable assumption we would at any rate be without recourse in the face of the broadband bias and aliasing that would arise unavoidably if it were violated. For simplicity x maps out a rectangle that can be sampled on an M × N ≈ 2 K grid given by In the non-rarified world of geophysical data analysis we will not be dealing with stochastic processes directly, rather with particular realizations thereof. These are our gravity and topography data, observed on finite domains, to which we continue to refer as H ( x ) and G ( x ) . The modified Fourier transform of these measurements, obtained after sampling and windowing with a certain function w K ( x ) , is In this expression W K ( k ) is the unmodified Fourier transform of the energy-normalized applied window, The spectral density or spectral covariance of continuous stationary processes is defined as the ensemble average (denoted by angular brackets) whereby we denote complex conjugation with an asterisk and δ ( k , k ' ) is the Dirac delta function. There can be no covariance between non-equal wavenumbers if the spatial covariance matrix is to be dependent on spatial separation and not location, as from eqs (3) and (7) In contrast to eq. (7), as follows readily from eqs (5) and (7), the covariance between the modified Fourier coefficients of the finite sample is Eqs (5) and (9) show that the theoretical fields d H ( k ) and their spectral densities S HH ( k ) are out of reach of observation from spatially finite sample sets. Spectrally we are always observing a version of the 'truth' that is 'blurred' by the observation window. Even if, or rather, especially when the windowing is implicit and only consists of transforming a certain rectangle of data, this effect will be felt. For example, whereas the true spectral density is obtained by Fourier transformation of the covariance at all lags, denoted by the summed infinite series a blurred spectral density is what we obtain after observing only a finite set, denoted by the summed finite series with | F K | 2 denoting Fej'er's kernel (Percival & Walden 1993). The design of suitable windowing functions (in this geophysical context, see, e.g., Simons et al. 2000, 2003; Simons & Wang 2011), is driven by the desire to mold what we can calculate from the observations into estimators of these 'truths' that are as 'good' as possible, e.g. in the minimum mean-squared error sense; we will keep the windows or tapers w K ( x ) and the convolution kernels W K ( k ) generically in all of the formulation. For the gravity observable, whose spectral density is denoted S GG , we find the modified Fourier coefficients and the spectral covariance, respectively, as Finally, we will need to sample H ( k ) , W K ( k ) , and G ( k ) on a grid of wavenumbers. Exploiting the Hermitian symmetry that applies in the case of real-valued physical quantities, for an M × N data set we select the half-plane consisting of the K = M × ( /floorleft N/ 2 /floorright +1) wave vectors The quantities H ( k ) , W K ( k ) , and G ( k ) are complex except at the dc wave vectors (0 , 0) and the Nyquist wave vectors (0 , π ) , ( -π, 0) and ( -π, π ) if they exist in eq. (14), which depends on the parity of M and N .", "pages": [ 4, 5 ] }, { "title": "2.2 Topography", "content": "As mentioned before, we apply the term 'topography', H , generically to any small perturbation of the Cartesian reference surface, which is assumed to be flat. Specifically, we need to distinguish between what we shall call 'initial', 'equilibrium' and 'final' topographies, respectively. In the classic multilayer loading scenario reviewed by, e.g., McKenzie (2003) and Simons et al. (2003), as the j th interface gets loaded by an initial topography, the singly-indexed quantity H j , a configuration results in which each of the interfaces expresses this loading by assuming an equilibrium topography, which is identified as the double-indexed quantity H ij . The first subscript refers to the interface on which the initial loading occurs; the second to the interface that reflects this process. The state of this equilibrium is governed by the laws of elasticity, as we will see in the next section. All of these equilibrium configurations combine into what we shall call the final topography on the j th interface, namely H · j , where the · is meant to evoke the summation over all of the interfaces that have generated initial-loading contributions.", "pages": [ 5 ] }, { "title": "6 Simons and Olhede", "content": "Thus, in a two-layer scenario, what in common parlance is called 'the' topography, i.e. the final, observable height of mountains and the depth of valleys expressed with respect to a certain neutral reference level, will be called H · 1 , and this then will be the sum of the two unobservable components H 11 and H 21 . In other words, the final 'surface' topography is Likewise, the final 'subsurface' topography, H · 2 , is given by the sum of two unobservable components H 12 and H 22 , totaling H · 2 = H 12 + H 22 . (16) This last quantity, H · 2 , is not directly observable but can be calculated from the Bouguer gravity anomaly, as we describe below. Both H 11 and H 12 refer to the same geological loading process occurring on the first interface but being expressed on the first and second interfaces, respectively. In a similar way, H 21 and H 22 refer to the process loading the second interface which thereby produces topography on the first and second interfaces, respectively. While postponing the discussion on the mechanics to the next section it is perhaps intuitive that a positive height perturbation at one interface creates a negative deflection at another interface: 'mountains' have 'roots', as has been known since the days of Airy (Watts 2001). The initial-loading topography, then, is given by the difference between these two equilibrium components. At the first and second interfaces, respectively, we will have for the initial topographies at the surface and subsurface, respectively, The sum of all of the equilibrium topographies, at all of the interfaces in this system and thus requiring two subscripts · , is given by H ·· = H 11 + H 12 + H 21 + H 22 , which is a quantity that we can only access through the free-air gravity anomaly that it generates, as we shall see.", "pages": [ 6 ] }, { "title": "2.3 Flexure", "content": "Mechanical equilibrium exists between H 11 and H 12 on the one hand, and H 21 and H 22 on the other. The equilibrium refers to the balance between hydrostatic driving and restoring stresses, which depend on the density contrasts, and the stresses resulting from the elastic strength of the lithosphere. Introducing the flexural rigidity D , in units of Nm, we obtain the biharmonic flexural or plate equation (Banks et al. 1977; Turcotte & Schubert 1982) as follows on the first (surface) interface: and at the second (subsurface) level, we have The mechanical constant D is the objective of our study: geologically, this yields to what is commonly referred to as the 'integrated strength' of the lithosphere, which can be usefully interpreted under certain assumptions as an equivalent or 'effective' elastic thickness. This quantity, T e , in units of m, relates to D by a simple scaling involving the Young's modulus E and Poisson's ratio, ν , as is well known (e.g. Ranalli 1995; Watts 2001; Kennett & Bunge 2008). Here we follow these authors and simply define Much has been written about what T e really 'means' in a geological context (Lowry & Smith 1994; Burov & Diament 1995; Lowry & Smith 1995; McKenzie & Fairhead 1997; Burov & Watts 2006). This discussion remains outside of the scope of this study. Moreover, eqs (20) are the only governing equations that we shall consider in this problem. It is not exact (e.g. McKenzie & Bowin 1976; Ribe 1982), it is not complete (e.g. Turcotte & Schubert 1982), and it may not even be right (e.g. Karner 1982; Stephenson & Lambeck 1985; McKenzie 2010). For that matter, a single, isotropic D may be an oversimplification (Stephenson & Beaumont 1980; Lowry & Smith 1995; Simons et al. 2000, 2003; Audet & Mareschal 2004; Swain & Kirby 2003b; Kirby & Swain 2006). However, the neglect of higher-order terms, additional tectonic terms in the force balance, time-dependent visco-elastic effects and elastic anisotropy remain amply justified on geological grounds. It should be clear, however, that any consideration of such additional complexity will amount to a change in the governing equations (20), which we reserve for further study. At the surface, eq. (20a) is solved in the Fourier domain as where we have defined the dimensionless wavenumber-dependent transfer function baptized by Forsyth (1985) At the subsurface, eq. (20b) has the solution (19) with the dimensionless filter function All of the physics of the problem is contained in the equations in this section. As a final note we draw attention to the assumption that the interfaces at which topography is generated and those on which the resulting deformation is expressed coincide: this is the first of the important simplifications introduced by Forsyth (1985). This assumption, though not universally made (e.g. McNutt 1983; Banks et al. 2001), is broadly held to be valid. Finding D in this context is the estimation problem with which we shall concern ourselves.", "pages": [ 6, 7 ] }, { "title": "2.4 Gravity", "content": "Every perturbation from flatness by topography generates a corresponding effect on the gravitational acceleration when compared to the reference state. We relate the gravity anomaly to the disturbing topography by the density perturbation ∆ j and account for the exponential decay of the gravity field from the depth z j ≤ 0 where it was generated. The 'free-air' gravitational anomaly (Hofmann-Wellenhof & Moritz 2006) from the topographic perturbation at the j th interface that results from the i th loading process is given in the spectral domain by where G is the universal gravitational constant, in m 3 kg -1 s -2 , not to be confused with the gravity anomaly itself. Once again this equation is inexact in assuming local Cartesian geometry (Turcotte & Schubert 1982; McKenzie 2003) and neglecting higher-order finite-amplitude effects (Parker 1972; Wieczorek & Phillips 1998), but for our purposes, this 'infinite-slab approximation' will be good enough. The observable free-air anomaly is the sum of all contributions of the kind (26), thus in the two-layer case The Bouguer gravity anomaly is derived from the free-air anomaly by assuming a non-laterally varying density contrast across the surface interface. It thus removes the gravitational effect from the observable surface topography (Blakely 1995), and is given by = -2 πG ∆ 2 e kz 2 [ d H 11 ( k )∆ 1 ∆ -1 2 ξ -1 ( k ) -d H 22 ( k ) ] . (31) In this reduction, we have used eqs (26)-(27), (16) and (22). For simplicity we shall write the Bouguer anomaly as defining one more function, which acts like a harmonic 'upward continuation' operator (Blakely 1995), At this point we remark that topography and gravity, in one form or another, are the only measurable geophysical quantities to help us constrain the value of D . The Bouguer anomaly G · 2 is usually computed from the free-air anomaly G ·· and the topography H · 1 , assuming a density contrast ∆ 1 . Any estimation problem that deals with any combination of these variables should thus yield results that are equivalent to within the error in the estimate (Tarantola 2005), though whether the free-air or the Bouguer gravity anomaly is used as the primary quantity in the estimation process could have an effect on the properties of the solution depending on the manner by which it is found - a paradox that this paper will eliminate.", "pages": [ 7 ] }, { "title": "2.5 Observables, deconvolution, and loading", "content": "We are now in a position to return to writing explicit forms for the theoretical observables from whose particular realizations (the data), ultimately, we desire to estimate the flexural rigidity D . These are the final 'surface' topography, given by combining eqs (15) and (24) as By analogy we shall write for the final 'subsurface' topography that which we can obtain by 'downward continuation' (Blakely 1995) of the Bouguer gravity anomaly. From eq. (32), or combining eqs (16) and (22) this quantity is then The dependence on the parameter of interest, the flexural rigidity D , is non-linear through the 'lithospheric filters' φ and ξ . While both H · 1 and H · 2 can thus be 'observed' (or at least calculated from observations) we are for the moment taciturn about the complexity caused by the potentially unstable inversion of the parameter χ (see also Kirby & Swain 2011). We return to this issue in Section 5. Combining eqs (17)-(18) with eqs (22)-(24) and then substituting the results in eqs (34)-(35) yields the equations that relate the observed topographies on either interface with the applied loads. Without changing from the expressions first derived by Forsyth (1985) these have come to be called the 'load-deconvolution' equations (Lowry & Smith 1994; Banks et al. 2001; Swain & Kirby 2003a; P'erez-Gussiny'e et al. 2004; Kirby & Swain 2008a,b). They a re usually expressed in matrix form as with the inverse relationships given by It should be noted that when D = 0 , in the absence of any lithospheric flexural strength, thus in the case of complete Airy isostasy, φξ = 1 at all wavenumbers, and no such solutions exist. In that case the problem of reconstructing the initial loads has become completely degenerate. Armed with these solutions we can solve for the equilibrium loads. Combining eqs (17)-(18) with eqs (22)-(24) returns usable forms for H 11 and H 22 , and substituting the results back into eqs (22)-(24) returns H 12 and H 21 , all in terms of the initial loads H 1 and H 2 , as To complete this section we formulate the initial-loading stresses, in kgm -1 s -2 , at each interface as (40) All variables that we have introduced up to this point are listed in Table 1, to which we further refer for units and short descriptions. We are now also in the position of further interpreting Fig. 1, once again drawing the readers' attention to the heart of the problem, which is the estimation of the single parameter, the flexural rigidity D , which is responsible for generating, from the initial loads (left), the equilibrium topographies (middle) whose summed effects (right) we observe in the form of 'the' topography and the (Bouguer) gravity anomaly.", "pages": [ 7, 8 ] }, { "title": "2.6 Admittance and coherence", "content": "Modeled after eq. (7), the Fourier-domain relation between the theoretical observable quantities that are the surface topography H · 1 and the Bouguer gravity anomaly G · 2 is encapsulated by the complex-valued theoretical Bouguer admittance, which we define as A quantity whose expression eliminates the dependence on the location of the first interface contained in the term χ of eq. (32) is the real-valued Bouguer coherence-squared, the Cauchy-Schwarz bounded quantity As illustrated by eqs (9)-(13), similarly, the values of either ratio when calculated using actual observations H · 1 and G · 2 or H · 2 , with or without explicit windowing, will be estimators for eqs (41) and (42), but will never manage to recover more than a blurred version of the true cross-power spectral density ratios that they are, and with an estimation variance that will depend on how the required averaging is implemented (Thomson 1982; Percival & Walden 1993). Despite the various attempts by many authors (Diament 1985; Lowry & Smith 1994; Simons et al. 2000, 2003; Kirby & Swain 2004, 2011; Audet & Mareschal 2007; Simons & Wang 2011) to design optimal data treatment, wavelet or (multi-)windowing procedures, with the common goal to minimize the combined effect of such bias or leakage and estimation variance, in the end this may result in a well-defined (non-parametric) estimate for coherence and admittance, but the actual quantity of interest, the flexural rigidity, D , still has to be determined from that. As we wrote in the Introduction, understanding the statistics of the estimators for D derived from estimates of coherence or admittance depends on fully characterizing their distributional properties: a daunting task that, to our knowledge, has never been successfully attempted. Without this, however, we will never know which method is to be preferred under which circumstance. Moreover, we will never be able to properly characterize the standard errors of the estimates except by exhaustive trial and error (see, e.g., P'erez-Gussiny'e et al. 2004; Cr osby 2007; Kalnins & Watts 2009) from data that are synthetically generated. This is no trivial task (Macario et al. 1995; Ojeda & Whitman 2002; Kirby & Swain 2008a,b, 2009); we return to this issue later. We have hereby reached the essence of this paper: our goal is to estimate flexural rigidity D from observed topography H · 1 and gravity G · 2 ; estimates based on inversions of estimated admittance and coherence have led to widely different results, a general lack of understanding of their statistics, and thus a failure to be able to judge their interpretation. We must thus abandon doing this via the intermediary of admittance Q · and coherence γ 2 · , and rather focus on directly constructing the best possible estimator for D from the data. This realization is not unlike that made in the last decade by the seismological community, where the inversion of (group velocity? phase velocity?) surfacewave dispersion curves or individual-phase travel-time measurements has made way for 'full-waveform inversion' in its many guises (e.g. Tromp et al. 2005; Tape et al. 2007). There too, the model is called to explain the data that are actually being collected by the instrument, and not via an additional layer of measurement whose statistics must remain incompletely understood, or modeled with too great a precision. In cosmology, the power-spectral density of the cosmic microwave background radiation (Dahlen & Simons 2008) is but a step towards the determination of the cosmological parameters of interest (e.g. Jungman et al. 1996; Knox 1995; Oh et al. 1999).", "pages": [ 8 ] }, { "title": "3 T H E S T A N D A R D M O D E L", "content": "The essential elements of a geophysical and statistical nature as they had been broadly understood by the late 1970s were reintroduced in the previous section in a consistent framework. In this section we discuss the important innovations and simplifications brought to the problem by Forsyth (1985). In a nutshell, in his seminal paper, Forsyth (1985) made a series of model assumptions that resulted in palatable expressions for the admittance and the coherence as defined in eqs (41) and (42), neither of which would otherwise be of much utility in actually 'solving' the problem of flexural rigidity estimation from gravity and topography. The first two of these were already contained in eq. (20): loading and compensation occur discretely at one and the same set of interfaces, and the constant describing the mechanical behavior of the system is a scalar parameter that does not depend on wavenumber nor direction. The first assumption might be open for debate, and indeed alternatives have been considered in the literature (e.g. Banks et al. 1977, 2001), but reconsidering it would not fundamentally alter the nature of the problem. The second: isotropy of the lithosphere, which is certainly only a null hypothesis (see, e.g. Stephenson & Beaumont 1980; Bechtel 1989; Simons et al. 2000, 2003; Swain & Kirby 2003b; Kirby & Swain 2006, and many observational studies that work on the premise that it must indeed be rejected), does require a treatment that is to be revisited but presently falls outside the scope of this work. To facilitate the subsequent treatment we restate the equations of Section 2.5 in matrix form.", "pages": [ 9 ] }, { "title": "3.1 Flexure of an isotropic lithosphere, revisited", "content": "We shall consider the primary stochastic variables to be the initial-loading topographies H 1 and H 2 , respectively, and describe their joint properties, and their relation to the theoretical observable final topographies H · 1 and H · 2 by defining the spectral increment process vectors Subsequently, we express the process by which lithospheric flexure maps one into the other in the shorthand notation", "pages": [ 9 ] }, { "title": "10", "content": "where the real-valued entries of the non-symmetric lithospheric matrices M D ( k ) and M -1 D ( k ) can be read off eqs (36)-(37) and the functional dependence on the scalar constant flexural rigidity D is implied by the subscript. We now define the (cross-)spectral densities between the individual entries in the initial-topography vector d H ( k ) as in eq. (7) by writing and form the spectral matrix S ( k ) from these elements using the Hermitian transpose as Lithospheric flexure transforms the spectral matrix of the initial topographies, S ( k ) , to that of the final topographies, S · ( k ) , defined as via the mapping implied by eqs (44) through (46). We specify We can now see that the theoretical admittance and coherence of eqs (41)-(42) can equivalently be written as which explains why so many authors before us have focused on admittance and coherence calculations as a spectral estimation problem. /negationslash To be valid spectral matrices of real-valued bivariate fields, the complex-valued S ( k ) and S · ( k ) only need to possess Hermitian symmetry, that is, invariance under the conjugate transpose, and be positive-definite, that is, have non-negative real eigenvalues. The spectral variances of the initial and final topographies at the individual interfaces, S 11 ( k ) ≥ 0 and S 22 ( k ) ≥ 0 , both arbitrarily depend on k , but without dependence between k = k ' . The only additional requirements are that S 12 ( k ) = S ∗ 21 ( k ) and |S 12 ( k ) | 2 ≤ S 11 ( k ) S 22 ( k ) . The general form of S ( k ) as a stationary random process can be rewritten with the aid of a coherency or spectral correlation coefficient, r ( k ) , which expresses the relation between the components of surface and subsurface initial topography as Should we make the additional assumption of joint isotropy for all of the loads, the spectral matrices would both be real and symmetric, S ( k ) = S ( k ) and S · ( k ) = S · ( k ) . In keeping with the notation from eq. (8), we would require a spatial covariance matrix to only depend on distance, not direction. With θ the angle between k and x -x ' we would have the real-valued with J 0 the real-valued zeroth-order Bessel function of the first kind. With S real, the spectral variances and covariances between top and bottom loading components would all be real-valued and so would the correlation coefficient r ( k ) = r ( k ) . It is important to note that the isotropy of the fields individually does not imply their joint isotropy. Two such fields can be spatially slipped versions of one another, but with slippage in a particular direction the fields may remain marginally isotropic but their joint structure will not.", "pages": [ 10 ] }, { "title": "3.2 Correlation between the initial loads", "content": "Statistically, eqs (45) and (49) imply that the initial-loading topographies on the two interfaces are related spectrally as whereby H ⊥ 1 ( x ) , the zero-mean orthogonal complement to H 1 ( x ) , is uncorrelated with it at all lags. The interpretation of what should cause a possible 'correlation' between the initial-loading topographies must be geological (McGovern et al. 2002; McKenzie 2003; Belleguic et al. 2005; Wieczorek 2007; Kirby & Swain 2009). Erosion (e.g. Stephenson 1984; Aharonson et al. 2001) is typically amenable to the description articulated by eq. (52), though much work remains to be done in this area to make it apply to the most general of settings. Under isotropy of the loading, the implication is that the initial subsurface loading H 2 ( x ) can be generated from the initial surface loading H 1 ( x ) by a radially symmetric convolution operator p ( x ) , By selecting the initial loads H j as the primary variables of the flexural estimation problem, and not the equilibrium H ij or final loads H · j , we now have the correlation r between the initial loads to consider in the subsequent treatment. Geologically, this puts us in a bit of a quandary, since if eq. (52) holds, this can only mean that one loading process 'follows the other in time', 'reacting to it'. However, the temporal dimension has not entered our discussion at all, and if it did, it would certainly make sense to choose the correlation between the equilibrium load on one and the initial load on the other interface as the one that matters. The linear relationship (44) between the loads renders these two viewpoints mathematically equivalent. Our definition of eq. (49) is chosen to be mathematically convenient because it is most in line with the choices to be made in the next section. Forsyth (1985) deemed correlations between surface and subsurface loads to be potentially important but he did not make the determination of the correlation coefficient (49) part of the estimation procedure for the flexural rigidity D , which was instead predicated on the assumption, his third by our count, that r ( k ) = 0 . He did recommend computing the correlation coefficient between the initial loads via eq. (44), after the inversion for D , and using the results to aid with the interpretation (see, e.g., Zuber et al. 1989). Studies by Macario et al. (1995), Crosby (2007), Wieczorek (2007) and Kirby & Swain (2009) have since shed more light on how to do this more quantitatively, but to our knowledge no-one has actually attempted to determine the best-fitting correlation coefficient as part of an inversion for flexural rigidity.", "pages": [ 10, 11 ] }, { "title": "3.3 Proportionality between the initial loads", "content": "Forsyth (1985) introduced the 'loading fraction' as the subsurface-to-surface ratio of the power spectral densities of the initial-loading stresses I 2 and I 1 , and thus from eqs (39)-(40) and (45) we can write This definition is fairly consistently applied in the literature (e.g. Banks et al. 2001), though McKenzie (2003) has preferred to parameterize by the fraction each of the loads contributes to the total, which is handy for situations with multiple interfaces (see Kirby & Swain 2009) and subsurface-only loading. Eq. (54) is a statement of proportionality of the power spectral densities of the initial loads, S 2 and S 1 . With this constraint, which we identify as his fourth assumption, Forsyth (1985) was able to factor S 11 out of the spectral matrix S in eq. (45), which as we recall from the previous section, by his third assumption had no off-diagonal terms, to arrive at simplified expressions for S · of eq. (46), which acquires off-diagonal terms through eq. (47), and ultimately for the admittance Q · and coherence γ 2 · in eq. (48). We revisit these quantities in the next section but conclude with the general form of the initial-loading spectral matrix that is implied by the definition of proportionality, which is With what we have obtained so far: flexural isotropy of the lithosphere, M D ( k ) , correlation of the initial-loading processes, r ( k ) , and proportionality of the initial-loading processes, f 2 ( k ) , the spectral matrix (47) of the final topographies - those we measure - is given by where we have defined the auxiliary matrices We define both T and ∆T so that we can easily revert to a model of zero correlation, in which case ∆T = 0 . Note that we are silent about the dependence on wavenumber by using the shorthand notation ξ and φ for the lithospheric filters (23) and (25), but have kept the full forms of the correlation coefficient r ( k ) and the loading ratio f 2 ( k ) to stress that they are in general functions of the wave vector as defined by eqs (49) and (54). In general r will be complex and of magnitude smaller than or equal to unity, and f 2 (and f ) will be real and positive.", "pages": [ 11 ] }, { "title": "3.4 Admittance and coherence for proportional and correlated initial loads", "content": "Via eqs (56)-(58) we have explicit access to the (cross-)spectral densities between the individual elements in the final-topography vector d H · , as required to evaluate eq. (46). We shall now consider those for the special case where both r ( k ) = r and f 2 ( k ) = f 2 are constants, no longer varying with the wave vector. Then, following eq. (48), we obtain simple expressions for the admittance and coherence that we shall further specialize to a few end-member cases for comparison with those treated in the prior literature. We hereby complete Table 1 to which we again refer for a summary of the relevant notation. The Bouguer-topography admittance, for correlated and proportional initial loads with constant correlation r and proportion f 2 , is Spectrally, this is a function of wavenumber, k , only, since the power spectra of the loading topographies, which both may vary (similarly, because of their proportionality) with the wave vector k have been factored out. This admittance can be complex-valued since the load correlation may be, unless the power spectra of the loading topographies are isotropic. At k = 0 the admittance yields the density contrast ∆ 1 . Assuming that the loads are uncorrelated but proportional simplifies the Bouguer-topography admittance to the familiar expression In scenarios where only top or only bottom loading is present, we get the original expressions (Turcotte & Schubert 1982; Forsyth 1985) where, as expected and easily verified, The Bouguer-topography coherence, for correlated and proportional initial loads with constant correlation r and proportion f 2 , is When the initial loads are uncorrelated but proportional the Bouguer-topography coherence is, as according to Forsyth (1985), simply which, as the admittance, is a function of wavenumber k regardless of the power spectral densities of the loading topographies. Unlike the admittance it has lost the dependence on the depth to the second interface, z 2 , and it is always real, 0 ≤ γ 2 · ≤ 1 . Fig. 2 displays the individual effects that varying flexural rigidity, loading fraction and load correlation have on the expected admittance and coherence curves. Regardless of the fact that much of the literature to this date has been concerned with the estimation of the admittance and coherence from the available data, and regardless of the justifiably large amount of attention devoted to the role of windowing and tapering to render these estimates spatially selective and spectrally free from excessive leakage; regardless, in summary, of any practicality to the actual methodology by which admittance and coherence are being estimated and how the behavior of their estimates affects the behavior of the estimated parameter of interest, the flexural rigidity, D , we show these curves to gain an appreciation of the complexity of the task at hand. No matter how well we may be able to recover the 'true' admittance and coherence behavior, the issue remains that they need to be interpreted - inverted - for a model that ultimately needs, or can, return an estimate for D but also of the initial-loading fraction, f 2 , and also of the correlation coefficient, r . Each of these have distinct sensitivities but overlapping effects on the predicted behavior of the measurements: selecting one end-member model (top-loading or bottom-loading only, for example, or disregarding the very possibility of load correlation, or imposing a certain non-vanishing value on the loading fraction or load-correlation coefficient) remains but one choice open to alternatives, and constraining all three is a task that, thus far, nobody has successfully attempted. Fig. 2 serves as a visual reminder of the limitations of admittance- and coherence-based estimation. However much information these statistical summaries of the gravity and topography data contain, it is not easily accessible for navigation in the three-dimensional space of D , f 2 and r . where β = ∆ 2 2 + 2 f ( ∆ 2 2 -∆ 1 ∆ 2 ) + f 2 ( ∆ 2 1 +∆ 2 2 +4∆ 1 ∆ 2 ) -2 f 3 ( ∆ 1 ∆ 2 -∆ 2 1 ) + f 4 ∆ 2 1 . In the paper by Simons et al. (2003) eq. (66) appears with a typo in the leading term, which was briefly the cause of some confusion in the literature (Kirby & Swain 2008a,b).", "pages": [ 11, 12 ] }, { "title": "3.5 Load correlation, proportionality and the standard model", "content": "The expressions in the previous section show how difficult it is to extract the model parameters D , f 2 and r individually from admittance or coherence. Forsyth (1985) argued that coherence depends on f 2 much more weakly than admittance, but what is important for the estimation problem is how the three parameters of interest vary together functionally: whether they occur in terms by themselves or as products, in which variations of powers, and so on. The geometry of the objective functions used to estimate the triplet of parameters, together with the distribution of any random quantities the objective functions contain, determine the properties of the estimators. We return to the question of identifiability after we have presented the new maximum-likelihood estimation method. For that matter, Forsyth (1985) suggested ignoring the load correlation, setting r = 0 , and finding an estimate for the flexural rigidity D using a constant initial guess for the loading fraction f 2 and the coherence modeled as γ 2 f in eq. (65), and then using eqs (37), (39)-(40) and (54) to compute a wavenumber-dependent estimate of f 2 , which can then be plugged back into eq. (65) as a variable, and iterating this procedure to convergence. However, this allows for as many degrees of freedom as there are 'data', thereby running the risk that an ill-fitting D can be reconciled with the data by adjustment with a very variable f 2 . It is unclear in this context what 'ill-fitting' or 'very variable' should mean, and thus it is hard to think of objective criteria to accomplish this. McKenzie (2003) showed misfit surfaces for the (free-air) admittance for varying D and varying f 2 held constant over all wavenumbers. These figures show prominent trade-offs, suggesting a profound lack of identifiability of D and f 2 with such a method.", "pages": [ 12 ] }, { "title": "14 Simons and Olhede", "content": "Even more importantly, McKenzie (2003) emphasized the possibility of non-zero correlations between the initial loads, deeming those prevalent in many areas of low-lying topography, on old portions of the continents: precisely where the discrepancy between estimates for elastic thickness derived from different methods has been leading to so much controversy. As an alternative to the Forsyth (1985) method, McKenzie & Fairhead (1997) suggested estimating D and f 2 from the free-air admittance in the wavenumber regime where surface topography and free-air gravity are most coherent. The rationale for this procedure is that there might be loading scenarios resulting in gravity anomalies but not (much) topography, a situation not accounted for in the Forsyth (1985) model that can, however, be described by initial-load correlation. Kirby & Swain (2009), most recently, discussed the differences between both approaches, only to conclude that neither estimates the complete triplet ( D,f 2 , r ) of parameters (rigidity, proportionality, correlation) without shortcuts. Once again the statistical understanding required to evaluate whether either of these techniques results in 'good' estimators is lacking. That the cause of 'internal loads without topographic expression' can indeed be attributed to correlation in the sense of (49) can be readily demonstrated by considering what it takes for the final, observable, surface topography H · 1 to vanish exactly. Solving eq. (36) or eq. (44) and using eqs (23) and (25) returns the conditions that the first and second initial topographies are related to each other as which, using eqs (45), (54) and (49), implies the following equivalent relations between them: This set of equations together with our model very strongly constrain both fields. Thus, as noted by McKenzie (2003) and others after him (Crosby 2007; Wieczorek 2007; Kirby & Swain 2009), a situation of internal loading that results in no net final topography may arise when the initial-loading topographies are perfectly correlated, balancing one another according to eqs (67)-(68). We can find a more complete condition for this scenario by equating eqs (67) and (52), which returns an expression for the orthogonal complement d H ⊥ 1 ; when this is required to vanish non-trivially we obtain the seemingly more general condition Requiring that the final surface topography have a vanishing variance S · 11 , substituting eqs (56)-(58) into eq. (46), we need to satisfy The correlation coefficients in eqs (69)-(70) must be real-valued since all of the other quantities involved are. Both eq. (69) and eq. (70) should be equivalent, and together they imply eq. (68). We are thus left to conclude that for the observable surface topography to vanish, the correlation between initial surface and subsurface loading must be perfect and positive, r = 1 . Solving the quadratic equation (70) for f yields real-valued results only when | r | 2 -1 ≥ 0 , thus r = 1 for positive but non-constant f , as expected. The above considerations have put perhaps unusually strong constraints on the spectral forms of the final topography H · 1 ( k ) or S · 11 ( k ) . From eq. (3) we learn that in doing so, the spatial-domain observables H · 1 ( x ) can never be non-zero. On the other hand, an observed H · 1 ( x ) could be zero over a restricted patch without its Fourier transform or its spectral density vanishing exactly everywhere. Alternatively, it can be very nearly zero, and this may also practically hamper approaches based on admittance or coherence which contain (estimates of) the term S · 11 ( k ) in the denominator (see eq. 48). When the observed topography becomes small, higher-order neglected terms may become prominent. Furthermore, there may be mixtures of loads with and without topographic expression (McKenzie 2003). Speaking quite generally, there will be areas with some correlation between the initial loads, and we should take this into account in the estimation. Either one of the load correlation or load fraction may vary with wavenumber. What emerges from this discussion is that the isotropic flexural rigidity D , the initial-load correlation r ( k ) , and the initial-load proportionality f 2 ( k ) should all be part of the 'standard model' of flexural studies. The last two concepts were introduced by Forsyth (1985), even though he did not further discuss the case of non-zero correlation. As we wrote in the first paragraph in this section, Forsyth's first assumption was that the depth of compensation and the depth of loading in fact coincide. He writes that the assumption of collocation of these hypothetical interfaces and their precise location at depth in Earth may well be the largest contributor to uncertainty in the estimates for flexural strength, but also that there may be a priori , e.g. seismological, information to help constrain the depth z 2 . Thus, much like the density contrasts ∆ 1 and ∆ 2 , we will not include the depth to the second interface z 2 as a quantity to be estimated directly. Rather, we will consider them known inputs to our own estimation procedure and evaluate their suitability after the fact by an analysis of the likelihood functions and of the distribution of the residuals.", "pages": [ 14 ] }, { "title": "4 MAXIMUM-LIKELIHOOD THEORY", "content": "Measurements of 'gravity' and 'topography', which we consider free from observational noise, can be interpreted as undulations, H · 1 and H · 2 , of the surface and one subsurface density interface, with density contrasts, ∆ 1 and ∆ 2 , located at depths z 1 = 0 at z 2 in Earth, respectively. Geology and 'tectonics' produce initial topographic loads, H 1 and H 2 , on these previously undisturbed interfaces. These are treated as a zero-mean bivariate, stationary, random process vector, d H , fully and most generally described by a spectral matrix, S ( k ) , under the assumption that the higher-order moments of H ( x ) are not too prominent (Brillinger 1975). For this paper we assume isotropy of the loading process, S = S ( k ) . The lithosphere is modeled as a coupled set of differential equations, whose action is described by the spectral-domain matrix M D , which depends on a single, scalar parameter of interest, the flexural rigidity D . Since our observations have experienced the linear mapping d H · = M D d H , their spectral matrix is S · ( k ) = M D ( k ) S ( k ) M T D ( k ) , and the objective is to recover D , we are led to study S · ( k ) . This includes its off-diagonal terms, which depend on the correlation coefficient of the loads at either interface, -1 ≤ r ( k ) ≤ 1 , recall r ( k ) ∈ R , and, under the assumption of proportionality of the initial-loading spectra, on a loading fraction, f 2 ( k ) . As part of the estimation we will thus also recover information about the loading process S . All previous studies in the geophysical context of lithospheric thickness determination have first estimated admittance and coherence, ratios of certain elements of S · whose estimators have joint distributions that have not been studied. These were then used in inversion for estimates of D whose statistics have remained unknown. In the remainder of this paper we construct a maximum-likelihood estimator sensu Whittle (1953), directly from the data 'gravity' and 'topography', and the 'known' parameters ∆ 1 , ∆ 2 , and z 2 . The unknowns are D , r and f 2 , and, as we shall see shortly, three more parameters by which we guarantee isotropy of the loading process S through a commonly utilized functional form. That this is more ambitious than the original objectives by Forsyth (1985) and the modifications by McKenzie (2003) is because the reduction of the data to admittance or coherence obliterates information that we are able to recover in some measure. Westudy the properties of the new estimators and derive the distributions of the residuals. When the procedure is applied to actual data, these should tell us where to adjust the assumptions used in designing the model.", "pages": [ 14, 15 ] }, { "title": "4.1 Choice of spectral parameterization, σ 2 , ν, ρ", "content": "In the above we have seen that the primary descriptor of what causes the observed behavior is the spectral matrix S ( k ) from which the initial interface-loading topographies are being generated. After the assumption of spectral proportionality of the loading at the two interfaces, the expressions for admittance and coherence no longer contain any information about this particular quantity, though of course the deviations of the observed admittance and coherence from the models discussed in Section 3.4 still might. However, this information is no longer in an easily accessible form. Furthermore, coherence and admittance are typically estimated non-parametrically: the infinitely many, or rather, 2 K = M × N dimensions of the data are reduced to a small number of wavenumbers at which they are being estimated, thus there is a loss of O ( K ) degrees of freedom. At the low frequencies, most tapering methods experience a further reduction in resolution, which is detrimental especially in estimating the value of thick lithospheres from relatively small data grids, as is well appreciated in the geophysical literature. Here, we will simply parameterize the initial loading using a 'red' model, thereby avoiding such a loss. We may consult Goff & Jordan (1988, 1989), Carpentier & Roy-Chowdhury (2007) or Gneiting et al. (2010) for such models. Here we do, however, make the very strong assumption of isotropy. This is unlikely to be satisfied in real-world situations, as spectral-domain anisotropy is part and parcel of all geological processes (Goff et al. 1991; Carpentier & Roy-Chowdhury 2009; Carpentier et al. 2009; Goff & Arbic 2010). Relaxing the isotropic loading assumption introduces considerable extra complications. Our reluctance to handle anisotropic loading situations stems from the fact that their estimation might be confused statistically with a possible anisotropy in the lithospheric response: we can thus not easily study one without studying the other. At this point we collect the parameters that we wish to estimate into a vector. To begin with, the 'lithospheric' parameters, flexural rigidity D , loading ratio f 2 and load correlation r are Wedenote a generic element of this vector as θ L . For the spectrum of the initial-loading topographies we choose the isotropic Mat'ern spectral class, which has legitimacy in geophysical circles (Goff & Jordan 1988; Stein 1999; Guttorp & Gneiting 2006). We specify whose parameters we collect in the set with generic element θ S . The third parameter, ρ , is distinct from the mass density, as will be clear from the context. The full set of parameters that we wish to estimate problem is contained in the vector whose general element we denote by θ . For future reference we define the parameter vector that omits all consideration of the correlation as ˜ θ = [ D f 2 σ 2 ν ρ ] T . (75) Fig. 3 shows a number of realizations of isotropic Mat'ern processes with different spectral parameters. As can be seen the parameters σ 2 ('variance') and ρ ('range') impart an overall sense of scale to the distribution while ν ('differentiability') affects its shape (Stein 1999; Paciorek 2007).", "pages": [ 15 ] }, { "title": "4.2 The observation vectors, d H and H", "content": "In Section 2 we introduced the standard statistical point of view on stationary processes (Brillinger 1975; Percival & Walden 1993). We specified how this applies to a finite set of geophysical observations that can be defined in a two-layer system, which we revealed to be the various types of 'topography' and 'gravity', and which are mapped into one another by the differential equations describing 'flexure'. Subsequently, we introduced the matrix formalism that describes the connections between the various geophysical observables and the initial driving forces that produce them, which we used extensively in Section 3 to discuss the standard approach of determining the unknown parameters of the flexural differential equation and the relative importance and correlation of the loading processes acting across either layer interface, which are of geophysical interest (e.g. Forsyth 1985; McKenzie 2003). To address the problem of how to properly estimate these northing & easting (km) northing & easting (km) northing & easting (km) unknowns and their distribution, we now return to the statistical formalism espoused in Section 2.1 in order to clarify how the 'theorized' geophysical observables, i.e. the spectral processes describing the various kinds of topography d H ( k ) and gravity anomalies d G ( k ) are being shaped into the 'actual' observations. Those are the windowed Fourier transforms H ( k ) and G ( k ) of particular realizations of topography and gravity as we can calculate from finite spatial data sets H ( x ) and G ( x ) measured in nature. In the spectral domain we continue to distinguish by the choice of font the theory (calligraphic) from what we can actually calculate (italicized). In the spatial domain, there is no need to define anything but H ( x ) or G ( x ) .", "pages": [ 15, 16 ] }, { "title": "4.2.1 In theory: infinite length and continuous", "content": "We recall that the spectral matrix S · ( k ) , given by eq. (56), of the vector of final, observable, topographies d H · ( k ) defined in eqs (43)-(47), is separable in the sought-after parameter vectors θ S and θ L by the factoring of the spectral density S 11 ( k ) of the initial-loading topographies, In writing eq. (76) we emphasize the wavenumber-only dependence of the 'spectral' matrix S 11 ( k ) , which is isotropic, but keep the full wavevector dependence of the 'lithospheric' matrices T ( k ) and ∆T ( k ) to make sure they have the same dimensions as the data. However, in the case of isotropic loading both T ( k ) and ∆T ( k ) will also only depend on wavenumber, and they will both be real. We thus rewrite eqs (57)-(58) with the dependencies φ ( k ) , ξ ( k ) , r ( k ) and f 2 ( k ) implicit in this sense, The Cholesky decomposition reverts to the Cholesky decomposition of T ( k ) when r = 0 . Explicit expressions appear in Appendix 9.1. Because of the above relationships the transformed quantities elevation (km) elevation (km) 8 0 -8 -16 8 0 -8 -16 0 800 1600 2400 0 800 1600 2400 distance (km) 0 800 1600 2400 0 800 1600 2400 distance (km) have a spectral matrix that is the 2 × 2 identity,", "pages": [ 16, 17 ] }, { "title": "4.2.2 In actuality: finite length and discretely sampled", "content": "We now define the vector of Fourier-transformed observations, derived from the actual measurements in eq. (5) and in (13), through eq. (35), With W K ( k ) the Fourier transform of the applied window defined in eq. (6), and by comparison with eqs (9)-(13), the covariance In comparison to eq. (46) and eqs (56) or (76), the finite observation window introduces spectral blurring, the loss of separability of the spectral and lithospheric portions, and small correlations between wave vectors. These we ignored when writing the last, approximate equality, introducing the blurred quantity (for a specific window w K , as opposed to eqs 10-11 where we first used the overbar notation) such that the transformed variable has unit variance (87) (82) distance (km) distance (km) H 1 H 2 H 11 H 12 H 21 H 22", "pages": [ 17 ] }, { "title": "4.2.3 In simulations: how to go from the continuous to the discrete formulation", "content": "Correctly generating a data set H · that is a realization from a theoretical spectral process d H · with the prescribed spectral density S · requires ensuring that when we observe a finite sample of it, and we form the (tapered) periodogram of this, we get the correctly blurred spectral density (Percival 1992; Chan & Wood 1999; Dietrich & Newsam 1993, 1997; Thomson 2001; Gneiting et al. 2006) in our case eq. (84). Stability considerations require that should we simulate data on one discrete grid and then extract a portion on another discrete grid, we replicate the correct covariance structure everywhere in space and always produce the correct blurring upon analysis. Failure to acknowledge the grid properly at the simulation stage can lead to severely compromised results as will be readily experienced but has not always been consciously acknowledged in the (geophysical) literature (Peitgen & Saupe 1988; Robin et al. 1993). The method that we outline here is variously known as Davies & Harte (1987) or circulant embedding (Wood & Chan 1994; Craigmile 2003). Let us assume that we have a spatial grid x as in eq. (4), and a half-plane Fourier grid k as in eq. (14). On the K entries of the latter we generate (complex proper) Gaussian variables Z · ( k ) and then transform these as suggested by eqs (86)-(87),", "pages": [ 18 ] }, { "title": "H · ( k ) = ¯ L · ( k ) Z · ( k ) ,", "content": "whereby ¯ L · is the Cholesky decomposition expressed on the grid k , of eq. (84) calculated on a much finer grid k ' . In other words, whereby | F ( k ) | 2 is the unmodified periodogram of the spatial boxcar function that defines the simulation grid. The convolution in eq. (89) is to be implemented numerically, with care taken to preserve the positive-definiteness of the result. We now define the discrete inverse Fourier transform of this particular set of variables for this fixed set of wave vectors k to be equal to the integral that we introduced in eq. (3), (88) which holds, in fact, for any x ∈ R 2 , and is consistent with eq. (5) which holds for the area of interest picked out by the boxcar window. We generate synthetic data sets H · ( x ) via eqs (88)-(90): by this procedure the covariance between any two points x and x ' in any portion of space identified as our region of interest is now determined to be which follows from eqs (90), (46) and (83) with the small correlations between wave vectors neglected, and using the notation introduced in eq. (51). Now eq. (91) is equal to the universal expression in eq. (8), consistent with eqs (10)-(12), and since the dependence is only on the separation x -x ' , stationarity is guaranteed. With x = x ' eq. (91) states Parseval's theorem: at every point in space the variance of H · is equal to all of its spectral energy. Of course in the isotropic case considered here, C 0 ( x -x ' ) = C 0 ( ‖ x -x ' ‖ ) , depending only on distance. Should we now take the finite windowed Fourier transform of such synthetically generated spatial data H · ( x ) on a different spatial patch (e.g. a subportion from the master set), while using any arbitrary window or taper w K ' ( x ) , we will be seeing the correctly blurred version of the theoretical spectral density S · , as required to ensure stability. Indeed, when forming a new set of modified Fourier coefficients H ' · ( k ) , distinguished by a prime, their covariance now must be, as follows directly from eqs (92), (91) and (6), the blurred quantity which is exactly as we have wanted it to be consistent with eq. (83). We will continue to neglect the small correlations between wave vectors, but fortunately this will have limited impact (Varin 2008; Varin et al. 2011). Fig. 4 shows a realization of a simulation produced with the method just described. In contrast to Fig. 1 we now show the result of the case where the initial-loading topographies are indeed (negatively) correlated. Evidence for the loading correlation is not apparent to the naked eye.", "pages": [ 18 ] }, { "title": "4.3 The log-likelihood function, L", "content": "Conditioned upon higher-order moments of the space-domain data being finite (Brillinger 1975), their Fourier components are near-Gaussian distributed, and for stationary processes, there are no correlations between the real and imaginary parts of the Fourier transform, which are independent. Writing N for the Gaussian and N C for the proper complex Gaussian distributions (Miller 1969; Neeser & Massey 1993), and dropping more wave vector dependencies as arguments than before, the observation vectors H · ( k ) in eq. (82) and the rescaled Z · ( k ) of eq. (86) are thus characterized at each wave vector k by the probability density functions As we have noted at the end of Section 2.1, at the Nyquist and zero wave numbers these quantities are real with unit variance. In so writing the observation vector is treated as a random variable, but we are interested in the likelihood of observing the particular data set at hand given the model, which for us means an evaluation at the data in function of the deterministic parameters σ 2 , ρ , ν , D , f 2 , r . This quantity, ¯ L ( θ ) , receives contributions from each wave vector k that, once the number K of considered wave vectors is large enough, can be considered independent from one another (Dzhamparidze & Yaglom 1983). The log-likelihood is thus, up to a constant, given by the standard form While we know that there is in fact correlation between the terms ¯ L k ( θ ) , only at very small sample sizes K will this produce inefficient estimators, as the accrued effects of the correlation diminish in importance with increasing sample sizes. At moderate to large sample sizes there is considerable gain in computational efficiency and no loss of statistical efficiency due to the fast spectral decay of the blurring kernel functions involved. Our objective function, the log-likelihood, remains simply the average of the contributions at each wave vector in the half plane. Of course eqs (96)-(97) contain the blurred spectral forms ¯ S · ( k ) that we defined in eq. (84), in acknowledgment of the fact that the variance experiences the influence from nearby wave vectors: the approximation made asymptotically is that of eq. (83), but eq. (84) is exact. While we cannot ignore this blurring for finite sample size and for the particular data tapers used to obtain the windowed Fourier transforms, for very large data sets and well-designed, fast-decaying, window functions (e.g. Simons & Wang 2011) the observation vectors H · will converge 'in law' (Ferguson 1996) to random variables H ' · that are distributed as complex proper Gaussian with an unblurred variance, in which case we would simply write Working with this distribution is mathematically more convenient since all of the subsequent calculations can be done analytically, and, per eq. (76), separably in the lithospheric and spectral parameters, so we will adhere to it until further notice. In this case the log-likelihood is While algorithms for simulation and data analysis will be based on eq. (97), we will use eq. (100) to study the properties of the solution, ultimately (in Section 6 and Appendix 9.8) demonstrating why such an approach is justified. On par with eq. (100) we introduce an equivalent likelihood in whose formulation the correlation coefficient r does not appear, with the notation of eqs (74)-(75) and eqs (76)-(78), namely", "pages": [ 18, 19 ] }, { "title": "4.4 The maximum-likelihood estimator, ˆ θ", "content": "The gradient of the log-likelihood, the score function, is the vector with generic elements, never to be confused with the coherence functions (64)-(65), that we shall denote as Following standard theory (Pawitan 2001; Davison 2003) we define the maximum-likelihood estimate as that which maximizes L ( θ ) , thus ˆ θ is the vector of the maximum-likelihood estimate of the parameters, for which Contingent upon the requisite second order conditions being satisfied (Severini 2001), this is also assumed to be the global maximum of (100) in the range of parameters that θ is allowed to take. We now let θ 0 be the vector containing the true, unknown values, and have a certain θ ' lie somewhere inside a ball of radius ‖ ˆ θ -θ 0 ‖ around it. Then we may expand the score with a multivariate Taylor series expansion, using the Lagrange form of the remainder, to arrive at the exact expression The random matrix F is the Hessian of the log-likelihood function, with elements defined by and an expected value -F , the Fisher 'information matrix', Hence the name 'observed Fisher matrix' which is sometimes used for the Hessian. If it is invertible we may rearrange eq. (105) and write ˆ θ = θ 0 - F ( θ ) γ ( θ 0 ) . (108) For this exponential family of distributions the random Hessian converges 'in probability' to the constant Fisher matrix This is more than a statement about means: the fluctuations of F about its expected value also become smaller and smaller. Thus, no matter where we evaluate the Hessian, at θ ' or at θ 0 , both tend to the constant matrix F . The distributional properties of the maximumlikelihood estimator ˆ θ can be deduced from eqs (108)-(109), which are also the basis for Newton-Raphson iterative numerical schemes (e.g. Dahlen & Simons 2008). We thus need to study the behavior of γ , F , and F . The symbols of the statistical apparatus that we have assembled so far are listed in Table 2.", "pages": [ 19, 20 ] }, { "title": "4.5 The score function, γ", "content": "Per eqs (102)-(104) the derivatives of the log-likelihood function L vanish at the maximum-likelihood estimate ˆ θ . With our representation of the unknowns of our problem by the parameter sets θ L and θ S we are in the position to calculate the elements of the score function γ explicitly. We remind the reader that these are not for use in the optimization using real data sets where the blurred likelihood ¯ L is to be maximized instead. In that case the scores of ¯ L will need to be calculated numerically. However, the scores of the unblurred likelihood L that we present here will prove to be useful in the calculation of the variance of the maximum-blurred-likelihood estimator. Combining eqs (100) through (103) we see that the general form of the elements of the score function will be given by For the lithospheric and spectral parameters, respectively, we will have 1 2 L ∂ ln(det T ∂θ L L ∂ T - 1 · ∂θ L The explicit expressions can be found in Appendices 9.2-9.3. For completeness we note here that ∂S -1 11 /∂θ S = -m θ S S -1 11 . To determine the sampling properties of the maximum-likelihood estimation procedure we use eqs (99)-(103) to make the identifications to obtain the standard result that the expectation of the score over multiple hypothetical realizations of the observation vector vanishes, as In the treatment that is to follow (Johnson & Kotz 1973), we will need to perform operations on multiple similar forms as in eq. (110), namely γ θ ( k ) = - 2 m θ ( k ) -S - 1 11 H H · A θ H · . (115) To facilitate the development for the second term in eq. (115) we use eq. (88), but again without the complications of spectral blurring, see eq. (80), and proceed by eigenvalue decomposition of the symmetric matrices L T · A θ L · to yield (118) Since the matrix P θ is orthonormal, Z θ and ˜ Z θ are identically distributed and thus we find through eq. (96) that eq. (117) is a weighted sum of independent random variables, each exponentially distributed, χ 2 2 / 2 , with unit mean and variance. In summary, we have the convenient form for the contributions to the score (110) from each individual wave vector, · ) - 1 ' m θ ( k ) = , A θ = , (111) Since m θ is nonrandom we thus have an expectation for the contributions to the score that confirms eq. (114), namely and a variance given by Eq. (121) gave us the variance of the derivatives of the log-likelihood function with respect to the parameters of interest, which was written in terms of the eigenvalues of the non-random matrix L T · A θ L · . More specifically, for the variances of the scores in the lithospheric parameters θ L in θ L = [ D f 2 r ] T , we will find whereas for the variances of the scores in any of the three spectral parameters θ S in θ S = [ σ 2 ν ρ ] T , judging from eq. (112), we will need the sum of the squared eigenvalues of -m θ S L T · T -1 · L · and since L · is the Cholesky decomposition of T · , we have T -1 · = L -T · L -1 · and As to the covariance of the scores in the different parameters we use eqs (113)-(114) to write and thereby manage to equate the variance of the score to the expectation of the negative of its derivative, which should of course specialize to verify eq. (121), giving us two calculation methods for the variance terms. We do not consider any covariance between the scores at non-equal wave vectors. From eqs (110) and (119) we have learned that the full score γ θ is a sum of random variables γ θ ( k ) or indeed the | ˜ Z ± θ ( k ) | 2 , which belong to the exponential family. Between those we consider no correlations at different wave vectors, and eqs (120) and (126) have given us their mean and covariance, respectively. Lindeberg-Feller central limit theorems apply (Feller 1968), and so the distribution of the score γ θ will be Gaussian with mean zero and covariance Using eqs (126), (100) and (106)-(107) we can rewrite the above expression in terms of the diagonal elements of the Fisher matrix, Wecan summarize all of the above by stating that, for K sufficiently large, ignoring wave vector correlations, and through the LindebergFeller central limit theorem, the vector with the scores in the individual parameters converges in law to what is distributed as", "pages": [ 20, 21 ] }, { "title": "4.6 The Fisher information matrix, F", "content": "From the definition in eq. (107) we have that the elements of the Fisher matrix F are given by the negative expectation of the elements of the Hessian matrix F , which themselves are the second derivatives of the log-likelihood function L with respect to the parameters of interest θ . Per eq. (128) the Fisher matrix scales to the covariance of the score γ , and by combining eqs (123)-(124) with eq. (110) or, ultimately, eqs (121) and (127), we thus find a convenient expression for the diagonal elements of the Fisher matrix, namely which, for the spectral parameters specializes to the more easily calculated expression For the cross terms, rather than combining eqs (119) and (127), we proceed via eq. (128) and thus require expressions for the elements of the Hessian. From eqs (106) and (110) we derive that the general expression for the elements of the symmetric Hessian matrix are Unless we use it in the numerical optimization of the log-likelihood we only need the negative expectation of eq. (132), the Fisher matrix where we have used eq. (122). Of course, when θ = θ ' , the general eq. (133) specializes to the special case (130) discussed before. Ultimately this equivalence is a consequence of eq. (126) which held that in expectation, the product of first derivatives of the log-likelihood is equal to its second derivative. The explicit forms are listed in Appendix 9.4, but looking ahead, we will point to two special cases that result in simplified expressions. It should be clear from the separation of lithospheric and spectral parameters achieved in eq. (76) and from eqs (111)-(112) that the mixed derivatives of one lithospheric and one spectral parameter, ∂ θ L m θ S = ∂ θ S m θ L = 0 and ∂ θ L S 11 = 0 , both vanish, and that we thereby have Finally, we also easily deduce that where we have used the previously noted special case of eq. (122) by which 〈S -1 11 H H · T -1 H · 〉 = tr ( L T · T -1 · L · ) = 2 . The previously encountered eq. (131) is again a special case of eq. (135) when θ S = θ ' S . Both expressions (134) and (135) are of an appealing symmetry. Between them they cover the majority of the elements of the Fisher matrix, which will thus be relatively easy to compute.", "pages": [ 21, 22, 23 ] }, { "title": "4.7 Properties of the maximum-likelihood estimate, ˆ θ", "content": "We are now ready to derive the properties of the maximum-likelihood estimate given in eq. (108), which we repeat here, as (136) From eq. (129) we know that the score γ converges to a multivariate Gaussian, and from eq. (109) we know that the Hessian F converges in probability to the Fisher matrix F . A Taylor expansion allows us to replace θ ' by θ 0 as in standard statistical practice (Cox & Hinkley 1974). Thus, by Slutsky's lemma (Severini 2001; Davison 2003) the distribution of ˆ θ is also a multivariate Gaussian. Its expectation will be showing how our maximum-likelihood estimator is unbiased. Its covariance is From eq. (128) we retain that K cov { γ ( θ 0 ) } = F ( θ 0 ) and with F = F T a symmetric matrix, we conclude that the covariance of the maximum-likelihood estimator is given by In summary, we have shown that which allows us to construct confidence intervals on the parameter vector θ . Denoting the generic diagonal element of the inverse of the Fisher matrix evaluated at the truth θ 0 as J θθ ( θ 0 ) , this equation shows us that each element of the parameter vector is distributed as As customary, we shall replace the needed values θ 0 with the estimates ˆ θ and quote the 100 × α %confidence interval on θ 0 as given by where z α is the value at which the standard normal reaches a cumulative probability of 1 -α , i.e. z α/ 2 ≈ 1 . 96 for a 95% confidence interval. These conclusions, which are exact for the case under consideration, will hold asymptotically when in practice we use the blurred likelihood (97) instead of eq. (100). In the blurred case and for all numerical optimization procedures, we expect to have to amend eqs (137) and (138) by correction factors on the order of K -1 and K -2 , respectively. Eq. (142) would receive extra correction terms starting with the order K -1 , which would be immaterial given the size of the confidence interval. In some sense, eq. (142) concludes the analysis of our maximum-likelihood solution to the problem of flexural-rigidity estimation. It makes the important statement that each of the estimates of flexural rigidity D , initial-loading ratio f 2 , and load correlation coefficient r , will be normally distributed variables centered on the true values and with a standard deviation which will scale with the inverse square-root of the physical data size K . Obtaining the variance on the estimates of effective elastic thickness T e from the estimates of D will be made through eq. (21) via the 'delta method' (Davison 2003). This implies that the estimate of the effective elastic thickness is approximately distributed as", "pages": [ 23 ] }, { "title": "4.8 Analysis of residuals", "content": "Once the estimate ˆ θ = [ ˆ θ L ˆ θ S ] T has been found, we may combine it with our observations, and through eq. (86), form the variable ˆ Z 0 ( k ) = ¯ L -1 · ( k ) ∣ ∣ ˆ θ H · ( k ) , (144) which should be distributed as the standard complex proper Gaussian N C ( 0 , I ) . Equivalently, and as a special case of eqs (97) and (117), 0 and inspect the quantile-quantile plot (Davison 2003) whereby the X ( j ) 0 , for all j = 1 , . . . , K , are plotted versus the inverse cumulative density function of the χ 2 4 / 2 distribution, evaluated at the argument j/ ( K + 1) . If, apart from at very low and very high values of j , this graph follows a one-to-one line, there will be no reason to assume that our model is bad for the data. This can then further be formalized by a chi-squared test (Davison 2003), but a plot of the residuals as a function of wave vector will be more informative to determine how the model is misfitting the data. In particular it may diagnose anisotropy of some form, or identify particular regions of spectral space that poorly conform to the model and for which the latter may need to be revised. Fig. 5 illustrates this procedure on a recovery simulation under correlated loading. If the method holds up to scrutiny of this type, then because ours is a maximum-likelihood estimator, it will be asymptotically efficient, with a mean-squared error that will be as small or smaller than that of all other possible estimators, converging to the optimal estimate as the sample size grows to infinity.", "pages": [ 23, 24 ] }, { "title": "4.9 Admittance and coherence return, briefly", "content": "The theoretical admittance Q · and coherence γ 2 · are nothing but one-to-one functions of our parameters of interest. Consequently (Davison 2003), maximum-likelihood estimates for either Q · or γ 2 · are obtained simply by evaluating the functions (59) or (64) at the maximumlikelihood estimate of the parameters. The equivalence is easy to appreciate by expanding the score in the desired function, e.g. γ 2 · , as a total derivative involving the parameters D , f 2 and r , The score in γ 2 · vanishes when ∂ L /∂D = ∂ L /∂f 2 = ∂ L /∂r = 0 as long as each of ∂γ 2 · /∂D , ∂γ 2 · /∂f 2 and ∂γ 2 · /∂r are non-zero. Thus the maximum-likelihood estimates Q · and γ 2 · are obtained at the maximum-likelihood values ˆ D , f 2 and ˆ r , and are computed without difficulty, as we will illustrate shortly. See Appendix 9.5 for a few additional considerations.", "pages": [ 24 ] }, { "title": "5 T E S T I N G T H E M O D E L", "content": "In the previous section we discussed the question whether the 'model' to which we have subscribed is at all 'valid' in very general terms. Here, we will address two possible concerns more specifically. The main ingredients of our model are the flexural equations (20), correlation (49) and proportionality (54) of the initial topographies, and the isotropic spectral form (72) that we assumed for the loading terms. Other than that, we have introduced a certain fixed two-layer density structure ∆ 1 , ∆ 2 and z 2 , and an approximate way of computing gravity anomalies by way of eq. (26). When working within this framework, we showed in Section 4.8 how to assess the quality of the data fit, and in Section 4.9 how to hindcast the traditional observables of admittance and coherence. However, what we have not addressed is the relative merits of alternative models. How appropriate is the Mat'ern class, especially in its isotropic form? How different would an analysis that does not consider correlated loading be from one that does? What would be the effect of modifying or adding additional terms to the flexural equations, as could be appropriate to consider more complex tectonic scenarios, elastic non-linearities, elastic anisotropy, or alternative rheologies (as, for example, Stephenson & Beaumont 1980; Stephenson & Lambeck 1985; Ribe 1982; Swain & Kirby 2003a; McKenzie 2010)? We cannot, of course, address all of these questions with any hope for completeness, but in this section we introduce two specific considerations that will speak to these issues. The first, detailed in Appendix 9.6, involves a stand-alone methodology to recover the spectral parameters in the Mat'ern form given univariate multi-dimensional data. This will help us build well-suited data synthetics; it will also enable the study of terrestrial and planetary surfaces per se , e.g. to measure the roughness of the ocean floor or the lunar surface (e.g. Goff & Arbic 2010; Rosenburg et al. 2011). Even more broadly, it is an approach to characterize texture (Haralick 1979; Cohen et al. 1991) in the context of geology and geophysics. Although our chosen parameterization (72) permits a wide variety of spectral shapes, we are of course limiting ourselves by only considering isotropic loading models. In future work, anisotropic spectral shapes for the loading terms will be considered. The second, in Appendix 9.7, is a worked example of how, specifically, the inclusion or omission of the initial-loading correlation coefficient, r , may influence the confidence that we should have in our maximum-likelihood estimates obtained with or without it. We might construct a likelihood L ( θ ) , as in eq. (100) with all terms (76)-(78) present, or instead we might force the initial-loading correlation to r = 0 . This would result in a simpler form that we have called ˜ L ( ˜ θ ) in eq. (101), whereby the parameter r is lacking altogether from the vector to be compared with the expression for θ in eq. (74). Since ˜ θ ⊂ θ , both models are 'nested': the less complicated model can be obtained by imposing constraints on the more complicated model, so that the simpler model is a special case of the more complicated one. In that case the likelihood-ratio test (Cox & Hinkley 1974; Severini 2001) that we describe in Appendix 9.7 is applicable. It is inappropriate to compare models using likelihood ratios if they are not nested, even if special exceptions exist to that rule (see, e.g., Vuong 1989; Fan et al. 2001). What we have not done is incorporate the effect of downward continuation in eq. (35) into the analysis. The 'data' that we will generate and analyze in our synthetic experiments will have been 'perfectly' downward continued to the single 'appropriate' interface at depth, from 'noise-free' gravity observations, which remains a very idealized situation. Some problems anticipated with numerical stability might be remediated through dedicated robust deconvolution methods, but more generally, giving up this level of idealization for real-world data analysis will cause complications that require special treatments. Absent these, our theoretical error estimates will be minimum bounds. Keeping in mind that the complications of this kind are shared by other gravity-based methods, we feel justified in not exhaustively discussing all of our options here. Nevertheless, we can look ahead at addressing the downward continuation of the gravity field within the framework of our maximum-likelihood method by considering what would happen if we took the surface topography and the gravity anomaly as the primary observables, rather than the surface and (deconvolved) subsurface topography as we now have, in eq. (43). We would, essentially, continue to carry the factors χ ( k ) from eq. (35) throughout the development. In the application of the blurred data analysis (89) those factors would appear inside the convolutional integrals, to appear in Appendix 9.8, of the kind (236), and their appearance there would no doubt regularize the gravity deconvolution by stabilizing the inverse (237) and its derivatives (238) as actually used by the optimization algorithm. However, the variance expressions for the maximum-likelihood estimates, which we derive based on the unblurred likelihoods, would presumably be farther from their blurred equivalents once the deconvolution is also part of the estimation in this way, and it would require much detailed work to arrive at a complete understanding of such a procedure. At the end of the day, we would still not have remediated the geophysical problems of measurement and data-reduction noise in obtaining the Bouguer gravity anomalies, nor handled possible departures from the two-layer model that may exist in the form of internal density anomalies. The list of caveats is long but again shared among other gravitybased methods, over which the maximum-likelihood method has a clear advantage, as we have seen, theoretically, above, and are about to show, via simulation, in what follows.", "pages": [ 25 ] }, { "title": "6 N U M E R I C A L E X P E R I M E N T S", "content": "Numerical experiments are straightforward. We generate synthetic data using the procedure established in Sections 4.2.1-4.2.3, and then employ an iteration scheme along the lines of eqs (108)-(109): starting from an initial guess we proceed through the iterations k = 0 , . . . as until convergence. In practice any other numerical scheme, e.g. by conjugate gradients, can be used, the only objective being to maximize (or minimize the negative) log-likelihood (97) by whichever iteration path that is expedient, and for which canned routines are readily available. The important points to note are, first, that we do need to implement the convolutional blurring step (89) in the generation of the data, so as to reference them to a particular generation grid while keeping the flexibility to subsample, section, and taper them for analysis as in", "pages": [ 25 ] }, { "title": "26", "content": "the real-world case. Second, we do need to maximize the blurred log-likelihood (97) and not its unblurred relatives (100) or (101). The datageneration grid and the data-inversion grid may be different. If these two stipulations are not met, an 'inverse crime' (Kaipio & Somersalo 2005, 2007; Hansen 2010) will be committed, leading to either unwarranted optimism, or worse, spectacular failure - both cases unfortunately paramount in the literature and easily reproduced experimentally. From the luxury of being able to do synthetic experiments we can verify, as we have, the important relations derived in this paper, e.g., the expectation of the Hessian matrices of eq. (107), the distribution of the scores in eq. (128), of the residuals in eq. (145), of the likelihood ratios in eq. (235) of the forthcoming Appendix 9.8, and of course virtually all of the analytical expressions listed in the Appendices. We can furthermore directly inspect the morphology of the likelihood surface (97) for individual experiments and witness the scaled reduction of the confidence intervals with data size predicted by eq. (142). Via eq. (147) we can compare coherence (and admittance) curves with those derived from perfect knowledge, and contrast them with what we might hope to recover from the traditional estimates of the admittance and coherence. We do stress again that even if we did have perfect estimates of admittance and coherence, the problem of estimating the parameters of interest from those would be fraught with all of the problems, encountered in the literature, that led us to undertake our study in the first place. Most importantly, we can check how well our theoretical distributions match the outcome of our experiments. After all, in the real world we will only have access to one data set per geographic area of interest, and will need to decide on the basis of one maximum-likelihood estimate which confidence intervals to place on the solution, and which trade-offs and correlations between the estimated parameters to expect. We were able to derive the theoretical distributions only by neglecting the finite-sample size effects, basing our expressions on the 'unblurred' likelihood of eq. (100) when using eq. (97) would have been appropriate but analytically intractable. In short, we can see how well we will do under realistic scenarios, and check how much we are likely to gain by employing our approach in future studies of terrestrial and planetary inversions for the effective elastic thickness, initial-loading fraction and load-correlation coefficient. Figs 1 and 3-5 were themselves outputs of genuine simulations to which the reader can refer again for visual guidance. Here we limit ourselves to studying the statistics of the results on synthetic tests with simulated data. In Figs 6-9 we report on two suites of simulations: one under the uncorrelated-loading scenario for two different data sizes in Figs 6-7, and one under correlated loading for two different data sizes in Figs 8-9. Histograms of the outcomes of our experiments are presented in the form of diffusion-based non-parametric 'kernel-density estimates' (Botev et al. 2010), which explains their smooth appearance. The distributions of the estimators are furthermore presented in the form of the quantile-quantile plots as introduced in eq. (146), which allows us to identify outlying regions of non-Gaussianity. Figures of the type of Fig. 5 should help identify problems with individual cases. For the uncorrelated-loading experiments shown in Figs 6-7 there are few meaningful departures between theory and experiment. The predicted distributions match the observed distributions very well, and the parameters of interest can be recovered with great precision. Indeed, Fig. 6 shows us that an elastic thickness T e = 43 . 2 km on a 1260 × 1260 km 2 grid can be recovered with a standard deviation of 2 . 9 km, with similarly low relative standard deviations for the other parameters. Fig. 7, whose data grid is twice the size in each dimension, yields standard deviations on the estimated parameters that are half as big, in accordance with eq. (142). What is remarkable is that both theory and experiment, shown in Fig. 10, predict that the flexural rigidity D and the initial-loading ratio f 2 can be recovered without appreciable correlation between them, and with little trade-off between them and the spectral parameters σ 2 , ν and ρ , even though the trade-off between the spectral parameters themselves is significant. This propitious 'separable' behavior is not at all what the entanglement of the parameters through the admittance and coherence curves shown in Fig. 2 would have led us to believe, and it runs indeed contrary to the experience with actual data as reported in the literature. The likelihood contains enough information on each of the parameters of interest to make this happen; the very act of reducing this information to admittance and coherence curves virtually erases this advantage by the collapse of their sensitivities. For the correlated-loading experiments shown in Figs 8-9 the agreement between theory and experiment is equally satisfactory. The introduction of the load-correlation coefficient r contributes to making the maximum-likelihood optimization 'harder'. In our example we are nevertheless able to estimate an elastic thickness T e = 17 . 8 km on a 1260 × 1260 km 2 grid with a standard deviation of only 0 . 7 km, as shown in Fig. 9. In contrast, Fig. 8, whose data grid is half the size in each dimension, yields standard deviations on the estimated parameters that are about twice as big, in accordance with eq. (142). Fig. 11 shows the normalized covariance of the estimators. In all of our experiments as reported here we implemented the finite-sample size blurring in the data analysis, but made predictions based on the unblurred likelihoods, as discussed before. The figures discussed in this section serve as the ultimate justification for the validity of this approach, with further heuristic details deferred to Appendix 9.8. When omitting the blurring altogether the agreement between theory and practice becomes virtually perfect. As we have argued, though, in those cases we commit the inverse crime of analyzing the data on the same grid on which they have been generated, which is unrealistic and needs to be avoided. We also note that in designing practical inversion algorithms, care should be taken in formulating an appropriate stopping criterion. The exactness of the computations should match the scaling of the variances with the data size, which we showed goes as 1 /K in eq. (128). This is difficult to tune, and some synthetic experiments might inadvertently trim or 'winsorize' the observed distributions by setting too stringent a convergence criterion. Figs 12 and 13, to conclude, show the distribution of estimates of the admittance and coherence for the entire set of experiments about which we have reported here. The maximum-likelihood estimates agree very well with the theoretical curves, although the effect of varying data size on the spread is understandably noticeable. Our initial misgivings about the traditional admittance and coherence estimates (obtained by Fourier transformation and averaging over radial wavenumber annuli) are well summed up by their behavior, which shows significant bias and large variance. While the bias can be taken into account in comparing measurements with theoretical curves, as it has been by various authors (Simons et al. 2000; P'erez-Gussiny'e et al. 2004, 2007, 2009; Kalnins & Watts 2009; Kirby & Swain 2011), the high variance remains an issue. Multitaper methods (Simons et al. 2003; Simons & Wang 2011) reduce this variance but expand the bias. The estimation of admittance and coherence is subservient to the estimation of the lithospheric and spectral parameters that are of geophysical value, and all methods that use admittance and coherence estimates, no matter how good, as a point of departure for the inversion for the 0 0 0 geophysical parameters, will be deprived of the many benefits that a direct maximum-likelihood inversion brings and that we have attempted to illustrate in these pages. 0", "pages": [ 26, 27, 29 ] }, { "title": "7 C O N C L U S I O N S", "content": "In this paper we have not answered the geophysical question 'What is the flexural strength of the lithosphere?' but rather the underlying statistical question 'How can an efficient estimator for the flexural strength of the lithosphere be constructed from geophysical observations?'. Our answer was constructive: we derived the properties of such an estimator and then showed how it can be found, by a computational implementation of theoretical results that also yielded analytical forms for the variance of such an estimate. We have stayed as close as possible to the problem formulation as laid out in the classical paper by Forsyth (1985) but extended it by fully considering correlated initial loads, as suggested by McKenzie (2003). The significant complexity of this problem, even in a two-layer case, barred us from considering initial loads with anisotropic power spectral densities, wave vector-dependent initial-loading fractions and load-correlation coefficients, anisotropic flexural rigidities, or any other elaborations on the classical theory. However, we have suggested methods by which the presence of such additional complexity can be tested through residual inspection. The principal steps in our algorithm are as follows. After collecting the Fourier-transformed observations (82) into a vector H · ( k ) we form the blurred Whittle likelihood of eq. (97) as the average over the K wavenumbers in the half plane, the Gaussian quadratic form whereby ¯ S · is the blurred version, per eq. (84), of the spectral matrix formulated in eqs (76)-(78). The likelihood depends on the lithospheric parameters of interest, namely the flexural rigidity D , the initial-loading ratio f 2 , and the load-correlation coefficient r , and on the spectral parameters σ 2 , ν , ρ of the Mat'ern form (72) that captures the isotropic shape of the power spectral density of the initial loading. Maximization of eq. (150) then yields estimates of these six parameters. To appraise their covariance, we turn to the unblurred Whittle likelihood of eq. (100), its first derivatives (the score), its second derivatives (the Hessian), and their expectation (the Fisher matrix), whose inverse relates to the variance of the parameter estimates as With this knowledge we construct 100 × α %confidence intervals The problem of producing likely values of lithospheric strength, initial-loading fraction and load correlation for a geographic region of interest required positing an appropriate model for the relationship between gravity and topography. The gravity field had to be downward continued (to produce subsurface topography), and the statistical nature of the parameter recovery problem had to be acknowledged. There are many methods to produce estimators, and depending on what can be reasonably assumed, different estimators will result, all with different bias and variance characteristics. In general one wishes to obtain unbiased and asymptotically efficient estimators, i.e. estimators whose variance is competitive with any other method for increasing sample sizes. Our goal in this work has been to whittle down the assumptions, while keeping the model both simple and realistic. If the parametric models that we have proposed are realistic then we are assured of good estimation properties. Maximum-likelihood estimators are both asymptotically unbiased and efficient (often with minimum variance, see, e.g., Portnoy 1977). Should we use another method, with more parameters, or even non-parametric nuisance terms, unless those extra components in the model are necessary, we will literally waste data points on estimating needless degrees of freedom, and accrue an increased variance. Modeling the initial spectrum nonparametrically is such an example, of wasting half of the data points on the estimation. Producing the coherence or admittance estimate as a starting point for a subsequent estimation of the lithospheric parameters of interest is also highly suboptimal, and for the same reason. If the parametric models that we have assumed are not realistic then we will be able to diagnose this problem from the residuals, and this will be a check on the methods we apply. Hence, if the parametric models stand up to tests of this kind, then because of the properties of maximum-likelihood estimators, asymptotically, no other estimator will be able to compete in terms of variance. In that case the confidence intervals that we have produced in this paper are the best that could be produced.", "pages": [ 32 ] }, { "title": "8 A C K N O W L E D G M E N T S", "content": "This work was supported by the U. S. National Science Foundation under grants EAR-0710860, EAR-1014606 and EAR-1150145, and by the National Aeronautics and Space Administration under grant NNX11AQ45G to F.J.S., by U. K. EPSRC Leadership Fellowship EP/I005250/1 to S.C.O. She thanks Princeton University and he thanks University College London for their hospitality over the course of many mutual visits. In particular also, F.J.S. thanks Theresa Autino and Debbie Fahey for facilitating his visit to London via Princeton University account 195-2243 in 2011, and S.C.O. thanks the Imperial College Trust for funding her sabbatical visit to Princeton in 2006, where and when this work was commenced. We acknowledge useful discussions with Don Forsyth, Lara Kalnins, Jon Kirby, Mark Wieczorek and Tony Watts, but especially with Dan McKenzie. Two anonymous reviewers and the Associate Editor, Saskia Goes, are thanked for their helpful suggestions, which improved the paper. All computer code needed to reproduce the results and the figures in this paper is made freely available on www.frederik.net .", "pages": [ 33 ] }, { "title": "R E F E R E N C E S", "content": "Forsyth, D. W., 1985. 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Potential anomalies on a", "pages": [ 36 ] }, { "title": "9.1 The spectral matrices T , ∆T and T ·", "content": "We restate eqs (56)-(58) or eqs (76)-(78), without any reference to the dependence on wave vector or wavenumber, as T · = T + ∆T , (157) The Cholesky decomposition (79) of T · evaluates to For general reference we note the Cayley-Hamilton theorem (Dahlen & Baig 2002) for an invertible 2 × 2 matrix A , The determinants and inverses of T · , T and ∆T are given by det 2 From these relationships we conclude that", "pages": [ 36 ] }, { "title": "9.2 The score γ in the lithospheric parameters D , f 2 and r", "content": "The first derivative of the log-likelihood function (100) is given by the expression (110). The elements of the score function γ θ L for a generic 'lithospheric' parameter θ L ∈ θ L = [ D f 2 r ] T are We obtain these via eq. (111), seeing that we will need the derivatives of the (logarithm of the) determinant and the inverse of T · . We compute these from their defining expressions or via the identities for symmetric invertible matrices (Strang 1991; Tegmark et al. 1997) We will thus also write that sphere: Applications to the thickness of the lunar crust, J. Geophys. Res. , 103 (E1), 1715-1724. Wilks, S. S., 1938. The large-sample distribution of the likelihood ratio for testing composite hypotheses, Ann. Math. Stat. , 9 (1), 60-62. Wood, A. T. A. & Chan, G., 1994. Simulation of stationary Gaussian processes in [0, 1] d, J. Comput. Graph. Stat. , pp. 409-432. Young, G. A. & Smith, R. L., 2005. Essentials of statistical inference , vol. 16 of Cambridge Series on Statistical and Probabilistic Mathematics , Cambridge Univ. Press. Zuber, M. T., Bechtel, T. D. & Forsyth, D. W., 1989. Effective elastic thicknesses of the lithosphere and mechanisms of isostatic compensation in Australia, J. Geophys. Res. , 94 (B7), 9353-9367. σ", "pages": [ 36, 37 ] }, { "title": "Maximum-likelihood estimation of flexural rigidity 37", "content": "From the above we then find that the expressions required by eq. (111) to calculate the score in the lithospheric parameters are A r = (1 2 r - r 2 ) 2 ( T - 1 + r 2 ∆T ) . (178) Since the score vanishes at the estimate, in the uncorrelated case we can solve eq. (168) for the estimate ̂ f 2 directly. Using eqs (174) and (177) for the case where r = 0 , we can thus write, with the help of the matrix V defined in eq. (171), an expression for the estimate In principle this would allow us to define a profile likelihood (Pawitan 2001), but such a procedure and its properties remain outside of the scope of this text. 9.3 The score γ in the spectral parameters σ 2 , ν and ρ The elements of the score function γ θ S for a generic 'spectral' parameter θ S ∈ θ S = [ σ 2 ν ρ ] T are To compute these via eq. (112) we need the derivatives of the Mat'ern form. Thus, directly from eq. (72), we obtain in particular, m 2 = 1 σ 2 , As above in eq. (179), we pick up one direct solution, namely where it is to be noted from eq. (72) that ( S 11 /σ 2 ) is indeed no longer dependent on σ 2 . With eq. (179) this would enable us to conduct a profile-likelihood estimation in a reduced parameter space (Pawitan 2001), but once again the details are omitted here.", "pages": [ 37 ] }, { "title": "9.4 The Hessian F and the Fisher matrix F", "content": "The Hessian or second derivative of the log-likelihood function (100), and its negative expectation or the Fisher information matrix, are given by the expressions (132) and (133), respectively. Both of these contain the terms (173)-(178) and (181)-(186) that we have just derived, which renders them eminently calculable analytically. In its raw form eq. (133) does not provide much insight, but in Section 4.6 we also introduced special formulations for elements of the Fisher matrix that involve at least one spectral variable, in which case the expressions (131), (134) and (135) for F θ S θ S , F θ L θ S and F θ S θ ' S , respectively, are of a common form. We do not foresee needing the expressions for the Hessian: while optimization procedures might benefit from those, even in eq. (149) the Fisher matrix could be substituted (Cox & Hinkley 1974). We are thus left with determining the entries of the Fisher matrix F θ L θ ' L when only lithospheric variables are present. The diagonal terms F θ L θ L are obtained via eq. (130), which we repeat here specifically for this case as - 1 - 1 2", "pages": [ 37 ] }, { "title": "38", "content": "Only to obtain the cross terms involving different lithospheric parameters do we need the full expression (133). Even this case simplifies since, owing to eq. (72), ∂ θ L S 11 = 0 , thereby yielding the expression /negationslash where we recall from eq. (111) that ∂ θ L A θ ' L = ∂ θ L ∂ θ ' L T -1 · . When θ L = θ ' L , as is seen from eqs (173)-(175), the first term ∂ θ L m θ ' L = 0 . When θ L = θ ' L , eqs (188)-(189) are exactly each others' equivalent, and either expression can be used. We will not really need the eigenvalues of the quadratic forms: their sums of squares (in eq. 188) or sums (in eq. 189) suffice to calculate the elements of the Fisher matrix. The specific eigenvalues are only required if we should abandon the normal approximations and develop an interest in calculating the distributions of eq. (117) exactly. Beginning with the flexural rigidity, we obtain 2 K F DD = ∑ k k 8 ∆ (1 - 2 1 - r ∆ 2 - 2 2 )( φξ - 2 - f - 2 1) 2 ( 2 f ∆ 1 ∆ 2 [ f - 3 r f - rf - r ] For the loading ratio, we obtain for the sum of squares of the eigenvalues Finally, for the load-correlation coefficient we conclude that For the cross terms that remain, we find, at last,", "pages": [ 38 ] }, { "title": "9.5 Properties of admittance and coherence estimates - and 'Cram'er-Rao lite' for the maximum-likelihood estimate", "content": "Let us consider how the uncertainty on the parameters ˆ θ estimated via the maximum-likelihood method propagates to estimates of the coherence and the admittance, ̂ γ 2 · and ̂ Q · , should we desire to construct those. Since Section 4.7 we have known that our estimate ˆ θ , which is based on the likelihood (97) and thus ultimately on the data H · ( k ) , is centered on the truth θ 0 as per We know the distributional properties of Y as having a mean of zero and a variance that is proportional to the inverse of the Fourier-domain sample size K . Taking the Bouguer-topography coherence as an example, we can again use the delta method to write for its estimate γ 2 · ( ˆ θ ) = γ 2 · ( θ 0 ) + [ ∇ γ 2 · ( θ 0 ) ] T Y , (197) from which easily follows that at identical wavenumbers k , and a statement similar in form to eq. (199) for the covariance of the coherence estimate between different wavenumbers k and k ' . With these we know the relevant statistics of maximum-likelihood-based admittance and coherence estimates. The 'traditional' methods use estimates of coherence and admittance to derive estimates of the parameters θ . Regardless of how the former are computed (via parameterized maximum-likelihood techniques as in this paper, or non-parametrically using multitaper or other spectral techniques), we know one important thing about their statistics. No alternative estimate for the parameters that is unbiased will beat the variance of our maximum-likelihood estimate. Let us imagine defining another unbiased estimator which would be given by another function of the data, generically written g 2 2 k and let us study the covariance of this hypothetical estimate with the zero-mean score of the maximum-likelihood (97), defined in eq (110): To obtain eq. (201) we followed an argument as in eqs (113)-(114) while continuing to assume the independence of the Fourier coefficients and using Leibniz' product rule of differentiation. We now know from Cauchy-Schwartz that and thus, combining eq. (204) with eqs (128) and (139), we find that The maximum-likelihood estimate is asymptotically efficient: no other unbiased estimate has a lower variance.", "pages": [ 38, 39 ] }, { "title": "9.6 Retrieval of spectral parameters", "content": "Were we to observe a single random field H ( x ) , distributed as an isotropic Mat'ern random field with the parameters θ = θ S , we would have Its parameters could also be estimated using maximum-likelihood estimation. Following the developments in Section 4.3 the blurred loglikelihood of observing the data under the model (206) would be written under the assumption of independence as When the spectral blurring is being neglected, the likelihood becomes, more simply, The scores in this likelihood are then which is only slightly different from the forms that they took in the multivariable case, eqs (110) and (112). In deriving the variance of the score in the multivariate flexural case, eq. (127), we neglected the complications of spectral blurring, as we do here, and we also neglected the slight correlation between wavenumbers, as we have here also. The simple form of eq. (209) allows us to re-examine the effect that wavenumber correlations will have on the score by bypassing the development outlined in eqs (116)-(117) and writing instead that { } ∑ ∑ k ' H ( k ) S 2 , S Previously we wrote expressions for the covariance of the finite-length spectral observation vector that took into account the blurring but not the correlation, e.g. in approximating eq. (9) by eq. (83), which we restate here for the univariate case as We shall now approximate this under slow variation of the spectrum, relative to the decay of the window functions W K , as Using Isserlis' theorem (Isserlis 1916; Percival & Walden 1993; Walden et al. 1994), we then have for the covariance of the periodograms | | | | { } 2 { } 2 S ( k ) 1 K 2 cov {| | H ( k ' ) | ' ) 2 | } cov ( γ S ) θ S , ( γ S ) θ ' S = m θ S ( k ) m θ ' S ( k ) ( k . (210)", "pages": [ 39 ] }, { "title": "40", "content": "since the first term, the pseudocovariance or relation matrix vanishes in the half-plane for the complex-proper Gaussian Fourier coefficients (Miller 1969; Thomson 1977; Neeser & Massey 1993) of real-valued stationary variables. We may thus conclude that the covariance of the scores suffers mildly from wavenumber correlation, However, for very large observation windows or custom-designed tapering procedures, we may write From eq. (128) we then also recover the entries of the Fisher matrix for this problem as exactly half the size of the multivariate equivalent that we obtained in eq. (135), as expected, which are to be used in the construction of confidence intervals for the parameters σ 2 , ρ and ν of the isotropic Mat'ern distribution as determined by this procedure. The expressions for m θ S were listed in Appendix 9.3. Refer again also to Table 2, which we have only now completed filling.", "pages": [ 40 ] }, { "title": "9.7 Testing correlation via the likelihood-ratio test", "content": "We seek to evaluate the null and alternative hypotheses /negationslash Our definition of the log-likelihood L ( θ ) in eq. (100) included the correlation coefficient r between initial-loading topographies as a parameter to be estimated from the data. In contrast, the log-likelihood ˜ L ( ˜ θ ) = L ([ ˜ θ T 0] T ) of eq. (101) did not. The Hessian of L is F and that of ˜ L is ˜ F , and from eq. (109) we know that F converges in probability to the negative Fisher matrix -F and, similarly, ˜ F converges to the constant -˜ F . This gives us the elements to evaluate the different scenarios. Should we evaluate 'uncorrelated data' using a 'correlated model', we need a significance test for the addition of the correlation parameter. Since the hypotheses (217) refer to nested models, ˜ θ containing some of the same entries as θ , see eqs (74)-(75), otherwise put θ = [ ˜ θ T r ] T , (218) standard likelihood-ratio theory (Cox & Hinkley 1974) applies. Let the truth under H 0 be given by the parameter vector and let us consider having found two maximum-likelihood estimates, /negationslash Note that L ( ˆ θ ) ≤ ˜ L ( ˆ ˜ θ ) and and that the estimates of 'everything-but-the-correlation-coefficient' are different from the full estimates depending on whether the correlation coefficient is included as a parameter to be estimated or not. We now define the maximum-log-likelihood ratio statistic from the evaluated likelihoods whereby we have used that, evaluated at the truth under H 0 , the likelihood values ˜ L ( ˜ θ 0 ) = L ( θ 0 ) , and defined the auxiliary quantities X X 1 = 2 K [ L ( ˆ θ ) -L ( θ 0 ) ] , and [ ] By Taylor expansion of the log-likelihoods around the truth, to second order and with the first-order derivatives vanishing, we then have 2 = 2 K ˜ L ( ˜ θ u ) - ˜ L ( ˜ θ 0 ) . (223) where we have used the limiting behavior (109). For more generality, we consider maximum-likelihood problems with a partitioned parameter vector whereby θ × may contain any number of extra parameters, θ × = [ r ] being the case under consideration. Introducing notation as we go along, the Fisher matrix for such problems partitions into four blocks (see also Kennett et al. 1998) such that we can write, ˆ The submatrices F × and F · contain the negative expectations of the second derivatives of the likelihood L , with respect to at least one of the 'extra' parameters θ × ∈ θ × , suitably arranged with the mnemonic subscripts × and · . The corner matrices ˜ F and ˜ F contain the second derivatives of the likelihood ˜ L in only the 'simpler' subset of parameters ˜ θ ∈ ˜ θ . The inverse of the Fisher matrix is given by thereby defining the auxiliary matrices, and, via the Woodbury identity, their inverses, as This yields the variances of the vectors partitions. Recalling from eq. (140) that we may use eqs (227)-(228) to express the marginal distribution of the partition θ × under the null hypothesis, (231) In this general framework we rewrite likelihood-ratio statistic (222) with the help of eqs (224)-(225) as In order to figure out the properties of the likelihood-ratio test we now need to understand the properties of the difference between the 'correlated' and 'uncorrelated' estimates ˆ ˜ θ -ˆ ˜ θ u of eqs (220)-(221). We may note directly from Cox & Hinkley (1974) that", "pages": [ 40, 41 ] }, { "title": "ˆ ˜ θ u = ˆ ˜ θ + ˜ F -1 F × ˆ θ × .", "content": "Inserting this relation into eq. (232) the limiting behavior of the likelihood-ratio test statistics becomes where we have used eq. (229). From eq. (231) then follows that the distribution of X is the sum of squared zero-mean Gaussian variates divided by their variance, i.e., chi-squared with as many degrees of freedom as the difference in number of parameters between the alternative models described by θ and ˜ θ , a conclusion first reached by Wilks (1938). For a derivation rooted in the geometry of contours of the likelihood surface, see Fan et al. (2000). In our particular case, the only complementary variable is the correlation r between the two initial-loading terms, and the likelihood-ratio test statistic of eq. (222) becomes which is how we may test the alternative hypotheses of initial-load correlation and absence thereof.", "pages": [ 41 ] }, { "title": "9.8 A posteriori justification for the behavior of the synthetic tests", "content": "We owe the reader a short theoretical justification of why using the unblurred likelihoods L of eq. (100) for the variance calculations (the black curves in Figs. 6-9) accurately predicts the outcome of experiments (the grey-shaded histograms) conducted on the basis of the blurred likelihoods ¯ L of eq. (97). The blurring enters through the spectral term, which is ¯ S · instead of S · as we recall from eq. (84), and it affects the likelihood (97) through its determinant and inverse. Instead of the purely numerical evaluation of the convolutions of the type (89) and conducting all subsequent operations on the result, which is how we construct ¯ L in the numerical experiments, in principle, in the notation suggested by eqs (45)-(46), we could attempt to explicitly evaluate, though this would be cumbersome, Of course, should the spectral windows be delta functions, eqs (236)-(237) would reduce to S 2 11 det T · and S -1 11 T -1 · (see eqs 166-167), as expected on the basis of eq. (76). With these expressions, we could proceed to forming the first and second derivatives of the blurred likelihood (see eqs 168-169). For example, for the score in the blurred likelihood we would then have (233)", "pages": [ 41 ] }, { "title": "42 Simons and Olhede", "content": "and then the derivatives of eq. (239) would be needed to determine the variance of the maximum-blurred-likelihood estimate in a manner analogous to eqs (128) and (139). In short, a full analytical treatment would be very involved, and a purely numerical solution would not give us very much insight. How then can we understand that we can approximate the variance of our maximum-blurred-likelihood estimator by replacing the second derivatives of the blurred likelihood with those of its unblurred form? We can follow Percival & Walden (1993) and regard the blurring as introducing a bias given by, to second order in the Taylor expansion, where we have used the hermiticity and periodicity of both the spectral density S · and the spectral window | W K | 2 , and the evenness and energy normalization of the latter. For more general (e.g. non-radially symmetric or non-separable) windows the equations will change, but not the conclusions. The first factor in eq. (242) is a measure of the bandwidth of the spectral window, which we shall call β 2 ( W ) , and the second is a measure of the spectral variability via the curvature of the spectral matrix. Thus the blurred spectral matrix is the sum of the unblurred spectral matrix and a second term which decays much faster with wavenumber than the first:", "pages": [ 42 ] }, { "title": "¯ 2 T", "content": "The matter that concerns us here is how the blurring affects the derivatives of the blurred spectrum and thus the derivatives of the blurred likelihood. What transpires is that the differentiation with respect to the parameters θ does not change the relative order of the terms in eq. (243), in the sense that the correction terms are only important at low values of the wavenumber k . Since the mean score is zero, by virtue of eq. (114), the correction term becomes important, which leads to a bias of the estimate. But since the variance of the score is not zero, see eq. (128), the correction term is dwarfed by the contribution from the unblurred term. Hence we should, as we have, use the blurred likelihood (97) to conduct numerical maximum-likelihood experiments on finite data patches, but we can, as we have shown, predict the variance of the resulting estimators using the analytical expressions based on the unblurred likelihood (100).", "pages": [ 42 ] } ]

[ { "title": "Abstract", "content": "This is the first of two related papers analising and explaining the origin, manifestations and parodoxical features of the quantum potential (QP) from the non-relativistic and relativistic point of view. QP arises in the quantum Hamiltonian, under various procedures of quantization of the natural systems, i.e. the Hamilton functions of which are the positive-definite quadratic forms in momenta with coefficients depending on the coordinates in ( n -dimensional) configurational space V n endowed so by a Riemannian structure. The result of quantization may be considered as quantum mecanics (QM) of a particle in V n in the normal Gaussian system of reference in the globally-static spacetime V 1 ,n . Contradiction of QP to the Principles of General Covariance and Equivalence is discussed. It is found that actually the historically first Hilbert space based quantization by E. Schrodinger (1926), after revision in the modern framework of QM, also leads to QP in the form that B. DeWitt had been found 26 years later. Efforts to avoid QP or reduce its drawbacks are discussed. The general conclusion is that some form of QP and a violation of the principles of general relativity which it induces are inevitable in the non-relativistic quantum Hamiltonian. It is shown also that Feynman (path integration) quantization of natural systems singles out two versions of QP, which both determine two bi-scalar (indepedendent on choice of coordinates) propagators fixing two different algorithms of path integral calculation. In the accompanying paper under the same general title and the subtitle 'The Relativistic Point of View' , relation of the non-relativistic QP to the quantum theory of the scalar field non-minimally coupled to the curved space-time metric is considered. Keywords: Riemannian space-time; Quantization; Path Integration; Quantum Potential; Principle of Equivalence; General Covariance; Problem of Measurement.", "pages": [ 1 ] }, { "title": "I. Non-Relativistic Origin, History and Paradoxes", "content": "(Joint Institute for Nuclear Research, Dubna 141980, Russia, [email protected]) May 31, 2021", "pages": [ 2 ] }, { "title": "1 Introduction", "content": "In the present paper, the different procedures of the Hilbert space based quantization 1 of the non-relativistic natural mechanical systems will be analyzed and compared. The (classical) natural systems (the term originated by E. Whittaker and re-animated by V. N. Arnold and A. B. Givental [3]) are those whose Hamilton functions are non-uniform quadratic forms in momenta p a with coefficients ω ab ( q ) depending on coordinates q a , a, b, · · · = 1 , . . . , n of configurational space V n : It provides V n by a Riemannian structure (Henceforth, subscripts ( ω ) and (g) will denote objects related to metric tensors ω ab and g αβ , α, β, · · · = 0 , 1 , . . . , n of V n and of n + 1 -dimensional space-time V 1 ,n respectively) Thus, H (nat) determines the dynamics of a natural system as a motion of a point-like particle in V n , to which a potential V (ext) ( q ) acts in addition. So, the actual motion of a neutral point-like particle in the external gravitation (including description of the motion in curvilinear coordinates without gravitation) treated general-relativistically as a curved space is a representative case of a natural system. However, there is a principal alternative way to construct the non-relativistic QM of this simplest physical system coupled to the geometrized gravitation. Namely, to extract it from the the general-relativistic quantum theory of scalar field as the non-relativistic ( c -1 = 0) asymptotic of its one-particle sector. The one-particle subspace in the particle-interpretable Fock representation of the canonically quantized field can be defined (in the asymptotical sense) even when the metric is time-dependent, i.e. ω ab = ω ab ( t, q ) . This approach will be considered in the companion paper [II] under the same title and subtitle 'The Field-Theoretic Point of View' and compared with conclusions of the present paper. It should be noted at once that these two approaches lead, in general, to QM's which do not coincide completely . This is one of interesting results of the work as a whole. Despite that this basic problem may be considered as of little 'practical' interest for physics, there is an important aspect of it. The theory, which is more general and geometrically transparent than the standard QM in a potential field, can serve as an instrument for a deeper insight on foundations of QM. E. Schrodinger was guided just by such an idea when he had proposed a method of construction of a quantum Hamiltonian for the generic natural system in the third [4] of his five papers [5] of 1926, by which the wave mechanics had been founded. Apparently, it was the first attempt of quantization in the sense which is close to the modern meaning of the term in theoretical physics. While this step, which had not received a deserving attention, Schrodinger did not even mentioned gravitation or general relativity at all. In the present paper, we shall, on contrary, analyze whether the two basic general-relativistic principles - the Principle of Equivalence (referred further as PE) and the Principle of General Covariance, which hold on the classical level for any natural system, are satisfied in a sense in the corresponding QM. It is a paradox that the both of the principles does not hold in the ordinary sense in QM constructed by quantization of the natural systems, but satisfy them in a restricted sense in QM extracted from the general-relativistic quantum theory of scalar field. All various procedures of quantization of Hamiltonian systems with finite degrees of freedom are ambiguous or problematic to be mathematically rigorous. Therefore, it seems more correct to speak on a paradigm of quantization rather than on an well-established theory. See, e.g., a discussion of the topic by M. J. Gotay in [1]. As concerns the level of mathematical rigor of the present discussion, the best way to characterize it is the following amusing citation taken from [6]: '...as Sir Michael Atiyah said in his closing lecture of the 2000 International Congress in Mathematical Physics,..., Mathematics and Physics are two communities separated by a common language.' Then, the present work is from the side of the Physics community. We will be mainly concerned with the so called the Hilbert-space based canonical quantization, which is meant here as a map where all 'hatted' objects are assumed to have representation as differential operators in the Hilbert space L 2 ( V n ; C ; ω 1 / 2 d n q ) and { ., . } is the Poisson bracket. The quantum Hamilto- ian ˆ H is assumed to be constructed of the 'primary' quantum observables ˆ q a , ˆ p b by some substantiated way. In the standard canonical quantization, it is found by the straightforward substitution (3)into H (nat) ( q, p ) and some Hermitizing ordering. Specifically, conclusions of analysis of the following approaches to QM of the natural systems by the present author will be exposed below descriptively or, at least, noted: Some intermediate conclusions from this first part of the analysis is given in Section 8 and further they will be compared in the accompanying paper [II] with the asymptotic in c -1 → 0 of the quantum theory of scalar field in the general globally static Riemannian space-time V 1 ,n and the proper frame of reference in which the metric form of V 1 ,n is [20, 19]:", "pages": [ 2, 3, 4 ] }, { "title": "2 Variational quantization of natural systems by Schrodinger", "content": "Schrodinger [4] searched a wave theory which plays the same role w.r.t. the Hamilton mechanics, that the Wave Theory of Light does w.r.t. the Geometrical Optics. In [4], entitled 'On relation of the Heisenberg-Born-Jordan quantum mechanics to the one of mine' , the third of the seminal papers [5], he constructed a wave (quantum) counterpart for the natural Hamilton function H ( ω ) ( q, p ) as an extremal of the following functional (Schrodinger considered ω ab ≡ ω ab ( q ) ): with the additional condition It is important to note that Schrodinger considered here the real wave functions ψ ( q ) . Variation of J (Sch) { ψ } results in an equation for eigenavalues E of a differential operator in the space of functions ψ ( q ) , which may be called the quantum Hamiltonian : It looks as satisfying to the conditions which are implied by GR: it is generally covariant, i.e. a scalar w.r.t. point transformations q a → ˜ q a ( q ) and satisfies to PE which sounds in the formulation by S. Weinberg [21] as follows: '... at every space-time point in an arbitrary gravitational field it is possible to choose 'a locally inertial coordinate system', such that within a sufficient small region of the point of question, laws of nature take the same form as in an unaccelerated Cartesian coordinate system.' According to eq.(5), ω ab may be, in particular, a relic of a general-relativistically treated gravitation and, in this sense, PE can be applied to ˆ H (Sch) . A more fine question is: are the Schrodinger equation (7) and its time-dependent and, further, general-relativistic generalizations are 'laws of nature' which must satisfy PE? We shall return to it in Sec.9. and in the companion paper [II] It should be emphasized also that Schrodinger himself by no means related his quantization of the natural systems to gravitation or GR. He considered it as an instrument to investigate the quantization procedure itself by application it to mathematically more general classical cases than the simple potential ones. Just this is our main aim but for a more wide variety of quantization procedures and in relation with GR.", "pages": [ 4, 5 ] }, { "title": "3 Revision of Schrodinger approach in framework of modern quantum mechanics", "content": "In the modern QM , ψ ( q ) are actually complex functions from a pre-Hilbertian space L 2 ( V n ; C ; ω 1 / 2 d n q ) with the scalar product The physical sense of Schrodinger's functional (6) is the mean value of the energy of the system in the state ψ ( q ) . Instead, today we should take the matrix elements of energy: where ˆ q a def = q a · ˆ 1 are the operators of coordinates in the configurational space V n and ˆ p a are the operators of momentum canonically conjugate to ˆ q a : They should be Hermitean(!) w.r.t. the scalar product ( ψ 1 , ψ 2 ) . Hermitean momentum operators for V n were introduced first by W.Pauli [22] in 1933: where 'cdot' denotes the operator product. Then, substitution of this expression into J (modern) { ψ, ψ } and Schrodinger's variational procedure give the eigenvalue equation similar to eq.(7) but with a different quantum hamiltonian ˆ H (DW) The term V (qm) was discovered for the first time by DeWitt [7] in 1952 in a different formalism of quantization, who called it the quantum potential , see Section 3 below. Surprising is that it depends on choice of coordinates q a (i.e., is not a scalar w.r.t. transformations of q a ). Also, it violates PE if eq.(13) is taken as a quantum 'law of Nature' for a particle in the gravitational field described by ω ab since in the quasi-Cartesian (normal Riemannian) coordinates y a with the origin at the point q under consideration. Thus, the dogmas of GR and QM are in conflict here! Moreover, non-covariance of the quantum potential implies that the energy spectrum and, after transition to time-dependent version of the Schrodinger equation, the dynamics depend on choice of coordinates in QM so constructed! A heretical thought comes here. Perhaps, it was a good fortune for the early stage of QM that Schrodinger did not realize the conflict: one may suppose, he and his successors in development of the wave mechanics (see [23], Sections 5.3, 6.1) would be embarrassed to proceed! Returning to expression of ˆ H (DW) with account of eqs. (8) and (14) we see that the zero-order term in the quasi-Cartesian coordinates y a having been taken separately is 'value' of a scalar object in these particular coordinates by its geometrical sense . However, in the full Hamiltonian ˆ H (DW) , they are not scalars because the non-invariance of the residual term tangles the situation in other coordinates. We shall call the such terms quasi-scalars in the theory under consideration.", "pages": [ 5, 6 ] }, { "title": "4 Discovery of quantum potential by DeWitt and generalization of his approach", "content": "26 years after Schrodinger's result, B. S. DeWitt [7] had come to the hamiltonian ˆ H (DW) by a procedure which may be called the canonical quantization; it is a map: Here, the von Neumann rule for definition of the operator corresponding to a function of classical observables, the Poisson brackets of which vanish, is applied for definition ˆ ω ab ( q ) . As a differential operator in L 2 ( V n ; C ; ω 1 / 2 d n q ) , Thus, the revised version of the Schrodinger quantization coincides with DeWitt's canonical quantization! Evidently, DeWitt himself did not know the original Schrodinger work [4]. DeWitt's result is related to the particular ordering of non-commuting operators ω ab (ˆ q ) , ˆ p a . Other (Hermitean) orderings (Weyl, Rivier, et all ) are well known. Then, on the our level of quantization, why not to consider Hermitean linear combinations of different orderings? The simplest class of Hamiltonians so obtained form an one-parametric family: DeWitt's ordering corresponds to ν = 2 . In the quasi-Cartesian coordinates y a Thus, there is an ordering with ν = 0 for which the zeroth-order short distance term vanishes, but the non-zero residual term retains; it means that PE is satisfied in the QM if one considers ˆ H ( ν =0) (because there is no curvature term at the point of the particle localization) but it is still not covariant. Besides, it will be seen in [II] that ν = 0 does not agree with the requirements of PE to the relativistic propagator.", "pages": [ 7 ] }, { "title": "5 Quasi-classical quantization", "content": "DeWitt did not take notice of the non-invariance of V (qm; ν ) , referring to possibility to transform it from one coordinate system to another, which is, of course, not invariance. However, evidently he had not been satisfied by the result of the canonical quantization. In 1957, DeWitt [8] determined quantum Hamiltonian as a differential operator in L 2 ( V n ; C ; ω 1 / 2 d n q ) through construction of quasi-classical propagator G ( q '' , t '' | q ' , t ' ) : by generalization of the Pauli construction for a charge in e.m. field [9] to the case of natural systems: and S ( q '' , t '' | q ' , t ' ) is a solution of the Hamilton-Jacobi equation for H ( nat ) ( q, p ) . Using the Hamilton-Jacobi equation DeWitt had found that, in a small neighborhood of space-time point { q ' , t ' } , the propagator G ( q ' t ' | q ' , t ' ) 'nearly satisfies the Schrodinger equation' . (Henceforth, V (ext) ≡ 0 is taken for simplicity.) where So we see that ˜ V (qm) looks as a scalar and yet as violating PE . Actually, ˜ V (qm) ( q ' , t ' | q ' , t ' ) is a bi-scalar and thus depends on choice of the line connecting points q ' and q ' . If the geodesic lines are chosen, then, in the asymptotic q ' → q ' , it is equivalent to fixation of q a as the quasi-Cartesian coordinates y a and, thus, the non-invariance of the quantum potential remains.", "pages": [ 7, 8 ] }, { "title": "6 Geometric quantization of natural systems", "content": "Geometric quantization is oriented to consider V n with non-trivial topologies, see e.g. the monograph by J. Sniatycki [10] and the paper [11]. In the latter paper, expansion by c -2 of the Hamilton function for the the relativistic particle in the proper system of reference: is considered using the Blattner-Kostant-Sternberg formalism. /negationslash The zero-order quantum potential is V (qm) ( q ) = /planckover2pi1 2 2 m 1 6 R ( ω ) ( q ) , that is a scalar but the geometric quantization is a locally asymptotic theory by construction, and, thus, merely supports DeWitt's and revised Schrodinger's local asymptotic quantum potential. This paper is interesting also in that the second order term in the asymptotic expansion in c -2 , which is quartic in the momenta, is considered. The corresponding potential is a rather complicate scalar expression including derivatives and quadratic expressions of the curvature tensor. Thus, ̂ H (nat) 2 = { ˆ H (nat) } 2 and consequently, the von Neumann rule does not work for the polynomials of H (nat) . An interesting problem to study.", "pages": [ 8, 9 ] }, { "title": "7 Feynman quantization of natural systems", "content": "There are many papers devoted to construction of the quantum propagator for natural systems by path integration so that the short-time asymptotic of the propagator would reproduce Schrodinger's original (invariant) Hamiltonian, However, it requires some deformation of the classic Lagrangean with which the path integration starts usually, see, e.g., [14, 15] and, as a method of quantization is equivalent, on my opinion, to mere postulation of the Schrodinger original Hamiltonian. Instead, I shall return to the original idea of Feynman on path integration [12], but with use results of the generalized canonical quantization (Section 4 above) and admit, if necessary, QP in the quantum Hamiltonian generating the original form of the Feynman propagator. Fixation of QP is, in fact, quantization (in the sense accepted here) of the natural mechanics under consideration. The Feynman propagator G ( F ) ( q, t | q 0 , t 0 ) is constructed by division of finite time interval t -t 0 by N →∞ intervals [ t I , t I +1 ] , I = 0 , 1 , . . . , N -1 , t N = t of duration /epsilon1 = ( t -t 0 ) /N as follows: where A def = (2 πi /planckover2pi1 /epsilon1 ) 1 2 n , q I def = q ( t I ) , ω I def = ω ( q I ) . A question arises at once what is the effective Lagrangean L (eff) ? For the natural systems Feynman's choice would be i.e. the Lagrangean of geodesic motion in V n (the case of V ( q ) ≡ 0 is taken for simplicity and straightforward comparison with the relativistic field theory in [20]). Then each integration on interval [ t I , t I +1 ] is taken along a geodesic connecting q I and q I +1 . However, to have the desired Schrodinger's result according to [8, 15] et al., the choice should be to compensate the quantum potential term. But then the virtual classical motion between q I and q I +1 will be not geodesical. Also, other ambiguities arise in the process of calculation of a Hamilton operator from the path integral (26). Instead of reviewing them, further I expose briefly main points of a special approach the initial idea of which is taken from paper by D'Olivo and Torres [16] but essentially modified in [19] and consists of the following steps: where q K = q ( t K ) . Thus, our task is to find V (eff) ( q ) which provides the hamiltonian form of propagator (30) with the Lagrangean form (26) so that L (eff) ≡ L (classic) . Here ( ˜ √ ω ) ( ν ) ( q J -1 , q J ) , ˜ L ( ν ) eff ( q J -1 , q J , ∆ q J //epsilon1 ) are the kernels of the corresponding operators in configurational representation. They are expressed, respectively, through functions √ ω ( q ) and along the rule: which follows from the general rules of quantization of Beresin and Shubin, [17], Chapter 5, in terms of the kernels of operators. where, in the exponent, the value of some effective Lagrangian L ' (eff) ( q, ˙ q ) (in general, it differs from L ( ν ) (eff) ) stands, which is taken at a point q ' J ∈ [ q J -1 , q J ] chosen so that to represent the exponent as L (classic) . To obtain the representation, all functions of q J -1 , q J , ¯ q J under the product in J should be expanded into the Tailor series near the point q ' J up to terms quadratic in ∆ q J , since only such terms contribute to the integral in eq.(33). Further, one should include the contribution of the pre-exponential factor to the exponent in a form of an additional QP. Referring to more details of rather complicate calculation in [19], now only final conclusions will be given, which are important for further discussion. In the both cases, we come again to noninvariant quantum potentials V (eff) ( q ) which we will denote V (eff;A) ( q ) and V (eff;B) ( q ) . In the case A), transformation of the Hamiltonian form (30) of the Feynman propagator G (F) to its Lagrangean form (26) with L (eff) ≡ L (classic) is possible only if ν = 2 and then the generally non-invariant potential V (eff) ( q ) has the form and, in the quasi-Cartesian coordinates y a , it is That is, in the case A) the quantization map H 0 → ˆ H (eff;A) 0 , which may be called the Feynman quantization , coincides remarkably with the result of the revised Schrodinger and 'canonical' DeWitt quantizations in the zeroth order of local asymptotic. In the complete form, it differs from the latter, what is worth of a further study. In the case B) the same approach leads to (the result does not depend on the parameter µ ) Asymptotic local expressions (40) and (42) for quantum potentials V (eff) A and V (eff) B (but not the complete potentials (39) and (41)) follow also from the study of the Pauli-DeWitt and Feynman quantizations by G. Vilkovysskii [24]. He worked in the framework of the formalism of proper time in relativistic quantum mechanics V 1 ,n (so that his expressions for potentials include R ≡ R g , the scalar curvature of V 1 ,n ). From his argumentation one may conclude that just the case A) is the preferred one. However, if the set of points q J is considered as a lattice in V n , then the case A ) corresponds to evaluation of the integrand as the arithmetic mean of its values on the adjacent nodes of the lattice. The case B ) corresponds to evaluation at the mean point of the edges. Thus, these two cases fix two different ways of lattice calculation of the path integral (26) with same integrand L (eff) = L (classic) , that give, in principle, different propagators. More important is that the complete expressions for QPs (39) and (41) are defined for any coordinates q whereas the asymptotic expressions (40) and (42) are suited only for quasiCartesian coordinates according to argumentation of Section 4. But the most impotant, though paradoxical, conclusion is that QPs in the both cases are not scalars of the general transformation ˜ q a = ˜ q a ( q ) Thus, we encountered again with the non-invariance of quantum Hamiltonian despite that the Lagrangean form (26) of the propagator is invariant.", "pages": [ 9, 10, 11, 12 ] }, { "title": "8 Intermediate conclusion", "content": "Firstly, we see a deep conflict in QM between the requirements of observability (hermiticity) and invariance with respect to transformations ˜ q a = ˜ q a ( q ) (general covariance). Apparently, the non-invariance of QM seems to be a general property of the standard quantization approaches based on a Hilbert space of states. It looks very strange, but conceptually it can be explained by that the quantum operators of observables and, particularly, of coordinates ˆ q a = q a · ˆ 1 , imply some concrete classical measurments over the quantum natural system which is an open system and depend on choice of them, while the classical q a are considered as any of abstract arithmetizations of space points. Further, this conception probably is related to the ideas that information on the open quantum system always includes more or less information on the apparatus which provides an information on the system. Discussion of these ideas may be found in papers by M. Mensky [26] and C. Rovelli [27]. However, our consideration perhaps leads further: not only information on the state of object but also information on its dynamics contains a mixture of information on the measuring device (of the particle's position, in our case) through the additional terms in the Hamilton operators. This thought is supported by that QP is not a specifity of curvature of the space only and related to choice of curvilinear coordinate systems in the flat space , too. Of course, this issue needs a deep analysis and still I can give nebulous speculations encouraged by the following words of Leon Rosenfeld concerning QM: '... inclusion of specifications of conditions of observation into description of phenomena is not an arbitrary decision but a necessity dictated by the laws themselves of evolution of the phenomena and mechanism of observation them, which makes of these conditions an integral part of the decsription of the phenomena' . 2 From this point of view, propagator G ( ν ) ( q '' , t '' | q ' , t ' ) , eq.(33), is the amplitude of transition of the particle from point q ' to point q '' , the position of which is subjected to continual observation by means of local coordinates y a on each infifnitesimal section of possible trajectory. To this end, it is sufficient to take the (quasi-)scalar terms of the quasi-classical approximation in the Hamiltonian and therefore the amplitude is a bi-scalar. (Apparently, this paradox is an analog of the one known as 'the continually observed kettle boils never' , see, e.g. [28].)So, the local quasi-Cartesian coordinates y a add no information except the scalar curvature. On contrary, any version of the full Hamiltonian ˆ H is used to prepare a state ψ ( q ) with use of curvilinear coordinates q a . Their coordinate lines are determined by n curvatures which are equal to zero only in the case of the Cartesian coordinates which exist only in E n . They affect as forces on motion of the particle, see, e.g., [29], Chapter 1 , which are different for different choice of the coordinates q a . The difference between the versions of quantum Hamiltonian originally formulated in the Cartesian coordinates and thereupon transformed to the the spherical ones and the one which is immediately formulated in the latter coordinates had been noted by B.Podolsky in 1928 [25]. The way out of the problem, which he had proposed, is a particular case of the following more general postulate: However, this is equivalent to the direct postulation of the desired invariant result while Schrodinger wanted to find the quantum Hamiltonian from a general variational principle. Further, why one may not to dispose any appropriate degrees of ˆ ω ( q ) between multipliers of ˆ H (DW) or even take normalized Hermitean linear combinations of such disposals? There is a continuum of such generalizations of ˆ H (Pod) with zero and as well as non-zero QPs. An answer may be that ˆ H (Pod) and apparently the the mentioned disposals of degrees of ˆ ω with the resulting zero QPs are discriminated by their invariance with respect to transformations of coordinates. Nevertheless, their multiplicity causes some dissatisfaction and needs of further study. Besides, as it will be seen in the companion paper [II], all these versions of the theory do not satisfy to the Principle of Equivalence from the general-relativistic point of view. Another way to invariant quantum Hamiltonian for the natural systems, which is based on use of non-holonomic coordinates, have been proposed by H. Kleinert in his monumental monograph [18] on path integration and by M. Mensky [30]. However, this interesting nonmetric approach will be out yet of the scope of the present paper where we hold only the metric approaches. Summing up, the considered or just now mentioned approaches to quantization of the natural systems discriminate three preferred classes of the quantum Hamiltonians which are characterized by the values of the constant ξ in the quasi-scalar term of the local asymptotics: These values and fomalisms which fix them are: ξ = 1 6 ←- { canonical and quasi-classical quantization by DeWitt; revised Schrodinger variational approach and Feynman quantization in present author's version along the evaluation rule on the lattice for functions in integrands: ξ = 1 3 ←- { quasi-classical quantization by Vilkovysky, Feynman quantization present author's version along the evaluation rule on the lattice for functions in integrands: ξ = 0 ←- { Schrodinger's original approach (no QP), Rivier ordering in canonical quantization (there is QP but the quasi-scalar term vanishes), the Podolski postulate and its generalizations, quantization in non-holonomic coordinates [18] } . There are very interesting problems to study related to each of the three versions of quantization. For example, it would be important to extend them for the algebra of polynomials of momenta p with coefficients depending on canonically conjugate q . This would transfer the theoreticphysical conception of quantization, which is adopted in the present paper and the most of references therein, to a more mathematically refined level. It should not be forgotten also that the (revised) Schrodinger variational approach to quantization can apparently be of interest for application to topologically non-trivial cases of V n , while, in the present work, the triviality is intrinsically implied. However, still, we shall follow the more pragmatic way and consider the possible values of ξ from a point of view of general-relativistic quantum theory of the linear scalar field, of which the nonrelativstic asymptotic ( c -1 → 0 ) of the one-particle sector of which should produce quantum mechanics of the natural systems. In particular, it will be shown there that just the theory with ξ = 1 / 6 is in accord with the Principle of Equivalence as it formulated by S. Weinberg, see Sec.3 and supported by the conformal symmetry if n = 3.", "pages": [ 12, 13, 14, 15 ] }, { "title": "9 Acknowledgement", "content": "Thanks to Professors P. Fiziev, V. V. Nesterenko and S. M. Eliseev for useful discussions nd consultations.", "pages": [ 15 ] } ]

[ { "title": "Abstract", "content": "This is the second of the two related papers analysing origins and possible explanations of a paradoxical phenomenon of the quantum potential (QP). It arises in quantum mechanics'(QM) of a particle in the Riemannian n -dimensional configurational space obtained by various procedures of quantization of the non-relativistic natural Hamilton systems. Now, the two questions are investigated: 1)Does QP appear in the non-relativistic QM generated by the quantum theory of scalar field (QFT) non-minimally coupled to the space-time metric? 2)To which extent is it in accord with quantization of the natural systems? To this end, the asymptotic non-relativistic equation for the particle-interpretable wave functions and operators of canonical observables are obtained from the primary QFT objects. It is shown that, in the globally-static space-time, the Hamilton operators coincide at the origin of the quasi-Euclidean space coordinates in the both altenative approaches for any constant of non-minimality ˜ ξ , but a certain requirement of the Principle of Equivalence to the quantum field propagator distinguishes the unique value ˜ ξ = 1 / 6 . Just the same value had the constant ξ in the quantum Hamiltonians arising from the traditional quantizations of the natural systems: the DeWitt canonical, Pauli-DeWitt quasiclassical, geometrical and Feynman ones, as well as in the revised Schrodinger variational quantization. Thus, QP generated by mechanics is tightly related to non-minimality of the quantum scalar field. Meanwhile, an essential discrepancy exists between the nonrelativistic QMs derived from the two altenative approaches: QFT generate a scalar QP, whereas various quantizations of natural mechanics, lead to PQs depending on choice of space coordinates as physical observables and non-vanishing even in the flat space if the coordinates are curvilinear.", "pages": [ 1 ] }, { "title": "E. A. Tagirov", "content": "Joint Institute for Nuclear Research, Dubna 141980, Russia, [email protected] December 6, 2018", "pages": [ 2 ] }, { "title": "1 Introduction", "content": "In the accompanying paper under the same general title and the subtitle 'I. Non-Relativistic Origin, History and Paradoxes.' , to which I shall refer as ( I ), the main formalisms of quantization of the natural Hamilton systems were analyzed with interesting and sometimes paradoxical conclusions. The natural systems are those whose Hamilton functions are non-uniform quadratic forms in momenta p a with coefficients ω ab ( q ) depending on coordinates q ( a ) , a, b, · · · = 1 , . . . , n of configurational space V n : Here and further, the notation is used, which is standard in General Relativity (GR). An important physical representative of this class of systems is the particle moving in an external static gravitational field defined general-relativistically as the metric form of an n + 1-dimensional (Lorentzian) space-time V 1 ,n in the normal Gaussian system of coordinates { x 0 ≡ ct, q ( a ) } : Then, this construction is a foliation of V 1 ,n ( frame of reference) by the normal geodesic translations of any space-like hypersurface the interior geometry of which is that of V n . If a metric tensor ω ab does not depend on t , V 1 ,n is a globally static space-time. Analysis of various quantization procedures of the generic natural system in [ I ] has shown that the resulting non-relativistic QMs of a particle do not reconcile with the basic principles of GR , namely, the Principles of General Covariance and of Equivalence, owing to inevitable appearance of QPs in the Hamilton operators or propagators. In the formers, these QPs are not invariant (not scalars) with respect to general transformations of coordinates q ' a = q ' a ( q ) and they single out persistently the potential term: at the origin of the quasi-Euclidean (normal Riemannian) coordinates y a , where R ( ω ) ( q ) is the scalar curvature of V n . It contradicts formally to the Principle of Equivalence (PE) in S. Weinberg's formulation [9] quoted also in [ I ], Section 3. In view of this paradoxes, we shall consider now an alternative approach to construction the non-relativistic QM in the globally static V 1 ,n , which starts from the general-relativistic quantum theory of a neutral scalar field and produces a non-relativistic QM as the limit for c -1 → 0 of the one-quasi-particle sector of an appropriate Fock representation. The initial theory is general-covariant and extraction of QM from it is covariant with respect to transformations of the spatial Gaussian coordinates q a . As concerns PE in quantum theory, the field-theoretical approach shows, in which sense it is satisfied on the relativistic level, and originates the term (4) in the non-relativistic QM. The paper is organized as follows. In Section 2 , a brief exposition of the classical theory of scalar field in V 1 ,n non-minimally coupled to the metric is given. In Section 3 and relation of the energy-momentum tensor in the conformal covariant version of the theory to the Dirac scalar-tensor theory of gravitation is shown. In Sections 4 - 6, the Fock representations of quantum theory of the field is constructed and and relation to PE of the structure on the light conoid of the propagator is considered. Restriction to the time-independent (globally static) case, which is necessary for comparison with conclusions of [ I ], is considered in Sections 7-8. A logical chain of conclusions of the both papers is given in Section 9.", "pages": [ 2, 3 ] }, { "title": "2 Scalar Field in Riemannian space-time, conformal covariance and Principle of Equivalence", "content": "Thus, we start with the (classical) real scalar field ϕ ( x ) , x ∈ V 1 ,n , which satisfies to the so called non-minimal generalization of the standard Klein-Gordon-Fock equation: Notation here and in sequel is Among the arbitrary values of ˜ ξ , there is a distinguished value for which eq.(5) is asymptotically conformal covariant for m → 0 , that is, if ϕ ( x ) , x ∈ V 1 ,n is a solution of eq.(5) with m = 0, then ˜ ϕ ( x ) def = Ω 1 -n 2 ( x ) ϕ ( x ) , x ∈ ˜ V 1 ,n , is a solution of the same equation in ˜ V 1 ,n whose metric tensor is ˜ g αβ ( x ) = Ω 2 ( x ) g αβ ( x ) and Ω( x ) is an arbitrary sufficiently smooth function. 1 Conformal covariance ensures conformal invariance of eq.(5) and corresponding conservation laws if V 1 ,n under consideration admits a group of conformal isometries (motions). /negationslash The term ˜ ξ R ( g ) ( x ) in eq.(5) again, as in the Schrodinger equation with QP, causes the question on PE (see the formulation by S. Weinberg [9] reproduced also in [ I ]) since the term does not disappear in the quasi-Cartesian coordinates with the origin at x if R ( g ) ( x ) = 0. Some answer on the question gives an investigation of structure of singularities of the Green functions for the field equation (5). First, in 1974, S. Il'in and the present author [11] had shown that for where ¯ G V 1 , 3 ( x, x ' ; ˜ ξ ) is the classical Green function in V 1 , 3 and Γ( x, x ' ) is the geodesic interval between x, x ' . Thus, singularities of ¯ G V 1 , 3 ( x, x ' ; ˜ ξ ) on the light conoid Γ( x, x ' ) = 0 (the locus of isotropic geodesics, emanated from x ' ) are the same as in the tangent space E 1 , 3 , 'a locally inertial coordinate system' in Weinberg's formulation of PE, see [ I ]. Thus, PE is satisfied in this sense in the classical field theory with ˜ ξ = 1 6 and n = 3 (The direct recalculation in V 1 ,n shows that the same property takes place also for arbitrary n ). Unfortunately, the authors of [11] had not recognized sufficiently the significance of their result for justification of PE for eq.(5). Therefore, it is not suprising that much later, Sonego and Faraoni [12] have reproduced, in fact, the same result but as a verification of PE. Generalization of this verification to the quantum theory given by A. A. Grib and E. A. Poberii [19] will be noted in Section 6 after quantization of field ϕ .", "pages": [ 3, 4 ] }, { "title": "3 Energy-momentum tensor and Dirac scalar-tensor theory", "content": "Eq.(5) is the unique linear covariant scalar field equation if one introduces no new dimensional constant into the theory [8]. It follows from variation of ϕ in the functional of action Its variation by g αβ ( x ) gives the (metric) energy-momentum tensor For solutions of eq.(5), one has and consequently i.e., it has the property which is inherent also for fields with spin 1/2 and 1 and which provides all these fields with the asymptotic conservation laws corresponding to conformal isometries (if any) when m → 0 . Note also, that T αβ ( x ; ˜ ξ ) = T αβ ( x ; 0) even in E 1 ,n if ˜ ξ = 0 . /negationslash /negationslash Tensor T αβ ( x ; ˜ ξ (conf) (3)) has been re-discovered later and called 'a new energy-momentum tensor' by Callan, Coleman and Jackiv [16]. They had postulated T αβ ( x ; ˜ ξ (conf) (3)) in the form of eq.(9) for the particular case of E 1 , 3 and generalized it afterwards for V 1 , 3 . Their reasoning is evidently an inversion of the straightforward generalrelativistic approach with the requirement of the conformal symmetry in [4]. More interesting is that, in 1973, Dirac[13] formulated a scalar-tensor theory of gravitation in relation with his famous hypothesis on large numbers. For n = 3 and ˜ ξ = 1 / 6 , the integral A is just the gravitational (geometrical) part of the action integral of the Dirac theory [13], formula (5.2) there. (The full Dirac action integral includes also the electromagnetic F µν F µν and non-linear const · ϕ 4 terms.) Therefore, our T αβ ( x ; 1 / 6) is just the left-hand side of the scalar-tensor Dirac equation . In fact, Dirac had been motivated by simplicity of the trace T ( x ; ˜ ξ ) , eq.11, when ˜ ξ = ˜ ξ ( conf ) (3) ≡ 1 6 . However, we see that the same reasoning is correct for any n and, thus, the Dirac theory can be generalized to any V 1 ,n as a conformalcovariant one. In fact, the theory based on the action integral A{ g .. ( . ) , ϕ ( . ); ˜ ξ (conf) (3) } is used for construction of so called conformal cosmology, an altenative to the standard model, and applied to fit recent data on distant supernovae taken as standard candles, [14] and references therein. Thus, determination of value of ˜ ξ acquires a 'practical' interest.", "pages": [ 5 ] }, { "title": "4 Quantization of the scalar field in the general Riemannian space-time", "content": "Now, the quantum theory of the field ϕ ( x ) , x ∈ V 1 ,n (denoted as QFT in sequel) will be formulated to extract from it a structure similar to the non-relativistic QM considered in [ I ]. The program of construction of a particle-interpreted Fock representation for quantum field ˇ ϕ ( x ) , x ∈ V 1 ,n , has been fulfilled in [15] with use of formulations from [17], Chapter 2, and [18], Chapter 3, ( 'check' over symbols will denote operators in the Fock spaces F ). Here, the main points of that program with some improvements including a consideration of PE in QFT will be reproduced in the following four sesections for a consecutive statement of the problem and conclusions. The program starts with complexification Φ c = Φ ⊗ C , of the space Φ of solutions to eq.(5) and a subspace Φ ' c ⊂ Φ c such that where Φ ± are supposed to be mutually complex conjugate spaces. They are selected so that the conserved (i.e. independent on choice of Σ ) Hermitean sesquilinear form be positive (negative) definite in Φ + (Φ -) , where dσ α is the normal volume element of a Cauchy hypersurface Σ induced by the metric of V 1 ,n and determined for an arbitrary vector field f α ( x ) and arbitrary interior coordinates q a on Σ by relation ∣ ∣ The form (13) can thus be considered as an inner product in Φ -providing the last with a pre-Hilbert structure. It is clear that bi-partition (12) of Φ c can be done by an infinite set of ways. In E 1 ,n and the globally static V 1 ,n , there is a discriminated bi-partition by the positive- and negativefrequency solution owing to existence of the conserved positive definite observable of energy. However, for a time being, the generically time-dependent V 1 ,n makes sense to be considered. Let, further, { ϕ ( x ; A } ⊂ Φ + be a basis enumerated by a multi-index A , which has values on a set { A } with a measure µ ( A ) , and orthonormalized with respect to the inner product (13). Then, with the operators ˇ c + ( A ) and ˇ c -( A ) of creation and annihilation of the field modes ϕ -( x ; A ) ∈ Φ -( or, of the quasi-particles ), which satisfy the canonical commutation relations for any appropriate function f ( A ) . They act in the Fock space F with the cyclic vector | 0 > ( the quasi-vacuum ) defined by equations The conservation property of the 'scalar product' (13) allows to consider the basis as defined on the space of the Cauchy data on a concrete hypersurface Σ , but the different choices of Σ determine different Fock spaces F which are, in general, unitarily uneqvivalent , see, e.g., [17]. Correspondingly, | 0 > ≡ | 0; Σ > and F ≡ F{ Σ } . Then, operators of the basic observables in F{ Σ } can be defined as follows. The operator of number of quasi-particles The operator of projection of momentum of field ˇ ϕ ( x ) on a given vector field K α ( x ) : where and in sequel the colons denote the normal product of operators c ± Σ . To define a QFT- prototype ˇ Q ( a ) { ˇ ϕ ; Σ } , a, b, · · · = 1 , . . . n of non-relativistic QM position operators ˆ q a which played a basic role in [ I ], introduce first n position-type functions q ( a ) ( x ) , x ∈ V 1 ,n which are defined in [15], Section 2, in terms of fibre bundles. Consideration in the present paper is restricted by the traditional conjecture in theoretical physics that V 1 ,n is a trivial manifold. (It is equivalent in physics to assumption that only local manifestations of the curvature are taken into account.) Then, it is sufficient to introduce q ( i ) Σ ( x ) are scalar functions of x α w.r.t. general transformations ˜ x α = ˜ x α ( x ) , which satisfy the conditions ∣ ∣ So, they define a point on the Cauchy hypesurface Σ = { x ∈ V 1 , 3 | Σ( x ) = const } . Their restrictions on Σ can serve as internal coordinates on it. Assuming that the corresponding QFT-operators ˇ Q ( i ) { ˇ ϕ ; Σ } have the same structure as the operators ˇ N and ˇ P K introduced above, let us impose the following conditions on them : These conditions lead apparently to the following unique set of n operators ˇ Q ( i ) on F : This definition, in a certain sense, leads to a generalization for V 1 , 3 of the known NewtonWigner operator of the Cartesian coordinate operators as it is shown in [15], Section 6.", "pages": [ 6, 7, 8 ] }, { "title": "5 One-quasi-particle subspace of Fock space", "content": "A normalized one-quasi-particle state vector in F{ Σ } is It determines the field configuration Obviously < ϕ | ϕ > = 1. Consider matrix elements of operators ˇ N ( ˇ ϕ ; Σ) , ˇ P K ( ˇ ϕ ; Σ) and ˇ Q a { ˇ ϕ ; Σ } between two such states | ϕ 1 > and | ϕ 2 > . Simple calculations with use of Eqs.(16), (9), (19) and (20) give: where where and These matrix elements are sesquilinear functionals of two functions ϕ 1 ( x ) , ϕ 2 ( x ) ∈ Φ -which are obviously Hermitean in the sense that, given a functional Z ( ϕ 1 , ϕ 2 ; Σ) , the following equality takes place:", "pages": [ 8, 9 ] }, { "title": "6 Principle of Equivalence in quantum field theory", "content": "Representation (14) of the quantum field ˇ φ allows to obtain the causal Green function (or, the propagator of the quasi-particle). where T denotes the chronological product and G (1) is the Hadamard elementary solution for the field equation (5) which is determined up to a regular solution of (5) w ( x, x ' ) satisfying the initial condition w ( x, x ' ) → 0 for x → x ' . Since, in general, the definition of quasi-particles and quasi-vacuum depend on choice of the initial Cauchy hypersurface Σ 0 , the bi-scalar w ( x, x ' ) does, too, according to definition (27), and determines creation and annihilation of the newly determined quasi-particles when Σ 0 (system of reference) is changed. Contrary to [11], A. A. Grib and E. A. Poberii [19] studied both terms in eq.(28) together and have obtained that Thus, they have shown directly that the quantum Green function supports PE if ξ = 1 / 6 . All the works mentioned above are restricted by the case of n = 3 but re-calculation for abitrary n leads to the same result amd therefore we come to an important conclusion that the (asymptotic) conformal covariance and PE are in accord only for n=3 and thus the dimensionality of our real space is distinguished by that .", "pages": [ 9 ] }, { "title": "7 From quasi-particles to a quantum point-like particle", "content": "Now, our main aim is to extract a counterpart to non-relativistic QM of the natural mechanical systems, that had been considered in [ I ], from the ambiguous relativistic one-quasi-particle structure just described, and to compare these two QMs. The space Φ -so discriminated could be interpreted on a sufficient physical basis as the space of wave functions of particles instead of the ambiguous notion of a quasi-particle. In E 1 , 3 and globally static space-times, there exists an unique decomposition (12) such that an irreducible representation of the spacetime symmetry is realized on Φ -Σ but, even in these exceptional cases, one should restore the quantum-mechanical operators on L 2 ( V n ; C ; d σ ) of canonical observables of coordinates q a and of momenta p a cojugate to them; this is not a completely evident task. In sequel the operators in L 2 ( V n ; C ; d σ ) and its analogs are denoted by 'hat' on top ; and the superscript '(ft)' denotes objects of the field-theoretical origin. All 'hatted' operators act along the hypersurface Σ /owner x or its normal geodesic translations S Σ = const are expressed in terms of projections of covariant derivatives ∇ α onto these hypersurfaces: (i.e. h αβ is the tensor of projection on S Σ ). I recall that, up for a time being, we consider non-static V 1 ,n for generality. Our first task is to construct a map so that eq.(5) would generate Schrodinger -DeWitt-type equation, eq.(17) in [ I ] in terms of ψ ( x ) ∈ L 2 (Σ; C ; ω 1 / 2 d n q ) so that the inner product in the latter were induced by the scalar product (13). In the generic V 1 ,n , map (30) can be constructed only as the quasi-non-relativistic asymptotic(i.e. for c -2 → 0 ). In [15], the space Φ -N { S Σ } of the following asymptotic in c -2 solutions of eq.(5) is taken as Φ -: The objects Σ , S Σ , ψ , and ˆ V ( x ) are: the superscript (ft) denotes the field-theoretical origin of the object. Operators ˆ h n ( x ) are determined by recurrent relations starting with ˆ h 0 ≡ ˆ H (ft) 0 ; their concrete form is not essential for purposes of the present paper because, finally, it will be concentrated on exactly non-relativistic case of N = 0. Wave functions ψ ( x ; N ) ∈ L 2 ( S Σ ; C ; d σ S ) ( d σ S being defined as in eq.(16) with Σ ∼ S Σ ) in the following asymptotic sense: All 'hatted' operators act along the hypersurface S /owner x that is they are differential operators containing only the covariant derivatives D α along S . Eq.(36) provides Φ -N { S } with the structure of L 2 ( S ; C ; d v S ) and ψ by the standard Born probabilistic interpretation in each configurational space S = const , i.e. | ψ ( x ) | 2 is the probability density to observe the field configuration which may be called 'a particle' at the point x ∈ S . At least, this field configuration satisfies an intuitive idea of the quantum particle as a localizable object. Further, let ˇ O signifies any of the QFT-operators of observables in the Fock representation determined by the space Φ -N { S Σ } , which have been introduced above in Section 3. Then, owing to relation (26), the corresponding asymptotically Hermitean quasi-non-relativistic QMoperator ˆ O is determined up to an asymptotic unitary transformation by the following general relation: again, ˆ o n are differential QM-operators along S determined by recurrence relations starting with ( ˆ O ) 0 . The simplest example of the relation is and hence the operator of the number of particles ˆ N ( ˆ ϕ ; Σ) is represented in the space Ψ N { S Σ } ∼ L 2 ( S Σ ; C ; d σ S ) by the unity operator as it should be in quantum mechanics of a single stable particle . In the same way, one could determine the asymptotic QM-operators of particle position ˆ q a ( x ) and of projection of momentum on a vector field K α ( x ) acting on Ψ N { S Σ } and along the hypersurface S Σ ( x ) = const . The formulae in their generality are somewhat lengthy and I refer for them to [15]. Instead, having in view as the main aim, comparison of the present asymptotic structure of the field-theoretic origin with QM in [I] obtained by quantization of the conservative natural mechanics, I give here a summary of the operators for the case when V 1 ,n is a globally static space-time . In this case, coordinates x can be chosen as { x a } ∼ { t, q a } so that the metric of V 1 ,n acquires the form (2), S = ct and Then, the asymptotic expansions of the QM-operators of observables can be represented as the formal closed expressions [15]: These formulae are of interest for separate investigation when c -1 > 0 . For example, it is seen that operators of coordinates ˆ q ( i ) S ( x ) do not commute except the case of S ∼ E n and q ( i ) S ( x ) ≡ y a , the Cartesian coordinates. However, I shall not dwell on these interesting questions here and pass directly to the non-relativistic QM resulting from this asymptotic structure in the limit c -1 = 0.", "pages": [ 9, 10, 11, 12 ] }, { "title": "8 Non-relativistic Quantum Mechanics generated by Quantum Field Theory", "content": "It is seen that the expressions (41 - 45) are invariant as w.r.t. the point transformations x α -→ ˜ x α ( x ) as well as w.r.t. the choice of classical position-type observables q ( i ) S ( x ) -→ ˜ q ( i ) S ( x ) generated by the chosen initial Σ . The expressions for quantum observables for c -1 = 0 in terms of arbitrary coordinates q a on foliums S of V 1 ,n are the following differential operators acting on ψ ( t, q ) ∈ L 2 ( V n ; C ; ω 1 / 2 d n q ) : where ˆ O 1 · ˆ O 2 denotes the operator product of these operators, ∇ ( ω ) a is the covariant derivative in S ∼ V n (i.e., i.e. w.r.t. the metric tensor ω bc ) and ω def = det ‖ ω bc ‖ ; Recall that q ( i ) are scalar functions of x α and, thus, of q a , which are subordinated to conditions (18). Thus, operators ˆ q ( i ) ( q ) , ˆ p K ( q ) , ˆ H (f t ) 0 ( q ) , are independent on choice of q a but depend on choice of scalars q ( i ) and the vector field K a . In particular, n vectors form a basis in the tangent spaces of V n determined by q ( i ) . Then, if the values of the latter scalars are taken as coordinates q a , i.e. Then, K ( i ) a = δ ( i ) a and the brackets in the superscript ( i ) may be omitted. Finally, we come Pauli's expression (12) in [ I ] : Though it looks as non-invariant operator w.r.t. transformations of q a , actually it is tightly related to choice of canonically conjugate q a the values of which are fixed by the scalar functions q ( i ) ( x ) | Σ and cannot be transformed. Thus, there is no sense to ask, is it an 1-form or not. Actually, it is a form-invariant : if we take another set of scalars ˜ q ( i ) that formalizes measurement of position in the configurational space Σ by a complete set of operators ˆ ˜ q a , then other set of momentum operators ˆ ˜ p a should be taken in the form of eq.(51). Consequently, returning to canonical quantization as in Section 3 of [ I ] with these changed basic observables gives different QP related to the the canged scalars q ( i ) ( x ) formalising observation of a particle position on a folium S .", "pages": [ 12, 13 ] }, { "title": "9 Conclusion", "content": "Summarizing the main results of the both papers we come to the following logical chain. Moreover, the mentioned latter three quantizations as well as the (revised) Schrodinger variational and canonical DeWitt quantizations give that is the formula (4 ) Generalization of the canonical quantization general ([ I ], Section 4) can give any value of ξ and some form of non-invariant QP persists to appear. though ˆ H (f t ) 0 ( q ) differs from hamiltonians ˆ H ( q ) in ([ I ]) obtained by quantization of the natural mechanics by that the former does not contain the part of QP depending on choice of coordinates q , that is the terms that are hid in the residual term O ( y ) in eq.(53). Meanwhile, there is a difference between QMs in V 1 ,n in that quantization of the natural systems generates a more complicate QP which does not vanish even in the Euclidean spacetime E 1 ,n if curvilinear coordinates are taken as the position observables q a . I have attempted in [ I ] to interpret this phenomenon as intervention of information on the (speculative) classical position detecting device. into the quantum Hamiltonian. The relativistic theory cannot include information on such a device in principle and takes into account only the local QP in (46). The difference between the two approaches is not a discrepancy, in my opinion, but different particular manifestations of a more deep quantum physics still unknown for us completely but apparently related to the problem of measurement. Recall also that some essential considerations related to the problem are given in the last section of [ I ].", "pages": [ 13, 14, 15 ] }, { "title": "10 Acknowledgement", "content": "The author is thankful to Professors P. Fiziev, V. V. Nesterenko and S. M. Eliseev for useful discussions nd consultations.", "pages": [ 15 ] } ]

[ { "title": "Trajectories in a space with a spherically symmetric dislocation", "content": "Alcides F. Andrade ∗ and Guilherme de Berredo-Peixoto † Departamento de F'ısica, ICE, Universidade Federal de Juiz de Fora Campus Universit'ario - Juiz de Fora, MG Brazil 36036-330 Abstract. We consider a new type of defect in the scope of linear elasticity theory, using geometrical methods. This defect is produced by a spherically symmetric dislocation, or ball dislocation. We derive the induced metric as well as the affine connections and curvature tensors. Since the induced metric is discontinuous, one can expect ambiguity coming from these quantities, due to products between delta functions or its derivatives, plaguing a description of ball dislocations based on the Geometric Theory of Defects. However, exactly as in the previous case of cylindric defect, one can obtain some well-defined physical predictions of the induced geometry. In particular, we explore some properties of test particle trajectories around the defect and show that these trajectories are curved but can not be circular orbits. Keywords: Dislocation, Spherical Symmetry, Linear Elasticity Theory, Gravity.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The topological defects attract great interest due to the applications to condensed matter physics (see, e.g., [1, 2] for an introduction and recent review). Another elegant approach in this area is the geometric theory of defects [3, 4] (see also [5] for the introduction), which is formulated in terms of the notions originally developed in the theories of gravity. In this framework, we can cite basically two kinds of defects, described in the view of Riemann-Cartan geometry: disclinations and dislocations. This means that the curvature and torsion tensors, respectively, are interpreted as surface densities of Frank and Burgers vectors and thus linked to the nonlinear, generally inelastic deformations of a solid. Recently, the qualitatively new kind of geometric defect has been described in [6] (see also [7]), corresponding to a tube dislocation (with cylindrical symmetry). Nevertheless, one can find others well known types of dislocations. In Elasticity Theory, the dislocations and their physical effects are a matter of great concern, specially the screw dislocation [8]. In this paper, we consider a ball dislocation (or sphere dislocation), which is the same type of defect studied in [6], translated to spherical symmetry. Is is remarkable that such a problem was not investigated yet. For our purposes, the approach will be limited to linear elasticity theory (using Riemann-Cartan geometry as a tool). The ball dislocation can be understood by the illustration in Figure 1. It is possible to treat this defect as a point defect in the limiting case where ρ is much smaller than the dimensions of the considered physical system. Of course, the ball dislocation provides a more general picture. This paper is organized in the following way. In Section 2, we treat mathematically the ball dislocation, find the solution of the variable which describes the defect, and with the help of Gravity Theory methods, we study the trajectories of test particles in Section 3. Finally we proceed to our conclusions.", "pages": [ 1, 2 ] }, { "title": "2 Ball dislocation in linear elasticity theory", "content": "Let us describe the ball dislocations in linear elasticity theory. Consider a homogeneous and isotropic elastic media as a three-dimensional Euclidean space R 3 with Cartesian coordinates x i , y i , where i = 1 , 2 , 3. The Euclidean metric is denoted by δ ij = diag(+ + +). The basic variable in the elasticity theory is the displacement vector of a point in the elastic media, u i ( x ), x ∈ R 3 . In the absence of external forces, Newton's and Hooke's laws reduce to three second order partial differential equations which describe the equilibrium state of elastic media (see, e.g., [9]), Here ∆ is the Laplace operator and the dimensionless Poisson ratio σ ( -1 ≤ σ ≤ 1 / 2) is defined as The quantities λ and µ are called the Lame coefficients, which characterize the elastic properties of media. Raising and lowering of Latin indices can be done by using the Euclidean metric, δ ij , and its inverse, δ ij . Eq. (1) together with the corresponding boundary conditions enable one to establish the solution for the field u i ( x ) in a unique way. Let us pose the problem for the ball dislocation shown in Figure 1. This dislocation can be produced as follows. We cut out the thick spherical sector of media located between two concentric spherical surfaces of radii r 1 and r 2 ( r 1 < r 2 ), move symmetrically both cutting surfaces one to the other and finally glue them. Due to spherical symmetry of the problem, in the equilibrium state the gluing surface is also spherical, of radius ρ which will be calculated below. Within the procedure described above and shown in Figue 1, we observe the negative ball dislocation because part of the media was removed. This corresponds to the case of r 1 < r 2 . However, the procedure can be applied in the opposite way by addition of extra media to R 3 . In this case, we meet a positive ball dislocation and the inequality has an opposite sign, r 1 > r 2 . Let us calculate the radius of the equilibrium configuration, ρ . This problem is naturally formulated and solved in spherical coordinates, r, θ, φ . Let us denote the displacement field components in these coordinates by u r , u θ , u φ . In our case, u θ = u φ = 0 due to the symmetry of the problem, so that the radial displacement field u r ( r ) can be simply denoted as u r ( r ) = u ( r ). The boundary conditions for the equilibrium ball dislocation are ∣ ∣ The first two conditions are purely geometrical, and the third one means the equality of normal elastic forces inside and outside the gluing surface in the equilibrium state. The subscripts 'in' and 'ex' denote the displacement vector field inside and outside the gluing surface, respectively. Let us note that our definition of the displacement vector field follows [5], but differs slightly from the one used in many other references. In our notations, the point with coordinates y i , after elastic deformation, moves to the point with coordinates x i : The displacement vector field is the difference between new and old coordinates, u i ( x ) = x i -y i . Indeed, we are considering the components of the displacement vector field, u i ( x ), as functions of the final state coordinates of media points, x i , while in other references they are functions of the initial coordinates, y i . The two approaches are equivalent in the absence of dislocations because both sets of coordinates x i and y i cover the entire Euclidean space R 3 . On the contrary, if dislocation is present, the final state coordinates x i cover the whole R 3 while the initial state coordinates cover only part of the Euclidean space lying outside the thick sphere which was removed. For this reason the final state coordinates represent the most useful choice here. The elasticity equations (1) can be easily solved for the case of ball dislocation under consideration. Using the Christoffel symbols in order to evaluate the expressions for the differential operators (Laplacian and divergence), equations (1) reduce to the only one non-trivial equation One can remember that only the radial component differs from zero. The angular θ and φ components of equations (1) are identically satisfied. The general solution for (4) is given by which depends on the two arbitrary constants of integration α and β . Due to the first two boundary conditions (2), the solutions inside and outside the gluing surface are The signs of the integration constants correspond to the negative ball dislocation. For positive ball dislocation, both integration constants have opposite signs. Using the solution (5) and the third boundary condition (2), one can determine the radius of the gluing surface, One can see that ρ is not the mean between r 1 and r 2 , as it is for the cylindrical symmetry defect [6]. On the contrary, the gluing surface is located closer to the external radius r 2 . After simple algebra, the integration constants can be expressed in terms of ρ and the thickness of the sphere l = r 2 -r 1 : It is straightforward also to get Observe that as r 1 is positive, we must have always ρ > 2 l/ 3. Finally, within the linear elasticity theory, eq. (6) with the integration constants (8) yields a complete solution for the ball dislocation in linear elasticity theory, valid for small relative displacements, when l/r 1 /lessmuch 1 and l/r 2 /lessmuch 1. It is remarkable that the solution obtained in the framework of linear elasticity theory does not depend on the Poisson ratio of the media. In this sense, the ball dislocation is a purely geometric defect which does not feel the elastic properties. In order to use the geometric approach, we compute the geometric quantities of the manifold corresponding to the ball dislocation. From the geometric point of view, the elastic deformation (3) is a diffeomorphism between the given domains in the Euclidean space. The original elastic media R 3 , before the dislocation is made, is described by Cartesian coordinates y i with the Euclidean metric δ ij . An inverse diffeomorphic transformation x → y induces a nontrivial metric on R 3 , corresponding to the ball dislocation. In Cartesian coordinates, this metric has the form We use curvilinear spherical coordinates for the ball dislocation and therefore it is useful to modify our notations. The indices in curvilinear coordinates in the Euclidean space R 3 will be denoted by Greek letters x µ , µ = 1 , 2 , 3. Then the 'induced' metric for the ball dislocation in spherical coordinates is where · g ρσ is the Euclidean metric written in spherical coordinates. We denote spherical coordinates of a point before the dislocation is made by { y, θ, φ } , where y without index stands for the radial coordinate and we take into account that the coordinates θ and φ do not change. Then the diffeomorphism is described by a single function relating old and new radial coordinates of a point, y = r -u ( r ), where It is easy to see that this function has a discontinuity u ext -u int | r = ρ = l at the point of the cut. Therefore a special care must be taken in calculating the components of induced metric. It is useful to express u ( r ) in a way simultaneously valid in both domains, r < ρ and r > ρ . We have then where H ( r -ρ ) is the Heaviside step function. As H ' ( r -ρ ) = δ ( r -ρ ), one achieves where By direct calculation of induced metric, by (10), one can write the corresponding line element as It is clear that the above expression, besides discontinuous, contains also a δ -function thanks to u ' . In order to avoid further conceptual consequences coming from a δ -function in the line element 1 , we shall drop it, and adopt u ' ( r ) → v ( r ). In other words, let us consider the line element Notice that the interior space is conformally flat, with a constant scale factor, while the exterior metric is not so. Both metrics are flat (as follows by direct calculation of Riemann tensor), as they should be (because they were obtained by coordinate transformations starting from the Euclidean metric). Nevertheless, the whole space is non-trivial since curvature is non-trivial exactly in the gluing surface. Next, we are going to investigate the consequences for trajectories of test particles around the defect.", "pages": [ 2, 3, 4, 5, 6 ] }, { "title": "3 Trajectories of test particles around the defect", "content": "What are the trajectories of test particles in a space with such a defect? Of course, the trajectories without defect would be straight lines, so we expect deviation from straight lines in the actual path. How these trajectories can be described? Are there any possible closed path around the defect? In order to answer these questions, let us consider the geodesic equations for both metrics, in the interior and in the exterior of gluing surface. The geodesic equations read where U µ = dx µ /dτ = (1 , ˙ r, ˙ θ, ˙ φ ) ( c = 1 and dot means derivative with respect to τ ) and Let us remmember that if only dislocations are present, then only torsion (without Riemannian curvature) is found in the geometric approach - only the Burgers vector is non-trivial. But for practical purposes, one can treat the problem in the reverse way, considering only Riemannian curvature, because both approaches are equivalent. This equivalence is very well-known in telleparalelism (see, e.g., [10]). Inside the defect, the geodesics are straight lines and thus we shall consider only the exterior metric. By direct calculation, the geodesic equations, outside the gluing surface, can be written as The denominator appearing on (17) is always positive, because from condition ρ > 2 l/ 3, one gets 3 r 3 -2 ρ 2 l > 3 r 3 -3 ρ 3 > 0. An interesting feature that we can understand is that, if ˙ r = 0, then the test particle should be necessarily at rest. This follows from equation (17). This means that if the test particle is moving, so its radial coordinate must be changing: there is no possible trajectory confined in a spherical surface. In other words, one can say that such a geometrical defect can not serve as an alternative description of gravitating objects (around which we know there are permitted circular orbits). However, other kinds of effects, in condensed matter physics, for example, can not be ruled out. A test particle can follow also a radial path, defined as any trajectory described by ˙ θ = ˙ φ = 0. To see that, let us consider ˙ θ = ˙ φ = 0, such that the radial equation (17) reads which can be solved as where K is an arbitrary integration constant. This trajectory is a straight line, and the particle's radial velocity is such that it has greater absolute values near the defect. This effect is illustrated in Figure 2, where we plot the coordinate r ( t ) against t (the coordinate system is centered at the defect), based on the integration of (20), where a and b are integration constants. We see in Figure 2 the effect of decreasing the absolute velocity as particle gets away from the defect (if there was no defect, the curve would be a straight line). Notice that as much the particle is away from the defect, its velocity approaches a constant value, as one should expect (for r >> ρ , kinematics is the same from a space without defect). To conclude, let us consider θ = π/ 2 ( ˙ θ = 0) and ˙ φ = 0. The geodesic equations read /negationslash The last equation can be integrated and we obtain where A is some integration constant. Far away from the defect, we see that ˙ φ is proportional to r -2 . Substituting (23) into (21), we have The above equation can be numerically integrated, and we can learn that the behavior of radial velocity is very similar to the one described by (20), which can be seen in Figure 3 (both drawn with the help of MAPLE software). One can extract an interesting information from (23), together with the natural assumption that the absolute velocity decreases in time (as suggested by (20)). As the solution ˙ φ = ( A/ 9) r -2 corresponds to straight lines (geodesics in flat space without defect), the actual derivative ˙ φ given by (23) decays more slowly comparing to the straight line case. This means that the actual path must deviate from a straight line, curving to the side of the defect. In other words, the path of a test particle is deflected around a defect in a similar way of the gravitational deflection.", "pages": [ 6, 7, 8 ] }, { "title": "Conclusions", "content": "We study a new kind of defect, which we call ball dislocation, using geometrical methods in linear elasticity theory. Whenever the displacement vector (whose discontinuity characterizes the defect) is small comparing to natural dimensions of some physical system, the linear elasticity theory is suitable, and the formalism of Geometric Theory of Defects can be disconsidered. Moreover, we consider a single defect and no other complicated configurations, as a continuous distribution of defects (for which the Geometric Theory of Defects is required and well-suited). Nevertheless, it is interesting to investigate the formulation of Geometric Theory of Defects for our problem. In doing so, we find that a direct (naive) application of this formulation are faced to ambiguity problems, in contrast to other kinds of defects (see [6]). The corresponding calculations are given in the Appendix. Some interesting properties can be seen in the trajectories of free classical particles which follow the geodesic equations in the presence of spherical defect. Among the properties of such motion, we show that any orbit (around the defect) confined on a sphere is forbidden. The circular orbit is a particular case. In the same time we know that circular orbits are permitted in gravitating systems; thus, according to at least this feature, the kinematical effects of a defect should not be completely identified with gravitational effects. On the other hand, all trajectories are deflected near the defect, in an analogous way of gravitating systems. One can ask if such a defect could describe some real condensed matter system, where other effects than gravity are dominating. In this case, we have a geometric description which mimics condensed matter effects from electrodynamics. This question is open, but the present article is a first step in studying the issue. It would be natural to identify each atom with a defect, and the effects on quantum particles (e.g., Dirac fermions) will be an interesting problem addressed to future works.", "pages": [ 8, 9 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "AFA acknowledges the CAPES for the schoolarship support and GBP thanks FAPEMIG and CNPq for financial support. We would like to express our gratitude to Ilya Shapiro (UFJF) for useful discussions.", "pages": [ 9 ] }, { "title": "Appendix", "content": "In order to consider the Geometric Theory of Defects, one should start from the induced metric, as derived in linear elasticity theory, calculate the corresponding curvature tensors, and identify the Einstein tensor with the energy-momentum tensor in the geometric dynamical equations (which is the Einstein equations) [5] (see also [6]). On can write the line element (15) in the form and calculate the components of the curvature tensor, given by so with δ ' ( r -ρ ) = dδ ( r -ρ ) /dr . The components of the Ricci tensor are given by: Hence, for the line element (24): The scalar curvature R is given by: Hence Notice that the curvature is non-trivial only in the gluing surface. Moreover, these quantities are also ambiguous because of the appearance of the product of δ -function for discontinuous functions, and the ambiguity is not cancelled in the calculation of Einstein tensor.", "pages": [ 9, 10 ] } ]

[ { "title": "A.C. Fabian 1", "content": "1 Institute of Astronomy, Madingley Road. Cambridge CB3 0HA UK email: [email protected] Abstract. Most of the X-ray emission from luminous accreting black holes emerges from within 20 gravitational radii. The effective emission radius is several times smaller if the black hole is rapidly spinning. General Relativistic effects can then be very important. Large spacetime curvature causes strong lightbending and large gravitational redshifts. The hard X-ray, powerlaw-emitting corona irradiates the accretion disc generating an X-ray reflection component. Atomic features in the reflection spectrum allow gravitational redshifts to be measured. Time delays between observed variations in the power-law and the reflection spectrum (reverberation) enable the physical scale of the reflecting region to be determined. The relative strength of the reflection and power-law continuum depends on light bending. All of these observed effects enable the immediate environment of the black hole where the effects of General Relativity are on display to be probed and explored. Keywords. black holes, X-ray astronomy, active galactic nuclei", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Black holes are a common feature of the Universe. They have long been suspected to be responsible for the prodigious powers of quasars (Lynden-Bell 1969) and Galactic X-ray binaries such as Cygnus X-1 (Tananbaum et al 1972). In these cases it is accretion of matter into the black hole which makes these objects luminous. We do not of course 'see' the black hole here, but the matter swirling around it in the accretion flow which is heated by the gravitational energy released. This process is the most efficient known, in terms of the fraction of rest mass released, after matter-antimatter annihilation, with a typical value of ten per cent, which is about 20 times higher than hydrogen fusing into helium. Much of the energy is released in the strong gravity regime very close to the black hole at only a few gravitational radii ( r g = GM/c 2 ) enabling the effects of general relativity to be probed by observation. Strongly curved spacetime leads to large gravitational redshifts, strong light bending and if the black hole is spinning, dragging of inertial frames, revealed through the Innermost Stable Circular Orbit or ISCO. We concentrate here on luminous accreting black holes and in particular on their X-radiation. The observed rapid variability seen in some Active Galactic Nuclei (AGN) has long shown that the Xray emission originates from a physically small region. Relativistically blurred reflection components and reverberation now show just how small a region this can be.", "pages": [ 1 ] }, { "title": "2. X-ray Reflection", "content": "Matter accreting onto a black hole is most unlikely to fall in radially but will have sufficient angular momentum to go into orbit about it. Viscosity then causes the matter to spiral inward while the angular momentum is transferred outward. This forms an accretion disc which will be dense, optically thick and physically thin provided that the gravitational energy released is radiated locally. The emitted spectrum is a quasiblackbody of temperature of about 10 7 K for a luminous disc around a stellar mass black hole and drops to about 10 5 K for a billion solar mass black hole. The accreting gas will therefore be a hot dense plasma with the differential rotation winding up the magnetic fields (which provides the viscosity). In a manner similar to the production of coronal magnetic structures on and above the Sun, we can expect that magnetic structures will occur above the inner accretion disc. Magnetic reconnection in this corona can accelerate particles in the corona and, through inverse Compton scattering of soft disc photons, produce a hard power-law continuum. Such a configuration (Fig. 1, left) explains the basic spectral components of an Active Galactic Nucleus (AGN); a big blue bump of quasi-blackbody emission from the disc itself and a hard power-law of X-rays extending to hundreds of keV. The coronal power-law emission can be rapidly variable due to its magnetic nature. The rapid X-ray variability seen in many sources shows that the power-law source in most accreting black holes is compact in size. Clear evidence of this compactness is provided by microlensing studies of quasars which are multiply imaged by an intervening galaxy (e.g. Morgan et al 2012). The half-light radius for the X-ray emission for Q 0158-4325, shown in Fig. 1, is less than 6 r g . The irradiation of the dense disc by the coronal power-law continuum provides a further spectral component, X-ray reflection (Fig. 2). This is just the back-scattered emission plus fluorescence, recombination and bremsstrahlung (see Fabian & Ross 2011 for a review). It consists of a hard Compton hump together with a soft excess of re-emission including emission lines. The strongest such line is usually iron K α at 6.4-6.95 keV, depending on the ionization state of iron in the disc. It is likely that the irradiation will be intense enough to control and raise the ionization state above that expected from the hot disc alone. Having emission lines produced from the disc is very important, since if they are observed then we can measure the Doppler shifts and thus the velocity of the accretion flow, which can be up to half the speed of light. We can also measure the gravitational redshift, which can tell us the radius at which the emission originates (Fig. 6, Fabian et al 1989; Laor 1991). We expect an accretion disc to extend down to the ISCO within which the matter plunges on a ballistic orbit into the black hole. Since the radius of the ISCO depends on the spin of the black hole, measurement of the largest gravitational redshift translates to a measurement, or at least a lower limit on, the black hole spin. The net observed spectrum from the inner parts of an accretion disc around a black hole therefore consists of a power-law continuum, a reflection spectrum blurred by Doppler shifts and gravitational redshifts, a soft excess below 2 keV, a broad iron line from 47 keV and a Compton hump peaking around 30 keV (Fig. 4). The broad iron line was first seen from an AGN (MCG-6-30-15) with the Japanese-US satellite mission, ASCA (Tanaka et al 1995). A recent version of the spectrum from this object using Suzaku is shown in Fig. 5 (Miniutti et al 2007). The line is so broad that indicates a high spin ( a > 0 . 95), with the ISCO well within radius r = 2 r g , where GR effects must be very strong. Broad iron lines have been found from a range of Seyfert 1 AGN (Nandra et al 2007, Brenneman & Reynolds 2009), Galactic Black Hole Binaries (BHB, Miller 2007) and neutron star systems (Cackett et al 2008). An example from a BHB is shown in Fig. 6 where the broad line of XTE J1752-223 appears not to change between the intermediate and hard state during its 20? outburst (Reis et al 2010). Fitting the spectrum with a reflection model reveals the black hole spin to be about 0.55. The discussion has so far ignored absorption and outflows (e.g. fast and slow warm absorbers) which clearly occur in some luminous accreting black holes. Some (e.g. Miller et al 2008) argue that a dense outflow completely obscures all emission from within tens to hundreds r g , hides the strong gravity regime from view. The absorber then mimics the relativistically blurred reflection features described above. How such mimicry works for both AGN and BHB is not explained. Many BHB are bright enough that any such extreme obscuration would be obvious. It is far more likely that absorption and outflows are just a complication that can be corrected for in spectral models. Microlensing, rapid X-ray variability and reverberation results all point to the strong gravity regime being directly observable in most Type I AGN.", "pages": [ 2, 3, 4, 5 ] }, { "title": "3. Light Bending", "content": "The iron line in MCG-6-30-15 is stronger than expected and, although variable, does not show the same level of variability as the power-law continuum (Fabian & Vaughan 2003). These are likely to be due to strong light bending (Strength: Martocchia & Matt 1996, 2002; Variability: Miniutti & Fabian 2004). Strong gravity close to the black hole bends the light towards it, causing it to be focussed on the disc. In so doing it makes an intrisically isotropic continuum source appear anistropic to the outside observer (Fig. 7). Fewer photons escape to infinity and more strike the disc, or fall into the hole as the corona lies closer to the black hole. Good evidence for light bending was noted in the BHB XTEJ1650-300 by Rossi et al (2005). A recent reworking of the data on that source by Reis et al (2012) shows this result more clearly (Fig. 7). Further evidence in support of extreme light bending emerged from early in 2011 when colleagues discovered that 1H0707-495 was dramatically reducing in flux and going into a low state (Fig. 8). The soft flux from the source reduced by over an order of magnitude in January and February before recovering in March. We triggered an observation of the object with XMM, under an accepted programme of Norbert Schartel for studying low states in AGN. The spectrum of the source now looked similar in shape to when brighter, although of course much reduced in flux and most interestingly shifted to lower energies. The corona had dropped to within 2 r g of the black hole (Fig. 8). The shape of the emissivity profile of reflection (deduced from the shape of the broad iron line) required the very strong gravitational light bending expected very close to the black hole (Wilkins & Fabian 2011, 2012a). Several other AGN have been seen to drop to a reflection-dominated phase, explainable by extreme light bending due to the corona collapsing to the centre (e.g. PG2112, Schartel et al 2007; Mkn 335, Gallo et al 2012; PHL 1092, Miniutti et al 2012).", "pages": [ 5 ] }, { "title": "4. Reverberation", "content": "Recently, we have seen both the iron K α and L α lines in the AGN 1H 0707-495 (Fig. 5). This object is a very highly variable type of AGN known as a Narrow Line Seyfert 1 galaxy (NLS1). The detection of the L α line is possible here since the abundance of iron is particularly strong. The result is from a very long (500 ks) XMM-Newton exposure which has also enabled the detection of X-ray reverberation for the first time (Fig. 9, Fabian et al 2009). This means that the reflection-dominated emission below 1 keV lags behind the power-law which dominates the spectrum above 1 keV, owing to the difference in light paths taken by the direct power-law and by reflection (Fig. 2). The lag is about 30 s which corresponds to about 2 r g for a 2 × 10 6 M /circledot black hole, such as is suspected in 1H0707-495. The results imply that most of the primary coronal X-ray source is very compact and centred close to the black hole above the inner accretion disc. We have now obtained similar lag results from another NLS1, IRAS13224-3809 (also plotted in Fig. 9; Fabian et al 2012). A key test of the reverberation/reflection picture is the energy dependence of the lags. Reflection occurs over a wide range of radii in the disc. Variations in the primary continuum should be rapidly followed by variations in the inner reflection, where the gravitational redshift is strongest, so appearing in the red wing of the broad line. The outer parts of the disc, giving the blue horn of the line, respond slower and give larger lags (Fig. 10 left). This is indeed what is observed in the first reverberation results from the iron-K band obtained in the X-ray bright AGN, NGC 4151 (Fig. 10 right, Zoghbi et al 2012). Iron K α lags are clearly seen with the most rapid (higher temporal frequency) variations showing shorter lags ( ∼ 1000s) at more redshifted energies (4-5 keV) than the slower (lower temporal frequency) variations showing larger lags ( ∼ 2500s) close to the rest energy of 6-7 keV. Iron K α lags are now also seen in the in the deepest observations of 1H 0707-495 and IRAS13224-3809 (Kara et al 2012a,b). A study of 32 AGN by Barbara De Marco et al (2012) lists 15 more AGN (9 above a significance of 3 σ ) showing rapid reverberation in soft X-rays (Fig. 9). The lag timescales correlate with mass and show that most of the reverberation originates from within a few gravitational radii. This naturally leads to the conclusion that soft X-ray emission from many AGN originates close to the ISCO around moderately to highly spinning black holes. Finally, the similarity between the geometry of AGN and BHB is emphasised by the discovery of ms timescale lags in the BHB by Uttley et al (2011). Note that there is a bias in any flux-limited sample of AGN towards highly-spinning objects if the distribution of mass accretion rate at large radii is the same for all spins (Brenneman et al 2011). The mass to radiation conversion efficiency of an accretion disc increases by a factor of 3 or more as the spin is increased. A detailed discussion of the energy and frequency development of the lags, including the Shapiro delay, is given by Wilkins & Fabian (2012b).", "pages": [ 5, 6, 7, 8 ] }, { "title": "5. Summary", "content": "We now have very good observational evidence of the strong gravity regime around black holes in AGN and BHB. X-ray observations enable us to probe General Relativistic effects such as large gravitational redshifts, strong light bending and an ISCO at radii implying dragging of inertial frames. The overall picture obtained in this way is consistent with GR but does not yet test it. That may come about through testing models in which GR is modified (e.g. Johanssen & Psaltis 2012) or, possibly, through a combination of more precise light bending (i.e. space) and reverberation (time) measurements.", "pages": [ 8 ] }, { "title": "6. Acknowledgements", "content": "I am grateful to the Conference Organisers for the opportunity to talk at this interesting meeting. Thanks to my many collaborators, including Dan Wilkins, Erin Kara, Dom Walton, Abdu Zoghbi, Rubens Reis, Phil Uttley, Ed Cackett, Jon Miller, Luigi Gallo, Giovanni Miniutti, Chris Reynolds and Randy Ross, and to George Chartas for Fig. 1 (right).", "pages": [ 8 ] }, { "title": "References", "content": "Miniutti G., Fabian A.C., 2004, MNRAS, 349, 1435 Miniutti G et al 2007 PASJ 59, 315 Miniutti G., Brandt W.N., Schneider D.P., Fabian A.C., Gallo L.C., Boller T., 2012, MNRAS, 425, 1718 Morgan C.W. et al 2012, ApJ, 756, 52 Nandra K, O'Neill PM, George IM, Reeves JN, 2007, MNRAS, 382, 194 Reis R.C., et al 2011, MNRAS, 410, 2497 Reis R.C., Miller J.M., Reynolds M.T., Fabian A.C., Walton D.J., Cackett E., Steiner J.F., 2012, arXiv:1208.3277 Ross R.R., Fabian A.C., 2005, MNRAS, 358, 211 Rossi S., Homan J., Miller J.M., Belloni T., 2005, MNRAS, 360, 763 Schartel, N., Rodrguez-Pascual P.M., Santos-Lle M., Ballo L., Clavel J., Guainazzi M., Jimnez- Bailn E., Piconcelli E., 2007, A&A, 474, 431 Tanaka Y. et al 1995 Nature 375 659 Tananbaum H. Gursky H. Kellogg E. Giacconi R. Jones C. 1972, ApJ, 177, L5 Uttley P., Wilkinson T., Cassatella P., Wilms J., Pottschmidt K., Hanke M., Bck, M., 2011, MNRAS, 414, L60 Wilkins D.R., Fabian A.C., 2011, MNRAS, 414 1269 Wilkins D.R., Fabian A.C., 2012a, MNRAS, 424 1284 Wilkins D.R., Fabian A.C., 2012b, MNRAS, submitted Zoghbi A., Fabian A.C., Reynolds C.S., Cackett E,M., 2012, MNRAS, 422 129", "pages": [ 9 ] } ]

[ { "title": "ABSTRACT", "content": "Feeding Compact Objects: Accretion on All Scales Proceedings IAU Symposium No. 290, 2012", "pages": [ 1 ] }, { "title": "On the connection between accreting X-ray and radio millisecond pulsars", "content": "T.M. Tauris 1 , 2 , ∗ 1 Argelander-Institut fur Astronomie, Universitat Bonn, Germany 2 Max-Planck Institut fur Radioastronomie, Bonn, Germany ∗ email: [email protected] Abstract. For many years it has been recognized that the terminal stages of mass transfer in a low-mass X-ray binary (LMXB) should cause the magnetosphere of the accreting neutron star to expand, leading to a braking torque acting on the spinning pulsar. After the discovery of radio millisecond pulsars (MSPs) it was therefore somewhat a paradox (e.g. Ruderman et al. 1989) how these pulsars could retain their fast spins following the Roche-lobe decoupling phase, RLDP. Here I present a solution to this so-called 'turn-off problem' which was recently found by combining binary stellar evolution models with torque computations (Tauris 2012). The solution is that during the RLDP the spin equilibrium of the pulsar is broken and therefore it remains a fast spinning object. I briefly discuss these findings in view of the two observed spin distributions in the populations of accreting X-ray millisecond pulsars (AXMSPs) and radio MSPs. Keywords. stars: neutron, pulsars: general, X-rays: binaries, stars: rotation", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Theoretical modelling of accretion torques with disk-magnetosphere interactions have been performed for nearly four decades (e.g. Ghosh & Lamb 1979). Likewise, detailed calculations of binary evolution have demonstrated that both the duration of the RLO in LMXBs and the amounts of matter transfered are adequate to fully recycle a pulsar (e.g. Tauris & Savonije 1999; Podsiadlowski et al. 2002). However, these previous studies did not combine numerical stellar evolution calculations with computations of the resulting accretion torque at work. For further details on these issues, and more general discussions of my results, I refer to the journal paper, Tauris (2012).", "pages": [ 1 ] }, { "title": "2. The magnetosphere-disk interactions", "content": "The interplay between the neutron star magnetic field and the conducting plasma in the accretion disk is a rather complex process. The physics of the transition zone from Keplerian disk to magnetospheric flow is important and determines the angular momentum exchange from the differential rotation between the disk and the neutron star (e.g. Spruit & Taam 1993). It is the mass transfered from the donor star which carries this angular momentum which eventually spins up the rotating neutron star once its surface magnetic flux density, B is low enough to allow for efficient accretion. The gain in neutron star spin angular momentum can approximately be expressed as: ∆ J /star = √ GMr mag ∆ Mξ , where ξ /similarequal 1 is a numerical factor which depends on the flow pattern (Ghosh & Lamb 1979), ∆ M = ˙ M · ∆ t is the amount of mass accreted in a time interval ∆ t with average mass accretion rate ˙ M and r mag is the magnetospheric boundary, roughly located at the inner edge of the disk; i.e. r mag = φ · r Alfven , where the magnetospheric coupling parameter, φ = 0 . 5 -1 . 4, Wang (1997); D'Angelo & Spruit (2010). For the numerical calculations of the RLDP, I included the effect of additional spin-down torques, acting on the neutron star, due to both magnetic field drag on the accretion disk (Rapapport et al. 2004) as well as magnetic dipole radiation, although these effects are usually not dominant. The total spin torque can be written as: where n ( ω ) = tanh((1 -ω ) /δ ω ) is a dimensionless function, depending on the fastness parameter, ω = Ω /star / Ω K ( r mag ) = ( r mag /r co ) 3 / 2 , which is introduced to model a gradual torque change in a transition zone near the magnetospheric boundary.", "pages": [ 1, 2 ] }, { "title": "3. Roche-lobe decoupling phase (RLDP)", "content": "The rapidly decreasing mass-transfer rate during the RLDP results in an expanding magnetosphere which causes a significant braking torque to act on the spinning pulsar. Thus it forces the rotational period to increase, as shown in Fig. 1. At some point the spin equilibrium is broken. Initially, the spin can remain in equilibrium by adapting to the decreasing value of ˙ M . Further into the RLDP, however, r mag increases on a timescale faster than the spin-relaxation timescale, t torque at which the torque can transmit the effect of deceleration to the neutron star and therefore r mag > r co , leading to P < P eq at all times onwards. The results is that AXMSPs in LMXBs typically lose ∼ 50 % of their rotational energy (see Fig. 2), depending on the B-field of the pulsar, the duration of the RLDP and the assumptions governing the disk-magnetosphere interactions.", "pages": [ 2 ] }, { "title": "4. The spin-relaxation timescale and the duration of the RLDP", "content": "To estimate the spin-relaxation timescale one can simply consider: t torque = J/N : In intermediate-mass X-ray binaries (IMXBs) the mass-transfer phase from a more massive companion is relatively short, causing the RLDP effect to be negligible. The reason for the major difference between the RLDP effect in LMXBs (often leading to the formation of radio MSPs with helium white dwarf companions) and IMXBs (primarily leading to mildly recycled pulsars with carbon-oxygen/oxygen-neon magnesium white dwarfs) is the time duration of the RLDP relative to the spin-relaxation timescale, i.e. the ratio of t RLDP /t torque . In LMXBs, t RLDP /t torque /similarequal 0 . 3. However, in IMXBs this ratio is often smaller by a factor of 10 which causes the spin period to 'freeze' at the original value of P eq , see Tauris et al. (2012) for further details and examples of model calculations.", "pages": [ 3 ] }, { "title": "5. Spin distribution of AXMSPs and radio MSPs", "content": "The RLDP effect discussed here can explain why the recycled radio MSPs (observed after the RLDP) are significantly slower rotators compared to the more rapidly spinning AXMSPs (observed before the RLDP), see Fig. 3. Only for radio MSPs with B > 10 8 G can the difference in spin periods be partly understood from regular magnetic dipole and plasma current spin-down over a radio MSP lifetime of several Gyr. When comparing radio MSPs and AXMSPs one should be aware of differences in their binary properties and observational biases, see SOM in Tauris (2012). Observationally, the RLDP effect can be verified if future surveys discover a significant number of AXMSPs and radio MSPs confined to an interval with similar orbital periods and companion star masses. Although AXMSPs are believed to be progenitors of radio MSPs, all of the 14 observed AXMSPs have orbital periods less than one day whereas fully recycled radio MSPs are observed with orbital periods all the way up to a few hundred days. This is a puzzle. Some radio MSPs are born (recycled) with B /similarequal 1 × 10 7 G and they would most likely not be able to channel the accreted matter sufficiently to become observable as AXMSPs. Future modelling of the RLDP should ideally include irradiation effects, accretion disk instabilities and cyclic accretion (e.g. Spruit & Taam 1993; D'Angelo & Spruit 2010).", "pages": [ 4 ] }, { "title": "References", "content": "D'Angelo, C.R., & Spruit, H.C. 2010, MNRAS , 406, 1208 Ghosh, P., & Lamb, F.K. 1979, ApJ , 234, 296 Podsiadlowski, Ph., Rappaport, S.A., & Pfahl, E.D. 2002, ApJ , 565, 1107 Rappaport, S.A., Fregeau, J.M., & Spruit, H.C. 2004, ApJ , 606, 436 Ruderman, M., Shaham, J., & Tavani, M. 1989, ApJ , 336, 507 Spruit, H.C., & Taam, R.E. 1993, ApJ , 402, 593 Tauris, T.M. 2012, Science , 335, 561 Tauris, T.M., & Savonije, G.J. 1999, A&A , 350, 928 Tauris, T.M., Langer, N., & Kramer, M. 2012, MNRAS , 425, 1601 Wang, Y.-M. 1997, ApJ , 475, L135", "pages": [ 4 ] } ]

[ { "title": "W/suppresslodzimierz God/suppresslowski 1 , Agnieszka Pollo 2 , 3 and Jacek Golbiak 4", "content": "1 , Institute of Physics, Opole University, Oleska 48, 45-052 Opole, Poland email: [email protected] 2 Astronomical Observatory, Jagellonian University, Orla 171, 30-244 Krakow, Poland 3 National Centre for Nuclear Research, Hoza 69, 00-689 Warszawa email: [email protected] 4 Institute of Philosophy of Nature and Natural Sciences, John Paul II Catholic University of Lublin;, Al. Racawickie 14, 20-950 Lublin, Poland email: [email protected]", "pages": [ 1 ] }, { "title": "Abstract.", "content": "We present the preliminary analysis of clustering of a sample of 1157 radio-identified galaxies from Machalski & Condon (1999). We found that for separations 2 -15 h -1 Mpc their redshift space autocorrelation function ξ ( s ) can be approximated by the power law with the correlation length ∼ 3 . 75 h -1 Mpc and slope γ ∼ 1 . 8. The correlation length for radiogalaxies is found to be lower and the slope steeper than the corresponding parameters of the control sample of optically observed galaxies. Analysis the projected correlation function Ξ( r ) displays possible differences in the clustering properties between active galactic nuclei (AGN) and starburst (SB) galaxies. Keywords. radiogalaxies, autocorrelation function Clustering of the radiogalaxies was first detected by Peacock & Nicholson (1991) in the redshift survey of 329 galaxies with z < 0 . 1 and radio fluxes S (1 . 4GHz) > 500mJy (hereafter PN91). They found that the redshift space correlation function of these radiogalaxies could be fitted by ζ = [ s/ 11 h -1 Mpc] -1 . 8 where s is the galaxy-galaxy separation in the redshift space. Later Peacock (1997) analyzed the sample of 451 radio identified galaxies selected from the LCRS (Shectman et al. (1996)) and NVSS (Condon et al. (1998)) surveys. Using the projected correlation function Ξ( r ) = ∫ ξ [( r 2 + x 2 ) 0 . 5 ] dx he found that for the optical galaxies the correlation length was ∼ 5 h -1 Mpc while the correlation length for the radio-loud subsample was ∼ 6 . 5 h -1 Mpc . Peacock (1997) suggested that these differences resulted from the fact that his sample was dominated by starburst (SB) galaxies while the majority of the PN91 sample were luminous active galactic nuclei (AGNs). In the present paper we analyze clustering of radiogalaxies using the sample of 1157 galaxies selected from the LCRS and NVSS by Machalski & Condon (1999). We measured angular and spatial autocorrelation functions w ( θ ) and ξ ( s ) both for optical galaxies and radiogalaxies The measurement for optical galaxies remains very similar to that obtained by Tucker et al. (1997). The angular correlation function of radiogalaxies is characterized by the slope /epsilon1 = γ -1 = 0 . 97 ± 0 . 10 for ALL the sample, with /epsilon1 = 1 . 13 ± 0 . 14 for AGNs and /epsilon1 = 0 . 86 ± 0 . 14 for SB galaxies. The w ( θ ) deviates from the power law on a scale 0 . 37 o , which is lower than the value obtained for optical LCRS galaxies (0 . 54 o ). The redshift space correlation function for radiogalaxies can also be approximated by a power law; the measured correlation lengths are 3 . 75 h -1 Mpc ± 0 . 4 for all classes of radio sources (ALL, AGN and SB). However, slopes are different in all cases: we obtain the value of γ = 1 . 76 ± 0 . 11 for a general sample of radiogalaxies, 2 . 39 ± 0 . 34 for AGNs and 1 . 66 ± 0 . 15 for SB galaxies The redshift space correlation function displays the same feature as the angular correlation function: a different slope for optical and radiogalaxies, and inside a radio-loud sample - a different slope for AGNs and SB galaxies. In general, the value of γ for radiogalaxies is higher than that found for optical galaxies: γ = 1 . 52 ± 0 . 04 (see Fig. 1). The analysis of the projected autocorrelation function Ξ( r ) suggests that the correlaton length is higher for AGNs than for SB galaxies. In the same time, the measured correlation length for AGNs is lower than that obtained for optical LCRS galaxies. In our opinion, the differences between all the samples are a mixed effect of two factors: a selection bias (mainly different redshift distributions for AGNs and SB galaxies, see Fig. 2) and different environments of AGN and SB galaxies. Our main result is that for separations between 2 -15 h -1 Mpc autocorrelation function ξ ( s ) for radiogalaxies can be approximated by the power law with slope γ ∼ 1 . 8 and correlation length ∼ 3 . 75 h -1 Mpc. The measurements of the projected autocorrelation function Ξ suggest that the correlation length for AGNs is higher than that for SB galaxies. All approaches to the measurement of the correlation function show differences in the slope coefficients between AGN and SB radiogalaxies. This result can be interpreted as following: AGN are radio sources of type FRI and FRII. We have no information whether a particular AGN belongs to the type FRI or FRII. However, it is clear that a significant number of the radio sources classified as AGNs is connected to elliptical galaxies located in the centers of galaxy clusters. In contrast, SB are mostly spiral galaxies, and they are located more often in the outer parts of clusters or even on the borders of filaments where processes of galaxy formation related to the starburst processes are stronger in the local Universe.", "pages": [ 1, 2 ] }, { "title": "References", "content": "Condon J.J., et.al. 1998, AJ, 115, 1693 Machalski J., Condon J.J. 1999 ApJS, 123, 21 Peacock J.A., Nicholson D. 1991, MNRAS, 253, 307 Peacock J. A. in 'The Most Distant Radio Galaxies', Amsterdam, 1997, Royal Netherlands Academy of Arts and Sciences, eds H. J. A. Rttgering, et al. 1999, p. 377. Shectman S.A, et al. 1996 ApJ, 470, 172 Tucker D. L., et al. 1997, MNRAS, 285, L5", "pages": [ 2 ] } ]

[ { "title": "Andrea Kotrlov'a, Gabriel Torok, Eva ˇ Sr'amkov'a and Zdenˇek Stuchl'ık", "content": "Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezruˇcovo n'am. 13, CZ-74601 Opava, Czech Republic email: [email protected]", "pages": [ 1 ] }, { "title": "Abstract.", "content": "The black hole mass and spin estimates assuming various specific models of the 3 : 2 high frequency quasi-periodic oscillations (HF QPOs) have been carried out in Torok et al. (2005, 2011). Here we briefly summarize some current points. Spectral fitting of the spin a ≡ cJ/GM 2 in the microquasar GRS 1915+105 reveals that this system can contain a near extreme rotating black hole (e.g., McClintock et al., 2011). Confirming the high value of the spin would have significant consequences for the theory of the HF QPOs. The estimate of a > 0 . 9 is almost inconsistent with the relativistic precession (RP), tidal disruption (TD), and the warped disc (WD) model. The epicyclic resonance (Ep) and discoseismic models assuming the c- and g- modes are instead favoured. However, consideration of all three microquasars that display the 3 : 2 HF QPOs leads to a serious puzzle because the differences in the individual spins, such as a = 0 . 9 compared to a = 0 . 7, represent a generic problem almost for any unified orbital 3:2 QPO model. Keywords. accretion, accretion disks; X-rays: binaries; black hole physics", "pages": [ 1 ] }, { "title": "1. The spin implied by individual models", "content": "Assuming the Kerr geometry, the Keplerian and epicyclic orbital frequencies ( ν K , ν r and ν θ ) for a given radius depend only on mass M and spin a of the black hole. It is therefore possible to infer the black hole spin or mass from the observed 3 : 2 frequencies and concrete orbital QPO models. The 3: 2 QPO frequencies in GRS 1915+105 are given by Assuming relation (1.1) and the well known formulae for the orbital frequencies, we calculate the implied mass-spin functions for the models associating the 3 : 2 QPOs with common radii by means of the definition relations given in Table 1. Following Torok et al. (2005) and taking into account the estimated range of the mass of GRS 1915+105, we infer the expected ranges of the spin. The results are presented in Table 1. The above considered models assume that both of the observed 3 : 2 frequencies are produced by the same mechanism and excited at a certain (common) preferred radius. For the discoseismic modes the individual observed QPOs correspond to different modes located at different radii. The frequencies of these modes depend on the black hole spin and the speed of sound in the accreted gas, and scale as 1 /M . The mass ranges implied by combinations of the fundamental discoseismic modes overlap with those observationally determined only for the model relating the 3 : 2 QPOs to the c-mode (corrugation vertically incompressible waves near the inner edge of the disk) and g-mode (inertialgravity waves that occur at the radius where ν r reaches its maximum value) provided that 0 . 90 /lessorequalslant a /lessorequalslant 0 . 94. Details and references are given in Torok et al. (2011).", "pages": [ 1, 2 ] }, { "title": "2. Conclusions", "content": "The internal (epicyclic) resonance and the discoseismic model (dealing with c- and gmodes) are favoured in the case of GRS 1915+105 provided that a > 0 . 9. On the contrary, the TD, WD, RP, and RP2 models are disfavoured. This statement was inferred assuming that ν K , ν r , and ν θ were the exact geodesic frequencies. Analysis including the influence of non-geodesic effects would require a very detailed study. A rough estimate of their possible relevance can be done assuming the relative non-geodesic correction ∆ ν (Torok et al., 2011), which is needed to match the observations of GRS 1915+105 with a given model for a certain spin. For a ∈ (0 . 9 , 1) and the RP model, it changes from -40% to -60%. The same is roughly true for the TD and WD models, while for the RP2 model the required correction is even higher. Thus, the above result is justified, except when very large non-geodesic corrections are taken into account. Only the RP1 model can survive with corrections of | ∆ ν | up to ∼ 20%, but the present physical interpretation of this model is unclear (see Torok et al., 2011 for references). Torok et al. (2005) pointed out that since the 3 : 2 QPO frequencies in microquasars scale roughly as ν U . = 2 . 8 ( M/ M /circledot ) -1 kHz, their spins implied by a given resonance model should not much vary among them. If very different spins in GRS 1915+105, GRO J1655 -40 and XTE J1550 -564 were confirmed, the difficulty of matching all the observed 3 : 2 frequencies would clearly be rather generic for most of the orbital QPO models. Because of the generic 1 /M scaling, the above difficulty also arises for a unified 3 : 2 QPO model assuming fundamental discoseismic modes. Acknowledgements. The authors acknowledge the research grant GA ˇ CR 209/12/P740 and the project CZ.1.07/2.3.00/20.0071 - 'Synergy' supporting international collaboration of the Institute of Physics at SU Opava.", "pages": [ 2 ] }, { "title": "References", "content": "McClintock, J. E., Narayan, R., Davis, S. W., et al. 2011, CQG , 28, 114009 Torok, G., Abramowicz, M. A., Klu'zniak, W., & Stuchl'ık, Z. 2005, A&A , 436, 1 Torok, G., Kotrlov'a, A., ˇ Sr'amkov'a, E., & Stuchl'ık, Z. 2011, A&A , 531, 59, arXiv:1103.2438 [astro-ph.HE]", "pages": [ 2 ] } ]

[ { "title": "Unveiling the super-orbital modulation of LS I + 61 · 303 in X-rays", "content": "Jian Li 1 , 2 Diego F. Torres 2 , Shu Zhang 1 , Daniela Hadasch 2 , Nanda Rea 2 , G. Andrea Caliandro 2 , Yupeng Chen 1 , and Jianmin Wang 1 1 Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, China, email: [email protected] 2 Institut de Ci'encies de l'Espai (IEEC-CSIC), Campus UAB, Torre C5, 2a planta, 08193 Barcelona, Spain", "pages": [ 1 ] }, { "title": "Abstract.", "content": "We found evidence for the super-orbital modulation in the X-ray emission of LS I +61 · 303 from the longest monitoring date by the RXTE . The time evolution of the modulated fraction in the orbital light curves can be well fitted with a sinusoidal function having a super-orbital period of 1667 days. However, we have found a 281.8 ± 44.6 day shift between the super-orbital variability found at radio frequencies and our X-ray data. We also find a super-orbital modulation in the maximum count rate of the orbital light curves, compatible with the former results, including the shift. Keywords. X-rays: binaries, X-rays: individual (LS I +61 · 303)", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "LS I +61 · 303 is one of the elite γ -ray binaries. Its nature is still under debate, with rotationally powered pulsar-composed systems (see Maraschi & Treves 1981; Dubus 2006) and microquasar jets (see Bosch-Ramon & Khangulyan 2009 for a review) being discussed. Long-term monitoring of the source is a key ingredient to disentangle differences in behavior which could point to the underlying source nature. Here, we report on the analysis of RXTE/PCA monitoring observations of LS I +61 · 303 and the possible superorbital modulation of the X-ray emission.", "pages": [ 1 ] }, { "title": "2. Observations and Results", "content": "Our data set includes 473 RXTE/PCA pointed observations from 2007 August 28 to 2011 September 15. The analysis is performed using the standard RXTE/PCA criteria. Only PCU2 has been used for the analysis. Our count rate values are given for an energy range of 3-30 keV. In order to remove the influence of several kilosecond-long flares, we cut all observations that presented a larger count rate than three times the average. Given a six-month time bin, we take the peak X-ray flux in orbital lightcurve and compute the modulated flux fraction. The latter is defined as ( c max -c min ) / ( c max + c min ), where c max and c min are the maximum and minimum count rates in the 3-30 keV orbital lightcurve of that period. Results are shown in Figure 1. Table 1 presents the values of the reduced χ 2 for fitting different models to the modulation fraction and the peak flux in X-rays. It compares the results of fitting a horizontal line, a linear fit, and two sinusoidal functions. One of the latter has the same period and phase of the radio modulation (from Gregory 2002, labeled as Radio in Table 1, dotted line in Figure 1). The other sine function has the same period as in radio but allowing for a phase shift from it (a solid line in Figure 1, labeled as Shifted in Table 1). It is clear that there is variability in the data and the sinusoidal description with a phase shift is better than the linear one. The phase shift derived by fitting the modulated fraction is 281.8 ± 44.6 days, corresponding in phase to ∼ 0.2 of the 1667 ± 8 day super-orbital period. The phase shift derived by fitting the maximum flux is 300.1 ± 39.1 days, which are compatible with the former.", "pages": [ 1, 2 ] }, { "title": "3. Conclusion", "content": "We have found evidence of super-orbital modulation in the X-ray emission from LS I +61 · 303. We show that there is a ∼ 0 . 2 phase shift between the radio and the X-ray super-orbital modulation. Torres et al. 2012 has proposed that LS I +61 · 303 could be subject to a flip-flop behavior. The superCorbital modulation is possibly due to the cyclic change of the circumstellar disk (Li et al. 2012; see also Papitto et al. 2012). In this context, multi-wavelength super-orbital modulation is expected and confirmed in radio, optical, X-ray and hinted in TeV (Li et al. 2012).", "pages": [ 2 ] }, { "title": "References", "content": "Bosch-Ramon V., & Khangulyan D. 2009, International Journal of Modern Physics , D18, 347 Dubus, G 2006, A & A , 456, 801 Gregory P. C. 2002, ApJ , 575, 427 Li, J., et al. 2012, ApJ , 744, 13 Maraschi L. & Treves A 1981, MNRAS , 194, 1 Torres D. F., et al. 2012, ApJ , 744, 106 Papitto, A., Torres, D. F. & Rea, N. ApJ , 756, 188", "pages": [ 2 ] } ]

[ { "title": "Zdenˇek Stuchl'ık, Andrea Kotrlov'a and Gabriel Torok", "content": "Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezruˇcovo n'am. 13, CZ-74601 Opava, Czech Republic email: [email protected] , [email protected] Abstract. Using known frequencies of the twin peak high-frequency quasiperiodic oscillations (HF QPOs) and known mass M of the central black hole, the black-hole dimensionless spin a can be determined assuming a concrete version of the resonance model. However, large range of observationally limited values of the black hole mass implies a low precision of the spin estimates. We discuss the possibility of higher precision of the black hole spin a measurements in the framework of multi-resonance model inspired by observations of more than two HF QPOs in some black hole sources. We determine the spin and mass dependence of the twin peak frequencies with a general rational ratio n : m assuming a non-linear resonance of oscillations with the epicyclic and Keplerian frequencies or their combinations. In the multi-resonant model, the twin peak resonances are combined properly to give the observed frequency set. We focus on the special case of duplex frequencies, when the top, bottom, or mixed frequency is common at two di ff erent radii where the resonances occur giving triple frequency sets. Keywords. accretion, accretion disks - X-rays: binaries - black hole physics", "pages": [ 1 ] }, { "title": "1. Multi-resonance models with Keplerian and epicyclic oscillations", "content": "The standard orbital resonance model assumes non-linear resonance between oscillation modes of an accretion disc orbiting a central object, here considered to be a rotating Kerr black hole. The frequency of the oscillations is related to the Keplerian frequency ν K (orbital frequency of tori), or the radial ν r and vertical νθ epicyclic frequencies of the circular test particle motion.", "pages": [ 1 ] }, { "title": "1.1. More resonances sharing one specific radius", "content": "This special case allows existence of so called strong resonant phenomena when two (or more) versions of resonance could occur at the same radius allowing cooperative e ff ects between the resonances (Stuchl'ık et al. 2008). Of course, such a situation is allowed for black holes with a specific spin only. Of special interest seems to be the case of the 'magic' spin a = 0 . 983, when the Keplerian and epicyclic frequencies are in the ratio ν K : νθ : ν r = 3:2:1 at the common radius x ≡ r / M = 2 . 395, see Fig. 1.", "pages": [ 1 ] }, { "title": "1.2. Resonances occurring at two specific radii - triple frequency sets", "content": "In general, we can expect the oscillations to be excited at two di ff erent radii of the accretion disc and to enter the resonance in the framework of di ff erent versions of the resonance model (i.e., four frequency set is observed generally). In special cases, for some specific values of the black hole spin, two twin peak QPOs observed at the radii xn : m and xn ' : m ' have the top (see Fig. 1), bottom or mixed (the bottom at the inner radius and the top in the outer radius, or vice versa) frequencies identical (Stuchl'ık et al. 2012). Such situations can be characterized by sets of only three frequencies (upper ν U, middle ν M and lower ν L) with ratio ν U : ν M : ν L = s : t : u . Let us consider a simple situation with the 'top identity' of the upper frequencies in two direct resonances between the radial ν r and vertical νθ epicyclic oscillations at two di ff erent radii xp , xp ' with p 1 / 2 = m : n , p ' 1 / 2 = m ' : n ' . The condition νθ ( a , xp ) = νθ ( a , xp ' ) is then transformed to the relation which uniquely determines the black hole spin a , since the radii xp and xp ' are related to the spin a by the resonance conditions for a θ/ r ( x , p ) and a θ/ r ( x , p ' ), respectively. When two di ff erent resonances are combined, we proceed in the same manner (Stuchl'ık et al. 2012). The black hole spin a is given by the types of the two resonances and the ratios p , p ' , quite independently of the black hole mass M . A detailed table guide across all the possible triple frequency sets and related values of the black hole spin a (limited by n /lessorequalslant 4) is presented in Stuchl'ık et al. (2012).", "pages": [ 1, 2 ] }, { "title": "2. Conclusions", "content": "The multi-resonance model of HF QPOs can be considered as a promising approach to understand the observational data from black holes sources. The special triple frequency set method determines the black hole spin precisely, but not uniquely, as in general the same frequency set could occur for di ff erent values of the spin within di ff erent versions of the resonance model. In such situations the black hole spin estimates coming from the spectra fitting and the line profile model could be relevant in determining the proper versions of the resonant model. When the black hole spin is found, its mass can be determined from the magnitude of the observed frequencies. The e ffi ciency of the black hole spin determination by using the triple frequency set ratios grows strongly with growing precision of the frequency measurements. The prepared new space X-ray mission LOFT proposes sensibility of the observational instruments high enough to reach data that could be precise enough to make application of the triple frequency set method realistic. Acknowledgements. The authors acknowledge the research grant GA ˇ CR 202 / 09 / 0772 and the project CZ.1.07 / 2.3.00 / 20.0071 'Synergy' supporting international collaboration of the Institute of Physics at SU Opava.", "pages": [ 2 ] }, { "title": "References", "content": "Stuchl´ık, Z., Kotrlov´a, A., & T¨or¨ok, G. 2008, AcA , 58, 441, arXiv:0812.4418 [astro-ph] Stuchl´ık, Z., Kotrlov´a, A., & T¨or¨ok, G. 2012, A & A , submitted", "pages": [ 2 ] } ]

[ { "title": "Gabriel Torok, Pavel Bakala, Eva ˇ Sr'amkov'a, Zdenˇek Stuchl'ık, Martin Urbanec, Kateˇrina Goluchov'a", "content": "Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezruˇcovo n´am. 13, CZ-74601 Opava, Czech Republic email: [email protected] Abstract. We consider twin-peak quasi-periodic oscillations observed in the accreting lowmass neutron star binaries and explore restrictions to central compact object properties that are implied by various QPO models. For each model and each source, the consideration results in a specific relation between the compact object mass M and the angular-momentum j rather than in their single preferred combination. Moreover, restrictions on the models resulting from observations of the low-frequency sources are weaker than those in the case of the high-frequency sources. Keywords. X-rays: binaries; stars: neutron; stars: fundamental parameters; stars: rotation", "pages": [ 1 ] }, { "title": "1. Aims and Scope", "content": "Twin-peak quasi-periodic oscillations (kHz QPOs) appear in the X-ray power-density spectra of several accreting low-mass neutron star (NS) binaries. Observations of the peculiar Z-source Circinus X-1 display unusually low QPO frequencies (Boutloukos et al., 2006). On the contrary, the atoll source 4U 1636-53 displays the twin-peak QPOs at very high frequencies (e.g., Barret et al., 2005; Belloni et al., 2007). In a serie of works - Torok et al. (2010, 2012) and Urbanec et al. (2010) - we consider these sources and explore restrictions to NS properties that are implied by various QPO models.", "pages": [ 1 ] }, { "title": "2. Main Findings", "content": "For each twin-peak QPO model and each source, the consideration results in a specific relation between the NS mass M and the angular-momentum j rather than in their single preferred combination. We also observe some differences in the χ 2 behaviour that represents a dichotomy between the high- and the low- frequency sources. In general, the low-frequency sources data are matched by the models better than those of the high-frequency sources. Based on the relativistic precession (RP) model introduced by Stella & Vietri (1999), we demonstrate that this dichotomy is related to strong variability of the model predictive power across the frequency plane implied by the radial dependence of the characteristic frequencies of orbital motion. As a consequence, restrictions on the models resulting from observations of the low-frequency sources are weaker than those in the case of the high-frequency sources. These findings are illustrated in Figures 1 and 2. For a particular non-geodesic modification of the RP model that we consider in Torok et al. (2012), the data of both classes of sources are well-matched (see Figure 2 for illustration). The same result is valid for some models assuming non-axisymmetric vertical and radial disc-oscillation modes.", "pages": [ 1 ] }, { "title": "2.1. Acknowledgements", "content": "The reported work has been supported by the Czech research grants GACR 209/12/P740, GACR 202/09/0772, MSM 4781305903 and the project CZ.1.07/2.3.00/20.0071 - 'Synergy' supporting international collaboration of the Institute of Physics at SU Opava. The authors further acknowledge the internal grant of SU Opava, SGS/1/2010.", "pages": [ 2 ] }, { "title": "References", "content": "Barret, D., Olive, J. F., Miller, M. C. 2005, Astronomical Notes , 326 Belloni, T., Homan, J., Motta, S., Ratti, E., Mendez, M. 2007, MNRAS , 379, 247 Boutloukos, S., van der Klis, M., Altamirano, D., Klein-Wolt, M., Wijnands, R., Jonker, P. G., Fender, R. P. 2006, ApJ , 653, 1435 Lin, Y. F., Boutelier, M., Barret, D. 2011, ApJ , 726 Stella, L., Vietri, M. 1999, Phys. Rev. Lett. , 82, 17 T¨or¨ok, G., Bakala, P., ˇ Sr´amkov´a, E., Stuchl´ık, Z., Urbanec, M. 2010, ApJ , 714, 748 T¨or¨ok, G., Bakala P., ˇ Sr´amkov´a E., Stuchl´ık Z., Urbanec M., Goluchov´a K. 2012, ApJ , 760, 138 Urbanec, M., T¨or¨ok, G., ˇ Sr´amkov´a, E., ˇ Cech, P., Stuchl´ık, Z., Bakala, P. 2010, A&A , 522, 72", "pages": [ 2 ] } ]

[ { "title": "Xin-Lin Zhou 1 and Roberto Soria 2", "content": "1 National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China email: [email protected] 2 International Centre for Radio Astronomy Research, Curtin University GPO Box U1987, Perth, WA 6845, Australia email: [email protected] Abstract. We discuss two methods to estimate black hole (BH) masses using X-ray data only: from the X-ray variability amplitude and from the photon index Γ. The first method is based on the anti-correlation between BH mass and X-ray variability amplitude. Using a sample of AGN with BH masses from reverberation mapping, we show that this method shows small intrinsic scatter. The second method is based on the correlation between Γ and both the Eddington ratio L bol /L Edd and the bolometric correction L bol / L 2 -10keV . Keywords. accretion, accretion disks, X-rays:binaries, X-rays: galaxies", "pages": [ 1 ] }, { "title": "1. X-ray variability amplitude", "content": "The XVA σ 2 rms (also known as 'excess variance') is the variance of a light curve normalized by its mean squared after correcting for experimental noise (Nandra et al. 1997; Turner et al. 1999). For a light-curve segment with N bins: where X i and σ i are count rates and uncertainties in each bin, and µ is the average of the count rates. There is an empirical (anti)correlation between XVA and BH masses in AGN (Lu & Yu 2001, O'Neill et al. 2005). To constrain this correlation, we selected and studied two (largely overlapping) AGN samples with X-ray observations longer than 40 ks: one sample with BH masses derived from reverberation mapping, and the other from the M BH -σ ∗ relation (see Zhou et al. 2010 for details). We found that the intrinsic dispersion of the M BH -σ 2 rms relation for the reverberation-mapped AGN is quite small, no larger than the uncertainties in the BH masses: A similar result was independently obtained by Ponti et al. (2012). We used this relation to determine the BH mass in the Seyfert galaxies MCG-6-30-15 and 1H0707 -495, using archival XMM-Newton data. We obtained BH masses of (2 . 6 ± 0 . 5) × 10 6 M /circledot and (6 . 8 ± 0 . 7) × 10 5 M /circledot , respectively (Zhou et al. 2010). The XVA derived from multiple XMM-Newton observations changes by a factor of 2-3. This means that the uncertainty in the BH mass from a single observation is slightly worse than that from the reverberation mapping or the stellar velocity dispersion methods. However, if the XVA randomly scatters around the true value for the power spectral density (PSD), the mean XVA of many data segments reduces the error (Vaughan et al. 2003). We conclude that the XVA might be a better BH mass estimator than the virial method. The M BH -σ 2 rms relation is explained by a shift of the high-frequency break f b in the PSD, to lower frequencies for higher BH masses. f b scales approximately as ˙ m/M BH , where ˙ m is the dimensionless Eddington accretion rate (McHardy et al. 2006); however, we found that there is no or very weak correlation between XVA and ˙ m , confirming the findings of O'Neill et al. (2005). This suggests that the normalization of the PSD varies with ˙ m in a way that compensates for the break-frequency dependence on ˙ m . Finally, we point out that our sample of AGN is skewed towards BHs with low mass and high Eddington rates. In forthcoming work, we will explore: where the relation saturates at the low-mass end (Ai et al. 2011); how it may extend to very-low-luminosity nuclear BHs in normal galaxies; and whether it may be used to estimate BH masses in ultraluminous X-ray sources (ULXs).", "pages": [ 1, 2 ] }, { "title": "2. BH mass estimates from Γ and L 2 -10keV", "content": "X-ray spectral studies of accreting BHs show a correlation between the photon index Γ of the power-law component and the Eddington ratio. Specifically, Γ varies from ≈ 2 . 5 for L bol ∼ L Edd to ≈ 1 . 5 for L bol ∼ 10 -2 L Edd (for AGN: Shemmer et al. 2008, Gu & Cao 2009; for stellar-mass BHs: Wu & Gu 2008). (At even lower luminosities, there is evidence that Γ increases again, but with a much weaker correlation: Gu & Cao 2009, Corbel et al. 2008). In Zhou & Zhao (2010), we refined this correlation by choosing a sample of 29 low-redshift ( z < 0 . 33) AGN in the luminosity range 10 -2 ∼ L bol /L Edd ∼ 1. We determined Γ by fitting their XMM-Newton /EPIC spectra in the 2-10 keV range. We selected only radio-quiet AGN, because beaming in radio-loud sources may affect measurements of the intrinsic value of Γ. All sources have BH masses from reverberation mapping, and L bol estimated from simultaneous X-ray, UV and optical data. We obtain: The correlation (2.2) appears stronger than (2.1) (see also Jin et al. 2012). In forthcoming work, we shall compare these correlations with those inferred from high-redshift AGN, to check for evolutionary effects in AGN spectral properties. Assuming no evolutionary effect, we can use (2.1,2.2) to determine BH masses in AGN for which we know the X-ray luminosity. We estimate a mean uncertainty in the BH mass of a factor of 2 or 3 (Shemmer et al. 2008). We shall also explore the Γ versus L bol /L Edd correlation for L bol /L Edd /greaterorsimilar 1, with possible applications to ULXs and quasars.", "pages": [ 2 ] }, { "title": "References", "content": "Ai, Y. L., Yuan, W., Zhou, H. Y., Wang, T. G., & Zhang, S. H. 2011, ApJ , 727, 31 Corbel, S., Kording, E., & Kaaret, P. 2008, MNRAS , 389, 1697 Gu, M.,& Cao, X. 2009, MNRAS , 399, 349 Jin, C., Ward, M., & Done, C. 2012, MNRAS , 425, 907 Lu, Y.,& Yu, Q. 2001, MNRAS , 324, 653 McHardy, I.M., Koerding, E., Knigge, C., Uttley, P., & Fender, R.P. 2006, Nature , 444, 730 Nandra, K., George, I.M., Mushotzky, R.F., Turner, T.J., & Yaqoob, T. 1997, ApJ , 476, 70 O'Neill, P.M., Nandra, K., Papadakis, I.E., & Turner, T.J. 2005, MNRAS , 358, 1405 Ponti, G., et al. 2012, A&A , 542, 83 Shemmer, O., Brandt, W. N., Netzer, H., Maiolino, R., & Kaspi, S. 2008, ApJ , 682, 81 Turner, T.J., George, I.M., Nandra, K., & Turcan, D. 1999, ApJ , 524, 667 Vaughan, S., Edelson, R., Warwick, R.S., & Uttley, P. 2003, MNRAS , 345, 1271 Wu, Q. & Gu, M. 2008, ApJ , 682, 212 Zhou, X.L., Zhang, S.N., Wang, D.X., & Zhu, L. 2010, ApJ , 710, 16 Zhou, X.L., & Zhao, Y.H. 2010, ApJ , 720, L206", "pages": [ 2 ] } ]

[ { "title": "New results from LOFAR", "content": "1", "pages": [ 1 ] }, { "title": "Vladislav Kondratiev , on behalf of Ben Stappers 2 and the LOFAR Pulsar Working Group", "content": "1 Jodrell Bank Center for Astrophysics, School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, UK email: [email protected] ASTRON, the Netherlands Institute for Radio Astronomy, Postbus 2, 7990 AA Dwingeloo, The Netherlands email: [email protected] 2 Abstract. The LOw Frequency Array, LOFAR, is a next generation radio telescope with its core in the Netherlands and elements distributed throughout Europe. It has exceptional collecting area and wide bandwidths at frequencies from 10 MHz up to 250 MHz. It is in exactly this frequency range where pulsars are brightest and also where they exhibit rapid changes in their emission profiles. Although LOFAR is still in the commissioning phase it is already collecting data of high quality. I will present highlights from our commissioning observations which will include: unique constraints on the site of pulsar emission, individual pulse studies, observations of millisecond pulsars, using pulsars to constrain the properties of the magneto-ionic medium and pilot pulsars surveys. I will also discuss future science projects and advances in the observing capabilities. Keywords. pulsars: general, telescopes: LOFAR", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "LOFAR is an interferometric array of dipole antenna stations distributed over the Netherlands and a few countries in Europe, that operates at the very low radio frequencies from 10 to 250 MHz. It consists of 24 core stations with the central part of about 300 m across occupied by 6 stations, which is called 'Superterp'. The Superterp provides a collecting area comparable to that of the 100-m Green Bank Telescope with a beam size of ∼ 0 . 5 · at 140 MHz. For the full core with a size of about 2 km across, it is already Arecibo-like collecting area and beam size of ∼ 5 ' . At the moment there are also 9 remote stations and 8 international stations included in the array. The latter are twice as big as Dutch stations and they can operate independently from the rest of the array. They are very powerful telescopes in their own right, each with collecting area comparable to that of the 64-m Parkes radio telescope. LOFAR's frequency range spans from 10 to 250 MHz which is achieved by using two types of dipoles, low-band antennas (LBA) at 10-90 MHz and high-band antennas (HBA) at 110-250 MHz. The HBA dipoles have bow-tie shape and grouped into tiles of 16 dipoles each. There are 48 tiles in Dutch stations and 96 in the international stations. LOFAR operates at the lowest radio frequencies visible from the Earth as at 10 MHz there is a cut-off due to ionospheric reflection. Operating at these low frequencies, LOFAR covers the lowest 4 octaves of the radio window , that makes it a very unique telescope, the only one working at such low radio frequencies with huge instantaneous fractional bandwidth.", "pages": [ 1 ] }, { "title": "2. LOFAR Capabilities", "content": "All of the beam-formed modes, capabilities and many commensal results are described in detail in our first LOFAR paper (Stappers et al. 2011). Here we highlight a few of those, focusing on new commissioning advancements. Multiple station beams. LOFAR is a very flexible, electronically steered aperture array. It is possible to form multiple station beams on the sky and observe several pulsars simultaneously (Hessels et al. 2010; Stappers et al. 2011). This technology will be crucial for the SKA. By trading off bandwidth for beams, we can have as a standard up to 8 widely separated field-of-views (FOVs) and optionally up to 244. Tied-array beams. Within each of the station beams we can also form multiple tiedarray (TA) beams, applying proper phase delays between stations while adding them coherently. At the moment we can form TA beams only using 12 Superterp sub-stations, as they share the single clock. Figure 1 shows the hexagonal pattern of 127 TA beams formed around the pulsar B2217+47. The signal-to-noise ratio of the other TA beams is about one order of magnitude smaller than the beam at the location of the pulsar. With 127 tied-array beams we can cover the whole station beam which is shown by the white circle. The FOV of the station beam is large enough to cover the entire Andromeda galaxy and then we can map it with TA beams in one single observation. Moreover, we can form a few station beams to cover a larger FOV and customize individual narrow TA beams pointing at different targets. The highest number of TA beams formed in the commissioning observations so far was 217 (8 rings plus a central beam). The list and description of all beam-formed modes that are well-tested and currently available to the wider community can be found on the LOFAR web-pages † . There are many possible modes or configurations, and the system is very flexible, to match different science goals. Different data products can be recorded, namely total intensity, full Stokes † http://astron.nl/radio-observatory/observing-capabilities/ depth-technical-information/major-observing-modes/beam-form parameters, or complex voltage data. All data are written in HDF5 ‡ format and we are already working to read it directly with DSPSR ¶ and PRESTO ‖ pulsar software. RFI. The RFI environment is very clean, much better than anticipated. The reasons for this is that a) we are using 12-bit ADCs at the station level, so the dynamic range is high; and b) the dipoles are located very low to the ground and do not pick up a much terrestrial interference. Typically, we flag about 1-2% of data in HBA, and 3-4% in LBA range. Below 30 MHz, however, the data get very contaminated by RFI. Full-core single-clock. We are currently working on expanding the number of stations that use the single clock, from six stations on the Superterp to the whole core of 24 stations within ∼ 1 km radius. The work is ongoing and will be finished by the end of October 2012. This will further increase the raw sensitivity of the system by a factor 4!", "pages": [ 2, 3 ] }, { "title": "3. LOFAR Highlights", "content": "Here I present some of our recent pulsar results; some are published or will be submitted soon. Pulsar timing. LOFAR is very capable of, and we have already started doing, observations of millisecond pulsars (MSPs), as shown in Figure 2. LOFAR MSP profiles show a very high quality at such a low frequency in comparison with previously acquired data using the Puschino BSA phased-array at 103 MHz. These are the highest-quality detections of these pulsars ever made below 200 MHz. For the MSP J0034 -0534 comparison of LOFAR profiles with the 376-MHz WSRT profile shows that a small scattering ‡ http://www.hdfgroup.org/HDF5/ ¶ http://dspsr.sourceforge.net ‖ http://www.cv.nrao.edu/~sransom/presto/ tail becomes visible at lower frequencies (more apparent at 112-124 MHz), along with a slight change in the relative amplitudes of the two profile components. We have already started timing observations of MSPs to test the system and pipeline, and with the fullcore single-clock we will start the real campaign of timing MSPs. LOFAR pulsar timing observations will be very important to get a handle on dispersion measure and pulse profile evolution crucial for high-precision timing at high radio frequencies. Ionospheric Faraday rotation calibration. We started Faraday rotation monitoring to be able to measure accurately pulsar rotation measures (RMs). Figure 3 (Sotomayor-Beltran et al. 2012) presents the observed Faraday depths for three pulsars together with the model predictions based on the total electron content (TEC) maps from the Royal Observatory of Belgium (ROB) and the International Geomagnetic Reference Field (IGRF11). It can be clearly seen that our measurements (circles) match the model (red triangles) very well. We are now getting down to very robust and precise RM measurements of about 0.1 rad m -2 . The observations presented used only 1/6 of the LOFAR's available bandwidth, thus showing a great potential for even better RM measurements using the full bandwidth especially in the LBA band. Dispersion measure vs. Profile variations. Hassall et al. (2012) studied dispersion measure (DM) and profile variations for four pulsars, B0329+54, B0809+74, B1133+16, and B1919+21 using simultaneous wideband observations with the LOFAR LBA and HBA at 40-190 MHz, the Lovell telescope at 1.4 GHz, and Effelsberg radio telescope at 8 GHz. We found that the dispersion law is correct to better than 1 part in 10 5 across our observing band. We also put unique constraints on emission heights for these pulsars using aberration/retardation arguments and show that, for instance, in the case of the pulsar B1133+16 all radio emission comes from a small region less than 59 km in altitude at a height of less than 110 km above the neutron star surface (only 0.2% of the light cylinder). We found no evidence for the super-dispersion delay previously reported at low frequencies (Shitov & Malofeev 1985; Kuzmin 1986) and suggest it could be caused by pulse profile evolution or a wrong fiducial point. We show that profile evolution has a siginificant impact on high-precision pulsar timing and should be taken into account. Low-frequency single-pulse studies. Figure 4 (left) shows the remarkable profile evolution of the pulsar B0809+74 from 62 down to 15 MHz. We performed a thorough single-pulse analysis for the pulsars B0809+74 and B1133+16 that show quite interesting results. For more details about single-pulse studies of the pulsar B0809+74 see by Kondratiev et al. (these proceedings). Pilot pulsar surveys. We have already finished two pilot pulsar surveys with LOFAR. For more details about the survey setup, search pipelines and results, see Coenen et al. (these proceedings). Low-frequency pulsar profiles. Some of the examples of LOFAR pulsar profiles at HBA and LBA bands were already shown in Stappers et al. (2011b). Currently we have already detected more than 110 pulsars in the HBA and 12 pulsars in the LBA. We expect these numbers to significantly increase in the very near future with the full-core single-clock, when the LOFAR raw sensitivity will be quadrupled. We are working on the ultimate LOFAR pulsar profile paper, and in particular on profile alignment with the high-frequency WSRT and Jodrell Bank data. Scattering studies. LOFAR's low-frequency range and huge fractional bandwidth is ideal for pulsar scattering studies. Figure 4 (right) shows the benefits of the LOFAR's huge fractional bandwidth where you can see a remarkable scattering tail from the pulsar B2111+46 changing across the band. This allows us to study precisely the frequency dependency of scattering parameters of this and other pulsars.", "pages": [ 3, 4, 5, 6 ] }, { "title": "4. Future advancements", "content": "LOFAR commissioning work is continuing and there are significant improvements which are coming by the end of Fall 2012, namely:", "pages": [ 6 ] }, { "title": "5. Conclusions", "content": "The results presented here have already proven the exceptional capabilities of the LOFAR and opened up the whole new window of comprehensive studies of pulsars at low frequencies. We have published first, intriguing results, with additional papers in preparation. The forthcoming implementation of the full-core single-clock, with the fourfold increase in sensitivity, will further enhance the LOFAR pulsar capabilities.", "pages": [ 6 ] }, { "title": "Acknowledgements", "content": "The LOFAR facilities in the Netherlands and other countries, under different ownership, are operated through the International LOFAR Telescope foundation (ILT) as an international observatory open to the global astronomical community under a joint scientific policy. In the Netherlands, LOFAR is funded through the BSIK program for interdisciplinary research and improvement of the knowledge infrastructure. Additional funding is provided through the European Regional Development Fund (EFRO) and the innovation program EZ/KOMPAS of the Collaboration of the Northern Provinces (SNN). ASTRON is part of the Netherlands Organization for Scientific Research (NWO).", "pages": [ 6 ] }, { "title": "References", "content": "Hassall, T. E., Stappers, B. W., Hessels, J. W. T., et al. 2012, A&A , 543, 66 Hessels, J. W. T., Stappers, B., Alexov, A., et al. 2010, ISKAF2010 Science Meeting , p. 25, arXiv:1009.1758 Kuzmin, A. D. 1986, Soviet Astronomy Letters , 12, 325 Shitov, Y. P. & Malofeev, V. M. 1985, Soviet Astronomy Letters , 11, 39 Sotomayor-Beltran, C., Sobey, C., et al. 2012, A&A , submitted Stappers, B. W., Hessels, J. W. T., Alexov, A., et al. 2011, A&A , 530, 80 Stappers, B., Hessels, J., Alexov, A., et al. 2011b, AIP Conf. Proc. , 1357, 325", "pages": [ 6 ] } ]

[ { "title": "ABSTRACT", "content": "Proceedings IAU Symposium No. 291, 2012 J. van Leeuwen, ed.", "pages": [ 1 ] }, { "title": "E. F. Keane", "content": "Max Planck Institut fur Radioastronomie, Auf dem Hugel 69, D-53121, Bonn, Germany. email: [email protected]", "pages": [ 1 ] }, { "title": "Abstract.", "content": "Pulsars are potentially the most remarkable physical laboratories we will ever use. Although in many senses they are extremely clean systems there are a large number of instabilities and variabilities seen in the emission and rotation of pulsars. These need to be recognised in order to both fully understand the nature of pulsars, and to enable their use as precision tools for astrophysical investigations. Here I describe these effects, discuss the wide range of timescales involved, and consider the implications for precision pulsar timing. Keywords. pulsars: general", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "A textbook pulsar emits a beam of radio emission from just above its magnetic poles. The mis-alignment of the spin and magnetic axes then results in a light-house effect as the star rotates. Those pulsars whose radio beams cut across the Earth are observed as a string of sharp pulses in the signals detected by radio telescopes. The signal is straight forward to model with a simple slow-down law consisting (usually) of just spin frequency, its derivative and (if applicable) some binary parameters. The regularity of the signal means that these pulses act as the ticks of an extremely precise clock. Furthermore, pulsars are often to be found in extreme environments, which we are able to study by utilising this clock-like nature. The moniker of 'super clocks in space' is well earned. A real-life pulsar deviates from the ideal in a number of ways. This is due to a number of instrumental, propagation and intrinsic effects, many of which are not well understood. In § 2 we discuss the wide range of variable behaviour observed in pulsar signals. In § 3 we consider how pulsars actually work before asking why this is of interest to pulsar astronomers in § 4. Finally, in § 5, we present conclusions and discussions.", "pages": [ 1 ] }, { "title": "2. What do we see?", "content": "The range of variability timescales in pulsars is remarkably wide, spanning all timescales on which it has been possible to measure. The fastest timescales to have been probed are nanoseconds. The voltage signals from radio telescopes are commonly Nyquist-sampled at rates of ∼ 1 GHz, but usually this time resolution is traded for frequency resolution, and furthermore data is integrated in time to increase the signal-to-noise ratio (S/N). However, in the case of the Crab pulsar this is not necessary in order to detect a signal, and Hankins et al. (2003) have observed kJy pulses lasting 2 ns, showing that its well-known ∼ µ s 'giant pulses' are in fact composed of a large number of such 'shots'. These shots appear to be the quanta of pulsar radio emission. They are not resolved - indeed, we might expect this, i.e. an intrinsic timescale of /lessorsimilar 100 ps, given the uncertainty principle and the observations that pulsars emit over bandwidths of several tens of GHz (Maron et al. 2000; Camilo et al. 2007). The actual mechanism is unknown but the brightness temperature of T B = 10 37 K (for the Crab pulses) implies, using the well-known expression for the maximum possible brightness temperature T B , max = 6 × 10 9 N ( γ -1) K, a coherence factor of N ≈ 10 27 /γ . Clearly the mechanism is coherent, most likely involving particles emitting in bunches, a plasma instability or some kind of maser, but despite much effort (Ginzberg & Zheleznyakov 1970; Asseo et al. 1990; Lyutikov et al. 1999; Melrose 2004) the details are not known. The duration of a time sample in most pulsar observations is usually /greatermuch 100 ps so that a large number of shots are incoherently added within each time sample. The Poisson distribution of the shots then approaches that of a Gaussian, and it is common to model the pulsar signal as amplitude-modulated noise (Rickett 1975). This model is insufficient however, as single pulse studies show non-Gaussian variations on µ s -ms scales, e.g. the 'giant micropulses' seen in Vela by Johnston et al. (2001), and we see dramatic variations from one pulse period to the next, on ms -s scales, e.g. we see changes in intensity (by factors of /greaterorsimilar 10 3 ), phase, pulse shape and the number of components. Extremely organised behaviour is seen on second to minute timescales, in the form of sub-pulse drifting. Here, a 'Joy Division plot' reveals that the pulses drift periodically (both earlier and later) in pulse phase in regular 'bands' as a function of time with typical repetition periods of tens of spin periods. A standard explanation for this behaviour has been the 'carousel model' (Ruderman & Sutherland 1975) where disparate emission spots above the stellar surface are induced to rotate by E × B drift. Lately however it has been shown that this model does not explain the drifting seen in PSR B0809+74 (Hassall et al., in prep.). In a study of 187 pulsars, using the Westerbork Synthesis Radio Telescope, Weltevrede et al. (2006) showed that at least one third exhibited drifting. On timescales of seconds to minutes, and even up to hours we see further organised behaviour in the form of 'moding' - the changing of the pulse profile to one of a small number of different profiles. If there is no detected radio emission from one of these 'modes' the phenomenon is commonly termed 'nulling'. A quantitative analysis of the pulse amplitude distributions of a large number of pulsars has recently been performed by Burke-Spolaor et al. (2012). This work looked at the single pulse statistics of 315 pulsars with detectable single pulses in the High Time Resolution Universe survey. The authors classify the pulse amplitude distributions as either Gaussian (7 sources, 2%), log-normal (84 sources, 27%), multi-peaked (18 sources, 6%) or unimodal (24 sources, 8%). Unfortunately the majority (182 sources, 58%) did not fit within these classifications. While we might suggest testing for more complex distributions for the unclassified sources, this is not possible due to a paucity of detected pulses. Of the unclassified 182 sources, only 92 had more than 20 detected pulses during the 9-minute survey pointings, and a single pulse was all that was detected for 22 of the sources. Moding is also observed on timescales of hours to weeks, or even months. The first realisation of this came when Kramer et al. (2006) discovered that PSR B1931+24 is detectable as a radio pulsar only for periods of ∼ 5 -10 days before 'turning off' and remaining undetectable for ∼ 25 -35 days in a quasi-periodic fashion. Crucially this moding is accompanied by a ∼ 50% change in the spin-down rate, with ˙ ν hi / ˙ ν lo = 1 . 5. Since then two more 'intermittent pulsars' have been reported - PSRs J1841 -0500 (Camilo et al. 2012) and J1832+0029 (Lorimer et al. 2012). These sources have 'on' and 'off' timescales ∼ 20 -30 times longer than B1931+24 and spin-down rate ratios of 2 . 5 and 1 . 8 respectively. Lyne et al. (2010) presented results of several decades of Lovell Telescope observations of 17 pulsars where correlated quasi-periodic changes in ˙ ν and pulse profile were clearly observed. The changes in spin-down rate ranged from 0 . 3%to 13%. More examples of such behaviour continue to be identified (see e.g. Karastergiou, these proceedings). We stress that it is not simply switching between two states that is seen in moding pulsars (see Fig. 5 from Burke-Spolaor & Bailes (2010) or Fig. 1 from Esamdin et al. (2012) for some excellent examples). Furthermore we note that such moding is seen on all timescales ranging from several years down to one rotation period (Keane & McLaughlin 2011). On the shorter time-scales changes in ˙ ν cannot be measured - Young et al. (2012) point out that for modes persistent for less than a day spin-down rate switching of a few percent would never be detectable. The associated profile changes are also often quite subtle and obviously cannot be discerned from pulse-to-pulse variations when the persistence of the mode is less than the required duration to surpass the stable profile threshold (see § 4). It would seem that switching between a number of stable states, often with some quasi-periodicity, is a generic feature of pulsars . Of course we must not forget that the emitted broadband signal from a pulsar is subject to the transfer function of the interstellar medium (which is also time variable on a number of scales) and that of the telescope-receiver system itself (which will also have a number of systematic contributions). There is an equally long list of these effects which must be accounted for in modelling the pulsar signal but which I will not elaborate upon here (but see e.g. Cordes & Shannon 2010). Table 1 gives an incomplete list of timescales on which pulsars are known, or expected, to be variable.", "pages": [ 1, 2, 3 ] }, { "title": "3. How do they work?", "content": "Assuming that the propagation and instrumental effects can be understood (whether or not they can be removed) there are still a wide range of transient behaviours seen in pulsars. This leads us to a big question: How do we get erratic radio emission from a PSR with a particular timescale, and periodic switching? For force-free magnetospheres (see below) it has been shown that a number of stable solutions exist with the closed magnetosphere not necessarily extending to the light cylinder radius (Contopoulos et al. 1999; Spitkovsky 2006). It has further been shown that perturbing these solutions can result in a rapid switch from one magnetospheric configuration to another (Contopoulos 2005). However, these perturbations are put in 'by hand' and the underlying reason for the switching remains unknown. Furthermore, why this would occur with a periodicity is unknown. That the switching is quasi-periodic, rather /negationslash than strictly periodic, must also be explained. Recently Seymour & Lorimer (2012) have suggested that the quasi-periodicity resembles that seen on 'the route to chaos' and detect chaotic behaviour in PSR B1828 -11, one of the Lyne et al. (2010) sample. The timescales for the erratic behaviour are wide (see Table 1), so much so that it is difficult to see what the decisive variables are. If the moding is simply a result of the magnetospheric switching (Timokhin 2010) the timescales for both phenomena are obviously one and the same. This raises the question of whether pulsars with large pulse-to-pulse modulation on much faster timescales than the intermittent pulsars are changing magnetospheric configuration constantly. This would suggest a picture of highly unstable and frequent fast changes on the scale of the entire magnetosphere. If this is not what is occurring in these cases it is unclear on which timescales this ceases, as there seems to be a continuum of moding/switching timescales observed (Keane 2010a). We are forced to abandon our big question entirely in favour of a more tractable one: What does a PSR magnetosphere even look like? There are two approaches to answering this question - the first is to solve Maxwell's equations for a rapidly-spinning strongly-magnetic highly-conductive ball; the second is to try to determine the geometry of the system from observations of the polarisation characteristics of pulsar emission. Both of these approaches should result in the same answer, but both are fraught with many difficulties. Here I briefly describe the first approach, but refer the reader to the works of Radhakrishnan, Cooke, Kramer, Karastergiou, Johnston, Weltevrede, Rankin, Wright and Noutsos for information on the geometrical approach. When calculating Maxwell's equations in the vicinity of the neutron star it is found that there are trapping surfaces for charges of opposite sign above the poles, and in the equatorial plane. Particles get ripped from the stellar surface and are simply trapped in these 'electrospheres' with no pulsar-like behaviour (see e.g. Fig. 2.4, Keane 2010b). One then would assume that either the initial conditions do not represent reality, i.e. in the violent supernova explosion wherein the neutron star was born there was abundant plasma provided from the offset so that the electrosphere scenario never arises, or, that the electrosphere solution is in fact unstable (e.g. to the diochotron instability, see Spitkovsky, these proceedings) and breaks down after some time. Regardless of the reason some authors have pressed on assuming 'a sufficiently large charge density whose origin we do not question' (Contopoulos et al. 1999) and solved 'the pulsar equation' (Michel 1973) for the first time. The results of this work show current flows in the magnetosphere coincident with the 'gap regions' for emission derived by the geometric approach so that it seems that progress is being made towards understanding pulsar magnetospheres. Table 2 summarises some of the knowns and unknowns.", "pages": [ 3, 4 ] }, { "title": "4. Who cares?", "content": "'I don't care, I just want to do timing.' Anonymous. Some astronomers may not be very concerned with how pulsars actually work, and only interested in pulsars for their use as clocks, e.g. to use in pulsar timing arrays (PTAs). In this case the only question that matters is whether or not pulsar profiles are stable for typical PTA observations. Fortunately this can be measured, and one such method involves calculating ρ , the cross-correlation coefficient of the observed pulse profile with a template profile. If 1 -ρ ∝ N -1 , where N is the number of periods folded into the observed profile, then the profile is stable (Liu et al. 2012). Longer integrations improve the profile's S/N only and not its stability. While the value of N where the exponent transitions to -1 denotes the stability timescale, different exponents reveal other timescales at work, e.g. nulling/moding timescales if present (Keane 2010b). Although the stability of pulsar profiles is implicitly assumed † in pulsar timing, it is not clear whether this has been systematically confirmed for all PTA pulsars. The received wisdom is that 10 4 periods gives you a stable profile but Liu et al. (2012) found this to be dependent upon the pulsar with values of up to 10 5 periods required in some cases. It is important to note that a high S/N does not imply stability (based solely on S/N we can time pulsars using their single pulses, but this is not precision pulsar timing, see Keane et al. (2011) for details). If one used pulsar profiles which were not stable then there would be no justification for expecting a good fit to the timing model, with χ 2 red = 1. Oddly enough there is a practice (which is admittedly dying out) to assume that the best fit model, i.e. the one with the lowest χ 2 red value, is the correct model, and to then scale the errors so as to make χ 2 red = 1. The errors in this case are scaled by an 'EFAC' quantity. This is very bad practice for several reasons (see § 3.2.1 of Andrae (2010) for more details), e.g. it assumes that: the error distribution is Gaussian; the model is linear in all of its parameters; the model used is correct (also completely negating the point of using the chi-squared test ). Pulse jitter is another contribution to errors in pulse time-of-arrival measurements which is usually ignored. Jitter is only evident in pulsar profiles when the S/N of single pulses are /greaterorsimilar 1. Currently, for PTA sources, this is only relevant for PSR J0437 -4715. For SKA-era sensitivity this must be accounted for in all PTA pulsars, but fortunately this is possible, as has been demonstrated for J0437 -4715 (Liu et al. 2012).", "pages": [ 5 ] }, { "title": "5. Conclusions & Discussion", "content": "Pulsar emission and rotation is variable on a wide range of timescales. It is vital to gain a full understanding of these things in order to (a) understand pulsars; and (b) perform precision pulsar timing. The author's bias suggests to him that it may be difficult to achieve the latter with first making significant inroads into achieving the former. For example the observed behaviour (described in § 2) suggest a number of questions which the pulsar timing community should be thinking about: Is there any reason why there would not be (perhaps periodic or quasi-periodic) spin-down rate switching occurring in many/all pulsars? Is there any reason why there would not be (perhaps periodic or quasi-periodic) spin-down rate switching in many/all millisecond pulsars? Are there other (perhaps deterministic) timing instabilities yet to be identified? The planned upcoming studies of large pulsar timing databases (S. Johnston, private communication) will no † It is assumed that the observed profile is a shifted scaled version of a smooth (sometimes analytic) template with additive noise. doubt shed valuable light on what the answers to these questions are, and bring us a few steps closer to understanding those super clocks in space.", "pages": [ 5, 6 ] }, { "title": "Acknowledgements", "content": "EK would like to thank the SOC for the invitation to speak at the General Assembly, and the LOC for their hospitality throughout the conference. EK is grateful to Lijing Shao for pointing out a very helpful reference paper, and to Mark Purver for valuable comments on this text.", "pages": [ 6 ] }, { "title": "References", "content": "Andrae, R. 2010, 'Error estimation in astronomy: A guide', astro-ph/1009.2755. Asseo, E., Pelletier, G. & Sol, H. 1990, MNRAS , 247, 529. Burke-Spolaor, S. & Bailes, M. 2010, MNRAS , 402, 855. Burke-Spolaor, S. et al. 2012 MNRAS , 423, 1351. 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[ { "title": "ABSTRACT", "content": "A.G. Kosovichev, E.M. de Gouveia Dal Pino & Y. Yan, eds.", "pages": [ 1 ] }, { "title": "Axel Brandenburg 1 , 2", "content": "1 Department of Astronomy, Stockholm University, SE-10691 Stockholm, Sweden Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden 2 Abstract. An update is given on the current status of solar and stellar dynamos. At present, it is still unclear why stellar cycle frequencies increase with rotation frequency in such a way that their ratio increases with stellar activity. The small-scale dynamo is expected to operate in spite of a small value of the magnetic Prandtl number in stars. Whether or not the global magnetic activity in stars is a shallow or deeply rooted phenomenon is another open question. Progress in demonstrating the presence and importance of magnetic helicity fluxes in dynamos is briefly reviewed, and finally the role of nonlocality is emphasized in modeling stellar dynamos using the mean-field approach. On the other hand, direct numerical simulations have now come to the point where the models show solar-like equatorward migration that can be compared with observations and that need to be understood theoretically. Keywords. MHD - turbulence - Sun: magnetic fields", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The objective of IAU Symposium 294 is to provide an update since IAU Symposium 157 on 'The Cosmic Dynamo' in Potsdam, 1992. The title of the present talk reflects the common thinking at that time that nonlinearity and chaos tend to come together. This is highlighted by the realization that a simple mean-field dynamo model for poloidal and toroidal fields needs to be supplemented by a third equation to produce chaos, and the idea that this third equation is the equation of magnetic helicity conservation. These developments have contributed to the unfortunate perception of mean-field dynamo theory being just as a toy rather than a quantitatively predictive theory. With the advent of numerous computer simulations of hydromagnetic turbulence exhibiting large-scale dynamo action, a new field of computer astrophysics has emerged where the objective is to understand simulated dynamos, where one has a chance to resolve all time and length scales. This approach has helped making mean-field theory quantitatively reliable and predictable. Various predictions have emerged and have been tested. Firstly, dynamos must transport magnetic helicity to escape catastrophic quenching. They do this through coronal mass ejections and through turbulent exchange across the equator. The resulting field is bi-helical, with opposite signs of magnetic helicity at large and small length scales. The signs depend on the sign of kinetic helicity and on the relative importance of turbulent diffusion. This has meanwhile been confirmed observationally using Ulysses data. Secondly, mean-field theory also predicts the formation of local magnetic flux concentrations as a result of strong density stratification. Simulations have now confirmed this remarkable theoretical prediction. This has opened the floor for suggestions that active regions and sunspots might be shallow phenomena operating near the surface at some 40 Mm depth. There are also several observations that do not yet have a satisfactory explanation. One of them concerns the dependence of the observed cycle period of late-type stars on their activity and another one is the equatorward migration of toroidal magnetic flux belts during the solar cycle: is it caused by meridional circulation, the migratory properties of the dynamo wave, or something else that we do not know about yet?", "pages": [ 1, 2 ] }, { "title": "2. Dynamo regimes", "content": "The graph of the ratio of stellar cycle to rotational frequencies versus magnetic activity or stellar age shows two branches (Brandenburg et al. 1998, hereafter BST); see Figure 1. These branches correspond to active (A) and inactive (I) stars and are separated by what is known as the Vaughan-Preston gap. In addition, there is a third branch of superactive (S) stars (Saar & Brandenburg 1999). How can we understand the origin of these branches? The BST diagram is not just a way of representing the non-dimensional cycle frequency in a two-dimensional diagram, it might represent some deeper physics. In a globally quenched α Ω-type dynamo, i.e., a model where only the smallest wavenumbers have a significant contribution (as found in simulations; see Brandenburg et al. 2008; Rheinhardt & Brandenburg 2012), the cycle frequency ω cyc is proportional to √ α Ω ' , where α represents the α effect responsible for reproducing the large-scale magnetic field (Moffatt 1978; Krause & Radler 1980) and Ω ' is the radial differential rotation. Both are functions of the angular velocity Ω (which has been subsumed into a factor Ω q with a poorly constrained exponent q ; see BST). The essential point is that ω cyc can be regarded as a proxy of α . Furthermore, the angular velocity normalized by the turnover time τ , which gives the inverse Rossby number Ro -1 = 2Ω τ , can be regarded as a measure of the non-dimensional magnetic field strength, B/B eq , where B eq is the equipartition field strength. This is a relationship that is independent of the existence of different branches, i.e., inactive and active stars lie on the same curve (BST). We can therefore imagine the BST diagram being really a representation of α ( B ), and that the two rising branches describe therefore an antiquenching of α with B/B eq . Indications of this, and a corresponding anti-quenching of the turbulent magnetic diffusivity, have been found in simulations of magneto-buoyancy (Chatterjee et al. 2011). On the other hand, for faster rotation there is a third branch of super-active stars (Saar & Brandenburg 1999), for which our proxy of α declines with that of B . These considerations are still as exciting today as they were back in 1998, but now we have realistic global simulations of convection that reflect a qualitative leap from earlier work in that we now find for the first time cyclic large-scale dynamo action with equatorward migrating activity belts (Kapyla et al. 2012). One needs to check, however, whether perhaps all of the dynamo solutions obtained so far are representative of the superactive branch, and that the physics behind the active and inactive ones remains still to be discovered. To make progress in understanding the different modes of cyclic stellar activity, one also needs to analyze why those models produce long cycle periods and equatorward migration, both of which are also seen in the Sun, but are theoretically not understood; is it related to the possible dominance of the magnetic α effect (Pouquet et al. 1976) (which is inversely proportional to the density and therefore important near the surface), to the tensorial structure of α ( α ij ) and turbulent diffusion ( η ijk ), to meridional circulation, or to subtleties in the differential rotation? Such understanding should be accomplished by deriving and solving suitable mean-field models that reproduce the behavior seen in DNS and Large Eddy Simulations (LES) of the Sun. It is unlikely that differences in the cycle period are the only criterion distinguishing stars on the two branches of the evolutionary BST diagram. Surface magnetic field structures as well as their spatio-temporal correlations are now becoming accessible to detailed scrutiny. Quantifying the nature of magnetic fields using observed correlations among the Stokes parameters might help to distinguish different types of behaviors and to associate them with different branches in the BST diagram, which may reflect different underlying dynamo modes. Progress can be made by considering turbulent dynamo simulations at different rotation rates, as has recently been done by Kapyla et al. (2013). We return to this issue in our conclusions and turn attention to recent simulation results that concern the magnetic surface activity of the Sun, such as small-scale or local dynamos and the evidence of helical magnetic fields from the global dynamo.", "pages": [ 2, 3 ] }, { "title": "3. Small-scale and local dynamos", "content": "In recent years the action of two separate dynamos in the Sun has become popular; one that governs the 11 year cycle and one that produces the small-scale field of the quiet Sun (Cattaneo 1999; Cattaneo et al. 2003; Vogler & Schussler 2007). This idea is supported by the fact that the observed small-scale field of the Sun is essentially uncorrelated with that of the 11 year cycle (Lites 2002; Ishikawa & Tsuneta 2009; Danilovic et al. 2010; Auri'ere et al. 2010; Stenflo 2012). A potential problem is the fact that the critical magnetic Reynolds number R m , crit grows larger as one decreases the value of the magnetic Prandtl number, Pr M = ν/η , to more realistic values (Schekochihin et al. 2005). Early work of Rogachevskii & Kleeorin (1997) did already predict an increased value of R m , crit in the limit of small values of Pr M . Boldyrev and Cattaneo (2004) argue that the reason for an increased value of R m , crit is connected with the 'roughness' of the velocity field. Iskakov et al. (2007) found that R m , crit has a local maximum at Pr M = 0 . 1, and that it decreases again as Pr M is decreased further. The reason for this is that near Pr M = 0 . 1 the resistive wavenumber is about 10 times smaller than the viscous one and thus right within the 'bottleneck' where the spectrum is even shallower than in the rest of the inertial range, with a local scaling exponent that corresponds to turbulence that is in this regime rougher still, explaining thus the apparent divergence of R m , crit . In the nonlinear regime the magnetic field affects the flow in such a way that the bottleneck effect tends to be suppressed, so the divergence in the roughness disappears and there is a smooth dependence of the saturation field strength on the value of Pr M ; see Brandenburg (2011) for details. In Figure 2, we show spectra compensated with /epsilon1 -2 / 3 k 5 / 3 . For Pr M = 0 . 02 and 0.01, the kinetic energy spectra show a clear bottleneck effect, i.e., there is a weak uprise of the compensated spectra toward the dissipative subrange (Falkovich 1994; Kaneda et al. 2003; Dobler et al. 2003). The compensated magnetic energy spectra peak around k = 20 k 1 , where k 1 = 2 π/L is the smallest wavenumber in a domain of size L . Both toward larger and smaller values of k there is no clear power-law behavior. The slopes of the k -11 / 3 spectrum of Golitsyn (1960) and Moffatt (1961) and the scale-invariant k -1 spectrum (Ruzmaikin & Shukurov 1982; Kleeorin & Rogachevskii 1994; Kleeorin et al. 1996) are shown for comparison. We recall that we have used here the strategy of generating low-Pr M solutions by gradually decreasing ν , and hence increasing the value of Re. As in the case of helical dynamos (Brandenburg 2009), the fact that a turbulent self-consistently generated magnetic field is present helps reaching these low-Pr M solutions. However, the presence of the magnetic field also modifies the kinetic energy spectrum and makes it decline slightly more steeply than in the absence of a magnetic field; see Figure 2. This suggests that the velocity field would be less rough than in the corresponding case without magnetic fields. Following the reasoning of Boldyrev and Cattaneo (2004), this should make the dynamo more easily excited than in the kinematic case with an infinitesimally weak magnetic field. In other words, there is the possibility of a subcritical bifurcation where the dynamo requires a significantly larger value of Pr M to bifurcate from the trivial B = 0 solution than the value needed to sustain a saturated dynamo.", "pages": [ 3, 4 ] }, { "title": "4. Solar surface activity", "content": "The theory of stellar structure explains that the outer 200 Mm of the Sun's radius are convectively unstable, resulting in fully developed turbulent convection. Numerical simulations of turbulent flows predict that part of the convective kinetic energy is converted to magnetic energy through dynamo action. If we did not have observations, would we have predicted that the Sun's magnetic field would choose to manifest itself in the form of spots? The answer might well be yes, but perhaps not for the reasons offered in text books. Standard thinking focuses on the tachocline, which is a strong shear layer at the bottom of the convection zone. Strong shear can produce a strong magnetic field in the form of thin flux tubes (Cline et al. 2003; Guerrero & Kapyla 2011). The magnetic pres-su in these tubes expels gas, and so, being less dense than their surrounding, they rise. If a segment of a tube pierces the surface of the Sun, the footpoints of the resulting arch appear as sunspot pairs of opposite polarity (as the magnetic field in the tube has a definite direction; see Figure 3). Simulations, on the other hand, predict turbulent magnetic fields with a diffuse large-scale component throughout the convection zone (Brown et al. 2010; Kapyla et al. 2010; Ghizaru et al. 2010), and this scenario can also reproduce the observed bipolar spots at the surface (Brandenburg 2005). It might become possible to use local helioseismology to distinguish between the scenarios sketched in the left and right hand panels of Figure 3. Unlike global helioseismology, local helioseismology is an advanced technique that uses correlations of measured Doppler shifts at the solar surface for different time intervals corresponding to sound travel times for rays down to a given depth, as is seen in the left-hand panel of Figure 4. This technique can provide detailed information on the structure of magnetic fields (Ilonidis et al. 2011) nearby and even inside a sunspot (Kosovichev 2009). In a particular case, some type of local activity has been detected at a depth of ∼ 60 Mm, which corresponds to 1/3 of the depth of the convective zone. If this was caused by a rising flux tube, as sketched in Figure 4, one would have expected a wider elongated feature. On the other hand, the observed activity might correspond to signatures of magnetic structures formed by the so-called negative effective magnetic pressure instability (NEMPI, Brandenburg et al. 2011a). Coronal mass ejections play a major role in shedding small-scale magnetic helicity from the dynamo to alleviate an otherwise catastrophic quenching of the dynamo (Blackman & Brandenburg 2003). Meanwhile, models have made contact with unexpected phenomena taking place in the solar wind. A striking example is the sign reversal of small-scale magnetic helicity away from the Sun. This surprising result was first obtained by analyzing data from the Ulysses spacecraft (Brandenburg et al. 2011b), see Figure 5, but the interpretation was greatly aided by similar results from simulations of Warnecke et al. (2011). It now seems that the reason for this is an essentially turbulent-diffusive transport down the local gradient of magnetic magnetic helicity density - even in the wind (Warnecke et al. 2012). While this work has focussed on parameter studies exploring the conditions for plasmoid ejections from helically forced turbulence as well as rotating convection, the physical realism of the model remained poor. The density contrast between dynamo region and corona is much bigger in reality, see for example Pinto et al. (2011). Significant improvements are possible with only modest increase of numerical resolution, as has been shown by Bingert & Peter (2011) using Pencil Code simulations with a realistic setup. One may envisage important follow-up diagnostics by producing visualizations of helical magnetic fields in the corona (see the left-hand panel of Figure 6) and to compute cases in which the field is generated either self-consistently by a dynamo beneath the surface, as in Warnecke et al. (2011, 2013), or the field is injected as a twisted flux tube in a deeper layer and let to emerge at the surface. Simulations without shear have successfully produced twisted magnetic field lines from a self-consistently generated bipolar sheet (see middle panel of Figure 6), but this has not yet been attempted in simulations where more localized bipolar regions are produced. An example of the formation of such regions has been seen in dynamo simulations with strong shear (Brandenburg 2005) leading to the occasional formation of bipolar regions when opposite polarities can be drawn apart by latitudinal differential rotation; see the right-hand panel of Figure 6. Observational evidence for such a process has been provided by Kosovichev & Stenflo (2008). Recent work using a simple model with a galactic wind has shown, for the first time, that shedding magnetic helicity by fluxes may indeed be possible. We recall that the evolution equation for the mean magnetic helicity density of fluctuating magnetic fields, h f = a · b , is where we allow two contributions to the flux of magnetic helicity from the fluctuating field F f : an advective flux due to the wind, F f w = h f U w , and a turbulent-diffusive flux due to turbulence, modelled here by a Fickian diffusion term down the gradient of h f , i.e., F f diff = -κ h ∇ h f . Here, E = u × b is the electromotive force of the fluctuating field. The scaling of the terms on the right-hand side with Re M has been considered before by Mitra et al. (2010) and Hubbard & Brandenburg (2010). In Figure 7 we show the basic result of Del Sordo et al. (2013). As it turns out, below Re M = 100 the 2 ηµ 0 j · b term dominates over ∇ · F f , but because of the different scalings (slopes being -1 and -1 / 2, respectively), the ∇ · F f term is expected to become dominant for larger values of Re M (about 3000). Surprisingly, however, ∇ · F f becomes approximately constant for Re M > ∼ 100 and 2 ηµ 0 j · b shows now a shallower scaling (slope -1 / 2). This means that that the two curves would still cross at a similar value. Our data suggest, however, that ∇ · F f may even rise slightly, so the crossing point is now closer to Re M = 1000. We have mentioned above some surprising behavior that has been noticed in connection with the small-scale magnetic helicity flux in the solar wind. Naively, if negative magnetic helicity from small-scale fields is ejected from the northern hemisphere, one would expect to find negative magnetic helicity at small scales anywhere in the exterior. However, if a significant part of this wind is caused by a diffusive magnetic helicity flux, this assumption might be wrong and the sign changes such that the small-scale magnetic helicity becomes positive some distance away from the dynamo regime. In Figure 8 we reproduce in graphical form the explanation offered by Warnecke et al. (2012).", "pages": [ 4, 5, 6, 7, 8 ] }, { "title": "5. Conclusions and further remarks", "content": "In this review we have put emphasis on the appearance of magnetic helicity at and above the surface of the dynamo. Other important diagnostics may come from local helioseismology to distinguish between shallow and deeply rooted dynamo scenarios. As mentioned above, simulations by various groups all produce distributed dynamo action where the magnetic field is present throughout the convection zone. A major breakthrough has been achieved through the recent finding of equatorward migration of magnetic activity belts in the course of the cycle (Kapyla et al. 2012); see Figure 9. These results are robust and have now been reproduced in extended simulations that include a simplified model of an outer corona (Warnecke et al. 2013). Interestingly, the convection simulations of other groups produce cycles only at rotation speeds that exceed those of the present Sun by a factor of 3-5 (Brown et al. 2011); see also Racine et al. (2011) for recent cyclic models at solar rotation speeds. Both lower and higher rotation speeds give, for example, different directions of the dynamo wave (Kapyla et al. 2012). Different rotation speeds correspond to different stellar ages (from 0.5 to 8 gigayears for rotation periods from 10 to 40 days), because magnetically active stars all have a wind and are subject to magnetic braking (Skumanich 1972). In addition, all simulations are subject to systematic 'errors' in that they poorly represent the small scales and emulate in that way an effective turbulent viscosity and magnetic diffusivity that is larger than in reality; see the corresponding discussion in Sect. 4.3.2 of Brandenburg et al. (2012) in another context. In future simulations, it will therefore be essential to explore the range of possibilities by including stellar age as an additional dimension of the parameter space. In future work it will be important to understand the results of simulations using simpler mean-field models. A potential problem is the fact that the turbulent eddies often have sizes comparable with the size of the domain. In that case, scale separation in space or time is poor and the mean-field α effect and turbulent diffusivity have to be replaced by integral kernels by which the dependence of the mean electromotive force on the mean magnetic field becomes nonlocal. In Figure 10 we show results for the Fourier transformed integral kernels ˜ α ( k ) and ˜ η t ( k ). Both ˜ α and ˜ η t decrease monotonously with increasing | k | . The two values of ˜ α for a given k/k f but different k f /k 1 and Re M are always very close together. The functions ˜ α ( k ) and ˜ η t ( k ) are well represented by Lorentzian fits of the form In Figure 10 we show the kernels ˆ α ( ζ ) and ˆ η t ( ζ ) obtained numerically. Observationally, similar results have been obtained by Abramenko et al. (2011). The results presented in Figure 10 show no noticeable dependencies on Re M . Although we have not performed any systematic survey in Re M , we interpret this as an extension of the above-mentioned results of Sur et al. (2008) for α and η t to the integral kernels ˆ α and ˆ η t . Of course, this should also be checked with higher values of Re M . Particularly interesting would be a confirmation of different widths for the profiles of ˆ α and ˆ η t . The challenge in solar and stellar dynamo theory is nowadays not just the understanding of the nature and origin of magnetic fields in observed stars and in the Sun, but also the understanding of simulated dynamos. Here we have a clear chance in achieving oneto-one agreement because the magnetic Reynolds numbers are still manageable. Only when such agreement has been achieved will we be able to address in a meaningful way solar and stellar dynamos.", "pages": [ 8, 9, 10, 11 ] }, { "title": "Acknowledgements", "content": "This research was supported in part by the European Research Council under the AstroDyn Research Project 227952 and the Swedish Research Council under the grants 621-2011-5076 and 2012-5797. The computations have been carried out at the National Supercomputer Centre in Ume˚a and at the Center for Parallel Computers at the Royal Institute of Technology in Sweden.", "pages": [ 11 ] }, { "title": "References", "content": "Phys. Rev. Lett., 98, 208501 398", "pages": [ 11, 12 ] } ]

[ { "title": "Sperello di Serego Alighieri 1 & members of the HeViCS team", "content": "1 INAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy email: [email protected] Abstract. We have searched for dust in an optical sample of 910 Early-Type Galaxies (ETG) in the Virgo cluster (447 of which are optically complete at m pg /lessorequalslant 18 . 0), extending also to the dwarf ETG, using Herschel images at 100, 160, 250, 350 and 500 µ m. Dust was found in 52 ETG (46 are in the optically complete sample), including M87 and another 3 ETG with strong synchrotron emisssion. Dust is detected in 17% of ellipticals, 41% of lenticulars, and in about 4% of dwarf ETG. The dust-to-stars mass ratio increases with decreasing optical luminosity, and for some dwarf ETG reaches values similar to those of the dusty late-type galaxies. Slowly rotating ETG are more likely to contain dust than fast rotating ones. Only 8 ETG have both dust and HI, while 39 have only dust and 8 have only HI, surprisingly showing that only rarely dust and HI survive together. ETG with dust appear to be concentrated in the densest regions of the cluster, while those with HI tend to be at the periphery. ETG with an X-ray active SMBH are more likely to have dust and vice versa the dusty ETG are more likely to have an active SMBH. Keywords. galaxies: ISM; galaxies: elliptical and lenticular, cD; ISM: dust, extinction", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Most of the baryons in a cluster of galaxies are in the hot intracluster medium, some of which is associated with the most massive galaxies. The hot gas interacts in many ways with the cold phases of the interstellar medium of these galaxies, and these interactions have a fundamental effect on the evolution of the galaxies themselves. In order to understand the evolution of massive galaxies (the topic of this Symposium), particularly in clusters, it is therefore important to study also the coldest phases of their interstellar medium. The most massive galaxies are Early-Type Galaxies (ETG), and, since they may have formed by merging or accretion of smaller ones, it is useful to include in the study also the dwarf ETG. In di Serego Alighieri et al. (2007) we have systematically studied the neutral atomic gas (HI) content of a large and complete sample of ETG in the Virgo cluster, using the Arecibo Legacy Fast ALFA 21-cm survey (Giovanelli et al. 2007). HI is found in very few massive ETG, where the cold gas could have a recent external origin, and in a few peculiar dwarf galaxies at the edge of the ETG classification. The Herschel Space Observatory (Pilbratt et al. 2010) is giving us the opportunity to study the dust content of the same sample of Virgo ETG. We have done so within the Herschel Virgo Cluster Survey (HeViCS, Davies et al. 2010), an open-time Key Programme for a confusion-limited imaging survey of a large fraction of the Virgo cluster in 5 bands: at 250, 350 and 500 µ m with SPIRE (Griffin et al. 2010) and at 100 and 160 µ m with PACS (Poglitsch et al. 2010). We describe here the main results of this work. A more detailed and complete account has been submitted to A&A (hereafter dSA12).", "pages": [ 1 ] }, { "title": "2. Input sample, analysis and results", "content": "We start with a sample of Virgo ETG selected in the optical from the GOLDMine compilation (Gavazzi et al. 2003), mostly based on the Virgo Cluster Catalogue (VCC, Binggeli et al. 1985), to be ETG (i.e. equal to or earlier than S0a) and excluding those with v hel < 3000 km/s. With these selection criteria 925 ETG are within the 4 HeViCS fields and constitute our input sample. Out of these, 447 are brighter than the VCC completeness limit ( m pg /lessorequalslant 18 . 0) and form the optically complete part of our input sample. Out of the input sample, 287 ETG have inaccurate positions in the literature, based only on the original work of Binggeli et al. (1985), insufficient to find reliable counterparts in the HeViCS images. Using r-band SDSS images, we then remeasured the position for these ETG, except for 15 (all with m pg > 19 . 0), for which the identification is unsure. We have looked for a reliable far-IR counterpart in the HeViCS 250 µ m mosaic image for all the 910 Virgo ETG with accurate coordinates, and found one for 52 of them at S/N > 6. For these sources we measured the flux in each of the 5 HeViCS bands using an aperture of 30 arcsec radius, large enough to contain the PSF also at 500 µ m. For 12 sources, which have far-IR emission exceeding this aperture, we used larger apertures, up to 78 arcsec radius. 46 far-IR counterparts have F 250 /greaterorequalslant 25 . 4 mJy, which is our completeness limit at 250 µ m. We detect dust above the synchrotron component in the 4 ETG with radio emission, including M87. Given the large number of background sources present in the 250 µ m images, following the methods of Smith et al. (2011), we estimate that on average 1.5 ETG (most likely dwarfs), out of our input sample of 910, have a far-IR counterpart which is a background source. We have used the distance given in GOLDMine, which distinguishes various components in the Virgo cluster at 17, 23 and 32 Mpc (Gavazzi et al. 1999). Dust appears to be very concentrated, much more than stars. The only ETG with a considerable amout of off-nuclear dust is M86, where it appears to be mostly in a filament at 2 arcmin (about 10 kpc in projection) to the South-East (Gomez et al. 2010). Dust masses and temperatures have been estimated for the 52 ETG with a far-IR counterpart by fitting a modified black-body to the measured far-IR fluxes, assuming a spectral index β = 2 and a MW emissivity, and taking into account colour and aperture corrections. We have also estimated stellar masses with the methods of Zibetti et al. (2009), using the available optical/IR broad-band photometry. The dust temperature ranges between 15 and 30 K, and correlates with the stellar mass and with the B-band average surface brigthness within the effective radius (dSA12). The latter correlation suggests that most of the dust heating is due to radiation produced by stellar sources.", "pages": [ 2 ] }, { "title": "3. Discussion", "content": "Dust detection rates for the complete samples (i.e. 43 far-IR counterparts with F 250 /greaterorequalslant 25 . 4 mJy out of the 447 input ETG with m pg /lessorequalslant 18 . 0 and accurate position) are 9.6% for all ETG, 17.1% for ellipticals, 41.4% for lenticulars and 3.7% for dwarf ETG. The latter rate becomes 3.6%, if we take into account that 1.5 of the assumed far-IR counterpart of dwarf ETG are in fact counterparts of background sources (see the previous section), and that about 8 of the dwarf ETG of the input sample without a measured radial velocity are likely background galaxies. These rates are smaller than those previously measured on samples of bright ETG (Knapp et al. 1989, Temi et al. 2004, Smith et al. 2012), as can be expected since our sample extends to faint galaxies and the dust detection rate correlates strongly with optical luminosity (dSA12). The dust-to-star mass ratio varies over almost 6 orders of magnitude, anticorrelates with the optical luminosity, and for some dwarf ETG reaches very high values (around a few 10 -2 ), as high as for the dusty late-type galaxies. This is surprising, also given that the dusty ETG do not show signs of star formation. In fact the colours of the dusty ETG are not bluer than those of the non-dusty ones (Fig. 1). The distinction between fast and slow rotators appears to be an important one for ETG (Emsellem et al. 2011, and references on the ATLAS 3 D project). For the ETG of our input sample the detailed kinematical information necessary for this distinction is available only for the 49 ones, which are in common with the ATLAS 3 D sample. Since we detect dust in 69 ± 23% of the slow rotators (in 9 out of 13) and in 28 ± 9% of the fast ones (in 10 out of 36), it appears that the former are considerably more likely to have dust. This is the opposite to what is seen for molecular gas in the whole of the 260 ETG of the ATLAS 3 D sample. In fact Young et al. (2011) find that the CO detection rate is 6 ± 4% in slow rotators and 24 ± 3% in fast ones. This is surprising, since dust and molecular gas are thought to be closely associated (Draine et al. 2007, Corbelli et al. 2012); in fact, for the dust-detected ETG of our sample which have information on the molecular gas content, the dust-to-molecular-gas mass ratio is always 2 × 10 -2 within a factor of two, and lower limits are consistent with this range. We suggest that a possible explanation of this difference could be an environmental effect, since most of the ATLAS 3 D galaxies are outside of the Virgo cluster. In fact, of the 19 dust-detected ETG, which we have in common with the ATLAS 3 D sample, molecular gas is detected in 3 slow rotators and in 5 fast ones, a much more balanced situation than found by Young et al. (2011) in the whole ATLAS 3 D sample, and all slow rotators with molecular gas in this whole sample are actually in the Virgo cluster. The difference we find could be due to the presence of kinematically peculiar objects among the dusty slow rotators in the Virgo cluster, like galaxies with counter-rotating components mimicking slow rotation. We can exclude this possibility, since the brightest and most regular ellipticals and lenticulars in the Virgo cluster like M49, M84, M86, M87, M89, NGC 4261 and NGC 4526 are among the dusty slow rotators, reinforcing our suggestion about an environmental effect. We have also looked at the relationship between dust and HI for the Virgo ETG, updating the work done by di Serego Alighieri et al. (2007) on the HI content of Virgo ETG to include the 4-8 degrees declination strip, which has become available in the ALFALFA HI survey since their work (Haynes et al. 2011). We find an intriguing incompatibility between dust and HI in Virgo ETG: we detect both dust and HI in only 8 ETG, while 39 ETG have dust but no HI, and 8 have HI but no dust. This dichotomy between dust and HI is reinforced by the position of the parent galaxies in the cluster. Dusty ETG appear to concentrate in the densest regions of the Virgo cluster, while HI-rich ETG tend to be at the periphery. While the presence of ETG with dust but no HI can be explained by the longer dust survival times and by the stronger effects of ram pressure stripping on HI, more difficult to understand is that there are ETG with HI but no dust, also given that these include two rather bright S0 galaxies: NGC 4262 and NGC 4270. Concerning the relationship between the presence of dust and that of an AGN in Virgo ETG, our input sample has 71 ETG in common with the sample of Virgo ETG for which Gallo et al. (2010) have looked for the presence of a supermassive black-hole (SMBH) by observing with Chandra the nuclear X-ray luminosity down to a few 10 38 erg/s. Out of the 71 common ETG, 25 (35%) have X-rays, most likely from a SMBH, and 14 (20%) have dust. Of the X-ray luminous ETG 36% have dust, and out of the dusty ETG 64% are X-ray luminous. It appears that ETG with an X-ray active SMBH are more likely to have dust, and that ETG with dust are more likely to have an X-ray active SMBH.", "pages": [ 2, 3 ] }, { "title": "References", "content": "Binggeli, B., Sandage, A. & Tammann, G.A., 1985, AJ , 90, 1681 Corbelli, E., Bianchi, S., Cortese, L., et al., 2012, A&A , 542, A32 Davies, J.I., Baes, M., Bendo, G.J., et al., 2010, A&A , 518, L48 di Serego Alighieri, S., Gavazzi, G., Giovanardi, C., et al., 2007, A&A , 474, 851 Draine, B.T., Dale, D.A., Bendo, G., et al., 2007, ApJ , 663, 866 Emsellem, E., Cappellari, M., Krajnovi'c, D., et al., 2011, MNRAS , 414, 888 Gallo, E., Treu, T., Marshall, P.J., et al., 2010, ApJ , 714, 25 Gavazzi, G., Boselli, A., Scodeggio, M., Pierini, D. & Belsole, E., 1999, MNRAS , 304, 595 Gavazzi, G., Boselli, A., Donati, A., Franzetti, P. & Scodeggio, M., 2003, A&A , 400, 451 Giovanelli, R., Haynes, M.P., Kent, B.R., et al., 2007, AJ , 133, 2569 Gomez, H.L., Baes, M., Cortese, L., et al., 2010, A&A , 518, L45 Haynes, M.P., Giovanelli, R., Martin, A.M., et al., 2011, AJ , 142, 170 Knapp, G.R., Guhathakurta, P., Kim, D.-W. & Jura, M., 1989, ApJSS , 70, 329 Pilbratt, G.L., Riedinger, J.R., Passvogel, T., et al., 2010, A&A , 518, L1 Smith, D.J.B., Dunne, L., Maddox, S.J., et al., 2011, MNRAS , 416, 857 Smith, M.W.L., Gomez, H.L., Eales, S.A., et al., 2012, ApJ , 748, 123 Temi, P., Brighenti, F., Mathews, W.G. & Bregman, J.D., 2004, ApJSS , 151, 237 Young, L.M., Bureau, M., Davis, T.A., et al., 2011, MNRAS , 414, 940 Zibetti, S., Charlot, S. & Rix, H.-W., 2009, MNRAS , 400, 181", "pages": [ 4 ] } ]

[ { "title": "Jared Gabor 1", "content": "1 CEA Saclay Bat. 709, Gif-sur-Yvette, 91191 France email: [email protected] Abstract. I highlight three results from cosmological hydrodynamic simulations that yield a realistic red sequence of galaxies: 1) Major galaxy mergers are not responsible for shutting off star-formation and forming the red sequence. Starvation in hot halos is. 2) Massive galaxies grow substantially ( ∼ × 2 in mass) after being quenched, primarily via minor (1:5) mergers. 3) Hot halo quenching naturally explains why galaxies are red when they either (a) are massive or (b) live in dense environments. Keywords. galaxies: evolution, galaxies: interactions", "pages": [ 1 ] }, { "title": "1. Major mergers are not responsible for quenching", "content": "The physical mechanism(s) responsible for shutting down star-formation and leading to the red sequence are still debated (Bower et al. 2006, Croton et al. 2006, Hopkins et al. 2008, Somerville et al. 2008). Two leading pictures have emerged: a) major mergers trigger starbursts and possibly AGNs, whose combined gas consumption and feedback can rid the galaxy of gas to fuel star-formation; and b) hot gas coronae form in massive halos, shock-heating any infalling gas, and some heating process (such as a radio AGN) prevents that gas from cooling. Often, these are respectively called 'quasar' mode and 'radio' mode AGN feedback, alluding to the possible importance of AGN. I have independently tested simplistic versions of these two mechanisms in cosmological hydrodynamic simulations. For merger quenching, I identify mergers on-the-fly during the simulation and eject all the gas from remnants in a 1000 km s -1 wind. For hot halo quenching, I identify galaxies whose halos are dominated by hot gas ( T > 10 5 . 4 K; Kereˇs et al. 2005), and continuously add thermal energy to the circum-galactic gas around them. The results of these simulations are shown in Figure 1 (cf. Gabor et al. 2011) as color-magnitude diagrams and luminosity functions. Merger quenching fails to yield a significant red sequence at z = 0, whereas hot halo quenching forms red galaxies in numbers consistent with observations. In the context of cosmological models, major mergers are neither necessary nor sufficient to explain the red sequence. They are not sufficient because galaxies are constantly accreting new gas from the cosmic web, even after mergers. Since the accreted gas provides fuel for star-formation, this accretion must be stopped to ensure that galaxies become red and stay red. Mergers are not necessary because an alternative quenching mechanism - hot halo quenching - appears to produce enough red galaxies. Note that there are enough major mergers to explain the numbers of red galaxies, if only the galaxies stopped accreting after the merger (Gabor et al. 2010). 4 r-band absolute mag", "pages": [ 1, 2 ] }, { "title": "2. Massive galaxies grow substantially after quenching", "content": "In the hot halo quenching simulation, massive galaxies typically turn red at z > 0 . 5, and grow significantly in mass after being quenched. The mass growth is shown in the left panel of Figure 2. Since star-formation is negligible in these galaxies, there are only two ways to change in stellar mass: mass loss from stellar evolution, and galaxy mergers. By number, most galaxies do not change much in mass - these are mostly recently-quenched satellites which are unlikely to merger with other satellites. Massive galaxies, on the other hand, grow by factors up to 3, with a large scatter driven by variations in merger history. These massive galaxies are typically central galaxies accreting their small satellites in minor mergers. The right-hand panel of Figure 2 shows the mean merger mass ratio (for those galaxies with at least one merger) as a function z = 0 stellar mass. Here I have weighted each merger event by the mass of the smaller galaxy. Thus the plot shows the mass ratio which has been most important for adding mass to galaxies in each bin of stellar mass. In all cases, the characteristic merger is below the typical 1:3 major merger threshold, with a typical value of 1:5. Minor mergers dominate the mass growth of quenched galaxies. This result implies strong growth in the sizes of quenched galaxies at high redshift (Gabor et al. 2012, Oser et al. 2012).", "pages": [ 3 ] }, { "title": "3. Hot gas quenching explains both 'mass quenching' and 'environment quenching'", "content": "In the hot halo quenching model, any galaxies that live in a hot corona are starved of incoming fuel for star-formation. A galaxy has two alternative paths to live inside a hot halo: a) it is the central galaxy in a halo of > 10 12 M /circledot , where a hot coronae is likely to form; or b) it is a satellite galaxy in such a halo. Case (a) can be thought of as 'mass quenching' or 'central quenching', and case (b) can be thought of as 'environment quenching' or 'satellite quenching.' These two modes of quenching are apparent in Figure 3, inspired by Peng et al. (2010) and taken from Gabor & Dav'e (2012). The fraction of red galaxies increases with overdensity (i.e. in denser environments) and with stellar mass. Moreover, the 'boxy' shape of the contours suggests that these modes are independent. In our simulation, they both result from hot gas cutting off the fuel supply for star-formation. 'Central quenching' and 'satellite quenching' can be explained naturally in this model. In hydrodynamic simulations, halos above ∼ 10 12 M /circledot are all dominated by hot gas (Birnboim & Dekel 2003, Kereˇs et al. 2005, Gabor et al. 2010). Furthermore, in the absence of quenching, stellar mass closely tracks halo mass. So a star-forming galaxy will increase its stellar mass as its halo mass increases due to accretion. Then, when the halo reaches ∼ 10 12 M /circledot (i.e. stellar mass reaches ∼ 10 10 . 5 M /circledot ), a hot corona will form which quenches star-formation. This manifests as a strong increase in red galaxy fraction at stellar masses /greaterorsimilar 10 10 . 5 M /circledot - mass quenching. Once a massive galaxy is quenched in this way, its satellite galaxies will also be quenched since they live in the same hot halo. The dark matter halo will continue to grow as additional galaxies fall in. Such infalling galaxies typically start out as starforming centrals, but after becoming satellites they will be quenched by the hot gas halo. This is satellite quenching. Satellites become quenched regardless of their masses, as long as they live in sufficiently dense environments where hot gas dominates.", "pages": [ 3 ] }, { "title": "4. Summary", "content": "Despite its simplicity, the hot halo quenching model does a remarkable job of matching basic observables. Although hot halos do not tell the whole story, they appear to be the dominant factor in forming the red sequence. Mergers must play some role in the formation of today's red ellipticals, but halting the inflow of new gas is crucial.", "pages": [ 4 ] }, { "title": "References", "content": "Birnboim, Y. & Dekel, A. 2003, MNRAS , 345, 349 Bower, R.G. et al. 2006, MNRAS , 370, 645 Croton, D. et al. 2006, MNRAS , 365, 11 Gabor, J. M., Dav´e, R., Finlator, K., & Oppenheimer, B.D. 2010, MNRAS , 407, 749 Gabor, J. M., Dav´e, R., Oppenheimer, B.D., & Finlator, K. 2011, MNRAS , 417, 2676 Gabor, J. M. & Dav´e, R. 2012, MNRAS in press, arXiv:1202.5315 Hopkins, P.F., Cox, T.J., Kereˇs, D., & Hernquist, L. 2008, ApJS , 175, 390 Kereˇs, D., Katz, N., Weinberg, D.H., Dav´e, R. 2005, MNRAS , 363, 2 Oser, L., Naab, T., Ostriker, J.P., & Johansson, P.H. 2012, ApJ , 744, 63 Peng, Y.-j. et al. 2010 ApJ , 721, 193 Somerville, R.S., Hopkins, P.F., Cox, T.J., Robertson, B.E., & Hernquist, L. 2008, MNRAS , 391, 481", "pages": [ 4 ] } ]

[ { "title": "Karl Glazebrook 1", "content": "1 Swinburne University of Technology, PO Box 218, Hawthorn, Vic 3122, Australia email: [email protected] Abstract. Future prospects in observational galaxy evolution are reviewed from a personal perspective. New insights will especially come from high-redshift integral field kinematic data and similar low-redshift observations in very large and definitive surveys. We will start to systematically probe the mass structures of galaxies and their haloes via lensing from new imaging surveys and upcoming near-IR spectroscopic surveys will finally obtain large numbers of rest frame optical spectra at high-redshift routinely. ALMA will be an important new ingredient, spatially resolving the molecular gas fuelling the high star-formation rates seen in the early Universe. Keywords. galaxies: evolution, galaxies: formation, galaxies: high-redshift, telescopes, instrumentation: miscellaneous", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "I would like to thank the organisers for their kind invitation to review the future observational prospects in galaxy evolution, and in particular for massive galaxies, the theme of this Symposium. I am going to attempt to look forward about five years, this seems a sensible time frame on which to make predictions of what will be the most highly impactful observations. If we review the last five years for comparison, it is quite startling to see the unexpected discoveries and developments that came about. Here are the ones that stick most in my mind (and references are intended to be illustrative not complete!): Given the recent history of unexpected developments in galaxy evolution this seems to make predicting the next five years fairly perilous! One thing that makes it slightly easier is that no major new telescopes will be commissioned during the period, indeed the new generation of Extremely Large Telescopes (ELTs) won't arrive until at least 2018. Other new large facilities such as the Large Synoptic Survey Telescope and the Square Kilometre Array are destined for the 2020's. In this look forward I am going to focus on three major areas that I have picked on due to upcoming new capabilities: (i) galaxy structures and kinematics, (iii) highredshift imaging and spectroscopic surveys and (iii) the imminent revolution in sub-mm astronomy from the Atacama Large Millmetre Array (ALMA).", "pages": [ 1, 2 ] }, { "title": "2. Galaxy Structures and Kinematics", "content": "Integral Field Spectroscopy (IFS) has revolutionised the study of the kinematics of high-redshift star-forming galaxies and we now have about 100-200 high-quality observations of galaxies at z glyph[greaterorsimilar] 1 from various surveys and nicely reviewed in S. Wuyt's talk at this Symposium. At these redshifts we see a picture where galaxy kinematic classes appear three-way split into (i) rotating objects with clearly disk-like velocity fields (ii) objects with kinematic structures but no uniform disk-like pattern (sometimes said to be 'mergers') and (iii) objects with no kinematic structure (sometimes referred to as 'dispersion dominated', Law et al. 2007). The split here is around 20-40% in each class but this is sensitive to the particular survey and selection function and the fraction of disks seems to increase towards higher stellar masses (Førster-Schreiber et al. 2009). Objects with disk kinematics seem to follow a Tully-Fisher relation in that they have the tightest scatter around a luminosity (or stellar mass) vs circular velocity line with a similar slope to, but a small offset from, the local Tully-Fisher relation (Puech et al. 2008, Cresci et al. 2008). A particular development at this symposium is a nice Tully-Fisher relation at z ∼ 1 . 2 form the MASSIV survey (P. Amran talk), a redshift in which there was previously somewhat of a gap. One key upcoming development is the advent of the KMOS IFS (Sharples et al. 2004) which is to be commissioned on the Very Large Telescope at the end of 2012. This offers the first near-IR multiplexed IFS on a large telescope and IFS observations of up to 24 galaxies can be performed simultaneously. This will enable two important advances: first, and obviously, much larger high-redshift IFS kinematic samples will be obtainable allowing statistical trends to be studied. Secondly the large multiplex means it will be efficient to study much fainter galaxies with longer exposure times. Current IFS surveys are restricted to studying the more luminous (in emission lines) objects, typically around ∼ L ∗ in H α at z ∼ 2, thus being able to tackle even small numbers of sub-glyph[suppress]L ∗ objects will allow selection biases to be studied. One open question, in my mind, to be tackled by future surveys is the evolution of the galaxy merger rate. IFS surveys typically identify 20-30% of galaxies as mergers via kinematics at 0 . 5 < z < 2 (Yang et al. 2008, Lopez-Sanjuan et al. 2012, Førster-Schreiber et al. 2009) which is in stark contrast to the local value of ∼ 4%. Are the merger rates identified via kinematics consistent with those measured by close-pair counts (e..g Y. Peng, this Symposium)? Can we even objectively identify mergers in kinematic maps? Pioneering work in this latter topic was done using kinemetry by Shapiro et al. (2008) but needs to be further developed, especially with respect to local calibration samples. In this Symposium P. Amran showed a new and different approach to quantitively identifying mergers. This is an excellent area for the future development of parametric and non- ametric statistics. A related question is can we go to the next step and measure mass ratios and merger timescales from IFS maps? I believe the other key development will be the carrying out of large-scale local IFS surveys, a 'kinematic SDSS'. Current local IFS samples are of order several hundred galaxies, diversely selected and with heterogeneous data. This is analogous to the situation for imaging and 1D spectroscopic surveys before the 2dF Galaxy Redshift Survey and the Sloan Digital Sky Survey (SDSS). The next five years will see surveys of several thousand, perhaps tens of thousands of local galaxies done with multiplexed IFS instruments. Projects actively building instruments and planning observational campaigns in the near term are the SAMI consortium (Croom et al. 2012), who will use the AngloAustralian Telescope, and the MANGA team (P.I. Kevin Bundy) planning to use the SDSS telescope. These instruments typically deploy ∼ 20 integral field units in a 2-3 · field-of-view. This will allow the statistical study of the distribution of resolved kinematic structures in the local Universe and other meta-questions (Figure 1). In particular we will move away from scaling relations such as Tully-Fisher to the study of true kinematic distribution functions where space-density plays a key role in comparing with theoretical models. These surveys will also provide a cornerstone for quantitative comparison with high-redshift surveys, for example by providing a high-quality merger sample where mergers are identified by kinematics and photometry (e.g. tidal tails and other low surface brightness features that may not be visible at high-redshift). They can also be used to find rare local analogues of high-redshift galaxies: because they are nearby they can then be followed up in exquisite detail to see what makes the tick astrophysically. One example of this is the work of Green et. al 2010 where we identified candidate local turbulent disks with high star-formation rates. We are currently engaged with HST, Gemini IFS and other facilities to prove if they are indeed analogues and how the star-formation is driven. We have also seen some nice work presented in this symposium on the kinematics and structures of red galaxies from high to low redshift. The so-called 'two-phase model' for the assembly of red galaxies (Forbes et al. 2011, Figure 2) is becoming popular where red galaxies start out as compact and very dense primordial 'red nuggets' † and then accrete a stellar halo via minor mergers as the core loses density. This allows a considerable amount of evolution of effective size per unit stellar mass increase and seems to be the emerging consensus explanation of size evolution in red galaxies. This does beg the question as to how the initial red nugget forms, is it via dissipative monolithic collapse and rapid starburst of a primordial gas cloud? Or the quenching or merging of highredshift disks? Is this consistent with the axial ratios and Sersic indices being found at high redshift? (e.g. Damjanov, this symposium, Chevance et al. 2012.) We now have a limited number of velocity dispersion measurements, from absorption lines, of the most massive high-redshift ellipticals which seem to supper the minor-merger hypothesis (e.g. I. Trujillo's review in these proceedings). What we do not yet have is resolved kinematic measurements, for example are the red nuggets very rapidly rotating disks? Absorption line measurements are very difficult but future deep IFS observations such as those of KMOS can address this question. So will deep imaging using multi-conjugate adaptive optics (AO) which will deliver resolution 2-3 × that of HST (McGregor et al. 2004). At low redshift it remains to be seen if the two-phase model can reproduce the distribution of elliptical galaxies between slow and fast rotators which has now been measured in the field and in very dense environments (R. Davies, these proceedings). Does the real cosmological merger history deliver the right final angular momentum distribution? This is a challenge for theory as well as observers (e.g. Burkert et al. 2008). Surveys such as MANGA and SAMI will deliver much better statistics but hydrodynamic simulations of massive galaxies embedded in large cosmological volumes remains supercomputerintensive. One final question that is perhaps unlikely to be answered in the next five years is the nature of the dispersion dominated compact star-forming galaxies that seem to constitute almost a third of the population. These are lower mass ( < 5 × 10 10 M glyph[circledot] ) so may not be", "pages": [ 2, 3, 4 ] }, { "title": "Two Phase galaxy formation?", "content": "related to the red nuggets even though they are a similar size ( ∼ 1-2 kpc). Are they purely dispersion dominated or do these conceal very compact disks that are unresolved even with AO IFS? This may require AO on ELTs to resolve, though spectroastrometry (Gnerucci et al. 2011) may allow information to be gleaned in the nearer term.", "pages": [ 5 ] }, { "title": "3. High-Redshift Imaging and Spectroscopic Surveys", "content": "In the last five years it has become routine for deep optical imaging surveys ( AB ∼ 2627) to cover tens to hundreds of square degrees. At these depths galaxies are surveyed to z ∼ 6. In the next five years even more gigapixels on sky will allow surveys such as the Dark Energy Survey (Flaugher 2005) and the Hype Suprime-Cam survey (Takada 2010) to cover thousands of square degrees at these depths. VISTA will similarly allow deep and wide near-IR surveys (McPherson at al. 2004). As outlined by D. Capozzi in these proceedings these imaging surveys will contribute to galaxy evolution studies via accurate measurements of photometric redshifts, luminosity functions, galaxy clustering, etc. However at the risk of some controversy I predict that the most important applications to galaxy evolution from the new imaging surveys will come from the use of galaxy lensing enabled by such large areas. Weak lensing will enable the direct statistical measure of dark matter in galaxy and cluster haloes - some very nice work along these lines using the CFHT Legacy Survey was presented by M. Hudson in these proceedings showing a good correlation between dark halo mass and stellar mass fraction in red and blue galaxies very suggestive of possible physical mechanisms. Strong lensing is also very powerful especially when combined with kinematic data (e..g. T. Treu talk in this symposium) as it allows mass structures and the IMF to be measured in the lensing galaxy. It is also very good for studying the lensed galaxy due to the large magnification of the light, making it brighter but also allowing smaller spatial scales to be resolved if the lens model can be inverted. The prospects of wider imaging surveys contributes to both weak lensing, via better statistics, and to strong lensing allowing more of these rare phenomena to be found. In spectroscopy the instrument that I am personally most excited about is MOSFIRE, the near-IR multislit spectrograph commissioned on Keck in mid-2012. This cryogenic instrument operates from 0.9-2.4 µ m and allows slit spectroscopy of up to 46 targets simultaneously (McLean et al. 2011). In my view it offers the first combination of three key features required to make near-IR spectroscopy succeed for faint high-redshift targets: (i) sufficient spectral resolution ( R = 3300) to well-resolve the airglow OH background out and 'get between the lines'. (ii) low scattered light and thermal background meaning it is truly dark between the sky lines; the measured interline background of MOSFIRE is very dark and comparable to the measurements of Maihaira et al.(1993). (iii) low readout noise and (iv) high instrument throughput 30-40%. Other similar instruments exist (such as F2 on Gemini) but do not offer the same spectral resolution for the one arcsec slit sizes required and have yet to be demonstrated on sky. The performance of MOSFIRE is shown by the detection of H α in normal Lyman Break Galaxies at z ∼ 2 in exposure times as short as 30 minutes! † The key science area which will be tackled by MOSFIRE is the routine continuum spectroscopy of normal galaxies at high-redshift in large numbers in the rest-frame optical for detailed comparison with low redshift surveys such as SDSS. These spectra will measure spectroscopic redshifts, stellar populations, metallicities and velocity dispersions † See 'first light presentation' on http://irlab.astro.ucla.edu/mosfire/ for homogenous samples. Without an instrument such as MOSFIRE this has been very difficult and most work in the last decade has relied on photometric redshifts. Even the very simplest product - redshift - should not be ignored as it allows clusters, environments and larger scale structures to be defined at high-redshift. These are the context of high-redshift galaxy evolution and current spectroscopic samples are highly biassed towards subsets of the population such as Lyman Break Galaxies. Photometric redshifts do not have the accuracy to measure such 3D environments though the most accurate ones, with medium band filters, do start to identify large scale structures and clusters (Spitler et al. 2012, Labb'e talk this symposium) but require spectroscopy to confirm. The prospects for MOSFIRE surveys are excellent with high-quality very deep high-quality near-IR imaging data for selection already available from HST (the CANDELS survey, Grogin et al. 2011, Koekemoer et al. 2011) and from the ground with medium bands. Because of this nexus we will now see a renaissance in high-redshift spectroscopy. It is interesting to note that this capability was in fact a key original science goal of 8m class telescopes and in the next five years we will finally see it delivered. Towards the end of the five year forecast we may see the Subaru Prime Focus Spectrograph arrive (Ellis et al. 2012) offering a 50-fold increase in optical near-IR multiplex and field-of-view over current systems (though being non-cryogenic will operate at wavelengths < 1 . 5 µ m). This will open the exciting prospect of using galaxies at z >> 1 for cosmology as well as galaxy evolution.", "pages": [ 5, 6 ] }, { "title": "4. The Age of ALMA", "content": "As I write one very significant new telescope is being commissioned: ALMA (Hills & Beasley 2008). Virtually no ALMA results were presented at this symposium as very few people actually have any ALMA data. † So far no more than about 1000 hours of ALMA science time has been available to the community. However if we have a conference such as this in five years time I fully expect ALMA results to dominate the conference. Why do I say this? Today high-redshift is dominated by optical and near-IR observations which are mainly sensitive to stars and hot ionised gas (e.g. from star-formation or AGN). However we need to consider the fuel as well as the fire. We know from current sub-mm observations that the molecular gas fractions of massive galaxies rises from a mere 5-10% at z = 0 to ∼ 50% at z ∼ 2 (Daddi et al. 2010, Tacconi et al. 2010). This probably accounts for the high prevalence of unstable, clumpy, turbulent disks (e.g. Genzel et al. 2008) and necessitates high inflow rates of cosmic material to sustain them (Dekel et al. 2009). However current sub-mm telescopes barely resolve high-redshift galaxies with 0.5-1 arcsec beams and require many hours of integration per target. ALMA will improve this by factors of ten and enable kpc-resolution morphology and kinematics of molecular gas and dust in normal star-forming galaxies to be routinely made. We predict the clumpy disks to be gas rich and thick. Will we see thick cold molecular gas disks co-rotating and aligned with the young stars seen by the near-IR IFS observations? Will we see supergiant molecular clouds associated with the giant star-forming regions see in the UV? I predict we will! A particularly important question for ALMA's spatial resolution is the nature of the star-formation law relating gas density to star-formation rate, a critical theoretical ingredient of galaxy formation simulations (the 'sub-grid physics'). Around 80% of the stars in the Universe formed at z > 1 but we have seen throughout this conference that galaxies † A show of hands at the symposium revealed at most 2-3 hands up in the audience. in the the high-redshift Universe are very different to today. Will the star-formation law be the same or quite different? The classical Kennicutt-Schmidt law (Kennicutt 1998) simply relates surface densities of gas and star-formation via a power law. Even locally there are many variations on this theme (a topic extensively discussed in Symposium 292 the previous week), for example there may be 'thresholds' or a volumetric relation may be more appropriate (Krumholz, McKee & Tomlinson 2009). At high-redshift Daddi et al. (2010) suggested there are in fact two relations - a 'sequence of starbursts' and a 'sequence of disks' but which may be unified by introducing a dynamical time in to the formulation. ALMA will bring a highly superior set of data to bear on this problem and I will predict some surprises! Finally one interesting prediction that could perhaps be tested by ALMA is the existence of dark turbulent disks (Elmgereen & Burkert 2010). The prediction is that turbulence in gas disks starts initially in an accretion driven phase lasting for ∼ 180 Myr before star-formation turns on. The gas would be cold and molecular - the visibility of such objects to ALMA has not yet been calculated, but would make for an interesting paper.", "pages": [ 6, 7 ] }, { "title": "5. Final Words", "content": "Some firm predictions for the next five years: Finally it is amusing to note that at this conference we saw Carlos Frenk (doyen of semianalytic modelers) saying that 'galaxy formation is complicated' and Simon Lilly (the archetypal observer) saying 'galaxy formation is simple'! This appears to be a reversal of the theory-observer dichotomy of ten years ago to my memory, however I will dare to suggest that they are both in fact wrong! I think in the next 5-10 years we will see basic physical questions of star-formation and quenching (i.e. the formation of the red sequence) ironed out through better spatially-resolved observations as described above and there will be less need for 'recipes' in both camps. I speculate these observations will reveal new simplicities but also more complexity then the over-simplified picture that has arisen from large surveys with integrated spectra.", "pages": [ 7 ] }, { "title": "References", "content": "Baldry, I. K., Balogh, M. L., Bower, R. G., Glazebrook, K., Nichol, R. C., Bamford, S. P., Budavari, T., 2006, MNRAS , 373, 469 Balogh, M. L., Baldry, I. K., Nichol, R., Miller, C., Bower, R., Glazebrook, K., 2004, ApJ , 615, L101-L104 Bournaud F., Elmegreen B. G., 2009, ApJ , 694, L158 Burkert, A., Naab, T., Johansson, P. H., & Jesseit, R. 2008, ApJ, 685, 897 Chevance M., Weijmans A.-M., Damjanov I., Abraham R. G., Simard L., van den Bergh S., Caris E., Glazebrook K., 2012, ApJ , 754, L24 Cimatti, A., Cassata, P., Pozzetti, L., et al. 2008, A&A, 482, 21 Cresci et al. 2009, ApJ (2009) vol. 697 pp. 115 Conselice, C. J., Mortlock, A., Bluck, A. F. L., & Gruetzbauch, R. 2012, MNRAS , in press, arXiv:1206.6995", "pages": [ 7 ] } ]

[ { "title": "Escape and propagation of UHECR protons and neutrons from GRBs, and the cosmic ray-neutrino connection", "content": "MAURICIO BUSTAMANTE, PHILIPP BAERWALD, WALTER WINTER Institut fur Physik und Astrophysik, Universitat Wurzburg, 97074 Wurzburg, Germany [email protected] Abstract: We present a model of ultra-high-energy cosmic ray (UHECR) production in the shock-accelerated fireball of a gamma-ray burst. In addition to the standard UHECR origin from neutron escape and decay into protons, our model considers direct proton emission through leakage from the edges of the accelerated baryonloaded shells that make up the fireball. Depending on the optical thickness of the shells to photohadronic interactions, the source falls in one of three scenarios: the usual, optically thin source dominated by neutron escape, an optically thick source to neutron escape, or a 'direct escape' scenario, where the main contribution to UHECRs comes from the leaked protons. The associated neutrino production will be different for each scenario, and we see that the standard 'one neutrino per cosmic ray' assumption is valid only in the optically thin case, while more than one neutrino per cosmic ray is expected in the optically thick scenario. In addition, the extra direct escape component enhances the high-energy part of the UHECR flux, thus improving the agreement between the predictions and the observed flux. Keywords: UHE, cosmic ray, neutrino, GRB", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The hypothesis of a common hadronic origin of UHECRs, neutrinos, and photons postulates that photohadronic interactions occur between magnetically-confined UHE protons, shock-accelerated to ∼ 10 12 GeV, and a dense background of photons in the source. The leading contribution is given by the D + ( 1232 ) resonance, i.e. , In the standard picture, the neutrons thus created escape the source and beta-decay as n → p + e + ¯ n e , with the protons eventually reaching Earth as UHECRs. The charged pions decay and generate UHE neutrinos through while the neutral pions decay as p 0 → g + g to generate the gamma-ray signals observed at Earth. Hence, neutrinos are created in the ratios n e : nm : nt = 1 : 2 : 0, and flavour mixing during propagation transforms this into 1 : 1 : 1 at Earth. We refer to the paradigm of 'one neutrino (of each flavour) per cosmic ray' as the 'standard case'. The standard case hinges on two assumptions: Recently, Refs. [1, 2] put the standard case to test and found that GRBs cannot be the sole source of the UHECR protons. However, GRB energetics arguably make them one of the most attractive potential sources of UHECRs and UHE neutrinos. Accordingly, we have explored a generalised emission model in which assumptions (1) and (2) are violated, which we will presently summarise. Further details on our model can be found in Ref. [3].", "pages": [ 1 ] }, { "title": "2 The GRB fireball model", "content": "In the fireball model, p g interactions occur when relativistically expanding baryon-loaded matter ejecta from a compact emitter collide among themselves, a process in which a fraction of the kinetic energy of the matter is radiated away as protons, photons, and neutrinos. We adopt here a simplified description of the internal collisions in the fireball, following Ref. [4]. From the measured variability timescale of a GRB, tv (in the observer's frame), we assume that the central engine emits spherical shells of thickness D r glyph[similarequal] ctv / ( 1 + z ) in the source frame, for a burst at redshift z . During the first stage of the burst, shells are accelerated by the energy transfer of photons to baryons, eventually reaching a maximum value of the Lorentz factor G , after which the shells coast, each with its own constant maximum velocity. Asssuming fluctuations of the Lorentz factor of the order DG / G ∼ 1, collisions among shells are expected to start at a radius r C glyph[similarequal] 2 G 2 ctv / ( 1 + z ) from the central engine, signaling the onset of the 'prompt phase', in which we will focus. The physics of individual collisions are described in detail in Refs. [5, 6, 7]. We compute the secondary ( e.g. , neutrons, neutrinos) injection 1 Q ' ( E ' ) (in units of GeV -1 cm -3 s -1 ) coming from photohadronic interactions from the photon and proton densities (GeV -1 cm -3 ) as where x ≡ E ' / E ' p is the energy fraction going to the secondary, y ≡ E ' p e ' / ( mpc 2 ) , and R ( x , y ) is a 'response function', which describes the outcome of the interaction as a function of the energies of the incident proton, photon, and secondary, taking into account several p g channels ( D -resonance, higher resonances, direct pion production) [8]. A broken power law is adopted for the photon density: N ' g ( e ' ) GLYPH<181> ( e ' / e ' g , break ) -kg , with kg = ag ≈ 1 when e ' g , min = 0 . 2 eV ≤ e ' ≤ e ' g , break , kg = bg ≈ 2 when e ' g , break ≤ e ' ≤ e ' g , max = 300 × e ' g , break , and N ' g = 0 otherwise. The break energy e ' g , break = O ( keV ) . For the proton density, Fermi shock acceleration is assumed to generate a non-thermal power-law spectrum of the form N ' p ( E ' p ) GLYPH<181> ( E ' p ) -a p × exp [ -( E ' p / E ' p , max ) k ] , with a p ≈ 2 and k = 2. The maximum proton energy E ' p , max is determined by comparing the acceleration timescale t ' acc ( E ' ) = E ' / ( h ceB ' ) to the dominant loss timescale, i.e. , where t ' dyn ≡ D r ' / c is the dynamical timescale, t ' syn ( E ' ) = 9 m 4 / ( 4 ce 4 B ' 2 E ' ) is the timescale due to synchrotron losses, t ' p g ( E ' ) is the timescale due to photohadronic interactions, computed numerically as in Ref. [8], and h is the acceleration efficiency. The photon and proton densities can be determined from the observed radiative flux F g of the GRB (in units of GeV cm -2 s -1 ), with which the isotropic equivalent radiative energy per collision (per shell) can be calculated as E sh iso glyph[similarequal] 4 p d 2 L F g tv / ( 1 + z ) in the source frame, with dL the luminosity distance to the source, and E ' sh iso = E sh iso / G in the SRF. Thus, the densities can be normalised through where V ' iso = 4 p r 2 C D r ' is the volume of the interaction region assuming isotropic emission, and f -1 e , the 'baryonic loading', is the ratio between the energy in protons and in electrons. Additionally, the magnetic field can be calculated by assuming that a fraction e e and e B of energy is carried, respectively, by electrons and by the magnetic field: B ' = √ 8 p ( e B / e e ) ( E ' sh iso / V ' iso ) . Putting all of this together, the injection spectrum Q ' of neutrinos and neutrons can be computed, and the corresponding fluence per shell (in units of GeV -1 cm -2 ) is obtained as Flavour mixing is implemented for neutrinos, assuming a normal mass hierarchy and the best-fit values of the mixing parameters from Ref. [9]. The total fluence of the burst is obtained by multiplying F sh by the number N glyph[similarequal] T 90 / tv of identical collisions, with the burst duration, T 90, defined as the time during which 90% of the photon signal is recorded.", "pages": [ 1, 2 ] }, { "title": "3 Optically thin and thick sources, and direct proton escape", "content": "We assume that particles are isotropically distributed within an expanding shell, and that the number of particles that escape the shell is proportional to its volume. Particles from within a shell of thickness l ' mfp are able to escape without interacting ('mfp' refers to the mean 11 11 free path of particles), so that the fraction of escaping particles [3] is f esc ≡ V ' direct / V ' iso glyph[similarequal] l ' mfp / D r ' . The mean free paths for protons and neutrons are determined by l ' p , mfp ( E ' ) = min [ D r ' , R ' L ( E ' ) , ct ' p g ( E ' ) ] and l ' n , mfp ( E ' ) = min [ D r ' , ct ' p g ( E ' ) ] , respectively, with R ' L ( E ' ) = E ' / ( eB ' ) the Larmor radius. The fluence of directly-escaping protons can then be obtained by multiplying the fluence calculated in equation (6) by the fraction f esc. The source can be characterised by the optical thickness to neutron escape, defined as t n ≡ ( t '-1 p g / t '-1 dyn ) | Ep , max . If t n glyph[greaterorsimilar] 1, neutrons (and protons) may interact multiple times in the source and remain confined inside of it ( optically thick source ), while the opposite occurs if t n < 1 ( optically thin source ). Note that, since t '-1 p g increases with energy, t n will be maximum at Ep , max; at lower energies, escape from the source is easier. 2 - cm GLYPH<215> GeV GLYPH<144> sh F 2 E 1 - s GLYPH<144> 1 - ' t z z For the optically thin source, the maximum proton energy is set by the dynamical timescale, and direct proton escape dominates at Ep , max, with most protons being able to escape due to the acceleration efficiency of h = 1. As a consequence, the associated neutrino fluence is low. For the optically thick source, in comparison, it is the photohadronic timescale which determines the maximum proton energy, and therefore neutron production is enhanced. However, only neutrons lying close to the shell edges are able to escape, which is why the dashed curve cannot be exceeded. On the other hand, neutrinos created everywhere inside the shell are able to free-stream out of it, which leads to a substantially larger neutrino fluence compared to the optically thin case. Three emission regimes are hence identified: Optically thin to neutron escape regime. The standard emission scenario: protons are magnetically confined in the source and photohadronic interactions produce neutrons which are able to escape the source and decay into UHE protons. The charged pion decays lead to the 'one (muon-)neutrino per cosmic ray' result. Direct escape regime. Directly escaping protons from the borders dominate the UHECR flux, at least at the highest energies. Neutron production is subdominant, and the 'one neutrino per cosmic ray' paradigm is no longer valid, since more cosmic rays (protons) will be emitted. Optically thick to neutron escape regime. Neutrons and protons in the bulk of the shell are trapped due to multiple photohadronic interactions, and only those on the borders are able to escape. Neutrino production is enhanced due to the larger number of p g interactions, since neutrinos can escape from anywhere in the shell.", "pages": [ 2, 3 ] }, { "title": "4 UHECR observations", "content": "To explore the effect on the observed UHECR flux of adding the directly-escaping proton component, we have adopted the proton injection spectrum from a sample burst whose parameters make the direct proton escape component dominate at high energies, and the component from neutron decays dominate at lower energies. The transport of protons from the sources to Earth is performed by numerically solving a kinetic equation for the comoving proton density Y (in units of GeV -1 cm -3 ) [10, 11]: The first term on the rhs takes care of the energy dilution due to the adiabatic cosmological expansion, with H ( z ) the Hubble parameter. The second term considers proton energy losses due to pair production on the cosmological microwave (CMB) and infrared/optical photon backgrounds, i.e. , p + g → p + e + + e -, while the third term considers losses due to photohadronic interactions. In general, the energy loss rate b = dE / dt . The last term, L CR, injects protons from the sources at each redshift step, and takes care of the evolution of the source number density with z . We have assumed that identical bursts (in the comoving source frame) are distributed in redshift following the GRB rate from Refs. [12, 13]. By evolving the kinetic equation from z max = 6 down to z = 0, the proton flux is obtained as J ( E ) = ( c / 4 p ) Y ( E , z = 0 ) . dip in the spectrum due to pair production on the CMB, as well as the ankle, for a p glyph[greaterorsimilar] 2 . 5. However, large values of a p are difficult to motivate from Fermi shock acceleration. The 'transition model' can reproduce the ankle using a p = 2, but fails to fit the observations at lower energies, where an extra component, possibly of galactic origin, becomes necessary. Both of these models consider only neutron escape. Finally, our two-component model, shown here for a p = 2 . 5, is able to fit the observations at lower and higher energies, closely reproducing the spectrum at the dip and ankle. Comparing the curves for a p = 2 . 5 corresponding to the two-component model and to the dip model, it is clear that the effect of adding the direct proton escape component is to enhance the high-energy part of the spectrum.", "pages": [ 4 ] }, { "title": "5 Summary and conclusions", "content": "We have introduced a model of UHE neutron and proton emission from GRBs in which, depending on the relative dominance of the energy-loss timescales (dynamical, synchrotron, photohadronic), the source can be optically thin or thick to neutron and proton escape, or the emission can be dominated by direct proton escape from the borders of the expanding matter shells. We have tested the validity of the 'one (muon-)neutrino per cosmic ray' paradigm and found that it is valid only in the optically thin regime, while in the optically thick and direct-escape dominated regimes, either more or fewer neutrinos are created, and this relationship no longer holds. Finally, we have calculated the expected local UHECR flux from our two-component emission model, and found that the addition of the direct-escape component at high energies improves the fit to the experimental data, compared to models with only neutron escape. Acknowledgments: Work supported by the GRK 1147 'Theoretical Astrophysics and Particle Physics', FP7 Invisibles network, Helmholtz Alliance for Astroparticle Physics, and DFG grant WI 2639/4-1. MB acknowledges support from the ICRC organisers.", "pages": [ 4 ] } ]

[ { "title": "Cosmic-ray spectral anomaly at GeV-TeV energies as due to re-acceleration by weak shocks in the Galaxy", "content": "SATYENDRA THOUDAM 1 , ∗ AND J ORG R. H ORANDEL 1 , 2 Abstract: Recent cosmic-ray measurements by the ATIC, CREAM and PAMELA experiments have found an apparent hardening of the energy spectrum at TeV energies. Although the origin of the hardening is not clearly understood, possible explanations include hardening in the cosmic-ray source spectrum, changes in the cosmicray propagation properties in the Galaxy and the effect of nearby sources. In this contribution, we propose that the spectral anomaly might be an effect of re-acceleration of cosmic rays by weak shocks in the Galaxy. After acceleration by strong supernova remnant shock waves, cosmic rays undergo diffusive propagation through the Galaxy. During the propagation, cosmic rays may again encounter expanding supernova remnant shock waves, and get re-accelerated. As the probability of encountering old supernova remnants is expected to be larger than the young ones due to their bigger size, re-acceleration is expected to be produced mainly by weaker shocks. Since weaker shocks generate a softer particle spectrum, the resulting re-accelerated component will have a spectrum steeper than the initial cosmic-ray source spectrum produced by strong shocks. For a reasonable set of model parameters, it is shown that such re-accelerated component can dominate the GeV energy region while the nonreaccelerated component dominates at higher energies, explaining the observed GeV-TeV spectral anomaly. Keywords: cosmic rays, diffusion, supernova remnants.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Recent measurements of cosmic rays by the ATIC [1], CREAM [2], and PAMELA [3] experiments have found a spectral anomaly at GeV-TeV energies. The spectrum in the TeV region is found to be harder than at GeV energies. The spectral anomaly is difficult to explain under the standard models of cosmic-ray acceleration and their propagation in the Galaxy, which predict a single power-law spectrum over a wide range in energy. Various explanations for the spectral anomaly have been proposed. These include hardening in the cosmic-ray source spectrum at high energies [4, 5, 6, 7], change in the cosmicray propagation properties in the Galaxy [8, 9], and the effect of nearby sources [10, 11, 12, 13]. In this contribution, we discuss the possibility that the anomaly can be an effect of re-acceleration of cosmic rays by weak shocks in the Galaxy. This scenario was also shortly discussed recently by Ptuskin et al. 2011 [14]. After acceleration by strong supernova remnant shock waves, cosmic rays escape from the remnants and undergo diffusive propagation in the Galaxy. The propagation can be accompanied by some level of re-acceleration due to repeated encounters with expanding supernova remnant shock waves [15, 16]. As older remnants occupy a larger volume in the Galaxy, cosmic rays are expected to encounter older remnants more often than the younger ones. Thus, this process of re-acceleration is expected to be produced mainly by weaker shocks. As weaker shocks generate a softer particle spectrum, the resulting re-accelerated component will have a spectrum steeper than the initial cosmic-ray source spectrum produced by strong shocks. As will be shown later, the re-accelerated component can dominate at GeV energies, and the non-reaccelerated component (hereafter referred to as the 'normal component') dominates at higher energies. Cosmic rays can also be re-accelerated by the same magnetic turbulence responsible for their scattering and spatial diffusion in the Galaxy. This process, which is commonly known as the distributed re-acceleration, has been studied quite extensively, and it is known that it can produce strong features on some of the observed properties of cosmic rays at low energies. For instance, the peak in the secondary-to-primary ratios at ∼ 1 GeV/n can be attributed to this effect [17]. Earlier studies suggest that a strong amount of re-acceleration of this kind can produce unwanted bumps in the cosmic-ray proton and helium spectra at few GeV/n [18, 19]. However, it was later shown that for some mild re-acceleration which is sufficient to reproduce the observed boron-to-carbon ratio, the resulting proton spectrum does not show any noticeable bumpy structures [17]. In fact, the efficiency of distributed reacceleration is expected to decrease with energy, and its effect becomes negligible at energies above ∼ 20 GeV/n. On the other hand, for the case of encounters with old supernova remnants mentioned earlier, the re-acceleration efficiency does not depend on energy. It depends only on the rate of supernova explosions and the fractional volume occupied by supernova remnants in the Galaxy. Hence, its effect can be extended to higher energies compared to that of the distributed re-acceleration, as also noted in Ref. [14]. In the present study, we will first determine the maximum amount of re-acceleration permitted by the available measurements on the boron-to-carbon ratio. Then, we will apply the same re-acceleration strength to the proton and helium nuclei, and check if it can explain the observed spectral hardening for a reasonable set of model parameters.", "pages": [ 1 ] }, { "title": "2 Transport equation with re-acceleration", "content": "Following Ref. [20], the re-acceleration of cosmic rays in the Galaxy is incorporated in the cosmic-ray transport equation as an additional source term with a power-law momentum spectrum. Then, the steady-state transport equation for cosmic-ray nuclei undergoing diffusion, re-acceleration and interaction losses can be written as, where we use cylindrical spatial coordinates with the radial and vertical distance represented by r and z respectively, p is the momentum/nucleon of the nuclei, N ( r , p ) represents the differential number density, D ( p ) is the diffusion coefficient, and Q ( r , p ) represents the source term. The first term on the left-hand side of Eq. (1) represents diffusion. The second and third terms represent losses due to inelastic interactions with the interstellar matter and due to reacceleration to higher energies respectively, where ¯ n represents the averaged surface density of interstellar atoms, v ( p ) the particle velocity, s ( p ) the inelastic collision crosssection, and x corresponds to the rate of re-acceleration. The fourth term with the integral represents the generation of particles via re-acceleration of lower energy particles. It assumes that a given cosmic-ray population is instantaneously re-accelerated to form a power-law distribution with an index s . We neglect ionization losses and the effect of convection due to the Galactic wind as these processes are important mostly at energies below ∼ 1 GeV/nucleon. The present study concentrates only at energies above 1 GeV/nucleon. The cosmic-ray propagation region is assumed as a cylindrical region bounded in the vertical direction at z = ± H , and unbounded in the radial direction. Both the matter and the sources are assumed to be uniformly distributed in an infinitely thin disk of radius R located at z = 0. This assumption is based on the known high concentration of supernova remnants, and atomic and molecular hydrogens near the Galactic plane. For cosmic-ray primaries, the source term is taken as Q ( r , p ) = ¯ n Q ( p ) , where ¯ n denotes the rate of supernova explosions (SNe) per unit surface area on the disk. The source spectrum is assumed to follow a power-law in total momentum with a high-momentum exponential cut-off. In terms of momentum/nucleon, it can be expressed as where A and Z represents the mass number and charge of the nuclei respectively, Q 0 is a constant related to the amount of energy f injected into a cosmic ray species by a single supernova event, q is the source spectral index, and pc is the high-momentum cut-off for protons. In writing Eq. (2), we assume that the maximum total momentum (or energy) for a cosmic-ray nuclei produced by a supernova remnant is Z times that of the protons. We further assume that the source spectrum has a low-momentum/nucleon cut-off at p 0 which also serves as the lower limit in the integral in Eq. (1). Moreover, the diffusion coefficient as a function of particle rigidity is assumed to follow D ( r ) = D 0 b ( r / r 0 ) a , where r = Apc / Ze is the particle rigidity, b = v / c , and c is the velocity of light. In the present model, since the re-acceleration of cosmic rays is considered to be produced by their encounters with supernova remnants, it follows that re-acceleration occurs only in the Galactic disk. If V = 4 p ' 3 / 3 is the volume occupied by a supernova remnant of radius ' , then in Eq. (1), x = h V ¯ n , where h is a correction factor for V we have introduced to take care of the unknown actual volume of the supernova remnants that re-accelerate cosmic rays. We keep h as a parameter, and we take ' = 100 pc which is roughly the typical radius of a supernova remnant of age 10 5 yr expanding in the interstellar medium with an initial shock velocity of 10 9 cm s -1 . The solution of Eq. (1) is obtained using the standard Green's function technique. For sources uniformly distributed in the Galactic disk, the solution at r = 0 is obtained as, where J1 is a Bessel function of order 1, and, the function A is given by Considering that the position of our Sun is very close to the Galactic plane, the cosmic-ray density at the Earth can be calculated from Eq. (3) taking z = 0. The first term on the right-hand side of Eq. (5) is the normal cosmic-ray component which have not suffered re-acceleration, and the second term is purely the reaccelerated component. For a given diffusion index, the high-energy spectra of the two components are shaped by their respective injection indices q and s . As re-acceleration takes out particles from the low-energy region and puts them into the higher energy part of the spectrum, for re-acceleration by weak shocks for which s > q , the reaccelerated component might become visible as a bump or enhancement in the energy spectrum at a certain energy range. In the case of re-acceleration by strong shocks which produces a harder particle spectrum, say s = q , the effect of re-acceleration will be hard to notice as both the components will have the same spectra in the Galaxy. These have been extensively discussed in Ref. [20]. For cosmic-ray secondaries, their equilibrium density N 2 ( r , p ) in the Galaxy is obtained following the same procedure as for their primaries described above, but with the source term replaced by where v 1 represents the velocity of the secondary nuclei, s 12 represents the total fragmentation cross-section of the primary to the secondary, and N 1 is the primary nuclei density. The subscripts 1 and 2 have been introduced to denote primary and secondary nuclei respectively. The secondary-to-primary ratio can be calculated by taking the ratio N 2 / N 1. For the case of no re-acceleration x = 0, it can be checked that Eq. (3) reduces to the standard solution of pure-diffusion equation (see e.g., [21]), and also that the secondary-to-primary ratio becomes inversely proportional to the diffusion coefficient at high energies. It can be mentioned that a steeper re-acceleration index s > q will produce an enhancement in the ratio at lower energies, and a harder index s = q will result into significant flattening of the ratio at high energies [16, 20]. Thus, the effect of re-acceleration on cosmic-ray properties in the Galaxy depends strongly on the index of re-acceleration. In the present study, since we assume that re-acceleration is produced mainly by the interactions with old supernova remnants, we will only consider the case of s > q with s glyph[greaterorsimilar] 4. This value of s corresponds to a Mach number of ∼ 1 . 7 of the shocks that re-accelerate the cosmic rays.", "pages": [ 2, 3 ] }, { "title": "3 Results and discussions", "content": "For the interstellar matter density, we consider the averaged surface density on the Galactic disk within a radius equivalent to the halo height H . We take H = 5 kpc for our study, and the averaged surface density of atomic hydrogen as ¯ n = 7 . 24 × 10 20 atoms cm -2 [11]. We assume that the interstellar medium consists of 10% helium. The inelastic interaction cross-sections are taken as the same used in the calculation in Ref. [11]. We take the size of the source distribution R = 20 kpc, the proton low and high-momentum cut-offs as p 0 = 100 MeV/c and pc = 1 PeV/c respectively, and the supernova explosion rate as ¯ n = 25 SNe Myr -1 kpc -1 . The latter corresponds to a rate of ∼ 3 SNe per century in the Galaxy. The cosmic-ray propagation parameters ( D 0 , r 0 , a ) , the reacceleration parameters ( h , s ) and the source parameters ( q , f ) are taken as model parameters. They are determined based on the measured data. We first determine ( D 0 , r 0 , a , h , s ) based on the measurement data for the boron-to-carbon ratio, and the spectra for the carbon, oxygen, and boron nuclei. Their values are found to be D 0 = 9 × 10 28 cm 2 s -1 , r = 3 GV, a = 0 . 33, h = 1 . 02, s = 4 . 5. These values correspond to the maximum amount of re-acceleration permitted by the available boronto-carbon data, while at the same time produces a reasonable good fit to the measured primary and secondary spectra. Figure 1 shows the result on the boron-to-carbon ratio (solid line) along with the measurement data. For comparison, we have also shown the result for the case of pure diffusion (dashed line) with no re-acceleration ( h = 0 ) taken from Ref. [11]. The good fit carbon and oxygen source parameters are found to be q C = 2 . 24 , f C = 0 . 024%, and q O = 2 . 26, f O = 0 . 025% respectively, where the f 's are given in units of 10 51 ergs. The present calculation assumes a force-field solar modulation parameter of f = 450 MV. Using the same values of ( D 0 , r 0 , a , h , s ) obtained above, we calculate the spectra for the proton and helium nuclei. The results are shown in Figure 2, where the top panel represents proton and the bottom panel represents helium. The lines represent our results, and the data are the same as used in Ref. [11]. The source parameters used are qp = 2 . 21 , fp = 6 . 95% for protons, and qHe = 2 . 18 , fHe = 0 . 79% for helium, and we use the same solar modulation parameter as given above. It can be seen that our results are in good agreement with the measured data, and explain the observed spectral anomaly between the GeV and TeV energy regions. Below ∼ 200 GeV/n, our model spectrum is dominated by the re-accelerated component while above, it is dominated by the normal component. The effect of re-acceleration is stronger in the case of protons than helium which is due to the larger inelastic collision losses for helium. This result into more prominent spectral differences in the GeV-TeV region for protons than for helium. For heavier nuclei for which the inelastic cross-sections are much larger, the re-acceleration effect is expected to be negligible. Figure 3 shows our result for iron nuclei. The calculation assumes qFe = 2 . 28, and fFe = 4 . 9 × 10 -3 %. As expected, the re-acceleration effect is hard to notice in Figure 3, and the model spectrum above ∼ 20 GeV/n follow approximately a single power-law unlike the proton and helium spectra.", "pages": [ 3, 4 ] }, { "title": "4 Conclusions", "content": "We have shown that the spectral anomaly at GeV-TeV energies, observed for the proton and helium nuclei, can be an effect of re-acceleration by weak shocks associated with old supernova remnants in the Galaxy. The re-acceleration effect is shown to be important for light nuclei, and negligible for heavier nuclei such as iron. Our prediction of decreasing effect of re-acceleration with the elemental mass can be checked by future sensitive measurements of heavier nuclei at TeV/n energies.", "pages": [ 4 ] }, { "title": "References", "content": "arXiv:1304.1400", "pages": [ 4 ] } ]

[ { "title": "Testing models of new physics with UHE air shower observations", "content": "JEFFREY D. ALLEN 1 AND GLENNYS R. FARRAR 1 1 Center for Cosmology and Particle Physics, Department of Physics, New York University, NY, NY 10003, USA [email protected], [email protected] Abstract: Several air shower observatories have established that the number of muons produced in UHE air showers is significantly larger than that predicted by models. We argue that the only solution to this muon deficit, compatible with the observed Xmax distributions, is to reduce the transfer of energy from the hadronic shower into the EM shower, by reducing the production or decay of π 0 s. We present four different models of new physics, each with a theoretical rationale, which can accomplish this. One has a pure proton composition and three have mixed composition. Two entail new particle physics and suppress π 0 production or decay above LHC energies. The other two are less radical but nonetheless require significant modifications to existing hadron production models - in one the changes are only above LHC energies and in the other the changes extend to much lower energies. We show that the models have distinctively different predictions for the correlation between the number of muons at ground and Xmax in hybrid events, so that with future hybrid data it should be possible to discriminate between models of new physics and disentangle the particle physics from composition. Keywords: muon deficit, hadronic interactions, composition, models", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Measurements of the density of muons at ground in ultrahigh energy (UHE) air-showers performed by hybrid observatories, first at HiRes-MIA [1] and more recently at the Pierre Auger Observatory [2, 3, 4, 5], have revealed that there is a significant deficit of muons in Monte Carlo (MC) simulations of air showers. The number of muons in the data is greater than that predicted using even iron initiated air-showers. Explaining this is made more challenging by the measurements of the distribution of the depth of shower maximum, Xmax . Measurements performed at the Pierre Auger Observatory (PAO) and Telescope Array (TA) both show that, depending on the hadronic model used to interpret the Xmax data, the mean mass at 10 EeV is light to intermediate [6, 7, 8, 9]. In the present study, we investigate potential resolutions to this discrepancy. We begin by exploring how generic properties of hadronic interactions are constrained by independent measurements of the density of muons at 1000 m from the shower core, N µ , and Xmax . We then present four schematic models of hadronic interactions which represent different methods to simultaneously fit measurements of both the mean Xmax and N µ . Each model can be tuned to reproduce the observed Xmax distribution and mean N µ . Fortunately, air shower observatories can differentiate the four models by studying the correlation between the Xmax and the N µ of hybrid air showers. Measurements of this nature should be feasible at both the PAO and TA, especially with upgrades to the muon sensitivity.", "pages": [ 1 ] }, { "title": "2 Constraints on Hadronic Interactions", "content": "N µ and Xmax are sensitive to several properties of hadronic interactions. Some properties, such as the primary cosmic ray mass composition and multiplicity of secondary particles, impact both these observables, while other properties impact only one or the other. By studying how MC predictions for N µ and Xmax behave under modifications to var- us hadronic interaction properties, we identify potential modifications which could resolve the muon deficit. The mean N µ is primarily sensitive to the multiplicity, the π 0 energy fraction (the fraction of incident energy carried by π 0 s in hadronic interactions), and the primary mass. The mean Xmax is primarily sensitive to the cross-section, elasticity, multiplicity, and primary mass. This dependence appears in the hadronic extension of the Heitler model [10], and has been studied quantitatively [11, 12]. Table 1 summarizes the qualitative impact that changing each property has on N µ and Xmax . To explore how changes in these properties affect the mean N µ and Xmax , we modify the secondary particles of the hadronic interaction model in the MC simulations. Modifications are made in a similar manner to that in [11]. The simulations are performed using the SENECA [13] air shower simulation with EPOS 1.99 [14] as the underlying hadronic event generator (HEG), although any other HEG could be used as the starting point. The primary energy in all simulations in this paper is 10 19 eV; except as noted modifications are performed at energies above 10 17 eV and become stronger with increasing energy. As expected, increasing the multiplicity or primary mass decreases the Xmax and increases N µ . However these mechanisms can play at most a partial role in solving the muon deficit, because Xmax becomes too shallow before N µ is sufficiently increased. The best hope for resolving the muon deficit is in decreasing the π 0 energy fraction, because that is the only modification which increases the mean N µ without encountering any restrictions from the Xmax observations 1 .", "pages": [ 1, 2 ] }, { "title": "3 Description of New Models", "content": "We have developed four schematic models which rely primarily upon modifying the fraction of energy which is transfered to decaying π 0 s in order to fit both the mean N µ and Xmax observed at the Auger Observatory. The models are implemented by modifying the secondary particles of EPOS 1.99. The energies of the initial interactions of an ultra-high energy air-shower are well above those achieved at accelerators. The center of mass energy of a 10 EeV proton incident upon a nucleon in the atmosphere is 137 TeV, and many secondary interactions are above the energy of the LHC. This justifies taking considerable freedom in exploring potential, new physics scenarios. We investigate new physics scenarios to uncover signatures of new physics and to explore a broad range of mechanisms which have the potential to solve the muon deficit.", "pages": [ 2 ] }, { "title": "3.1 Chiral Symmetry Restoration", "content": "In the Chiral Symmetry Restoration (CSR) model [15], we imagine that at the energy densities achieved in some hadronic interactions in UHE air showers, chiral symmetry is restored and the production of pions becomes greatly suppressed. In the CSR model, the primary cosmic rays are protons in order to achieve the highest energy density. The energy density of an interaction is determined by the impact parameter. We use the elasticity of the interaction as a tracer of the impact parameter; when the elasticity of the interaction is below a certain threshold, the interaction enters the CSR phase. More modifications are necessary to realize an acceptable pure-proton scenario than simply a reduction of the production of pions; the average Xmax predicted by EPOS using proton primaries is deeper than observations by both the Auger Observatory and TA. In the CSR phase, the multiplicity is increased and elasticity decreased. The cross section is rapidly increased at high energies to reduce the RMS of the predicted Xmax distribution. The tunable parameters of the CSR model include the strength of the pion production suppression, the elasticity threshold for entering the CSR phase, the elasticity of CSR interactions, and the increase in the proton-Air cross-section. Through suitable adjustment of the energy dependence of the parameters, an acceptable mean Xmax and RMS ( Xmax ) can be found for all energies (Fig. 2); see [15] for details. The CSR model thus provides an example of a proton-only model which can fit air shower observables [15].", "pages": [ 2 ] }, { "title": "3.2 Pion decay suppression", "content": "Pion decays in air shower MCs are treated as if they take place in a vacuum. However, in the rest frame of high en- ergy pions, the atmosphere is in fact a very dense medium. We postulate that pion decay could be suppressed through interactions with the dense air medium [16]. In the pion decay suppression (PDS) model, pion decay is suppressed at high energies. The impact this has on air shower development is similar in effect to simply decreasing pion production: since the π 0 s do not decay, they do not feed the electromagnetic shower. There is only one tunable parameter of the model, which is the energy above which pion decay is suppressed for a reference air density. The primary mass composition is assumed to be mixed in order to provide freedom to fit the Xmax distribution.", "pages": [ 2, 3 ] }, { "title": "3.3 Pion production suppression", "content": "The popular HEGs, such as QGSJET-II [17], SIBYLL [18] and EPOS, predict a wide range of π 0 energy fractions. For example, QGSJET-II-03 predicts that 25% of all the energy in secondary particles are carried by π 0 s and η s, while for EPOS 1.99 and SIBYLL 2.1 this is closer to 20%. This difference in the models persists at all energies. It is thus possible that the fraction of energy carried by pions in hadronic interactions may be less than any of the models currently predict. We consider two variants of a pion production suppression model: i) (PPS) at all energies, pions in the forward direction, chosen to be above a pseudorapidity of 5 at LHC energy, are converted to baryons, and all pions, regardless of pseudorapidity, are converted to kaons, with a common probability that is a tunable parameter, and ii) (PPS-HE) the same modification is made but only for interactions with incident energy above 10 17 eV. The high-energy variant is introduced to explore the impact of performing the pion production suppression in different energy regimes of shower development and would be similar to modifications to the string percolation probabilities [19], or if heavy flavor production is enhanced in kinematic regimes where quark masses may be insignificant. As in the pion decay suppression model, the primary mass composition is assumed to be mixed.", "pages": [ 3 ] }, { "title": "4 The N µ -X max Plane", "content": "As demonstrated in Sec. 3, we can construct models which fit the muon ground density and Xmax data using various deviations from standard HEGs. Fortunately, the four models can be discriminated by observing correlations between the depth of shower maximum and the number of muons at ground for an ensemble of showers. To discriminate between the models, the distribution of showers in the N µ -Xmax plane must be observed. Any observable which is related to the total number of muons in the showers is suitable, generically called N µ here; for definiteness we continue to use the density of muons at 1000 m. The correlation of N µ and Xmax is primarily sensitive to two basic properties of hadronic interactions: the mass composition, and the energy threshold for the suppression of pion decay and production. The number of muons produced in air showers is determined, in part, by the number of generations between the energy at which pion production begins and the energy at which pions decay, since in each generation energy is lost to the electromagnetic sub-shower [20, 10]. In general, iron and other heavy primaries have fewer generations and thus produce more muons. This causes a large negative correlation between N µ and Xmax when the composition is mixed, and a weak correlation in the case of a proton-only composition [21]. However, when pion production is suppressed above an energy threshold, this tends to equalize the number of generations between initial pion production and decay, and thus produce a similar number of muons. This causes the strength of the correlation to decrease as the degree of the pion production/decay suppression is increased. The four models span a wide range of primary compositions and mechanisms for suppressing pion production or decay and, thus, have different correlation strengths between N µ and Xmax . Fig. 5 shows the average N µ as a function of Xmax and its variance in ensembles of 800 simulated events, for 10 EeV showers at a zenith angle of 38 o , for each of the four schematic models. These simulations were done at a fixed zenith angle and energy, but the same statistical power could be achieved by normalizing showers to a fiducial energy and angle. The negative correlation between N µ and Xmax is strongest when the modification is made at all energies, and weaker when the modification is applied only at high energy. The PPS and PPS-HE models show a negative correlation while the CSR model shows almost no correlation. Finally, the PDS model actually shows a positive correlation: showers with a shallow Xmax produce fewer muons than showers with a deep Xmax . Thus the models can be discriminated at high significance with presently realizable datasets. The behavior of the PDS and PPS models demonstrates that the correlation is sensitive to the energy threshold of the pion modifications. This is made explicit in Fig. 6, which compares the average N µ and Xmax of iron, carbon, and proton initiated showers, for different degrees of pion production suppression at high energy. As pion production suppression is increased, the relative difference between the mean N µ in iron and proton showers decreases.", "pages": [ 3, 4 ] }, { "title": "5 Conclusion", "content": "We argue that the muon deficit in simulations of air showers indicates that the hadronic models are incorrectly predicting the fraction of energy which is transfered to the electromagnetic sub shower. Changing the π 0 energy fraction or suppressing pion decay are the only modifications which can be used to increase the number of muons at ground without coming into conflict with the Xmax observations. We have developed four schematic models of hadronic interactions, all of which are capable of correctly describing the Xmax distributions and number of muons at ground. They utilize both pure proton and mixed primary compositions, which demonstrates the need for models to correctly describe all air shower observables before they are used to interpret the primary mass composition. The four models can be distinguished by observations of the correlation between the number of muons N µ and Xmax , for an ensemble of hybrid events. This correlation provides a crucial new observable for determining the nature of UHE air showers. Existing hybrid datasets may already be large enough to rule out some explanations of the muon excess. Acknowledgements: The authors are members of the Pierre Auger Collaboration and acknowledge with gratitude innumerable valuable discussions with colleagues. This research was supported by NSF-PHY-1212538.", "pages": [ 4 ] }, { "title": "References", "content": "China, 2 (2011) 95, arxiv:1107.4809", "pages": [ 4 ] } ]

[ { "title": "Gamma-ray constraints on decaying Dark Matter", "content": "E. MOULIN 1 , M. CIRELLI 2 , P. PANCI 3 , P. SERPICO 4 , A. VIANA 1 , 5 [email protected] Abstract: New bounds on decaying Dark Matter (DM) are derived from the γ -ray measurements of (i) the isotropic residual (extragalactic) background by Fermi and (ii) the Fornax galaxy cluster by H.E.S.S. . We find that those from (i) are among the most stringent constraints currently available, for a large range of DM masses and a variety of decay modes, excluding half-lives up to ∼ 10 26 to few 10 27 seconds. In particular, they rule out the interpretation in terms of decaying DM of the e ± spectral features in PAMELA , Fermi and H.E.S.S. , unless very conservative choices are adopted. We also discuss future prospects for CTA bounds from Fornax which, contrary to the present H.E.S.S. constraints of (ii), may allow for an interesting improvement and may become better than those from the current or future extragalactic Fermi data. Keywords: Dark Matter, Cosmic rays, Gamma rays, BSM physics", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The possibility that Dark Matter (DM), which constitutes most of the matter in the Universe, consists of a particle that actually decays on a very long time scale has attracted much attention lately. This is because the decay time scale τ dec can be taken to be 'short' enough that the decay products give signals in current high energy cosmic ray experiments. Namely, if τ dec /similarequal few · 10 26 sec, decaying DM can be invoked to explain the excesses in the fluxes of positrons and electrons measured by PAMELA , Fermi and H.E.S.S. , see Ref. [1] and references therein. On the other hand, this value of τ dec is so much longer than the age of the Universe that the slow decay does not make a dent in the overall cosmological DM abundance and does not spoil the agreement with a number of astrophysical and cosmological observations [2]. From the phenomenological point of view, the main feature of decaying DM with respect to the 'more traditional' annihilating DM is that it is less constrained by neutral messenger probes (essentially γ -rays, but also neutrinos) originating from dense DM concentrations such as the galactic center, the galactic halo or nearby galaxies. The reason is simple and well-known: while the signal originating from annihilating DM is proportional to the square of the DM density, for decaying DM the dependence is on the first power; as a consequence, dense DM concentrations shine above the astrophysical backgrounds if annihilation is at play, but remain comparatively dim if DM is decaying. Decaying DM 'wins' instead, generally speaking, when large volumes are considered. This is why in the following we will focus on targets as large as galaxy clusters or, essentially, the whole Universe. On the observational side, the Fermi and H.E.S.S. telescopes are making unprecedented progress in the field of γ -ray astronomy, producing measurements of many differ- nt targets including those of interest for decaying DM. To this aim, we will compute the predicted signal from decaying DM, for a variety of decay channels and compare it to the γ -ray measurements, deriving constraints on the decay half-life. Here, we make use of two distinct probes: (i) The isotropic residual γ -ray flux recently measured by Fermi [3], which now extends from about 200 MeV up to 580 GeV. (ii)The recent observation in γ -rays of the Fornax galaxy cluster by H.E.S.S. [4] and the sensitivity of the upcoming large ˇ Cerenkov Telescope Array ( CTA ) [ ? ]. Our analysis presented here benefits from the release of new data on charged cosmic rays, the inclusion of EW corrections in all our computation of DM-generated fluxes, and an improved propagation scheme for e ± in the Galaxy. See Ref. [1] for more details. The rest of this paper is organized as follows. In Sec. 2 we update charged CR fits. In Sec. 3 we discuss the calculation of the constraints from the isotropic residual γ -ray flux and those from the Fornax galaxy cluster. In Sec. 5 we present the combined results. In Sec. 6 we present our conclusions.", "pages": [ 1 ] }, { "title": "2 Update of the decaying DM fits to charged CR anomalies", "content": "The anomalous PAMELA , Fermi and H.E.S.S. data in e + and ( e + + e -) have been interpreted in terms of DM decay. We use the following data sets: (i) PAMELA positron fraction [5]; (ii) Fermi positron fraction [6]; (iii) Fermi ( e + + e -) total flux [7], provided in the low energy (LE) and high energy (HE) samples; (iv) H.E.S.S. ( e + + e -) total flux [8, 9], also provided in a lower energy portion and a higher energy one; (v) MAGIC ( e + + e -) total flux [10]; (vi) PAMELA ¯ p flux [11]. We perform the fit to these data using the DM generated fluxes as provided in [12]. In looking for the best fitting regions, we scan over the propagation parameters of charged cosmic rays and over the uncertainties on the slope and normalization of the astrophysical electron, positron and antiproton background. We have assumed a NFW profile for the galactic DM halo. Positron fraction, e + e -spectrum, antiprotron spectrum and the isotropic γ -ray flux from the above-mentioned datasets are fitted for various decaying channels. The typical decay time scales that are required for the global fit are of the order of 10 26 to 10 27 seconds. Only 'leptophilic' channels allow a global fit: for the quark and gauge boson channels, the few TeV decaying DM needed by ( e + + e -) is in conflict with ¯ p data. See Ref. [1] for more details. The impact of the new data and the improved analysis tools on the identification of the best fit DM properties is shown in Fig. 1 for the µ + µ -channel.", "pages": [ 1, 2 ] }, { "title": "3 Isotropic γ -ray flux", "content": "The measurements in [3] by the Fermi satellite correspond to the (maximal) residual, isotropic γ -ray flux present in their data. Its origin can be in a variety of different phenomena, both in the form of unresolved sources and in the formof truly diffuse processes (see [3] and reference therein). DMdecays can also contribute to this isotropic flux, with two terms: 1) an extragalactic cosmological flux, due to the decays at all past redshifts; 2) the residual emission from the DM halo of our Galaxy. The former is of course truly isotropic, at least as long as one neglects possible nearby DM overdensities. The latter is not, but its minimum constitutes an irreducible contribution to the isotropic flux. In formulæ, the predicted differential DM flux that we compare with Fermi isotropic diffuse γ -ray data is therefore given by The extragalactic flux is given, in terms of the Earthmeasured photon energy E γ , by where Γ dec = τ -1 dec is the decay rate. Here the Hubble function H ( z ) = H 0 √ Ω M ( 1 + z ) 3 + ΩΛ , where H 0 is the present Hubble expansion rate. Ω DM, Ω M and ΩΛ are respectively the DM, matter and cosmological constant energy density in units of the critical density, ρ c , 0. The γ -ray spectrum dN / dE γ , at any redshift z , is the sum of two components: (i) the high energy contribution due to the prompt γ -ray emission from DM decays and (ii) the lower energy contribution due to Inverse Compton Scatterings (ICS) on CMB photons of the e + and e -from those same decays. Using Eq. (2), the extragalactic flux can therefore be computed in terms of known quantities for any specified DM mass M DM and decay channel. The factor e -τ ( E γ , z ) in Eq. (2) accounts for the absorption of high energy γ -rays due to scattering with the extragalactic UV background light. The flux from the galactic halo, coming from a generic direction d Ω , is given by the well known expression i.e. as the integral of the decaying DM density piling up along the line of sight (los) individuated by the direction d Ω . ρ halo is the DM distribution in the Milky Way, for which we take a standard NFW [13] profile with parameters rs = 24 . 42kpcand ρ s = 0 . 184GeV/cm 3 [12]. The coordinate r , centered on the GC, reads r ( s , ψ ) = ( r 2 /circledot + s 2 -2 r /circledot s cos ψ ) 1 / 2 , where r /circledot = 8 . 33 kpc and ψ is the angle between the direction of observation in the sky and the GC. As indicated in Eq. (1), we need to determine the minimum of the flux in Eq. (3). We choose to locate the minimum always at the anti-GC. d Φ isotropic / dE γ is compare with the Fermi data of [3]. The DM signal does not agree in shape with the data, which are instead well fit by a simple power law [3], and we are driven to derive constraints on the maximum DM signal, and therefore the minimum τ dec, admitted by the data. There are however several possible ways to compute such constraints. Our procedure uses 'DM signal + powerlaw background' to obtain the fiducial constraints shown in Fig. 1. This procedure is 'fiducial' also in the sense that it matches the analysis we do for charged CR anomalies (see Sec. 2) and therefore fit regions and constraints are essentially consistent with each other in Fig. 2. For discussion on this procedure and alternative ones, see [1].", "pages": [ 2, 3 ] }, { "title": "4 Fornax cluster γ -ray flux", "content": "Galaxy clusters are the largest gravitationally bound structures in the universe, 80% of their total mass being in the form of DM. Although they are located at much larger distances than other popular targets such as dwarf satellites of the Milky Way, they turn out to be attractive environments to search for DM due to their predicted high DM luminosity. A few galaxy clusters have been observed by H.E.S.S. : the most attractive of them for DM searches is Fornax [4]. The predicted DM γ -ray flux from Fornax can be easily obtained by integrating Eq.(3) over the observational solid angle ∆Ω , and replacing ρ halo with ρ Fornax. Various DM halo models for Fornax have been considered in [4] and varying over these profiles, the difference in the flux factor is less than a factor of 3 for a given opening integration angle. We take as 'fiducial' the X-ray based determination of the DMprofile, hereafter referred to as the RB02 profile [4]. Decaying DM searches largely benefit from an optimization of the opening angle to guarantee the highest signal-to-noise ratio. As it is straightforward from Eq.(3), the luminosity scales with the size of the solid integration angle. On the other hand, background is increasing as well. We find that the optimization of the signal-to-noise ratio for the RB02 profile yields a solid angle of ∆Ω = 2 . 4 × 10 -4 sr. In order to extract exclusion limits on the DM lifetime from γ -ray astronomical observation with IACTs, the background needs to be determined to constrain the decaying DM luminosity. The background is calculated in a region referred to as the OFF region, and the signal region as the ON region. Both ON and OFF regions depend on the observation mode and are specific to the IACT instrument. As for the background level, it is taken simultaneously, i.e. in the same data-taking observing conditions, to the signal events in order to allow for the most accurate estimate in the so-called template method procedure [4]. H.E.S.S. has observed Fornax for a total of 14.5 hours at low zenith angle to allow for best sensitivity to low DM masses.", "pages": [ 3 ] }, { "title": "5 Results and discussion", "content": "regions for the b ¯ b channel. As illustrated in the example in Fig. 1, adopting the more conservative constraint procedure may marginally reallow a portion of the fit regions, for the DM → µ + µ -, but leaving a clear tension. Keeping only data published in 2010 in [14] still allows to exclude the CR fit regions. The constraints from Fermi rise gently as a function of the mass, essentially as a consequence of the fact that the measured flux rapidly decreases with energy. They also depend (mildly, a factor of a few at most) on the decay channel [1]. The constraints from H.E.S.S. Fornax remain subdominant, roughly one order of magnitude below the Fermi ones. However, for the case of the DM → τ + τ -channel, the bound also reaches the CR fit region and essentially confirms the exclusion (Fig. 2). They do not look competitive with respect to Fermi even for larger masses. With respect to the isotropic γ -ray constraints of [15], the bounds derived here are stronger by a factor of 2 to 3. The reasons of this essentially amount to: updated datasets, more refined DM analyses and the adoption of a more realistic constraint procedure. With respect to the work in [16], our constraints from the Fermi isotropic background are somewhat stronger than their corresponding ones. In addition, [16] presents bounds from the observation of the Fornax cluster by Fermi : these are less stringent than our Fermi isotropic background constraints but more powerful than our H.E.S.S. -based constraints at moderate masses. At the largest masses, our H.E.S.S. -based constraints pick up and match theirs (for the b ¯ b channel).The bounds from clusters other than Fornax are less powerful, according to [16]. On the other hand, the preliminary constraints shown (for the b ¯ b channel only) in [17], obtained with a combination of several clusters in Fermi , exceed our bounds by a factor of 2. Bounds from probes other than the isotropic flux and clusters do not generally achieve the same constraining power. Within the context of observations performed by IACT, we note that the decay lifetime constraints obtained with galaxy clusters are stronger than those from dwarf galaxies. Even for the ultra-faint dwarf galaxy Segue 1, which is believed to be the most promising dwarf in the northern hemisphere, the constraints are 2 orders of magnitude higher for a DM particle mass of 1 TeV [18]. We also estimate that the constraints that we derive are stronger than those that can come from neutrino observation by Icecube of the Galactic Center (see for instance [19]). In summary: with the possible exception of the preliminary bounds from a combination of clusters by Fermi for the b ¯ b channel, the constraints that we derive from the isotropic γ -ray flux are the most stringent to date. Currently the constraints on decaying DM from the Fermi satellite are dominant with respect to those from H.E.S.S. . However, while the former may increase its statistics by at most a factor of a few, for the latter there are prospects of developments in the mid-term future. The next-generation IACT will be a large array composed of a few tens to a hundred telescopes [20]. The goal is to improve the overall performances of the present generation: one order-of-magnitude increase in sensitivity and enlarge the accessible energy range both towards the lower and higher energies allowing for an energy threshold down to a few tens of GeV. From the actual design study of CTA , the effective area will be increased by a factor ∼ 10 and a factor 2 better in the hadron rejection is expected. The cal- lation of 95% C.L. CTA sensitivity is detailed in [1] and given by where N 95%C . L . γ is the limit on the number of γ -ray events, A CTA is the CTA effective area and T obs the observation time. Figure 2 shows the 95% C.L. sensitivity of CTA on the decay lifetime for the RB02 halo profile for 50h observation time and ∆Ω = 2 . 4 × 10 -4 sr. To conclude, we also mention that a technique which could allow for significant improvements in the exploration of the parameter space of decaying DM using clusters is the one of stacking the observation of a large number of different clusters, as recently discussed in [21]. The authors find that improvements of up to 100 can be theoretically achieved, albeit this factor is ∼ 5 for more realistic background-limited instruments.", "pages": [ 3, 4 ] }, { "title": "6 Conclusions", "content": "Decaying DM has come to the front stage recently as an explanation, alternative to annihilating DM, for the anomalies in CR cosmic rays in PAMELA , Fermi and H.E.S.S. . But, more generally, decaying DM is a viable possibility that is or can naturally be embedded in many DM models. It is therefore interesting to explore its parameter space in the light of the recent observational results. We discussed the constraints which originate from the measurement of the isotropic γ -ray background by Fermi and of the Fornax cluster by H.E.S.S. , for a number of decaying channels and over a range of DM masses from 100 GeV to 30 TeV. We improved the analysis over previous work by using more recent data and updated computational tools. We found that the constraints by Fermi rule out decaying half-lives of the order of 10 26 to few 10 27 seconds. These therefore exclude the decaying DM interpretation of the charged CR anomalies, for all 2-body channels, at least adopting our fiducial constraint procedure. The constraints by H.E.S.S. are generally subdominant. For the DM → τ + τ -channel, they can however also probe the CR fit regions and essentially confirm the exclusion. With one possible exception for the DM → b ¯ b channel, the constraints that we derive from the isotropic γ -ray flux are the most stringent to date. We also discussed the prospects for the future ˇ Cerenkov telescope CTA , which will be able to probe an even larger portion of the parameter space.", "pages": [ 4 ] }, { "title": "References", "content": "arXiv::0905.0105 Astrophys. J. 462 (1996) 563 (2012) 042 [17] Talk by S. Zimmer and J. Conrad at the FERMI Symposium 2011. [18] E. Aliu, et al. [VERITAS Coll.], Phys. Rev. D 85 (2012) 062001 [19] R. Abbasi, et al. [IceCube Coll.], Phys. Rev. D 84 (2011) 022004 [20] The CTA Consortium (M. Actis et al.), Exper. Astron. 32 (2011) 193 [21] C. Combet, et al. , Phys. Rev. D 85 , 063517 (2012)", "pages": [ 4, 5 ] } ]

[ { "title": "Solar neutrino analysis of Super-Kamiokande", "content": "HIROYUKI SEKIYA 1 , 2 FOR THE SUPER-KAMIOKANDE COLLABORATION. [email protected] Abstract: Super-Kamiokande-IV data taking began in September of 2008, and with upgraded electronics and improvements to water system dynamics, calibration and analysis techniques, a clear solar neutrino signal could be extracted at recoil electron kinetic energies as low as 3.5 MeV. The SK-IV extracted solar neutrino flux between 3.5 and 19.5 MeV is found to be (2.36 ± 0.02(stat.) ± 0.04(syst.)) × 10 6 /(cm 2 sec). The SK combined recoil electron energy spectrum favors distortions predicted by standard neutrino flavour oscillation parameters over a flat suppression at 1 σ level. A maximum likelihood fit to the amplitude of the expected solar zenith angle variation of the elastic neutrino-electron scattering rate in SK, results in a day/night asymmetry of -3 . 2 ± 1 . 1(stat.) ± 0.5(syst.)%. The 2.7 σ significance of non-zero asymmetry is the first indication of the regeneration of electron type solar neutrinos as they travel through Earth's matter. A fit to all solar neutrino data and KamLAND yields sin 2 θ 12 = 0 . 304 ± 0 . 013, sin 2 θ 13 = 0 . 031 + 0 . 017 -0 . 015 and ∆ m 2 21 = 7 . 45 + 0 . 20 -0 . 19 × 10 -5 eV 2 . Keywords: Solar neutrino, neutrino oscillation, matter effects.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Solar neutrino flux measurements from Super-Kamiokande (SK) [1] and the Sudbury Neutrino Observatory(SNO) [2] have provided direct evidence for solar neutrino flavor conversion. However, there is still no clear evidence that this solar neutrino flavor conversion is indeed due to neutrino oscillations and not caused by any other mechanism. Currently there are two testable signatures unique to neutrino oscillations. The first is the observation and precision test of the MSW resonance curve [3]. Based on oscillation parameters extracted from solar neutrino and reactor antineutrino measurements, there is an expected characteristic energy dependence of the flavor conversion. The higher energy solar neutrinos (higher energy 8 B and hep neutrinos) undergo complete resonant conversion within the sun, while the flavor changes of the lower energy solar neutrinos (pp, 7 Be, pep, CNO and lower energy 8 B neutrinos) arise only from vacuum oscillations, which limits the average electron flavor survival probability to exceed 50%. The transition from the matter dominated oscillations within the sun, to the vacuum dominated oscillations, should occur near 3 MeV, making 8 B neutrinos the best choice when looking for a transition point within the energy spectrum. A second signature unique to oscillations arises from the effect of the terrestrial matter density on solar neutrino oscillations. This effect is tested directly by comparing solar neutrinos which pass through the Earth at nighttime to those which do not during the daytime. Those neutrinos which pass through the Earth will in general have an enhanced electron neutrino content compared to those which do not, leading to an increase in the nighttime electron elastic scattering rate (or any charged-current interaction rate), and hence a negative 'day/night asymmetry'. SK detects 8 B solar neutrinos over a wide energy range in real time, making it a prime detector to search for both solar neutrino oscillation signatures. In this Presentation, the energy spectrum results of SKIV, the combined SK day/night asymmetry analysis, and an oscillation analysis of SK data and a global analysis which combines the SK results with other relevant experiments are presented.", "pages": [ 1 ] }, { "title": "2 Improvements of Super-Kamiokande IV", "content": "Super-Kamiokandeis a large, cylindrical, water Cherenkov detector consisting of 50,000 tons of ultra pure water located underground, 1000 m underneath Mount Ikenoyama, in Kamioka City, Japan. The SK detector is optically separated into a 32.5 kton cylindrical inner detector (ID) surrounded by a 2.7 meter active veto outer detector (OD). The structure dividing the detector regions contains an array of photo-multiplier tubes (PMTs). In October of 2006, with 11,129 inner and 1,885 outer PMTs, data taking restarted as the SK-III phase [5]. The fourth phase of SK (SK-IV) began in September of 2008, with new front-end electronics for both the inner and outer detectors, and continues to run. Improving the front-end electronics, the water circulation system, calibration techniques and the analysis methods have allowed the SK-IV solar neutrino measurement to be made with a lower energy threshold and with a lower systematic uncertainty, compared to SK-I, II and III. The new front-end electronics called QBEEs were installed, allowing for the development of a new online data acquisition system. The essential components on the QBEEs, used for the analog signal processing and digitization, are the QTC (high-speed Charge-to-Time Converter) ASICs [6], which achieve very high speed signal processing and allow the recording of every hit of every PMT. The resulting PMT hits information are sent to online computers where a software trigger searches for timing coincidences within 200 ns to pick out events. The energy threshold of this software trigger is only limited by the speed of the online computers. Ultra-pure water is continuously supplied from the bottom of the detector and drained from the top, as it is circulated through the water purification system with a flow rate of 60 ton/hour. If a temperature gradient exists within the detector and the supply water temperature is different from the detector temperature, convection can occur throughout the detector volume and radioactive radon gas, which is usually produced by decays from the U/Th chain near the edge of the detector, can make its way into the center of the detector. Radioactivity coming from the decay products of radon gas, most commonly 214 Bi beta decays, can mimic the lowest energy solar neutrino events. In January of 2010, a new automated temperature control system was installed, allowing for control of the supply water temperature at the ± 0 . 01 K level. By controlling the water flow rate and the supply water temperature (within 0.01 K), convection within the tank can be kept to a minimum and the background level in the central region of the fiducial volume has since become significantly lower. In addition to hardware improvements, a new analysis method had been developed. Even at the low energies of the 8 B solar neutrinos, it is possible to use the PMT hit pattern of the Cherenkov cones to reconstruct the multiple Coulomb scattering of the resultant electrons. Very low energy electrons will incur more multiple scattering than higher energy electrons and thus have a more isotropic PMT hit pattern. The radioactive background events such as 214 Bi beta decays generally have less energy than 8 B solar neutrinos. To characterize this hit pattern anisotropy, a 'direction fit ' goodness is used. This goodness is constructed by first projecting 42 · cones from the vertex position, centered around each PMT that was hit within a 20 ns time window (after time of flight subtraction). Pairs of such cones are then used to define 'event direction candidates', which are vectors taken from the vertex position to the intersection points of the two projected cones on the detector surface. Only cone pairs which intersect twice are taken as 'event direction candidates'. Clusters of these candidates are then found by forming vector sums which are within 50 · of a 'central event direction'. Once an 'event direction candidate' has been used in the formation of a cluster, it is then not used as a 'central event direction' and is skipped in further vector sums. Further iterations of this process will use the vector sums as the 'central event directions', serving to maximize and center the clusters. After a couple of iterations, the vector sum with the largest magnitude is kept as the 'best fit direction'. The multiple scattering goodness (MSG) is then defined by taking the magnitude of the largest vector sum (the 'best fit direction') and normalizing it by the number of 'event directions' which would result using all hit PMTs within the 20 ns time window. For example, a MSG value of 0.4 would mean that 40% of all 'event directions' based on hit PMTs within 20 ns are included in the vector sum.", "pages": [ 1, 2 ] }, { "title": "3 Analysis Results", "content": "The start of physics data taking occurred on October 6th, 2008, with this paper including data taken until December 31st, 2012. The total livetime is 1306.3 days. The entire data period was taken using the same low energy threshold, with 84% triggering efficiency at 3.5-4.0 MeV, 99% at 4.04.5 MeV and 100% above 4.5 MeV kinetic energy. In the case of ν -e interactions of solar neutrinos in SK, the incident neutrino and recoil electron directions are highly correlated. Fig.1 shows the cos θ sun distribution for events between 3.5-19.5 MeV as well as the definition of cos θ sun. In order to obtain the number of solar neutrino interactions, an extended maximum likelihood fit is used. This method is also used in the SK-I [1], II [4], and III [5] analyses. The solid line of Fig.1 is the best fit to the data. The dashed line shows the background component of that best fit. SK-IV has N bin = 22 energy bins; 19 bins of 0.5 MeV width between 3.5-13.5 MeV, two energy bins of 1 MeV between 13.5 and 15.5 MeV, and one bin between 15.5 and 19.5 MeV. Below 7.5 MeV, each bin is split into three sub-samples of MSG, with boundaries set at MSG=0.35 and 0.45. These three sub-samples are then fit simultaneously to a single signal and three independent background components. The combined systematic uncertainty of the total flux in SK-IV is found to be 1.7% as the quadratic sum of all components. This is the best value seen throughout all phases of SK, much improved over 2.2% in SK-III. The main contributions to the reduction come from improvements in the uncertainties arising from the energy-correlated uncertainties (energy scale and resolution), the vertex shift, trigger efficiency and the angular resolution. SK-III data below 6.0 MeV recoil electron kinetic energy has only about half the livetime as the data above, while SK-IV's livetime is the same for all energy bins. As a consequence, the energy scale and resolution uncertainties lead to a smaller systematic uncertainty of the flux in SK-IV than in SK-III. The addition of the 3.5 to 4.5 MeV data lessens the impact of energy scale and resolution uncertainty on the flux determination even further. The number of solar neutrino events (between 3.5 and 19.5 MeV) is 25 , 222 + 252 -250 ( stat . ) ± 429 ( syst . ) . This number corresponds to a 8 B solar neutrino flux of Φ 8 B = ( 2 . 36 ± 0 . 02 ( stat . ) ± 0 . 04 ( syst . )) × 10 6 / ( cm 2 sec ) , assuming a pure ν e flavor content. Fig.3 shows the resulting SK-IV energy spectrum, where below 7.5 MeV MSG has been used and above 7.5 MeV the standard signal extraction method without MSG is used. To test the expected 'upturn' distortion below ∼ 6 MeV from the MSW resonance effects, energy-dependent parameterized functions of the ν e survival probability ( Pee ) were fitted to all the SK-I to SK-IV spectra like SNO performed [7, 8]. The fitting result shows that SK spectra disfavors flat suppression based on the constant Pee by ∼ 1 σ and favors the 'upturn' with also ∼ 1 σ significance. The SK-IV livetime during the day (night) is 626.4 days (679.9 days). The solar neutrino flux between 4.5 and 19.5 MeV and assuming no oscillations is measured as Φ D = ( 2 . 29 ± 0 . 03 ( stat . ) ± 0 . 05 ( sys . )) × 10 6 / ( cm 2 sec ) during the day and Φ N = ( 2 . 42 ± 0 . 03 ( stat . ) ± 0 . 05 ( sys . )) × 10 6 / ( cm 2 sec ) during the night. A more sophisticated method to test the day/night effect is given in [1, 9]. For a given set of oscillation parameters, the interaction rate as a function of the solar zenith angle is predicted. Only the shape of the calculated solar zenith angle variation is used, the amplitude of it is scaled by an arbitrary parameter. The extended maximum likelihood fit to extract the solar neutrino signal is expanded to allow time-varying signals. The likelihood is then evaluated as a function of the average signal rates, the background rates and the scaling parameter which is called the 'day/night amplitude'. The equivalent day/night asymmetry is calculated by multiplying the fit scaling parameter with the expected day/night asymmetry. In this manner the day/night asymmetry is measured more precisely statistically. Because the amplitude fit depends on the assumed shape of the day/night variation, it necessarily depends on the oscillation parameters, although with very little dependence expected on the mixing angles (in or near the large mixing angle solutions and for θ 13 values consistent with reactor neutrino measurements [10]). The day/night asymmetry coming from the SK-I to IV combined amplitude fit can be seen as a function of recoil electron kinetic energy in Fig.4, for ∆ m 2 21 = 4 . 89 × 10 5 eV 2 , sin 2 12 = 0 . 314 and sin 2 13 = 0 . 025. The day/night asymmetry in this figure is found by multiplying the fitted day/night amplitude from each energy bin, to the expected day/night asymmetry (red distribution) from the corresponding bin.", "pages": [ 2, 3 ] }, { "title": "4 Oscillation Analysis", "content": "We analyzed the SK-IV elastic scattering rate, the recoil electron spectral shape and the day/night variation to constrain the solar neutrino oscillation parameters. The combination of SK-I, II, III and IV solar neutrino data measure the solar mixing angle to sin 2 θ 12 = 0 . 341 + 0 . 029 -0 . 025 and the solar neutrino mass splitting to ∆ m 2 21 = 4 . 8 + 1 . 8 -0 . 9 × 10 -5 eV 2 . We then combined the SK-IV constraints with those of previous SK phases, as well as other experiments. The allowed contours of all solar neutrino data (as well as KamLAND's constraints) are shown in Fig.6 and 7. In Fig.6 the contours from the fit to all solar neutrino data are almost identical to the ones of the SK+SNO combined fit. In figures some tension between the solar neutrino and reactor anti-neutrino measurements of the solar ∆ m 2 21 is evident. This tension is mostly due to the SK day/night measurement. Even though the expected amplitude agrees well within 1 σ with the fitted amplitude for any ∆ m 2 21 in either the KamLAND or the SK range, the SK data somewhat favor the shape of the variation predicted by values of ∆ m 2 21 that are smaller than KamLAND's. In Fig.7, the significance of non-zero θ 13 from the solar+KamLAND data combined fit is about 2 σ .", "pages": [ 4 ] }, { "title": "5 Conclusion", "content": "In the fourth phase of SK we measured the solar 8 B neutrino-electron elastic scattering rate with the highest precision yet, (2.36 ± 0.02(stat.) ± 0.04(syst.)) × 10 6 /(cm 2 sec). We find a 2.7 σ indication for the existence of a solar day/night effect in the SK solar neutrino data, measured as the solar neutrino elastic scattering day/night rate asymmetry of -3 . 2 ± 1 . 1(stat.) ± 0.5(syst.)%. SK's solar zenith angle variation data results in the world's most precise measurement of ∆ m 2 21 = 4 . 8 + 1 . 8 -0 . 9 × 10 -5 eV 2 , using neutrinos rather than antineutrinos. A fit to all solar neutrino data and KamLAND yields sin 2 θ 12 = 0 . 304 ± 0 , 013, sin 2 θ 13 = 0 . 031 + 0 . 017 -0 . 015 and ∆ m 2 21 = 7 . 45 + 0 . 20 -0 . 19 × 10 -5 eV 2 . This value of θ 13 is in agreement with reactor neutrino measurements.", "pages": [ 4 ] } ]

[ { "title": "Upper Limits From Five Years of VERITAS Blazar Observations", "content": "MATTEO CERRUTI 1 , FOR THE VERITAS COLLABORATION. 1 Harvard-Smithsonian Center for Astrophysics; 60 Garden Street, Cambridge, MA 02138, USA [email protected] Abstract: The VERITAS array of Cherenkov telescopes was used to observe ≈ 130 blazars from 2007 to 2012. Of these, 25 were detected as very-high-energy (VHE; E > 100 GeV) sources. We present here the results of the analysis of 65 VERITAS non-detected blazars, including upper limits on their VHE flux. Results from a stacked analysis of the entire data set and of smaller sub-sets (defined as a function of the redshift and the blazar class) are presented and discussed. Keywords: Gamma-rays: observations; Galaxies: active; Blazars", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The extra-galactic sky observed at very-high-energies (VHE; E > 100 GeV) is dominated by blazars, a class of active galactic nuclei (AGN). In the framework of the AGN unified model [1], blazars are considered as radioloud AGN whose relativistic jet is aligned to the observer's line of sight. The observational properties of blazars are an optical/UV spectrum dominated by a non-thermal continuum, a high degree of polarization, extreme temporal variability and a spectral energy distribution (SED) composed of two distinct bumps, peaking in millimiter-to-X-rays and γ -rays, respectively. The blazar class is divided into the two sub-classes of FSRQs (flat-spectrum radio quasar) and BL Lac objects, depending on whether emission lines are observed (in FSRQs) or not (in BL Lacs) in the optical spectrum [2]. The origin of the lower-frequency SED component is ascribed to synchrotron emission by a population of nonthermal electrons in the jet. The high-energy component is generally associated, in leptonic models, with inverseCompton scattering of low energy photons (the synchrotron photons themselves, or an external photon field, see [3, 4]) off the electrons in the emitting region. The frequency of the synchrotron peak is used to further classify blazars (see e.g. [5]): if ν peak < 10 14 Hz, the object is classified as low-synchrotron-peaked blazar (LSP), while if ν peak > 10 15 Hz, it is classified as high-synchrotron peaked blazar (HSP). Sources with 10 14 ≤ ν peak ≤ 10 15 Hz are classified as intermediate-synchrotron-peaked blazars (ISP). While FSRQs are essentially all LSPs, BL Lac ob- jects show a variety of synchrotron peak frequencies. Interestingly, the population of known VHE blazars is not homogeneous, being dominated by HSP blazars 1 . Another characteristic of VHE blazars is related to absorption on the extra-galactic background light (EBL, see [6]). VHE photons interact with EBL photons to pair-produce electron-positron pairs. This absorption effect defines a horizon, above which the VHE flux from the source is significantly reduced. Up to now, the most distant VHE blazars detected have a redshift of z ≈ 0.6 (see [7, 8]). The VERITAS array has observed more than one hundred blazars since its beginning of operation in 2007. Among them, only 25 have been detected as VHE emit- ters (using the standard significance threshold of σ > 5). We present here the preliminary results of the analysis of 65 non-detected blazars. The sample includes different blazar sub-classes (HSP vs ISP/LSP blazars) in different redshifts, several unidentified Fermi sources as well as two Seyfert 1 galaxies (type-1 radio-quiet AGN).", "pages": [ 1 ] }, { "title": "2 VERITAS observations", "content": "VERITAS is an array of four 12-m diameter atmospheric Cherenkov telescopes, located at the Fred Lawrence Whipple Observatory (FLWO) in southern Arizona (31 40N, 110 57W, 1.3km a.s.l.). It is sensitive to γ -rays with an energy above 85 GeV up to 30 TeV, with an energy resolution of 15-20% and an angular resolution R 68% < 0 . 1 · . The field of view of VERITAS is roughly 3 . 5 · . For more details see [10]. The interaction of the γ -ray with the Earth's atmosphere triggers a particle shower, which emits Cherenkov photons. The images of the Cherenkov flashes are used to determine the direction and energy of the incoming photon. In the data analysis, different cuts are applied to the parameters of the Cherenkov images to differentiate between γ -rayinduced showers, and cosmic-ray-induced showers, which represent the main background. The cuts are optimized for sources with different spectral indexes. For our analysis we have used medium and soft cuts, optimized for sources with an index Γ = -2 . 4 and -4 . 0, respectively (see e.g. [11]). The observations have been performed using the standard wobble observation configuration, where the telescopes are pointed 0 . 5 · away from the source to allow a simultaneous estimation of the background. The results presented in the next Section have been cross-checked using an independent analysis.", "pages": [ 1 ] }, { "title": "4 Conclusions", "content": "For each observed source we computed the significance of the detection for both medium and soft cuts. Given that none of the sources is detected ( σ < 5, and indeed all the sources have σ < 4), for each source we computed the correspondent upper limit on the VHE flux as well. Upper limits are computed at the 99% confidence level, following [12].", "pages": [ 3 ] }, { "title": "3.1 Significance distributions", "content": "In Figure 1, we show the significance distributions for all our sources and for both cuts. Above the histograms we plot the gaussian distribution expected from a randomly distributed sample, with mean equal to zero, and variance equal to one. Both distributions deviate from this expectation: the best-fit gaussian functions have a mean of 0 . 34 and 0 . 46, with variance equal to 1 . 08 and 1 . 26, for medium and soft cuts, respectively. We studied two different properties of the sample: the distribution of significances as a function of the distance (top plots of Figure 1), and of the blazar class (medium plots). The major contribution to the positive values of the significance distributions is provided by sources with z < 0 . 6, and by HSP blazars, respectively. The first result agrees with the expected absorption of TeV photons by the EBL, which strongly reduces the flux from distant (z > 0.6) blazars; the latter is consistent with the current population of AGN detected by IACTs, which is dominated by HSP blazars. In the bottom plots of Fig. 1 we report the distribution of the integral upper limits, estimated above the corresponding energy threshold of each source. For the computation of the upper limits, we assumed an index of Γ = -2 . 5 for medium cuts, and Γ = -3 . 5 for soft cuts. The upper limits obtained are, for most of the sources, the first upper limits ever produced at VHE.", "pages": [ 3 ] }, { "title": "3.2 Stacked analysis", "content": "To estimate the significance of the excess that can be seen in Figure 1, we performed a stacked analysis of our results. We summed all the excesses observed, and their correspondent uncertainties, computing the significance of the stacked excess. We made this evaluation for the entire data set and for dedicated subsets. The fact that the significance distributions are skewed towards positive values is confirmed by the significance of the stacked excess: for the entire data set, the stacked significance is σ = 3 . 5, for medium cuts, and σ = 3 . 9, for soft cuts, corresponding to an excess of 368 and 1411 γ -rays, respectively. The significance is larger when considering the most favorable of the subsets: HSPs located at z < 0 . 6. The stacked analysis on this subset yields σ = 3 . 9 and σ = 4 . 1 for medium and soft cuts, respectively. On the other hand, if we consider another subset, composed by all sources apart from the HSPs, at a redshift higher than 0 . 6, we do not see any excess: in this case we have σ = 0 . 7 and σ = 0 . 4 for the two cuts. We have presented here the preliminary results from the analysis of 65 extra-galactic sources, observed by VERITAS from 2007 to 2012. The sample includes mainly blazars, plus several unidentified Fermi sources, plus two AGN classified as Seyfert 1 galaxies. None of these sources are detected, all having a significance lower than 4. We have shown the significance distributions of the sample, divided by redshift and by AGN class, as well as the distributions of the integral upper limit values. The significance distributions appear skewed towards positive values when compared to the expectation from a randomly distributed sample. The same trend is shown by a stacked analysis, that has been done for the entire data set, and smaller sub-sets. The entire data set has a stacked significance of 3 . 5 and 3 . 9 σ for the two cuts used in the analysis. The excess is clearly associated with nearby HSPs, as confirmed by the stacked analysis of the relative data. Non-detection of VHE emission from a candidate source, and the estimation of the correspondent upper limit on its flux, is important for several reasons. Firstly, upper limits can constrain the modeling of the γ -ray emission, especially for sources detected by Fermi at lower energies, but not seen by Cherenkov telescopes. Secondly, given the extreme variability observed in blazars, if one of these sources will be detected in the future during a flaring state, previous upper limits are fundamental to constrain the variability properties of the object. Finally, all these estimations will be useful in the perspective of observations with the next-generation VHE observatory, CTA. Acknowledgment: This research is supported by grants from the U.S. Department of Energy Office of Science, the U.S. National Science Foundation and the Smithsonian Institution, by NSERC in Canada, by Science Foundation Ireland (SFI 10/RFP/AST2748) and by STFC in the U.K. We acknowledge the excellent work of the technical support staff at the Fred Lawrence Whipple Observatory and at the collaborating institutions in the construction and operation of the instrument.", "pages": [ 3 ] } ]

[ { "title": "Using Raster Scans of Bright Stars to Measure the Relative Total Throughputs of Cherenkov Telescopes", "content": "SEAN GRIFFIN 1 , AND DAVID HANNA 1 1 Department of Physics McGill University Montreal, QC H3A 2T8, Canada [email protected] Abstract: Gamma-ray astronomy at energies in excess of 100 GeV is carried out using arrays of imaging Cherenkov telescopes. Each telescope comprises a large reflector, of order 10 m diameter, made of many mirror facets, and a camera consisting of a matrix of photomultiplier pixels. Differences in the total throughput between nominally identical telescopes, due to aging of the mirrors and PMTs and other effects, should be monitored to reduce possible systematic errors. One way to directly measure the throughput of such telescopes is to track bright stars and measure the photocurrents produced by their light falling on camera pixels. We have developed such a procedure using the four telescopes in the VERITAS array. We note the technique is general, however, and could be applied to other imaging Cherenkov experiments. For this measurement, a raster scan is performed on a single star such that its image is swept across the central pixels in the camera, thus providing a statistically robust set of measurements in a short period of time to reduce time-dependent effects on the throughput. Photocurrents are measured using the starlight-induced baseline fluctuations of the pixel outputs, as recorded by the standard readout electronics. In this contribution we describe details of the procedure and report on feasibility studies carried out during the 2012-2013 observing season. Keywords: Cherenkov Telescopes, Calibration", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Very-high-energy (VHE) gamma-ray astronomy makes use of arrays of imaging atmospheric Cherenkov telescopes (IACTs). Observations made with more than one telescope achieve better background rejection, improved energy and angular resolution, and are immune to the effects of local muons. An issue that arises when using multiple telescopes is that of their relative calibration. Differences between nominally identical telescopes can come about from differential aging of mirror facets and photomultiplier tubes (PMTs) or from different maintenance or upgrade schedules. A simple parameter that can be used to correct for the overall effect of such changes is the relative total throughput of a telescope. This parameter can be estimated using a variety of techniques, such as the inclusive rate for cosmic-ray showers [1], analysis of shower-image sizes [2], signals from local muons [3] acquired using special triggers, and observations of scattered light from a distant laser beam [4, 5]. A solid understanding of a telescope's calibration will result in the same number emerging from each of the techniques and the origin of changes to the number can be determined by examining data from component-specific calibration procedures such as light-pulse PMT calibration [6] and wholedish mirror reflectivity measurements [10]. In this contribution we describe a total-throughput measurement procedure based on using photocurrents induced by the image of a bright star falling on PMTs in a telecope's camera. The method was developed for the VERITAS array and we report here on initial tests made with that instrument. However the method is quite general and can be used for other arrays. Initial tests were conducted using magnitude 7 stars with spectra very different from the standard Cherenkov spectrum relevant to air-shower detection. We are currently exploring the use of ultraviolet filters, already acquired for observing under bright moonlight, to extend this technique to shorter wavelengths.", "pages": [ 1 ] }, { "title": "2 The VERITAS IACT Array", "content": "VERITAS comprises an array of four IACTs located at the Whipple Observatory at the base of Mount Hopkins in southern Arizona [7, 8]. Each of the telescopes is based on a 12-m diameter Davies-Cotton reflector focussing light onto a 499-pixel camera made from close-packed Hamamatsu R10560 PMTs coupled to conical light concentrators. Each reflector is made up of 345 identical mirror facets, the alignment of which is such that the on-axis point-spreadfunction is smaller than a pixel diameter [9].", "pages": [ 1 ] }, { "title": "3 Raster Scanning", "content": "The total throughput of one of the telescopes could be measured by tracking a single star and measuring the photocurrent from the camera's central pixel. However it is statistically more powerful to illuminate several pixels, in sequence, in order to average out effects such as differences in the wavelength dependence of quantum efficiency in different PMTs. The standard tracking software for VERITAS is designed to map a telescope's nominal pointing direction onto the central pixel and to change that to an arbitrary pixel required modifications. Such modifications were implemented as part of the VERITAS mirror alignment scheme [9] whereby a raster scan over a grid centred on a bright star is performed. We have adapted this scanning technique to cause the image of a star to sweep over the central pixels of the camera in a controlled fashion and with a grid step that is small enough to obtain data with the star image very close to the centre of a given pixel. For the data reported on here we used a 25-by-25 array of pointings with a step size of 0.03 degrees. A VERITAS pixel has a field-of-view diameter of 0.15 degree so the grid should cover a square array of order 25 pixels. The mismatch between the hexagonal nature of the PMT positions and the square scan grid reduces this to 23 pixels (Figure 1).", "pages": [ 1, 2 ] }, { "title": "4 Photocurrent Measurement", "content": "The readout electronics for a VERITAS pixel consist of a preamp in the PMT base followed by an amplifier and a 500 MS/s FADC located in the electronics shed under the telescope. The signals are AC coupled; a capacitor before the preamp blocks the DC photocurrent. However, a resitive path to ground upstream of the capacitor is provided for purposes of monitoring this current. Low-resolution (0.5 m A step size) measurements are available, mainly to allow for switching off the high-voltage to a PMT in case of excess currents. For our purposes we need finer granularity. To obtain this we make use of the fact that baseline fluctuations (pedestal variations) as recorded by the FADC readout can be used as a proxy for photocurrent. This can be motivated by a simple model that posits current as coming from a stream of single-photoelectron pulses approximated as narrow digital pulses. The empirical proof of the correlation is shown in Figure 2. Data for this figure were extracted from standard observing runs made under partial moonlight but while the moon was setting so the currents vary over an interesting range. In the upper left panel we plot the current readings for an arbitrary pixel as a function of time in minutes. In the upper right panel we plot the corresponding smoothed baseline variances. For every event (approximately 300 times per second for these runs; the rate is random and dominated by cosmic ray triggers) a 16-sample FADC trace is recorded for each pixel and the variance can be calculated. Most traces are empty except for fluctuations due to night sky background photons (the scale of these fluc- tuations is much less than the scale of Cherenkov pulses). However, since the data were acquired under normal trigger conditions, there are some traces with Cherenkov pulses in them and these cause long tails in the variance distributions. To deal with this we use a 'mean of medians' technique. First, we divide the data stream into groups of 300 events. These are further subdivided into 60 subgroups of 5 and we average the 60 medians from each 5-member subgroup, obtaining an average variance estimate approximately once per second. In the lower left panel we plot the average variances as a function of current. The linear correlation is evident and justifies the use of baseline variance as a proxy for current in the following. The slopes of the fitted line from plots like that in Figure 2 are used to convert the increment in baseline variance due to the effect of the star to an increment in photocurrent.", "pages": [ 2 ] }, { "title": "5 Test Results", "content": "Raster-scan data for this study were acquired on two separate occasions. For each run a magnitude 7 star was chosen as the target and the telescopes were slewed to its coordinates. The raster scan was then performed and at the end the telescopes returned to their nominal tracking directions. The scan was carried out with a one-second dwell time at each pointing and one second between points for slewing and settling so the 625-point scan took just over 20 minutes. The target star was selected to be rising and close to transit; even though the run was reasonably short, we wanted to avoid systematic effects due to changes in flux due to atmospheric absorption. Data were acquired by externally triggering the array at a fixed rate of 300 Hz. Results from the central pixel in one of the telescopes are shown in Figure 3 where we plot baseline variance (current proxy) as a function of time in seconds. The elevated currents at the beginning and end of the run are due to the tracking of the target star such that its image is contained within the central pixel. This feature is absent in Figure 4 where data from an off-centre pixel are plotted. In both figures one sees structure resulting from the scan where currents rise and fall as the star image is swept across the pixel field-of-view and the maximum of each peak rises and falls as the distance of the scan line from the centre of the pixel varies. In the following, we use the amplitude of the largest peak to make an estimate of the current that would result if the star's image were exactly centred on the pixel. In Figure 5 we plot the peak currents achieved on one night vs the peak currents from the previous night for a single telescope. It is clear that the results obtained are reproducible over the short term and that the statistical errors are understood. The peak currents from this telescope for a single night ( i.e. the x projection of data plotted in Figure 5) are plotted in Figure 6. The dispersion is relatively large; the RMS is slightly more than 10% of the mean. The reasons for this are under study but may be partly due to the fact that the star's image does not cross over the exact center of every camera pixel. Similar results have been obtained from all telescopes in the VERITAS array and we are currently evaluating the level of systematic errors to be expected. Already the statistical errors indicate that we can expect to measure differences in relative throughput of a few percent.", "pages": [ 2, 3 ] }, { "title": "6 Conclusions", "content": "We have tested a method for measuring the net throughputs of different telescopes in an array. The procedure requires no specialized equipment and can be carried out in less than 30 minutes, possibly during periods where moonlight or non-optimal weather lessen the competition for observing time. The initial results are very encouraging and we expect to pursue this in the future to look at long-term stability and possible improvements to the precision of the method. Acknowledgments: We warmly thank our colleagues in the VERITAS collaboration for their support of this work and for assistance with data acquisition. VERITAS research is supported by grants from the U.S. Department of Energy Office of Science, the U.S. National Science Foundation and the Smithsonian Institution, by NSERC in Canada, by Science Foundation Ireland (SFI 10/RFP/AST2748) and by STFC in the U.K. We acknowledge the excellent work of the technical support staff at the Fred Lawrence Whipple Observatory and at the collaborating institutions in the construction and operation of the instrument.", "pages": [ 3 ] } ]

[ { "title": "Detection prospects for short time-scale transient events at VHE with current and next generation Cherenkov observatories", "content": "S. LOMBARDI 1 , 2 , A. CAROSI 1 , 2 , L.A. ANTONELLI 1 , 2 1 INAF-Osservatorio Astronomico di Roma, Via Frascati 33, 00040 Monte Porzio Catone (RM), Italy [email protected] Abstract: In the current view of Gamma-Ray Burst (GRB) phenomena, an emission component extending up to the very-high energy (VHE, E > 30 GeV) domain is though to be a relatively common feature at least in the brightest events. This leads to an unexpected richness of possible theoretical models able to describe such phenomenology. Hints of emission at tens of GeV are indeed known since the EGRET observations during the '90s and confirmed in the Fermi -LAT data. However, our comprehension of these phenomena is still far to be satisfactory. In this respect, the VHE characterization of GRBs may constitute a breakthrough for understanding their physics and, possibly, for providing decisive clues for the discrimination among different proposed emission mechanisms, which are barely distinguishable at lower energies. The current generation of Cherenkov observatories, such as the MAGIC telescopes, have opened the possibility to extend the measurement of GRB emission, and in general to any short time-scale transient phenomena, from few tens of GeV up to the TeV energy range, with a higher sensitivity with respect to γ -ray space-based instruments. In the near future, a crucial role for the VHE observations of GRBs will be played by the Cherenkov Telescope Array (CTA), thanks to its about one order of magnitude better sensitivity and lower energy threshold with respect to current instruments. In this contribution, we present a method aimed at providing VHE detection prospects for observations of GRB-like transient events with Cherenkov telescopes. In particular, we consider the observation of the transient event GRB 090102 as a test case for the method and show the achieved detection prospects under different observational conditions for the MAGIC telescopes and CTA. Keywords: icrc2013, VHE, Cherenkov telescopes, short time-scale transient GRB-like events.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Time domain astrophysics is going to play a key role in our understanding of different kind of cosmic sources. In particular, the discovery of high-energy γ -rays from an unexpected large variety of transient events with time-scale ranging from millisecond up to days poses a new series of theoretical problems [1]. The list of γ -ray band transient sources comprises both local phenomena, as terrestrial and solar γ -ray flares, as well as galactic and extra-galactic transient events. Furthermore, short time-scale variability has long been observed in active galaxies, especially for blazars-class objects [2]. The extension, when possible, of the multi-wavelength coverage up to very high energy (VHE, E > 30 GeV) can provide powerful diagnostic tools to understand the nature of these objects and discriminate among the different proposed interpretative scenarios. In particular, Gamma-Ray Bursts (GRBs) have long been seen as the transient events per excellence. At their peak activity, GRBs become the most luminous objects of the Universe releasing enormous amounts of energy from 10 52 erg to 10 54 erg of isotropic-equivalent energy over brief periods of 0 . 01 - 1000 s. They usually show their phenomenology mainly in the 10 keV - 1 MeV energy band with extremely rapid and irregular variability (see e.g. [3] for a review). However, recent results from the Fermi -LAT(Large Area Telescope) have showed that, at least for the brightest events, a GeV emission from GRBs is a relatively common phenomenon [4]. Interestingly, in the majority of the LAT GRBs, GeV emission occurs with a significant delay with respect to the MeV and sub-MeV emission and it last- onger than the emission detected by the Fermi -GBM (Gamma-ray Burst Monitor). While sub-MeV GRB emission can often be explained by electron synchrotron processes, the theoretical framework of a possible second emission component in the highenergy regime is still less clear. Furthermore, the recent detection of GRB 130427A with photons up to 94 GeV [5] also indicates the possibility of an observable VHE counterpart. Several authors have derived predictions for the VHE emission from the GRB prompt and afterglow phase taking into account non-thermal leptonic and hadronic processes [6] as well as photospheric up-scattered emission [7]. VHE observations with sufficiently high sensitivity may definitively solve, or at least strongly constrain, the mechanisms for prompt and early afterglow emission through broader energy band coverage. At the same time, and also for other GRB-like transient phenomena, such as Tidal Disruption Events [8], VHE data may throw light on some physic aspects which are still poorly understood, including the determination of the bulk Lorentz factor of the outflow, the dynamics of particle acceleration, and the jet formation. The capabilities of a significant detection at VHE from such a kind of events are strongly related, on the one hand, to the scientific performance of the instrument, and, on the other hand, to the short-time scale features of the γ -ray signal. The strong time dependence of the emission hence makes the usual considerations based on the sensitivity of the instrument not suitable for providing detection prospects for transient events at VHE. In this work, we illustrate a method for evaluating the detectability of GRB-like events with the MAGIC stereoscopic system and the next generation Cherenkov Telescope Array (CTA) based on the time evolution of the significance of the observation. As a test case, we consider the particular event GRB 090102 [9] (which was observed by the MAGIC-I telescope [10]) and show its detection prospect results.", "pages": [ 1, 2 ] }, { "title": "2 Observations of GRB-like events with current and next generation Cherenkov telescopes", "content": "Despite the remarkable results of the Fermi -LAT, the number of detected photons above few tens of GeV remains rather limited, motivating follow-up observations with much better sensitivity in the VHE band with the use of Imaging Atmospheric Cherenkov Telescopes (IACTs) [11]. Thanks to the technical evolution of such kind of instruments, in the last decade, intense studies have been performed on GRB science with IACTs to explore possible VHE band emission for these enigmatic events. Since already several years, current IACTs, such as MAGIC[12], H.E.S.S.[13], and VERITAS[14], despite their reduced duty cycle, started observational programs on GRB follow-up, making the ∼ 100GeV - TeV energy range accessible to GRB observations. As a matter of fact, several attempts to observe GRB emission have been reported by current IACT collaborations (e.g. [15, 16, 17]). In all cases only upper limits have been derived. However, it is well known that the flux above ∼ 100GeV is affected by the attenuation by pair production with the lower energetic (optical/IR) photons of the diffuse Extragalactic Background Light (EBL) [18]. The consequent Universe opacity heavily affect Cherenkov observations, almost hindering the detection for relatively high redshift ( z > 0 . 5) sources. This is the case for GRBs which have long been known to have redshift slightly larger than 2. This basically implies that the expected detection rate for current Cherenkov telescopes is estimated to be around 0 . 1 0 . 2 GRBs/year and should significantly improve only with the coming CTA, for which GRBs will be among primary targets [19]. The CTA project [20] aims at developing the next generation ground-based instrument dedicated to the observations in the VHE γ -ray band. In the current layout of CTA, the arrays will consist of three types of telescopes with different main mirror sizes in order to cover the full energy range from few tens of GeV up to a hundred of TeV. The lowest energy band (i.e., where GRBs are mainly foreseen to show their activity) will be covered by few 24-m Large Size Telescopes (LSTs). With respect to current IACT facilities, CTA will mainly benefit from a lower energy threshold (down to ∼ 20 GeV), a much larger effective collection area, particularly in the few tens of GeV energy range, and a sensitivity about one order of magnitude better in the whole energy range [21, 22]. Furthermore, LSTs are conceived to have rapid slewing capability with a repositioning time of around 180 · azimuthal rotation in 20s (i.e. comparable to the performance achieved by the MAGIC telescopes [23]). In some cases, this will permit GRBobservations during prompt emission phase while the majority of the events can be observed at early afterglow stage. Estimate based on different LST performance and GRB statistics currently foresees a still limited detection rate of few bursts per year [19]. However, CTA high sensitivity will permit the collection of enough VHE photons to perform time-resolved studies of the observed events.", "pages": [ 2 ] }, { "title": "3 Detection prospects for transient events at VHE", "content": "The basic quantities that are normally taken into account for evaluating the detectability at VHE of a given γ -ray source with IACTs are the sensitivity 1 of the instrument and the flux level of the source. However, these quantities are useful for detection considerations under the hypothesis of a steady γ -ray emission. In case of transient γ -ray events, whose flux is strongly time dependent, a different approach is therefore needed. In this respect, a more useful quantity that can be considered is the significance of the observation ( σ ) as a function of time, provided an emission model for the transient source. In this way, in fact, it is possible to evaluate whether the typical detection condition σ > 5 is achieved or not (in a certain energy interval and for different observational conditions). The commonly used definition of σ for IACT observations is given in Eq. 17 of [24]: where Non and Nof f are the number of events in the signal region of the ON and OFF 2 data sets, and α is the ON -OFF normalization factor expressed (for real observations) as the ratio between the effective time of the ON and OFF data sets, which implies that the expected amount of irreducible background in the ON data set is N bkg = α Nof f . Since Non and Nof f refer to a given energy interval ∆ E and are functions of time, the significance is energy and time dependent. In addition, in case of transient event observations, the starting time of observation T s = T0 + ∆ T (where T0 is the time of the transient event burst) must be taken into account to define the initial time at which the source emission must be considered. In order to evaluate how the significance of a given short time-scale transient event observation evolves with time, in a given energy interval ∆ E 3 , and for a given starting time of observation T s , the following quantities must be taken into account: where A ∆ E i ef f 4 is the (average) effective collection area of the instrument in the i -th energy bin, and d Φ /d E is the differential energy spectrum model of the given transient event emission as a function of energy and time. The effect of the γ -ray attenuation by pair production with EBL photons [18] must be taken into account in the spectrum model. where d N ∆ Ei bkg / d t is the background rate of the instrument in the i -th energy bin. In the present work, this quantity is assumed to be independent of time and of the telescope azimuthal pointing. The significance of a given transient event observation as a function of time, in a given energy interval ∆ E , and for a given starting time of observation T s , is thus given by where N [ ∆ E i , T s ] γ and N [ ∆ E i , T s ] bkg are defined in Eq. 2 and Eq. 3, respectively, and α is equal to 1 for MAGIC [21] and 0 . 2 for CTA [22]. In Tab. 1, the main quantities needed for the calculation of the significance as a function of time (provided an emission model for the transient source) for MAGIC [21] and CTA (candidate array I) [22], in 5 logarithmic energy bins between 10 1 . 6 GeV and 10 2 . 6 GeV 5 , are shown. All quantities refer to point-like source observations. For completeness, the differential sensitivities S 5 σ , 50h of the MAGIC telescopes and CTA (candidate array I) are also reported.", "pages": [ 2, 3 ] }, { "title": "4 Test case: GRB 090102", "content": "As a test case for the detection prospect method presented in Sec. 3, we consider the GRB 090102 event. This GRB was detected and located by the Swift satellite on January 2 nd , 2009, at 02:55:45 UT [9]. The MAGIC-I telescope observed GRB 090102 after ∼ 1100s from the event burst, deriving flux upper limits above ∼ 50GeV [10]. The prompt light curve was structured in four partially overlapping peaks for a total T90 of 27 . 0 ± 2 . 0s. The moderate measured redshift of z = 1 . 547 implies an isotropic energy value of Eiso = 5 . 75 × 10 53 erg. According to the relativistic blast-wave model [6], we use the Synchrotron SelfCompton (SSC) mechanism to derive the expected VHE emission during the afterglow in the IACT energy range. The Spectral Energy Distribution (SED) of the event can be expressed as where φ 0 = 0 . 78 × 10 -6 TeV cm -2 s -1 is the normalization constant at 1 TeV and 1 s, p = 2 . 29 is the index of the electrons power-law distribution [25], and e -τ ( E , z ) is the EBL absorption factor evaluated for z = 1 . 547 using the model by [18]. In Fig. 1 the modeled SED of GRB 090102, at three different times after the event burst, is shown. Using Eq. 1, Eq. 4, the quantities reported in Tab. 1, and the SED emission model defined in Eq.5, we can estimate how the significance of the GRB 090102 observation would be with the MAGIC stereoscopic system and CTA as a function of time, for different observational conditions. In Fig. 2, we present the achieved results in the energy interval 63 . 1 < E [ GeV ] < 158 . 5 (where, from our estimates, the GRB 090102 detection prospects turn out to be the most favourable) and for three different starting times of observation: T s = T0 + 180 , 600 , 1100s. A systematic error of 50% on the effective collection area values is taken into account in the significance calculations. As expected, the starting time of observation T s (in case of SSC emission model) is a crucial parameter: the earlier the IACT observation starts after the transient event burst, the higher is the possibility to detect a γ -ray signal from the source. Furthermore, it is interesting to point out how the CTA performance would allow a significant detection of the event even up to T s /similarequal T 0 + 1ks, while, in case of MAGIC observation, the source would be detectable only for starting times of observation T s < T0 + 180s.", "pages": [ 3, 4 ] }, { "title": "5 Conclusions", "content": "One of the primary goals for current IACTs, like the MAGIC telescopes, and for future Cherenkov Telescope Array is to catch VHE signal from GRBs. In this contribution, we presented a method aimed at providing detection prospects for short time-scale transient events at VHE (provided their emission model), and considered the particular event GRB 090102 as a test case. Our estimates show that, for this particular event, MAGIC follow-up observations made within a couple of minutes from the event onset would have the potential to detect the VHE component or at least to derive constraining upper limits. In fact, the steep time decay of the source (as t -1 . 1 ÷ 1 . 2 ) makes a MAGIC detection at later starting times (T s > T0 + 200s) unlikely, while interesting prospects for an afterglow significant detection in the VHE domain at such later times are possible within the CTA context. The possibility to extend our method to other classes of variable and transient sources is going to be investigated producing reliable detection prospects at VHE for the coming age of time domain astrophysics.", "pages": [ 4 ] } ]

[ { "title": "High confidence AGN candidates among unidentified Fermi-LAT sources via statistical classification", "content": "M. DOERT 1 , 3 , M. ERRANDO 2 [email protected], [email protected] Abstract: The second Fermi -LAT source catalog (2FGL) is the deepest survey of the gamma-ray sky ever compiled, containing 1873 sources that constitute a very complete sample down to an energy flux of ∼ 10 -11 erg cm -2 s -1 . While counterparts at lower frequencies have been found for a large fraction of 2FGL sources, active galactic nuclei (AGN) being the most numerous class, 576 gamma-ray sources remain unassociated. In these proceedings, we describe a statistical algorithm that finds candidate AGNs in the sample of unassociated 2FGL sources by identifying targets whose gamma-ray properties resemble those of known AGNs. Using two complementary learning algorithms and intersecting the high-probability classifications from both methods, we increase the confidence of the method and reduce the false-association rate to 11%. Our study finds a highconfidence sample of 231 AGN candidates among the population of 2FGL unassociated sources. Selecting sources out of this sample for follow-up observations or studies of archival data will substantially increase the probability to identify possible counterparts at other wavelengths. Keywords: AGN, gamma rays, Fermi-LAT, 2FGL catalog, statistical classification", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The second Fermi -LAT source catalog (2FGL) characterizes 1873 gamma-ray sources detected in the energy range of 0.1 to 100 GeV [1]. The catalog covers the whole sky with little observational bias, although the sensitivity is not uniform mainly due to the intensity of the diffuse Galactic gamma-ray emission. A total of 127 sources from the 2FGL catalog are firmly identified through simultaneous variability (periodic or episodic) or common morphology with their multiwavelength counterparts. An additional 1170 sources are reliably associated with counterparts from a-priori selected catalogs of candidate gammaray emitting source classes. The remaining 576 sources for which no counterpart was identified are left unassociated. There are fourteen classes of gamma-ray sources represented in the 2FGL catalog with at least one source count. A complete list of source types can be found in Table 1. The most numerous class are active galactic nuclei (AGN), representing 60% of the catalog. Gamma-ray emitting pulsars (4.4%), pulsar wind nebulae and supernova remnants (adding up to 3.8%) are other well-represented source classes. The rest of the catalog is distributed in unassociated sources (31%), and sources belonging to source classes with small number counts. With this work, we aim to identify unassociated sources in the 2FGL catalog whose gamma-ray properties are similar to those of gamma-ray emitting AGNs. To do that, we train two different classification algorithms on the gamma-ray properties of the known AGNs in the 2FGL catalog and apply them to the population of unassociated sources.", "pages": [ 1 ] }, { "title": "2 Source classes in the 2FGL catalog", "content": "The main goal for this work is the identification of highconfidence AGN candidates. We approach the classifica- tion of unassociated 2FGL sources as a two-class problem, where each source can either be labeled as 'AGN' or 'nonAGN'. Following this approach, we assigned one of these labels to each of the fourteen source classes represented in the 2FGL catalog, as shown in Table 1. The most abundant classes which contribute to the AGN sample are blazars (both BL Lac and FSRQs), as well as AGN of uncertain type. Most of the 'non-AGN' sources are pulsars, pulsar wind nebulae and supernova remnants. The total number of identified and associated sources is 1297, out of which we label 1092 elements as 'AGN' and 205 as 'non-AGN'. The unassociated sample comprises 576 sources. After the classification, each of these sources will be labelled as either 'AGN', 'non-AGN' or 'unclassified'. The latter applies to sources for which the confidence for a correct classification is below a certain threshold which we specify during the optimization. This way, an optimal purity of the sample of sources tentatively labeled as AGN is achieved.", "pages": [ 1, 2 ] }, { "title": "3 Selection of attributes", "content": "In the 2FGL catalog, the gamma-ray properties measured by the Fermi -LAT are reported for every source. Among these attributes are the flux values F for five energy bands with boundaries at 0.1, 0.3, 1, 3, 10 and 100 GeV, the spectral index obtained from a power law fit to the energy spectrum, parameters quantifying the flux variability and spectral curvature , and the significance associated to the detection of each source. Figure 1 shows distributions of some parameters directly extracted from the 2FGL catalog for sources in the associated sample. Here the AGNs show distinct properties compared to other source classes. To train our learning algorithms, we explored the set of attributes present in the 2FGL catalog as well as physically meaningful combinations of those, many of them already introduced in [2]. The best separation power between the populations of 'AGN' and 'non-AGN' sources in the catalog was found when using the following attributes: where HRij describes the hardness ratio between the energy fluxes measured in two contiguous spectral bands: where Fi and Ei are respectively the flux and mean energy in the i -th spectral energy band, with i = 1 being the lowest spectral band reported in the 2FGL catalog. A hardness slope parameter was also defined as which presents a powerful handle to separate possible AGN candidates from pulsar-like sources, as pulsars generally show a spectral cut-off around these energies. Additionally, we use a normalized variability defined as We do not use any variable that is directly related to the overall flux of the sources detected by Fermi -LAT as associated sources have on average higher fluxes and detection significances than the sources in the unassociated sample (see Figure 2, middle-left panel). To avoid an influence of the different ranges of the individual parameters on the classification, we renormalized all attribute distributions to the range between 0 and 1.", "pages": [ 2 ] }, { "title": "4 Analysis tools and Methods", "content": "We perform the complete classification process within the data mining framework Rapid Miner. This is an opensource software originally developed at Technische Universitat Dortmund under the name 'YALE' and now maintained and distributed by Rapid-i [3]. It offers attribute selection, combination and filtering tools, as well as a variety of built-in classification methods. Before starting the learning process, we split the associated 2FGLsources (1297 elements) into a training sample (70% of the sources) and a test sample (30%) using stratified sampling. The training sample is used to train the learning algorithms and optimize their performance, while the test sample is set aside and only used after the algorithms have been trained and optimized to evaluate their performance. We investigated a variety of supervised statistical classification methods. To achieve a good estimate of the suitability of each method, we performed a coarse optimization and assessed the performance using ten-fold crossvalidation on the training sample. Here, the sample is iteratively trained on 90% of the training sample and tested on the remaining 10%, and this process is repeated 10 times until the entire training sample has been tested. Based on their robustness and the obtained performance, we chose the random forest method (RF) and neural networks (NN) as the two algorithms to use in our study. The RF is a very powerful classification algorithm based on the fractional number of entries construction of a number of decision trees, where the attributes used for separation in each node are randomly selected [4]. The NN is a learning method which is based on different layers of interconnected nodes, so-called neurons , and has been developed as an artificial replication of the human nervous system [5].", "pages": [ 2, 3 ] }, { "title": "5 Optimization of the algorithms", "content": "Weoptimized the two selected methods further on the training sample using cross-validation, evaluating the achieved performance in terms of the fraction of sources incorrectly classified as AGNs and the fraction of true AGNs which are correctly classified as such. The two classification methods can be optimized by tuning some key parameters. For RF we used a number of trees = 100 and the depth of trees = 10, while for NN we selected a number of cycles = 1000, learning rate = 0 . 2, and momentum = 0 . 1. Each method provides a confidence value for each classified item which gives the probability for this item to be correctly labeled. We introduced additional thresholds for both confidence values of sources assigned as 'AGN' and adjusted them such that we achieved a fraction of misclassified sources in the final sample of ∼ 10%. The distributions of confidence values for true AGN and non-AGN in the test sample are shown in Fig. 3, together with the chosen thresholds for each classification method. The final classification algorithm consists in assigning the 'AGN' label only to sources that have been classified as 'AGN' by both classification algorithms independently (NN ANDRF) and passed the confidence thresholds which we defined for each of the methods. Out of all tested methods, the combination of RF and NN methods was found to have the smallest overlap of the populations of misclassified sources, thus reducing the number of wrongly classified sources. A combination of three or more methods did not deliver a considerable gain in performance compared to the loss of correctly classified sources in the final sample.", "pages": [ 3 ] }, { "title": "6 Estimation of the performance", "content": "We assess the performance of the combined algorithms and thresholds in terms of the false-association rate , i.e. the fraction of sources incorrectly classified as AGNs, and the recall , which is defined as the fraction of true AGNs which are correctly classified and appear in the final sample. The populations of associated and unassociated 2FGL sources have different distributions in significance and galactic latitude (see Fig. 2 left panels). Weaker sources with lower significance have larger location errors and are less likely to be present in counterpart catalogs or cannot be identified unequivocally, and are therefore more abundant in the unassociated sample. The same happens for sources near the galactic plane, where the bright galactic diffuse gamma-ray background makes the location of gamma-ray sources more uncertain, and some counterpart catalogs are incomplete due to the foreground and extinction on the plane at other wavelengths. Low significance and low latitude sources also pose a challenge to the classification methods used in this study. They offer less firm information about, e.g., the spectral shape or the variability, and they are influenced by the higher levels of diffuse emission. During the optimization of the algorithms, we saw that weak and low-latitude sources were much more likely to be misclassified, as shown in the right panels of Figure 2. Therefore, testing the performance of our classification method on the test sample would lead to an over-optimistic estimation of the false association rate, as the sample of unassociated sources contains a larger fraction of low-significance and low-latitude sources that our algorithms are more likely to misclassify. To get a realistic estimation of the false-association rate, we assign weights to each source in our training and test samples according to their significance and galactic latitude, so that the weighted distributions match that of the population of unassociated sources. Then, the false-association rate is estimated by dividing the sum of the weights of the sources misclassified as AGNs by the total sum of weights. Table 2 shows the performance of each individual algorithm and of the combined algorithm which requires for each source to be labeled as 'AGN' by both methods (NN AND RF) in order to be classified as such. The combined algorithm is expected to recognize 80% of the AGNs present in the sample of unassociated sources, while having an estimated contamination of sources incorrectly labeled as AGNs of 11%.", "pages": [ 3 ] }, { "title": "7 Results", "content": "After evaluating the performance on the test sample, we apply the combined classification method (NN AND RF) to the sample of unassociated sources. From a total of 576 1 1 sources, 231 are labeled as AGNs. According to the performance estimated on the test sample, up to 26 of the 231 tentative classifications are expected to be non-AGNs whose gamma-ray properties are similar to those of the known 2FGL AGNs. The sky distribution of the AGN candidates, together with the sources that were not conclusively labeled as candidate AGNs, is shown in Figure 4.", "pages": [ 3, 4 ] }, { "title": "8 Conclusions", "content": "We have used two independent classification algorithms to find objects in the unclassified sample of the 2FGL catalog whose gamma-ray properties resemble those of gammaray emitting AGNs, the most numerous source class detected by Fermi -LAT. Our work identifies 231 AGN candidates based on their gamma-ray properties, which constitute 40% of the 2FGL unassociated source population. The final list of sources is available upon request. By weighting the sample of test sources to have a similar significance and galactic latitude distribution as the unassociated sample, we calculate a realistic false-association rate of 11%. This fraction is significantly higher than the 0 . 05% reported in [2], which was estimated on the fraction of wrong associations in the test sample. However, the value in [2] is likely to be an over-optimistic estimate, since most misclassifications occur for sources with low galactic latitude or small detection significance, which are more abundant in the unassociated source sample than in the test sample. In a similar work, a false-association rate of 2.3% was reported in [6] estimated using cross-validation. That estimate is less sensitive to low-significance and lowlatitude sources as the authors excluded the galactic plane from their study. However, cross-validation is known to give optimistic performance estimates as the same population of sources is used for training and testing. Studying and identifying the selected AGN candidates through multi-wavelength studies is likely going to extend the population of gamma-ray emitting AGNs to lower gamma-ray fluxes and therefore lower luminosities, having a potential impact on population studies and the estimates of the contribution of unresolved AGNs to the extragalactic diffuse gamma-ray emission. In addition, our classification method can also help in targeting unassociated AGNs close to the galactic plane, where counterparts are more difficult to identify due to galactic extinction and diffuse foreground emission at low galactic latitudes.", "pages": [ 4 ] }, { "title": "Acknowledgments", "content": "Wegratefully acknowledge support by DFG (SFB 823, SFB 876), DAAD (PPP USA) and HAP. ME acknowledges support from NASA grant NNX12AJ30G. We thank Sabrina Einecke, Brian Humensky, Reshmi Mukherjee, Daniel Nieto and Ann-Kristin Overkemping for feedback on the manuscript.", "pages": [ 4 ] } ]

[ { "title": "Muon Identification with VERITAS using the Hough Transform", "content": "JONATHAN TYLER 1 , FOR THE VERITAS COLLABORATION. 1 McGill University, Department of Physics [email protected] Abstract: Imaging atmospheric Cherenkov telescope (IACT) arrays such as VERITAS are used for groundbased very high-energy gamma-ray astronomy. This is accomplished by the detection and analysis of the Cherenkov light produced by gamma-ray-initiated atmospheric air showers. IACTs also detect the Cherenkov light emitted by individual muons. Identification of these muons is useful because their Cherenkov light can be used to calibrate the telescopes. Muons create characteristic annular patterns in the cameras of IACTs, which may be identified using parametrization algorithms. One such algorithm, the Hough transform, has been successfully used to identify muons in VERITAS data. Details of this technique are presented here, including results regarding its effectiveness. Keywords: icrc2013, VERITAS, Hough, transform, muons.", "pages": [ 1 ] }, { "title": "1 VERITAS", "content": "VERITAS (shown in figure 1) is an array of four 12m imaging atmospheric Cherenkov telescopes (IACTs) located at the base of Mount Hopkins in southern Arizona. Very high energy gamma rays interact with nuclei in the atmosphere, producing extensive air showers. The particles in the air showers move faster than the speed of light in air, emitting Cherenkov light. This light is collected and focused with large segmented reflectors onto arrays of light sensitive photomultiplier tubes (PMTs) called cameras. The PMTs produce signals that are proportional to the amount of light detected. These signals are converted into digital information, providing images of the air showers. The directions of the incident gamma rays are determined by the geometry of the images. The energies of the gamma rays are related to the geometry of the images and the amount of light detected [1]. In order to measure the energies of the incident gamma rays, the intensity of the signals recorded must be related to the amount of light detected. This relationship is a poorly constrained parameter in the energy calibration of IACTs. Muons can be used as calibrated light sources to better determine this relationship [2].", "pages": [ 1 ] }, { "title": "2 Muons", "content": "Muons emit Cherenkov light in a cone at a nearly constant angle as they propagate through the lower atmosphere. This creates annular patterns in IACT cameras, with radii determined by the Cherenkov angles of the muons [2]. Muons propagating parallel to the optical axis (on-axis /negationslash muons) that hit the center of the reflector (impact parameter b = 0) appear as complete rings with azimuthally symmetric light distributions in the cameras. On-axis muons with b = 0 but less than the radius of the reflector R appear as complete rings with azimuthally dependent light distributions in the cameras as shown in figures 2 and 3. Distant on-axis muons with b > R appear as incomplete rings (arcs) in the cameras. Muons propagating at angles not parallel to the optical axis appear as rings or arcs with centers offset from the center of the cameras. The amount of light produced by muons with known Cherenkov angles and impact parameters is well understood. Therefore muons can be used to calibrate the energy measurements of the telescopes. Since muons produce characteristic patterns, these images can be parametrized using feature finding algorithms such as the Hough transform.", "pages": [ 1 ] }, { "title": "3 Hough transform", "content": "The Hough transform is an algorithm used for parametrizing assumed shapes (in this case, circles) in digital images. This algorithm converts the problem of iteratively fitting a circle in image space into a problem of finding the best parameters for a circle in a parameter space. Each parameter must have a limited range and resolution defined by the user. Therefore, each pixel in a digital image is intersected by a finite number of circular parametrizations. Each of the circles describes a point in a 3D parameter space P ( x , y , r ) , where ( x , y ) is the center of the circle and r is the radius [3]. The Hough transform employs a histogram called an accumulator array to accumulate votes in the parameter space. This is accomplished by adding the intensity values of each pixel in the image to the bins of the accumulator array that correspond to the circles that pass through those pixels. Once the accumulator array has been filled, the best circular parametrization of the image corresponds to the coordinates of the bin with the highest number of votes [3]. A lookup table was constructed consisting of lists of circular parametrizations associated with each pixel of the VERITAS cameras. This was accomplished by determining which pixels were intersected by various circular parametrizations. The locations of the centers of each PMT in the VERITAS cameras were used as the location of the centers of the circles, and the radii of the circles consisted of values from 3 PMT diameters to 11 PMT diameters, incremented by a third of a PMT diameter. This choice of parametrizations resulted in 12475 distinct circles being used to generate the lookup table. This lookup table was used to perform the Hough transform on VERITAS events.", "pages": [ 2 ] }, { "title": "4 Muon identification parameters", "content": "The AP parameter: the value of the bin of the accumulator array with the most votes divided by the average non-zero bin value. Specifically: The AP parameter can be thought of as a measure of the strength or signal to noise ratio of the best parametrization of the event. Since muon events produce sharp peaks in the accumulator array, they should have large AP values. The TD parameter: the sum of the distances in the parameter space between the three best parametrizations of the event. Specifically, if ( x 1 , y 1 , r 1 ) , ( x 2 , y 2 , r 2 ) and ( x 3 , y 3 , r 3 ) represent the best, second best and third best parametrizations of the event, then: where, The TD parameter can be thought of as a measure of the unanimity of the parametrizations, or the continuity of the parameter space distribution. Since the three best parametrizations are similar for muon events, these events should have small TD values. The Npix parameter: the number of pixels with non-zero values after standard image processing is applied.", "pages": [ 2 ] }, { "title": "5 Cuts and results", "content": "In order to test the effectiveness of the parameters described in the previous section for muon identification, 22774 events from the same run were visually inspected and categorized. These events were labeled muons, nonmuons or ambiguous. Events with fewer than 10 hit pixels were not categorized due to the fact that circular patterns were difficult to identify in those images. 1516 events were categorized as muons, 17027 events were categorized as non-muons and 4231 events were categorized as ambiguous. The cuts on the muon identification parameters were optimized on the first half of the visually categorized events so that no non-muon events passed. The cuts that resulted in the greatest number of muons were found to be:", "pages": [ 3 ] }, { "title": "6 Conclusions", "content": "The Hough transform was found to be effective at parametrizing the circular pixel patterns produced by muons in the VERITAS cameras. The cuts on the muon identification parameters obtained from the accumulator array were optimized using the visually categorized events and found to produce highly pure muon samples when applied to other runs. This technique is currently being implemented in the VERITAS offline analysis software. Upon completion, the technique will be used to identify muons for calibration work. Future research will involve improving the technique by investigating the efficiencies of different muon identification parameters as well as assessing the usefulness of the Hough transform algorithm for event reconstruction. Acknowledgments: The author gratefully acknowledges the help of Ken Ragan, David Hanna, Andrew McCann, Micheal McCutcheon, Gernot Maier, Roxanne Guenette, Sean Griffin, Gordana Tesic, Simon Archambault, David Staszak, JeanFrancois Rajotte and Paul Mercure. This research is supported by grants from the U.S. Department of Energy Office of Science, the U.S. National Science Foundation and the Smithsonian Institution, by NSERC in Canada, by Science Foundation Ireland (SFI 10/RFP/AST2748) and by STFC in the U.K. We acknowledge the excellent work of the technical support staff at the Fred Lawrence Whipple Observatory and at the collaborating institutions in the construction and operation of the instrument.", "pages": [ 3 ] } ]

[ { "title": "High-energy neutrino production from photo-hadronic interactions of gamma rays from Active Galactic Nuclei at source", "content": "J.C. ARTEAGA-VEL 'AZQUEZ 1 , ANGELO MART'INEZ 2 [email protected] Abstract: Recent astronomical observations reveal that Active Galactic Nuclei (AGN) are sources of highenergy radiation. For example, the Fermi-LAT and Hess telescopes have detected gamma-ray emissions from the cores of several types of AGN's. Even more, the Pierre Auger observatory has found a correlation of ultrahigh energy cosmic ray events with the position of Active Galactic Nuclei, such as Centaurus A. In this way, according to particle physics, a flux of high-energy neutrinos should be expected from the interactions of cosmic and gamma-rays with the ambient matter and radiation at the source. In this work, estimations of the diffuse neutrino flux from AGN's arising from interactions of the gamma radiation with the gas and dust of the sources will be presented. Keywords: Diffuse Neutrino flux, Gamma ray emission, Photo-hadronic interactions, Active Galactic Nuclei", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Active Galactic Nuclei are among the most luminous extragalactic objects in the universe. Their emission covers all the electromagnetic spectrum, from radio up to the gammaray wavelengths and seems to have its origin in gravitational potential energy of matter falling inwards a supermassive black hole hidden deep at the source. It is around these supermassive objects where suitable conditions for particle acceleration may be found. Nowadays, it is believed that this process is behind the origin of the γ -ray emission seen from AGN's. In fact, there are two known particle physics scenarios for the production of γ -rays in AGN's, the so called hadronic and leptonic models. The former involves the acceleration and production of protons and atomic nuclei, whose interactions with the matter and radiation at the source lead to the production of secondary hadrons, which decay in γ -photons and neutrinos [1]. In this case, active galactic nuclei could also be important sources of highenergy neutrinos and cosmic rays. On the other hand, in the framework of the leptonic model, electrons and positrons are accelerated up to the relativistic regime producing gamma rays by radiative processes [2]. Although, no hadron acceleration up to the highest energies occurs, neutrinos are still produced in this scenario by means of secondary mechanisms induced by γ -ray collisions with the material and radiation of the source. In any case, no matter the mechanism of γ -ray production, the sole presence of this radiation implies the existence of at least a feeble flux of highenergy neutrinos coming from gamma-ray interactions at the source and its surroundings. In reference [3], the diffuse flux of neutrinos from FR I and BL Lac type galaxies produced by photo-hadronic interactions of gamma rays during their way out the source and the host galaxy was investigated. In that work, in fact, it was shown that this particular flux is out of the reach of modern neutrino telescopes. Even more, in case that bigger neutrino detectors could be built, the detection of this diffuse ν flux would be difficult to achieve due to the presence of the strong atmospheric ν background. However, it was learned that high-energy neutrino emission is not absent in AGN's at all and that, if neutrinos are ever detected from FRI and BL Lac galaxies, they would come from hadronic scenarios. In the present contribution, the research is extended to FR II galaxies. These objects have a bigger intrinsic absorption in X-rays than FR I galaxies [4], therefore, they offer more target material for ν -production by photo-hadronic interactions. Besides, from the FermiLAT surveys, it results that FR II galaxies with MeV -TeV emission are more γ -ray luminous than FRI type AGN's, which can also enhance neutrino emission by γ -ray collisions in those environments. However, their contribution to the diffuse flux of high-energy neutrinos in the universe could be strongly compromised due to the fact that the population of FR II seems to escape from the Fermi-LAT detection [5], which may be caused by an anisotropic emission from the source. If true, this phenomenon would favor the detection of FR II type galaxies with small jet inclination angles with respect to the observer's line of sight in the Fermi-LAT data [5]. The paper is organized as follows: First, a brief description of the calculation of the diffuse flux of neutrinos from individual sources is presented. Then, the γ -ray spectral luminosity of FR II objects is shown along with the associated neutrino flux for a single source. Next, a model for the matter distribution of the powerful radio galaxy Cygnus A is described. This is the closest FR II object to the Earth. And finally, the diffuse flux of neutrinos from photo-hadronic interactions of γ -photons from FR II galaxies is given.", "pages": [ 1 ] }, { "title": "2 The diffuse neutrino flux", "content": "The extragalactic flux of neutrinos (in units of s -1 · sr -1 · TeV -1 · cm -2 ) detected at Earth is estimated from the following expression: where L ν [ L γ , E ν ] is the neutrino spectral luminosity of a FR II type source localized at redshift z and characterized by an integrated γ -ray luminosity L γ in the interval from 100MeV to 10GeV. Here, E ν = E · ν ( 1 + z ) represents the neutrino energy at source and E · ν , the redshifted energy as measured at Earth. On the other hand, ργ ( L γ , z ) is the gamma-ray luminosity function (GLF) of FR II sources per comoving volume dVc and interval d ( log 10 L γ ) , as given by [6]. The Hubble parameter at z is represented by H ( z ) . Along the paper, a Λ CDM cosmology is assumed with ΩΛ = 1 -Ω m = 0 . 74. Integral limits are zmax = 5 [6] and L min ( max ) γ = 43 ( 50 ) ergs / s -1 (based on observations from FR II and FSRQ type objects performed by Fermi -LAT [5].", "pages": [ 1, 2 ] }, { "title": "3 Luminosities of FR II galaxies", "content": "As in reference [5], we assumed a power-law spectrum for the photon spectral luminosity at source, L γ ( E γ ) = dN γ / dtdE γ = L · E α γ . Measurements of the spectral slope, α , in the interval 100MeV -10GeV were provided in [5] by the Fermi-LAT collaboration for a set of four FR II galaxies (see table 1). For our calculations, we adopted a standard FR II source with α = -2 . 57 ± 0 . 18, which corresponds to the mean value of the spectral index of the set of data in table 1 (the error represents the corresponding standard deviation of the data). We also add an energy cutoff around E = 100TeV, assuming conservatively that the leptonic model is at work. On the other hand, we will adopt the hypothesis that the γ -ray emission from FR II type galaxies is anisotropic to explain the small number of FR II objects detected by Fermi-LAT [5]. The main idea behind this hypothesis is that the γ -ray flux is born in the AGN jet as a result of Compton scattering of external photons by electrons, which produces a strong Doppler boosting and a narrow beaming cone of emission [5]. From the discussion at the beginning of this section and following [8, 9], the photon spectral luminosity per solid angle interval at the reference frame of the source galaxy will have the following form: where i is the angle between the jet direction of the AGN and the line of sight to the observer and θ , the angle between the jet axis and the direction of the emitted photon. Here, N is a normalization factor chosen in such a way that for an observer with an angle of view i , the measured integrated luminosity per interval of solid angle at source that would be measured in that direction is L γ / 4 π . On the other hand, with δ D ( θ ) = 1 / [ Γ ( 1 -µθβ )] , the Doppler factor, and µθ = cos ( θ ) . Similar expressions apply for µ i and δ D ( i ) . In the above equation, Γ is the Lorentz factor of the material in the jet and β = √ 1 -1 / Γ 2 . We will take Γ = 5 and 10. Finally, a = -α -1. For the calculation of the neutrino luminosity, only photo-hadronic interactions are taken into account. Contributions from µ -pair production in γ -ray interactions with matter and photons will be neglected due to their lowest cross-section. The target for the gamma radiation will be the nucleons of the gas and dust of the AGN and its host galaxy. We will assume that this material is composed of protons with energies well below 100MeV. In this way, they will be considered at rest during the calculations of the γ P collisions. The neutrino spectral luminosity per solid angle interval along a given direction θ from the jet axis is given by the expression when the observer has an angle of view, i , with respect to the jet direction. Here, Σ H ( θ ) is the column density of target protons in the direction θ , σγ P ( E γ ) is the γ P crosssection at a photon energy E γ and Y γ P → ν ( E γ , E ν ) is the ν yield, i.e., the number of neutrinos produced with energy around dE ν during a collision of a γ -ray with energy in the interval dE γ with a proton at rest. The cross-section σγ P ( E γ ) was evaluated according to [7] and the yield of neutrinos was taken from [3], where it was calculated with the Monte Carlo program SOPHIA v2.01 [10]. Using equation 2 in expression 4, summing over all directions θ and averaging on the observer's view angle, i , we arrive to the formula where In the above formula, we have put a constraint to the direction i of the observer. This restriction is obtained when the luminosity of FR II galaxies, observed from the direction i with photon spectral luminosity L γ , is limited to be smaller than 10 50 ergs / s along the jet axis. This upper limit comes from the observations of FSRQ galaxies with the Fermi-LAT telescope [5], which are just FR II AGN's observed along the jet direction according to the unified AGN model [11]. In this way, we integrate the angle i only inside a limited solid angle interval δΩ ( L γ ) for which the aforementioned condition is valid. Finally, in equation 5, integration is performed from E γ = 10 -0 . 8 (just above the γ -energy threshold for pion photo-production) to 10 6 GeV.", "pages": [ 2 ] }, { "title": "4 Column density", "content": "To estimate the column density of the gas and dust at the source, first, we assume that the birth place of the gamma radiation observed from FR II type galaxies is found at the nucleus of the AGN and coincides with the radio core location. In fact, combined multi-wavelength observations of the radio galaxy M87, which hosts a FR I type object, point out that the gamma-ray production site could be located at the nucleus of the AGN [12]. On the other hand, we take Cygnus A as a model for FR II galaxies. The advantage of this choice is that this powerful FR II galaxy has been well studied by different astronomical instruments due to its proximity to the Earth ( z = 0 . 056) [13]. Based on reported astronomical observations, we built a simple model of Cygnus A to describe the matter distribution of the main structures of the galaxy (see table 2). To start with, for the radius of the gamma-source at the nucleus, we take the limit derived from [14] for the compact radio source in Cygnus A. The density was assumed to be 10 6 cm -3 , which agrees with estimations at the nucleus of AGN's [15]. Observations against the nucleus indicates the presence of a strong X-ray absorber, which could be due to material from the BLR region and a dusty torus. In [16], the torus was modeled with a clumpy circumnnuclear disk. Its geometry and density were restricted with radio and IR data. For the torus, we will use the disk model of [16] with a constant radial density, only depending on the angle β respect to the equator of the galaxy, a half opening angle σ = 20 · (measured from the equatorial plane), an internal radius equal to 0 . 6pc, an outer radius of 130pc and with axis oriented along the jet direction. We will assume that the BLR region is an inner extension of the torus [17]. Therefore, the BLR region will be described also with a disk with the same geometrical parameters, but with smaller dimensions, particularly with an external radius of 0 . 6pc. Density is found in such a way that for β = 10 · , which corresponds to the viewing angle of an observer at the Earth [16], the total column depth along the BLR and torus region is NH = 2 × 10 23 cm -2 , the value reported by the Chandra telescope for the X-ray absorber [21]. We will assume that this column density is equally divided between the BLR region and the torus, since it is unknown exactly what is the exact contribution from each region to the Xray absorption. A second disk, tilted 21 · with respect to the equatorial plane of the host galaxy, a density nH = 10 4 cm -3 , a radius of 80pc and an opening angle of 14 · was also added to our model [18]. It was detected with VLBA HI absorption studies of the core region of Cygnus A [18]. Observations at IR wavelengths also reveal an edge-on oriented biconical structure, which is likely caused by a kpc-scale dust lane characterized by a disk geometry and funnels along the jet axis [19]. In the model of [19], the funnels have an opening angle of 116 · , while the disk axis is aligned within 15 · with the jet direction. Here, we will incorporate the dust lane with the above configuration in our model. Besides, for simplicity, we will consider that the disk lies on the equatorial plane of the galaxy. The external diameter and the width of the disk are set to 3kpc [20] and 1 . 5kpc [21]. On the other hand, we will assume that the inner frontier of the disk coincides with the external radius of the torus. To calculate the density, we take the column density estimated in [21] for the dust lane along the direction to Earth. Taking into account the geometry of the disk and the orientation of the observer, and assuming an homogeneous distribution of the material inside the dust lane, we estimate a density of 5 . 6 × 10 -1 cm -3 . The distribution of the interstellar matter up to 2kpc will be described using the same function derived in [22], from optical and IR observations, for the stellar density profile in Cygnus A within the 2kpc region around its nucleus. As in [22], we will assume that matter is spherically distributed in this zone. Observations suggest that the above structures are surrounded by a cocoon and an external shell with shocked matter [23]. The shapes and densities of these structures were modeled in [24]. Here, we will take the model number 3 of [24], which seems to be in better agreement with observations. According to [24], these structures have a prolate spheroidal shape. The inner (outer) radius of the shell is of the order of 60 ( 62 . 5 ) kpc along the semi-major axis, according to the above model, while along the semi-minor axis is 15 ( 25 ) kpc. On the other hand, in model 3 of [24], the cocoon has a volume of 2 . 8 × 10 4 kpc 3 and encloses a mass of 1 . 4 × 10 8 M /circledot . From this data the mean cocoon's density is estimated. It is worth to mention that the density of the interstellar region at 2kpc is normalized in such a way that it reproduces the density of the cocoon. An average density of 1 . 4 × 10 -2 cm -3 for the shocked region was estimated from the 2D density plots of [24]. Beyond the cocoon and the shell, we found the galactic halo, which is composed of hot and low density gas. This structure has been detected in X-rays. Its emission extends up to 720kpc according to [13]. The electron density has been modeled in [13] from 60kpc to 500kpc from X-ray observations of the halo of Cygnus A. We will use this profile for our proton density, assuming that the proton and electron densities are similar inside the halo. We will apply such an expression only up to 500kpc.", "pages": [ 2, 3 ] }, { "title": "5 Results and Discussion", "content": "The ξ ( L γ ) factor is presented in figure 1 for different Γ and α parameters. We notice that the ξ ( L γ ) factor decreases as the solid angle δΩ ( L γ ) gets smaller for high Γ values and L γ luminosities. As we will see, that will imply low d- ino fluxes from FR II galaxies for high Lorentz factors. The neutrino background flux as observed at Earth from FR II objects, after ν oscillations, is shown in figure 2. The error band of this flux, calculated by varying the photon index ( α ) of formula 2 in the interval -2 . 57 ± 0 . 18 is also presented. The spectrum is compared with the background of atmospheric neutrinos and the result for FR I type sources [3]. Two fluxes derived from hadronic models [29, 30] and two experimental upper bounds from ICECUBE [25] and Antares [26], respectively, are also shown. The results show that, although γ -ray FR II emitters seems to be less abundant than the corresponding FR I population, the higher Doppler and beaming factors of FR II galaxies can make their corresponding ν background flux higher than the one for FR I objects. However this difference is not big. Fig. 2 shows that, the investigated neutrino flux from FR II galaxies is too small in comparison with the experimental bounds and the atmospheric background of neutrinos. Neither this feeble flux nor the flux from FR I objects can explain the observed neutrino events detected recently by ICECUBE around 1PeV [31]. Therefore, if these events comes from FR I and FR II objects, their most probable origin would be found at hadronic mechanisms originated by cosmic-ray acceleration.", "pages": [ 3, 4 ] }, { "title": "6 Conclusions", "content": "We have shown that in addition to FR I AGN's, FR II type galaxies characterized by γ -ray emission also produce a small neutrino diffuse flux produced by photo-hadronic interactions by γ radiation with the gas and dust at the sources. Both fluxes, in general, are too low to be detected by the modern neutrino telescopes and to explain the last PeV neutrino events detected recently by the ICECUBE observatory. Acknowledgment: This work was partially supported by the Consejo de la Investigaci'on Cient'ıfica of the Universidad Michoacana (Project CIC 2012-2013) and CONACYT (Project CB2009-01 132197).", "pages": [ 4 ] }, { "title": "References", "content": "[17] M. Nenkova et al., ApJ.685 (2009) 160-180; Erratum-ibid.723 (2010) 1827.", "pages": [ 4 ] } ]

[ { "title": "Ko Sanders ∗", "content": "Enrico Fermi Institute, University of Chicago 5640 South Ellis Avenue, Chicago, IL 60637, USA", "pages": [ 1 ] }, { "title": "Abstract", "content": "The linear scalar quantum field, propagating in a globally hyperbolic spacetime, is a relatively simple physical model that allows us to study many aspects in explicit detail. In this review we focus on the thermal equilibrium (KMS) states of such a field in a stationary spacetime. Our presentation draws on several existing sources and aims to give a unified exposition, while weakening certain technical assumptions. In particular we drop all assumptions on the behaviour of the time-like Killing field, which is important for physical applications to the exterior region of a stationary black hole. Our review includes results on the existence and uniqueness of ground and KMS states, as well as an evaluation of the evidence supporting the KMS-condition as a characterization of thermal equilibrium. We draw attention to the poorly understood behaviour of the temperature of the quantum field with respect to locality. If the spacetime is standard static, the analysis can be done more explicitly. For compact Cauchy surfaces we consider Gibbs states and their properties. For general Cauchy surfaces we give a detailed justification of the Wick rotation, including the explicit determination of the Killing time dependence of the quasi-free KMS states.", "pages": [ 1 ] }, { "title": "Contents", "content": "∗ E-mail: [email protected]", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "For a quantum mechanical system with a Hilbert space H , a thermal equilibrium state can be described by the density matrix for the Gibbs grand canonical ensemble, where H is the Hamiltonian operator of the system, N the particle number operator, β the inverse temperature and µ the chemical potential. 1 Z is a normalization factor, which ensures that the trace Tr ρ ( β,µ ) = 1. For this to be well defined we need to know that e -β ( H -µN ) is a trace-class operator, a condition which can often be established in explicit models, especially when the system is confined to a bounded region of space. For physical purposes it is of some interest to study thermal equilibrium in much more general situations than for quantum mechanical systems, such as for a quantum field propagating in a given gravitational background field. In these cases one immediately encounters three well known problems: in a general curved spacetime there is no clear notion of particle, no clear choice of a Hamiltonian operator and, even if there were, the exponentiated operator in Eq. (1.1) might not be of trace-class. Additional problems arise if one wants to use the technique of Wick rotation, which has important computational advantages in the quantum mechanical case, but which requires a preferred choice of a well behaved time coordinate. In this review paper we treat the problems above for the explicit example of a linear scalar quantum field propagating in a globally hyperbolic spacetime. We combine results and arguments from several sources into a unified exposition and we take the opportunity to show that some of the technical conditions made in the earlier literature may be dropped or weakened. It is well known how to formulate a linear scalar quantum field theory in all globally hyperbolic spacetimes [1, 2, 3, 4]. A notion of particle and Hamiltonian can be introduced whenever the spacetime is also stationary [3]. We will therefore focus on stationary spacetimes, in which case the notion of global thermal equilibrium is (in principle) well understood [5, 6]. Under suitable positivity assumptions on the field equation we first give a full characterization of all ground states on the Weyl algebra and we describe in detail a uniquely preferred ground state [7]. More precisely, our assumptions are that the field should satisfy the (modified) Klein-Gordon equation with a smooth, real-valued potential V which is stationary and strictly positive everywhere. Unlike Ref. [7] we do not insist that the ground state should have a mass gap, which allows us to drop the restrictions that the norm and the lapse function of the time-like Killing field be suitably bounded away from zero. This is of some importance in certain physical applications, e.g. when the stationary spacetime is the exterior region of a stationary black hole [8, 9]. In that case the norm of the Killing field may become arbitrarily small. Gibbs states as in Eq. (1.1) have a certain property, first noticed by Kubo [10] and Martin and Schwinger [11] and now known as the KMS-condition. This property was proposed as a defining characteristic for thermal equilibrium states by Ref. [12], even when the Gibbs state is no longer defined, on the grounds that it survives the thermodynamic (infinite volume) limit under general circumstances for systems in quantum statistical mechanics in Minkowski spacetime. Further support for this proposal comes from an investigation of the second law of thermodynamics for general C ∗ -dynamical systems [13] and from the study of explicit models in quantum statistical mechanics [14]. In addition to its physical context, the KMS-condition has also become important in the abstract theory of operator algebras, where it is related to Tomita's modular theory [15]. In the case of a standard static spacetime (see Sec. 3 for the definition) with a compact Cauchy surface we will see that the Gibbs state of Eq. (1.1) makes sense. In the case of a general stationary spacetime we will give a full characterization of all KMS states on the Weyl algebra and we describe uniquely preferred KMS states at any temperature [6]. Unfortunately, the arguments of Ref. [12] concerning the thermodynamic limit fail to work for quantum field theories. This indicates that the behaviour of the temperature of a quantum field, with respect to locality, is presently rather poorly understood, even in a spacetime with a favourable background geometry. With a view to physical applications, e.g. in cosmology, an improved understanding would be highly desirable. (At this point we would also like to point out that Refs. [16, 17] have recently proposed a notion of local thermal equilibrium in general curved spacetimes, but the full merit of this new approach is as yet unclear and a review of these recent developments is beyond the scope of this paper.) When we study the Wick rotation we will restrict attention to spacetimes which are standard static. Under these geometric circumstances there is a preferred Killing time coordinate and it is well understood how KMS states can be obtained from a Wick rotation [5, 18]. We show that any technical assumptions are automatically verified for the systems under consideration. After complexifying the Killing time coordinate we obtain an associated Riemannian manifold and we compactify the imaginary time coordinate to a circle of radius R . We then show that there exists a uniquely distinguished Euclidean Green's function, which can be analytically continued back to the Lorentzian spacetime. We will find the explicit Killing time dependence of this Green's function and on the Lorentzian side we recover the two-point distribution of the preferred KMS state with inverse temperature β = 2 πR . The contents of this paper are organised as follows. Section 2 below considers some basic features of thermal equilibrium states in an abstract, algebraic setting. The main aim is to elucidate the structure of the spaces of all ground and KMS states on the Weyl algebra under minimal assumptions. Section 3 provides a review of recent geometric results on stationary, globally hyperbolic spacetimes and the subclass of standard static ones. In addition, it introduces the spacetime complexification procedure needed to perform the Wick rotation. After these algebraic and geometric preliminaries we describe in Section 4 the linear scalar field under consideration, with an emphasis on those results that depend on the presence of the time-like Killing field. This section also contains a discussion of the two-point distributions of thermal equilibrium states. Section 5 considers the space of ground states and the GNS-representation of the uniquely preferred ground state. It also includes a discussion of the renormalised stress-energy-momentum tensor. Section 6 considers thermal equilibrium states at non-zero temperature, from several perspectives. It contains existence results of Gibbs states, under suitable assumptions, and it discusses the motivations to use the KMS-condition to characterize thermal equilibrium. Furthermore, it characterizes all KMS states, including a uniquely preferred one, and in the static case it provides a rigorous justification of the Wick rotation. A number of useful results from functional analysis, needed for Sections 2, 4 and 6, are collected in A, so as not to hamper the flow of the presentation. These results concern strictly positive operators and the relation between operators in Hilbert spaces and distributions.", "pages": [ 2, 3, 4 ] }, { "title": "2 Equilibrium states in algebraic dynamical systems", "content": "Much of the structure of dynamical systems can be conveniently described in an abstract algebraic setting, which subsumes a great variety of physical applications. In this section we provide a brief overview of a number of notions and results relating to equilibrium states for such systems and some more specialised results pertaining to Weyl C ∗ -algebras. (For a detailed treatment of Weyl C ∗ -algebras we refer to Ref. [19] and references therein.) Note that we generally do not assume any continuity of the time evolution, so our results must remain more limited than those for C ∗ -dynamical systems or W ∗ -dynamical systems [20, 14]. This is in line with our physical applications later on, where we will consider the Weyl C ∗ -algebra of certain pre-symplectic spaces. As it turns out, for these systems the time evolution will not be norm continuous in the given algebra, but there will be continuity at the level of the symplectic space. To accommodate for such situations, the results in this section will only make ad hoc continuity assumptions in suitable representations.", "pages": [ 4 ] }, { "title": "2.1 Algebraic dynamical systems and equilibrium states", "content": "We begin with the following basic definition: Definition 2.1 An algebraic dynamical system ( A , α t ) consists of a ∗ -algebra A with unit I , together with a one-parameter group of ∗ -isomorphisms α t on A . The algebra A is interpreted as the algebra of observables and α t describes the time evolution. A state ω on A is a linear functional ω : A → C which is normalised, ω ( I ) = 1, and positive, ω ( A ∗ A ) ≥ 0 for all A ∈ A . Every state gives rise to a unique (up to unitary equivalence) GNStriple [14] ( π ω , H ω , Ω ω ), where H ω is a Hilbert space and π ω is a representation of A on H ω , in general by unbounded operators, such that the vector Ω ω is cyclic for π ω ( A ), i.e. π ω ( A )Ω ω = H ω , and ω ( A ) = 〈 Ω ω , π ω ( A )Ω ω 〉 . We will denote the space of all states on A by S ( A ). It is a convex set in the (algebraic) dual space A ' , which is closed in the weak ∗ -topology. We will call a state pure if for any decomposition ω = λω 1 +(1 -λ ) ω 2 with ω 1 , ω 2 ∈ S ( A ) and 0 < λ < 1 we must have ω 1 = ω 2 = ω . For dynamical systems, the following class of states are of special interest: Definition 2.2 An equilibrium state ω for an algebraic dynamical system ( A , α t ) is a state ω on A such that α ∗ t ω := ω · α t = ω for all t ∈ R . We denote the space of all equilibrium states by G ( A ) (suppressing the dependence on α t ). Note that G ( A ) is a closed convex subset of S ( A ). In the GNS-representation space of an equilibrium state ω the time evolution α t is implemented by a unitary group U t via The group U t is uniquely determined by the additional condition that U t Ω ω = Ω ω (cf. Ref. [14] Cor. 2.3.17). If the group U t is strongly continuous, it has a self-adjoint generator by Stone's Theorem (Ref. [21] Thm. VIII.8), so we may write U t = e ith , where the self-adjoint operator h is called the Hamiltonian.", "pages": [ 4 ] }, { "title": "2.1.1 Ground states", "content": "Definition 2.3 A ground state ω on an algebraic dynamical system ( A , α t ) is an equilibrium state for which U t = e ith is strongly continuous and the Hamiltonian h satisfies h ≥ 0 . We denote the space of all ground states by G 0 ( A ) . Note that pure ground states are always extremal. Furthermore, we have the following result, which is essentially due to Borchers [22]: Theorem 2.1 A non-degenerate ground state ω on an algebraic dynamical system ( A , α t ) with A a C ∗ -algebra is pure. Proof: The strongly continuous unitary group U t on H ω defines a group of automorphisms on the von Neumann algebra R := π ω ( A ) '' . (A ' denotes the commutant of an algebra and '' the double commutant [15].) The result of Ref. [22] is that U t ∈ R for all t ∈ R . Now any unit vector ψ of the form ψ = X Ω ω with X ∈ R ' satisfies hψ = Xh Ω ω = 0. Because Ω ω is cyclic for R , it is separating for R ' , so ψ = λ Ω ω if and only if X = λI . Hence if ω is nondegenerate, then R ' = C I , which means that ω is pure (Ref. [15] Thm. 10.2.3). /square In the case that A is commutative, ground states have a special property which is worth singling out. The proof involves analytic continuation arguments which are typical for the study of ground and KMS states: Lemma 2.1 Let ω be a state on an algebraic dynamical system ( A , α t ) with A a commutative ∗ -algebra. Then the following statements are equivalent: Proof: Suppose that ω is a ground state. For arbitrarily given A,B ∈ A we consider the function f ( t ) := ω ( Aα t ( B )) = ω ( α t ( B ) A ). Because h ≥ 0 (by definition of ground states) we may use Lemma A.8 to define a bounded, continuous function F + ( z ) on the upper half plane { z := t + iτ | τ ≥ 0 } by which is holomorphic on τ > 0 and satisfies F + ( t ) = f ( t ) for τ = 0. Similarly we can define a bounded continuous function F -( z ) on the lower half plane by which is holomorphic for τ < 0 and which again satisfies F -( t ) = f ( t ) for τ = 0. It follows from the Edge of the Wedge Theorem [23] that there is an entire holomorphic function F which extends both F + and F -. Since F must be bounded as well it is constant by Liouville's Theorem [23]. Restricting to τ = 0 we find f ( t ) = f (0), i.e. ω ( Aα t ( B )) = ω ( AB ). Now suppose that the second item holds for ω . Then ω is an equilibrium state (taking A = I ) and using the group properties of α t one easily shows that ω ( Aα t ( B ) C ) = ω ( ABC ) for all t ∈ R and A,B,C ∈ A . This implies that π ω ( α t ( B )) = π ω ( B ) and hence that U t = I for all t ∈ R . Finally, U t = I implies h = 0, so ω is a ground state. /square Lemma 2.1 allows us to give a nice description of all ground and equilibrium states on those algebraic dynamical system ( A , α t ) for which A is a commutative C ∗ -algebra. For this we make use of the classic structure theorem for commutative C ∗ -algebras (cf. Ref. [15] Thm. 4.4.3), which tells us that there is a compact Hausdorff space X , unique up to homeomorphism, and a ∗ -isomorphism α : A → C ( X ), where C ( X ) is the C ∗ -algebra of continuous, complex-valued functions on X in the suppremum norm. The one-parameter group of ∗ -isomorphisms β t := α · α t · α -1 on C ( X ) is then given by β t ( F ) = Ψ ∗ t F , where Ψ t is a (uniquely determined) one-parameter group of homeomorphisms of X . We define the set of fixed points X 0 := { x ∈ X | Ψ t ( x ) = x for all t ∈ R } , which is closed in X and hence compact. Theorem 2.2 Using the notations above, the following statements are true for an algebraic dynamical system ( A , α t ) with A a commutative C ∗ -algebra: /negationslash Note in particular that pure equilibrium states are automatically ground states. Proof: We only prove statement (v), as the others follow from standard results on cummutative C ∗ -algebras and the definitions above [15]. By Lemma 2.1, ω µ is a ground state if and only if ∫ X dµ F (Ψ ∗ t G -G ) = 0 for all F, G ∈ C ( X ). Because Ψ ∗ t G -G = 0 on X 0 this is certainly the case when supp( µ ) ⊂ X 0 (cf. Ref. [15] Remark 3.4.13). Conversely, for any x ∈ X c 0 in the complement of X 0 we can find a t ∈ R and an open set U ⊂ X such that x ∈ U and Ψ t ( U ) ∩ U = ∅ . (In detail: we may first choose a t ∈ R such that y := Ψ t ( x ) = x . As X is Hausdorff we may find an open set V ⊂ X such that x ∈ V and y /negationslash∈ V . Taking U := V \\ Ψ -t ( V ) will do.) By Urysohn's Lemma [24] there is a G ∈ C ( X ) with G ( x ) = 1 which vanishes on X \\ U . Note that G Ψ ∗ t G = 0, so if ω µ is a ground state we have ∫ X dµ | G | 2 = -∫ X dµ G (Ψ ∗ t G -G ) = 0. As G ( x ) = 1 this entails that x /negationslash∈ supp( µ ), so supp( µ ) ⊂ X 0 . /square", "pages": [ 5, 6 ] }, { "title": "2.1.2 KMS states", "content": "In physical applications, thermal equilibrium states can be characterised by the KMS-condition: Definition 2.4 A state ω on an algebraic dynamical system ( A , α t ) is called a β -KMS state for β > 0 , when it satisfies the KMS-condition at inverse temperature β , i.e. when for all operators A,B ∈ A there is a holomorphic function F AB on the strip S β := R × i (0 , β ) ⊂ C with a bounded, continuous extension to S β such that We will denote the space of all β -KMS states by G ( β ) ( A ) . A β -KMS state ω is called extremal if for any decomposition ω = λω 1 +(1 -λ ) ω 2 with ω 1 , ω 2 ∈ G ( β ) ( A ) and 0 < λ < 1 we must have ω 1 = ω 2 = ω . When A is a topological ∗ -algebra and ω is a continuous state, then it suffices to require the existence of F AB for A,B in a dense sub-algebra of A , as we will see in Proposition 2.1 below. When ( A , α t ) is a C ∗ -dynamical system one may also drop the requirement that F AB is bounded (Ref. [14] Prop. 5.3.7). The motivations behind this condition will be discussed in some detail in Section 6, in the context of our physical applications to the linear scalar quantum field. Note, however, that a ground state satisfies a similar condition with β = ∞ , when we identify S β , respectively S β , with the open, respectively closed, upper half plane. (This may be seen by the same methods as used in the proof of Lemma 2.1.) The following general result again relies on analytic continuation arguments: Proposition 2.1 Let ω be a β -KMS state on an algebraic dynamical system ( A , α t ) . Then the following hold true: Proof: For any B the function F IB ( z ) satisfies F IB ( t ) = F IB ( t + iβ ). Let F ( z ) be the periodic extension of F IB ( z ) in Im( z ) with period β . Then F is continuous and bounded on C and it is holomorphic, even when Im( z ) ∈ β Z , by the Edge of the Wedge Theorem [23]. F must then be a constant by Liouville's Theorem [23], so F IB ( t ) = F IB (0), i.e. ω ( α t ( B )) = ω ( B ) and ω is in equilibrium. For any operators A,B ∈ A the corresponding function F AB on S β satisfies by the boundedness of F AB and Hadamard's Three Line Theorem (Ref. [21], Appendix to IX.4). The second statement then follows from the first, and the Cauchy-Schwarz inequality. /square For commutative algebras a state ω is a β -KMS state if and only if it is a ground state (cf. Lemma 2.1).", "pages": [ 6, 7 ] }, { "title": "2.2 Weyl C ∗ -algebras", "content": "For our physical applications to linear scalar quantum fields we will make use of an algebraic formulation involving Weyl C ∗ -algebras. In preparation for those applications we will now briefly review some fundamental aspects of these algebras [19], especially in relation to thermal equilibrium states. We consider a pre-symplectic space ( L, σ ), which means that L is a real linear space and σ is an anti-symmetric bilinear form. We call ( L, σ ) a symplectic space if σ is non-degenerate, which means that σ ( f, f ' ) = 0 for all f ' ∈ L implies f = 0. For each pre-symplectic space ( L, σ ) there is a unique C ∗ -algebra generated by linearly independent operators W ( f ), f ∈ L , subject to the Weyl relations [19] This is the Weyl C ∗ -algebra, which we will denote by W ( L, σ ). By construction, the linear space generated by all W ( f ), but without taking the completion in the C ∗ -norm, is also ∗ -algebra, which we will denote by · W ( L, σ ) and which is a dense subset of W ( L, σ ). Every state on W ( L, σ ) restricts to a state on · W ( L, σ ), but we even have the following stronger result: Lemma 2.2 The restriction map r : S ( W ( L, σ )) → S ( · W ( L, σ )) is an affine homeomorphism for the respective weak ∗ -topologies. This follows from Theorem 3-5 and Lemma 3-3a) of Ref. [19] and the fact that the weak ∗ -topology on a bounded set in the continuous dual space W ( L, σ ) ' is already determined by the dense set · W ( L, σ ) ⊂ W ( L, σ ). The Weyl C ∗ -algebra W ( L, 0) is commutative, so there is a ∗ -isomorphism α : W ( L, 0) → C ( X ), where we may identify X as the space of pure states S ( W ( L, 0)). Alternatively we may identify X with the dual group ˆ L of L , viewed as an additive group [19]. Elements of ˆ L are characters of L , i.e. group homomorphisms from L (as an additive group) to the unit circle S 1 (as a multiplicative group). The bijection between pure states ρ ∈ X and characters χ ∈ ˆ L is given by ρ ( W ( f )) = χ ( f ) (cf. Ref. [15] Prop. 4.4.1). Remark 2.1 For any pure state ρ ∈ S ( W ( L, 0)) we can define a ∗ -isomorphism η ρ : W ( L, σ ) → W ( L, σ ) by continuous linear extension of η ρ ( W ( f )) := ρ ( W ( f )) W ( f ) [19]. The ∗ -isomorphisms η ρ are sometimes known as gauge transformations of the second kind . We will denote the gauge transformations on the commutative Weyl algebra W ( L, 0) by ζ ρ . The state space S ( W ( L, 0)) contains a special state, 2 ρ 0 , defined by ρ 0 ( W ( f )) = 1 for all f ∈ L . This state is pure, because its GNS-representation is one-dimensional. It is easy to verify that ρ = ζ ∗ ρ ρ 0 for all pure states ρ ∈ S ( W ( L, 0)) . The algebras W ( L, λσ ), 0 ≤ λ ≤ 1, may be viewed as a strict and continuous deformation [25] of the commutative algebra W ( L, 0). It will be interesting for us to compare the state space of the Weyl C ∗ -algebra W ( L, σ ) with that of the commutative Weyl C ∗ -algebra W ( L, 0): Lemma 2.3 For every ω ' ∈ S ( W ( L, σ )) there is a unique weak ∗ -continuous, affine map λ ω ' : S ( W ( L, 0)) → S ( W ( L, σ )) which is given by λ ω ' ( ρ ) = η ∗ ρ ω ' on pure states. For any pure state ρ ' on W ( L, 0) we have λ ω ' · ζ ∗ ρ ' = η ∗ ρ ' · λ ω ' and λ ω ' is injective when ω ' ( W ( f )) = 0 for all f ∈ L . /negationslash Proof: For pure states we have Because every state in S ( W ( L, 0)) is a weak ∗ -limit of finite affine combinations of pure states, λ ω ' extends uniquely to a weak ∗ -continuous, affine map from S ( W ( L, 0)) to S ( W ( L, σ )), which is given by the same formula. The injectivity of λ ω ' under the stated assumptions is immediate from this formula and Lemma 2.2. The intertwining relation with the gauge transformations of the second kind is a straightforward exercise. /square", "pages": [ 7, 8 ] }, { "title": "2.2.1 Quasi-free and C k states", "content": "On any Weyl C ∗ -algebra there is a special class of states, called quasi-free states, which are distinguished by their algebraic form. They are obtained from the following well known result: Theorem 2.3 Let ( L, σ ) be a pre-symplectic space. A sesquilinear form ω 2 on the complexification L ⊗ C defines a state ω on W ( L, σ ) by continuous linear extension of if and only if for all f, f ' ∈ L ⊗ C : We will call ω 2 a two-point function, even though it is generally not a function of two points x, y ∈ M . The two-point function ω 2 can be characterised alternatively in terms of a one-particle structure [7]: Definition 2.5 A one-particle structure on a pre-symplectic space ( L, σ ) is a pair ( p, K ) consisting of a complex linear map p : L ⊗ C →K into a Hilbert space K such that Given a one-particle structure, one can define an associated two-point function by ω 2 ( f, f ' ) := 〈 p ( f ) , p ( f ' ) 〉 . Conversely, a two-point function ω 2 determines a unique one-particle structure ( p, K ) such that the above equality holds, by similar arguments as used in the GNS-construction. This we call the one-particle structure associated with ω 2 . A wider class of states which will be of interest is the following: /negationslash Definition 2.6 A state ω on the Weyl C ∗ -algebra W ( L, σ ) is called C k , k > 0 , when the maps are well defined on C ∞ 0 ( M ) × n for all 1 ≤ n ≤ k . The ω n are linear maps and they are called the n -point functions. A state is called C ∞ , when it is C k for all k > 0 . When ω is a quasi-free state, it is C ∞ and all higher n -point functions can be expressed in terms of the two-point function ω 2 via Wick's Theorem. For such states it only remains to analyze the two-point functions ω 2 . A physical reason why quasi-free states are of interest is the following (see also Theorems 5.1 and 6.2 below): Theorem 2.4 Let ( L, σ ) be a pre-symplectic space and let ω be a C 2 state on W ( L, σ ) . ω 2 , as defined in Definition 2.6, defines a unique quasi-free state ω ' by Theorem 2.3 and a one-particle structure ( p, K ) . Then, Proof: The claim that ω 2 satisfies the assumptions of Theorem 2.3 is a standard exercise. The characterization of pure quasi-free states in terms of their one-particle structures was established in Ref. [8], Lemma A.2, for the symplectic case. The generalization to the pre-symplectic case is straightforward. The fact that this implies that ω = ω ' is a theorem due to Ref. [26], for the symplectic case. This result and its proof carry over to the pre-symplectic case without modification. /square A related result in the commutative case is the following characterisation of the state ρ 0 : Proposition 2.2 If ρ ∈ S ( W ( L, 0)) is a C 1 pure state, then ρ ( W ( f )) = e iρ 1 ( f ) for all f ∈ L . In particular, if ρ 1 = 0 , then ρ = ρ 0 . Proof: Given any f ∈ L we consider F ( t ) := ρ ( W ( tf )). Because ρ is pure and W ( L, 0) is commutative, F ( t + t ' ) = F ( t ) F ( t ' ) (cf. [15] Prop. 4.4.1) and hence ∂ t F ( t ) = F ( t ) ∂ t F (0) = F ( t ) iρ 1 ( f ). Hence, F ( t ) = e itρ 1 ( f ) and the results follow. /square", "pages": [ 8, 9 ] }, { "title": "2.3 Quasi-free dynamics on Weyl C ∗ -algebras", "content": "A pre-symplectic isomorphism T of ( L, σ ) is a real-linear isomorphism T : L → L which preserves the pre-symplectic form, σ ( Tf,Tf ' ) = σ ( f, f ' ). Each pre-symplectic isomorphism gives rise to a unique ∗ -isomorphism α T of W ( L, σ ) such that α T ( W ( f )) = W ( Tf ) (see Ref. [19], or also Ref. [14] Thm. 5.2.8). Hence, a one-parameter group of pre-symplectic isomorphisms T t gives rise to a one-parameter group α t of ∗ -isomorphisms on W ( L, σ ). Not every one-parameter group of ∗ -isomorphisms on W ( L, σ ) arises in this way, but the time evolution that we will be interested in for our physical applications does. Definition 2.7 A one-particle dynamical system ( L, σ, T t ) is a pre-symplectic space ( L, σ ) with a one-parameter group of pre-symplectic isomorphisms T t . The associated algebraic dynamical system ( W ( L, σ ) , α t ) with α t ( W ( f )) = W ( Tf ) is called quasi-free . An equilibrium one-particle structure ( p, K ) on a one-particle dynamical system ( L, σ, T t ) is a one-particle structure on ( L, σ ) for which there is a one-parameter unitary group ˜ O t on K such that ˜ O t p = pT t . A ground one-particle structure is an equilibrium one-particle structure ( p, K ) for which the unitary group ˜ O t = e itH is strongly continuous and H ≥ 0 . A KMS one-particle structure at inverse temperature β > 0 is an equilibrium one-particle structure ( p, K ) , with associated two-point function ω 2 , such that for all f, f ' ∈ L there exists a bounded continuous function F ff ' on S β , holomorphic on its interior, satisfying An equilibrium one-particle structure is called non-degenerate when ˜ O t = e itH is strongly continuous and 0 is not an eigenvalue for H . Note that a quasi-free state ω with two-point function ω 2 is in equilibrium for a quasi-free dynamical system if and only if the associated one-particle structure ( p, K ) is in equilibrium. Furthermore, we have Proposition 2.3 Let ω be a C 2 equilibrium state on a quasi-free algebraic dynamical system ( W ( L, σ ) , α t ) . Let ( p, K ) be the one-particle structure associated to ω 2 and assume that ω 1 = 0 . When ω is quasi-free, the converses of these statements are also true. /negationslash Proof: We may identify K as a closed linear subspace of the GNS-representation space H ω , spanned by the vectors p ( f ) := Φ ω ( f )Ω ω := -i∂ s π ω ( W ( sf ))Ω ω | s =0 . This derivative is well defined, because ω is C 2 . The unitary group U t on H ω restricts to a unitary group ˜ O t on K , because the dynamics is quasi-free, and the generator h of U t restricts to the generator H of ˜ O t . Also note that K is perpendicular to Ω ω , because ω 1 = 0. It is then clear that when ω is a (non-degenerate) ground state, then H is (strictly) positive and ( p, K ) is a (non-degenerate) ground one-particle structure. When ω is a β -KMS state and f, f ' ∈ L , we may take A ( s ) := s -1 ( W ( sf ) -I ) and B ( s ) := s -1 ( W ( sf ' ) -I ) for any s = 0 to find functions F A ( s ) B ( s ) . Because ω is C 2 , the functions ω ( A ∗ ( s ) A ( s )) and ω ( A ( s ) A ∗ ( s )) have well defined limits as s → 0, and similarly for B . We may then use Proposition 2.1 to take the uniform limit of -F A ( s ) B ( s ' ) as s, s ' → 0, which yields the desired function F ff ' . This proves both items. If ω is quasi-free, its GNS-representation is a Fock space, H ω = ⊕ ∞ n =0 P + ,n K ⊗ n , where P + ,n is the projection onto the symmetrised n -fold tensor product. U t is the second quantization of ˜ O t and h is the second quantization of H . For the converse of the first statement we note that ω is a (non-degenerate) ground state iff the restriction of h to each n -particle space with n ≥ 1 is (strictly) positive. If ( p, K ) is a (non-degenerate) ground one-particle structure, then H is (strictly) positive. The restriction h n of h to P + ,n K ⊗ n is given by H n P + ,n , where H n is defined to be the operator H n := ∑ n j =1 I ⊗ j -1 ⊗ H ⊗ I ⊗ n -j on the algebraic tensor product D ( H ) ⊗ n of the domain D ( H ) of H . By Nelson's Analytic Vector Theorem (Ref. [21] Thm. X.39), H n is essentially selfadjoint (because H is). The closure of each summand in H n is a (strictly) positive operator (by Lemma A.3), and hence so is H n (by Lemma A.6). Therefore, h n is (strictly) positive for n ≥ 1 and ω is a (non-degenerate) ground state. Now we turn to the converse of the second statement. One may use the Weyl relations and the quasi-free property to find Using F ff ' in the exponent yields the desired F W ( f ) W ( f ' ) . For finite linear combinations of Weyl operators the desired property is now clear and for general operators in W ( L, σ ) one appeals to Proposition 2.1 and a limiting argument. /square One of the nice aspects of quasi-free dynamical systems is that we may view T t also as a presymplectic isomorphism of ( L, 0), so we may compare the corresponding quasi-free dynamics on W ( L, σ ) and on W ( L, 0). In this context we prove the following result (adapted from Ref. [27]): Proposition 2.4 Let ( L, σ, T t ) be a one-particle dynamical system and consider the corresponding quasi-free dynamical systems ( W ( L, σ ) , α t ) and ( W ( L, 0) , β t ) . /negationslash Proof: First consider the KMS case. It follows from Lemma 2.3 that λ ( β ) defines a continuous affine map from G 0 ( W ( L, 0)) to S ( W ( L, σ )), which is injective because ω ( β ) ( W ( f )) = e -1 2 ω ( β ) 2 ( f,f ) = 0. If ρ ∈ G 0 ( W ( L, 0)), then ω := λ ( β ) ( ρ ) is invariant under α t , because ω ( β ) and ρ are equilibrium states for α t and β t , respectively, and these one-parameter groups are quasi-free with the same underlying T t . For any A = ∑ n i =1 c i W ( f i ) and B = ∑ n i =1 d i W ( f ' i ) in · W ( L, σ ) we have by a short computation involving the Weyl relations and the properties of ρ established in Lemma 2.1. A similar computation for ω ( α t ( B ) A ) and the KMS-condition for ω ( β ) now imply the existence of a function F AB as needed for the KMS-condition for ω . For the operators in the C ∗ -algebraic completion W ( L, σ ) one uses Proposition 2.1. Hence ω is a β -KMS state. For ground states, Eq. (2.3) (with ω 0 instead of ω ( β ) ) implies that the unitary group U t that implements α t in the GNS-representation of ω is weakly continuous and hence strongly continuous. The dense domain π ω ( W ( L, σ )))Ω ω is invariant under the action of U t and one may show that U t = e ith has strong derivatives there, because the same is true for ω 0 . Hence this domain is a core for the Hamiltonian h (see e.g. Thm. VIII.10 of Ref. [21]). Taking the derivative with respect to t of Eq.(2.3) and taking A = B shows that h ≥ 0, by Schur's Product Theorem (cf. Ref. [28] Ch.6 Sec.7 or Ref. [29]). This proves that ω is a ground state. We now turn to surjectivity. Given any ω ∈ G ( β ) ( W ( L, σ )) we may define the linear map ρ on · W ( L, 0) by ρ ( W ( f )) := ω ( W ( f )) ω ( β ) ( W ( f )) for all f ∈ L . Given any f, f ' ∈ L we now let F ( β ) W ( -f ) W ( f ' ) ( z ) and F W ( -f ) W ( f ' ) ( z ) be the functions on S β , obtained from the KMS-condition for ω ( β ) and ω , respectively. Note that F ( β ) W ( -f ) W ( f ' ) ( z ) = C exp( -F -f,f ' ( z )), by the one-particle KMS-condition for ω 2 (cf. Proposition 2.3), where C := exp( -1 2 ( ω ( β ) ( f, f ) + ω ( β ) ( f ' , f ' ))). Hence, defines a bounded and continuous function on S β which is holomorphic in its interior. Furthermore, G ( t ) = ρ ( W ( -f ) β t ( W ( f ' ))) and G ( t + iβ ) = ρ ( β t ( W ( f ' )) W ( -f )). As ρ is defined on a commutative C ∗ -algebra it then follows that G ( z + iβ ) = G ( z ) and we may extend G periodically to a bounded continuous function on C , which is entire holomorphic by the Edge of the Wedge Theorem [23]. Hence, G is constant (by Liouville's Theorem [23]) and ρ ( W ( -f ) β t ( W ( f ' )) = ρ ( W ( -f ) W ( f ' )) for all t ∈ R . A similar argument holds for the case of ground states. For any A = ∑ n i =1 c i W ( f i ) we have For some t > 0 we now let F M i := ∑ M -1 m =0 1 M T mt f i for any M ∈ N . Using the previous paragraph one shows that ρ ( W ( -F M i ) W ( F M j )) = ρ ( W ( -f i ) W ( f j )), from which we find However, as the one-particle structure ( p, K ) associated to ω ( β ) 2 is non-degenerate, we see from von Neumann's Mean Ergodic Theorem (Ref. [21] Thm. II.11) that lim M →∞ p ( F M i ) = 0. The exponential term will then converge to 1 as M →∞ , leading to the conclusion that ρ is positive. The unique extension of ρ to a state on W ( L, 0) is a ground state by the result of the previous paragraph and Lemma 2.1. The same argument works for the case of ground states. Finally, to see that λ ( β ) (resp. λ 0 ) is a homeomorphism it suffices to note that the inverse map ω ↦→ ρ is weak ∗ -continuous from G ( β ) ( · W ( L, σ )) (resp. G 0 ( · W ( L, σ ))) to G 0 ( · W ( L, 0)), by construction. /square Remark 2.2 In the setting of Proposition 2.4 we note that the space G 0 ( W ( L, 0)) of classical ground states always contains the pure state ρ 0 and that ω ( β ) = λ ( β ) ( ρ 0 ) . For any other pure classical ground state ρ ∈ G 0 ( W ( L, 0)) we consider the gauge transformations of the second kind η ρ of W ( L, σ ) and ζ ρ of W ( L, 0) (cf. Remark 2.1). We then have ρ = ζ ∗ ρ ρ 0 and λ ( β ) · ζ ∗ ρ = η ∗ ρ · λ ( β ) . Thus every extremal β -KMS state can be obtained from ω ( β ) by a gauge transformation of the second kind. The same holds for extremal ground states and ω 0 . In particular, all extremal ground states are pure.", "pages": [ 9, 10, 11, 12 ] }, { "title": "3 Review of geometric results", "content": "Before we consider the details of the linear scalar quantum field it is in order to study the spacetime in which it propagates. In the paragraphs below we will describe the class of stationary, globally hyperbolic spacetimes and the subclass of standard static spacetimes. For the latter case we also introduce the complexification and Euclideanization that are necessary in order to perform a Wick rotation. Most of our exposition here is a brief review of recent results of Refs. [30] and [31]. We assume that the reader is already familiar with the following standard terminology, which will be used throughout (cf. the reference book [32]): Definition 3.1 A spacetime M = ( M , g ) is a smooth, connected, oriented manifold M of dimension d ≥ 2 with a smooth Lorentzian metric g of signature ( -+ . . . +) . A Cauchy surface Σ in M is a subset Σ ⊂ M that is intersected exactly once by every inextendible time-like curve in M . A spacetime is said to be globally hyperbolic when it has a Cauchy surface. For a spacetime M we note that the manifold M is automatically paracompact [33]. We are mainly interested in spacetimes that are globally hyperbolic, because they allow us to formulate the linear field equation as an initial value (or Cauchy) problem. We will only consider Cauchy surfaces that are space-like, smooth hypersurfaces [34]. A globally hyperbolic spacetime is automatically time-orientable and we will assume that a choice of time-orientation has been fixed. It follows that any Cauchy surface is also oriented. Our notions and notations for causal relations, the Levi-Civita connection, etc. follow standard usage [32]. We will let h denote the Riemannian metric on a Cauchy surface Σ induced by the Lorentzian metric g on M , and we let ∇ ( h ) denote the corresponding Levi-Civita connection on Σ. Spacetime indices a, b, . . . are chosen from the beginning of the alphabet and run from 0 to d -1, whereas spatial indices are denoted by i, j, . . . and run from 1 to d -1.", "pages": [ 12 ] }, { "title": "3.1 Stationary spacetimes", "content": "Stationary spacetimes come equipped with a preferred notion of time-flow, which is mathematically encoded in the presence of a time-like vector field. To be precise: Definition 3.2 A stationary spacetime ( M,ξ ) is a spacetime M together with a smooth, complete, future-pointing, time-like Killing vector field ξ on M . Here completeness means that the corresponding flow Ξ: R ×M→M , defined by Ξ(0 , x ) = x and d Ξ( t, x ; ∂ t , 0) = ξ (Ξ( t, x )), is well defined for all t ∈ R . This flow is interpreted physically as the flow of time and following standard usage we write Ξ t : M→M for the map Ξ t ( x ) := Ξ( t, x ). ξ is a Killing vector field if it satisfies Killing's equation, ∇ ( a ξ b ) = 0, where the round brackets in the subscript denote symmetrization as an idempotent operation. Equivalently, it means that the metric is invariant under the time flow of ξ , Ξ ∗ t g = g for all t ∈ R . Example 3.1 Standard stationary spacetimes: Examples of stationary spacetimes are easily obtained by the following construction. Let S be a manifold of dimension d -1 , let h be a Riemannian metric on S , let v > 0 be a smooth, strictly positive function on S and let w be a smooth one-form on S such that h ij w i w j < v 2 . One now defines M := R × S with canonical projection map π : M→ S and the canonical time coordinate t : M→ R is the canonical projection onto the first factor. A stationary spacetime M = ( M , g ) is then obtained by defining where ⊗ s is the symmetrised tensor product. We will always choose adapted local coordinates on M , i.e. coordinates ( t, x i ) such that the x i are local coordinates on S , unless stated otherwise. Note that g indeed has a Lorentz signature and that the canonical vector field ∂ t on R gives rise to a Killing vector field ξ on M . On S 0 := { 0 }× S we can write ξ a = Nn a + N a , where n a is the future pointing unit normal vector field to S 0 ⊂ M and n a N a = 0 . The function N is known as the lapse function and N a as the shift vector field. They are related to v and w by where we use the fact that N a is tangent to Σ , so the component for a = 0 vanishes (in adapted local coordinates). The inverse of the metric takes the form where h ij is the inverse of the Riemannian metric h . Definition 3.3 A stationary spacetime of the form of Example 3.1 is called a standard stationary spacetime . Note that a standard stationary spacetime M is uniquely determined by the data ( S, h, v, w ). However, different data may give rise to the same spacetime, because there is a lot of freedom in the choice of the surface S ⊂ M . This is another way of saying that a stationary spacetime has a preferred time-flow, given by the Killing vector field, but it does not have a preferred time coordinate, because we can choose different canonical time coordinates which vanish on different spatial hypersurfaces. Although not all stationary spacetimes are standard, 3 they are the only ones of interest to us because of the following result: Proposition 3.1 Let M be a stationary spacetime which is globally hyperbolic. Then M is isometrically diffeomorphic to a standard stationary spacetime. This is Proposition 3.3 of Ref. [31]. The proof is elegant and short and we include it here for completeness: Proof: Fix a Cauchy surface Σ ⊂ M and use the flow Ξ of the Killing vector field to define a local diffeomorphism ψ : R × Σ → M by ψ ( t, x ) = Ξ( t, x ), The curves t ↦→ ψ ( t, x ) are time-like and inextendible, because ξ is assumed to be complete. This means that they intersect Σ exactly once, proving that ψ is both injective and surjective and hence a diffeomorphism. We define M ' := ( R × Σ , ψ ∗ g ) and it remains to show that M ' is standard stationary. This follows easily from the fact that ψ ∗ ξ = ∂ t , where t is the canonical time-coordinate on M ' , together with the fact that ∂ t ψ ∗ g = 0, which is Killing's equation. /square A more complicated issue is the converse question, whether a standard stationary spacetime is globally hyperbolic. A full characterization of those data ( S, h, v, w ) that give rise to a standard stationary spacetime M which is globally hyperbolic was recently given by Ref. [30]. It should be noted that S need not be a Cauchy surface, even if M is globally hyperbolic. A full characterization of those data for which S is a Cauchy surface was also given in Ref. [30]. To close this section we will sketch the main ingredients of this analysis and state the main results, although they will not be needed in the remainder of this paper. Let s ↦→ γ ( s ) := ( t ( s ) , x ( s )) be a smooth, time-like curve in a standard stationary spacetime M with data ( S, h, v, w ). The fact that γ is time-like can be stated as the quadratic inequality where ˙ denotes a derivative w.r.t. s . If γ is future pointing this leads to whereas for past-pointing γ we find F and ˜ F are smooth, strictly positive functions on TS \\ 0, where 0 denotes the zero section. (In fact, F and ˜ F define Finsler metrics on S of Randers type. We refer the interested reader to Ref. [30] for a brief introduction or to Ref. [35] for a full exposition on Finsler geometry.) It turns out that the questions concerning the causality of the standard stationary spacetime with data ( S, h, w, v ) can be determined entirely from the properties of S with respect to F and ˜ F . As for a Riemannian metric, we can use F to define the length of a smooth curve γ : [0 , 1] → S as l F ( γ ) := ∫ 1 0 F ( ˙ γ ( s )) ds and from that we can define a generalised distance function where C ( p, q ) is the set of all piecewise smooth curves from p to q . d satisfies all properties of a distance function, except symmetry. Indeed, if ˜ γ ( s ) := γ (1 -s ) we have l F (˜ γ ) = l ˜ F ( γ ), which in general differs from l F ( γ ). However, taking the ordering into account one can still define notions of forward and backward Cauchy sequences and corresponding notions of forward and backward completeness for the pair ( S, F ) [35, 30]. We now state without proof the results on the causality of standard stationary spacetimes (Thm. 4.3b, Thm. 4.4 and Cor. 5.6 of Ref. [30]). Theorem 3.1 Let M be a standard stationary spacetime with data ( S, h, v, w ) . (iii) If M is globally hyperbolic, then ( S, ˜ h ) is a complete Riemannian manifold with We record for completeness that the inverse metric of ˜ h is given by ˜ h ij = v 2 h ij -v 2 N -2 N i N j = v 2 g ij , where g ij is expressed in adapted coordinates.", "pages": [ 13, 14, 15 ] }, { "title": "3.2 Standard static spacetimes", "content": "We have seen that stationary spacetimes have a preferred time flow, but no preferred time coordinate. This is different for standard static spacetimes, which we will describe now. For a fuller discussion of static spacetimes we refer the reader to Ref. [31] and references therein. Definition 3.4 A static spacetime M = ( M , g, ξ ) is a stationary spacetime with a Killing vector field ξ that is irrotational. The property that ξ is irrotational means that the distribution of vectors orthonogal to ξ is involutive, i.e. [ X,Y ] a ξ a = 0 when X a ξ a = Y a ξ a = 0. This can be expressed equivalently as where the square brackets in the subscript denote anti-symmetrization as an idempotent operation. By Frobenius' Theorem (Ref. [32] Thm. B.3.2) ξ is irrotational if and only if M can be foliated by hypersurfaces orthogonal to ξ . If x i , i = 1 , . . . d -1, are local coordinates on a ( d -1)-dimensional hypersurface H ⊂ M orthogonal to ξ we can (locally) supplement them by the parameter t appearing in the flow Ξ t to define coordinates on a portion of M . When used like this, we call t a Killing time coordinate. Note that the surfaces of constant t remain orthogonal to ξ = ∂ t , because they are the image of H under Ξ t . Remark 3.1 Although the definition of a (local) Killing time coordinate depends on the choice of the hypersurface H , any two Killing time coordinates on the same open set differ at most by a constant, because both are constant on the hypersurfaces orthogonal to ξ . In this sense, static spacetimes have a preferred time coordinate up to a constant, which we will often call the Killing time coordinate, with some slight abuse of language. In the local coordinates ( t, x i ) the metric can be expressed as with 1 ≤ i, j ≤ d -1 and the smooth coefficient functions v, g ij are independent of t . We introduce a special name for the class of static spacetimes for which this form of the metric can be obtained globally: Definition 3.5 A standard static spacetime M = ( M , g, ξ ) is a standard stationary spacetime with a vanishing shift vector field, i.e. M/similarequal R × S , ξ = ∂ t and where the Killing time coordinate t is the projection on the first factor of R × S , π is the projection on the second factor, h is a Riemannian metric on S and N is a smooth, strictly positive function on S . The data ( S, h, N ) determine a unique standard static spacetime, which is the standard stationary spacetime with data ( S, h, v = N,w = 0). The canonical time coordinate of the latter coincides with the Killing time coordinate. Unlike the stationary case, there is only a limited freedom in the choice of data that describe a fixed standard static spacetime M . Indeed, suppose that ( S, h, v ) and ( S ' , h ' , v ' ) determine the same standard static spacetime M and consider the hypersurfaces S 0 = { 0 }× S and S ' 0 = { 0 }× S ' in M . By Remark 3.1 there is a T ∈ R such that the diffeomorphism Ξ T of M has S ' 0 = Ξ T ( S 0 ), Ξ ∗ T h ' = h and Ξ ∗ T v ' = v . For our applications to Wick rotations we are particularly interested in spacetimes which are both standard static and globally hyperbolic. To determine whether a standard static spacetime is globally hyperbolic we quote from Theorem 3.1 in Ref. [31]: Theorem 3.2 For a standard static spacetime M with data ( S, h, v ) the following are equivalent: This is in fact a special case of Theorem 3.1, when w = 0. In the ultra-static case v ≡ 1, it essentially reduces to Proposition 5.2 in Ref. [7]. Note, however, that ( S, h ) itself need not be a complete Riemannian manifold in general. Remark 3.2 The metric ˜ h is also called the optical metric [36], because geodesics of ˜ h are the projections onto Σ of light-like geodesics in M . To see this we first note that the light-like geodesics of M = ( M , g ) coincide with those of ˜ M := ( M , v -2 g ) after a reparametrization (cf. Ref. [32] Appendix D). Because ˜ M is ultra-static, the geodesic equation for a curve γ ( s ) = ( t ( s ) , x ( s )) decouples into the geodesic equation for x in ( S, ˜ h ) and ∂ 2 s t = 0 . (Ref. [36] also uses the term optical metric in the stationary case for the metric N -2 h , although the motivation is less convincing in that case. It might be more appropriate to refer to the Finsler metrics F, ˜ F of Section 3.1 as optical metrics.) When the spacetime M is both globally hyperbolic and static, it is automatically a standard stationary spacetime by Proposition 3.1. However, it may yet fail to be a standard static spacetime. A simple counter-example, taken from Ref. [37] (see also Ref. [31]), is the cylinder spacetime M = ( R × S 1 , g ) with the metric g := -dt ⊗ 2 + dθ ⊗ 2 + 2 dt ⊗ s dθ . This is a globally hyperbolic spacetime with Cauchy surfaces diffeomorphic to the circle S 1 . The vector field ξ = ∂ t is a time-like Killing field, which is irrotational on dimensional grounds. However, hypersurfaces orthogonal to ξ must be diffeomorphic to R , as they wind around the cylinder. A complete characterization of which static, globally hyperbolic spacetimes are standard static is given by Proposition 3.2 Let ( M,ξ ) be a static, globally hyperbolic spacetime. Then M is isometrically diffeomorphic to a standard static spacetime if and only if it admits a Cauchy surface that is Killing field orthogonal. Proof: If M is isometrically diffeomorphic to a standard static spacetime, the existence of a Killing field orthogonal Cauchy surface follows from Theorem 3.2. Conversely, if such a Cauchy surface exists we may choose this surface in the proof of Proposition 3.1, which simultaneously shows that M is isometrically diffeomorphic to a standard stationary spacetime M ' and that the metric g ' has no cross terms involving w . Hence, M ' is standard static. /square", "pages": [ 15, 16 ] }, { "title": "3.3 Spacetime complexification", "content": "To conclude our geometric considerations we now define complexifications and Riemannian manifolds associated to any given standard static spacetime. With a view to our applications to thermal states it is necessary to consider the case where the domain of the imaginary time variable is compactified. For this purpose we let R > 0 and we define the cylinder This equivalence relation compactifies the imaginary axis of C to a circle S 1 R of circumference 2 πR . C ∞ := C can be taken as a degenerate case with R = ∞ and S 1 ∞ := R . Let M be a standard static spacetime with data ( S, h, N ). For any R > 0 we define the complexification M c R as the real manifold M c R = C R × S endowed with the symmetric, complexvalued, tensor field where z = t + iτ is the coordinate on C R . M can be embedded into M c R as the τ = 0 surface and g c R is the analytic continuation of g in z . Furthermore, we define the Riemannian manifold M R := { ( z, x ) ∈ M c R | t = 0 } endowed with the pull-back metric of g c R Note that M R /similarequal S 1 R × S as a manifold and since S = M ∩ M R in M c R , we can identify S also as the { τ = 0 } surface in M R . M R has a Killing field ξ R = ∂ τ , which can be viewed as the analytic continuation of ξ = ∂ t . The constructions above do not depend on any freedom in the choice of S , because this freedom boils down to a Killing time translation (see Remark 3.1) which has a unique analytic continuation to M c R . It is also unnecessary for S to be a Cauchy surface at this stage. Note that in the standard stationary case there is more freedom to choose canonical time coordinates, so it would be unclear whether an analogous construction can be made independent of the choice of such a coordinate. Besides, any cross terms w ⊗ dt in the metric would spoil the real-valuedness of the restriction g R of the analytically continued metric, so it would not be Riemannian. Whereas the Killing time coordinate on M is used to define the complexifications M c R and the Riemannian manifolds M R , it may be a bad choice of coordinate to analyze the behaviour near the edge of S . This will be the case e.g. if M is the right wedge of a static black hole spacetime with a bifurcate Killing horizon and we wish to study the behaviour near the bifurcation surface. 4 Anticipating these problems we now consider Gaussian normal coordinates near S , instead of the Killing time coordinate, and we study the properties of the complexification procedure above with respect to these new coordinates. Proposition 3.3 Let M be a standard static spacetime, let R > 0 and let x i denote local coordinates on a portion U of S . Let x = ( x 0 , x i ) be the corresponding Gaussian normal coordinates on a portion of M , containing U , and let x ' = (( x ' ) 0 , x i ) be Gaussian normal coordinates on a portion of M R , containing U . We may express the metrics g and g R in these coordinates as and we then have for all n ≥ 0 : In the ultra-static case we have x 0 = t , which means that the metric g is real-analytic in x 0 and its analytic continuation satisfies g ab ( ix 0 , x i ) = ( g R ) ab ( x 0 , x i ), This immediately implies Eq. (3.1), by the Cauchy-Riemann equations and the reality of g and g R . In the general case, the Proposition can be interpreted as saying that g is 'infinitesimally holomorphic' in z := x 0 + i ( x ' ) 0 . Proof: The form of the metrics follows from the construction of Gaussian normal coordinates, as is well known [32]. The idea is now to use the fact that the geometries of M and M R are entirely determined by ( S, h, N ). The number of coefficients in ( h ij , ξ a ) equals d ( d +1) 2 , which is exactly the number of components of Killing's equation. We may write out Killing's equation in the chosen local coordinates, for which the Christoffel symbol vanishes when two or more indices are 0. The (00)-component of Killing's equation is then ∂ 0 ξ 0 = 0, which means that ξ 0 ( x ) = N ( x i ). Substituting this back in the remaining equations yields 5 All normal derivatives of ξ i and h ij are uniquely determined by the initial data, as can be shown by induction, taking successive normal derivatives of the equations above. In the Riemannian case we find similarly ξ 0 R ( x ' ) = N ( x i ) and Note the change of sign in the first equation when compared to the Lorentzian case. One now proves by induction on n ≥ 0 that 6 For n = 0 these equalities are true, because they just express the equality of the initial data. (Note in particular that ξ i | U = 0 = ξ i R | U .) Now suppose they hold true for all 0 ≤ l ≤ n . We use Killing's equation and ∂ 0 N = ∂ ' 0 N = 0 to compute where the induction hypothesis was used in the second equality. Similarly, by the binomial formula, where we used that fact that . As h ij is invertible, the result for n + 1 follows, completing the proof by induction. The statement of the proposition is then immediately clear, because both h ij and h ' ij are real-valued. /square Corollary 3.1 For a smooth curve γ : [0 , 1] → S the following are equivalent: Proof: We express the geodesic equation in M in terms of local coordinates x i on S and a Gaussian normal coordinate x 0 near S ⊂ M . Using the notation γ a := x a · γ , with γ 0 = 0, the components form exactly the geodesic equation in ( S, h ). The remaining equation is which is true by Proposition 3.3. This proves the equivalence of the first and second statements. The equivalence of the first and third statement is shown in a similar manner. /square", "pages": [ 16, 17, 18 ] }, { "title": "4 The linear scalar quantum field", "content": "It is well understood how to quantize a linear real scalar field on any globally hyperbolic spacetime [1, 2, 3, 4]. In this section we will present this quantization, with a special focus on the case where the spacetime is stationary [7]. This extra structure allows one to obtain additional results concerning e.g. ground states for the Killing flow. As a matter of convention we will identify distributions on M, M R and Σ with distribution densities, using the natural volume forms determined by the metrics. To unburden our notation we will often leave the volume form implicit, which should not lead to any confusion. However, we point out that the volume form is important when restricting to submanifolds, because in that case a change in volume form is involved. We will also make use of the natural Hilbert spaces of squareintegrable functions on the various spacetimes and Riemannian manifolds, where integration is performed with respect to the volume forms determined by the metrics. This understood we may leave the volume forms implicit in our notation, writing e.g. L 2 ( M ), L 2 (Σ) instead of L 2 ( M,d vol g ) and L 2 (Σ , d vol h ).", "pages": [ 19 ] }, { "title": "4.1 The classical scalar field in stationary spacetimes", "content": "The classical theory of a linear scalar field on a spacetime M is described by the (modified) Klein-Gordon equation for φ ∈ C ∞ ( M ), where /square := ∇ a ∇ a denotes the Laplace-Beltrami operator and the potential V is a smooth, realvalued function. V is often chosen to be of the form with mass m and scalar curvature coupling c . In any globally hyperbolic spacetime, the KleinGordon equation has a well posed initial value formulation (see e.g. Ref. [2] Ch.3 Thm. 3.). To formulate it we introduce the space of initial data as a topological direct sum, where each summand carries the test-function topology. Theorem 4.1 Let Σ ⊂ M be a Cauchy surface in a globally hyperbolic spacetime M with future pointing normal vector field n a . For each ( φ 0 , φ 1 ) ∈ D (Σ) there is a unique φ ∈ C ∞ ( M ) such that Moreover, supp( φ ) ⊂ J (supp( φ 0 ) ∪ supp( φ 1 )) and the linear map S : D (Σ) → C ∞ ( M ) which sends ( φ 0 , φ 1 ) to the corresponding solution φ of Eq. (4.2) is continuous, if C ∞ (Σ) is endowed with the usual Fr'echet topology. It follows from Theorem 4.1 that the Klein-Gordon operator K has unique advanced ( -) and retarded (+) fundamental solutions E ± and we define E := E --E + . The solution map S and the operator E will be used frequently to translate between the spacetime and the initial data formulations of the theory and we note that where Σ ⊂ M is any Cauchy surface and f, f ' ∈ C ∞ 0 ( M ). The kernel of E , acting on C ∞ 0 ( M ), is exactly KC ∞ 0 ( M ) [1] and for later use we introduce the real-linear space In a stationary, globally hyperbolic spacetime ( M,ξ ), the Killing vector field determines a natural time evolution. We fix a Cauchy surface Σ ⊂ M and use it to write M as a standard stationary spacetime (cf. Sec. 3.1). We will work throughout in adapted coordinates x a = ( t, x i ) and assume that the potential V is stationary, As the potential V is real-valued we may view K as a symmetric operator on the dense domain C ∞ 0 ( M ) in L 2 ( M ). We will now separate off the canonical time dependence of this operator and write the spatial dependence in terms of h ij , N , N i and V . The cleanest way to do so is by ensuring that we obtain symmetric operators in L 2 (Σ) for the spatial parts. For this reason it is convenient to consider the unitary isomorphism onto the Hilbert tensor product, where R is viewed as a Riemannian manifold with the standard metric dt . To see that U is indeed an isomorphism we use Schwartz Kernels Theorem, the diffeomorphism M /similarequal R × Σ and the fact that det g = -N 2 det h and d vol g = Ndt d vol h , which may be seen by choosing local coordinates on Σ that diagonalize h ij at a point. The symmetric operator UNKNU -1 can now be written as The computation that leads to this expression has been omitted, because it is straightforward. 7 Because ξ is a Killing field, the flow Ξ t preserves the Klein-Gordon equation: K Ξ ∗ t φ = Ξ ∗ t ( Kφ ) for all t ∈ R . Moreover, if Kφ = 0 and φ has compactly supported initial data on some Cauchy surface, then the same is true for Ξ ∗ t φ . This means that the time flow determines a time evolution on the initial data in D (Σ). Indeed, let S be the solution operator of Theorem 4.1 and let S -1 be its inverse, i.e. S -1 ( φ ) = ( φ | Σ , n a ∇ a φ | Σ ). We may define the time evolution maps T t on D (Σ) by T t := S -1 Ξ ∗ t S . The maps T t form a continuous (even smooth) one-parameter group for t ∈ R , by Theorem 4.1. The infinitesimal generator H cl of the group T t is the classical Hamiltonian: Lemma 4.1 The (classical) Hamiltonian operator H cl is given (in matrix notation on D (Σ) ) by Proof: The computation is simplified somewhat by defining X := ( I 0 N i ∇ ( h ) i N ) , with inverse the first row of XH cl X -1 is simply (0 -iI ) and the second row can be be found by writing ∂ 2 0 = N 1 2 ∂ 2 0 N -1 2 and by eliminating the second order time derivative using Eq. (4.4) and Kφ = 0. H cl is then obtained from a straightforward matrix multiplication. The details are omitted. /square Although the metric ˜ h has the advantage of being complete, it may be a less natural choice than h , especially when the spacetime M is isometrically embedded into a larger spacetime. For any solution φ ∈ C ∞ ( M ) of the Klein-Gordon equation one defines the stress-energymomentum tensor which is symmetric and because Kφ = 0. By Killing's equation ξ satisfies where we used the assumption that V is stationary. Note that energy-momentum is conserved, even though the stress-tensor may not be divergence free. On a Cauchy surface Σ with future pointing normal n a , the energy density is defined by If φ = S ( φ 0 , φ 1 ) for some ( φ 0 , φ 1 ) ∈ D (Σ), then we can also define the total energy on Σ by The conservation of P a ( φ ) implies that E ( φ ) is independent of the choice of Cauchy surface, by Stokes' Theorem. In particular, E (Ξ ∗ t φ ) = E ( φ ) for all t , because the left-hand side is the integral of ε Σ ' ( φ ) over the Cauchy surface Σ ' := Ξ t (Σ). Lemma 4.2 Viewing D (Σ) as a dense domain in L 2 (Σ) ⊕ 2 we have where the operator A is given by Proof: The form of E can be computed by expressing the energy density on Σ in terms of the initial data. The computation is straightforward, so the details are omitted. The final equality is then obvious from Lemma 4.1, whereas the final inequality follows from where the first term in the integrand is non-negative and may be dropped. When V > 0 everywhere, we may define an energetic inner product on L ⊗ C by setting where the inner product on the right-hand side is in L 2 (Σ) ⊕ 2 . Note that 〈 , 〉 e is indeed positive and non-degenerate, by the properties of A established in Lemma 4.2 and the positivity of V N and N -1 v 2 . Since V is stationary, the energetic inner product is independent of the choice of Cauchy surface, like the energy, because /square Definition 4.1 When V is stationary and V > 0 , the energetic Hilbert space H e is the Hilbert space completion of L ⊗ C in the energetic norm. H e can be interpreted as the space of all (complex) finite energy solutions of the Klein-Gordon equation (4.1). The following detailed description of the energetic Hilbert space is the main result of this section. The proof makes use of strictly positive operators and we have collected some basic results on such operators in A (see also Ref. [38]). Theorem 4.2 Let M be a stationary, globally hyperbolic spacetime with a Cauchy surface Σ and assume that V is stationary and V > 0 . Let ˆ A denote the Friedrichs extension of the operator A of Lemma 4.2. The linear map q cl : D (Σ) → L 2 (Σ) ⊕ 2 defined by q cl ( φ 0 , φ 1 ) := √ ˆ A ( φ 0 φ 1 ) is continuous, injective, has dense range, commutes with complex conjugation and satisfies ‖ q cl ( φ 0 , φ 1 ) ‖ 2 = 2 2 There is a unique, strongly continuous unitary group O t = e itH e on L 2 (Σ) ⊕ 2 such that O t q cl = q cl T t . Its infinitesimal generator is given by H e := 2 i √ ˆ Aσ √ ˆ A . iH e commutes with complex conjugation, H e and all its powers H n e , n ∈ N , are essentially self-adjoint on the range of q cl , H e is invertible and the range of q cl is a core for | H e | -1 . E ( S ( φ 0 , φ 1 )) . Hence, H e /similarequal L (Σ) ⊕ . The explicit characterization of H e in terms of L 2 (Σ) ⊕ 2 is often very useful, although it is less aesthetically appealing, because it requires the choice of an arbitrary Cauchy surface Σ. Proof: We first consider the Friedrichs extension ˆ A of A , which is a positive, self-adjoint operator. By Lemma A.7, D (Σ) is a core for ˆ A 1 2 . Furthermore, ˆ A ≥ B , where the operator B := 1 2 ( V N 0 0 N -1 v 2 ) is defined on D (Σ) (cf. Lemma 4.2). Note that B is essentially self- adjoint with a strictly positive closure, by Proposition A.1. Hence, ˆ A is also strictly positive, by Lemma A.6, and D (Σ) is in the domain of ˆ A -1 2 . Moreover, as D (Σ) is a core for ˆ A 1 2 , the latter has a dense range on D (Σ). Therefore, q cl is a well defined, injective linear map with dense range R and by Lemma 4.2, ‖ q cl ( φ 0 , φ 1 ) ‖ 2 = E ( S ( φ 0 , φ 1 )). As S is continuous, the last equation also entails the continuity of q cl . (Alternatively one may use Theorem A.1 of A.) Also note that A commutes with complex conjugation in L 2 (Σ) ⊕ 2 , hence the same is true for ˆ A 1 2 and for q cl . Because q cl is invertible we may define O t by O t = q cl T t q -1 cl on R . Note that the total energy ‖ O t q cl ( φ 0 , φ 1 ) ‖ 2 = E (Ξ ∗ t S ( φ 0 , φ 1 )) is independent of t , so each O t is a densely defined isometry, which extends uniquely to a unitary isomorphism on the entire Hilbert space, again denoted by O t . O -1 t = O -t and the continuity of f ↦→ T t f in the test-function topology entails the strong continuity of O t . Because the time-derivative of T t ( φ 0 , φ 1 ) converges in the test-function topology of D (Σ) and q cl is continuous, the infinitesimal generator of O t is well defined on the range R of q cl , where it is given by ˆ A commutes with complex conjugation, so it is clear that iH e also commutes with it. Furthermore, the map M := i 2 ˆ A -1 2 σ ˆ A -1 2 is well defined on R and it satisfies MH e = I there. Note that M is closable, because it is symmetric and densely defined. By Lemma A.1, H e must be invertible. Lemma A.4 implies that H -1 e is self-adjoint and invertible and a core is given by H e R ⊂ R . As M is a symmetric extension of H -1 e on this domain, we must have M = H -1 e and the domain R of M is a core for H -1 e and hence also for | H e | -1 , by the Spectral Calculus Theorem. /square because of the Lemmas 4.1, 4.2. Both H e and O t preserve R , so H e and all its powers are essentially self-adjoint on R by Lemma 2.1 in Ref. [39].", "pages": [ 19, 20, 21, 22 ] }, { "title": "4.2 The scalar quantum field in stationary spacetimes and equilibrium one-particle structures", "content": "We now study the quantised scalar field in a stationary spacetime, where the ground states play a similarly important role for the theory as the vacuum state in Minkowski spacetime. Because of the importance of quasi-free equilibrium states (cf. Sec. 2) we first focus on equilibrium one-particle structures, whereas the ground and equilibrium states (beyond their two-point distributions) will be discussed in Section 5 below. The well posedness of the Cauchy problem established in Theorem 4.1 remains true if we specify arbitrary distributional initial data, allowing distributional solutions and using distributional topologies [40]. In this setting it is natural to introduce local observables, associated to arbitrary f ∈ C ∞ 0 ( M ), which measure the distributional field φ by the formula φ ( f ) := ∫ φf . These observables φ ↦→ φ ( f ) can be regarded as functions on the space of classical solutions φ and we may use them to generate an algebra of observables. We choose to work with the Weyl C ∗ -algebra W cl := W ( L, 0), whose elements we interpret as e iφ ( f ) , which remains bounded when φ and f are real-valued. Interpreting the right-hand side of Eq. (4.3) in terms of initial values and momenta motivates the introduction of the symplectic space ( L, E ), so that the corresponding quantum theory is described by W := W ( L, E ). For each open subset O ⊂ M we will denote by W ( O ) the C ∗ -subalgebra generated by those W ( f ) with f supported in O (and similarly for W cl ( O )). In this way one obtains a net of local C ∗ -algebras [41, 4]. When ( M,ξ ) is a stationary, globally hyperbolic spacetime and V is stationary, ( L, 0 , T -t ) and ( L, E, T -t ) become one-particle dynamical systems. This follows from the fact that Ξ ∗ -t preserves the metric and that the E ± are unique, so the symplectic form E ( f, f ' ) := ∫ M fEf ' is preserved. We may consider the associated quasi-free dynamical systems ( W cl , α cl t ) and ( W , α t ), so that /negationslash for all f ∈ L . 8 α cl t and α t describe the Killing time flow at an algebraic level and we note that α t ( W ( O )) = W (Ξ t ( O )) and similarly in the classical case. However, neither α cl t nor α t is normcontinuous in t , as ‖ w ( f ) -w ( g ) ‖ = 2 for all f = g ∈ L (Ref. [19] Prop. 3-10). For this reason, general results on C ∗ -dynamical systems [14, 20] do not apply directly to our situation. (Nor can we view ( W , α t ) as a W ∗ -dynamical system, because W is not a W ∗ -algebra or von Neumann algebra.) In order to take advantage of the smoothness of the time evolution maps T t on D (Σ) we need the following definition. Definition 4.2 We call a state ω on the Weyl C ∗ -algebra W (or W cl ) D k , k > 0 , when it is C k (cf. Def. 2.6) and the maps are distributions on M × n for all 1 ≤ n ≤ k . The ω n are called the n -point distributions. A state is called regular , or D ∞ , when it is D k for all k > 0 . In our setting the distributional character of the ω n is natural and useful. Remark 4.1 An alternative description of the scalar quantum field uses the ∗ -algebra A , generated by the identity I and the smeared field operators Φ( f ) , f ∈ C ∞ 0 ( M ) , satisfying Although the algebras A and W are technically different, their relation can be understood from a physical point of view by formally setting W ( f ) = e i Φ( f ) . In suitable representations this can be made rigorous. This applies in particular to regular states ω on W , which give rise to a corresponding state on A .", "pages": [ 23, 24 ] }, { "title": "4.2.1 Two-point distributions", "content": "When ω is a D 2 state on W , we may identify the one-particle structure ( p, K ) of ω 2 as a map into a subspace of the GNS-representation space H ω , as in the proof of Proposition 2.3. A similar construction applies to the so-called truncated two-point distribution, ω T 2 ( x, x ' ) := ω 2 ( x, x ' ) -ω 1 ( x ) ω 1 ( x ' ), where we now take p ( f ) := π ω (Φ( f ) -ω 1 ( f ) I )Ω ω . Note that ω T 2 is indeed a twopoint distribution, (cf. Theorem 2.3) and that ω 2 = ω T 2 when ω 1 = 0, so in that case the two constructions coincide. When ω 2 is a distribution, the associated one-particle structure can be viewed as a K -valued distribution p which satisfies the Klein-Gordon equation [42]. (Conversely, when p is a distribution, the associated ω 2 is also a distribution.) For any Cauchy surface Σ, p is uniquely determined by its initial data, which form a continuous linear map q Σ : D (Σ) →K with dense range and such that (cf. Eq. (4.3)). Conversely, any such linear map q Σ determines a unique one-particle structure. Indeed, just like smooth solutions to the Klein-Gordon equation, two-point distributions are uniquely determined by their initial data on a Cauchy surface: Proposition 4.1 Let Σ ⊂ M be a Cauchy surface in a globally hyperbolic spacetime with future pointing normal n a and let ω be a distribution density in M × 2 . If K x ω ( x, y ) = K y ω ( x, y ) = 0 , then the restrictions are well defined distribution densities in Σ × 2 for all i, j ∈ N . Conversely, for any four distribution densities ω ij , 0 ≤ i, j ≤ 1 , on Σ × 2 , there is a unique distribution density ω on M × 2 such that Support and continuity properties analogous to Theorem 4.1 also hold, but we will not need them. We omit the proof of this basic result. There is a preferred class of D 2 states, called Hadamard states, which are characterised by the fact that their two-point distribution has a singularity structure that is of the same form as for the Minkowski vacuum state. These states are important, because the renormalised Wick powers and stress tensor of the quantum field have finite expectation values in them. To put it more precisely, ω 2 is of Hadamard form if and only if [43] /negationslash This condition is already implied by one of the following apparently weaker, and often more convenient, estimates on ω 2 or its associated one-particle structure ( p, K ): where V ± M ⊂ T ∗ M is the space of future (+) or past ( -) pointing causal co-vectors on M (cf. Ref. [42], Prop. 6.1). For any regular state (even if it is not quasi-free) the Hadamard condition allows one to estimate the singularity structure of all higher n -point distributions too [44], so that the state satisfies the microlocal spectrum condition of Ref. [45]. By the Propagation of Singularities Theorem and the fact that ω 2 solves the Klein-Gordon equation in both variables it suffices to check the condition in Eq. (4.6) on a Cauchy surface Σ: Unfortunately it is somewhat complicated to see whether a state ω 2 is Hadamard by inspecting its initial data on a Cauchy surface Σ. The initial data of ω 2 should be smooth away from the diagonal in Σ × 2 , so it suffices to characterize the singularities on the diagonal. However, for the singularities on the diagonal we are not aware of any argument that avoids the use of the Hadamard parametrix construction, which involves the Hadamard series for which Hadamard states were originally named.", "pages": [ 24, 25 ] }, { "title": "4.2.2 Equilibrium two-point distributions", "content": "An equilibrium one-particle structure ( p, K ) has some nice additional structure when p is a distribution: Lemma 4.3 If ( p, K ) is an equilibrium one-particle structure such that p is a distribution, then the unitary group ˜ O t on K defined by ˜ O t p = p Ξ ∗ -t (on C ∞ 0 ( M ) ) is strongly continuous, ˜ O t = e itH . Its strong derivative is well defined on the range of p , H is essentially self-adjoint on this range and Hp ( f ) = ip ( ∂ 0 f ) for all f ∈ C ∞ 0 ( M ) . Proof: The strong continuity of ˜ O t follows from the continuity of t ↦→ Ξ ∗ -t f in the test-function topology and the fact that p is a distribution. The formula for H on the range of p can be deduced from the continuity of p by a direct calculation: The essential self-adjointness of H on the range of p then follows from Chernoff's Lemma [39]. /square The next two results are the main results of this section. They are existence and uniqueness results for non-degenerate ground and β -KMS one-particle structures. For the existence of a nondegenerate ground we adapt a result of Ref. [7], which imposed additional restrictions on the potential V and on the Killing field in order to obtain such a ground one-particle structure with, in addition, a mass gap. For the existence of a non-degenerate β -KMS one-particle structure see Refs. [6, 27]. Theorem 4.3 Let M be a globally hyperbolic, stationary spacetime and consider a linear scalar field with a stationary potential V such that V > 0 . where P -is the spectral projection onto the negative part of the spectrum of H e and p cl ( f ) := q cl S -1 E ( f ) . The occurrence of P -, rather than P + , is in line with the footnote on page 23. Proof: We start with the H e -valued distribution p cl ( f ) := q cl S -1 E ( f ) and the unitary group O t determined by Theorem 4.2. Define p 0 ( f ) := √ 2 | H e | -1 2 P -p cl ( f ) and let the closed range of p 0 be denoted by K 0 . It is not hard to see that O t p 0 ( f ) = p 0 (Ξ ∗ t f ), so O t preserves K 0 and we may let ˜ O t := O -t | K . The generator H of this strongly continuous unitary group is the restriction of -H e , which is strictly positive there. The range of p 0 is in the domain of H and H -1 2 , by Theorem 4.2. If we let C denote the complex conjugation on L 2 (Σ) ⊕ 2 , then CH e C = -H e , so CP -C = P + , the spectral projection onto the positive part of the spectrum of H e . Thus, This proves that ( p 0 , K 0 ) is a non-degenerate ground one-particle structure. Viewing K ( β ) as a subspace of H ⊕ 2 e we note that O ⊕ 2 t preserves the range of p ( β ) , because O ⊕ 2 t p ( β ) ( f ) = p ( β ) (Ξ ∗ t f ). We can therefore define a strongly continuous unitary group ˜ O t on K ( β ) as the restriction of O ⊕ 2 -t . The generator H of ˜ O t is given by the restriction of | H e | ⊕ -| H e | and the range of p ( β ) is contained in D ( e -β 2 H ). One may then compute The formula for p ( β ) is well defined, because the range of p cl is in the domain of | H e | -1 by Theorem 4.2. It defines a K ( β ) -valued distribution with dense range, which solves the Klein-Gordon equation. Just like for the ground one-particle structure one may check that 〈 p ( β ) ( f ) , p ( β ) ( f ' ) 〉 -〈 p ( β ) ( f ' ) , p ( β ) ( f ) 〉 = iE ( f, f ' ), so ( p ( β ) , K ( β ) ) does indeed define a one-particle structure. This implies the one-particle KMS-condition, because for any f, f ' ∈ C ∞ 0 ( M, R ) the function is bounded and continuous on S β and holomorphic in its interior. The correct boundary conditions follow from Eq. (4.7). /square As ( p 0 , K 0 ) is non-degenerate, the associated quasi-free state is non-degenerate too (Proposition 2.3) and hence it is pure (by Borchers' Theorem 2.1). We then see from Theorem 2.4 that p 0 already has dense range on the real subspace. (Of course a direct proof of this fact is also possible.) Remark 4.2 Note that there is a connection between the classical energy and the Hamiltonian operator H 0 in the ground one-particle structure, which is given by as may be shown by the same techniques employed in the proof of Theorem 4.3. Next we establish a uniqueness result for non-degenerate ground and β -KMS one-particle structures [46, 6]. 9 Proposition 4.2 Let ( p 2 , K 2 , ˜ O (2) t ) be a ground, resp. β -KMS, one-particle structure (with β > 0 ) and let P 2 be the orthogonal projection onto the space of ˜ O (2) t -invariant vectors. Let ( p 1 , K 1 , ˜ O (1) t ) be the non-degenerate ground, resp. β -KMS, one-particle structure of Theorem 4.3. Then there is a unique isometry U : K 1 →K 2 such that ˜ O (2) t U = U ˜ O (1) t and Up 1 = ( I -P 2 ) p 2 . In particular, if P 2 = 0 , then U is an isomorphism. Let w := ω (2) 2 -ω (1) 2 denote the difference of the associated two-point functions ω ( i ) 2 . Then w is a real-valued, symmetric (weak) bi-solution to the Klein-Gordon equation which is of positive type and independent of the Killing time (in both entries). If p 2 is a distribution on M , then w ∈ C ∞ ( M × 2 ) . Proof: The proof follows Ref. [46] (see also Ref. [47]). For arbitrary f, f ' ∈ C ∞ 0 ( M, R ) the function is continuous (by the Definition 2.7 of ground and β -KMS one-particle structures) and real-valued on R . Suppose both one-particle structures satisfy the one-particle β -KMS condition at the same β > 0. There is then a bounded continuous extension ˜ F of F to S β , holomorphic in the interior. By repeatedly applying Schwarz' reflection principle [23], ˜ F extends to a bounded holomorphic function on all of C , which means that ˜ F and F are constant, by Liouville's Theorem [23]. Similarly, if both are ground one-particle structures, the positivity of the infinitesimal generators H i implies that there is a bounded, holomorphic function F + in the upper half plane, which has F as its boundary value. By Schwarz' reflection principle, F + can be extended to a bounded holomorphic function on the entire plane, which again means that F is constant. Note that the range of p 1 is in the domain of H 1 , because the strong derivative ∂ t ˜ O (1) t p 1 ( f ) | t =0 exists (cf. Theorem 4.3). The same is true for p 2 and H 2 , because ‖ ( ˜ O (2) t -I ) p 2 ( f ) ‖ 2 -‖ ( ˜ O (1) t -I ) p 1 ( f ) ‖ 2 ≡ 0, by the previous paragraph. The constancy of F implies ∂ 2 t F | t =0 = 0, i.e. This equality must hold for all f, f ' ∈ C ∞ 0 ( M ), by complex (anti-)linearity. We may therefore define linear maps X i := H i p i and we let V i := ker( X i ) denote their kernels. By the previous equation, V 1 = V 2 =: V , so the X i descend to linear injections ˜ X i : C ∞ 0 ( M ) /V →K i . We set U := ˜ X 2 ˜ X -1 1 between the ranges of the X i . It is obvious from the previous paragraph that U is an isometry, because UH 1 p 1 = H 2 p 2 . The non-degeneracy of the first one-particle structure implies that H 1 is injective, while the range of p 1 is a core for it. It follows that the map ˜ X 1 has a dense range, so U extends by continuity to an isometry from K 1 into K 2 . Note that U intertwines between the unitary groups, because ˜ O ( i ) t H i p i ( f ) = H i p i (Ξ ∗ -t f ). Hence UH 1 = H 2 U and P 2 UH 1 = ( P 2 H 2 ) U = 0, which means that P 2 U = 0, because H 1 has a dense range. Let R be the unique linear map such that RP 2 = 0 and RH 2 = I -P 2 . Then U = RH 2 U = RUH 1 and Up 1 = RUH 1 p 1 = RH 2 p 2 = ( I -P 2 ) p 2 . The uniqueness of U is then obvious, as p 1 has a dense range. By construction, w := ω (2) 2 -ω (1) 2 is a real-valued, symmetric bi-solution to the Klein-Gordon equation (in a weak sense). Moreover, as U is isometric and Up 1 = ( I -P 2 ) p 2 , so w is of positive type. For fixed f, f ' ∈ C ∞ 0 ( M ), w ( f, Ξ ∗ -t f ' ) = F ( t ) = w (Ξ ∗ t f, f ' ) is constant, as we saw in the first paragraph of this proof. If p 2 is a distribution on M , then w is a distribution on M × 2 and, in adapted coordinates, ∂ 0 w = ∂ ' 0 w = 0. The equation K x K x ' w = 0 then reduces to an elliptic equation on Σ × 2 , which implies that w is smooth (see e.g. Ref. [48] Thm. 8.3.1). /square Remark 4.3 Proposition 4.2 shows in particular that there is at most one non-degenerate ground one-particle structure and at most one non-degenerate β -KMS one-particle structure at any fixed β > 0 , up to unitary equivalence. These are the ones of Theorem 4.3. The degenerate ones may be classified in terms of w . In spacetimes with a compact Cauchy surface Σ we note that the only smooth function w with the stated properties is w = 0 . Indeed, for any fixed y ∈ Σ , v y ( x ) := w ( x, y ) solves Cv y = 0 for C := -∇ ( h ) i ( Nh ij -N -1 N i N j ) ∇ ( h ) i + V N . (This is because w solves the KleinGordon equation and is Killing time independent.) As 0 = 〈 v y , Cv y 〉 ≥ ‖ √ V Nv y ‖ 2 in L 2 (Σ) this implies v y = 0 and hence w = 0 .", "pages": [ 25, 26, 27, 28 ] }, { "title": "4.2.3 Simplifications in the standard static case", "content": "On a standard static spacetime M , the construction of the non-degenerate ground and β -KMS one-particle structures in the proof of Theorem 4.3 simplifies. For later convenience we formulate these results as a proposition [7]: Proposition 4.3 Let Σ ⊂ M be a Cauchy surface orthogonal to the Killing field of the standard static, globally hyperbolic spacetime M . Under the assumptions of Theorem 4.3 we have: Furthermore, the unitary group ˜ O t of Lemma 4.3 is given by ˜ O t = e it √ C . Furthermore, the unitary group ˜ O t of Lemma 4.3 is given by ˜ O t = e it C ⊕ e -it C . √ √ Here C is the closure of the partial differential operator defined on C ∞ 0 (Σ) . C 0 and all integer powers of it are essentially self-adjoint on the invariant domain C ∞ 0 (Σ) . Furthermore, C is strictly positive with C ≥ V N 2 and C ∞ 0 (Σ) is contained in the domain of C ± 1 2 for both signs. One may also write C in terms of the conformal metric ˜ h as on L 2 (Σ , d vol ˜ h ), where we used the footnote on page 20 and the fact that v = N in the static case. The completeness of ˜ h (Theorem 3.2) implies that all powers of -/square ˜ h are essentially self-adjoint on the test-functions. Proposition 4.3 shows, among other things, that the additional terms do not spoil this result. Proof: In the standard static case N i ≡ 0, so the operator A of Lemma 4.2 can be written as a diagonal matrix A = 1 2 ( α 0 0 N ) , where α := -∇ ( h ) ,i N ∇ ( h ) i + V N . Let ˆ α denote the Friedrichs extension of α , which is strictly positive by Lemmas A.7, A.6. We may then compute √ ˆ A and hence, on the range of √ ˆ A , Both √ ˆ α √ N and √ N √ ˆ α are closable operators, because H e is closeable. Furthermore, their closures are each others adjoints, because H e is self-adjoint. By the Polar Decomposition Theorem (Ref. [15] Thm. 6.1.11) there is then a partial isometry U such that √ ˆ α √ N = UC 1 2 and √ N √ ˆ α = C 1 2 U ∗ , where C = √ N ˆ α √ N = C 0 . Now H 2 e = ( √ ˆ αN √ ˆ α 0 0 C 0 ) on the range of √ ˆ A , which is invariant. The essential self-adjointness of all even powers of H e on this range (Theorem 4.2), restricted to the second summand of L 2 (Σ) ⊕ 2 , implies that all integer powers of C 0 are essentially self-adjoint on the range of √ N , which is just C ∞ 0 (Σ). The estimate C ≥ V N 2 follows from a partial integration, whereas strict positivity follows from Lemma A.6. That C ∞ 0 (Σ) is in the domain of C 1 2 is clear, because it is in the domain of C , and that it is in the domain of C -1 2 follows again from Lemma A.6. Finally, the domain and range of U are the entire L 2 (Σ), because C 1 2 and ˆ α 1 2 have dense ranges. This establishes all the claims concerning C . Returning to one-particle structures, we may write, after some short computations: where we introduced the unitary operator V := ( U 0 0 I ) . A comparison with the proof of Theorem 4.3 yields where we made use of the fact that P ± V = 1 2 ( U ∓ iI ) ( I ± iI ). As ‖ Uψ ⊕ ± iψ ‖ 2 = ‖ √ 2 ψ ‖ 2 , the first factors in each summand can safely be replaced by √ 2, leading to a unitary equivalent formulation, q ( β ) , Σ . Note that the range of q ( β ) , Σ is dense in L 2 (Σ) ⊕ 2 , because if ψ ⊕ χ is orthogonal to this range, then we may use the strict positivity of the operators ( I -e -β √ C ) -1 2 C ± 1 4 to show that ψ ± e -β 2 √ C χ = 0 for both signs and hence ψ = χ = 0. The proof of the fact that H = √ C ⊕-√ C is an easy exercise which we omit. The case of the ground one-particle structure is similar, but simpler. /square The result of Proposition 4.3 can be interpreted in terms of positive and negative frequency solutions [49]. Indeed, any solution φ = Ef ∈ S with initial data ( f 0 , f 1 ) can be decomposed into positive and negative frequency parts where f ± = 1 2 ( f 0 ∓ iN 1 2 C -1 2 N 1 2 f 1 ). In the ground state we have which vanishes when f 0 = iN 1 2 C -1 2 N 1 2 f 1 , which is the case precisely when f + = 0, i.e. when φ is a negative frequency solution. (The occurrence of negative, rather than positive, frequency solutions here is explained by the footnote on page 23.)", "pages": [ 28, 29 ] }, { "title": "5 Ground states and their properties", "content": "We are now ready to study the space G 0 ( W ) of ground states, under the assumptions of Theorem 4.3, and to consider some of their properties. These properties often generalize the special properties of the Minkowski vacuum. Note that a characterization of all classical equilibrium and ground states on the commutative Weyl C ∗ -algebra W cl can be given, in principle, using the results of Section 2.", "pages": [ 30 ] }, { "title": "5.1 The space of ground states", "content": "The following theorem gives a full description of the space G 0 ( W ) of all ground states. (This result may be compared to Theorem 2.2.) Theorem 5.1 Let M be a globally hyperbolic, stationary spacetime and consider a linear scalar field with a stationary potential V such that V > 0 . where O ⊥ := int( M \\ J ( O )) denotes the causal complement for any subset O ⊂ M . Recall that the Reeh-Schlieder property means that the ground state has many non-local correlations [50, 41]. In fact, the Reeh-Schlieder property is known for all quasi-free D ∞ equilibrium states [51]. Proof: Let ω 0 be the quasi-free state whose two-point distribution is associated to the nondegenerate ground one-particle structure ( p 0 , K 0 ) of Theorem 4.3. Then ω 0 is a non-degenerate and pure (and hence extremal) ground state, by Theorems 2.3 and 2.1. As ω 0 is quasi-free and ω 0 2 is a distribution (density), ω 0 is a regular state. Furthermore, the representation π 0 is irreducible, because ω 0 is pure, and it is faithful, because the space ( L, E ) is symplectic (by construction) and hence W is simple (Ref. [14] Thm. 5.2.8). Using Lemma 4.3 and the fact that ω 0 is quasi-free one may show that the strong derivatives of t ↦→ π 0 ( α t ( W ( f )))Ω 0 are well defined for all f ∈ L . The map λ 0 := λ ω 0 of Lemma 2.3 then restricts to the stated affine homeomorphism by Proposition 2.4. For regular ground states, the Hadamard property is known to hold [52] and the microlocal spectrum then follows [44, 45]. The Hadamard property for D 2 ground states then follows from the last statement of Proposition 4.2. From the definition of λ 0 we have As ω 0 is regular and quasi-free it follows that λ 0 ( ρ ) is C k (resp. D k ) if and only if ρ is C k (resp. D k ). Extremal ground states ω on W are of the form λ 0 ( ρ ) for an extremal ground state ρ on W cl . Such ρ are pure by Theorem 2.2, so by Lemma 2.3 this entails ω = η ∗ ρ ω 0 . Because η ∗ ρ preserves pure states it follows that every extremal ground state on W is pure (cf. Remark 2.2). Furthermore, η ∗ ρ preserves the local algebras W ( O ), so the extremal ground states have the Reeh-Schlieder property, because ω 0 does [51]. The statement on the regularity of extremal ground states follows directly from Proposition 2.2. This also proves the second uniqueness clause for ω 0 . The first uniqueness clause follows from Theorem 2.4. To prove the existence of the mass gap we note that, under the stated assumptions, ˆ A ≥ /epsilon1 2 I by Lemma 4.2. In the energetic Hilbert space we then use ( iσ ) ∗ = iσ to estimate Hence, | H e | ≥ /epsilon1I , H ≥ /epsilon1I and ‖ H -1 ‖ < /epsilon1 -1 . Finally, the fact that ω is pure entails Haag duality, at least when d = 4 (Ref. [53], Thm. 3.6), even for slightly more general regions O than used here. /square A few remarks concerning the interpretation of the results of this section and their implications are in order: Remark 5.1 The gauge transformations of the second kind, which appeared in the proof of Theorem 5.1, can be physically interpreted as field redefinitions. If ω 1 is a linear map on L , then χ := e -iω 1 is a character and ρ ( W ( f )) := e -iω 1 ( f ) defines a pure state on W cl . If we write (formally) W ( f ) = e i Φ( f ) we have In particular, if ω is any pure C 2 ground state with one-point distribution ω 1 and ρ is defined as above, then we must have η ∗ ρ ω = ω 0 by Theorem 5.1. Hence, ω ( W ( f )) = e iω 1 ( f ) ω 0 ( W ( f )) . Because pure states ρ of this exponential form are dense (Ref. [19] Lemma 4-2) we may argue on physical grounds that we may as well restrict attention to the pure ground state with vanishing one-point distribution, ω 0 . Remark 5.2 Because ω 0 is a uniquely distinguished ground state and π 0 is faithful we may perform the following standard modification of the original theory. For each bounded region O ⊂ M we define the von Neumann algebra R ( O ) := π 0 ( W ( O )) '' . This gives rise to a local net of von Neumann algebras in the spacetime M and we let the C ∗ -algebra R be their inductive limit. Each R ( O ) contains the corresponding W ( O ) , so that R ⊃ W . We may then consider the class of states on R which are locally normal, i.e. they restrict to normal states on each von Neumann algebra R ( O ) . Such states clearly restrict to a state on W and a state ω on W has at most one extension to R . This extension exists if and only if ω is locally normal w.r.t. ω 0 (by definition). This includes at least all quasi-free Hadamard states [54]. There are good physical reasons to consider only states on W that are locally normal with respect to ω 0 . For any self-adjoint operator A ∈ W ( O ) for any bounded region O , the algebra R ( O ) contains all the spectral projectors of A , so the operational question whether the measured value of A attains a value in some Borel set I ⊂ R corresponds to the same projection operator for all locally normal states. Another reason to restrict only to locally normal states is of a more technical nature. The action of the one-parameter group α t on W is not norm continuous, but the larger algebra R contains a C ∗ -algebra R 0 which is dense in R in the strong operator topology and on which α t is norm continuous (cf. Ref. [55] Sec. 4, or also Ref. [20] Thm. 1.18 for a closely related result). This means that a large number of results on C ∗ -dynamical systems can be brought to bear on ( R 0 , α t ) , and hence indirectly also on W , if one considers states that are locally normal [14, 20] with respect to ω 0 . Let us briefly describe the constructions of Ref. [55] (adapted to a stationary, globally hyperbolic spacetime and with a possibly non-compact Cauchy surface). The C ∗ -algebra R 0 may be generated by operators of the form where A ∈ W ( O ) for some bounded region O and f ∈ C ∞ 0 ( R ) . Then A f ∈ R ( O ' ) , where O ' is another bounded region that depends on O and on the support of f . Such operators form a ∗ -algebra which is invariant under the action of α t and on which α t is norm continuous. R 0 is the norm closure of this ∗ -algebra.", "pages": [ 30, 31, 32 ] }, { "title": "5.2 The ground state representation and the quantum stress-energymomentum tensor", "content": "As ω 0 is quasi-free, H 0 is a Fock space (cf. Sec. 3.2 of Ref. [8]) and we may introduce a particle interpretation for the field, based on creation and annihilation operators. Note that such an interpretation fails in general spacetimes, because there are many unitarily inequivalent Fock space representations and there is no generally covariant prescription to single out a preferred one [56, 3]. Following standard notations [14] we will write H 0 = ⊕ ∞ n =0 H ( n ) 0 , where the n -particle Hilbert space is H ( n ) 0 := P + ( K 0 ) ⊗ n , in which ( p 0 , K 0 ) is the one-particle structure associated to ω 0 2 and P + denotes the projection onto the symmetric tensor product. We write N for the number operator, so that N | H ( n ) 0 = nI . We will use the notation a ∗ ( ψ ) and a ( ψ ) for creation and annihilation operators, respectively, where ψ ∈ K 0 . As a ∗ ( ψ ) ∗ = a ( ψ ) we see that a is complex anti-linear in ψ , whereas a ∗ is linear. The field Φ is given by and is complex linear, as desired. We may introduce the initial value and normal derivative of the quantum field as so that Φ( f ) = Φ 1 ( f 0 ) -Φ 0 ( f 1 ), where ( f 0 , f 1 ) = S -1 Ef . This is in line with what one would get if Φ were a classical solution to the Klein-Gordon equation (cf. Eq. (4.3)). It will also be convenient to introduce the operators Because the classical stress-energy-momentum tensor played a significant role in the classical and quantum descriptions of the linear scalar field in a stationary spacetime, it seems fitting to also spend a few words on the quantum stress-energy-momentum tensor. If the field theory on M can be extended to all globally hyperbolic spacetimes in a locally covariant way [4], e.g. if V = cR + m 2 , then there is a generally covariant way to define the renormalised stress-energy-momentum tensor [57]. However, in our setting it will be advantageous not to renormalize the stress tensor in a generally covariant way, but instead to exploit the extra structure of the stationary spacetime. (Nevertheless, our presentation of the classical and quantum stress tensor is based on existing treatments that fit in a generally covariant framework, e.g. Ref. [58].) We may define a tensor field G ab on a sufficiently small neighborhood U ⊂ M × 2 of the diagonal ∆ := { ( x, x ) | x ∈ M } by the property that for any vector v b ∈ T x ' M , the vector g ac ( x ) G cb ( x, x ' ) v b ( x ' ) ∈ T x M is the parallel transport of v along a unique geodesic connecting x to x ' . (The uniqueness of the geodesic can be ensured by choosing U sufficiently small.) Using G ab and G ab ( x, x ' ) := g ac ( x ) g bd ( x ' ) G cd ( x, x ' ) we may write the classical stress-energy momentum tensor in terms of a differential operator as Instead of letting the operator T split ab act on the classical fields φ ⊗ 2 , we can let it act on the normal ordered quantum field, For any vector ψ ∈ π 0 ( A )Ω 0 we may define the H 0 -valued distribution (density) where δ n ∈ C ∞ ( M × 2 ) is a sequence of functions that approximates the delta distribution δ ( x, x ' ) and f ab is a compactly supported, smooth test-tensor [45]. The operator T ren ab ( f ab ) is densely defined and it is a symmetric operator when f ab is real-valued. Moreover, if V > 0 everywhere one can show that T ren ab ( χ a χ b ) is semi-bounded from below for real-valued test-vector fields χ a [58]. (Note that the method of proof in Ref. [58] is not affected by the presence of the non-negative potential energy term V in the equation of motion.) In analogy with the classical case we define the quantum energy-momentum one-form and the energy density by in the sense of H 0 -valued distributions, when acting on π 0 ( A )Ω 0 . One may check that T ren ab is symmetric in its indices a, b and that where the Wick square : Φ 2 : is the restriction of : Φ ⊗ 2 : to the diagonal ∆ ⊂ M × 2 . It follows from ∂ 0 V = 0 that ∇ a P ren a = 0, just like in the classical case. Remark 5.3 From a physical point of view it seems reasonable to expect that for real-valued f the operator /epsilon1 ren ( f 2 ) is semi-bounded from below, using the same motivation as for existing quantum inequalities [58]. However, the details of the argument require that we can write ξ a n b + n a ξ b = ∑ k j =1 χ a j χ b j for some finite number of (real) vectors χ a j . An easy exercise shows that this is possible if and only if we are in the static case, where ξ a = Nn a , in which case the single vector χ a = N -1 2 ξ a will suffice. Thus, in the static case, the results of Ref. [58] apply and /epsilon1 ren ( f 2 ) is semi-bounded from below. There is another result, however, which does work very nicely in the general stationary setting: Theorem 5.2 Under the assumptions of Theorem 5.1, let ω 0 be the unique ground state. For any real-valued test-tensor f ab , the operator T ren ab ( f ab ) is essentially self-adjoint on π 0 ( A )Ω 0 . A similar essential self-adjointness result for the smeared stress-energy-momentum tensor in general globally hyperbolic spacetimes is much harder to obtain by a direct proof (cf. Ref. [59] for partial results). Proof: It follows from Lemma 4.3 (and second quantization) that the Hamiltonian operator h is essentially self-adjoint on the dense, invariant domain π 0 ( A )Ω 0 and that for all ψ, ψ ' in that domain. (Here we have used the fact that ω 0 is an equilibrium state.) The idea is now to use the Commutator Theorem X.36' of Ref. [21] to prove essential self-adjointness of T ren ab ( f ab ). This means we need to prove that for any test-tensor f ab there is a C > 0 such that for all ψ, ψ ' ∈ π 0 ( A )Ω 0 . By polarization it suffices to take ψ = ψ ' . It also suffices to consider f ab to be supported in a convex normal neighborhood, by a partition of unity argument. Moreover, the antisymmetric part of f ab does not contribute and the symmetric part can be written as a finite sum of terms of the form χ a χ b , so it suffices to consider f ab = χ a χ b . Now consider the operators Π( f ) for f ∈ C ∞ 0 ( M ). [Π( f ) , Π( f ' )] = [Φ( f ) , Φ( f ' )] = iE ( f, f ' ), so for any ψ ∈ π 0 ( A )Ω 0 the distribution is a Hadamard two-point distribution. As for the field Φ( f ) one may introduce the normal-ordered product : Π( f )Π( f ' ) : := Π( f )Π( f ' ) -ω 0 2 ( f, f ' ) and following Ref. [58] one proves that the operator is semi-bounded from below. Hence, for some c > 0, The first term on the right-hand side is the second quantization of an operator T on H (1) 0 , for which we have for some c ' > 0, where we defined φ := ω 0 2 ( ., f ). On the other hand, because the classical energy is independent of the Cauchy surface, h satisfies (cf. Lemma 4.2) where τ ∈ C ∞ 0 ( R ) satisfies ∫ τ = 1. Choosing τ ≥ 0 and τ > 0 on the compact support of χ a and using the fact that Nh ij -N -1 N i N j is positive definite, the desired estimate Eq. (5.2) easily follows from Eq.'s (5.3, 5.4, 5.5). /square Note that [ T ren ab ( f ab ) , π 0 ( W ( f ' ))] = 0 whenever supp( f ' ) ∩ J (supp( f ab )) = ∅ . It follows from Haag duality that T ren ab ( f ab ) is affiliated to the local von Neumann algebra R ( D (supp( f ab ))). Lemma 5.1 Let Σ be Cauchy surface in a stationary, globally hyperbolic spacetime M . Let f ∈ C ∞ 0 ( M ) , τ ∈ C ∞ 0 ( R ) with ∫ τ = 1 and χ ∈ C ∞ 0 (Σ) such that χ ≡ 1 on supp( τ ) ∩ J (supp( f )) , where we view τ, χ as functions on M in adapted coordinates. Then Proof: We follow the computations in Ref. [55], Appendix A.2. Fix a vector ψ ∈ π 0 ( A )Ω 0 , so that φ ' := 〈 ψ, Φ( . ) ψ 〉 is a smooth function. Let φ := E ( ., f ) and note that ∂ 0 φ = E ( ., ∂ 0 f ), by the uniqueness of E ± . Using ω ([: Φ ⊗ 2 : ( x, x ' ) , Φ( f )]) = iφ ( x ) φ ' ( x ' ) + iφ ' ( x ) φ ( x ' ) we find after some algebra Using the Klein-Gordon equation and Eq. (4.3) we may then compute for any Cauchy surface Σ ' By polarization the desired operator equality now holds on the indicated dense domain. /square", "pages": [ 32, 33, 34, 35 ] }, { "title": "6 KMS states in stationary spacetimes", "content": "We now come to the thermal equilibrium states at non-zero temperature. We still consider a linear scalar field in a stationary, globally hyperbolic spacetime and we assume that the theory has a unique C 2 ground state ω 0 as in Section 5 and a Hamiltonian operator h . In Section 6.2 below we will review the states satisfying the KMS-condition, which exist for every inverse temperature β > 0. Afterwards, in Section 6.3, we show that their two-point distributions can be obtained from a Wick rotation, in case M is standard static (see also Ref. [5]). Before we come to this, however, we study the motivation to use the KMS-condition as a characterization of thermal equilibrium in Section 6.1. In particular we show that for a standard static spacetime M with compact Cauchy surfaces we may also define Gibbs states to describe thermal equilibrium and these Gibbs states satisfy the KMS-condition.", "pages": [ 35 ] }, { "title": "6.1 Gibbs states and the KMS-condition", "content": "Consider, then, a stationary, globally hyperbolic spacetime M and a linear scalar field satisfying the assumptions of Theorem 5.1. If, for some inverse temperature β > 0, the operator e -βh is of trace-class in the ground state representation π 0 , i.e. if it has a finite trace, one may define the thermal equilibrium state to be the Gibbs state Here we use the fact that the set of bounded trace-class operators on a Hilbert space forms a ∗ -ideal in the algebra of all bounded operators (Ref. [15] Rem. 8.5.6 or Ref. [21] Thm. VI.19). We now show that these Gibbs states are well defined whenever M is standard static and has compact Cauchy surfaces. Moreover, we explain that these Gibbs states satisfy the KMS-condition. Theorem 6.1 We make the assumptions of Theorem 5.1 with the additional assumptions that M is a standard static spacetime with compact Cauchy surfaces, so that the theory has a mass gap. For any β > 0 Proof: By Ref. [14] Proposition 5.2.27, the operator e -βh has a finite trace on H 0 if and only if e -βH has a finite trace on H (1) 0 /similarequal K and βH is strictly positive. The latter is satisfied by our assumptions, so we only need to show that e -βH has a finite trace. Our proof of this fact is adapted from the proof of nuclearity in Ref. [60]. We refer to Proposition 4.3 for a convenient formulation of the ground one-particle structure, with K /similarequal L 2 (Σ) and H = √ C . By assumption, the theory has a mass gap, so √ C ≥ /epsilon1I > 0. The exponential e -β √ C is bounded and may be written as C -n ( C n e -β √ C ) for any n ≥ 1, where both C -n and the product in brackets are bounded. Because trace-class operators form an ideal in the algebra of bounded operators, it suffices to prove that C -n is trace-class. The operator C is a partial differential operator, while C -2 n defines a distribution density u on Σ × 2 by Theorem A.1. We then have ( C n uC n )( x, y ) = δ ( x, y ). Note that C ⊗ C is an elliptic operator on Σ × 2 . Choosing n large enough, we can make u continuous. Because Σ is compact it follows that u ∈ L 2 (Σ × 2 ), which implies that it is Hilbert-Schmidt (Ref. [21] Thm. VI.23) and, by definition of Hilbert-Schmidt operators, C -n is trace-class. ω ( β ) is normal with respect to the ground state by definition. This completes the proof of the first item. The quasi-free property follows from Proposition 5.2.28 of Ref. [14]. For the KMS-condition we follow Ref. [12] and note that the function takes values in the bounded operators on H 0 for z = t + iτ ∈ S β , as 0 ≤ τ ≤ β . By Lemma A.8 it is continuous on S β and holomorphic on the interior S β . Moreover, f ( z ) is trace-class, because either e ( τ -β ) h or e -τh is trace-class. Using the fact that | Tr( CD ) | ≤ ‖ C ‖ Tr | D | for all bounded operators C and trace-class operators D , 11 we see that Tr f ( z ) is a bounded, continuous function on S β , which is holomorphic in the interior. Dividing by Tr e -βh proves the second item. /square We see that, under suitable physical (and technical) conditions, Gibbs states are well defined for systems in a finite spatial volume. In fact, we will see in Theorem 6.2 below that for given β > 0 it is the only β -KMS state on W satisfying some natural additional conditions. In general, however, the given exponential operator is not of trace-class and the definition of the Gibbs state does not make sense. In such cases one takes the KMS-condition to be the defining property of thermal equilibrium states. Theorem 6.1, together with the uniqueness result of Theorem 6.2 below, is a good indication that such a definition is justified. Further evidence comes from the analysis of Ref. [13], who investigated the second law of thermodynamics for general C ∗ -dynamical systems. They call a state ω of such a system completely passive, if it is impossible to extract any work from any finite set of identical copies of this system, all in the same state, by a cyclic process. They then showed, among other things, that a state is completely passive if and only if it is a ground state or a KMS state at an inverse temperature β ≥ 0. 12 This analysis applies to our situation, if we restrict attention to states which are locally normal with respect to the ground state (cf. Remark 5.2). We will see in Section 6.2 that quasi-free, D 2 KMS states do indeed satisfy this local normality condition, because they are Hadamard. A more general and detailed study of the relations between passivity, the Hadamard condition and quantum energy inequalities was made by Ref. [55]. Probably the most direct motivation in favor of the KMS-condition is an analysis of Ref. [12] (see also Ref. [14]) which shows, in the context of quantum statistical mechanics, that a thermodynamic (infinite volume) limit of Gibbs states satisfies the KMS-condition. Reformulated to our geometric setting, the idea is to approximate h by operators h O , where O ⊂ Σ has finite volume, such that e ith O ∈ R ( D ( O )) = π 0 ( W ( D ( O ))) '' for all t ∈ R , where D ( O ) ⊂ M denotes 12 If it is impossible to extract any work from only one copy of this system in the given state, the state is called passive. The set of passive states also contains convex combinations of the ground and KMS states. the domain of dependence. If e -βh O is a trace-class operator on H 0 ( O ) := π 0 ( W ( D ( O )))Ω 0 for some β > 0, then it gives rise to a Gibbs state ω ( β,O ) . The argument of Ref. [12] shows that, under some additional assumptions on the h O , one may show that the thermodynamic limit ω ( β ) := lim O → Σ ω ( β,O ) exists and is a β -KMS state. In the case of non-relativistic point-particles in Minkowski spacetime, an explicit construction of the approximate Hamiltonians h O and the corresponding limiting procedure is described in detail in Ref. [14] (see also the classic paper Ref. [61], where the thermodynamic limit of a non-relativistic free Bose gas was investigated in detail). For a quantum field it is tempting to choose h O to be of the form h O = /epsilon1 ren ( f ) for some suitable f ∈ C ∞ 0 ( D ( O )), in view of Theorem 5.2 and Lemma 5.1. However, the argument becomes more problematic for two reasons. Firstly, the restriction to a bounded open region O does not entail the desired reduction in the degrees of freedom, due to the Reeh-Schlieder property: if O is non-empty, the subalgebra R ( D ( O )) already generates the entire Hilbert space H 0 when acting on the ground state vector Ω 0 . Secondly, and more to the point, the operators e -βh O cannot be trace-class. In fact, R ( D ( O )) is a type III 1 factor (Thm. 3.6g) of Ref. [53]), so the only trace-class operator X ∈ R ( D ( O )) is X = 0. 13 This means that no h O can possibly satisfy the assumptions made in Ref. [12]. Even in a spacetime with a compact Cauchy surface Σ, the Reeh-Schlieder property of the ground state and the type of the local von Neumann algebras prevent us from finding appropriate Gibbs states to define thermal equilibrium states in any bounded region V ⊂ Σ which is strictly smaller than Σ. All this in spite of naive physical intuition and the positive results for quantum statistical mechanics. It is possible that other techniques, such as local entropy arguments [62], can be employed to elucidate the local aspects of thermal equilibrium for quantum fields, but we are not aware of a detailed treatment of this issue. We must therefore conclude that, even though it is still perfectly satisfactory to use the KMS-condition as the defining property of global thermal equilibrium, the local aspects of thermal equilibrium and temperature of a quantum field are presently not well understood.", "pages": [ 35, 36, 37 ] }, { "title": "6.2 The space of KMS states", "content": "We now give a full description of the space G ( β ) ( W ) of all β -KMS states in general stationary, globally hyperbolic spacetimes. (This result may be compared to Theorem 2.2 and 5.1.) Theorem 6.2 Let M be a globally hyperbolic, stationary spacetime and consider a linear scalar field with a stationary potential V such that V > 0 . Let β > 0 . /negationslash Proof: Let ω ( β ) be the quasi-free state whose two-point distribution is associated to the nondegenerate β -KMS one-particle structure ( p ( β ) , K ( β ) ) of Theorem 4.3. Then ω ( β ) is a β -KMS state, by Theorem 2.3. As ω ( β ) is quasi-free and ω ( β ) 2 is a distribution (density), ω ( β ) is a regular state. The representation π ( β ) is faithful, as in the proof of Theorem 5.1. The map λ ( β ) := λ ω ( β ) of Lemma 2.3 restricts to the stated affine homeomorphism by Proposition 2.4. For regular β -KMS states the Hadamard property is known to hold [52] and the microlocal spectrum then follows [44, 45]. The Hadamard property for D 2 β -KMS states then follows from the last statement of Proposition 4.2. The fact that λ ( β ) ( ρ ) is C k (resp. D k ) if and only if ρ is, is shown as in Theorem 5.1. Local quasi-equivalence of all quasi-free Hadamard states was proved in Ref. [54], which applies in particular to ω ( β ) and ω 0 . Extremal β -KMS states ω on W are of the form ω = η ∗ ρ ω 0 , as in Theorem 5.1, and the ReehSchlieder property for ω follows from that of ω ( β ) [51]. The statement on the regularity of extremal β -KMS states follows directly from Proposition 2.2. This also proves the uniqueness clause for ω ( β ) . Using Theorem 4.3 one may show that lim β →∞ ω ( β ) 2 ( f, f ) = ω 0 2 ( f, f ). Indeed, the range of p cl is in the domain of | H e | -1 by Proposition 4.2 and the functions F ( x ) := e -β 2 x √ x 1 -e -βx and G ( x ) := √ x 1 -e -βx -√ x converge uniformly to 0 on the positive half line as β →∞ . The explicit expression for p ( β ) and the Spectral Calculus Theorem for the functions F ( | H e | ) and G ( | H e | ) then prove the claim. It follows that lim β →∞ ω ( β ) ( W ( f )) = ω 0 ( W ( f )), because the ω ( β ) and ω 0 are quasi-free. Hence, lim β →∞ ω ( β ) = ω 0 . As ω ( β ) is locally normal w.r.t. ω 0 , it extends in a unique way to a locally normal state on R , which contains a dense, C ∗ -dynamical system R 0 (cf. Remark 5.2), for which ω is again a β -KMS state (by Proposition 2.1 and a limit argument). The GNS-representation π ω of ω on R restricts to the GNS-representations of R 0 and of W , which all generate the same Hilbert space H ω . The final item then follows from Ref. [20] Theorem 4.3.9. /square It is known that the state ω ( β ) is not pure, but it can be purified by extending it to a so-called doubled system [63]. This abstract procedure finds a natural interpretation in the setting of black hole thermodynamics [38]. Because ω ( β ) is not pure we cannot use Theorem 2.4 to obtain a uniqueness result, unlike the ground state case.", "pages": [ 37, 38 ] }, { "title": "6.3 Wick rotation in static spacetimes", "content": "In Section 4.2 we have shown the existence of unique non-degenerate β -KMS one-particle structures for a linear scalar quantum field on a stationary, globally hyperbolic spacetime, provided the interaction potential is stationary and everywhere strictly positive. In this section we will show that the corresponding two-point distributions can also be obtained by a Wick rotation, in case the spacetime is standard static. The geometric backbone of the argument was already presented in subsection 3.3, so in this section we may focus on the functional analytic aspects of the technique of Wick rotation. The results we describe correspond to those in Ref. [5], but our presentation focusses more on the operator theoretic language. The case of R = ∞ , which leads to a ground state, has already been described in some detail [49], so we will focus primarily on the case R < ∞ .", "pages": [ 38 ] }, { "title": "6.3.1 The Euclidean Green's function", "content": "For some R > 0 consider the complexification M c R and the associated Riemannian manifold M R of a standard static globally hyperbolic spacetime M . Because the Laplace-Beltrami operator /square on M is defined in terms of the metric and the potential V is assumed stationary, there is a natural corresponding Euclidean Klein-Gordon operator on M R , namely K R := -/square g R + V . Our first task is to find a preferred Euclidean Green's function, which will be the starting point for the Wick rotation that should lead to a two-point distribution on the Lorentzian spacetime M . Definition 6.1 A Euclidean Green's function is a distribution (density) G R on M × 2 R which is a fundamental solution, ( K R ) x G R ( x, y ) = ( K R ) y G R ( x, y ) = δ ( x, y ) , of positive type, G R ( f, f ) ≥ 0 for all f ∈ C ∞ 0 ( M R ) . Just like there are many (Hadamard) two-point distributions on M , there may be many Green's functions on M R . The common wisdom is to obtain a preferred one by the following method: the partial differential operator K R can be viewed as a positive, symmetric linear operator on the domain C ∞ 0 ( M R ) in L 2 ( M R ). Assuming K R is self-adjoint and strictly positive, it has a well defined inverse. We may then take G ( f, f ' ) := 〈 f, ( K R ) -1 f ' 〉 , whenever this is a distribution. In an attempt to substantiate this procedure we will analyze the operator K R in some more detail. For a standard static spacetime M we have N i ≡ 0 ≡ w , so Eq. (4.4) simplifies to where C 0 is the partial differential operator acting on C ∞ 0 (Σ) in L 2 (Σ) (cf. Proposition 4.3). Recall from Section 4.1 that the powers 3 2 and 1 2 of N to the left and right of K were chosen in such a way that C 0 is symmetric and at the same time the operator ∂ 2 0 appears without any spatial dependence. In the case at hand that completely separates the Killing time dependence from the spatial dependence. In a similar manner we may split off the imaginary Killing time dependence of K R . For this we will view the circle S 1 R of radius R as a Riemannian manifold in the canonical metric dτ 2 . In analogy to the Lorentzian case (cf. Sec. 4.1), there is a unitary isomorphism onto the Hilbert tensor product, because d vol g R = Ndτ d vol h . Then, N 3 2 K R N 1 2 = -∂ 2 τ + C 0 , with the same operator C 0 on Σ as in the Lorentzian case. More precisely, we have where the operator B R := -∂ 2 τ acts on the dense domain C ∞ 0 ( S 1 R ) in L 2 ( S 1 R ) and the operator on the right-hand side is defined on the algebraic tensor product of the domains of B R and C 0 . The properties of the operator B R are well known and we quote them without proof: Proposition 6.1 The operator B R := -∂ 2 τ is essentially self-adjoint on C ∞ 0 ( S 1 R ) in L 2 ( S 1 R ) . If R is finite, there is a countable orthonormal basis of eigenvectors ψ n ( τ ) := 1 √ 2 πR e inτ/R , n ∈ Z , with eigenvalues λ n := n 2 R 2 . This follows e.g. from Thm. II.9 in Ref. [21] by rescaling to R = 1. Note that for finite R the operator B R is positive, but not strictly positive. From now on we will use B R to denote the unique self-adjoint extension found in Proposition 6.1, to unburden our notation. Together with the results for C (Proposition 4.3), Proposition 6.1 implies Theorem 6.3 For any R > 0 the operator NK R N is essentially self-adjoint on C ∞ 0 ( M R ) in L 2 ( M R ) , its closure is strictly positive with NK R N ≥ V N 2 and the domain of ( NK R N ) -1 2 contains C ∞ 0 ( M R ) . Proof: By Theorem V III. 33 in Ref. [21] the sum B R ⊗ I + I ⊗ C is essentially self-adjoint on the algebraic tensor product D := C ∞ 0 ( S 1 R ) ⊗ C ∞ 0 (Σ), because both B R and C are essentially self-adjoint on the space of test-functions. By Eq. (6.3) the operator U R NK R NU -1 R extends B R ⊗ I + I ⊗ C and U R is unitary, so NK R N is already essentially self-adjoint on the smaller domain U -1 R D . In fact, because D ⊂ C ∞ 0 ( S 1 R ⊗ Σ) in L 2 ( S 1 R ⊗ Σ , dτ d vol h ) we have U R NK R NU -1 R = B R ⊗ I + I ⊗ C ≥ I ⊗ C ≥ I ⊗ V N 2 on D . It follows that NK R N ≥ V N 2 on U -1 R D and hence on C ∞ 0 ( M R ). The claim on the domain of ( NK R N ) -1 2 then follows from Lemma A.6 in A. /square In the ultra-static case, where N is constant, Theorem 6.3 (in combination with Theorem A.1) suffices to justify the procedure to define a Euclidean Green's function by G R ( f, f ' ) := 〈 ( K R ) -1 2 f, ( K R ) -1 2 f ' 〉 . In the general case, however, the study of the self-adjoint extensions of the operator K R is more complicated. 14 Nevertheless, we can define a Euclidean Green's function by a slight modification of the common procedure as using Theorem 6.3 and the fact that multiplication by N is a continuous linear map on C ∞ 0 ( M ). It is straightforward to verify that this satisfies all the requirements to be a Euclidean Green's function and we will see shortly that this choice of the Euclidean Green's function will indeed allow us to recover the KMS two-point distributions.", "pages": [ 38, 39, 40 ] }, { "title": "6.3.2 Analytic continuation of the Euclidean Green's function", "content": "We may now establish the explicit Killing time dependence of the Euclidean Green's function and its analytic continuation: /negationslash Theorem 6.4 Consider a standard static globally hyperbolic spacetime M . For each R < ∞ there is a unique continuous function G c R ( z, z ' ) from C × 2 R into the distribution densities on Σ × 2 , holomorphic on the set where Im( z -z ' ) = 0 , such that for all χ, χ ' ∈ C ∞ 0 ( S 1 R ) and f, f ' ∈ C ∞ 0 (Σ) we have with z = t + iτ . When Im( z -z ' ) ∈ [ -2 πR, 0] it is given by Proof: It suffices to check that the given formula for G c R satisfies all the desired properties, but let us first sketch a more constructive argument to see where the formula comes from. When we try to extract the Killing time dependence of G R , as defined in Eq. (6.4), we may make use of the fact that the inverse of the strictly positive operator NK R N can be found as a strongly converging integral of the heat kernel, for all ψ ∈ D (( NK R N ) -1 ). The importance of the heat kernel (i.e. the exponential function) is that it allows us to separate out the Killing time dependence. Indeed, for all α ≥ 0 there holds e -α ( NK R N ) = U -1 R e -αB R ⊗ e -αC U R , because of Trotter's product formula (Ref. [21] Thm. VIII.31). Now let λ n , n ∈ Z , denote the eigenvalues of B R and P n the corresponding orthogonal projections. Then we may perform the integral over the heat kernel to find U R ( NK R N ) -1 U -1 R P n = P n ⊗ ( C + λ n ) -1 . Summing over n we then expect the formula where we have written P n as an integral kernel on ( S 1 R ) × 2 and we substituted the values of λ n . The sum over n can be performed (cf. Ref. [65] formula 1.445:2) in the sense of the Spectral Calculus Theorem, leading to The analytic continuation is then obvious. Let us now verify that the given formula for G c R has the desired properties. First note that for each z, z ' with Im( z -z ' ) ∈ [ -2 πR, 0] it defines a distribution density on Σ × 2 by Theorem A.1, because multiplication by N is a continuous linear map from C ∞ 0 (Σ) to itself, C ∞ 0 (Σ) is in the domain of C -1 2 , by Proposition 4.3, and by the Spectral Calculus Theorem. Moreover, both exponential terms in the first factor of the last expression are bounded operators that depend holomorphically on z, z ' as long as Im( z -z ' ) ∈ ( -2 πR, 0). This proves the continuity and the holomorphicity claims. As the uniqueness of G c R is clear from the Edge of the Wedge Theorem [23], it only remains to prove that it restricts to G R . ' ∞ For any f, f ∈ C 0 (Σ) the function is continuous for τ -τ ' ∈ [ -2 πR, 0] and holomorphic in the interior. We may compute the derivatives in the distributional sense, which leads to Letting U R K R U -1 R = N -1 ( -∂ 2 τ + C 0 ) N -1 act on G 2 R ( iτ, iτ ' ; x, x ' ) from the left and right we find which shows that the restriction of G c R to ( S 1 R ) × 2 is indeed the Euclidean Green's function. /square The case R = ∞ can be treated using similar methods [49], now using Ref. [65] formula 3.472:5. The result is the distribution density-valued function Alternatively, this expression can be obtained as the limit for fixed f, f ' ∈ C ∞ 0 (Σ), using Lemma A.8.", "pages": [ 40, 41 ] }, { "title": "6.3.3 Wick rotation to fundamental solutions and thermal states", "content": "Using the analytic continuation G c R we now want to complete the Wick rotation by considering the restriction to real values z = t and z ' = t ' . Following Ref. [5] we show how the thermal two-point distribution and the advanced, retarded and Feynman fundamental solutions are obtained. Both for t > t ' and t < t ' we can approach the real axis from above, Im( z -z ' ) > -2 πR , and from below, Im( z -z ' ) < 0. This prompts us to define the following functions on R × 2 with values in the distribution densities on Σ × 2 : Note that the E ± and E F R are given by They give rise to distribution densities on M × 2 defined by and using Schwartz Kernels Theorem to extend the distribution to all test-functions in C ∞ 0 ( M ). (Note that the factors √ N are required to account for the change in integration measure and they can equivalently be written in terms of the unitary isomorphism U .) Proposition 6.2 E ± and E F R are left and right fundamental solutions for the Klein-Gordon operator K = -/square + V and we have E ± ( t, t ' ; f, f ' ) = E ∓ ( t ' , t ; f, f ' ) = E ± ( t, t ' ; f, f ' ) = E ± ( t, t ' ; f ' , f ) . Proof: The first sequence of equalities follows directly from Eq. (6.6) and the fact that L 2 (Σ) carries a natural complex conjugation which commutes with the operator C and any real-valued function of C . To see that the distribution densities are fundamental solutions we use Eq. (6.2) to find and we use the fact that (The differentiations can be carried out by going into the complex manifold M c , where G c R is holomorphic, and then extending by continuity to the boundary.) For the case of E ± ( t, t ' ) we then have, by Eq. (6.6): We account for the factors θ by restricting the domain of integration and then perform partial integrations, after which we are only left with the boundary terms, which immediately yield the result. By the symmetry properties of E ± , E ± is also a right-fundamental solution. The proof for E F R uses a similar computation. /square It follows from the support properties of the distribution densities E ± that they are the advanced ( -) and retarded (+) fundamental solutions, so our notation is consistent. As Eq. (6.6) shows, they are independent of R , in line with the uniqueness of these fundamental solutions. E F R is the Feynman fundamental solution, as can be inferred from the fact that the real axis of t -t ' is approached by a rigid rotation from the imaginary time axis in counterclockwise direction. It does depend on the choice of R and it defines a choice of two-point distribution as follows: Proposition 6.3 For 0 < R < ∞ the function G c R ( t, t ' ) = -i ( E F R -E -)( t, t ' ) on R × 2 has a corresponding distribution density ω ( β ) 2 := -i ( E F R -E -) where we set β := 2 πR . ω ( β ) 2 is the two-point distribution density of ω ( β ) (as defined in Theorem 6.2) and Proof: The equality G c R ( t, t ' ) = -i ( E F R -E -)( t, t ' ) follows directly from the definitions of G c R , E F R and E -, so it remains to check the properties of ω ( β ) 2 . ω ( β ) 2 is a bisolution to the KleinGordon equation because it is -i times a difference of two fundamental solutions (Proposition 6.2). Furthermore, comparison with Eq.'s (6.6, 6.7) shows that the anti-symmetric part of ω ( β ) 2 is given by i 2 ( E --E + ). Remembering that ∂ t = Nn a ∇ a and that the restriction of a distribution density from M to Σ incurs a factor N -1 we find that the initial data of ω ( β ) 2 are given by On the other hand, the non-degenerate β -KMS one-particle structure, which is described in Proposition 4.3 for the standard static case, defines a two-point distribution whose initial data coincide with those in Eq. (6.8), as one may verify by a short computation. This proves that ω ( β ) 2 , as defined above, is indeed the two-point distribution of ω ( β ) . /square Using similar techniques one may treat the case R = ∞ , which leads to the two-point distribution ω 0 2 of the ground state ω 0 of Theorem 5.1 [49].", "pages": [ 42, 43 ] }, { "title": "Acknowledgments", "content": "Parts of this paper are based on a presentation given at the mathematical physics seminar at the II. Institute for Theoretical Physics of the University of Hamburg, during a visit in 2011. I thank the Institute for its hospitality and the attendants of the seminar for their comments and encouragement. I would also like to thank Kartik Prabhu for proof reading much of this paper.", "pages": [ 43 ] }, { "title": "A Some useful results from functional analysis", "content": "In this appendix we collect some results from functional analysis, to make our review self-contained. Most of the proofs are omitted, because they are elementary or make use of standard methods. For more information we refer the reader to Refs. [15], [21] and to Ref. [38] for strictly positive operators. In particular these references contain a detailed formulation of the Spectral Calculus Theorem (Ref. [15] Sec. 5.6, or Ref. [21] Thm. VIII.6). If X : H 1 →H 2 is a linear operator between two Hilbert spaces H i , we denote the domain of X by D ( X ). We wish to record the following useful relation between operators on a Hilbert space and distributions. Theorem A.1 Let X : H 1 →H 2 be a closed, densely defined linear operator between two Hilbert spaces H i and let L : C ∞ 0 ( M ) →H 1 be an H 1 -valued distribution density. If the range of L is contained in D ( X ) , then f ↦→ XL ( f ) is an H 2 -valued distribution density. Proof: If X is a bounded operator this is immediately clear from ‖ XL ( f ) ‖ ≤ ‖ X ‖ · ‖ L ( f ) ‖ . If X is a self-adjoint operator on H 1 = H 2 we may use its spectral projections P ( -n,n ) onto the intervals ( -n, n ) to define bounded operators X n := P ( -n,n ) X for n ∈ N . Each X n L defines a distribution density and lim n →∞ X n L ( f ) = XL ( f ) for all f ∈ C ∞ 0 ( M ), because L ( f ) ∈ D ( X ). From the Uniform Bounded Principle (Ref. [21] Thm. III.9) we see that XL also defines an H 2 -valued distribution density. The general case now follows from the polar decomposition, Theorem 6.1.11 of Ref. [15], which allows us to write T = V ( T ∗ T ) 1 2 , where V is bounded and ( T ∗ T ) 1 2 is a self-adjoint operator on H 1 with the same domain as T . /square We now turn to injective (and therefore invertible) operators on a Hilbert space, starting with the following four general Lemmas: Lemma A.1 A densely defined, closable and injective operator X in a Hilbert space H has an injective closure X if and only if X -1 is closable. Lemma A.2 If X is a densely defined, injective operator with dense range, then X ∗ and ( X -1 ) ∗ are injective and ( X ∗ ) -1 = ( X -1 ) ∗ . Lemma A.3 A self-adjoint operator X is invertible if and only if it has a dense range on any core. Lemma A.4 If X is self-adjoint and invertible, then X -1 is self-adjoint and invertible, where the domain of X -1 is the range of X . If D is a core for X , then X D is a core for X -1 . These Lemmas can be proved using entirely elementary methods. As positive invertible operators are particularly useful we make the following definition. Definition A.1 A densely defined operator X in a Hilbert space H is called strictly positive if and only if X is self-adjoint and for any 0 = φ ∈ D ( X ) : 〈 φ, Xφ 〉 > 0 . /negationslash Several equivalent characterizations can be given as follows: Lemma A.5 For a positive, self-adjoint operator X the following are equivalent: (i) X is strictly positive, (ii) X is injective, (iii) X has a dense range on any core, (iv) X -1 is strictly positive. Proof: (i) is equivalent to (iv) by Lemma A.4, because 〈 φ, X -1 φ 〉 = 〈 Xψ,ψ 〉 when φ := Xψ . The implication (i) ⇒ (ii) is immediate and (ii) is equivalent to (iii) by Lemma A.3. To see that (ii) implies (i) one uses the Spectral Calculus Theorem and the fact that 〈 φ, Xφ 〉 = 0 implies X 1 2 φ = 0 and Xφ = 0. If X is injective, this means that φ = 0. /square The following estimate is often useful to find strictly positive operators, in particular in combination with Lemma A.7 below. Lemma A.6 Let X and Y be positive self-adjoint operators with X strictly positive and assume that Y ≥ X on a core for Y 1 2 . Then Y is strictly positive, D ( Y -1 2 ) ⊃ D ( X -1 2 ) and Y -1 ≤ X -1 on D ( X -1 2 ) . Proof: Let D denote the core for Y 1 2 on which the estimate holds. The estimate ‖ X 1 2 ψ ‖ ≤ ‖ Y 1 2 ψ ‖ for ψ ∈ D can be extended to the entire domain D ( Y 1 2 ). Because X is strictly positive the same must be true for Y by Lemma A.5. By Lemma A.4, ‖ X 1 2 Y -1 2 ψ ‖ ≤ ‖ ψ ‖ on D ( Y -1 2 ). Note in particular that the range of Y -1 2 is contained in D ( X 1 2 ). As X 1 2 Y -1 2 is bounded on D ( Y -1 2 ) we also find that the range of X 1 2 , which is D ( X -1 2 ), is contained in the domain of ( Y -1 2 ) ∗ = Y -1 2 . It now follows that ( X 1 2 Y -1 2 ) ∗ = Y -1 2 X 1 2 on D ( X 1 2 ). As ‖ X 1 2 Y -1 2 ‖ ≤ 1 we must also have ‖ Y -1 2 X 1 2 ‖ ≤ 1, which implies that ‖ Y -1 2 ψ ‖ ≤ ‖ X -1 2 ψ ‖ on D ( X -1 2 ) and the conclusion follows. /square Lemma A.7 Let X ≥ 0 be a densely defined, positive operator. Then the Friedrichs extension ˆ X is positive and D ( X ) is a core for ˆ X 1 2 . The following lemma concerns the heat kernel: Lemma A.8 Let X be a positive self-adjoint operator on H and let C + := { z ∈ C | Re( z ) > 0 } be the right half space. Then the function z ↦→ e -zX is holomorphic on C + with values in the bounded operators on H and for each ψ ∈ H the function e -zX ψ is continuous on C + . To close this appendix we provide some facts concerning multiplication operators on the L 2 space of a semi-Riemannian manifold: Proposition A.1 Let ( M,g ) be an orientable semi-Riemannian manifold, let w ∈ C ∞ ( M ) and let W be the corresponding multiplication operator in L 2 ( M,d vol g ) , defined on C ∞ 0 ( M ) by ( Wf )( x ) = w ( x ) f ( x ) . If | w | is bounded, then W is bounded. If w is real-valued, then W is essentially selfadjoint. W is (strictly) positive if and only if w is (strictly) positive (almost everywhere).", "pages": [ 43, 44, 45 ] } ]

[ { "title": "ABSTRACT", "content": "International Journal of Modern Physics A c © World Scientific Publishing Company", "pages": [ 1 ] }, { "title": "Initial Conditions for Numerical Relativity ∼ Introduction to numerical methods for solving elliptic PDEs ∼", "content": "Hirotada Okawa ∗ CENTRA, Departamento de F'ısica, Instituto Superior T'ecnico, Universidade T'ecnica de Lisboa - UTL, Av. Rovisco Pais 1, 1049 Lisboa, Portugal. [email protected] Numerical relativity became a powerful tool to investigate the dynamics of binary problems with black holes or neutron stars as well as the very structure of General Relativity. Although public numerical relativity codes are available to evolve such systems, a proper understanding of the methods involved is quite important. Here we focus on the numerical solution of elliptic partial differential equations. Such equations arise when preparing initial data for numerical relativity, but also for monitoring the evolution of black holes. Because such elliptic equations play an important role in many branches of physics, we give an overview of the topic, and show how to numerically solve them with simple examples and sample codes written in C++ and Fortran90 for beginners in numerical relativity or other fields requiring numerical expertise. Keywords : numerical relativity; numerical method; black hole. PACS numbers:", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Numerical relativity is now a mature science, the purpose of which is to investigate non-linear dynamical spacetimes. A traditional example of the application and importance of numerical relativity concerns the modelling of gravitational-wave emission and consequent detection. In order to detect gravitational waves(GWs) from black hole-neutron star (BH-NS), BH-BH or NS-NS binaries, one needs to accurately understand the waveforms from these sources in advance, because their signals are quite faint for our detectors. 1 To understand why the problem is so difficult, consider Newtonian gravity, as applied to systems like our very own Earth-Moon. In Newton's theory, binary systems can move on stable, circular or quasi-circular orbits. However, in binary systems heavy enough or moving sufficiently fast, the effects of General Relativity become important, and the notion of stable orbits is no longer valid: GWs take energy and angular momentum away from the system and energy conservation implies that the binary orbit shrinks until finally the objects merge and presumably form a final single object. Accordingly, the evolution of binary stars can be divided into an inspiral, merger and a ring-down phase. 2 In the inspiral phase, GW emission is sufficiently under control by using slowmotion, Post-Newtonian expansions because the stars are distant from each other and their gravitational forces can be described in a perturbation scheme. 3, 4 The ringdown phase describes the vibrations of the final object. Because of the uniqueness theorems, 5 GWs can be computed by BH perturbation methods. 6, 7 Advanced BH perturbation methods are reviewed in Ref. 8 Numerical Relativity enables us to obtain the GW form in all phases. 9-11 Furthermore, we note that techniques of numerical relativity are also available in a variety of contexts. For example, but by no means the only one, it became popular to investigate the nature of higher dimensional spacetimes, 12 most specially in the framework of large extra dimensions. 13-16 It was pointed out that a micro BH can be produced from high energy particle collisions at the Large Hadron Collider(LHC) and beyond, 17,18 and while some works use shock wave collisions 19-21 the full-blown numerical solution is clearly desirable to investigate the nature of gravity with high energy collisions in four dimensional spacetime. 22-25 If we consider spacetime dimensions higher than four in our simulations, larger computational resources will be required. However, it can be reduced to four dimensional problem with small changes assuming the symmetry of spacetimes. 26-31 As a result, to investigate the nature of higher dimensional spacetimes is in the scope of numerical relativity. 32-34 Fortunately, open source codes to evolve dynamical systems with numerical relativity are available. 35-38 All that one needs to do is to prepare the initial data describing the physics of the problem one is interested in. Here, we explain precisely how this is achieved, to prepare initial data for numerical relativity in this paper. Briefly, it amounts to solving an elliptic partial differential equation (PDE) and we explain how to solve the elliptic PDE from the numerical point of view of beginners in numerical studies.", "pages": [ 1, 2 ] }, { "title": "2. ADM formalism", "content": "In numerical relativity, we regard our spacetimes as the evolution of spaces. We begin by showing how to decompose the spacetime into timelike and spacelike components in the ADM formalism. 39-41 Then, we derive evolution equations from Einstein's equations along the lines of York's review. 42", "pages": [ 2 ] }, { "title": "2.1. Decomposition", "content": "First, we introduce a family of three-dimensional spacelike hypersurfaces Σ in fourdimensional manifold V . Hypersurfaces Σ are expressed as the level surfaces of a scalar function f and are not supposed to intersect one another. We can define a one-form Ω µ = ∇ µ f which is normal to the hypersurface. Let g µν be a metric tensor in four-dimensional manifold V . The norm of one-form Ω µ can be written by a positive function α as where α is called lapse function. We define a normalized one-form by The orthogonal vector to a hypersurface Σ is written by whose minus sign is defined to direct at the future and we note that this timelike vector satisfies n µ n µ = -1 by definition.", "pages": [ 2, 3 ] }, { "title": "2.1.1. Induced metric", "content": "The induced metric γ µν on Σ and the projection tensor ⊥ µ ν from V to Σ are given by the four-dimensional metric g µν , where one can show n µ γ µν = 0, which yields that timelike components of γ µν vanish and only spacelike components γ ij exist. The induced covariant derivative D i on Σ is also defined in terms of the four-dimensional covariant derivative ∇ µ . j D W = ⊥ ρ j ⊥ i λ ∇ ρ W , (7) where ψ and W λ denote arbitrary scalar and vector on Σ. By a straightforward calculation, one can show that the induced covariant derivative satisfies D i γ jk = 0.", "pages": [ 3 ] }, { "title": "2.1.2. Curvature", "content": "Riemann tensor on Σ is defined using an arbitrary vector W i by where [ ] denotes the antisymmetric operator for indices and R ijk/lscript denotes Riemann tensor on Σ. Ricci tensor is determined by the contraction of the induced metric and Riemann tensor on Σ. Ricci scalar is also determined by the contraction of the induced metric and Ricci tensor. We define the extrinsic curvature on Σ, which describes how the hypersurface is embedded in the manifold V . The extrinsic curvature is defined by where ( ) denotes the symmetric operator for indeces. One can also show that the extrinsic curvature is spacelike by multiplying the normal vector n µ in the same manner as γ µν . In addition, by the definition of the projection tensor, we obtain the i λ 4 following relation between the covariant derivative of the normal vector and their projection, where the relation n λ n λ = -1 is used in the last equation. Then, the extrinsic curvature can be rewritten by where £ n γ µν denotes the Lie derivative of the tensor γ µν along the vector n µ . The geometrical nature of the three-dimensional hypersurfaces can be determined by the induced metric and extrinsic curvature on Σ. K ij and γ ij must satisfy the following geometrical relations to embed Σ in V .", "pages": [ 3, 4 ] }, { "title": "2.1.3. Geometrical relations", "content": "We derive geometrical relations by the projection of the four-dimensional Riemann tensor to the hypersurface Σ. First, in order to obtain the relation between the four-dimensional Riemann tensor R µνρλ and the three-dimensional Riemann tensor R ijk/lscript , we rewrite the definition of R ijk/lscript with R µνρλ and the extrinsic curvature, Eq. (13) is called Gauss' equation. Secondly, we project the four-dimensional Riemann tensor contracted by an orthogonal normal vector n λ . Eq. (14) is called Codazzi's equation. Finally, we consider a Lie derivative of the extrinsic curvature to the time direction. We define a timelike vector t µ with a lapse function and a shift vector β µ which satisfies Ω µ β µ as Then, with the Lie derivative along t µ and β µ , we rewrite the four dimensional Riemann tensor contracted by two orthogonal normal vectors as which Eq. (16) is called Ricci's equation.", "pages": [ 4 ] }, { "title": "2.2. Decomposition of Einstein's equations", "content": "Let us now use the geometric relations to decompose Einstein's equations. Let us for convenience define the Einstein tensor G µν , where G denotes the gravitational constant and c denotes the speed of light and hereafter we set G = c = 1 for simplicity. We start by decomposing the energy momentum tensor as where ρ ≡ T µν n µ n ν , j µ ≡ - ⊥ ρ µ T ρλ n λ and S µν ≡⊥ ρ µ ⊥ λ ν T ρλ . We multiply Gauss' equation (13) by an induced metric γ ik and obtain In addition, Eq. (19) contracted by γ j/lscript gives twice as much as the Einstein's tensor contracted by two orthogonal normal vectors n µ and n ν . Then, we obtain Similarly, Codazzi's equation (14) contracted by γ jk results in Note that Eq. (20) and Eq. (21) are composed of only spacelike variables and should be satisfied on each hypersurface Σ because they do not depend on time. Therefore, Eq. (20) and Eq. (21) are called the Hamiltonian and momentum constraints, respectively. Finally, let us rewrite Ricci's equation (16). Einstein's equations (17) can also be expressed with the trace of the energy momentum tensor T ≡ g µν T µν as The projection of Einstein's equations (22) yields where S = γ ij S ij . Ricci's equation (16) is rewritten with Eq. (19) and Eq. (23) as", "pages": [ 5 ] }, { "title": "2.3. Propagation of Constraints", "content": "In ADM formalism, Einstein's equations are regarded as evolution equations in time with the geometrical constraints on each hypersurface. In general, it is numerically expensive to guarantee the constraints on each step because we must solve elliptic PDEs as described in Sec. 3.1. However, in principle, one does not have to solve the constraint equations if the initial data satisfy the constraints. 43-45 This works as follows. We first define the following quantities, where C = 0 corresponds to the Hamiltonian contstraint, C µ = 0 correspond to the momentum constraints and C µν = 0 denote the evolution equations in ADM formalism. Einstein's equations can be decomposed in terms of C, C µ and C µν as Thanks to the Bianchi identity which is a mathematical relation for the Riemann tensor, the covariant derivative of Einstein's tensor vanishes. Besides, the covariant derivative of the energy momentum tensor which appears in the right-hand-side of Einstein's equations denotes the energy conservation law, Let us project the covariant derivative of Einstein's equations to n µ direction and to the spatial direction with ⊥ ρ µ . where D µ denotes the covariant derivative with respect to the induced metric, noting that C µ and C µν are spatial. Thus, we show the propagation of constraints along the timelike vector as Therefore, we can keep the Hamiltonian and momentum constraints satisfied during evolution, as long as we evolve the initial data satisfying the constraints C = 0 and C µ = 0 on Σ by the evolution equation C µν = 0. It should be noted that the ADM evolution equations are numerically unstable and not suitable for numerical evolutions; instead one uses hyperbolic evolution equations for time integration, for example, BSSN and Z4 evolution equations. 26,46-49", "pages": [ 6 ] }, { "title": "3. Initial Condition for Numerical Relativity", "content": "As emphasized in Sec. 2.2, initial data cannot be freely specified, as it needs to satisfy the Hamiltonian and momentum constraints on a hypersurface Σ. In general, the problem of constructing initial data is called 'Initial Value Problem'. The standard method for solving an initial value problem is reviewed in Ref. 50,51 In this section, we derive the equations for the initial value problem and then introduce some examples as initial data for numerical relativity.", "pages": [ 7 ] }, { "title": "3.1. Initial Value Problem", "content": "There are twelve variables( γ ij , K ij ) as the metric part to be determined and four constraint equations in ADM formalism. One can obtain four variables by solving constraints after assuming eight variables by physical and numerical reasons.", "pages": [ 7 ] }, { "title": "3.1.1. York-Lichnerowicz conformal decomposition", "content": "To begin with, we introduce the conformal factor ψ as where we define det ˜ γ ij ≡ 1 and we have one degree of freedom in ψ and five degrees of freedom in ˜ γ ij . By the conformal transformation, the following relations between variables with respect to γ ij and ˜ γ ij are immediately given by where ˜ D i , ˜ Γ i jk , ˜ R and ˜ /triangle are respectively covariant derivative, Ricci scalar, Christoffel symbol and Laplacian operator defined by ˜ /triangle ψ = ˜ γ ij ˜ D i ˜ D j ψ with respect to ˜ γ ij .", "pages": [ 7 ] }, { "title": "3.1.2. Transverse-Traceless decomposition", "content": "As for the extrinsic curvature, we start by decomposing it into a trace and a tracefree part, where γ ij A ij = 0 and K = γ ij K ij and we have one degree of freedom in K and five degrees of freedom in A ij . Then, we also define the conformal transformation for A ij as According to the definition of derivatives with respect to γ ij and ˜ γ ij , we obtain the following relation, In addition, we decompose the conformal traceless tensor ˜ A ij into a divergenceless part and a 'derivative of a vector' W j part. Hereafter we assume the divergenceless part vanishes for simplicity. The conformal traceless extrinsic curvature is described by The covariant derivative of ˜ A ij is written by where we used the definition of Riemann tensor.", "pages": [ 7, 8 ] }, { "title": "3.1.3. Constraints as initial value problem", "content": "With the above conformal transformation, the Hamiltonian and momentum constraints are rewritten as", "pages": [ 8 ] }, { "title": "3.2. Schwarzschild Black Hole", "content": "Let us consider an exact solution of Einstein's equations as initial data for numerical relativity. The Schwarzschild BH is the simplest BH solution in static and spherically symmetric spacetimes. 52 The line element of the Schwarzschild BH in spherical coordinates (¯ r, θ, φ ) is given by where we define f 0 with BH mass M , Let us define the coordinate transformation by where r denotes the isotropic radial coordinate and we introduce a scalar function ψ 0 . Then, we solve ¯ r under the coordinate transformation and obtain ψ 0 and the relation between ¯ r and r as After straightforward calculations, the line element is rewritten by where η ij denotes the flat metric and we define α 0 . In the isotropic coordinates, all spatial metric components remain regular, in contrast to the ones in the standard Schwarzschild coordinates. The range [2 M < ¯ r < ∞ ] in the spherical coordinates corresponds to [ M 2 < r < ∞ ] in the isotropic coordinates. In addition, when we change to a new coordinate ˜ r ≡ ( M/ 2) 2 /r , we obtain the same expression as Eq. (50) with ˜ r instead of r . It yields that the range [ M 2 < ˜ r < ∞ ] corresponds to [0 < r < M 2 ]. The solution is inversion symmetric at r = M/ 2 and corresponds to the Einstein-Rosen bridge. 53 Obviously, the extrinsic curvature K ij of the Schwarzschild BH vanishes because the spacetime is static, and therefore the momentum constraints are trivially satisfied. Besides, the Hamiltonian constraint is also satisfied as because the Schwarzschild BH is an exact solution of Einstein's equations.", "pages": [ 8, 9 ] }, { "title": "3.3. Puncture Initial Data", "content": "One can analytically solve the momentum constraints with the following conditions, The derivative operator becomes quite simple assuming conformal flatness. We also note that Eq. (42) and Eq. (43) are decoupled with K = const. condition.", "pages": [ 9 ] }, { "title": "3.3.1. Single Black Hole", "content": "/negationslash Next, let us consider a BH with non-zero momentum( P i = 0), for which the momentum constraints become non-trivial. However, a solution to conditions (52) can still be found. In this case, the momentum constraints are given by We have a simple solution to satisfy Eq. (53) as where P i and S i are constant vectors corresponding to the momentum and spin of BH and n i ≡ x i /r denotes the normal vector. Then, we obtain the Bowen-York extrinsic curvature 54 by substituting Eq. (54) into Eq. (40), On the other hand, to satisfy the Hamiltonian constraint (42) we must in general solve an elliptic PDE, even if simple-looking, Let us define the function u as a correction term relative to the Schwarzschild BH, We can regularize the Hamiltonian constraint (56) when ru is regular at the origin. ˜ A ij is at most proportional to r -3 at and ψ is proportional to r -1 , so that the divergent behavior of ˜ A ij ˜ A ij is compensated by the ψ -7 term at the origin. Therefore, the Hamiltonian constraint yields", "pages": [ 9, 10 ] }, { "title": "3.3.2. Multi Black Holes", "content": "We can easily prepare the initial data which contains many BHs without any momenta under the condition (52) because the Hamiltonian constraint is the same as Eq. (51) and we know that the following conformal factor satisfies the Laplace equation. where M n and x n denote the mass and position of n-th BH, respectively. The initial data defined with ψ = ψ M and ˜ A ij = 0 is called Brill-Lindquist initial data. 55 As for BHs with non-zero momenta, we also use the Bowen-York extrinsic curvature and the same method for the Hamiltonian constraint as ψ ≡ ψ M + u , In principle, it is possible to construct initial data for multi BHs with any momenta and spins by solving an elliptic PDE. 56", "pages": [ 10 ] }, { "title": "3.4. Kerr Black Hole", "content": "It should be noted that we have the exact BH solution of a rotating BH for Einstein's equations and we can also use it as initial data. The Kerr BH is an exact solution of Einstein's equations in stationary and axisymmetric spacetime. 57 The line element of the Kerr BH in Boyer-Lindquist coordinates 58 is defined by where where M and a denote the mass and spin of BH respectively. ∆ vanishes when the radial coordinate r BL is at the radius of the inner or outer horizon r ± , which is a coordinate singularity. Let us introduce a quasi-isotropic radial coordinate in the same manner as for the Schwarzschild BH by Thus, the line element of the Kerr BH yields The spatial metric components in the quasi-isotropic coordinates also remain regular. 59, 60 One can show that the extrinsic curvature of the Kerr BH in the quasiisotropic coordinates is given by which comes from the shift vector β φ .", "pages": [ 10, 11 ] }, { "title": "4. Apparent Horizon Finder", "content": "Now we can perform long-term dynamical simulations containing BHs with numerical relativity. For the sake of convenience, we usually use the apparent horizon (AH) to define the BH and investigate the nature of BH during the evolution. In this section, we introduce the concept of AH and derive the elliptic PDE to determine the AH.", "pages": [ 11 ] }, { "title": "4.1. Apparent Horizon", "content": "The region of BH in an asymptotic flat spacetime is defined as the set of spacetime points from which future-pointing null geodesics cannot reach future null infinity. 61 To find the BH, one can use the event horizon(EH) which is defined as the boundary of such region. It is possible to determine the EH by the data of the numerical simulation because in principle, one can integrate the null geodesic equation for any spacetime points forward in time during the evolution, where x µ and λ denote the coordinates and the affine parameter. The numerical cost to find the EH is normally high, because we need global metric data. 62 We define a trapped surface on the hypersurface Σ as a smooth closed twodimensional surface on which the expansion of future-pointing null geodesics is negative. The AH is defined as the boundary of region containing trapped surfaces in the hypersurface and is equivalent to the marginally outer trapped surface on which the expansion of future-pointing null geodesics vanishes. 63 The EH is outside the AH if the AH exists. 61 We often use the AH to find the BH in the numerical simulation instead of the EH because the AH can be locally determined and then the numerical cost is lower compared with finding the EH. Let us introduce the normal vector s i to the surface and define the induced two-dimensional metric as where γ µν denotes the induced metric on the three-dimensional hypersurface Σ. The null vector is described with s i and the normal vector to Σ by Then, the following equation should be satisfied on the AH by definition. where Θ denotes the expansion of null vector and D i denotes the covariant derivative with respect to γ ij .", "pages": [ 12 ] }, { "title": "4.2. Apparent Horizon Finder", "content": "We can find the AH during the dynamical simulation by solving Eq. (73). 64-66 Let us define the radius of the AH by Thus, the normal vector s i can be described with h ( θ, φ ) by where ˜ s i is introduced for convenience and we raise their indeces of s i and ˜ s i by γ ij and ˜ γ ij respectively. Incidentally, the divergence of the normal vector is given by where γ denotes the determinant of γ ij . Therefore, we obtain the equation to determine the AH as the elliptic PDE consisting of first and second derivatives of h ( θ, φ ). Note that because the AH equation is originally a non-linear elliptic PDE, we change the AH equation to the flat Laplacian equation with non-linear source term, 64 which has the advantage of fixing the matrix with diagonal dominance mentioned in Sec. 5.2.2. Specifically, we solve the following equation. where ζ denotes a constant to be chosen by the problem and the source term is given by the flat laplacian term and the AH equation as where ξ ij ≡ ˜ γ ij -η ij .", "pages": [ 12, 13 ] }, { "title": "4.3. Mass and Spin of Black Hole", "content": "The area of the AH is defined by where S denotes the surface of AH. We also compute the quantities related to the AH, the polar and equatorial circumfential length( C p , C e ). If the BH relaxes to a stationary state during the evolution, the BH would be the Kerr BH because of no-hair theorem. The quantities related to the AH of the Kerr BH can be obtained by where M BH , a and r + denote the mass, spin and outer horizon radius defined by r + = 1 + √ 1 -a 2 and E ( z ) denotes an elliptic integral defined by", "pages": [ 13, 14 ] }, { "title": "5. Numerical Methods for solving elliptic PDEs", "content": "Constructing the initial data for numerical relativity is, in general, equivalent to solving the elliptic PDEs (42) and (43) with appropriate conditions. In order to solve a binary problem with high accuracy, the spectral method should be the standard method for solving an elliptic PDEs. In fact, there are useful open source codes, for example, TwoPuncture 67, 68 and LORENE. 69 Futhermore, we have to solve another elliptic PDE to find the BH in simulations within numerical relativity as described in Sec. 4. In this case, fast methods to solve the ellitptic PDE are preferred. Because there are many elliptic PDE solvers, the method has to be chosen according to the specific purpose. In this section, we introduce some classical numerical methods for beginners. It would also be the basis for Multi-Grid method mentioned in Appendix B.", "pages": [ 14 ] }, { "title": "5.1. Discretization", "content": "We should discretize our physical space to the computational grid space by finite difference method because we cannot take continuum fields into account on the computer. Consider first one-dimensional problems for simplicity, and introduce the grid interval ∆ x . Taylor expansion of a field Q ( x ) is given by Thus, the derivative of the field Q ( x ) can be expressed as  where Q j +1 , Q j and Q j -1 denote Q ( x +∆ x ) , Q ( x ) and Q ( x -∆ x ) respectively and both accuracies of the forward and backward difference method for derivatives are O (∆ x ). In addition, the central difference method whose accuracy is O (∆ x 2 ) can be defined by both Taylor expansions as Similarly, the second-order derivative of Q ( x ) is written by One can increase accuracy of the calculation by using many points. For example, using five values Q ( x +2∆ x ) , Q ( x +∆ x ) , Q ( x ) , Q ( x -∆ x ) and Q ( x -2∆ x ) around x , the fourth-order accuracy scheme are defined by noting that higher accuracy scheme can also be defined. We also note that we can discretize our space in more than two dimensions in the same way.", "pages": [ 14, 15 ] }, { "title": "5.2. Relaxation Method", "content": "Hereafter, let us focus on Poisson equations ( /triangle ψ = S ) with a field ψ and a source S as elliptic PDEs. These are a sufficiently general and complex class of problems that they embody all necessary elements to solve Poisson equation for constructing initial data for numerical relativity or finding an apparent horizon of BH. We explain how to solve general elliptic PDEs in Appendix A. One of the simple methods to solve elliptic PDEs, so-called relaxation method, 70-72 is described in this section. Let us introduce a virtual time τ to solve an elliptic PDE and our equation of elliptic type can be transformed to the equation of parabolic type as which denotes the original Poisson equation after ψ relaxes by iteration. We adopt Cartesian coordinates in three-dimensional spaces and discretize the Poisson equation with second-order accuracy as where the superscript n denotes the label of virtual time and the subscript j, k and l denote labels of x -, y -and z -direction, respectively. Therefore, the field in the next step of the iteration is determined by We continue to update the field ψ by the above expression until ψ relaxes and obtain the solution of the Poisson equation ( /triangle ψ = S ).", "pages": [ 15, 16 ] }, { "title": "5.2.1. Jacobi Method", "content": "In order to discuss method in practice, we consider one-dimensional Poisson equation and discretize it as We consider Eq. (98) as the equation to determine ψ j , where the superscript n denotes the label of time step and we attach the label J on the field of next time step in Jacobi's method. Thus, we repeat updating the field until ψ converges. In other words, the flowchart of Jacobi method is as follows. In addition, it is easy to extend to the three-dimensional Poisson equation as where we define ∆ h ≡ ∆ x = ∆ y = ∆ z for simplicity.", "pages": [ 16 ] }, { "title": "5.2.2. Matrix expression", "content": "Discretized elliptic PDEs can be expressed by matrices and vectors. Once we describe the elliptic PDE via a matrix expression, the problem involves solving the inverse of the matrix. For example, a matrix expression for Jacobi method is given as follows. We introduce a solution vector ψ I and source vector b I . Then, Eq. (98) can be expressed as where A IJ is the coefficient matrix corresponding to the Laplacian operator and the first and last rows of A IJ denote boundary conditions to be determined by physics. It is formally expressed by A IJ ψ J = b I and the problem leads to solving the inverse of coefficient matrix as ψ J = A -1 IJ b I . There are many methods to numerically solve the inverse of matrices. In general, Jacobi method for an arbitrary coefficient matrix is expressed as /negationslash /negationslash We note that other elliptic PDEs may not be solvable by the Jacobi method because the iteration is not always stable; this can be shown by the von Neumann numerical stability analysis. However, the Poisson equation is fortunately stable, which is equivalent to that the matrix is diagonally dominant.", "pages": [ 16, 17 ] }, { "title": "5.2.3. Gauss-Seidel Method", "content": "In iterative methods to solve Poisson equations as Jacobi method, it depends on a trial field ψ n how fast we obtain solutions. We usually expect that it becomes better solution as iteration step goes forward. In order to obtain a closer trial field to the solution, we should actively use updated values. Thus, by Gauss-Seidel method, we can determine a next trial field as where GS denotes that the field is determined by Gauss-Seidel method. In addition, the matrix expression for Gauss-Seidel method is given by", "pages": [ 17 ] }, { "title": "5.2.4. SOR Method", "content": "It turns out that the Poisson equation is faster to solve with Gauss-Seidel method than with the Jacobi method, as shown later. Although the speed with which the numerical solution converges depends on the trial field in iterative methods, the Gauss-Seidel method gives a 'better' field than Jacobi's in that respect. Thus, it is possible to accelerate convergence by specifying trial guess of the field more aggressively. This method is called Successive Over-Relaxation(SOR) method, which is defined by where the superscript S denotes the label of SOR method and ω denotes an acceleration parameter whose range is defined in 1 ≤ ω < 2 by the stability analysis. When we set the acceleration parameter as unity, SOR method is identical to the Gauss-Seidel method by definition.", "pages": [ 18 ] }, { "title": "6. Results", "content": "In this section, we introduce sample codes to solve Poisson equations using different methods. These codes are sufficiently general that they can be applied to other problems in physics, provided one slightly changes the source term and boundary conditions.", "pages": [ 18 ] }, { "title": "6.1. Code Tests", "content": "As tests for our codes, we use the following analytical solutions. We show numerical results of Poisson equations with simple linear source and sufficiently non-linear source. In addition, the code to find the AH of the Kerr BH is also shown as an example. Some sample codes parallelized with OpenMP are also available Appendix C.", "pages": [ 18 ] }, { "title": "6.1.1. Linear source", "content": "Let us consider simple source term for the Poisson equation as In numerical computation, we set the range as 0 ≤ x ≤ 1 and boundary conditions by ∣ Then, we obtain the analytical solution by integrating Eq. (106) twice with boundary conditions (107), Arbitrary initial guess for the solution can be given and we set ψ ( x ) = 1 at initial for those boundary conditions. We set the resolution of the computational grid as ∆ x = 1 / 100. Fig. 1 (a) shows the numerical solution by Jacobi method as compared to the analytical solution. We note that the accuracy of the numerical result depends on the computational resolution and how the accuracy increase with resolution depends on the scheme of discretization; Fig. (b) is compatible with second-order accuracy. We compare Poisson solvers in Fig. 1 (c) by the time steps needed to obtain the solution. Curves show the difference of methods and we choose the Jacobi method, Gauss-Seidel method, SOR methods with ω = 1 . 5 and ω = 1 . 9. The SOR method gives the solution about 10-100 times faster than Jacobi method, and depends on the acceleration parameter ω . ψ", "pages": [ 18, 19 ] }, { "title": "6.1.2. Non-linear source", "content": "Next, let us consider a weak gravitational field, namely Newtonian gravitational source. A gravitational potential Φ can be determined by the Poisson equation, where we omit the Newton constant by using G = 1 units. Suppose gravitational sources are distributed with spherical symmetry as where ρ 0 is a constant. Corresponding Poisson equation is rewritten by Thus, we obtain the analytical solution of the source (110) by solving the equations separately as the region ( r > 1) with the boundary condition Φ → 0 at infinity and ( r ≤ 1) with the regularity condition at the origin. We consider this analytical solution to test our code. The Poisson equation with spherical symmetry can be regarded as one dimentional Poisson equation with the non-linear source in our method, whose range to be considered as 0 ≤ x ≤ 10 and boundary conditions are set as ∣ where the Robin boundary condition is chosen because we expect Φ → r -1 at large distance. Fig. 2 (a) shows the source distribution and (b) shows the numerical result by solving the Eq. (113). The result is obtained with 400 grid points but shown with only 40 points.", "pages": [ 20 ] }, { "title": "6.1.3. Apparent Horizon of Kerr Black Hole", "content": "Let us apply our code for Poisson solver to solving the AH of Kerr BH. The AH equation (79) should be reduced to simpler equation with axisymmetry ∂ φ = 0. The normal vector s i is defined with axisymmetry and the normalization C is determined by the Kerr metric as To be concrete, we note that the non-trivial part of the AH equation with axisymmetry in isotropic coordinates can be written by where On the other hand, we can also solve the AH in Boyer-Lindquist coordinates, only to change the following part. In Fig. 3 (a), we show the surface of AH of the Schwarzschild BH in isotropic coordinates with the code 'sor AHF SBH ISO.f90' and show the three dimensional shape of the AH in 1/8 spaces of computational grid. Fig. 3 (b) shows the difference of the shape on x-z two dimensional plane among different spin parameters with the code 'sor AHF KBH ISO.f90'. The AH radius shrinks as the spin of BH increses.", "pages": [ 21, 22 ] }, { "title": "6.2. Kerr Black Hole and Single Puncture Black Hole", "content": "As the last example, let us compare Kerr BH to single puncture BH with a spin as initial data for numerical relativity. A Kerr BH in quasi-isotropic coordinates can be used as the initial data discussed in Sec. 3.4. A single puncture BH is obtained by solving the Hamiltonian constraint (58) without any momenta P i = 0 and with a spin S z in the Bowen-York extrinsic curvature (55). In order to check whether our AHF for this comparison works well, in Fig. 4 (a) we show the relation between AH area of the Kerr BH and AH radius in isotropic coordinates as a function of spin parameter. The blue line denotes the analytical AH area and red crosses denote numerical results by solving AH equation for Kerr BH. The green circles show the coordinate radii where the AHs with different are located. Much larger computational resources should be required to obtain the solution with a high BH spin because high resolution in the coordinate radius is required in this regime. We perform numerical relativity simulations with the initial data of single puncture BH and Kerr BH in Fig. 4 (b). The BSSN evolution equations which give stable dynamical evolution 46-48 are adopted in these simulations. The color difference shows the difference among spins and the type of lines denotes the difference between Kerr BH and single puncture BH. The spins of single puncture BHs settle down at late time, which shows BHs relax to almost the stationary state and one can compare results of Kerr BHs at late time. The single puncture BH with the higher spin does not reach at the spin which we expect. This is because we assume the conformal flatness for constructing puncture BH but Kerr BH should not be expressed by the conformal flat metric. However, it should be noted that the puncture BH can represent the small spin BH well and it is actually powerful to construct the initial data for multi BHs system.", "pages": [ 22, 23 ] }, { "title": "7. Conclusions", "content": "In these notes, we showed how to prepare the initial data for numerical relativity and how to obtain the apparent horizon of BHs, which are reduced to solving elliptic PDEs in general. We presented several BH solutions as initial data for numerical relativity and described several numerical methods to solve elliptic PDEs. In particular, sample codes to solve Poisson equations with linear and non-linear sources are available online to public users. It is worth noting that these simple, 'classical' methods are still powerful enough to be of use for current problems. In addition, we note that modern methods (e.g. Multi-Grid method) can help to eventually upgrade these classical methods in terms of numerical costs and consuming-time. Of course, one should carefully choose which method to use to solve elliptic PDEs, according to the problem at hand.", "pages": [ 23, 24 ] }, { "title": "Acknowledgments", "content": "The author would like to thank Vitor Cardoso for giving the opportunity to lecture on this school, and to the Organizers and Editors of the NR/HEP2: Spring School at Instituto Superior T'ecnico in Lisbon. The author would also thank an anonymous referee for a careful reading of the manuscript and many useful suggestions. The author is thankful to Ana Sousa who helps to improve English on this notes, S'ergio Almeida who maintains the cluster 'Baltasar-Sete-S'ois' and Takashi Hiramatsu who maintains the 'venus' cluster. Numerical computations in this work were carried out on the cluster of 'Baltasar-Sete-S'ois' at Instituto Superior T'ecnico in Lisbon which is supported by the DyBHo-256667 ERC Starting Grant and on the 'venus' cluster at the Yukawa Institute Computer Facility in Kyoto University. This work was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development & Innovation.", "pages": [ 24 ] }, { "title": "Appendix A. LDU decomposition", "content": "In this Appendix, we describe how to numerically solve only the Poisson equation. However, we can also solve general elliptic PDEs in principle, namely, in case except for the problem with diagonally dominant matrix. A system of linear equations can be expressed by the matrix described in Sec. 5.2.2 as Let us decompose a matrix A IJ into the lower and upper triangular matrices defined as L IJ and U IJ respectively, where D KK denotes the diagonal matrix. Then, the solution vector can be solved ψ J step by step as Thus, it is easy to obtain the solution as the following precedures. First, we obtain an auxiliary vector ξ K as solving Eq. (A.3) from ξ 1 to ξ N , Therefore, the solution vector ψ J is written by similarly solving Eq. (A.4) from ψ N to ψ 1 , As the last of this section, we note how we compute the lower and upper matrices from our matrix A IJ , which is the time-consuming part. The matrix A IJ is written with the diagonal, lower and upper triangular matrices by Although LDU decomposition allows us to numerically solve general elliptic PDEs, the large numerical costs will be required in many cases.", "pages": [ 24, 25, 26 ] }, { "title": "Appendix B. Multi-Grid method", "content": "Multi-Grid method is proposed by R. Fedorenko and N. Bakhvalov and developed by A. Brandt. 73-76 The SOR method as mentioned in Sec. 5.2.4 has the advantage of reducing the high frequency components of residual between the exact solution and numerical solution, because the values near the grid point to be updated are used for next trial guess during the iteration. On the other hand, it would take much time to reduce the low frequency modes of redisual with this iteration method. When we consider different resolution grids, however, the low frequency modes on the finer grid can be the high frequency modes on the coarser grid. The low frequency modes of residual on the finer grid can efficiently be reduced on the coarser grid. The Multi-Grid method is based on the concept of reducing different frequency modes of residual with different resolution grids. In fact, it was implemented by some groups. 77-79", "pages": [ 26 ] }, { "title": "Appendix B.1. Multi-Grid structure", "content": "Suppose we have different resolution grids and the level of different grids is labeled by k , which the larger k denotes the finer grid. One can solve the Poisson equation on the level k by any iterative methods described in Sec. 5 and obtain the numerical solution, where φ ( k ) is the numerical solution on the level k . We define the residual on the level k between φ ( k ) and the exact solution by", "pages": [ 26 ] }, { "title": "Appendix B.1.1. Lagrange interpolation", "content": "In general, the communication of the quantities such as the residual with different grid levels is needed. Now, we just use the Lagrange interpolation to communicate with each other level defined by /negationslash where F, x j , x and N denote the quantity to be interpolated, the coordinate on the level, the location to be interpolated, and the number of grid points to be used by the interpolation, respectively.", "pages": [ 27 ] }, { "title": "Appendix B.1.2. Restriction operator", "content": "After we obtain the solution on the finer grid k , we transfer the information of the solution from the finer grid k to the coarser grid k -1. Now we use the second-order discretization scheme and choose the third-order Lagrange interpolation. We define the modified source term on the coarser level k -1 with the information of the solution on the finer grid k by where R k -1 k denotes the restriction operator to the coarser grid k -1 and φ ( k -1) c denotes the smoothing solution by the restriction operator. Roughly speaking, the modified source term S ( k -1) consists of that on the level k with smoothing operation and the correction by the difference of Laplacian operator between two levels. Then, we obtain the numerical solution φ ( k -1) on the level k -1 to solve the Poisson equation with the modified source term.", "pages": [ 27 ] }, { "title": "Appendix B.1.3. Prolongation operator", "content": "The solution with the modified source term on the coarser level k -1 is to be brought back to the finer level k . Now the communication is also done by third- order Lagrange interpolation. where P k k -1 denotes the prolongation operator and φ ( k ) m denotes the solution on the level k modified by the coarser grid k -1. The modification is done by Eq. (B.11).", "pages": [ 27, 28 ] }, { "title": "Appendix B.1.4. Cycle of the Multi-Grid method", "content": "There are some ways of deciding the order of the level to compute. Fig. (5) shows the difference of such order between the methods of V-cycle and W-cycle as examples. Now we choose V-cycle because it is easier to implement to the code. We use the restriction operator before computing on the coarser level and the prolongation operator before computing on the finer level. This cycle is repeated until we obtain the expected error of the Poisson equation.", "pages": [ 28 ] }, { "title": "Appendix B.2. Code test", "content": "Let us consider the same test problem as Sec. 6.1.2. In the 3D problem, we impose the boundary conditions at large distance by We note that the boundary of the finer grid is given by the interpolation. Fig. 6 shows the results on the x-axis by solving the Poisson equation with the source (110) by Multi-Grid method. The solution including the boundary converges to the analytical solution by iterations.", "pages": [ 28, 29 ] }, { "title": "Appendix C. List of Sample codes", "content": "We have some sample codes for the lecture on NR/HEP2: Spring School at Instituto Superior T'ecnico in Lisbon and they are available online. In this section, we show the simplest code to solve an elliptic PDE and the sample code which is parallelized with OpenMP. One can see what is the parallel computing in Ref. 80 Here is the list of sample codes which are available in http://blackholes.ist.utl.pt/nrhep2/?page=material, This is the code for solving the problem described in Sec. 6.1.1 with Jacobi", "pages": [ 29 ] }, { "title": "(2) gs test1.f90", "content": "This is the code for solving the problem described in Sec. 6.1.1 with Gauss-Seidel method(See Sec. 5.2.3). This is the code for solving the problem described in Sec. 6.1.1 with SOR This is the code for solving the problem described in Sec. 6.1.1 with Jacobi method(See Sec. 5.2.1). 30", "pages": [ 29, 30 ] }, { "title": "(6) sor AHF KBH ISO.f90", "content": "This is the code for solving the AH of Kerr BH in isotropic coordinates described in Sec. 6.1.3 with SOR method(See Sec. 5.2.4).", "pages": [ 30 ] }, { "title": "(7) sor AHF KBH BL.f90", "content": "This is the code for solving the AH of Kerr BH in Boyer-Lindquist coordinates described in Sec. 6.1.3 with SOR method(See Sec. 5.2.4).", "pages": [ 30 ] }, { "title": "(8) jacobi openMP.f90", "content": "This is the code for solving the problem described in Sec. 6.1.1 with Jacobi method(See Sec. 5.2.1) using many processors with OpenMP.", "pages": [ 30 ] }, { "title": "(9) jacobi test1.C", "content": "This is the code written in C++ for solving the problem described in Sec. 6.1.1 with Jacobi method(See Sec. 5.2.1). This is the code written in C++ for solving the problem described in Sec. 6.1.1", "pages": [ 30 ] }, { "title": "(11) jacobi openMP.C", "content": "This is the code written in C++ for solving the problem described in Sec. 6.1.1 with Jacobi method(See Sec. 5.2.1) using many processors with OpenMP.", "pages": [ 30 ] }, { "title": "References", "content": "40", "pages": [ 40 ] } ]

[ { "title": "Vacuum Stability of the PT -Symmetric ( -φ 4 ) Scalar Field Theory", "content": "Abouzeid M. Shalaby ∗ Physics Department, Faculty of Science, Mansoura University, Egypt", "pages": [ 1 ] }, { "title": "Abstract", "content": "In this work, we study the vacuum stability of the classical unstable ( -φ 4 ) scalar field potential. Regarding this, we obtained the effective potential, up to second order in the coupling, for the theory in 1+1 and 2+1 space-time dimensions. We found that the obtained effective potential is bounded from below, which proves the vacuum stability of the theory in space-time dimensions higher than the previously studied 0+1 case. In our calculations, we used the canonical quantization regime in which one deals with operators rather than classical functions used in the path integral formulation. Therefore, the non-Hermiticity of the effective field theory is obvious. Moreover, the method we employ implements the canonical equal-time commutation relations and the Heisenberg picture for the operators. Thus, the metric operator is implemented in the calculations of the transition amplitudes. Accordingly, the method avoids the very complicated calculations needed in other methods for the metric operator. To test the accuracy of our results, we obtained the exponential behavior of the vacuum condensate for small coupling values, which has been obtained in the literature using other methods. We assert that this work is interesting, as all the studies in the literature advocate the stability of the ( -φ 4 ) theory at the quantum mechanical level while our work extends the argument to the level of field quantization. PACS numbers: 03.65.-w, 11.10.Kk, 02.30.Mv, 11.30.Qc, 11.15.Tk Keywords: non-Hermitian models, PT -symmetric theories, effective potential.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Among its wide range of applications, the subject of PT -symmetric theories has stressed the bounded-from-above ( -x 4 ) quantum mechanical potential [1-6]. The recipe for the calculations within such theories is to choose a specific contour in the complex x -plane and apply the quantization condition ( χ n → 0 as | x | → ∞ ) on the wave functions χ n . It is this boundary condition that renders the problem non-Hermitian and PT -symmetric as well. For the complete determination of the transition amplitudes within the PT -symmetric theories, the positive definite metric operator and thus the equivalent Hermitian Hamiltonian have to be obtained. This has been done for the PT -symmetric ( -x 4 ) theory in Ref. [5]. Remarkably, the equivalent Hermitian Hamiltonian is bounded from below. This gives no doubt that the spectrum of the PT -symmetric ( -x 4 ) theory is stable. However, in higher space-time dimensions i.e. for quantum field problems, the treatment of the theory on a complex contour is hard to follow, because it is possible to have complicated Jacobian factors [7]. To avoid the existence of complicated Jacobian factors within the study of a PT -symmetric theory on a complex contour, one may seek a way to modify the recipe used in quantum mechanical PT -symmetric problems in a manner that makes it applicable for PT -symmetric field theories. The usual recipe to study PT -symmetric quantum theories mentioned above has shown that the spectrum of the PT -symmetric ( -x 4 ) is bounded from below, although the classical potential ( -x 4 ) is unstable. Therefore, the mentioned algorithm can advocate the vacuum stability for the theory in 0 + 1 space-time dimensions. Before we go on, we need to assert that for PT -symmetric quantum field theories, there exists a lack of studies in the literature that discuss the vacuum stability for unstable classical potentials like that of the PT -symmetric ( -φ 4 ) field theory. In this work, we apply an algorithm which mimics the usual complex contour method, and it avoids the problems associated with its direct extension to quantum field problems. As we will show in this work, the algorithm we use can explain the stability of the vacuum of the the PT -symmetric ( -φ 4 ) field theory for which classical analysis prohibits vacuum stability. In fact, the algorithm can be applied to any quantum field theory but we use the PT -symmetric ( -φ 4 ) theory as an illustrative example. The algorithm we follow to study a PT -symmetric field theory is in the same spirit as the known complex contour method applied to the quantum mechanical cases [1-5]. In this algorithm, we follow the canonical quantization method in which the Hamiltonian determines the dynamics of the system [9]. Therefore, the non-Hermiticity of the theory can be realized. Moreover, the canonical quantization method employs two important features; (i) the equal time canonical commutation relations and (ii) the Heisenberg picture for the operators which leads the field to verify the Heisenberg equation of motion. These features let the amplitudes obtained through this algorithm to know about the metric [10, 11]. Accordingly, the algorithm we use avoids the calculation of the metric operator, which is hard to get for the PT -symmetric ( -φ 4 ) field theory [11]. To account for the complex contour in the method we apply, we shift the field φ to ψ + B , where B is a C -number representing the vacuum condensate. The field ψ is real and has a different mass, while the condensate is to be determined from the effective potential by constraining it to satisfy the following stability conditions; where M is the renormalized mass of the field ψ . Since the renormalized mass is always chosen to be real and positive, the effective potential as a function of the condensate B is bounded from below. However, as we will see later, in this case B ought to be imaginary, and thus the contour ψ + B is complex. Hence, the resulting effective theory is non-Hermitian but PT -symmetric, which secures the reality of spectrum. The conditions in Eq.(I.1) guarantee a bounded-from-below effective potential, and also agree with the known constraints applied to the effective potential [9, 12]. In fact, the condition ∂V eff ∂B = 0 is used to kill tad pole diagrams [9], while ∂ 2 V eff ∂B 2 = M 2 represents the mass renormalization condition [12]. To give an idea about how this algorithm mimics the famous complex contour method, we mention that in quantum mechanical studies we used to have localized wave functions ( χ → 0 as x → ∞ ) associated with bounded-from-below potentials. Apparently, the conditions ∂V eff ∂B = 0 and ∂ 2 V eff ∂B 2 = M 2 define a minimum in the effective potential. Therefore, the algorithm mimics the quantization condition χ → 0 as | x | → ∞ , applied in the complex contour method. Within this regime, the field shift φ → ψ + B with B imaginary resembles the choice of a complex contour. For some theories, the spectrum is sensitive to the boundary condition χ → 0 as | x | → ∞ , and thus the theory has different spectra for different contours. In this case, in our algorithm, the conditions; lead to different B solutions, and the theory will have different vacua defined by different condensate solutions. For a quantitative test for the algorithm mentioned above in the study of PT -symmetric problems, we refer to our previous work in Ref.[6]. There, we applied the effective field algorithm for the calculations within the quantum mechanical PT -symmetric ( -x 4 ) theory. We found reasonable results for the energy spectrum and the vacuum condensate compared to exact results. Also, we obtained the relations; for the vacuum condensate B and the effective mass of the massless PT -symmetric ( -x 4 ) theory. These relations have been reproduced by Jones in Ref. [11] using the SchwingerDyson equations [20]. Such kind of interesting results support the extension of the algorithm to quantum field theories (higher dimensions) which is our aim in this work. In fact, we will tackle the point of vacuum stability of the PT -symmetric ( -φ 4 ) theory, which has not been stressed before in the literature. However, since in higher dimensions there exist UV divergences in the calculations, one has to employ known tools to cure them. For that, the algorithm we apply starts by using a normal ordered theory. To eliminate divergences at the first order in the coupling, one normal order the theory with respect to another mass parameter. This technique has been used in the context of super renormalizable quantum field theories in Refs.[13, 14]. The paper is organized as follows. In Section II, the formulation of the effective field method is introduced. The calculation of the effective potential up to g 1 and g 2 order of approximations for the PT -symmetric ( -φ 4 ) field theory in 1 + 1 space-time dimensions is presented in Section III, while the 2 + 1 case is considered in Section IV. In Section V, the discussions and conclusions are introduced.", "pages": [ 2, 3, 4 ] }, { "title": "II. FORMULATION OF EFFECTIVE FIELD METHOD", "content": "In the absence of an external source, the effective potential is equivalent to the vacuum energy E ( E = 〈 0 | H | 0 〉 ). To illustrate the implementation of the above mentioned ideas for the calculation of the effective potential of the PT -symmetric ( -φ 4 ) theory, we start by the Hamiltonian density of the form; in which N m indicates that H is a normal-ordered form with respect to the vacuum of the field φ of mass m . In introducing the field shift φ → ψ + B , the Hamiltonian density takes the form; Also, in taking into account the relation [13]; one can obtain the resulting Hamiltonian normal-ordered with respect to the new mass parameter M of the effective field ψ . To show this, we first note that this relation can lead to the following set of relations; where ∆ is the free field two point function [13]. For the kinetic term, we can get the result; where Here D is the dimension of the space-time. Considering these forms, one can rewrite the Hamiltonian density H in Eq.(II.1) in the form; In fact, ∆ and E 0 might be divergent in space-time dimensions higher than one. The divergences can be eliminated as it was done by Coleman in Ref. [13], where the propagator of mass m is subtracted from that of the mass M of the effective field. In Ref. [15], this regularization method has been used also to regularize the sunset diagram. So, we shall use this regularization method even for contributions to the effective potential beyond the normal ordering result. The effective potential, or equivalently the vacuum energy can be obtained from Eq.(II.6) where normal-ordered fields result in zero vacuum expectation values, and thus do not contribute to the effective potential. In the formula above for the effective Hamiltonian, the quantities ∆ and E o depend on the dimension of the space-time. Accordingly, we will study the 1 + 1 and 2 + 1 cases individually.", "pages": [ 5, 6 ] }, { "title": "III. THE EFFECTIVE POTENTIAL OF THE PT -SYMMETRIC ( -φ 4 ) 1+1 FIELD THEORY", "content": "The Hamiltonian form in Eq.(II.6) includes the space-time dependent terms ∆ and E o . In 1 + 1 space-time dimensions, one can expand E o (Ω) in Eq.(II.5) as a power series in /epsilon1 = D -2 to get the result; and thus, we obtain the following form; This is exactly the result obtained in Ref.[13]. The vacuum energy is then given by; with ∆ = -1 4 π ln t and t = M 2 m 2 . This result has been obtained relying on the fact that the vacuum expectation values of the normal-ordered operators in Eq. (II.6) are certainly zero, and we are left with the field-independent terms (last line in Eq.(II.6)). To cure the divergences that appear in the calculations of ∆, we subtracted the propagator with mass m from that with M as in Ref. [13]. The above result for the vacuum energy accounts for the contribution of the one vertex Feynman diagram ( diagram (a) in Fig.1) to the effective potential. In the absence of external source, the effective potential is equivalent to the vacuum energy [9], and it has to satisfy the conditions [12]; In using the parameters redefinition; b 2 = 4 πB 2 , t = M 2 m 2 , and G = g 2 πm 2 , one gets; The condition ∂E ∂B = 0 leads to the relation /negationslash where for b = 0, it results in the solution t = exp ( 1 3 Gb 2 -2 G ) . In using the relation; ∂ 2 E ( M,B,g ) ∂B 2 = M 2 , one can show that b 2 = -t G , but in this case both conditions in Eq.(III.4) are used which means that the obtained parameters ( b and t ) define the minimum of the effective potential. Equivalently, we get the result; where W is the Lambert's W function defined by W ( x ) e W ( x ) = x . Note that W ( x ) = x + O ( x 2 ), for small values of the argument x . Therefore, we obtain the result; This exponential behavior for the dependence of the vacuum condensate on the coupling has been obtained before in Ref. [8], which constitutes a good test for our calculations. To advocate the vacuum stability, one can use the the relation t = exp ( 1 3 Gb 2 -2 G ) to plot the vacuum energy in Eq.(III.5). As shown in Fig. 2, the effective potential is bounded from below, and thus the plot shows the stability of the vacuum state. This result is pretty interesting, as it is the first time to show that the vacuum of the PT -symmetric ( -φ 4 ) scalar field theory is stable in 1 + 1 space-time dimensions. A note to be mentioned is that for imaginary b , the effective Hamiltonian obtained in Eq.(II.6) is non-Hermitian, but it is PT -symmetric. Also, the Bψ 3 term turns the theory well defined on the real line [3]. One can go beyond the above result for the vacuum energy and include the radiative corrections received from the sunset ( diagram (b) in Fig.1), and the watermelon ( diagram (c) in Fig.1) diagrams. These diagrams constitute the G 2 contribution to the effective potential, which then takes the form; with β = 3 . 155 and α = 1 2 ( Ψ ( 1 3 , 1 ) -Ψ ( 2 3 , 1 )) , while Ψ ( x, n ) = d n +1 dx n +1 ln Γ ( x ) ( see the appendix for the calculation of the Feynman diagrams). In applying the condition ∂E ∂B = 0 , the coefficient of ψ is always zero, and thus the above result does not include Feynman diagrams resulting from the ψ term in the Hamiltonian in Eq.(II.6). Accordingly, the stability requirement for which one always subject E to the condition ∂E ∂b = 0 then yields the result; /negationslash For b = 0, one can solve for t to have the form; where x = 2 -Gb 2 3 G . Again, when we substitute this result in E , and for b imaginary, we get the bounded-from-below effective potential plotted in Fig. 3 .", "pages": [ 7, 8, 9 ] }, { "title": "IV. THE EFFECTIVE POTENTIAL OF THE PT -SYMMETRIC ( -φ 4 ) 2+1 FIELD THEORY", "content": "For further confirmation of the stability of the vacuum of the PT -symmetric ( -φ 4 ) scalar field theory in other space-time dimensions, we consider the 2 + 1 dimensions case. In this case, ∆ in Eq.(II.6 ) takes the form; and E 0 (Ω) in Eq.(II.5) is given by; After substituting for the values of ∆ and E 0 in Eq.(II.6), we get; or, where G = g 4 πm , t = M m , and b = B √ 4 π m . Since the most important corrections to the effective potential come from logarithmic contributions, one has to include at least the G 2 corrections to obtain a reliable contribution to the effective potential. This leads to the result; Similar to the 1 + 1 case, in applying the condition ∂E ∂b = 0, we get; and for b = 0, we have /negationslash Again, in substituting this result into the form of E , we obtain the bounded- from-below effective potential shown in Fig.4. This is correct as long as b is kept imaginary. Also, one can follow the same argument led to Eq.(III.8) for the 1 + 1 case to show that the b value at the minimum of the effective potential behaves like; Such exponential behavior has also been obtained in Ref.[8], which constitutes a good check for the accuracy of our calculations.", "pages": [ 9, 10, 11 ] }, { "title": "V. DISCUSSIONS AND CONCLUSIONS", "content": "We employed the canonical quantization method for the calculation of the effective potential for the PT -symmetric ( -φ 4 ) scalar field theory. We considered the cases of the 1 + 1 and 2 + 1 space-time dimensions individually. We have done that although in the literature the effective potential is often studied within the path integral formulation of the theory [9]. In fact, the path integral formulation by itself is obtained via the canonical quantization of the theory, for which the equal time canonical relations of the form; are satisfied. Our point in following the canonical quantization method is that, with in this regime, the Hamiltonian operator determines the dynamics of the system. Therefore, the non-Hermiticity of the Hamiltonian operator for a PT -symmetric field theory can be realized easily. So, we find it more plausible to work with operators ( canonical quantization) than working with integration over classical functionals ( path integral). Note that, in the canonical quantization of a theory, one also employs the Heisenberg picture for the operators ( see the chapters in the first part in Ref.[9]). Thus, for the theory under consideration, the Heisenberg equation of motion is satisfied. Accordingly, the calculated amplitudes know about the metric [10, 11]. This means that the followed algorithm in our work avoids the calculation of the metric operator, which is hard to get for the theory under consideration. For the PT -symmetric ( -φ 4 ) scalar field theory, the classical potential is bounded from above. Consequently, the common classical analysis predicts an unstable vacuum. In our work, we have shown that the effective potential is bounded-from-below which shows that the vacuum state of the PT -symmetric ( -φ 4 ) scalar field theory is stable. This result tells us that classical analysis are not always reliable either quantitatively or qualitatively. The stability of the theory is constrained by the existence of an imaginary condensate. The imaginary value of the condensate renders the effective theory non-Hermitian but PT -symmetric. In fact, the effective theory is well defined on the real line because of the existence of the pure imaginary, Bψ 3 , term in the Hamiltonian. To test the accuracy of our results, we obtained the vacuum condensate at the minimum of the effective potential. The behavior of the condensate has been found to approach its zero value for small coupling in an exponential manner ( Eq.(III.8)&Eq.(IV.8)). This exponential behavior has been obtained before in Ref.[8], which represents a good test for the accuracy of our results. This work sheds light on some how a new strange behavior of the quantum world. It tells us that classical analysis does not always rule the quantum behavior of a quantum particle. The situation is very similar to the tunneling effect in quantum physics for which classical analysis totally prohibits tunneling from existence, while the quantum world admits it. Likewise, the vacuum stability is totally prohibited from a classical point of view for bounded- from-above potentials, while we have shown that the potential felt by the quantum particle is bounded from below, and thus allows a stable vacuum.", "pages": [ 11, 12 ] }, { "title": "Acknowledgments", "content": "We would like to thank M. Al-Hashimi for his help in revising the manuscript.", "pages": [ 12 ] }, { "title": "A. The Sunset Diagram", "content": "The sunset diagram (diagram (b) in Fig. .1) involves the integral; where n = 3. Also, we can obtain the following result, In using the Euclidean variable q E , such that q 0 E = -iq 0 , and q i E = q i , the integral over the internal momentum q will take the form; The result of the q -integration is then; Similarly, the integration over the internal momentum w can be obtained as; Therefore, I s in Eq.(A.1) takes the form; where The integrand I xy below can be simplified as; In 1 + 1 dimensions; where Ψ ( x, m ) is the polygamma function given by; Accordingly, the diagram contribution (∆ E s ) to the vacuum energy is; where we divided by a symmetry factor of 3! × 2. In 2 + 1 dimensions, the integral I xy can also be calculated, and we get; and thus; In the above result, we used the power series expansion for the Gamma function as; where /epsilon1 = D -3, and γ is the Euler number.", "pages": [ 12, 13, 14, 15 ] }, { "title": "B. The Watermelon Diagram", "content": "For diagram (c) in Fig. 1, one can follow the same steps used in the sunset diagram above to calculate its contribution. For this case, consider the integral; After introducing the Feynman parameters and the Euclidean variables q E such that q 0 E = -iq 0 and q i E = q i , one can get the result; where n = 4, and In 1 + 1 space-time dimensions, one can calculate the integral; numerically and get the diagram contribution (∆ E w ) to the effective potential as; In 2+1 space-time dimensions, although the diagram is finite from the dimensional analysis point of view, it does have a sub divergent diagram ( diagram (b)) and one has to be careful in dealing with such diagram calculations. This diagram has been calculated in Ref. [19] but in following the same regularization technique we used before ( subtracting the diagram with mass m from that with mass M ) we get, or Note that, we used the fact that the renormalization scheme should be fixed [18], which means that M ν = m µ = t , where ν and µ are of mass units introduced to have dimensionless logarithms. . .", "pages": [ 15, 16, 18, 19 ] } ]

[ { "title": "Gauged Lifshitz model with Chern-Simons term", "content": "Gustavo Lozano a , Fidel Schaposnik b ∗ and Gianni Tallarita b a Departamento de Física, FCEyN, Pabellón 1, Ciudad Universitaria Universidad de Buenos Aires, 1428, Buenos Aires, Argentina b Departamento de Física, Universidad Nacional de La Plata Instituto de Física La Plata C.C. 67, 1900 La Plata, Argentina", "pages": [ 1 ] }, { "title": "Abstract", "content": "We present a gauged Lifshitz Lagrangian including second and forth order spatial derivatives of the scalar field and a Chern-Simons term, and study non-trivial solutions of the classical equations of motion. While the coefficient β of the forth order term should be positive in order to guarantee positivity of the energy, the coefficient α of the quadratic one need not be. We investigate the parameter domains finding significant differences in the field behaviors. Apart from the usual vortex field behavior of the ordinary relativistic Chern-Simons-Higgs model, we find in certain parameter domains oscillatory solutions reminiscent of the modulated phases of Lifshitz systems.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "In their well-honored proposal to describe dual strings [1], Nielsen and Olesen stressed the connection between the Abelian Higgs model and the Ginzburg-Landau theory of superconductivity, relating the free energy in the latter with the action for static configurations of the former. In this way, the vortex filaments of type-II superconductors were identified with string-like classical solutions in a gauge theory with spontaneous symmetry breaking. More than 30 years ago Ginzburg proposed [2] a generalization of the Ginzburg-Landau functional by including higher derivative terms, this implying an anisotropic coordinate scaling, in order to describe superdiamagnets - a class of materials with strong diamagnetism but differing from conventional superconductors. Such generalization was then used to analyze [3] the properties of superconductors near a tricritical Lifshitz point, a point in the phase diagram at which a disordered phase, a spatially homogeneous ordered phase and a spatially modulated ordered phase meet. The study of Lifshitz critical points has recently attracted much attention, not only in connection with condensed matter systems (see [4] and references therein) but also in the analysis of gravitational models in which anisotropic scaling leads to improved short-distance behavior (see [5] and references therein). A link between these two issues was established in [6] within the framework of the gauge/gravity correspondence by searching gravity duals of nonrelativistic quantum field theories with anisotropic scaling, dubbed in [8] as 'Lifshitz field theories'. The question that we address in this work is whether one can find Nielsen-Olesen like solutions when anisotropic scaling is introduced in the Abelian Higgs model through the addition of higher order spatial derivatives. As a laboratory we consider a 2 + 1 dimensional model with a complex Higgs scalar coupled to a U (1) gauge field with a Chern-Simons (CS) action [9]. The topological character of the CS term avoids the possibility of including higher order derivatives for the gauge field action (as it would be case for the Maxwell action). When higher order derivative terms in the scalar Lagrangian are absent, the Chern-Simons-Higgs model has vortex-like finite energy solutions carrying both quantized magnetic flux Φ and non trivial electric charge Q = -κ Φ with κ the CS coefficient [10]-[11]. Moreover, for an appropriate sixthorder symmetry breaking potential, first order BPS equations [12]-[14] exist, which can be easily found by analyzing the supersymmetric extension of the model [15]. Our goal will be to determine whether this kind of solutions also exists in a 'Lifshitz Abelian Higgs' model and, in the affirmative, how they depend on the parameters associated to the Lagrangian scaling anisotropy. The plan of the paper is the following: we introduce in section 2 a (2 + 1) -dimensional LifshitzHiggs model with gauge field dynamics governed by a Chern-Simons term. In order to solve the classical equations of motion we make the same ansatz leading to vortex solutions in the ordinbary (relativistic) case. Then, in section 3 we analyze the asymptotic behavior of the gauge and scalar fields resulting from the equations of motion, showing the existence of four regions according to the values of the parameters of the model. We discuss in section 4 the properties of the solutions obtained numerically i giving a summary of results and a discussion on possible extensions of our work in section 5. We briefly describe in an appendix the linearized approximation we employed to determine the asymptotic behavior of the solutions in different parameter regions.", "pages": [ 2 ] }, { "title": "2 The Lagrangian", "content": "We consider a 2 + 1 dimensional model with Chern-Simons-Higgs Lagrangian with µ = 0 , 1 , 2 and i = 1 , 2 . The metric signature is (1 , -1 , -1) . We consider space and time coordinate units so that Accordingly, γ , β and κ are dimensionless and α has length dimensions [ α ] = -2 . Concerning the dimensions of the complex scalar φ and U (1) gauge field A µ one has [ φ ] = 0 , [ A i ] = -1 , [ A 0 ] = -2 . The Lagrangian (1) is a generalization of the one considered in [12]-[13] incorporating higher (forth) order covariant derivative terms for the scalar fields. For vanishing potential and at the 'Lifshitz point' α = 0 , the Lagrangian is invariant under anisotropic scaling with 'dynamical critical exponent' z = 2 Note that the choice of a Chern-Simons term ensures that scale invariance is preserved even in the presence of gauge fields (as opposed to what would happen with a standard Maxwell term). The covariant derivative D µ acts on the scalar field φ according to with [ e ] = 0 . The potential V [ φ ] is to be specified below. Given the Lagrangian (1) one gets Gauss's law by differentiating with respect to A 0 , where Defining one then has, using eq.(5), Inserting this result in eq.(6) one gets so that the usual Chern-Simons-Higgs model relation between charge Q and magnetic flux Φ holds The energy density ¯ E associated to Lagrangian (1) is A lower bound for the energy requires β to be positive while α can have any sign. As stated before, in the β = 0 , γ = 1 relativistic case and for a sixth order symmetry breaking potential this theory is known to have, at the classical level, self-dual vortex solutions both in the Abelian case [12]-[13] and in its non-Abelian extension [14]. In order to solve the Euler-Lagrange equations deriving from Lagrangian (1) we consider the static axially symmetric ansatz with n ∈ Z . Given this ansatz the magnetic and electric fields read The equations of motion take the form The potential V is in general chosen so as to allow for spontaneous symmetry breaking. In the relativistic 2 + 1 dimensional case the most general renormalizable self-interacting scalar potential is sixth order and in fact to find first order BPS equations it should be of this order and take the form [12]-[13] with v the Higgs field vev and the coupling constant τ has length dimensions [ τ ] = -2 . In the relativistic model first order self dual equations exist at a certain value τ = τ BPS which would correspond in the present Lifshitz case to τ BPS = 8 /α 2 . From here on, and in order to compare the Lifshitz model results with those arising in the relativistic case, we shall take V as given in (19) and τ = τ BPS .", "pages": [ 2, 3, 4 ] }, { "title": "3 Asymptotic behavior", "content": "We start by discussing the conditions that we shall impose at the origin and at the boundary. We choose as conditions at the origin those leading to regular solutions in the relativistic case (see for example [12]): Note that a constant term a 0 in the A 0 ( r ) expansion is included in order to achieve consistency of eq.(16) at the origin. Coefficients a 0 and d 0 are related according to Concerning large r , we write with h ( r ) , a ( r ) and a 0 ( r ) small fluctuations. We then linearize the equations of motion which reduce to where Eqs.(26)-(27) can be written as two decoupled second order equations First we deal with the scalar field behavior. After writing with h 0 a constant, the solutions are determined from the equation The asymptotic behavior of the scalar field is then given by From the results one can see that there is a critical value for β above which q 2 ± become imaginary. In the region α > 0 and β < β crit the solutions for q ± are real. In particular, for βσ glyph[lessmuch] α 2 Note that q -coincides with the standard relativistic case solution where it plays the role of the Higgs field mass [12]. Concerning the region α ≥ 0 and β > β crit one has which gives a complex solution. This region corresponds to underdamped oscillations of the Higgs field. We can write this as where The solution is therefore where where δ is a constant phase. We now consider the case of α < 0 . In this case for β < β crit we have that which is always negative leading to oscillatory solutions with wavenumbers | q ± | . Finally let us consider the β > β crit region where the solutions become leading for the scalar field behavior to a situation similar to the case of α > 0 with β > β crit . Let us now study the asymptotic behavior of the gauge fields. For α > 0 the consistent asymptotic behavior is Notice that in this region the asymptotic field behavior ensures finite energy and quantized magnetic flux as in the relativistic case In the α = 0 case linearization leading to eqs.(30)-(31) is no longer valid. Instead, writing a = √ rg ( r ) and using the gauge field equations of motion one gets a second order nonlinear equation for g compatible with bounded solutions at infinity. As will be discussed in next section, we do find a bounded numerical solution for α = 0 . Concerning the α < 0 region, one has with The oscillatory behavior of configurations satisfying (46) will require the introduction of appropriate boundary conditions at a finite radius R .", "pages": [ 4, 5, 6 ] }, { "title": "4 Solutions", "content": "We shall present in this section numerical solutions of eqs.(18) satisfying the asymptotic condition discussed above. For definiteness we take n = 1 and we shall fix γ = 1 (Since we are considering static solutions, changing gamma amounts to a redefinition of the scalar field coupling with A 0 ). In order to ensure positivity of the energy we shall take β > 0 . We shall separately consider α ≥ 0 and α < 0 regions. Following the discussion in the previous section, we shall distinguish regions with β ≶ β crit . The numerical procedure is based on a forth-order finite differences method applied in the interval ( glyph[epsilon1], R ) with glyph[epsilon1] close to the origin and R large, in combination with the behavior of fields close to the origin given by eq. (20).", "pages": [ 7 ] }, { "title": "4.1 The α ≥ 0 region", "content": "We start by studying the the α > 0 , β < β crit region. We give the results of our numerical calculation for E and B in figures 1 and the scalar field in figure 2. One can see that the profile of the fields in this region exhibit slight deviations to the relativistic case, originated by the fourth order derivative terms. It should be noted that as β grows we found numerically that the maximum magnitude of the electric and magnetic fields decrease. Concerning the Higgs field, it reaches its vacuum value exponentially according to eq.(34), as can be seen in figure 2 with a similar profile as that corresponding to the relativistic case as shown in figure 2. Let us now consider β > β crit range where the roots q ± are complex (37), this giving rise to underdamped oscillations in the Higgs the profile, as shown in figure (3). For a given value of α the magnetic and electric field solutions for β > β c rit are qualitatively the same as those shown in figure 1 for β < β c rit . We then conclude that in the α > 0 region the electric and magnetic field behavior is very similar to the ordinary relativistic CS-Higgs model. Concerning the scalar field, as one crosses from β < β crit to β > β crit , it changes from the usual to an underdamped approach to its vacuum expectation value. We have studied the β -dependence of the energy in this region finding a linear behavior for small β . As an example, we show in figure 4 a numerical calculation of the energy E as a function of β for α = 1 , β crit = 0 . 0625 . We find that E behaves approximately as E ≈ E 0 +0 . 25 β . E We end this subsection by discussing the α = 0 case for which, for vanishing potential, the Lagrangian is invariant under anisotropic scaling with 'dynamical critical exponent' z = 2 . In this case β crit = 0 so that for any β > 0 the Higgs field shows an underdamped behavior. We have numerically confirmed this result and also found bounded solutions for the gauge fields. The field profiles are qualitatively similar to those found for α > 0 , β > β crit .", "pages": [ 7, 8, 9 ] }, { "title": "The α < 0 region", "content": "One expects in this region a clearly different behavior compared to the relativistic CS-Higgs system since the negative sign of α in the | D i φ | 2 energy term implies not only a change of sign in the |∇ φ | 2 term but also in the gauge field 'mass term' that now has the 'wrong' sign. We start by studying the β > β crit region where the fields asymptotic behavior is given by eqs. (43)-(47). This behavior leads to an oscillatory energy density (and consequently to an in general unbounded energy). For example, the third term in expression (11) for the energy density takes the asymptotic form We show in figure 5 the electric and magnetic fields in the α < 0 , β > β crit region. Their profiles show the asymptotic oscillatory damped behavior consistent with eq.(46). The behavior of the scalar field is presented in figure 5. A zoom outside the vortex core shows damped oscillations consistent with equations (34)-(43). In the β < β crit region, the roots we found in section 3, eq.(42), lead to pure oscillatory solutions with no damping. The assumption of h in eq.(22) being asymptotically a small perturbation to the scalar vacuum expectation value v is then not self-consistent. We have not been able to find stable solutions of our 2 + 1 model with the ansatz (12)-(13). We indeed know that in the absence of dynamical gauge fields this range of parameters corresponds to the modulated ordered Lifshitz phase associated to spontaneous breaking of translations [19]. We then conclude that in this region a more detailed numerical study allowing the implementation of more general ansätze would be necessary.", "pages": [ 9, 10 ] }, { "title": "5 Summary and Discussion", "content": "We have proposed a gauged Lifshitz Lagrangian with higher (forth) order spatial derivatives of the scalar field and a CS term and studied numerically non-trivial solutions of the classical equations of motion. Notice that contrary to previous analysis of Lifshitz theories with CS term [21] with z = 2 we considered higher derivatives for the scalar field rather than for the gauge fields. As a consequence, the classical solutions of our model have a different character of the ones resulting from such model [22]. Coming back to the model we analyzed, let us recall that β , the coefficient of the forth order derivatives term, was taken positive in order to ensure positivity of the energy. In contrast, the α coefficient multiplying the ordinary second order derivative term could take both positive and negative values being α = 0 the Lifshitz point at which the model exhibits z = 2 anisotropic scaling in the absence of a potential term. In order to solve the equations of motion we have made the static axially symmetric ansatz that leads to vortex solutions in the relativistic case. For α > 0 we have found solutions with magnetic and electric fields qualitatively similar to those of the ordinary relativistic model. The magnetic flux is quantized and the usual relation between electric charge and magnetic field in CS systems holds. The difference with the standard relativistic case manifests more pronouncedly in the Higgs field behavior which for β > β crit approaches its vacuum expectation value with underdamped oscillations. The critical value is given by formula (35), β crit = α 2 / 4 σ 2 , showing a dependence on the coefficient of the quadratic derivative coefficient and on the parameters of the model (the value v of the Higgs field at the minimum, the gauge coupling e , the CS coefficient κ and the Higgs field self-interaction coupling constant τ ). For α = 0 the numerical solutions that we found are qualitatively similar to those found for α > 0 , β > β crit . The situation for the α < 0 region radically changes basically because of the change in sign of the gauge field mass term. The ansatz led to pure oscillatory solutions for the gauge fields with no damping. Concerning the scalar field one can again distinguish two situations depending on wether β is larger or smaller than β crit . In the former case we were able to find solutions exhibiting electric and magnetic field profiles with an asymptotic oscillatory behavior while the Higgs field profile shows damped oscillations. This behavior leads in general to an oscillatory energy density and an unbounded energy. In the β < β crit region, the proposed ansatz led to pure oscillatory solutions with no damping. We think that in the region α < 0 other terms in the Lagrangians, as those considered by Ginzburg for the free energy of superdiamagnets and superconductors [2] might become relevant. Also, more general ansätze, not purely relying in cylindrical symmetry should be considered in order to incorporate the possibility of asymptotic breaking of translational symmetry which is characteristic of modulated Lifshitz phases. We hope to come back to this problem in a future work.", "pages": [ 11 ] }, { "title": "Appendix: The Fröbenius Method", "content": "In this section we wish to apply Fröbenius's method to the linearized Higgs field equation of motion in order to determine its behaviour close to r = 0 , where f is assumed to be small (see eq.(20)) and the equation has a regular singular point. Following ref. [20] we recast the equation of motion (17) for the Higgs field close to r = 0 in simplified form as where we take the vorticity n = 1 and ignore the contribution from A ( r ) given that this vanishes at the origin. Note that higher order terms in f coming from the potential are to be ignored in the linearized analysis. We proceed to make a Fröbenius ansatz for the behaviour close to the origin of the form Upon substituting this ansatz in eq.(49) and looking at the lowest order in r one obtains the indicial equation of the system, hence we look at the equation at order r λ -4 where we obtain Therefore we have three distinct roots λ = 3 , 1 , -1 with multiplicities 1 , 2 , 1 respectively. We proceed to determine the coefficients a m by looking at higher orders in r . The equation at order r λ -3 implies that F 1 = 0 . The order r λ -2 equation leads to which gives solutions for both roots λ = 1 and λ = 3 as where a 2 and b 2 are coefficients extracted from eq.(52) with the appropriate choice for λ , and an ill-defined solution for λ = -1 which we will return to later. The solution f 2 corresponds to the behaviour used in eq.(20) at n = 1 . Both these solutions and their derivatives are well behaved at the origin. The next order coefficients can be extracted from the order r λ equation as with higher order F m 's for odd m vanishing. Being λ = 1 a multiplicity 2 root, we know that the λ derivative of this solution is also a solution of the equations of motion. In general if f ( λ ) is a solution of the form eq.(50), then which means that an independent solution is of the form This solution has a singular derivative at the origin. The solution of the linearized problem for λ = -1 takes the form where as before the sum extends over even m and one finds that B 0 and C 0 are non-vanishing. This solution of the linearized problem diverges at r = 0 and hence should not be taken into account for searching physically acceptable solutions. Acknowledgments: This work was supported by CONICET , ANPCYT , CIC, UBA and UNLP, Argentina.", "pages": [ 11, 12 ] } ]

[ { "title": "D. L. Bennett ∗", "content": "Brookes Institute for Advanced Studies, Bøgevej 6, 2900 Hellerup, Denmark", "pages": [ 1 ] }, { "title": "L. V. Laperashvili †", "content": "The Institute of Theoretical and Experimental Physics, Bolshaya Cheremushkinskaya, 25, 117218 Moscow, Russia", "pages": [ 1 ] }, { "title": "H. B. Nielsen ‡", "content": "The Niels Bohr Institute, Blegdamsvej 17-21, DK-2100 Copenhagen, Denmark", "pages": [ 1 ] }, { "title": "A. Tureanu §", "content": "Department of Physics, University of Helsinki, P.O. Box 64, FIN-00014 Helsinki, Finland We present several theories of four-dimensional gravity in the Plebanski formulation, in which the tetrads and the connections are the independent dynamical variables. We consider the relation between different versions of gravitational theories: Einstenian, dual, 'mirror' gravities and gravity with torsion. According to Plebanski's assumption, our world, in which we live, is described by the self-dual left-handed gravity. We propose that if the Mirror World exists in Nature, then the 'mirror gravity' is the right-handed anti-self-dual gravity with broken mirror parity. Considering a special version of the Riemann-Cartan space-time, which has torsion as additional geometric property, we have shown that in the Plebanski formulation the ordinary and dual sectors of gravity, as well as the gravity with torsion, are equivalent. In this context, we have also developed a 'pure connection gravity' - a diffeomorphism-invariant gauge theory of gravity. We have calculated the partition function and the effective Lagrangian of this four-dimensional gravity and have investigated the limit of this theory at small distances.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION. PLEBANSKI'S FORMULATION OF GRAVITY", "content": "The main idea of Plebanski's formulation of the 4-dimensional theory of gravity [1] is the construction of the gravitational action from the product of two 2-forms [1-7]. These 2-forms are constructed using the connection A IJ and tetrads θ I as independent dynamical variables. We consider a Lorentzian metric. The signature of the metric tensor denoted by the pair of integers ( p, q ) is chosen as the Lorentzian signature (1 , 3). The tetrads θ I are used instead of the metric g µν . Both A IJ and θ I are 1-forms: Here the indices I, J = 0 , 1 , 2 , 3 refer to the space-time with Minkowski metric η IJ : η IJ = diag(1 , -1 , -1 , -1). This is a flat space which is tangential to the curved space with the metric g µν . The world interval is represented as ds 2 = η IJ θ I ⊗ θ J , i.e. Considering the case of the Minkowski flat space-time with the group of symmetry SO (1 , 3), we have the capital latin indices I, J, ... = 0 , 1 , 2 , 3, which are vector indices under the rotation group SO (1 , 3). In the well-known Plebanski's BF-theory of general relativity (GR) [1], the gravitational action (with zero cosmological constant) is Below we use units κ = 1 (in these units M Pl = 1). S I ∧ S I . (10) is the covariant derivative. In the Plebanski BF-theory, the gravitational action with nonzero cosmological constant Λ is presented by the integral: √ . where κ 2 = 8 πG , G is the gravitational constant, and the reduced Planck mass is M red. Pl = 1 / 8 πG red. The 'dual sector' of gravity is described by the following integral [9]: where F ∗ IJ = 1 2 /epsilon1 IJKL F KL is the dual tensor. The paper is organized as follows. In Section I we review the main idea of Plebanski to construct the 4dimensional theory of gravity considering the integrand of the gravitational action as the product of two 2-forms where B IJ and F IJ are the following 2-forms: and Here the tensor F IJ µν is the field strength of the spin connection A IJ µ : which determines the Riemann-Cartan curvature: Now the question is how many different products of two simple 2-forms can be constructed in the space-time of the 4-dimensional GR. Then all of these 4-forms can be considered as terms of the integrand of the gravitational action. We have only a few possibilities for such 4-forms: their dual counterparts: and The 2-form contains the torsion S I µν [8]: where containing only tetrads and connections which are independent dynamical variables. The BF-theories are presented for the ordinary and dual gravity in the Minkowski flat space-time. In Section II we construct the self-dual left-handed and the anti-self-dual right-handed gravitational worlds. In Section III we investigate the 'mirror gravity' existing in the 'Mirror World', which describes the states of particle physics with opposite chirality. We assume that the mirror gravity which has to interact with the states of opposite 'right-handed' chirality can be described by the anti-self-dual right-handed action of gravity. Using modern astrophysical and cosmological measurements, we consider the broken mirror parity (MP), and discuss the communications between visible and hidden worlds. Section IV is devoted to gravity with torsion. It was shown that in the self-dual Plebanski formulation of GR, gravity with torsion coincides with the ordinary and dual versions of gravity. A new type of gauge transformation in Riemann-Cartan space-time is considered: Einstein's theory of gravity described only by curvature can be rewritten as Einstein's teleparallel theory of gravity described only by torsion. In Section V the equations of motion resulting from Plebanski's gravitational action are used to construct the action containing only the connection and auxiliary fields. Einstein's equations are investigated in terms of Plebanski's theory of gravity. Section VI is devoted to the diffeomorphism invariant gauge theory of gravity, which is a new type of gravitational theory with a 'pure connection' formulation of GR. The calculation of the partition function and the effective Lagrangian of this 4-dimensional gravity is presented, as well as field equations. The limits of this theory at small and large distances are investigated. In the asymptotic limit of this theory, we have gravity in the flat (Euclidean or Minkowski) space-time and the effective gravitational coupling constant is given by the bare cosmological constant Λ: g eff = Λ / 2. At large distances we predict a more complicated theory of gravity. The perspectives of the quantum theory of Plebanski's gravity are discussed in Subsections VI A and VI B. Section VII contains a summary and conclusions.", "pages": [ 1, 2, 3 ] }, { "title": "II. THE 'LEFT-HANDED' AND 'RIGHT-HANDED' GRAVITY", "content": "The next step of the study of the Plebanski BF-theory is the consideration of the decomposition of SO (1 , 3)group of GR on left- and right-handed sectors. For any antisymmetric tensor A IJ there exists a dual tensor given by the Hodge star dual operation on the indexes I, J of flat space: This antisymmetric tensor A IJ can be split into a self-dual component A + and an anti-self-dual component A -, according to the relation: As with any Lie group, the best way to study many aspects of the Lorentz group is via its Lie algebra. Since the Lorentz group is SO (1 , 3), its Lie algebra is reducible and can be decomposed into two copies of the Lie algebra of SL (2 , R ): As it was shown explicitly in [10], this is the Minkowski space analog of the SO (4) = SU (2) left × SU (2) right decomposition in a Euclidean space. The complex Lorentz algebra splits into two complex SO (3) algebra called the self-dual (left-handed) and anti-self-dual (right-handed) components [1, 2]: Because of this split, the curvature of the self-dual components of the connection is the self-dual component of the curvature. In particle physics, a state that is invariant under one of these copies of SO (3 , C ) is said to have chirality, and is either left-handed or right-handed, according to which copy of SO (3 , C ) it is invariant under. Self-dual tensors transform non-trivially only under SO (3 , C ) left and are invariant under SO (3 , C ) right . By this reason, they are called 'left-handed' tensors. Similarly, anti-self-dual tensors, non-trivially transforming only under SO (3 , C ) right , are called 'right-handed' tensors. These self-dual and anti-self-dual tensors A ± IJ have only three independent components given by IJ = 0 i, i = 1 , 2 , 3: which transform as adjoint vector components under the corresponding SU (2)-group. Such a decomposition shows (see Refs. [2-5]) that the actions I GR and I dual GR , given by Eqs. (14) and (15), respectively, can be represented in terms of the 'left-handed' and 'right-handed' gravity: and where F ≡ F + , ¯ F ≡ F -, Σ ≡ Σ + , ¯ Σ ≡ Σ -, and the left-handed and right-handed Σ ± i are given by: The choice of the second Killing form in the cosmological term is made so that the full Plebanski action, obtained by adding the so-called simplicity constraints to (21), is equivalent (in the appropriate sector) to ordinary gravity (see also [6]). The so-called simplicity constraints are introduced in the gravitational actions by means of the Lagrange multipliers ψ ij , which are considered in theory as auxiliary fields, symmetric and traceless. Finally, the resulting actions of the Plebanski gravity are [1-7]: and where The stationarity with respect to ψ ij provides the correct number of constraints, reducing the 36 degrees of freedom of (Σ + , Σ -) to the 16 degrees of freedom of tetrads θ I µ . Now we can distinguish the two worlds - two sectors of gravity: left-handed gravity and right-handed gravity. The self-dual left-handed gravitational world can be described by the action: while the anti-self-dual right-handed gravitational world is given by the following action: If the anti-self-dual right-handed gravitational world is absent in Nature ( ¯ F = 0 and ¯ Σ = 0), then gravity is presented only by the self-dual left-handed Plebanski action (28). The main assumption of Plebanski was that our world, in which we live, is a self-dual left-handed gravitational world described by the action (28). The same self-dual formulation of general relativity (GR) was developed later by Ashtekar [11]. If there are exist in Nature two parallel worlds with opposite chiralities - Ordinary and Mirror (see below, Section III) - then we must consider the left-handed gravity in the Ordinary world and the right-handed gravity in the Mirror world. It is not difficult to show [1] that the Plebanski action (28) corresponds to the Einstein-Hilbert action of gravity. In the Minkowski space background, the Einstein-Hilbert action is: where R is a scalar curvature, and Λ 0 is Einstein's cosmological constant. Here we have (see Subsection V A): According to Eqs. (25) and (26), in the Plebanski self-dual formulation of gravity we have the following equality: Both actions are given by the formula (28).", "pages": [ 3, 4, 5 ] }, { "title": "III. MIRROR WORLD AND MIRROR GRAVITY", "content": "Previously, in Refs. [12, 13] we have assumed that there exists in Nature a Mirror World (MW) [14, 15], which is a duplication of our Ordinary World (OW), or shadow Hidden World (HW) [16, 17], parallel to our Ordinary World (OW). This MW (or HW) can explain the origin of dark matter and dark energy. Postulating the existence of the Mirror World, we confront ourselves with a question: should the mirror gravity be the anti-self-dual right-handed gravity? We assume that we have this case, and the mirror gravitational action is given by Eq. (29) describing the anti-self-dual right-handed gravity. The MW is a mirror copy of the OW and contains the same particles and types of interactions as our visible world. Lee and Yang were the first [14] to suggest such a duplication of the worlds, which restores the left-right symmetry of Nature. The term 'Mirror Matter' was introduced by Kobzarev, Okun and Pomeranchuk [15]. They suggested the 'Mirror World' as the hidden sector of our Universe, which interacts with the ordinary (visible) world only via gravity or another very weak interaction. This assumption was considered in many papers. Considering only pure gravity, we can formulate mirror parity (MP) investigating the invariance of the Plebanski gravitational action under the dual symmetry, i.e. under the interchanges: Introducing projectors on the spaces of the so-called self- and anti-self-dual tensors, we obtain: where If we represent the gravitational action (14) as we can consider the relation: The dual symmetry gives α = 1. In this case the action is invariant under the interchanges (33). Considering the left-handed gravitational action (28) in the OW and the right-handed gravitational action (29) in the MW, we have an unbroken mirror parity (MP). In this case, the bare cosmological constants in the OW and MW are identical: Let us consider now the Universe with matter fields. Assuming the existence of the mirror world, we can enlarge the Standard Model (SM) gauge group G SM = SU (3) c × SU (2) L × U (1) Y to G SM × G ' SM , where the gauge group G ' SM = SU (3) ' c × SU (2) ' R × U (1) ' Y (see Refs. [18, 19] and review [20]) is a mirror of G SM with identical gauge In the units κ = κ ' = 1, we have: The vacuum energy densities ρ ( O,M ) vac are given by a trace of the stress-energy tensor of matter T ( O,M ) µν in the O- and M-worlds. The effective vacuum energy of the Universe is the sum: Since the ordinary and mirror worlds are not identical, we have: /negationslash With the aim to explain the tiny value of the dark energy density ρ DE = Λ eff = ρ vac.eff /similarequal (2 . 3 × 10 -3 eV) 4 , verified by astronomical and cosmological observations [21], we have several possibilities. For example: I) The Universe is described by the theory G SM × G ' SM with broken mirror parity [18-20]. We can assume that If the supersymmetry breaking scale is the same in O- and M-worlds, then ρ ( SM ) vac /similarequal ρ ( SM ' ) vac and Then the condensate of gravitational fields can explain the value of ρ DE . couplings, under which the matter contents switch their chiralities. Hence, the mirror parity is restored in the Universe where the visible and mirror worlds coexist in the same space-time. Astrophysical and cosmological observations [21] have revealed the existence of dark matter ( DM ) which constitutes about 23% of the total energy density of the present Universe. This is five times larger than all the visible matter, Ω DM : Ω M /similarequal 5 : 1. In parallel to the visible world, the mirror world conserves mirror baryon number and thus protects the stability of the lightest mirror nucleon. Mirror particles have therefore been suggested as candidates for the inferred dark matter in the Universe [22] (see also Refs. [18-20, 23, 24]. This explains the right amount of dark matter, which is generated via the mirror leptogenesis [13, 25], just like the visible matter is generated via ordinary leptogenesis [26]. If the ordinary and mirror worlds are identical, then O- and M-particles should have the same cosmological densities. But this is in conflict with recent astrophysical measurements [21]. Mirror parity (MP) is not conserved, and the ordinary and mirror worlds are not identical [12, 13, 18-20] in the sense that, although the chain of breakings of the gauge groups is the same in both worlds, the energy scales at which these breakings take place are different. Superstring theory also predicts that there may exist in the Universe another form of matter - hidden 'shadow matter', which only interacts with ordinary matter via gravity or gravitational-strength interactions [17]. According to the superstring theory, the two worlds, ordinary and shadow, can be viewed as parallel branes in a higher dimensional space, where O-particles are localized on one brane and hidden particles - on another brane, and gravity propagates in the bulk. In the presence of matter, the Einstein field equations: contain the energy momentum tensor of matter T µν , and all quantum fluctuations of the matter contribute to the vacuum energy ρ vac of the Universe. The resulting cosmological constant is the effective cosmological constant which is equal to /negationslash If the mirror parity is broken also in the gravitational sector ( α = 1), then we can distinguish bare cosmological constants of the O- and M-worlds: /negationslash /negationslash II) In Refs. [12, 13] we considered the theory of the E 6 unification with different types of breaking in the visible (O) and hidden (M) worlds. We assumed that at the first stage of the Universe we had unbroken mirror parity: Λ ( O ) 0 = Λ ( M ) 0 and E 6 = E ' 6 . Finally, E 6 was broken to G SM , and E ' 6 underwent the breaking to G ' SM × SU (2) ' θ , where SU (2) ' θ is the group whose gauge fields are massless vector particles, 'thetons' [27] . These 'thetons' have a macroscopic confinement radius 1 / Λ ' θ . The estimate given by Refs.[12, 13] confirms the scale Λ ' θ ∼ 10 -3 eV. We assumed that Λ ( O ) eff = Λ ( O ) 0 + ρ ( SM ) vac /similarequal 0, and Λ ( M ) eff = Λ ( M ) 0 + ρ ( SM ' ) vac + ρ ( θ ' ) vac = 0. Assuming the same supersymmetry breaking scale in O- and M-worlds, we considered: Λ ( M ) 0 + ρ ( SM ' ) vac /similarequal 0. Then: in accordance with cosmological measurements [21].", "pages": [ 5, 6, 7 ] }, { "title": "A. Communications between Visible and Hidden Worlds", "content": "Mirror particles have not been seen so far, because the communication between visible and hidden worlds is hard. There are several fundamental ways by which the hidden world can communicate with our visible world. I) It is necessary to assume that the self-dual gravity interacts not only with visible matter, but also with mirror matter, and anti-self-dual gravity also interacts with matter and mirror matter (see Subsection VI A). It is then to be expected that a fraction of the mirror matter exists in the form of mirror galaxies, mirror stars, mirror planets etc. These objects can be detected using gravitational microlensing [28]. II) There exists the kinetic mixing between the electromagnetic field strength tensors for visible and mirror photons: The photon-mirror photon mixing induces oscillations between orthopositronium and mirror orthopositronium [29]. Orthopositronium could then turn into mirror orthopositronium and then decay into mirror photons. Besides these interactions, the hidden world can communicate with our visible world by mass mixings between visible and mirror neutrinos [30] (neutrino-mirror neutrino oscillations). Also mirror neutrons can oscillate to ordinary neutrons [31]. It is expected the interaction between visible and mirror quarks, leptons and Higgs bosons, etc. (see [13, 18, 20, 24]). The search for mirror particles at LHC is discussed in Ref. [32]. Heavy Majorana neutrinos N a are singlets of G SM and G ' SM , and they can be messengers between visible and hidden worlds [13, 25]. The dynamics of the two worlds of our Universe, visible and hidden, is governed by the following action: where L grav is the gravitational (left-handed) Lagrangian in the visible world, and L ' grav is the gravitational righthanded Lagrangian in the hidden world, L M ( L ' M ) is the matter Lagrangian in the O(M)-world, and L mix is the Lagrangian describing all mixing terms (see [18, 20]). Mixing terms give very small contributions to physical processes.", "pages": [ 7 ] }, { "title": "IV. TORSION IN PLEBANSKI'S FORMULATION OF GRAVITY", "content": "The gravitational theory with torsion can be presented in the Plebanski formalism by the following integral: Using the partial integration and putting it is not difficult to show that According to Eqs. (15) and (49), we have: This means that in the self-dual Plebanski formulation of the gravitational theory the following sectors of gravity coincide: Now it is obvious that we can exclude torsion as a separate dynamical variable. Here we see a close analogy of the geometry of the curved Riemann-Cartan space-time with torsion [8], with a structure of defects in a crystal [33-35]. A crystal can have two different types of topological defects [33]. A first type of such defects are translational defects called dislocations : a part of a single-atom layer is removed from the crystal and the remaining atoms relax to equilibrium under the elastic forces. A second type of defects is of the rotation type and called disclinations . They arise by removing an entire wedge from the crystal and re-gluing the free surfaces. The geometry of the 4-dimensional Riemann-Cartan space-time is described by the direct generalizations of the translational and rotational defect gauge fields of the 3-dimensional crystal to the tetrads θ I µ and connections A IJ µ , which play the role of the translational and rotational defect gauge fields in the 4-dimensional Riemann-Cartan space-time [33]. The field strength of A IJ µ is given by the tensor (6), which describes the space-time curvature, and the field strength of θ I µ is the torsion given by Eq. (12). Torsion is presented by dislocations, and curvature by disclinations. But these defects are not independent of each other: a dislocation is equivalent to a disclination-antidisclination pair, and a disclination presents a string of dislocations. This explains why Einstein's theory of gravity described only by curvature can be rewritten as Einstein's 'teleparallel' theory of gravity [36] described only by torsion. In summary, we wish to emphasize that if the Einstein-Hilbert Lagrangian is expressed in terms of the translational and rotational gauge fields, the tetrads θ I µ and the connection A IJ µ , then the Cartan curvature can be converted to torsion and back, totally or partially, by a new type of gauge transformation in Riemann-Cartan space-time [33]. In this general formulation, Einstein's original theory is obtained by going to the zero-torsion gauge, while the 'teleparallel' theory is obtained in the gauge in which the Cartan curvature tensor vanishes. But any intermediate choice of the field A IJ µ is also allowed.", "pages": [ 7, 8 ] }, { "title": "V. EQUATIONS OF MOTION", "content": "The equations of motion resulting from Plebanski's action of gravity given by Eq. (28) (with I ≡ I ( left GR ) ) are: Eq. (54) states that A i is the self-dual part of the spin connection compatible with the 2-forms Σ i , where D is the exterior covariant derivative with respect to A i . Eq. (55) implies that the 2-forms Σ i can be constructed from tetrad one-forms giving (24), which fixes the conformal class of the space-time metric g µν = η IJ θ I µ ⊗ θ J ν defined by tetrads. Eq. (56) states that the trace-free part of the Ricci tensor vanishes [1-3]. The 2-form fields Σ i can therefore be integrated out of Eq. (28). Thus, we are led to Plebanski's gravity given by the form: discussed in Refs. [1-7], and and where", "pages": [ 8, 9 ] }, { "title": "A. Einstein's equations", "content": "Let us consider Einstein's equations in terms of the self-dual Plebanski's theory of gravity (we assume O-world). From Eq. (56) we obtain: Here, According to Ref. [3]: Taking into account that Tr ψ ij = 0, we have: what gives: The curvature is a 2-form, and can be split in the basis of self-dual Σ i and anti-self-dual ¯ Σ i : where: From Eq. (63)) we obtain: Calculating the matrices X ij , we obtain ten equations in Plebanski's variables, equivalent to the vacuum Einstein's equations: We have the following ten equations with matter fields, equivalent to Einstein's equations (40) : Here T is the trace of the stress-energy tensor of matter T µν . However, if we have two worlds OW and MW (or HW), then we have two Einstein's equations:", "pages": [ 9 ] }, { "title": "B. Newtonian gravity", "content": "In the non-relativistic limit we have A i 0 = ∂ i Φ( x ), where Φ( x ) is given by g 00 = 1 + 2Φ( x ). Then what leads to the Newtonian gravitational potential of the particle with mass M and G N = G : This result comes from Eq. (68).", "pages": [ 10 ] }, { "title": "VI. DIFFEOMORPHISM INVARIANT GAUGE THEORY OF GRAVITY", "content": "Gravity is not a gauge theory of the usual type. The carriers of force in a usual gauge theory are spin one particles. Moreover, in the electromagnetic field theory, for example, there are two types of charged objects, negatively and positively charged particles, which interact by exchange of carriers of force. As a result, particles can either repel, or attract. In contrast, there is only one type of charge in gravity, and we have only attraction. However, scattering amplitudes for gravitons can be expressed as squares of amplitudes for gluons (see for example [37, 38]): the closed string theory describes gravity, and the open string theory is a gauge theory. The relationship is not direct, but it exists, and it is not easy to find a Lagrangian version of the correspondence. In the Plebanski-Ashtekar formalism the gravity/gauge theory relationship was developed in Refs. [39, 40]. Finally, it was realized that Plebanski's formulation of GR [1] can be integrated out to obtain a 'pure connection' formulation of GR, where the only dynamical field is an SU (2) connection [40, 41]. The result is a completely new perspective on general relativity, in which GR was reformulated as a diffeomorphism invariant gauge theory . The pure connection formulation of GR [40] was further developed. It was shown in Ref. [42] that in this case we have not a single diffeomorphism invariant gauge theory, but an infinite parameter class of them. All these theories have the same number of propagating degrees of freedom (DOF). For any theory of this type the phase space is that of an SU (2) gauge theory. However, in addition to the usual SU (2) gauge rotations, there are also diffeomorphisms acting on the phase space variables, which reduce the number of propagating DOF from 6 of the SU (2) gauge theory to 2 of GR [43]. In the present paper we try to develop the gravity/gauge theory correspondence. Using the imaginary time x 0 = it , e.g. assuming the Euclideanized self-dual Plebansky action (57), we can calculate the partition function: Here, The Hodge-star operation in the curved space-time determines the following dual tensor: and we have: The requirement of self-duality in the curved space-time is absent in gravity: /negationslash Then we obtain the following partition function: In the limit of large F 2 /similarequal 1 ( M red Pl = 1 in our units), the effective charge g eff of gravitational fields is asymptotically small and equal to: which follows from Eq. (78): minimal g eff corresponds to ψ ij = 0. In such a regime (at small distances r ∼ λ Pl ) the space-time is Euclidean. This means that in our (visible) Universe, we have respectively the flat Minkowski space-time, and we can consider the condition of self-duality: F = F ∗ . Then the asymptotic gravitational Lagrangian is: However, at large distances the theory of gravity is more complicated and the effective Lagrangian depends on F i · F ∗ i (see for example Refs. [41]).", "pages": [ 10, 11 ] }, { "title": "A. Coupling to matter", "content": "A complete theory of gravity can be constructed only if all matter fields are incorporated into the theory. Then it is necessary to construct the Lagrangian of the Universe considered in Eq. (48). In the Plebanski-Ashtekar formulation, the fundamental objects are a rule for parallel transport with a connection in the curved space. An operator which compares fields at different points is an operator of the parallel transport between the points x and y : where P is the path ordering operator, C xy is a curve from point x till point y and with σ i being the Pauli matrices. The operator: is the well-known Wilson loop. The graviton is related to a closed string. In the case of a spinor field χ ( x ) interacting with the gauge field A i µ , we have an additional gauge invariant observable: In the theory with two worlds, ordinary and mirror (or hidden), the self-dual gravity with connection A i µ interacts with the left-handed spinors χ L and χ ' L of the visible and mirror worlds, respectively, while the anti-self-dual gravity with connection ¯ A i µ interacts with the right-handed spinors χ R and χ ' R of the O- and M-worlds, respectively. Due to CP violation, the following cross-sections with ordinary quarks q and mirror quarks q ' : /negationslash are different from each other, what is essential for the baryogenesis.", "pages": [ 11 ] }, { "title": "B. Quantum gravity and renormalization problem", "content": "In the framework of quantum field theory, and using the standard techniques of perturbative calculations, one finds that gravitation is non-renormalizable. The theory of Loop Quantum Gravity (LQG) is a way of quantizing the Plebanski-Ashtekar gravity. In LQG, space is represented by a spin network, evolving over time in discrete steps [44, 45]. The phase space version [42] of the new 'pure connection' viewpoint on GR in the Plebanski formalism has led to the approach of LQG [45]. This class of theories is closed under the renormalization [41]. In Refs. [44] and [45] it was argued that it is possible to use Wilson loops as the basis for a nonperturbative quantization of gravity. Explicit (spatial) diffeomorphism invariance of the vacuum state plays an essential role in the regularization of the Wilson loop states. An explicit basis of states of quantum geometry was obtained, and the geometry was shown to be quantized - that is, the (non-gauge-invariant) quantum operators representing area and volume have a discrete spectrum. In this context, spin networks arose as a generalization of Wilson loops. Plebanski's formalism is a starting point for 'spinfoam' models (see [45] and references therein). Should LQG succeed as a quantum theory of gravity, the known matter fields will have to be incorporated into the theory. Considering the problem of renormalizability of quantum gravity, one can construct a model of multi-gravitons (see for example [46, 47]) with N massive gravitons.", "pages": [ 11, 12 ] }, { "title": "VII. SUMMARY AND CONCLUSIONS", "content": "In this paper we have explained the main idea of Plebanski [1] to construct the 4-dimensional theory of gravity described by the gravitational action with an integrand presented by a product of two 2-forms, which are constructed from the tetrads θ I and the connection A IJ considered as independent dynamical variables. Both A IJ and θ I are 1-forms. The tetrads θ I µ were used instead of the metric g µν . We considered the Minkowski space with the group of symmetry SO (1 , 3). We have reviewed the well-known Plebanski BF-theory of general relativity (GR) and constructed the gravitational actions of the different theories of pure gravity: ordinary, dual and 'mirror' ones, as well as the gravity with torsion. We have considered the self-dual left-handed gravity of the Ordinary World (OW) and the anti-self-dual right-handed gravity of the Mirror World (MW) with broken mirror parity. We have shown that in the Plebanski self-dual formulation of gravity the ordinary and dual gravitational actions coincide. We reviewed the close analogy of geometry of space-time in GR with a structure of defects in a crystal [33]. We have considered the translational defects - dislocations, and the rotational defects - disclinations, in the 4-dimensional crystals. The crystalline defects represent a special version of the curved space-time - the Riemann-Cartan space-time with torsion [8]. The world crystal is a model for Einstein's gravitation which has a new type of gauge symmetry with zero torsion as a special gauge, while a zero connection (with zero Cartan curvature) is another equivalent gauge with nonzero torsion which corresponds to the Einstein's theory of 'teleparallelism' [36]. Here we showed that in the Plebanski formulation, the phase of gravity with torsion is equivalent to the ordinary or dual gravity, and we can exclude torsion as a separate dynamical variable. We have considered the equations of motion which follow from the Plebanski action of gravity with the tetrads, self-dual connection and auxiliary fields ψ ij . The vacuum Einstein's equations were obtained in the framework of the Plebanski theory of gravity. Integrating out the tetrads we constructed the gravitational action containing only the connection and the auxiliary fields. The integration of the action over the auxiliary fields ψ ij leads to a new type of formulation of the gravitational theory with a 'pure connection'. Here the diffeomorphism invariant gauge theory of gravity is developed where the only dynamical field is an SU (2) spin connection [40, 41]. This theory is a completely new perspective on GR. We have calculated the partition function and the effective Lagrangian of this 4-dimensional gravity. We have considered the asymptotic limit of this theory: the large values F 2 ∼ M 4 Pl , which correspond to small transPlanckian distances r ∼ λ Pl , where λ Pl is the Planck length. At these small distances, the connection fields A i µ exist in the flat (Euclidean or Minkowski) space-time, and the effective gravitational coupling constant is given by the cosmological constant Λ: g eff = Λ / 2. At large distances we envisage a more complicated theory of gravity. A complete theory of gravity has to be constructed only with couplings to matter. Finally, we recalled the role of Plebanski's formalism in the theory of Loop Quantum Gravity, which is a way of quantizing the Plebanski-Ashtekar theory of gravity.", "pages": [ 12 ] }, { "title": "Acknowledgments", "content": "We are grateful to Masud Chaichian for useful discussions. The support of the Academy of Finland under the Projects No. 136539 and No.140886 is acknowledged.", "pages": [ 13 ] } ]

[ { "title": "ABSTRACT", "content": "International Journal of Modern Physics A c © World Scientific Publishing Company", "pages": [ 1 ] }, { "title": "Toward a Supergravity Spectral Action", "content": "J. L. L'opez a ∗ , O. Obreg'on a † , M. P. Ryan b ‡ and M. Sabido a § a Departamento de F'ısica, DCI-Campus Le'on, Universidad de Guanajuato, A.P. E-143, C.P. 37150, Guanajuato, M'exico. b Instituto de Ciencias Nucleares Universidad Nacional Aut'onoma de M'exico, A.P. 70-543, M'exico D.F. 04510,M'exico. Present Address: 30101 Clipper Lane, Millington MD 21651 USA. A spectral action for a generalized bosonic sector corresponding to the Dirac operator of Euclidean supergravity is proposed. We calculate, up to a 4 , the Seeley-Dewitt coefficients in the expansion of the spectral action. It is in general not known how to construct a 'matter fermionic' supersymmetric partner to the spectral action. The action we propose provides the effective action to be completed to get, at any order of the expansion, the corresponding supersymmetric action. Keywords : Supergravity; Noncommutative geometry. PACS numbers:04.65.+e,02.40.Gh,11.10.Nx", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The equivalence principle and gauge invariance are the fundamental pillars of the two most successful theories in physics, general relativity and Yang-Mills theory. They lead to a greater understanding of the basic interactions in our Universe. However, these theories seem to be incompatible at the quantum level. This incompatibility might suggest that they are theories arising from some other more fundamental principle. One of the most interesting proposals in the literature is the spectral action of noncommutative geometry. It involves a spectral geometry consistent with the physical measurements of distances. The usual emphasis on the points x ∈ M on a geometric space is replaced by the spectrum Σ of the Dirac operator D and it is assumed that the spectral action depends only on Σ. This is the spectral action principle. The spectrum is a geometric invariant that replaces diffeomorphism invariance. By applying this basic principle to the noncommutative geometry defined by the standard model, it has been shown 1 that the dynamics of all interactions, including gravity, are given by the spectral action. Its heat kernel", "pages": [ 1 ] }, { "title": "2 J. L.L'opez, O. Obreg'on and M. Sabido", "content": "expansion in terms of the Seeley-DeWitt coefficients a n gives an effective action up to the coefficient considered. Assuming the Riemannian spin connection for the gravitational sector of the spectral action, the first three terms in the expansion correspond to a constant, the usual Einstein-Hilbert action plus Weyl gravity and a Gauss-Bonnet topological invariant. As is well known, if one would apply a Palatini formalism to this effective action one would get a non-Riemannian connection and this result will also be different at any desired order in the expansion, namely for each effective action. The Dirac operator is an essential element in the physical action of noncommutative geometry. It encodes, together with representation of the algebra of coordinates, both geometry and physics. It was stated in Ref. 2 that in the case of gravity one can consider the eigenvalues of the Dirac operator as observables if they satisfy certain constraints that restrict the phase space and the structure of the space-time manifold. The same type of analysis was later performed for Euclidean supergravity 3, 4 where also the eigenvalues of the appropriate Dirac operator can be understood as observables that must satisfy a set of generalized constraints. We were motivated by this last result and in this work we calculate the first three terms of the spectral action restricting our calculations to the Dirac operator of simple Euclidean supergravity. As is well known, the spin connection of this operator corresponds to the standard Riemannian connection, by means of which the gravitational sector of the spectral action is usually constructed, plus a contortion term due to the presence of the Rarita-Schwinger field. First, in Sec. 2 we review the well known 1 spectral action construction related to pure gravity based on the standard Riemannian connection. Then in Sec. 3 we present the calculation of the first three Seeley-DeWitt coefficients of the heat kernel expansion based on the trace of the square of the supergravity Dirac operator. This procedure, however, will not provide in general a supersymmetric effective action at any desired order in the expansion. 18 As is well known, a supersymmetric 'matter fermionic' partner to the spectral action is in general not known. We are providing the procedure to obtain the bosonic part of the spectral action, here we calculate it up to the a 4 coefficient. We will get an 'effective generalized bosonic action'. Based on our proposal at certain order in the expansion one could, in principle, supersymmetrize the effective action of interest to get the appropriate 'matter fermionic' terms. This is a complicated task and, not knowing a general 'matter fermionic' action, the calculation must be performed at each desired order in the expansion, namely for each effective action independently. We will not attempt to complete the supersymmetry, up to the a 4 Seeley-DeWitt coefficient, corresponding to the 'generalized spectral bosonic action' we propose. This is a specific particular task for each order in the expansion. It will be shown that the effective generalized bosonic action has the curvature term in the first order formalism and we know, at this level in the expansion, that we should add the Rarita-Schwinger action to get N = 1 supergravity. At any desired order in the expansion in the Seeley-De Witt coefficients, one can construct generalized bosonic actions based on the Dirac operator of N = 1 supergravity and the effective actions one is able to construct are the ones one should supersymmetrize, for each particular case, to get an effective supergravity spectral action. Then, we present the 'bosonic action' up to the a 4 term. Finally, Sec. 4 is devoted to conclusions and outlook.", "pages": [ 2, 3 ] }, { "title": "2. Spectral action of pure gravity", "content": "Instead of the well known geometry of space-time, the basic data of noncommutative geometry consists of an involutive algebra A of operators in a Hilbert space H , which plays the role of the algebra of coordinates, a self-adjoint operator of Dirac type D in H which plays the role of the inverse line element. A fundamental principle in the noncommutative approach is that the usual emphasis on points in space-time is replaced by the spectrum of the operator D . An operator is of Dirac type if its square is of Laplace type. Locally such operators can be expressed as for a unique endomorphism E acting on vector bundles of M . The spectral triple ( A , H , D ) encodes the geometry of every noncommutative space. A Riemannian spin manifod M is completely characterized by the algebra of smooth functions on M , A = C ∞ ( M ), the Hilbert space of square integrable spinors, H = L 2 ( M , S ) and the Dirac operator D of the Levi-Civita spin connection. On Riemannian manifolds D is an elliptic operator. The selfadjointness and ellipticity of D is essential for the construction of ( A , H , D ). The spectral action principle states that the physical action depends only on the spectrum of the Dirac operator. These ideas were the origin of the spectral action given in Ref. 1. The bosonic part of the spectral action is For a specific choice of the cutoff function f , the spectral action (2) is expressed up to the first three terms of its asymptotic expansion 5 where f 4 = ∫ ∞ 0 f ( u ) u 3 du , f 2 = ∫ ∞ 0 f ( u ) udu , f 0 = f (0), and the a n are the SeeleyDewitt coefficients of the heat kernel expansion of D . Every a n is function of geometric invariants of order n constructed from E , the field strength Ω µν and the", "pages": [ 3 ] }, { "title": "4 J. L.L'opez, O. Obreg'on and M. Sabido", "content": "Riemann tensor. 6 The relevant a n 's are For the gravitational Dirac operator D we have where ˜ ω µ is the spin connection on M , ˜ ω µ = 1 4 ˜ ω ab µ γ ab with ˜ ω µ related to e a µ by the vanishing of the covariant derivative ∇ µ e a ν = 0, this allows us to express the ˜ ω ab µ as functions of the tetrads as in standard Einstein tetradic gravity, applying the Riemannian torsion free condition. The coefficients (4) take the form where C µνρσ is the Weyl tensor of conformal gravity C µνρσ C µνρσ = R µνρσ R µνρσ -2 R µν R µν + 1 3 R 2 and the Euler characteristic χ E is given by, χ E = 1 32 π ∫ d 4 x √ gR ∗ R ∗ with R ∗ R ∗ = R µνρσ R µνρσ -4 R µν R µν + R 2 , the Gauss-Bonnet topological invariant. For this particular Dirac operator the spectral action is where α , β and γ are constants. The spectral action gives the Hilbert-Einstein action with corrections. The action above is of particular interest, because this same expression has been considered as a good candidate for a renormalizable and ghost free theory of gravity. 7, 8 In the next section we consider the Dirac operator of N = 1 Euclidean supergravity and calculate the Seeley-Dewitt coefficients associated with its spectral action. The calculation of this action gives a 'generalized effective bosonic action' being the natural supersymmetric extension of the spectral action for gravity, not including the appropriate matter fermionic terms that we would need to complete the supersymmetry. A procedure that one should perform at any desired order in the expansion due to the fact that we do not know a general expression for the 'matter fermionic' supersymmetric action corresponding to the spectral action. 18 As mentioned in the introduction, it has already been shown that under certain conditions 3, 4 the eigenvalues of the supergravity Dirac operator we use can be considered as observables of Euclidean supergravity.", "pages": [ 4, 5 ] }, { "title": "3. Spectral action and Euclidean Supergravity", "content": "Let M be a compact Riemannian spin manifold without boundary in four dimensions with metric g µν = e a µ e νa , the tetrad fields are labeled with greek space-time and latin internal indices respectively. The Dirac operator D given by the spin connection in (5), is an elliptic operator on M and is formally selfadjoint on H . Because M is compact, D admits a discrete spectrum of real eigenvalues and a complete set of eigenspinors D ψ n = λ n ψ n . The λ n 's define a discrete family of real valued functions on the phase space of smooth tetrad fields and as it was shown in Ref. 2, these eigenvalues are invariant under diffeomorphisms of M and under rotations of the tetrad fields, so they form a set of observables for general relativity. These ideas were extended in Ref. 3, 4 to achieve the geometric construction of Euclidean supergravity. This involving a supersymmetric partner of the graviton, the gravitino, and also imposing local supersymmetric invariance. The gravitino is represented by a Euclidean spinor vector ψ a µ defined by a Majorana condition, ¯ ψ = ψ T C . The phase space will be the space of all pairs ( e, ψ ) that are solution of the equations of motion modulo gauge transformations, which are the ones needed in the non supersymmetric case plus the transformations of local N =1 supersymmetry. The supersymmetric Dirac operator D SG is given by The difference between D SG and D is the additional ψ dependent term, 9, 10 K µab = -1 4 ( ¯ ψ µ γ b ψ a -¯ ψ a γ µ ψ b + ¯ ψ b γ a ψ µ ). D SG is an elliptic operator defined on the full spin bundle and it is possible to define an inner product such that D SG is formally selfadjoint. 3 Here we calculate the Seeley DeWitt coefficients of the square of D SG . The square of this Dirac operator can be expressed in the form of (1) with 6 J. L.L'opez, O. Obreg'on and M. Sabido where R is the curvature scalar of standard general relativity. The field strength is By this means, and in a similar procedure as in pure gravity, we get the spectral action related to this particular Dirac operator and the constrained geometry defined by it. The first non constant term is The term a 4 is a combination of terms quadratic in ψ and non trivial interactions between the graviton and the gravitino The cuadratic terms appearing in (12) are given by", "pages": [ 5, 6 ] }, { "title": "Toward a Supergravity Spectral Action 7", "content": "We recognize in the a 2 term the 'bosonic' part of the N =1 supergravity action written as a second-order formalism. Now, as is well known, we can write it in terms of the curvature scalar that is function of the spin connection including the torsion term Having modified the Dirac operator in a consistent way, summing the torsion term, the spectral action includes the geometric part of the N =1supergravity action. The full supersymmetric action is not known. There is not a general 'matter fermionic' supersymmetric partner of (2), in particular not one constructed with the Dirac operator (8). It is however possible, as we show, to provide at any desired order in the expansion of the action in the Seeley-DeWitt coefficients a 'generalized effective bosonic action' and at the order of interest one can, in principle based in our result, supersymmetrize that effective particular bosonic action. The bosonic action (up to", "pages": [ 7 ] }, { "title": "8 J. L.L'opez, O. Obreg'on and M. Sabido", "content": "a 4 ) is then a higher order theory represented by the spectral action with α , β , and γ constants. It is of interest to notice that the spectral action (2) for D = D 2 is a theory of gravitation. It includes in a natural way the Einstein-Hilbert action. A general higher order theory of gravity suffers from ghosts, a scalar and a spin-2 mode, both massive. However, the particular theory of gravitation that emerges in the non commutative geometry framework, up to the a 4 term, is of the type this form of the theory, without a cosmological constant, was considered first in Ref. 11 and it was argued that it is renormalizable. It was then reconsidered in Ref. 7 including Λ because in this case the scalar mode is absent and for a special value of α in terms of Λ, the massive spin-2 mode also disappears, leaving a theory consisting only of a massless graviton and is possibly renormalizable. The spectral action we propose (17) is the 'bosonic' part of the supergravity spectral action, that one should construct in a supersymmetric procedure at each order in the expansion in the Seeley-De Witt coefficients. It will correspond to the gravity action (18) in Ref. 8. The action (17) gives, up to a 2 , the usual simple supergravity by adding the Rarita-Schwinger action 〈 Ψ , D SG Ψ 〉 . The supersymmetric action up to a 4 could be written in the form of an Einstein-Weyl supergravity without the scalar and vector auxiliary fields and it could also be studied in relation to its renormalizability. On the other hand, the Seeley-Dewitt coefficients have been calculated, in particular, when totally antisymmetric torsion is present and by these means the associated spectral action. 12-17 The motivations of doing so are based on some physical arguments, for instance the coincidence of geodesics in both manifolds, one in which there is torsion and another where it is absent. Torsion in supergravity is of a more general kind and there is no reason, physical or mathematical, for not considering its associated Dirac operator as we have done.", "pages": [ 8 ] }, { "title": "4. Discussions", "content": "The spectral action allows us to construct a modified theory of gravity as well as the 'bosonic' part of a generalized supergravity at the desired order in the SeeleyDeWitt coefficients. The fact that makes this possible is that simple supergravity and pure gravity can both be viewed as theories with well defined, mathematically consistent, Dirac operators. This is not, in general, the case for extended supergravities. We were partially motivated by the result that the eigenvalues of the supergravity Dirac operator can, with certain constraints, be considered as observables. 3, 4 We were able to calculate the expansion up to a 4 of the spectral action (2) for the supersymmetric Dirac operator (8), and we have constructed a generalized bosonic effective action that provides at each order in the expansion of the SeeleyDeWitt coefficients an effective action that is the appropriate to be, in principle, supersymmetrized to get an effective supergravity action. The gravity action (7,18) has been constructed with the pure gravity spin connection (5). This gravity action was already known 8, 11 and may be considered to possibly be renormalizable. The corresponding supergravity action would be given by the supersymmetric completion of our action (17). It is a matter of further work to search for the calculation of this effective supergravity theory and look for its possible renormalizability.", "pages": [ 8, 9 ] } ]

[ { "title": "Hidden Supersymmetry in Dirac Fermion Quasinormal Modes of Black Holes", "content": "V. K. Oikonomou ∗ Max Planck Institute for Mathematics in the Sciences Inselstrasse 22, 04103 Leipzig, Germany November 15, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "We connect the quasinormal modes corresponding to Dirac fermions in various curved spacetime backgrounds to an N = 2 supersymmetric quantum mechanics algebra, which can be constructed from the radial part of the fermionic solutions of the Dirac equation. In the massless fermion case, the quasinormal modes are in bijective correspondence with the zero modes of the fermionic system and this results to unbroken supersymmetry. The massive case is more complicated, but as we demonstrate, supersymmetry remains unbroken even in this case.", "pages": [ 1 ] }, { "title": "Introduction", "content": "Black holes in equilibrium are generally speaking, simple objects due to the very few parameters that are needed to describe them [1-4]. However, it is physically impossible to have an isolated black hole in nature, due to the fact that matter always exists around them, interacting directly with the black hole. This leads us to the conclusion that a black hole is always in a some sort of perturbed state, with more parameters needed to describe it, than in the unperturbed state. Quasinormal modes [1-33] describe a long period of damped proper oscillations of gravitational waves, and are extremely important in various physical phenomena. The direct observation of black holes is actually based on quasinormal modes, with the most dominating one being the fundamental mode, namely the lowest frequency in the spectrum. The quasinormal modes are the characteristic sound of black holes and being such, matter field perturbations of such extreme gravitational backgrounds can reveal many important properties of black holes. The perturbation of a black hole can be achieved either by directly perturbing the gravitational background or by simply adding matter or gauge fields in the black hole spacetime [4]. In this paper we shall shall use the latter approach, in the linear approximation, which suggests that the field has no back-reaction on the metric. Particularly, we shall study Dirac fermion systems around various black hole and spacetime environments and study when the system possesses an unbroken hidden supersymmetry. Supersymmetry has been connected to quasinormal modes spectra in the past, but in a different context [34-38]. Most of these works studied bosonic quasinormal modes and their relation to supersymmetry. We study the zero modes of the fermionic system and directly relate these to the quasinormal modes. As we shall see, the zero modes and quasinormal modes have a bijective correspondence, a fact that can actually be very crucial for supersymmetry to be unbroken. The specific type of supersymmetry that we found is an N = 2 supersymmetric quantum mechanics [39-42, 44-47, 49-52] (shortened to SUSY QM hereafter) with zero central charge. The aforementioned supersymmetry is inherent to many gravitational systems [53,54]. In this paper the focus is on supersymmetries in gravitational backgrounds. We exploit the existence of the quasinormal modes in this backgrounds in order to establish the fact that the Witten index of the corresponding underlying supersymmetric algebra is zero, with the kernels of the corresponding operators being non-empty and consequently, supersymmetry is unbroken. This paper is organized as follows. In section 1 we study the supersymmetric underlying structure for the case of a Kerr black hole, with the latter being the most physically interesting black hole satisfying the Einstein's equations. In section 2, we study the massless and massive Dirac fermion in Kerr-Newman, Reissner-Nordstrom, Schwarzschild gravitational backgrounds. The massive case proves to be much more complicated compared to the massless case, but still, supersymmetry remains unbroken. In section 3, the focus is on three different spacetimes, namely, the D-dimensional de Sitter, Kerr-Newman-de Sitter and Reissner-Nordstrom-anti-de Sitter spacetimes, finding the same results as in the previous sections. In reference to the Reissner-Nordstrom-anti-de Sitter spacetime, we find that this fermionic system has two N = 2 SUSY QM algebras. In section 4, we present some physical and mathematical implications of the SUSY QM on the fermionic system and also to the fibre bundle structure of the spacetime. In addition, we study the impact of compact radial perturbations to the Witten index of the SUSY QM algebra, in the case the spacetime is maximally symmetric. Moreover, we address the question if there can be a higher extended supersymmetry underlying these N = 2 supersymmetries we found. The conclusions follow in section 5.", "pages": [ 1, 2 ] }, { "title": "1 Massless Fermion Quasinormal modes for a Kerr Black Hole", "content": "We shall study first the quasinormal modes of a Dirac fermion in a Kerr black hole background. Particularly, we focus on the massless fermion case. As we mentioned earlier, the Kerr black hole is one of the most important four dimensional black hole solutions, since realistic astrophysical black holes are rotating with negligible electric charge and in addition, the quasinormal modes stemming from such a background are very important for observations of gravitational waves [55-60] (and more likely gravity waves will come from such objects). Following references [25-32,61] and employing the Newman-Penrose formalism, the mass- less Dirac equation in the null tetrad basis reads: with the covariant derivative being equal to [62-65], The spin connection ω a µ b on the pseudo-Riemannian manifold, satisfies the following equation: The four dimensional Kerr background spacetime has the following metric: with, the parameters ρ 2 , ∆, a appearing above, being equal to: For later purposes, we introduce the parameters ¯ ρ, ¯ ρ ∗ to be: such that ρ 2 = ¯ ρ ¯ ρ ∗ . In addition, we define the following operators, which shall frequently be used in this section: with K = ( r 2 + a 2 ) ω + am and Q = aω sin θ + m sin θ , and also, Then, following [25-32] the Dirac equation (1) in the Kerr spacetime can be cast in the form: where D 1 2 and L 1 2 , can be deduced from Eq.(8). In order to make contact with the quasinormal modes spectrum, we separate the above functions into radial and angular parts, as follows: By substituting Eq.(10), to Eq.(9), we obtain the angular part of the Dirac equation: and the radial part of the Dirac equation: with R ' -1 2 ( r ) = √ 2 √ ∆ R -1 2 ( r ). Equations (11) and (12) correspond to the angular and radial part of the Dirac equation respectively. Eliminating S 1 2 or S -1 2 from the above equation, and upon defining u = cos θ , we can find that the angular equation can be written as (with s = ± 1 2 ): with A lm = λ 2 +2 maa -( aω ) 2 . The parameter λ is real, but our results are indifferent to whether λ is real or not. This would change slightly our notation, but the results would be the same. Equations (11) and (12), will be our starting point of our analysis. We shall see that the solutions of these equations are related to a supersymmetric Hilbert space. The above equation (12) (using the tortoise coordinate x , defined as d r d x = ∆ ω K ), can be transformed to the following Schroedinger like equation [25-32]: with Z ± = P 1 2 ± P -1 2 and V ± ( x ) = λ 2 ∆ ¯ K 2 ± λ d d x ( √ ∆ ¯ K ) . Equation (14) is the quasinormal modes master equation for a fermion field in Kerr spacetime. We are not interested in solving the master equation, there are quite rigorous techniques for doing that [132]. Our main interest is whether the spectrum in terms of ω is discrete or continuous. Quasinormal modes are solutions of the above equation, with the wave functions satisfying certain boundary conditions at the horizon and at infinity. In addition, the quasinormal modes corresponding to Kerr black holes form a countable set of discrete frequencies. The boundary conditions are very crucial in order to define a trace class operator, which be valuable to us in the following. Based on equations (11) and (12), we can construct an N = 2 supersymmetric quantum algebra acting on the fermionic solutions. We define the matrix D K , acting on the vector: Replacing the operators D 0 from equation (8), D K is equal to: We can easily obtain D † K which is equal to (note that K contains ω which is complex): acting on, Obviously, each quasinormal mode satisfies the zero mode equation of D K . Therefore, we could say that the quasinormal modes (which can be found from equation (14)) are in bijective correspondence to the zero modes of the operator D K . The same argument applies for the operator D † K , with the difference that in this case the quasinormal modes are the complex conjugates of the previous case. Making use of the operators D K and D † K we can form an N = 2 SUSY QM algebra. The supercharges of this algebra, Q K and Q † K are defined in terms of D K and D † K , Additionally, the quantum Hamiltonian can be cast in following diagonal form, These operators, corresponding to the the radial part of fermionic black hole system, are elements of an unbroken N = 2 SUSY QM algebra, as we now demonstrate. The operators (20) and (21), satisfy the d = 1 SUSY algebra: The Hilbert space of the supersymmetric quantum mechanical system, which we denote H , is a Z 2 graded vector space, with the grading provided by the operator W , the socalled Witten parity. The latter is an involution operator that commutes with the total Hamiltonian, and also, anti-commutes with the supercharges, In addition, the operator W being a projection operator, satisfies, The Witten parity W , spans the total Hilbert space into equivalent Z 2 subspaces. Therefore, the total Hilbert space of the quantum system is written: with the vectors corresponding to the two subspaces H ± , classified to even and odd parity states, according to their Witten parity: In addition, the corresponding Hamiltonians of the Z 2 graded spaces are: In the present case, the operator W , can be represented in the following form: In equation (27) the operator P , is defined in such a way, so that the eigenstates of P ± , which are, | ψ ± 〉 , satisfy: We call them positive and negative parity eigenstates. Using the representation (29) for the Witten parity operator, the parity eigenstates are represented by, with | ψ ± 〉 /epsilon1 H ± . Turning back to the fermionic system at hand, we write the fermionic states of the system (16) and (19) in terms of the SUSY QM algebra, that is: Hence, the corresponding even and odd parity SUSY quantum states | ψ + 〉 and | ψ -〉 , are written in terms of | φ -K 〉 and | φ + K 〉 : When Fredholm operators are used, supersymmetry is considered unbroken if the Witten index is a non-zero integer. In this paper we shall not make use of Fredholm operators. Therefore, we shall need a generalization of the Fredholm index (and of the corresponding Witten index). The heat-kernel regularized index, both for the operator A , that is ind t A and for the Witten index, ∆ t , is defined as follows [40,41]: In the above, t > 0, and additionally the tr ± stands for the trace in the subspaces H ± . The heat-kernel regularized index is defined for operators that are trace class (in our case, the operator tr( -W e -tA † A ) must be trace class), that is, they have a finite trace norm. This is independent of the orthonormal basis describing the Hilbert space. From the Banach space of all trace class operators, we shall be interested in the subspace spanned by the A and A † and their product AA † . We now turn our focus on the regularized Witten index corresponding to the case at hand. The equations of the quasinormal modes, D K | φ -〉 = 0 and it's conjugate D † K | φ + 〉 = 0 have complex conjugate solutions. Obviously we have a bijective correspondence between the quasinormal modes, given by equation D K | φ -〉 = 0, and their complex conjugate counterparts, given by D † K | φ + 〉 = 0. It is obvious that this bijective correspondence holds between the zero modes of the matrices D K and D † K . Therefore, /negationslash /negationslash which in turn implies ker D K D † K = ker D K D † K = 0. As a consequence, the following relation holds for the operators e -t D † K D K and e -t D K D † K Recall that tr ± stands for the trace in the subspaces H ± . As a consequence of relation (36), the regularized index of the operator D K is equal to zero. Consequently the regularized Witten index is also zero. Hence, since the relation (35) holds true, the fermionic system possesses an unbroken N = 2 SUSY QM algebra. Using the notation of relation (33) the equation D K | φ -〉 = 0 has a direct representative equation for the supercharge, namely, Q | ψ -0 〉 = 0. This implies that the zero mode eigenstate | ψ -0 〉 is a negative Witten parity eigenstate, and is equal to: In the same vain, the positive Witten parity vacuum eigenstate is, The angular case can be treated accordingly, following the same line of argument, as in the radial part of the Dirac equation. Adopting the notation of the previous paragraphs, the algebra can be built on the matrices D R θ and D † K θ . The matrix D K θ is defined to be: acting on which is a direct consequence of equation (11). It's conjugate equals to: Recall that the operators L 1 2 and L † 1 2 contain ω , which is a complex number. We must note that the situation at hand is much more complicated in comparison to the angular case. It is obvious that, since quasinormal modes exist, the operator D K θ certainly has a set of discrete zero modes, that belong to a countable set of complex numbers. However, we cannot argue that the same holds for the operator D † K θ . In the radial case, we could solve the zero mode problem of the two corresponding operators simultaneously, since the zero modes of the operators were complex conjugate (a proof for existence of solutions in a much more general setup see [66]). But in the angular case, we cannot use the same argument. Therefore, we can argue that in general, the number of zero modes of the two matrices are not equal. Hence, we could naively argue that supersymmetry is unbroken in this case, but for different reasons in comparison to the radial case. This naive argument is based on the fact that the operators have not the same number of zero modes, and therefore, supersymmetry is unbroken. This however would be true only in the case the operators were Fredholm, which are not (since dim kerD k θ →∞ ). Moreover, we cannot be sure whether the operator D † K θ is trace-class. Therefore, we conclude that only the radial part of a Dirac fermionic system in the Kerr black hole background can be associated to a N = 2 SUSY QM algebra. Before closing this section, we must note that up to date, the important theoretical issue that addresses the nature of neutrino, that is whether it is Dirac or Majorana, has not be answered successfully yet. Hence, any information on the effect of massless (if the neutrino can be considered massless) fermions in nature is invaluable. We studied a massless Dirac fermion in the most realistic curved four dimensional gravitational background and found an underlying supersymmetry. This supersymmetry is unbroken, a fact that is guaranteed by the existence of quasinormal modes of the fermion in the same background. It would certainly be interesting to study if supersymmetric structures exist when massive Dirac fermions and also when Majorana fermions are considered. Of course this would require a complete study of the quasinormal modes of massive Dirac fermions and of Majorana fermions in Kerr backgrounds.", "pages": [ 2, 3, 4, 5, 6, 7, 8, 9 ] }, { "title": "2.1 The Kerr-Newman Black Hole", "content": "In this section we further explore whether supersymmetric structures underlie any other fermion systems in curved gravitational backgrounds. We shall study first the KerrNewman black hole, which is the only asymptotically flat solution of the Einstein equations, with electrifying vacuum. We consider a massive fermion of Dirac type in such a black hole background. Following the line of research of the previous section and adopting the notation of reference [26], the fermionic equations of motion can be recast as, in reference to the angular part. The radial part can be written as: with R ' 1 2 = √ ∆ R + 1 2 and ∆ = r 2 + a 2 -2 Mr + Q 2 . The operators that appear in relations (42) and (43) are equal to: with K = ( r 2 + a 2 ) ω + am . Note that ' λ ' is the same as in the Kerr case, since the angular equation can be reduced to Eq. (13), corresponding to the Kerr case [26]. In the following paragraphs, we shall study both the massless and massive fermion case. As we shall see, the mass can introduce some complications to our initial arguments, but the final result is the same as in the massless case.", "pages": [ 9, 10 ] }, { "title": "2.1.1 Massless Fermion case", "content": "Consider the radial part of a massless Dirac fermion first. As in the previous section, from the equations of motion (43), we can construct the matrix D R , on which the supersymmetric quantum algebra can be built on. This matrix is defined to be: acting on the vector: Using the explicit form of the operators defined in equation (44), the operator D R equals to: We can easily obtain it's adjoint, D † R , which is equal to: acting on, The number of the zero modes of the operator D † R are bijectively related to the number of the zero modes of D R . This is because, the set of the zero modes of D R are in one-to-one correspondence to the quasinormal modes, corresponding to the equation (43). In the same vain, the zero modes of D † R are in one-to-one correspondence to the quasinormal modes, corresponding to complex conjugate of the equation (43). It is necessary to note that, in order to obtain consistent solutions for the complex conjugate of equation (43), the wave functions must obey the complex conjugate boundary conditions of the wave functions that correspond to equation (43). The existence of a solution for this case is obvious, but can be further justified by a theorem on second order differential equations [66]. We are not interested in the specific form of the solutions, but only in the fact that the zero modes of the operators D † R and D R have a bijective correspondence. The situation at hand is very similar to the massless Kerr fermion case of the previous section. Having found a correspondence between the zero modes, it is easy to verify that supersymmetry is unbroken, with the heat-kernel regularized Witten index being equal to zero. Let us see this in detail. The SUSY QM algebra can be built on the supercharges, and also the corresponding Hamiltonian, These satisfy {Q R , Q R † } = H R , Q 2 R = 0, Q † R 2 = 0 and [ W , H R ] = 0. Hence, an N = 2 SUSY QM algebra underlies the radial part of fermionic Kerr-Newman black hole system. In addition, as a result of the bijective correspondence of the two matrices zero modes, we have, /negationslash /negationslash which in turn implies, ker D R D † R = ker D R D † R = 0. As a consequence, the following relation holds for the operators e -t D † R D R and e -t D R D † R The above result holds only in the case the operator tr( -W e D R D † R ) is trace class, which is true, since the operators D R , D † R and consequently the operators D R D † R , D † R D R are trace class [40]. Hence, the regularized Witten index is zero, and this fact, in conjunction with equation (52), implies that the radial part of the fermionic system has unbroken supersymmetry. We omit the study of the angular part of the fermionic system since, as in the Kerr case, supersymmetry is broken, for the same reasons.", "pages": [ 10, 11 ] }, { "title": "2.1.2 A brief Discussion", "content": "We mentioned in the above that in order relation (53) holds true, the operator tr( -W e D R D † R ) must be trace class. We argued that this is true since the operators D R D † R and D † R D R are trace class. However, this is not prerequisite for our case, and the only requirement for tr( -W e D R D † R ) to be trace class is that D R D † R -D † R D R is trace class. Indeed, by virtue of a theorem (see for example [40], page 161, comments before Theorem 5.20 and also Theorem 5.22), if T and S are two self-adjoint operators, and T -S is trace class, then f ( T ) -f ( S ) is also trace class. The function f is a map f : R→R , satisfying: ∣ ∣ with ˆ f the Fourier transform of f . It is clear that in our case, the function f is the trace of the exponential of the operators D R D † R and D † R D R . Hence, the theorem applies perfectly in the present situation.", "pages": [ 11, 12 ] }, { "title": "2.1.3 Massive Fermion case", "content": "We extend the study in the case the Dirac fermion has mass m F . From the equations of motion (43) we can construct the matrix, which acts on the vector: The adjoint of the matrix D MR is equal to: /negationslash Unlike the massless case, it is not obvious if the zero modes of D † MR exist at all. One must solve the equation D † MR ψ = 0, subject to the complex conjugate boundary conditions corresponding to equation (43). The zero modes of the matrix D MR exist, and the countable set that these belong to is in one-to-one correspondence to the quasinormal modes for a massive fermion in the Kerr-Newman background. The latter were studied in detail in [26]. We are thus confronted with a non-trivial problem. In any case, there are two alternative situations that can occur, in reference to the zero modes of D † MR . Either dim ker D † MR = 0 or dim ker D † MR = 0, and we cannot be sure which case is true unless we solve explicitly the equation D † MR ψ = 0. Nonetheless, we can answer whether supersymmetry is broken or not if we make use of a theorem. First note that, we can write D MR = D R + C , with C the odd symmetric matrix [40], and also, the D R is the massless Kerr-Newman case matrix defined in equation (47). Now, if the operator tr W e -t ( D + C ) 2 is trace class, the following theorem holds (see for example [40] page 168, Theorem 5.28), /negationslash with C a symmetric odd operator and D any trace class operator. That is, the regularized index of the operator D + C is equal to the index of the operator D . In our case, this reads, ind t ( D R + C ) = ind t D R . Recall that the Witten index and the heat kernel regularized index of D R in the Kerr-Newman background are both zero, hence ind t D R = 0. By virtue of theorem (59), ind t ( D R + C ) = 0. Hence, in conjunction with the fact that ker D R = 0, we may argue that supersymmetry is unbroken even for the massive fermion case.", "pages": [ 12, 13 ] }, { "title": "2.2 The Reissner-Nordstrom and Schwarzschild Black Hole", "content": "Following the lines of argument of the previous sections, it can be easily proven that an unbroken N = 2 SUSY QM algebra underlies the radial part of massless and massive Dirac fermionic systems in Reissner-Nordstrom and Schwarzschild black hole backgrounds. For a detailed study of the corresponding fermionic quasinormal modes can be found in references [27] and [28], respectively. We shall not get into details, since the results are identical to those of the Kerr-Newman case. Indeed, the radial part of the Dirac equation in the Reissner-Nordstrom background can be reduced to the same form of equations, like Equation (43). In the Reissner-Nordstrom black hole, ∆ = r 2 -2 Mr + Q 2 and λ 2 = ( j + 1 2 ) 2 , with j positive. The charges of the supersymmetric quantum mechanical algebra are identical to those of the Kerr-Newmann case, with the appropriate replacement of ∆ and λ . In the case of the Schwarzschild black hole the same arguments hold, but we should replace everywhere ∆ = r 2 -2 Mr and λ 2 = ( j + 1 2 ) 2 .", "pages": [ 13 ] }, { "title": "3 The D-dimensional de Sitter Spacetime, Kerr-Newmande Sitter Black Hole and Reissner-Nordstrom-anti-de Sitter Spacetime", "content": "In this section we shall study whether the supersymmetric structures we found in the previous sections, also underlie de Sitter and anti-de Sitter related spacetimes. We start first with a D-dimensional de Sitter spacetime, in which we consider a massive Dirac fermion. We adopt the notation of reference [29]. See also references [30-32]. In general, the metric of an D -dimensional spherically symmetric spacetime, is of the form, with dΩ 2 D -2 , the metric of the D -2 dimensional unit sphere. In this spacetime, the Dirac equation of a fermion with mass m f can be cast as: with k = ± ( D -2 2 + n ). In the D-dimensional de Sitter spacetime background, the quantities U , V and W are equal to: In the above equation, L is related to the cosmological constant. Using the coordinate z = tanh( y L ), with r = zL , the Dirac equation (61) can be written: with ˜ ω = ωL and ˜ M = m f L and additionally, In the case ˜ M = 0, equation (63) becomes: The above equation gives the quasinormal mode spectrum of the fermionic system in this de-Sitter background. It is obvious that, also in this case, the quasinormal spectrum is in bijective correspondence to the zero modes of the operator: acting on the vector: and also D † dS : acting on As in the Kerr case, the operators can be used to form an unbroken N = 2 SUSY QM algebra. Without getting into details, the supercharges of this algebra, Q dS and Q † dS are, Moreover, the quantum Hamiltonian is, Since, which in turn implies, ker D dS D † dS = ker D dS D † dS . As a consequence, the following relation holds for the operators, e -tD † dS D dS and e -t D dS D † dS , Hence, supersymmetry is unbroken, just in the Kerr massless fermion case. The difference between the two problems however is that, in the de Sitter case, the whole system possesses this supersymmetry and in the Kerr case, only the radial part has this symmetry. The massive case can be also treated identically to the massive case of the Kerr-Newman black hole, yielding the result that the system possesses an unbroken N = 2 SUSY QM. Before closing this section, we mention that the system of a Dirac fermionic field in the KerrNewman-de Sitter black hole background has also an N = 2 SUSY QM, but we omit such a study since it is identical to the other cases we studied, with different supercharges and Hamiltonian of course. For a study of the fermionic quasinormal modes in a KerrNewman-de Sitter background, see reference [30].", "pages": [ 13, 14, 15 ] }, { "title": "A Brief Presentation of the Reissner-Nordstrom-anti-de Sitter Spacetime", "content": "An interesting situation, different from the ones we met in the previous sections, occurs in the case a Dirac fermion is considered in a Reissner-Nordstrom-anti-de Sitter spacetime. Particularly, as was established in reference [67], this fermionic gravitational system has two unbroken N = 2 SUSY QM algebras. We shall briefly present it since we are going to use it in the following (for a detailed analysis see [67]). The metric in a D -dimensional Reissner-Nordstrom-anti-de Sitter spacetime is given by: where f ( r ) is equal to: In the above equation, L is the AdS radius, Q is the black hole charge, and r 0 is related to the black hole mass M . The dΩ 2 D -2 ,k is the metric of constant curvature, with k characterizing the curvature. The value k > 0 characterizes the metric of an D -2 dimensional sphere, while the k = 0 describes R D -2 . Finally, when k < 0, it describes H D -2 . In the 4-dimensional case, the k = 0 Reissner-Nordstrom-anti-de Sitter metric is, As is demonstrated in [67], the first unbroken N = 2 SUSY QM algebra, which we denote as N 1 , can be constructed by using the following operator: The supercharges of the N 1 SUSY algebra, which we denote Q RN and Q † RN , are equal to: Moreover, the quantum Hamiltonian is equal to, The second unbroken N = 2 SUSY QM algebra, denoted as N 2 , can be built on the matrix: with the corresponding supercharges, The Hamiltonian of N 2 is Hence the Dirac fermionic system in an Reissner-Nordstrom-anti-de Sitter background, possesses an supersymmetric structure that is the direct sum of two N = 2 supersymmetries, namely:", "pages": [ 15, 16 ] }, { "title": "4.1 Extended Supersymmetric-Higher Representation Algebras", "content": "As we have demonstrated, in the case of the Reissner-Nordstrom-anti-de Sitter gravitational fermionic system, there are two N = 2 SUSY QM algebras. In view of this fact, a natural question that springs to mind, is whether the SUSY QM algebras that underlie the gravitational systems have an extended (with N = 4 , 6 ... ) one dimensional supersymmetry origin. This argument is further supported from the fact that [68] non-Abelian uplifted magnetic dilatonic black holes are connected to a N = 4 SUSY QM algebra. It is obvious that the theoretical frameworks are different in the two cases, but nevertheless the underlying SUSY QM structure motivates us to search for an extended supersymmetry structure. Unfortunately, there is no extended SUSY QM algebra underlying the gravitational systems. In fact, the extended SUSY QM algebra is always connected to a global spacetime supersymmetry in four dimensions, and clearly there is no such structure in the gravitational systems we studied. The only case for which there exists a more rich SUSY QM structure than the N = 2 SUSY QM, is in the case of the Reissner-Nordstrom-anti-de Sitter gravitational fermionic system. In that case, the two N = 2 supersymmetries can be combined to a higher representation of a single N = 2, d = 1 supersymmetry. Indeed, the supercharges of this representation, which we denote Q T and Q † T are equal to: Accordingly, the Hamiltonian of the combined quantum system, which we denote H T , reads, The operators (84) and (85), satisfy the N = 2, d = 1 SUSY QM algebra, namely: In this case, the Witten parity operator is equal to: Apart from this representation, there exist equivalent higher dimensional representations for the combined N = 2, d = 1 algebra, which can be obtained from the above algebra, by making the following sets of replacements: Moreover, a higher order reducible representation of the N = 2 SUSY QM algebra supercharges, that is equivalent to the one materialized in relation (84), is given by: In addition, the Hamiltonian of the corresponding quantum system, is, There exist other equivalent higher order representations, which we omit for brevity. In conclusion, there exists no extended N ≥ 3 supersymmetric quantum algebra underlying the fermionic systems we have studied in this paper, only higher order reducible representations of the N = 2 SUSY QM algebra, and this happens only in the case of the fermionic system in the Reissner-Nordstrom-anti-de Sitter spacetime.", "pages": [ 17, 18 ] }, { "title": "4.2 A Global R-symmetry of the Hilbert Space of Quantum States", "content": "Due to the N = 2 SUSY QM algebra, the Hilbert space of quantum states possesses an inherent global R-symmetry, which is actually a U (1). This is common to all systems we described in the previous sections, so we do not specify to a particular choice of supercharges. Suppose that the SUSY QM algebra is described by the supercharges Q G and Q † G . The N = 2 superalgebra is invariant under the transformations: Thus the quantum system is invariant under an R -symmetry of the form of a globalU (1). Obviously, the Hamiltonian of the SUSY QM algebra is invariant under the U (1)transformation, that is H ' M = H M . This R-symmetry has a direct impact on the transformation properties of the Hilbert states of the system. Recall that the total Hilbert space H , is Z 2 -graded. Let ψ + M and ψ -M denote the Hilbert states corresponding to the spaces H + M and H -M . The even and odd states, are transformed under the U (1) transformation, as follows: The parameters β + and β -are global parameters defined in such a way that a is equal to a = β + -β -. Therefore, the resulting quantum system possesses a continuous R-symmetry. Let us comment here that the breaking of this R-symmetry into a discrete one, can be a very interesting situation. An intriguing question is, how to break this continuous symmetry explicitly, because an explicit breaking could make us assume that a supercharge acquires a constant vacuum expectation value (assuming it's vacuum expectation value is it's trace over all vacuum eigenstates). Such a thing does not spoil the trace-class properties of the operators we studied and it's physically appealing since discrete symmetries are inherent to quantum system with fermionic condensates. Although interesting, such a task exceeds the purposes of this article.", "pages": [ 18, 19 ] }, { "title": "4.3 A Global two term Spin Complex Structure", "content": "In this subsection, we shall present a spin complex structure, inherent to every fermionic system in the gravitational backgrounds we described in the previous sections. As we have seen, the Witten parity W provides a Z 2 grading to the Hilbert space of the SUSY QM mechanics algebra. Hence, the Hilbert space of each gravitational fermionic system is split in the following way H ( M ) = H + ( M ) ⊕H -( M ), with M denoting the corresponding curved spacetime. Each of the H ± ( M ) contains the even and odd Witten parity states. This grading can be used to define a spin complex, with the aid of the superconnection, which as we will demonstrate later on in this section, is the supercharge. In order to see this, let us note an important implication of the SUSY QM algebra on the vectors that belong to the graded Hilbert spaces H ± ( M ). Recall the vectors | ψ + 〉 H + ( M ) and | ψ -〉 ∈ H -( M ), which are equal to, with | φ ± 〉 , the vectors corresponding to the matrices D i , defined in the previous sections. As can be easily checked, the supercharges Q , Q † of the SUSY QM algebra, act in the following way on the vectors | ψ ± 〉 : Therefore, the supercharges imply the following two maps: These two maps in turn, constitute a two term spin complex which has the following form: The index of this two term spin complex, is equal to the index of the operator D i , that is, ind t D i .", "pages": [ 19, 20 ] }, { "title": "4.4 The Krein Spectral Shift", "content": "From the Witten index of the fermionic gravitational systems we studied in this paper, we can directly compute other topological quantities related to the corresponding spectral problems. Particularly, we shall be interested in the Krein spectral shift ξ ( λ ) [40]. It worths recalling the definition of this function. Following [40], given two trace class operators T 1 and T 2 , the function ξ : /Rfractur →/Rfractur is defined in such a way such that: The regularized index of an operator Q , is written in terms of the Krein spectral shift as follows: Recall from relation (34), that the Witten index is the limit of the regularized index for t →∞ . It is proven that the Witten index is equal to ξ ( ∞ ) [40], that is: Since the Witten index for the fermionic gravitational systems we studied is zero, we easily obtain that the corresponding Krein spectral shift is zero, that is", "pages": [ 20 ] }, { "title": "4.5 Compact Radial Perturbations of Maximally SUSY QM and the Witten Index", "content": "In the previous sections we found that an N = 2 SUSY QM underlies fermionic systems when these are considered in curved gravitational backgrounds. As we will demonstrate, this SUSY QM algebra underlies any massless Dirac fermionic system when this is considered in a D -dimensional maximally symmetric spacetime. In addition, what we are mainly interested here, is to see what is the impact of compact radial perturbations to the Witten index of the SUSY QM algebra. We start first with the N = 2 SUSY QM issue. A D -dimensional maximally symmetric spacetime ( D ≥ 4), has the metric of the form [69]: The corresponding Dirac equations for this background are [69]: Following the lines of research of the previous sections, an unbroken N = 2 SUSY QM algebra underlies this fermionic system. The supercharges of this SUSY algebra are: and the Hamiltonian is with D M , the operator: In the above, κ stands for the eigenvalues of the Dirac operator in the D -2-dimensional manifold dΣ 2 D -2 . It can be easily shown that the Witten index of the SUSY QM algebra is ∆ = 0, with ker D M = ker D † M = 0. Hence supersymmetry is unbroken. A perturbation of this fermionic system is materialized by adding a function to the term W = iκ F ( r ) H ( r ) , of the form: We employ a simplified approach to perturbations, since we are not interested for the perturbations, but for the index of the operator D M . A much more detailed analysis on perturbations can be found in [69]. The function S ( r ) is considered to be fast convergent, so that the following operator is compact: Hence, since we can write D ' M = D M + C M , and the operator (107) is compact and odd, according to the theorem of section 2, the index of the operator D M , is invariant: Therefore, the Witten index does not change under compact odd perturbations of the fermionic system. This means that N = 2, d = 1 SUSY is unbroken in the case of compact perturbations. /negationslash", "pages": [ 20, 21 ] }, { "title": "4.6 Some Local Geometric Implications of the SUSY QM Algebra on the Spacetime Fibre Bundle Structure", "content": "In this section, we shall present in brief the local geometric implications of the SUSY QM algebra on the fibre bundle structure of the spacetime M . This local geometric structure is a common attribute of all the spacetimes we studied in the previous sections. Particularly, we shall demonstrate that locally, the spacetime manifold M (and locally means at an infinitesimally small open neighborhood of a point x ∈ M ), due to the N = 2 SUSY QM algebra, is rendered a supermanifold, with the supercharge of the SUSY QM algebra being the superconnection on this supermanifold, and the square of the supercharge being the corresponding curvature. For some important references on the mathematical issues we will use see [70-83]. The charged fermions that are defined on the spacetime M , are sections of the total U (1)-twisted fibre bundle P × S ⊗ U (1), where S is the representation of the Spin group Spin (4), which in four dimensions is reducible, and P , the double cover of the principal SO (4) bundle on the tangent manifold TM . Recall that a Z 2 grading on a vector space E , is a decomposition of the vector space E = E + ⊕ E -. A decomposition of an algebra A , to even and odd elements, A = A + ⊕ A -, such that: is called a Z 2 -grading of A , and the algebra A is called a Z 2 -graded algebra. Let the vector space E = E + ⊕ E -, and W an involution ∈ End( E ), that is, it belongs to the endomorphisms of E . In addition, it is assumed that W = ± 1 on the vectors of E , that is: This involution W , renders the algebra End( E ) a Z 2 algebra. In the case at hand, for the vector space H , the elements of End( H ) are matrices of the form: with g 1 , g 2 complex numbers in general. The N = 2 SUSY QM algebra, W , Q , Q † and particularly the involution W , generates the Z 2 graded vector space H = H + ⊕H -. The subspace H + contains W -even vectors while H -, W -odd vectors. This grading is actually an additional algebraic structure A on M , with A = A + ⊕ A -an Z 2 graded algebra. Particularly, A is a total rank m ( m = 2 for our case) sheaf of Z 2 -graded commutative R -algebras. Hence M becomes a graded manifold ( M, A ). The structure sheaf A is isomorphic to the sheaf C ∞ ( U ) ⊗ ∧ R m of some exterior affine vector bundle ∧H E ∗ = U ×∧ R m , with H E the affine vector bundle with fiber the vector The sheaf A contains the endomorphism W , W : H → H , with W 2 = I . Thereby, End( H ) ⊆ A . A is called a structure sheaf of the graded manifold ( M, A ), while M is the body of ( M, A ). space H . The structure sheaf A = C ∞ ( U ) ⊗ ∧H , is isomorphic to the sheaf of sections of the exterior vector bundle ∧H E ∗ = R ⊕ ( ⊕ m k =1 ∧ k ) H E ∗ . Without getting into much more detail to the sheaf structure of ( M, A ), let us see the local geometric implications of the aforementioned sheaf structure. In the case at hand, the sections of the bundle TM ∗ ⊗H are actually those sections of the fermionic bundle P × S ⊗ U (1), related to the SUSY QM algebra, which are the ones related to the radial part of the Dirac equation. A superconnection, denoted S , is an one form with values in End( E ) (or equivalently a section of TM ∗ ⊗∧H E ∗ ⊗H E ). The curvature of the superconnection, which we denote C , is a End( E )-valued two form on M , such that C = S 2 . Locally on M , this suggests that the superconnection is a section of TM ∗ ⊗ End( E ) odd , which are simply the odd elements of End( E ). Locally, the supercharge of the SUSY QM algebra is the superconnection, that is S = Q , and therefore, the curvature of the supermanifold is locally C = Q 2 . Finally, let us note that in addition to the local geometric implications we presented above, there are some global implications for the graded manifold ( M,A ), but due to the complexity of their structure are, by far, out of the scopes of this article. In conclusion, the SUSY QM renders each manifold M a graded manifold and locally a supermanifold, with superconnection Q and curvature Q 2 .", "pages": [ 22, 23 ] }, { "title": "4.7 Is There any Connection of SUSY QM and Spacetime Supersymmetry?", "content": "With the spacetime M being locally a supermanifold, it is unavoidable to ask whether there is some connection of the SUSY QM algebra with a global spacetime supersymmetry. The answer is no. When studying supersymmetric algebras in different dimensions we have to be cautious since the spacetime supersymmetric algebra (related to the superPoincare algebra in four dimensions) is four dimensional while the SUSY QM algebra is one dimensional. Furthermore, spacetime supersymmetry in d > 1 dimensions and SUSY QM, that is d = 1 supersymmetry, are different. There is however a connection, owing to the fact that extended (with N = 4 , 6 ... ) SUSY QM models describe the dimensional reduction to one dimension of N = 2 and N = 1 Super-Yang Mills models [84-92]. Still, the N = 2, d = 1 SUSY QM supercharges do not generate spacetime supersymmetry and therefore, SUSY QM does not relate fermions and bosons in terms of representations of the Poincare algebra in four dimensions. The SUSY QM supercharges provide a Z 2 grading on the Hilbert space of quantum states and also generate transformations between the Witten parity eigenstates. That is why the manifold M is globally a graded manifold and not a supermanifold.", "pages": [ 23 ] }, { "title": "5 Concluding Remarks", "content": "In this paper we studied the relation of the quasinormal modes to SUSY QM. In particular, we showed that an N = 2 SUSY QM algebra underlies Dirac fermionic systems when these are considered in various curved spacetimes backgrounds. We examined thoroughly the massless Dirac fermion Kerr black hole, the massless and massive fermion case in Kerr-Newman, Reissner-Nordstrom, Schwarzschild, D-dimensional de Sitter, and KerrNewman-de Sitter spacetimes. In all the cases we found a hidden N = 2, d = 1 unbroken supersymmetry in the radial part of the Dirac fermion system, and the non-breaking of supersymmetry was very closely connected to the operators being trace-class and to the existence of quasinormal modes. Actually the zero modes of the fermionic system in each case, were in bijective correspondence to the quasinormal modes of each system. Up to date, the important theoretical issue that addresses the nature of neutrino, that is whether it is Dirac or Majorana, has not be answered successfully yet. Hence, any information on the effect of massless (if the neutrino can be considered massless) fermions in nature is invaluable. We studied massless Dirac fermion in realistic curved four dimensional gravitational backgrounds and found an underlying one dimensional supersymmetry. This result provides us with valuable information, in view of the fact that the existence of quasinormal modes guarantees the unbroken supersymmetry. In view of gauge/gravity dualities, it would be interesting to search for such supersymmetries in quantum systems with AdS backgrounds. In addition, such a study would be complete if both Majorana and Dirac fermion zero modes are studied in such backgrounds, in reference to supersymmetric structures. Moreover, this would require the Majorana fermion quasinormal mode spectrum, in black hole backgrounds, and since the nature of the neutrino is not known (that is whether it is Majorana or Dirac), up to date, it would be interesting by it self to study this issue. Finally, it worths investigating whether there is any connection between the topology of the black holes and the N = 2 SUSY QM algebras we found. This connection to the topology is also advocated by the fact that, the Witten index (a topological property characterizing the various spin structures of the spacetime) of the algebra vanishes, but the kernels of the corresponding operators are non-empty.", "pages": [ 23, 24 ] } ]

[ { "title": "AdS ×", "content": "Cristhiam Lopez-Arcos a ∗ and Horatiu Nastase a † a Instituto de F'ısica Te'orica, UNESP-Universidade Estadual Paulista R. Dr. Bento T. Ferraz 271, Bl. II, Sao Paulo 01140-070, SP, Brazil", "pages": [ 1 ] }, { "title": "Abstract", "content": "We apply a physical principle, previously used to eliminate ambiguities in quantum corrections to the 2 dimensional kink, to the case of spinning strings moving in AdS 4 × CP 3 , thought of as another kind of two dimensional soliton. We find that this eliminates the ambiguities and selects the result compatible with AdS/CFT, providing a solid foundation for one of the previous calculations, which found agreement. The method can be applied to other classical string 'solitons'.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Quantum corrections to solitons have a long and complicated history, and it has proven difficult to find an algorithmic way to calculate them, due to regularization-dependent ambiguities. The most studied case, for being the simplest and easiest to analyze, is the kink in two dimensions. Studies of its quantum corrections started with [1] (see also [2, 3], and supersymmetric extensions started with [4-7]) and still go on (see [8] for basic techniques and references, and [9] for a review of recent results), due to the many subtleties present. In [10] a physical principle was proposed that eliminates the ambiguities and gives a quantum correction consistent (in the supersymmetric case) with supersymmetry. A seemingly different area that has received a lot of attention lately is quantum corrections to classical (long) strings moving in gravitational backgrounds. The reasons for that interest are usually related to AdS/CFT, since one application has been to systems which have a field theory dual admitting a Bethe ansatz for the dual to the string. 1 This is useful, since unlike other cases, when we need to invoke supersymmetry to match weak coupling field theory results to strong coupling gravity results, the Bethe ansatz allows one to have a prediction for the expected quantum correction at strong coupling. Then, provided we can trust AdS/CFT and the Bethe ansatz, we have a prediction for the expected quantum correction. Of course, the classical string is just a type of solitonic solution in a two dimensional field theory (the sigma model of the string moving in the gravitational background), and as such a priori suffers from the same ambiguities as the largely studied kink. From this point of view, one should not be surprised that early calculations for the corrections to a spinning string in AdS 4 × CP 3 gave different results, and apparently incompatible with AdS/CFT [11-14]. In [15], a calculation was proposed that matches with AdS/CFT expectations. 2 See [16-23] for later related works. However, the calculation in [15] still suffers from the same a priori ambiguities, and it amounts to a particular choice of regularization for them, whose only justification is a posteriori, through matching with AdS/CFT. It is the purpose of this paper to provide a justification for that calculation, by taking the physical principle of [10] and applying it to classical strings. We will show that its use for the model in [15] eliminates the ambiguities implicitly hidden there, and thus offers the possibility of extending the same method to other classical string solutions. The paper is organized as follows. In section 2 we explain our method based on a physical principle, in section 3 we review how it applies to the case of the two dimensional kink, in section 4 we apply it to the spinning string in AdS 4 × CP 3 , and in section 5 we conclude.", "pages": [ 2 ] }, { "title": "2 The method", "content": "As is well known, one-loop corrections to the energy of the vacuum are equivalent (via the exponentiation of one-loop determinants) to sums of zero-point fluctuations, for a bosonic mode ∑ 1 2 /planckover2pi1 ω B . If we have fermionic modes, they will contribute with -∑ 1 2 /planckover2pi1 ω F . These fluctuations give rise in particular to the Casimir energy, which is the difference in this zero-point energy between the infinite space and a space of finite size. Of course, this answer is a priori ambiguous ( ∞-∞ ), and moreover highly divergent. Generically, ω ∼ n implies a quadratic ( n 2 ) divergence. The same idea applies when we calculate the quantum mass of some soliton, generically denoted φ sol ( x ), with classical mass M . We have to calculate the fluctuations in the presence of the soliton, i.e. eigenmodes around φ sol ( x ), and subtract the fluctuations in its absence (in the vacuum). An extra factor to take into account is renormalization. In terms of Feynman diagrams, we know we have counterterms, which correspond to renormalizing the parameters of the theory, for instance a bare mass parameter m 0 becomes the renormalized mass m . When going to the fluctuation representation, a useful way of encoding the counterterm for the energy, δM , is by the variation of the classical mass M when expressed in terms of unrenormalized parameters like m 0 , vs. renormalized ones like m , with a result linear in m 0 -m = δm , where for δm we need to take the result of the one-loop Feynman diagramatic calculation. Therefore generically the one-loop contribution to the quantum mass of a soliton is given by where the ω B n , ω F n are the frequencies coming from the bosonic and fermionic parts of the action, respectively, labelled by an integer n , and the (0) refers to the vacuum, i.e. without the soliton solution. This expression contains ambiguities. The first type of ambiguities is due to the fact that we have generically the ∞-∞ difference of quadratic divergences (if at large n , ω n ∼ n , then ∑ ω n ∼ n 2 ), which a priori will be linearly divergent ambiguities, not even constant ambiguities. Here we should note that we would be tempted to say that if we have something like, say, M = ∑ n √ 1 + m 2 /n 2 -∑ n 1, this is the same as M = ∑ n m 2 / (2 n 2 ) which is finite. But this in fact amounts to a particular choice of regularization scheme. One needs more information to be precise about which regularization it is, but this would basically be part of the mode number regularization, if we would have instead of n , a k n together with a relation between n and k . Mode number regularization means that we identify each mode in a sum with another mode in the other sum, effectively giving the summation operator as a common factor. In general however, we are not allowed to make the ∑ n common if both sums are infinite. In terms of choosing a cut-off, there are always at least two ways to regularize, mode number cut-off (which corresponds to making the sum common) and energy/momentum cut-off. The second, choosing the same upper energy instead for the two sums (convert sum over k n to integrals over k and identify the variable k in the two sums) gives different results if the sums are infinite [24]. Note that even in usual quantum field theory divergent integrals we can have this situation, just that usually one doesn't think about it. For instance, if ∫ f and ∫ g are UV divergent, then ∫ Λ ( f -g ) automatically means that we take the same cut-off Λ for f and g , but we could in principle choose ∫ Λ f -∫ Λ+ a g , giving a different result. There might be situations where this is necessary. Yet another type of ambiguity is related to the existence of different types of possible boundary conditions, in turn determining different functions k n , or k ( n ) in the continuum limit. But we want a physically unambiguous way to determine the correct regularization and boundary conditions. In [10] a physical principle was used to fix both. The principle can be simply formulated by saying that the non-trivial topology of the soliton boundary does not introduce any extra energy. The first part of the method involves the notion of topological boundary condition, i.e. that the boundary condition should not introduce boundary-localized energy (surface effects), thus fixing one type of ambiguity. For scalar fields, the boundary conditions should be compatible with the classical solution (if the classical solution is antiperiodic, then so must the boundary condition for fluctuations), but for fermions and higher spins we need to be more careful. The method was described in detail in [10]: consider the symmetries of the action and the symmetries of the solution. For the kink, the action has a { φ → -φ, ψ → γ 3 ψ } symmetry and a ψ → -ψ symmetry, and the kink solution is antisymmetric in φ . Then e.g., the fluctuations around the kink solution have The second part is that when we take the classical soliton mass to zero, specifically by taking a relevant mass scale on which it depends, like the m above, to zero, the quantum mass of the soliton should also go to zero, such that there is no mass depending purely on topology, i.e. localized at the boundary. That in turn means that we can calculate instead of the soliton mass, its derivative with respect to the relevant mass scale m , thus reducing the UV divergence of the result, and obtaining a 'derivative regularization'. For instance, in the example above, ∂M/∂m = ∑ n m/ ( n 2 √ 1 + m 2 /n 2 ) is now indeed finite and unambigous. We should emphasize that it is not guaranteed that this procedure eliminates ambiguities in general, since taking only one derivative may not reduce the divergence sufficiently. Nevertheless, we hope that in many cases of interest, the result is unambiguous. Note that taking more derivatives will in general reduce further the divergence, but it is not clear if there is a physical principle that will correspond to this modified prescription. We can therefore define our procedure as follows: Find the soliton solution φ sol ( x ), find the frequencies of fluctuations around it, and the renormalization of the relevant mass parameters. Then find the relevant mass parameter to define derivative regularization with respect to it, and topological boundary conditions. Ideally, the resulting quantum mass should be well-defined and unambigous, and we should be able to calculate it. We will see that in the simple λφ 4 kink case it is indeed true, however the string soliton case is more complicated. We can prove that the resulting answer is unambiguous, but we will still need to employ the same procedure used in [15] to calculate it.", "pages": [ 3, 4, 5 ] }, { "title": "3 One-loop mass for the kink in φ 4 theory", "content": "We want first to understand how this method applies to the kink solution of the φ 4 theory in two dimensions. Here we review [10]. The theory has the Lagrangian There are two degenerate vacuum states (trivial solutions), φ = ± µ 0 / √ λ , and therefore topologically nontrivial, localized solutions (kinks) must tend to ± µ 0 / √ λ as x -→ ±∞ . We thus have two nontrivial, stable, finite energy solutions, the kink and anti-kink with classical mass M 0 = 2 √ 2 µ 3 0 / 3 λ . The eigenfrequencies of small fluctuations around the vacuum (trivial sector) are and the allowed values for k n come from the condition k n L = 2 πn , where L is the size of the one dimensional spatial box in which we put the system. The eigenfrequencies for small fluctuations around the kink (nontrivial sector) have the same expression as the trivial vacuum ( ω n = √ k 2 n + m 2 ), but the condition for the allowed values of k n has a different form where the explicit form of the phase shifts δ ( k ) is and is obtained from the explicit scattering solutions in the potential generated by the perturbation around the kink. In the case of fermionic fluctuations (for a supersymmetric version of the kink), which are 2 component vectors, there is a further phase shift θ ( k ) giving a e ± iθ ( k ) / 2 relative factor at ±∞ between the two components. As we mentioned in the previous section, the one-loop counterterm for the soliton mass M comes from varying the classical M under the renormalization δm = m 0 -m , and gives Now we have the necessary ingredients for the calculation of the one-loop correction to the energy. Substituting the frequencies in the expression for the 1-loop correction we obtain As we can see, even after the subtraction, the sum is linearly divergent. To apply our derivative regularization, we must find the mass parameter which takes the soliton mass to zero, and take a derivative with respect to it. In this case, it is obvious, namely the mass parameter is m . We then differentiate the energy with respect to m , perform the summation and integrate back with respect to m . That will get rid of both linearly and logarithmically divergent ambiguities. The physical principle then dictates that the constant of integration is zero. Taking the derivative, we obtain where √ The sums are less divergent now, and can be turned into integrals by taking into account the conditions for the allowed k n s (3.4), obtaining a finite result. Integrating back with respect to m we will get a constant of integration, but applying the physical principle, the constant is zero. Therefore finally the 1-loop energy correction is", "pages": [ 5, 6 ] }, { "title": "4.1 Applying the method", "content": "We now try to apply the same method to the classical (long) string on the background AdS 4 × CP 3 . This can be thought of just as another 2d field theory, with action As we have discussed, the first step is to understand the 2d vacuum and soliton solution. The (trivial) 'vacuum' corresponds to the point-like string, equivalent to φ = φ 0 =constant for the λφ 4 model. The nontrivial soliton whose mass we want to calculate is a spinning string solution, with nontrivial X µ ( σ, τ ). The computation of the 1-loop energy correction for this soliton was done in different ways, obtaining different results. The calculation of [15] gave the correct result matching the expectation from AdS/CFT, but there was no a priori reason why it should be correct, given the implicit choice of regularization scheme needed to obtain it. We will therefore try to identify the ambiguities as above. In order to apply our method, we note several complications with respect to the kink case. We note that (4.1) is a non-linear sigma model, and we have no potential, so formally it looks different from the kink. That however means we can avoid at least one ambiguity from the kink case. No potential means that the phase shifts δ ( k ) and θ ( k ) are not present, so at least the ambiguity of boundary conditions (related to non-zero δ ( k ) and θ ( k )) is not there. It is also lucky, since for the calculation of δ ( k ) and θ ( k ) we would need the full solutions, which as we will see are hard to find. The only ambiguity we still have is the UV divergence. To deal with that, we need to define our physical regularization. But instead of the mass parameter m of the kink, we will have several parameters (which come in the solution), and we have to carefully analyze which can be varied in order to relate the energies and use the derivative regularization. Note however that there are no parameters in the action (4.1) (other than R AdS which multiplies the whole action, so is not relevant), so the parameters of relevance will just characterize the vacuum solution. That also means that there are no counterterm contributions, since the only possible counterterm could be for R AdS , which is not renormalized. The parameter we want needs to be something that when equal to zero, takes the classical mass of the long string to zero, but also something that, like m for the kink, is normally non-zero in the vacuum. An extra complication will be, as we will see, that it is not possible to find the full solutions for the eigenfrequencies, only as an expansion in a large parameter ω . But then it matters how n is related to ω ; in particular, the expansion is not valid for n > ω , which corresponds to the UV divergence we want to analyze. So the only goal we will have is to show that the physical derivative regularization obtained as above selects the regularization implicit in [15]. In order to actually compute the quantum correction, we will still need to use the same procedure as in [15]. In the following sections we will perform first the classical analysis of the model, then we will find the frequencies, and finally apply the derivative regularization.", "pages": [ 6, 7 ] }, { "title": "4.2 The nontrivial soliton", "content": "In this subsection we will analyze the spinning string in AdS 4 × CP 3 . We will see that there are several parameters present in this nontrivial solution, but there are relations between them due to the Virasoro constraints, so our search for the parameter that is nonzero in the vacuum, but takes the soliton mass to zero when it equals zero (the analog of the mass parameter m for the kink), will be highly constrained. The conserved quantities, like the energy, which here has the meaning of 'soliton mass' modulo an additive constant, will be dependent on these parameters. An important technical detail is that the Virasoro constraints are complicated, so we can only solve them perturbatively in certain limits, hence the same will happen for the energy ('soliton mass'). The classical analysis for the string in this background have been done completely (see [15] or [11]), so here we will review the main points. The bosonic part of the action for the spinning string is the one in (4.1), which can be split as and the background metrics appearing in the nonlinear sigma model are Here we have factored out the scale of the metric which is very large (very large ¯ λ , though finite). The soliton we are interested in was found in [25]. It is a rotating string lying in an AdS 3 × S 1 subspace of AdS 4 × CP 3 , which from the point of view of the 2d worldsheet looks like a soliton with Unlike the kink case or usual quantum field theory, now we have also gravity on the worldsheet, which in the conformal gauge manifests itself in the presence of the Virasoro constraints T ab = 0. For the solution (4.6), we have an equation of motion and the Virasoro constraints reduce to They can be solved perturbatively, as done in [15], in a certain limit that we will define shortly. The charge densities are so that the classical energy, spin and the charges under the second and third Cartan generators of SO (6) are where r 0 = cosh ρ ∗ . The limit we use to solve the constraints (following [15]) and find some relations between the constants consists in taking large spin S and large angular momentum J , with their ratio u (and also k ) held fixed, i.e. For this solution, the expansion of the classical energy at large J = ω and thus large angular momentum J = √ ¯ λ J = √ ¯ λω is given by We will see later that this large ω limit is also needed to have a workable form for the eigenfrequencies around the classical solution. On top of this limit, in the next subsection we will use another perturbative expansion which will have as a limit a trivial sector ('vacuum'). We will later see that we need to be only a bit away from this new limit (i.e., to be in the perturbative expansion) in order to be able to use our regularization procedure.", "pages": [ 8, 9 ] }, { "title": "4.3 The vacuum solution", "content": "Since the two dimensional soliton we are interested in corresponds in spacetime to a long spinning string, it follows easily that the trivial solution ('vacuum') has to be a pointlike string. Guided by the BMN limit [26], where we also have perturbations around a state with large J , we know that the eigenvalues of the Hamiltonian, corresponding to perturbations around a BPS state, are the equivalent of the soliton mass, and therefore we look for states of lowest E -J as the vacuum. We then vary the parameters in the nontrivial solution to obtain such a vacuum. This Hamiltonian is, as we saw, E 0 -J from (4.12), where S ∼ r 1 and u ∼ m . The smallest value is then obtained for which implies in particular very small S as well (relative, since we formally took S → ∞ before, though note that ¯ λ is large in S = √ ¯ λr 2 1 w ), with everything else ( J, ω, k, κ ) kept fixed in this second limit. Then we obtain the 'soliton mass' in the vacuum E -J = 0, as we wanted. Taking these limits directly on the spinning solution we indeed get then the point-like string, the trivial solution we were looking for. Now that we have both solutions we can proceed to analyze quantum fluctuations around them.", "pages": [ 9, 10 ] }, { "title": "4.4.1 Bosons", "content": "To find the characteristics frequencies we expand the action (4.1) around the solution (4.6). For the bosonic fluctuations we have six scalars corresponding to motion on CP 3 : one is massless, other four degrees of freedom give the same result, and the last one gives From the scalars corresponding to motion in AdS space we find one massless degree of freedom, one massive one with and two fluctuations whose dispersion relation is given by the roots of the quartic equation We can find the explicit solutions to this equation (though they do not give much information), but only when we expand in large ω .", "pages": [ 10 ] }, { "title": "4.4.2 Fermions", "content": "For the fermionic part the spectrum contains four different frequencies, each being doublydegenerate. Two such pairs have frequencies while the frequencies of the other two pairs are solutions of the equation which can be solved in the same limit as in the bosonic case. With the bosonic and fermionic frequencies we can start to calculate the quantum corrections, formally defined as in (2.1). But in order to do that, we must apply a regularization technique, specifically the derivative regularization previously defined. For that, we need to find the parameter that plays the role of m for us.", "pages": [ 11 ] }, { "title": "4.5 Physical limit and regularization", "content": "We want the parameter to lead to E = J as it goes to zero, but be otherwise finite in the vacuum. Since we could try u or k only, as we have S = Ju . Note one subtlety here: we have E 0 = E 0 ( S, J, ¯ λ, u, k ), however u = S/J so there is an ambiguity in the split of E 0 (how to we isolate the S dependence, when we could always write any u as S/J ). We can consider that the S term is the one that is independent on k , which will be useful shortly. However, u is not a good parameter, since it becomes always zero in the vacuum. On the other hand, k stays fixed in the vacuum, yet k → 0 keeping everything else ( u, J, ω, κ ) fixed gives E 0 → S + J . That is then not enough, and we need to supplement our original definition of the nontrivial vacuum with u small, and therefore also w,S,m small, i.e. in the perturbative expansion away from the vacuum. Therefore k is the parameter that relates the two energies. One more subtlety to note is that, since we will use the large ω expansion, and since by doing the 1 /ω expansion first, we will not be able to match terms linear in u , as we will explain better later. We should note that it was crucial that there were at least two parameters, J and k : J to guarantee a long string, with large J giving a perturbation theory, and k to differentiate with respect to it. It is our hope that this is more general for long strings, with something like J guaranteeing a long string, and something like k giving the 'shape', allowing us to differentiate with respect to it. We are finally ready for the calculation of the quantum correction to the energy.", "pages": [ 11, 12 ] }, { "title": "4.6 Quantum correction to the energy", "content": "where H 2 is the Hamiltonian for the quadratic fluctuations, but subtleties arose that were not well appreciated. In order to understand what the issues are, we first review a few facts about previous calculations. First, previous calculations have not taken into account the trivial sector or 'vacuum' (cf. (2.1)), but considered only E 1 = 1 2 ∑ n ( ω B n -ω F n ). Of course, at the classical level that does not matter, but it does matter at the quantum one-loop level. As we will see, removing the contribution of the trivial sector from the sum will help to the cancellation of some ambiguities. Second, since one gets divergent sums in E 1 , a regularization scheme is necessary, and various calculations gave regularization-dependent results [11-15, 17]. In the calculations of [11-13] the sum was turned into an integral, after which a cut-off was introduced and the integral sign given as a common factor, effectively choosing a form of energy/momentum cut-off regularization, as we explained in section 2. In [14] a different regularization was chosen, where one combines a mode number cut-off with a certain grouping of terms: instead of ∑ n ( ω Bose n -ω Fermi n ), one forms combinations called ω heavy n and ω light n and then a certain n -dependent combination of ω light n is added to ω heavy n , and the resulting sum over n is turned into an integral. This regularization gave a different result from the previous one. More recently, in [17], a modification of the regularization in [14] was given, with different combinations of ω heavy n and ω light n . Yet another type of regularization was considered in [15], where a regularization method used successfully in the case AdS 5 × S 5 [28] was applied, together with a physically motivated redefinition of the coupling constant. The result of [15] is in agreement with AdS/CFT, so it was considered correct, but a priori we did not know which regularization scheme to choose to obtain an unambigous result, since as we saw different schemes can lead to different results, exactly as in the case of the 2d kink. We can use matching with AdS/CFT only as a kind of a posteriori check, exactly as one used the saturation of the BPS bound for the 2d supersymmetric kink (where both the mass and the central charge of the kink get renormalized in the same way). In what follows we take a large ω expansion for both trivial and nontrivial sectors, and we will focus on the leading order in this expansion. As mentioned above, this will force us to take a small u expansion as well, and we can only say something about the leading term in the u expansion. More importantly, in [15] it was explained that if we expand in 1 /ω , since we can allow any value of p 1 = n , we have two regions for the expansion: a) 1 /ω → 0 with n fixed, i.e. n /lessmuch ω , for which we still have a discrete sum; and b) n, ω → ∞ , with x = n/ω =fixed, for which we can replace the sum with an integral. It was then noticed that while both regions contain divergences, the divergence of one can be identified with the divergence of the other, and can be dropped, obtaining a finite result. What we want to show here is that the ambiguity inherent in this procedure is removed by our method. What we would have liked to do is take first the derivative with respect to k , and then do the sum over n , maybe with the same 1 /ω expansion, but this turns out to be prohibitively difficult, so we will be forced to follow the same analysis as [15] once we prove that our method eliminates the ambiguities. We will start by analyzing region a), where we have discrete sums, and where The trivial sector ('vacuum') is simpler, and illustrates the point well, so we will start with it. The sum of bosonic frequencies (bosonic summand) in the trivial sector is Replacing the perturbative solutions of the Virasoro constraints and expanding in ω , we get A similar procedure for the fermionic summand (minus the sum of fermionic frequencies) gives Taking the sum of the two expressions to obtain the summand, we get terms like n 2 -n 2 and ω -ω (since ω > n , these are of the same type), which are ambiguous, but they will be cancelled after taking the derivative with respect to k . After the derivative with respect to k , the trivial sector summand ˜ e ( n ) gives with no n 2 -n 2 and ω -ω ambiguities. Moving on to the nontrivial sector, the leading terms in the large ω expansion of the nontrivial sector summand are and it can be seen that again terms like n 2 -n 2 appear, but they are again cancelled by taking the derivative with respect to k . After ∂/∂k , the summand of the nontrivial sector gives We note that even at u = 0, there is a constant piece that would give a divergence when summed over n , however it is the same one as in the trivial sector summand (4.27), so by subtracting the two we get rid of the last potential ambiguity. We finally get It would seem that we still have a divergence after we take the sum, but we need to remember that u /lessmuch 1 /n , so these terms linear in u do not give rise to divergences in this limit (or another way of saying it is that they belong to the omitted higher order terms in 1 /ω < 1 /n ). The final result for the one-loop correction to the energy coming from region a) is the sum over (4.30), integrated over k (with zero constant of integration). There is a certain subtlety here, since in the end we want to calculate a correction to the energy that will turn out to have contributions linear in u , but as we mentioned, our only purpose (given our technical, i.e. calculational, limitations) is to show that the procedure of [15] becomes unambigous if we consider our physical principle. Let us now analyze the result of [15] and compare to what we get. Expanding (4.28), now called e sum ( n ) to emphasize that we are in region a), at large n we get where the first term becomes divergent when summed over n (singular piece) and the second term becomes regular. The divergence and hidden ambiguities implicit in (4.31) were eliminated in our result (4.30). On the other hand, in region b), with ω/n = x =fixed, the expansion of the summand, now denoted e int ( x ) gives [15]: Note that in computing this expression we have also assumed the cancellation of ∞-∞ terms that are a priori ambiguous, i.e. a priori the first term in the expansion would be ω , not 1 /ω , but its coefficient is of the type z -z and is k -independent, therefore disappears under our ∂/∂k . 3 Then we can check that at x → 0, the coefficient of the 1 /ω term becomes regular (constant), whereas from 1 /ω 3 on, we have inverse powers of x at x → 0, meaning a divergence in the integral ∫ 0 dx . Note that these singular terms all come multiplied by powers of u , so we cannot properly analyze them using our method, as u < 1 /ω for us (for technical reasons). However, we have as expected. Similarly, in e int ( x ) have terms with inverse powers of x , which become singular (divergent) when integrated, but we can easily verify that Due to this fact, in [15] it was proposed to just drop these singular terms, but this procedure hides a regularization ambiguity, since for instance we could expand in a slightly /negationslash /negationslash different parameter that ω and then by the same logic resolve to drop a different divergent piece from the total result. With our procedure, it becomes clear that result is unambigous and free of potential divergences, and we are in fact led to drop the singular terms of [15]. Indeed, the effect of summing over (4.30) and integrating over k with zero constant is (to leading order in u , which is what we can check) the same as just dropping the divergent terms in (4.31). In conclusion, we see that there were a priori ∞-∞ ambiguities that were hidden in the formal 1 /ω expansion procedure above, but we have checked that our physical principle just cancels them, and then we can continue with the same calculation as in [15]. Namely, the one-loop correction is now where E n =0 is the zero mode contribution. The terms giving odd powers of J are where agrees with the value of h ( λ ) argued in [15] to be predicted by AdS/CFT (though a direct calculation of quantum corrections to the dual to h ( λ ) is still lacking). Note however that changing both the h ( λ ) above and the energy correction simultaneously could maintain agreement (see e.g. [17,19]). Here we will assume, following [15], that the choice of h ( λ ) above is unambigous (at least as long as the number of modes summed over in various terms differs only by a finite amount; in the heavy-light prescriptions used for instance in [14], some terms are summed over twice as many modes than other terms, due to some unitarity prescription).", "pages": [ 12, 13, 14, 15, 16 ] }, { "title": "5 Conclusions", "content": "In this paper we have proposed to apply the physical principle developed in [10] for elimination of ambiguities in the quantum corrections to the energy of two dimensional solitons, to the case of classical (long) strings moving in gravitational backgrounds, taking as a primer the case of the spinning string in AdS 4 × CP 3 . In that case, it was found that there existed a certain regularization dependence, giving rise to different results (e.g [15] and [11]). A procedure was devised in [15] that gave a result consistent with AdS/CFT, but the regularization issue was hidden, without a clear physical principle to explain the choice. As the long history of the quantum corrections to the energy of two dimensional kinks has shown, just because a certain regularization choice seems natural is no guarantee that it is correct, and one needs some physical input to justify it. It was our goal to justify the choice in [15] by a physical principle which can be applied to other cases of long strings as well. We have found that technical reasons limit how far we can calculate with our method in this case, but we can check that to leading order in u our procedure eliminates the ambiguities, and therefore justifies the choice in [15], leading to the result consistent with AdS/CFT. We hope to apply the same methods to other long strings in the future. Acknowledgements . We would like to thank Radu Roiban for a careful reading of the manuscript and many useful comments and suggestions. The work of HN is supported in part by CNPq grant 301219/2010-9. CLA would like to thanks Humberto Gomez and Alexis Roa for reading the manuscript and CAPES for full support.", "pages": [ 16, 17 ] } ]

[ { "title": "S. Mignemi †", "content": "Dipartimento di Matematica, Universit'a di Cagliari viale Merello 92, 09123 Cagliari, Italy and INFN, Sezione di Cagliari", "pages": [ 1 ] }, { "title": "Abstract", "content": "We investigate the global properties of black brane solutions of a three-parameter EinsteinMaxwell model nonminimally coupled to a scalar with exponential potential. The black brane solutions of this model have recently been investigated because of their relevance for holography and for the AdS/condensed matter correspondence. We classify all the possible regular solutions and show that they exist only for a limited range of values of the parameters and that their asymptotic behavior either breaks hyperscaling invariance or has the form of a domain wall. We also write down some exact solutions in Schwarzschild coordinates.", "pages": [ 1 ] }, { "title": "1. The model", "content": "The Einstein-Maxwell model nonminimally coupled to a scalar field with Liouville potential (EML) has been studied in several contexts [1-5]. Models of this kind arise for example as low-energy limits of supergravity and superstring theories [6]. They are especially relevant in the context of holography [3-4], since they are dual to models exhibiting a breaking of the conformal symmetry leading to hyperscaling violations [7]. Because of the nonminimal coupling, the standard no-hair theorems [8] do not hold for these models, and their black hole solutions have unusual asymptotics and support nontrivial scalar fields. In particular, the spherically symmetric black hole solutions have been classified in ref. [1], where it was shown that all such solutions have non-standard asymptotic behavior. Because the potential for the scalar field does not admit local extrema, these models do not allow for AdS vacua. Hence, also the Reissner-Nordstrom-AdS black hole solutions are forbidden. It should be noticed that instead AdS 2 × S 2 solutions can exist. Particularly interesting are also the solutions exhibiting planar symmetry, especially black branes. From an analytical point of view, planar solutions have the advantage of being easier to find in closed form than spherically symmetric ones, since their calculation requires one less integration, but are especially interesting because of their relevance for holography and the AdS/condensed matter correspondence [3,4]. Although EML models do not allow for AdS vacua, they still could admit asymptotic solutions preserving scaling isometries. These solutions would be asymptotic to Lifshitz spacetimes [9,10], i.e. spacetimes presenting a scale isometry under which timelike and spacelike coordinates transform with different exponents. This property may be relevant for the holographic description of quantum phase transitions. In the general case, the scaling isometry is broken and the metrics only transform covariantly under scale transformations [5]. To this class belong in particular solutions for which the Poincar'e invariance of the brane is preserved. In the literature they are often referred to as domain wall solutions [11,4,12-14], in analogy with the domain wall solutions of supergravity theories. In a EML context they were studied in [4]. For a wide range of values of the the parameters, domain wall solutions are conformal to AdS and allow for an holographic interpretation [11,4]. It has been shown that the holographic interpretation of Einstein-Maxwell-scalar theories gives rise to a very rich phenomenology in the dual QFT. This includes phase transitions triggered by scalar condensates and non-trivial transport properties of the dual field theory [2-5,10,12,15]. From the holographic point of view the difficulty connected with the absence of an AdS vacuum, and hence of an ultraviolet fixed point in the dual QFT can easily be circumvented. In fact EML models can be considered as the near-horizon, near-extremal description of an Einstein-Maxwell-dilaton gravity model whose scalar field potential allows for a local maximum, and hence for an ultraviolet fixed point [16]. Very recently, it has been realized that the solutions of the EML theory represent the main example of scale covariant models that lead to hyperscaling violation in the dual field theory [7]. They are therefore a very promising framework for the holographic description of hyperscaling violation in condensed matter critical systems (e.g. Ising models [17]). Moreover they have been used for the description of Fermi surfaces and for the related area-law violation of the entanglement entropy [7,18]. Although planar solutions of the EML model have been already investigated and several partial results have been obtained [1-4], in this paper we systematize the previous findings and obtain general results for all values of the parameters of the model. In particular, we study the phase space of the dynamical system associated with the planar solutions of the model in four dimensions, using the same approach adopted in ref. [1] for the spherically symmetric case, and classify all the possible black brane solutions admitting regular horizon, according to their asymptotic behavior. It turns out that all solutions have nonstandard asymptotics, of domain wall or hyperscaling-violating form, but not of anti-de Sitter or Lifshitz form. Moreover, only for a very limited range of values of the parameters of the model regular black brane solutions are possible. We also obtain some exact solutions for special values of the parameters. Some of these solutions have already been obtained in [2], using a different parametrization of the metric, that gives rise to complicated field equations, and less clear interpretation. Since the main purpose of this paper is the classification of the black brane solutions of the EML theory with regular horizon and asymptotic regions, we shall not investigate other possible solutions (namely solutions containing naked singularities or cosmological horizons). We also do not discuss the physical applications of the model, for example to holography, except for some considerations on thermodynamics. A thorough investigation of these topics can be found in [3-5].", "pages": [ 2, 3 ] }, { "title": "2. Action and field equations", "content": "We consider the action where F µν is a Maxwell field, φ is a scalar field and g and h are two real parameters. Some special cases are well known: for h → ∞ one obtains the GHS model [19,20], for h = 0 the GHS model with a cosmological constant. For g → ∞ one gets a Liouville model, for g = 0 the minimally coupled Einstein-Maxwell theory with a Liouville potential. For h = -g = ± 1, the action can be derived from string theory [11], for h = -1 /g = ± √ 3 from the Kaluza-Klein reduction of a five-dimensional model. The field equations read We look for electrically charged solutions with planar symmetry. Magnetic solutions can be obtained by duality. More precisely, electrically charged solutions of the model with parameter g are magnetically charged solutions of the model with parameter -g [2]. It is useful to parametrize the metric as* where ν , ρ and φ are functions of the radial coordinate ξ . In these coordinates, the Maxwell equation is solved by with Q the electric charge, and the remaining field equations can then be put in the form where Writing the equations (2.5), (2.7) and (2.8) in terms of the independent variables η , χ and ρ , one obtains where The case α = 0 is singular, since (2.10) and (2.11) are no longer independent. In the following discussion, it will be important to know in what range of values of g and h the constants α , γ 1 and γ 2 are positive. This is shown in fig. 1, and we shall implicitly assume this result in the rest of the paper. In particular, it is useful to observe that γ 1 < 0 implies that α and γ 2 are negative and that if h 2 < 3, α and γ 1 are positive. The system (2.5)-(2.8) is invariant under the shift with constant parameter ε . This invariance, which is not present in the spherically symmetric case, facilitates the solution of the system. In fact, it implies the existence of a conserved quantity, that can be obtained combining (2.6), (2.10) and (2.11). Indeed, and hence with b an integration constant. Substituting in (2.12) one gets Eqs. (2.10) and (2.11) form a dynamical system for the variables η and χ , subject to the constraint (2.17). In view of the following discussion, it is useful to write down the functions ν ' and φ ' in terms of η ' and χ ' , taking into account (2.16):", "pages": [ 3, 4, 5 ] }, { "title": "3. Exact solutions", "content": "In a few special cases the field equations can be solved exactly. These solutions are useful for the understanding of the general case.", "pages": [ 5 ] }, { "title": "A. Neutral solutions, Q = 0", "content": "A simple case in which an exact solution can be found is when the electric charge vanishes, Q = 0. This is a minimally coupled gravity-scalar model with exponential potential. The standard no hair theorems [8] do not apply because the solutions are not spherically symmetric, and hence nontrivial solutions are possible. The field equations (2.5)-(2.8) reduce to In addition to (2.14), the system now enjoys a further symmetry, for ν → ν + κ , ρ → ρ + hκ , φ → φ +2 κ , with constant parameter κ , that permits to completely integrate the system. Assuming h 2 = 3, from (3.1) it follows that /negationslash with integration constants b , d , c , with c = ( b +2 d ) /h . Solving (2.10) with Q = 0 yields Substituting (3.3) and (3.4) into (3.2), one gets a constraint between the parameters of the solution, For ξ → 0, the radial function e ρ ∼ e ( η + dξ ) / (3 -h 2 ) goes to infinity if h 2 < 3, or to zero if h 2 > 3. In the first case, we identify this limit with spatial infinity. The other relevant limit is for ξ →±∞ . If the solution possesses a regular horizon, e ρ must be a nonvanishing constant in that limit. This request enforces the choice d = -a , and hence b = (2 -h 2 ) a , c = -ha . Integrating (3.3) and substituting the solution (3.4), one can obtain the explicit form of the metric functions ν , ρ and φ . Rather than pursuing with the present coordinates it is however convenient, in order to get a more transparent interpretation of the solutions, to write the metric in a Schwarzschild form, namely, where U ( r ) = e 2 ν , R ( r ) = e ρ , and F tr = Q R 2 . In this gauge, the field equations take the form The new coordinate r is related to ξ by where ≈ means modulo a constant factor. It follows that with µ a constant. From the solutions obtained above, one can finally write down the metric functions in terms of r and the new integration constants A , B , C , µ , as The constants can be fixed substituting (3.10) into (3.7). It turns out that µ is a free parameter, while two of the constants, say A and C , can be set to 1 by rescaling the coordinates. Then B = (1 + h 3 - 2 ) 2 2 h . These solutions have been found in [13] using a different method. It is easy to verify that the scalar curvature is proportional to and hence the only singularity is at r = 0. Therefore, for positive λ and h 2 < 3, the solutions (3.10) describe a one-parameter family of black brane solutions with horizon at r h = µ (1+ h 2 ) / (3 -h 2 ) and domain-wall asymptotics, that reduces to planar anti-de Sitter if h = 0. Solutions exist also for negative λ , if h 2 > 3. In this case the mass term dominates and an asymptotic region exists only if µ < 0, with U ∼ r -(1 -h 2 ) / (1+ h 2 ) . However, a naked singularity occurs at r = 0. We conclude that solutions with a regular horizon and regular infinity exist only if λ > 0 and h 2 < 3.", "pages": [ 6, 7 ] }, { "title": "B. Planar GHS solutions, λ = 0", "content": "Another case in which it is possible to obtain exact solutions is when the potential vanishes, λ = 0. This is the planar extensions of the well-known GHS solutions [19-20]. Also in this case a new symmetry is present that enforces complete integrability. The field equations (2.5)-(2.8) reduce to and the system is invariant under (2.14) and under ν → ν + κ , ρ → ρ , φ → φ -κ . λ From (3.11) follows that with c = -b/g , and eq. (2.11) is solved by Moreover, substituting in (3.12) one gets the condition In this case, e ρ ∼ e -χ + dξ and vanishes for ξ → 0. As in the previous case, a horizon is present if e ρ goes to a constant at ξ →∞ , i.e. if d = a . It follows that b = g 2 a , c = -ga . Defining the coordinate r as in (3.8), with a suitable choice of the integration constant one obtains Passing to Schwarzschild coordinates (3.6), Also in this case, substituting in (3.7), one can easily check that µ is free, A and C can be set to 1, and The scalar curvature is proportional to and hence diverges at r = 0. The properties of the solutions depend on the sign of the free parameter µ , which dictates the asymptotic behavior. An asymptotic region is present only if µ > 0, with U ∼ r -(1 -g 2 ) / (1+ g 2 ) . In this case, a horizon is present at r = -µ , but a naked singularity occurs at the origin, so these are not regular black brane solutions. For µ < 0, the horizon is of cosmological type, but a naked singularity is still present. Therefore, no regular black brane solutions exist in this case.", "pages": [ 7, 8 ] }, { "title": "C. Special case χ ' = η '", "content": "Finally, an exact special solution can be found for generic values of the parameters of the action, when χ ' = η ' , i.e. χ = η +log K , with constant K . In this case, with b an integration constant. Moreover, comparing (2.10) and (2.11), one gets and (2.12) becomes where with α , γ 1 and γ 2 given by (2.13). Solving (2.10), with Q 2 e 2 χ = γ 2 γ 1 λ e 2 η , one then obtains with a an integration constant. Substituting this result in (3.20), one gets b 2 = ( g + h ) 4 a 2 . The radial function is given now by e ρ = e [( g + h ) 2 η + bξ ] /α , and if α is positive diverges for ξ → 0, signalling the presence of an asymptotic region. The request that e ρ goes to a constant for ξ →-∞ , necessary to ensure the presence of a horizon, enforces instead the choice b = -( g + h ) 2 a . Defining then the radial coordinate r as in (3.8), one obtains Again, one can write the metric in the form (3.6). One has The constants can be determined substituting in (3.7). In particular, one of them, say C , can be set to 1, while it is easy to check that µ is a free parameter. The remaining constants result in These solutions have been obtained in different coordinates in [2]. Also in this case the scalar curvature is the sum of two terms, one proportional to r -4 h ( g + h ) /β , and the other to µr -(4+3( g + h ) 2 ) /β . While the second term can diverge only at r = 0, the first is singular either at r = 0 or at r = ∞ , depending on the value of the parameters g and h . The properties of the solutions depend on the values of the parameters g , h and λ . It turns out that solutions which are regular at infinity and possess a regular horizon are possible only if λ , α , γ 1 and γ 2 are all positive and h ( g + h ) > 0. In that case the solutions (3.24) describe charged black branes, with horizon at r h = µ β/α and hyperscaling-violating asymptotic behavior, that in the special case g = 2 /h -h is enhanced to that of a domain wall.", "pages": [ 8, 9, 10 ] }, { "title": "4. Thermodynamics", "content": "Before passing to the study of the global structure of the space of solutions of the system (2.10)-(2.12), we calculate the thermodynamical parameters of the exact black brane solutions found in the previous section. We omit the λ = 0 case, since it always contains naked singularities.", "pages": [ 10 ] }, { "title": "A. Q = 0 .", "content": "These solutions describe black branes if λ > 0 and h 2 < 3. In general, the mass density m can be defined as [10] ∣ where we have subtracted a term corresponding to the solution evaluated on the background with r h = 0. For the solutions (3.10) this gives The temperature T can be obtained calculating the periodicity of the Euclidean section, as In our case Finally, the entropy density s is simply obtained as Hence, It follows that dm = Tds and m = Ts 3 -h 2 . For m → 0, the entropy of the brane vanishes, while the temperature vanishes if h 2 < 1 and diverges if h 2 > 1. When h 2 = 1 the temperature is independent of the mass.", "pages": [ 10 ] }, { "title": "B. χ ' = η '", "content": "These solutions are valid if λ, α, γ 1 , γ 2 > 0 and h ( g + h ) > 0. Using the previous formulae, one obtains for the solutions (3.23) and It is easy to see that dm = Tds and m = 2( g + h ) 2 α Ts . Also in this case, for m → 0 the entropy vanishes, while the temperature vanishes if α > 2( g + h ) 2 or diverges otherwise.", "pages": [ 10, 11 ] }, { "title": "5. The dynamical system.", "content": "For arbitrary values of the parameters g and h , following the methods of ref. [21,1], eqs. (2.10),(2.11) and (2.17) can be put in the form of a dynamical system, by defining X = χ ' , Y = η ' , P = Q e χ , Z = √ | λ | e η , with The independent variables are X , Y and Z , while P 2 is defined as where α is given by (2.13) and /epsilon1 = +1 if λ > 0 or /epsilon1 = -1 if λ < 0. For each value of the free parameters b and Q , we can now discuss the structure of the phase space of the dynamical system. In the following discussion we shall not consider the limit cases Q = 0 and λ = 0, since these have already been examined in sect. 3. Their properties can however be useful in order to understand the general case. The details of the phase space depend on the location of the critical points. All the solutions of interest connect a critical point at finite distance with a critical point at infinity. We shall therefore undertake the study of these points. Since the system is invariant for Z →-Z , it is sufficient to discuss the Z > 0 portion of phase space. Moreover, the system is invariant for X →-X , Y →-Y , together with ξ →-ξ , and hence to each critical point with X > 0 corresponds a critical points with X < 0, with the direction of the trajectories reversed. Therefore, in the following we list only the critical points with positive X .", "pages": [ 11 ] }, { "title": "Critical points at finite distance", "content": "For positive X , the critical points at finite distance are attained in the limit ξ →-∞ . In the following discussion it will be important to study the behavior of the radial function e ρ near the critical point, since it indicates to what physical region these points correspond. While in the most common cases [21] the critical points at finite distance correspond to a finite value of e ρ and hence to a region of spacetime at finite distance, in the present case, for some values of the parameters, they may correspond to e ρ →∞ , and hence to spatial infinity. The critical points at finite distance lie at Z 0 = P 0 = 0, and hence their coordinates X 0 , Y 0 satisfy the equation If α > 0, this is the equation of a hyperbola in the Z = 0 plane, while for α < 0 it represents an ellipse. In the degenerate case α = 0, the set of critical points is given by a pair of straight lines. Another limit case is b = 0: also in this case the critical points lie on a pair of straight lines. The characteristic equation for small perturbations around these critical points has eigenvalues 0, 2 X 0 and Y 0 . Hence, for a given value of b , each point in the Z = 0 plane satisfying (5.3) with X 0 > 0, Y 0 > 0, repels a 2-dimensional bunch of solutions in the full 3-dimensional phase space. The points with X 0 > 0, Y 0 < 0 act instead as saddle points. The presence of a vanishing eigenvalue is due of course to the fact that there is a continuous set of critical points lying on a curve. Integrating (2.16) when α = 0, one gets the expression for the radial function, /negationslash Recalling that for ξ →-∞ , e χ ∼ e X 0 ξ and e η ∼ e Y 0 ξ , it is easy to check that the function e ρ may vanish or diverge as ξ →-∞ , depending on the sign of ρ 0 = γ 1 Y 0 -γ 2 X 0 -b . In the special case ρ 0 = 0, instead, e ρ goes to a constant value, signalling the presence of a horizon. Combining ρ 0 = 0 with (5.3), one obtains the only real solution X 0 = Y 0 = b/ ( g + h ) 2 . Therefore, the critical points satisfying that condition correspond to a horizon, and all the trajectories starting from there can describe black brane solutions. The other critical points with positive X 0 and Y 0 describe instead either naked singularities or spatial infinity and hence the trajectories starting from those points cannot describe black brane solutions. This is confirmed by the computation of the Ricci scalar R in the limit ξ →-∞ : it either vanishes or diverges, except when X 0 = Y 0 , in which case it takes a finite value.", "pages": [ 11, 12 ] }, { "title": "Critical points at infinity", "content": "To complete the analysis of the phase space it is necessary to investigate the nature of the critical points on the surface at infinity. These points are attained for a finite value of ξ [21], and like the critical points at finite distance may correspond to the asymptotic region of the physical solution or to a singularity at finite distance, depending on the value of the parameters. The analysis of the phase space at infinity can be performed defining new coordinates u , y , and z such that the surface at infinity is obtained in the limit u → 0, i.e. X →∞ : In these coordinates, eqs. (5.1) take the form where we have defined p = P/X and a dot denotes ud/dξ . The discussion of the properties of the critical points at infinity is complicated, because it strongly depends on the values of the parameters g and h . The limits α = 0 and g = -1 /h are degenerate. In particular, the latter is completely integrable and is solved in the appendix. It is convenient to discuss separately the case of positive and negative λ .", "pages": [ 12 ] }, { "title": "A. λ > 0", "content": "For positive λ , the critical points with u = 0 can be classified in four categories: The point L is the endpoint of the trajectories lying in the Z = 0 plane. The analysis of stability shows that it acts as an attractor on the trajectories coming from finite distance and, if γ 1 > 0, also on a two-dimensional bunch of trajectories lying on the surface at infinity (otherwise as a saddle point). This is the endpoint of the trajectories lying on the hyperboloid P = 0. The analysis of stability shows that, if 1+ gh > 0, it attracts the trajectories coming from finite distance and a one- or a two-dimensional bunch of those lying on the surface at infinity, depending on the sign of γ 2 . √ The point N is the endpoint of the hyperboloid (5.2) in the X = Y plane. All the eigenvalues containing are real and if α, γ 1 , γ 2 < 0 are negative, and N acts as an attractor both on the trajectories coming from finite distance and on the trajectories at infinity. In the other case, it acts as a saddle point for the trajectories at infinity. The points Q are the endpoints of the trajectories with P = Z = 0. They act as centers on the trajectories coming from finite distance, while their nature for the trajectories at infinity depends on the sign of 3 -h 2 and 1 + gh .", "pages": [ 13 ] }, { "title": "B. λ < 0", "content": "For negative λ , the critical points at u = 0 can again be divided in four categories:", "pages": [ 13 ] }, { "title": "Asymptotic properties of the solutions", "content": "As mentioned above, critical points at infinity correspond to the limit ξ → ξ 0 , where ξ 0 is a finite constant. It is easy to see that for ξ → ξ 0 , the functions χ and η behave as where v 0 ≡ (1 + gh ) /epsilon1z 2 0 +(1 + g 2 ) p 2 0 . In order to discuss the properties of the solutions, one must first of all investigate the behavior of the radial function e ρ , that at infinity behaves as | ξ -ξ 0 | -γ 1 y 0 -γ 2 αv 0 . Its behavior near the critical points is reported in table 1. It results that e ρ diverges at points M and at points N if α > 0, indicating the presence of an asymptotic region. In all other cases, e ρ vanishes at the critical points at infinity. In order to investigate the causal structure of the solutions, it is also useful to compute the behavior of the Ricci scalar R at the critical points. It can be shown that this is given by and is reported in table 1 as well. The curvature is always regular at points M , and at points N if α > 0 and g ( g + h ) > 0, otherwise it diverges. From (5.7) and (2.16), (2.18), one can also deduce the behavior of the metric functions ending at the critical points. It is useful to write them in terms of the Schwarzschild coordinate r defined as ∫ e 2 ν +2 ρ dξ in (3.8), whose asymptotic behavior for ξ → ξ 0 is given in table 1 and is analogous to that of e ρ . Straightforward algebraic manipulations then lead to the asymptotic behavior of the metric functions listed in table 2. The parameter β is defined in (3.23). | - | | - | | - | Although the critical points at infinity do not always correspond to asymptotic regions of the solutions, they always give the asymptotic behavior of the background solutions, i.e. of those solutions that do not depend on free parameters. For the general solutions instead, as seen in sect. 3, the mass term can dominate at infinity on the background term, modifying the asymptotic behavior. In fact, the behaviors of the generic solutions in table 2 coincide with those of the exact background solutions found in sect. 3: more precisely, point M corresponds to case A , point L to case B , and point N to case C . From the results listed above, we can deduce the global properties of the solutions in terms of the values of the parameters λ , g and h . The following picture of the phase space emerges: solutions with regular horizons are described by trajectories that connect the point of the hyperbola (or ellipse if α < 0) of critical points at finite distance such that X 0 = Y 0 = b/ ( g + h ) 2 , b > 0 with one of the critical points at infinity. Among these trajectories, only those that end at critical points at infinity for which e ρ → ∞ , R → 0 correspond to regular black branes. It follows that regular black brane solutions exist only if λ > 0, with asymptotic behavior of type M , if h 2 < 3, or N if α, γ 1 , γ 2 > 0 and g ( g + h ) > 0. Under these conditions the critical points at infinity attract either a 1-dimensional or a 2-dimensional bunch of trajectories in phase space. Each solution is associated with the two free parameters b (related to the mass) and Q (electric charge). In general, a third parameter may be necessary to parametrize the solutions. However, as emerges from the discussion of the exact special solution in the appendix, consistency of the thermodynamical interpretation may require that this parameter be related to the electric charge. Also interesting is the possibility that solutions presenting different asymptotic behaviors exist for given values of the parameters λ , g and h .", "pages": [ 14, 15 ] }, { "title": "6. Final remarks", "content": "Although in the general case it is not possible to obtain the planar solutions of the EML model in analytic form, we have discussed their global properties and classified all the possible regular black brane solutions in four dimensions. Although several possibilities may arise, depending on the values of the parameters λ , g and h that define the model, we have been able to show that regular black brane solutions can exist only for a very limited range of parameters. In particular, no regular solution exists if λ ≤ 0. Moreover, only two kinds of asymptotic behavior allowed: one of them is common with the limit of vanishing charge, while the other is characteristic of the general case. For given values of the parameters, solutions presenting both asymptotic behaviors can exist. No asymptotically flat, anti-de Sitter or Lifshits solutions arise, except in the trivial case h = 0. The general analytic solution can be found for vanishing charge or potential, and for special values of the parameters g and h . In addition, also some special exact solutions can be found for generic values of the parameters. Some of these solutions had already been obtained in the literature [2], but not in full generality and in awkward coordinates.", "pages": [ 15 ] }, { "title": "Acknowledgements", "content": "I wish to thank Mariano Cadoni for some useful discussions.", "pages": [ 15 ] }, { "title": "APPENDIX", "content": "When g = -h -1 , the system can be completely integrated. This has been partially done in [2], but with a choice of coordinates that obscures the structure of the solutions. Our discussion illustrates the possibility that for a given value of h solutions exhibiting different asymptotic behaviors exist, as it has been deduced in the general case from the study of the phase space. For g = -h -1 , the system (2.5)-(2.6) diagonalizes, and is solved by and with a , b , c and K integration constants. Substituting in the constraint (2.12), and requiring the existence of a regular horizon, one gets /negationslash Defining r = ∫ e 2 ν +2 ρ dξ ≈ (1 -e 2 aξ ) -(1+ h 2 ) / (1 -h 2 ) , the K = 1 solution can be put in the Schwarzschild form (3.6) with One must now distinguish the generic solutions with K = 1 from the special solutions with K = 1. where δ = (3 -h 2 ) / (1 + h 2 ) and µ is a free parameter. These solutions exist for h 2 < 3 if λ is positive and for h 2 > 3 if λ is negative. In the first case, they represent a black brane with domain wall asymptotics, parametrized by the mass density m = 1 h √ (1+ h 2 ) 5 (3 -h 2 ) 3 λQ 2 µ and the charge density Q . In the second case, a naked singularity is present for r →∞ . If K = 1, proceeding in the same way, one obtains /negationslash The interpretation of this solution is however not easy, in particular for what concerns its thermodynamics, because of the factor 1 / √ ν ( ν + µ ) in R 2 , that implies that there is no These solutions present a further free parameter ν = 0, that can be associated to the scalar charge. For µ, ν > 0, they describe a 3-parameter family of black branes with domain wall asymptotic behavior different from that of the K = 1 solutions, which are recovered in the singular limit ν →∞ . The curvature is finite at the point r = µ 1 /δ , that can be identified with a horizon, while it diverges at r = 0 and r = -ν , which are singularities of the metric. /negationslash ground state corresponding to ν = 0. A similar situation occurs for the neutral black branes discussed in [14]. In our case, the problem can be solved imposing that ν ( ν + µ ) = σQ 2 , with σ an arbitrary normalization, thus reducing the number of free parameters. While for spherically symmetric solutions this condition is dictated by the normalization of the volume element, in the planar case it must be imposed by hand. Notice that this choice implies that µ and ν must be positive, in order to avoid naked singularities. The special cases h 2 = 1 , 3 should be studied separately, since for these values of the parameter h some degeneracies appear in our calculations, but the generalization is straightforward, and we do not consider it in detail.", "pages": [ 15, 16, 17 ] } ]

[ { "title": "Matsuo S ato 1", "content": "Department of Natural Science, Faculty of Education, Hirosaki University Bunkyo-cho 1, Hirosaki, Aomori 036-8560, Japan", "pages": [ 1 ] }, { "title": "Abstract", "content": "We construct a Lie 3-algebra extended model of the IIB matrix model. It admits any Lie 3-algebra and possesses the same supersymmetry as the original matrix model, and thus as type IIB superstring theory. We examine dynamics of the model by taking minimal Lie 3-algebra that includes u(N) Lie algebra as an example. There are two phases in the minimally extended model at least classically. The extended action reduces to that of the IIB matrix model in one phase. In other phase, it reduces to a more simple action, which is rather easy to analyze.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The IIB matrix model [1] is a convincing candidate of non-perturbative type IIB superstring theory. It possesses the same space-time supersymmetry as the corresponding string theory. The existence of graviton in this model is guaranteed by the thirty-two supercharges that generate the N = 2 supersymmetry in ten dimensions. Although the IIB matrix model is known to give some perturbative and non-perturbative dynamics in string theory, it is difficult to examine the matrix model because of its numerous interactions. We need many ideas to study non-perturbative aspects of string theory. Extending the model is one way to provide new ideas to understand not only non-perturbative aspects of string theory but also the original matrix model. In this paper, we construct a supersymmetric model that is an extension of the IIB matrix model. 'Extension' means that the model possesses a 3-algebraic structure [2-47] that includes a Lie-algebraic structure 1 . The extended model admits any Lie 3-algebra, whose triple product is totally antisymmetric. It possesses the same supersymmetry as the IIB matrix model, namely as type IIB superstring theory. Therefore, the existence of graviton is guaranteed in the extended model. We also study dynamics of the extended model by taking the minimal Lie 3-algebra that includes u ( N ) algebra as a concrete example. In this model, we find that there are two phases at least classically. While it reduces to the original IIB matrix model in one phase, it reduces to a more simple model in another phase. This simple model is rather easy to examine. The organization of the rest of this paper is as follows. In section 2, we construct an extended IIB matrix Model that admits any Lie 3-algebra. In section 3, we choose the minimal Lie 3-algebra that includes u ( N ) algebra and study the model with that algebra as a concrete example. In section 4, we elucidate a phase structure of the extended model. In section 5, we conclude and discuss on the model.", "pages": [ 2 ] }, { "title": "2 Extended IIB Matrix Model", "content": "The IIB matrix model possesses the same supersymmetry as type IIB superstring theory, namely chiral N = 2 spacetime supersymmetry in ten dimensions [1]. The model also possesses u(N) gauge symmetry, which is conjectured to include diffeomorphism of the tendimensional spacetime. In this section, we construct an action that possesses the same supersymmetry as above. Let us consider the Lie 3-algebra-valued scalar Φ, SO(1,9) vector X M ( M = 0 , · · · 9) and SO(1,9) Majorana-Weyl fermion Θ that satisfies Γ 10 Θ = -Θ. We examine dynamical supertransformations, where Weconstruct a supersymmetric model by assuming no specific Lie 3-algebra but by using only totally antisymmetry of the three products, invariance of the metric and the fundamental identity, which is an analogue of Jacobi identity. This transformations give a supersymmetry algebra, The right-hand sides of second and third lines represent gauge transformation of X M and Θ, respectively. This algebra closes on-shell if we treat as the equation of motion of Θ. By transforming this equation under (2.1), we have the equation of motion of X M , These equations of motion are derived from This action is invariant under (2.1). There is no ghost in this model because this action does not include the elements of a center. There is another supersymmetry of the action (2.6), which is called kinematical supersymmetry, generated by and the other fields are not transformed. Let us summarize the supersymmetry algebra which (2 . 6) possesses. First, the commutator of the dynamical supersymmetry transformations gives on-shell and up to the gauge symmetry. The commutator of the dynamical ones gives The commutator of the dynamical and kinematical ones gives If we define we obtain and This result means that 32 supertransformations ∆ = ( δ ' , ˜ δ ' ) form the algebra of SO(1,9) N = 2 chiral supersymmetry, which is the same supersymmetry algebra of the IIB matrix model and thus of type IIB superstring theory.", "pages": [ 3, 4, 5 ] }, { "title": "3 Minimally Extended Model", "content": "In the previous section, we constructed an extended model that admits any Lie 3-algebra. In the following, we consider the minimal Lie 3-algebra that includes u ( N ) algebra. Non-zero commutators are where [ T i , T j ] is the Lie bracket. Non-zero components of the inverse of a metric g AB = < T A T B > are given by where h ij is Cartan metric of the Lie algebra. The action (2 . 6) with this minimal Lie 3-algebra can be written in a Lie-algebra manifest form, Let us consider backgrounds /negationslash where ¯ Φ 0 and ¯ X M 0 take arbitrary values. These backgrounds are BPS and thus stable. The fluctuations of Φ 0 , X M 0 and Θ 0 are zero modes around these backgrounds as one can see in (3.3). Therefore, we should treat each of these backgrounds as independent vacuum and fix the zero modes. We elucidate the theories around these vacua in this and next section. In this section, we consider Φ 0 = 0 phase. Next, we study gauge symmetry of the action. The 3-algebra manifest form is where X represents X M , Φ and Θ. We can rewrite it in the Lie algebra manifest form, where Λ (1) k i = 2Λ 0 j f jk i and Λ (2) i = Λ jk f jk i are independent gauge parameters. Λ (1) represents the u(N) gauge transformation, whereas Λ (2) represents a shift transformation. In /negationslash Φ 0 = 0 phase, by utilising the shift transformation, one can choose Φ i = 0 gauge. In this gauge, where Y and Z stand for any field. As a result, we have By field redefinitions of X M and Θ, we obtain the action of the IIB matrix model, /negationslash That is, in Φ 0 = 0 phase, the minimally extended model is equivalent to the IIB matrix model.", "pages": [ 5, 6, 7 ] }, { "title": "4 New Supersymmetric Action", "content": "In this section, we study Φ 0 = 0 phase. In this case, the vacuum can be chosen without loss of generality as where µ = 0 , · · · , 8 . In this vacuum, the action reduces to By redefining ¯ X 9 0 Φ as Φ and renaming Γ 9 Γ, we obtain This action is invariant under a dynamical supertransformation inherited from (2.1), where ˜ Γ µ = 1 2 (ΓΓ µ -Γ µ Γ). The supersymmetry algebra is given by This algebra closes on-shell. This dynamical and kinematical supersymmetries form a supersymmetry generated by 32 supercharges in the same way as in the full action. This supersymmetry is consistent with that of type IIB superstring. We can analyze this phase rather easily because (4.3) possesses a simple form.", "pages": [ 7, 8 ] }, { "title": "5 Conclusion and Discussion", "content": "In this paper, we have constructed the extended IIB matrix model. The extended model admits any Lie 3-algebra. It possesses the same supersymmetries as the original model, thus, as type IIB superstring theory. As an initial step to elucidate dynamics of the extended model, we have examined the model with the minimal Lie 3-algebra that includes u(N) algebra. There are BPS backgrounds (3.4) specified by arbitrary fields Φ 0 and X M 0 , and the fluctuations of Φ 0 , X M 0 and Θ are zero modes around these backgrounds. One can treat each of the BPS configurations as independent vacuum and fix the zero modes if the BPS backgrounds are stable. As a result, at least in the classical level, we have gotten two phases. In one phase, the action reduces to the original IIB matrix model, whereas in another phase, it reduces to the new action (4.3). Naively, one can expect that the BPS backgrounds are exactly stable because the model possesses the maximal supersymmetry. However, some BPS backgrounds can be unstable quantum mechanically in low-dimensional field theories because of infrared divergence [49]. We need to examine whether the BPS backgrounds are stable or not when quantum corrections are taken into account, by calculating an effective action of Φ 0 and X M 0 . If the backgrounds are exactly stable, the minimally extended model has dynamics that are not described by the IIB matrix model. If the backgrounds become unstable at some order of quantum corrections, the minimally extended model is equivalent to the IIB matrix model. Even in this case, we can obtain some important dynamics of the IIB matrix model rather easily up to that order, by studying (4.3). In both cases, it is important to study dynamics described by (4.3). It is rather easy because (4.3) has a simple form. We can discuss dimensionality of the space-time which (4.3) determines, by calculating an effective action of eigen values of X M [49] or expectation values of tr( X M X N ) [50, 51]. In this paper, we have studied dynamics of the minimally extended model, whereas we have obtained the extended model admitting any Lie 3-algebra. In general, the extended model cannot be equivalent to the IIB matrix model. Finite-dimensional indecomposable metric Lie 3-algebras with maximally isotropic center are categorized in [37]. These algebras give non-vanishing potentials for Φ α and X M α . Thus, the extended model should possess various different dynamics.", "pages": [ 8, 9 ] }, { "title": "Acknowledgements", "content": "We would like to thank T. Asakawa, K. Hashimoto, N. Kamiya, H. Kunitomo, T. Matsuo, S. Moriyama, K. Murakami, J. Nishimura, S. Sasa, F. Sugino, T. Tada, S. Terashima, S. Watamura, K. Yoshida, and especially H. Kawai and A. Tsuchiya for valuable discussions. This work is supported in part by Grant-in-Aid for Young Scientists (B) No. 25800122 from JSPS.", "pages": [ 9 ] } ]

[ { "title": "One Dimensional Supersymmetric Algebras in Color Superconductors and Reissner-Nordstrom-anti-de Sitter Gravitational Systems", "content": "V. K. Oikonomou ∗ Max Planck Institute for Mathematics in the Sciences Inselstrasse 22, 04103 Leipzig, Germany August 29, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "We study two fermionic systems that have an underlying supersymmetric structure, namely a color superconductor and Dirac fermion in a Reissner-Nordstrom-anti-de Sitter gravitational background. In the chiral limit of the color superconductor, the localized fermionic zero modes around the vortex form an N = 2 with zero central charge d = 1 quantum algebra, with all the operators being Fredholm. We compute the Witten index of this algebra and we find an unbroken supersymmetry. The fermionic gravitational system in the chiral limit too, has two underlying unbroken N = 2, d = 1 supersymmetric algebras. The unbroken supersymmetry in the later is guaranteed by the existence of fermionic quasinormal modes in the Reissner-Nordstrom-anti-de Sitter background. In this case the operators are not Fredholm and regularized indices are deployed.", "pages": [ 1 ] }, { "title": "Introduction", "content": "During the last decade the research towards the interrelation of gravity and condensed matter physical systems has received considerable attention, especially the study of holography in such systems [1-6]. This research stream was further enhanced by the experimental verification of certain condensed matter systems that have topological origin [7], an observation which was done very recently actually [8-10]. Particularly these where time reversal symmetric extensions of the famous topological originating quantum Hall effect in two [11,12] (topological quantum spin Hall effect) and three dimensions [13-15], known as topological insulators [7]. Of course, the quantum Hall effect is the most famous topologically non trivial state of matter in two dimensions [16]. A non-trivial charge of the single-particle Hamiltonian is intrinsic to a topological state of matter. Along with the topological insulators, the topological superconductors serve as another class of topological states of condensed matter, particularly the p x + ip y ones. Moreover gapless localized fermions appear around the vortex core of a vortex defect in topological superconductors. Such theoretical constructions are realized in the surface of three dimensional insulators [7,17]. The most extreme condensed matter states can occur at ultra high quark densities. Of course this is no ordinary matter since the quarks are deconfined, but we are talking about condensed matter physics of QCD. At ultra high densities the QCD coupling is relatively small, a situation that can physically occur in neutron stars for example. In these conditions (low temperature, high density) the quarks may form cooper-like pairs, thus breaking explicitly the gauge symmetry, and forming a so-called color superconductor. Bearing in mind that matter states can be topological in ordinary superconductors, it is natural to ask whether a color superconductor is topological. This questions has been answered and the answer is in the affirmative [7]. In relation to this cooper pair condensation, a color superconductor might be related to a condensation that fermionic fields might experience in involved AdS gravitational backgrounds. A similar phenomenon to a holographic superconducting phase transition [1-6] was studied in [18], but for an ReissnerNordstrom-anti-de Sitter black hole spacetime, and the results where consistent with the holographic superconducting phase transition. Particularly in [18] the order parameter is an Dirac fermion charged through a direct coupling to a Maxwell field. In this sense, this model of a charged Dirac fermion in the background of an Reissner-Nordstrom-anti-de Sitter black hole, could serve as a simple model of color superconductivity. The approach of the authors of [18] is done by computing the quasinormal modes of the charged Dirac fermion field in the aforementioned curved background. As is well known, quasinormal modes [19-22], describe a long lasting period of damped gravitational waves oscillations. The quasinormal modes are so to speak, the characteristic sound of black holes hence matter field perturbations of such gravitational backgrounds can be a useful tool to study black holes. Moreover the quasinormal modes depend only on a few physical parameters of the black hole, namely, the mass, angular momentum and charge of the black holes, thus rendering the spacetime parameters identification easier. The perturbation of a quantum field in a black hole background consists of three time evolution stages, that is, the wave burst, the quasinormal mode stage and the power-law tail [23]. For the calculation of a Dirac fermion quasinormal modes in a Reissner-Nordstrom-anti-de Sitter Spacetimes in D = 4 and D > 4 we refer the reader to references [24], and [25] respectively. In addition for Dirac quasinormal modes in curved backgrounds see also [26-28]. Owing to the vast number of applications and implications of many theoretical frameworks that embody quasinormal modes, research in this area has attracted a lot of attention. Firstly the existence of a black hole can be directly verified by observing it's fundamental quasinormal mode. Additionally the thermodynamic properties of loop quantum gravity (an appealing alternative to string theory) black holes can be further understood using the quasinormal modes. Moreover the quasinormal modes of anti de Sitter black holes have a dual physical correspondence to quantities of the dual conformal field theory via the well known AdS/CFT correspondence [34]. From astrophysical aspects, the most interesting spacetimes are the asymptotically flat ones, however the observation of the universe's expansion motivated the study of quasinormal modes in de Sitter [29]. Quasinormal modes can yield which gravitational systems are stable under dynamical perturbations. Actually a static or non-static solution describing a compact object is stable if all it's quasinormal modes are decaying in time, on the contrary even if one mode is growing, the gravitational system is unstable [22]. Owing to the fact that the charged Dirac fermion in the background of an ReissnerNordstrom-anti-de Sitter black hole could be a simple model of color superconductivity, we present in this paper that in both systems, namely the fermionic spectrum around the boundary vortex of a color superconductor and the fermion in the Reissner-Nordstromanti-de Sitter black hole spacetime, there exists a hidden N = 2 supersymmetric quantum mechanics algebra [30-33] (SUSY QM hereafter). Particularly, for the color superconductor system, the supersymmetric algebra occurs for the m = 0, p z = 0 (chiral case) case of the Bogoliubov-de Genne equation. In the case of the fermionic field in the curved gravitational background, the supersymmetric algebra is very closely related to the quasinormal modes spectrum, and the very existence of supersymmetry is guaranteed by the existence of quasinormal modes. In the case of color superconductivity, the supersymmetry is due to the vortex, which actually causes localized fermionic solutions around it. Supersymmetric structures of the same kind around defects where studied in [35, 36] where the case of a superconducting and a cosmic string where analyzed respectively. Supersymmetry in the case where fermionic quasinormal modes are studied in various gravitational backgrounds, was investigated in [37]. In the case we shall present in this paper, the fermionic system actually has two N = 2 d = 1 supersymmetries, the supercharges of which could be related to harmonic superspace extensions [38-46]. The supersymmetric quantum mechanics algebras are very important from physical and mathematical point of views, since these can be directly connected to harmonic superspace [38-46] and to d = 1 supersymmetric sigma models with very interesting target space geometries. Furthermore, these supersymmetric extensions provide superextensions of the Landau problem and of the quantum Hall effect [47-49]. In addition N = 2 d = 1 supersymmetry appears in condensed matter systems, like in graphene for example, see [50]. In addition, there is a close connection of photon, gravitino and graviton modes from extremal Reissner-Nordstrom black holes, which is expressed in terms of an isospectrality in the spectrum [51-53]. Although these works investigate systems in the context of supergravity, a supersymmetric quantum mechanics algebra could be a remnant of local supersymmetry. It is surprisingly interesting that the color superconductors and the fermionic system in Reissner-Nordstrom-anti-de Sitter black hole (a model that is believed to the gravitational description of color superconductivity) are linked via the same supersymmetric underlying pattern. However, these supersymmetries are different, owing to the fact that in the case of color superconductors, the operators are Fredholm, while in the case of the gravitational system that is not true. This paper is organized as follows: In section 1 we present the color superconductor model and the underlying N = 2 SUSY QM algebra. In section 2 we study the charged Dirac fermion in the background of an Reissner-Nordstrom-anti-de Sitter black hole and we present the structure of the resulting two SUSY QM. At the end of section 2 we present a global symmetry that the aforementioned fermionic system possess. The conclusions follow thereafter.", "pages": [ 1, 2, 3 ] }, { "title": "1 Superconductors and Vortices", "content": "In this section we study the underlying supersymmetry that the fermionic system that describes color superconductivity has. We start with the mean-field model of color superconductivity, it's benchmark of which is the Hamiltonian [7]: with α, β and γ 5 being equal to: In the above equation (1), the matrix C stands for the charge conjugation matrix, namely C = iγ 2 γ 0 , where γ i are the Dirac gamma matrices. The model that is described by the aforementioned Hamiltonian, contains three colors and three flavors, which are denoted by the letters a, b and f, g respectively, in the Hamiltonian (1). The pairing gap is described by ∆ ab,fg , in the Lorentz singlet and even parity channel ( J p = 0 + ). Its specific dependence on the color and flavor is described by: The Hamiltonian (1) after a orthogonal transformation in the color-flavor space, can be brought in the decoupled form, with H SCS = ∑ 9 i H j , with H j being equal to: /negationslash The case where ∆ = 0 describes a fully gapped color flavor locked phase. The Hamiltonian (4) possesses many symmetries, like the charge conjugation symmetry and the time reversal symmetry. Such Hamiltonian have symmetry properties that have been classified and tabulated formally, see for example [54, 55]. The single particle Hamiltonian H has the following charge conjugation symmetry: with C being equal to: Moreover when ∆( x ) is a real number and in addition has a uniform phase over the space, the Hamiltonian has the following transformation properties: where T stands for: It is a common fact in the superconductor literature that for an 2 Dp x + ip y superconductor, the non-trivial topological charge of the free space Hamiltonian is closely to a localized fermionic zero mode around a vortex line (see [7] and references therein). Same arguments hold for the Hamiltonian (4). We are interested in zero mode fermionic solutions around vortices, with a non-trivial pairing gap, in the even parity pairing case 1 . The theoretical context that underlies the calculation of the fermionic spectrum around a quantized vortex line is pretty much described by the Bogoliubov-de Genne equation, namely: where we employed polar coordinates to be our coordinate system. The above Hamiltonian describes the free space one particle Hamiltonian with pairing gap ∆( x ) = e iθ | ∆( r ) | and under the assumption that the vortex line extents in the z -direction and also that the pairing gap does not depend on z . Additionally, | ∆( r ) | is required to obey lim r →∞ | ∆( r ) | > 0, or in words it is required to have a positive non-vanishing asymptotic value. Hence any localized fermion solutions that we will find in the following, can be considered independent of the vortex, a fact that entails some sort of universality of the solutions (see also the comment at end of the present section). The zero modes we shall present have a purely topological origin [7] in contrast to other solutions describing bound fermions of Caroli-de Gennes-Matricon type with vortex dependent solutions [7,56]. The solution to the above equation (9) look like: We shall be mainly interested in the case m = 0 and p z = 0, and particularly in the zero energy Bogoliubov-de Genne equation at m = p z = 0. The last case is the so-called chiral limit, in reference to m = 0. The solutions of this equation will actually be the localized zero modes around the vortex line. We can classify the solutions of the zero mode (E=0) Bogoliubov-de Genne equation to left handed and right handed fermion solutions according to their γ 5 parity. These solutions are exponentially localized solutions around the vortex and are equal to [7]: in reference to the right handed one, while the left handed one takes the form [7]: The above fermionic system, which is based on the zero modes solutions of the Bogoliubovde Genne equation, namely: can constitute an N = 2 supersymmetric quantum mechanics algebra ( N = 2 SUSY QM hereafter). To see this, let us briefly present the basic features of an unbroken N = 2 SUSY QM algebra. The generators of the N = 2 algebra are the two supercharges Q 1 and Q 2 and a Hamiltonian H , which obey [30-33], The supercharges can be used to define the new supercharge, and the its adjoint, The new supercharges satisfy, and additionally, A very important element of the algebra is the Witten parity, W , defined as, which anti-commutes with the supercharges, Additionally W satisfies the following, The main utility of the Witten parity W , is that it spans the Hilbert space H of the quantum system to positive and negative Witten parity subspaces, that is, Hence, the quantum system Hilbert space H can be written H = H + ⊕H -. For the present case we shall choose a specific representation for the operators defined above, which for the general case can be represented as: with I the N × N identity matrix. Recalling that Q 2 = 0 and {Q , W } = 0, the supercharges can take the form, The N × N matrices A and A † , serve as annihilation and creation operators, with, A : H -→ H + and also A † as, A † : H + → H -. Based on relations (23), (24), (25) the Hamiltonian H , can take a diagonal form, Hence the total supersymmetric Hamiltonian H that describes the supersymmetric system, can be written in terms of the superpartner Hamiltonians, and Consequently, and also, For reasons that will be immediately clear, we define the operator P ± , the eigenstates of which, | ψ ± 〉 , satisfy the following relation: Therefore we call them positive and negative parity eigenstates, parity referring to the P ± operator. Representing the Witten operator as in (23), the parity eigenstates can be cast in the following representation, with | φ ± 〉 /epsilon1 H ± . Using the formalism we just exploited, we construct an N = 2 SUSY QM algebra using the fermionic system around the vortex. The Bogoliubov-de Genne equation can be written as: with D being equal to: Based on the above matrix, we can built a supersymmetric algebra. Indeed, the adjoint of D is equal to, The zero modes equation for this matrix is D † Φ ' ( x ) = 0. The supercharges of the SUSY QM algebra, Q and Q † can be defined in terms of D and D † as follows, Moreover, the quantum Hamiltonian of the SUSY QM system is, It is easy to see that the supercharges (36) and the Hamiltonian and (37), the following relations: But the most interesting feature of this color superconductor related supersymmetric quantum system is that the underlying N = 2 supersymmetric quantum mechanical system, has unbroken supersymmetry. Supersymmetry is unbroken for a quantum mechanical system if there exists at least one quantum state in the Hilbert space, | ψ 0 〉 , with vanishing and also, energy eigenvalue, that is H | ψ 0 〉 = 0. In turn, this entails that Q| ψ 0 〉 = 0 and Q † | ψ 0 〉 = 0. For a negative parity state this implies, or equivalently A | φ -0 〉 = 0. Moreover for a positive parity ground state we have, or equivalently A † | φ + 0 〉 = 0. Whether supersymmetry is unbroken or not, is very closely related to the number of zero modes of the system. Zero modes are perfectly described by the Witten index. Let n ± be the number of zero modes of H ± in the subspace H ± . For a finite number of zero modes, n + and n -, we define the Witten index of the system to be, /negationslash In the case the Witten index is an non-zero integer, supersymmetry is unbroken for sure. The case for which the Witten index is zero is much more involved. Indeed, if the Witten index is zero, it and if n + = n -= 0 supersymmetry is broken. Conversely, if n + = n -= 0 the system retains an unbroken supersymmetry. The definition for the Witten index we just gave, holds true for Fredholm operators only. An operator A is Fredholm, if it has discrete spectrum, a fact that is ensured if dim ker A < ∞ . By the same reasoning, if an operator is trace-class, this embodies the Fredholm feature for this operator [31]. Accordingly, the Fredholm index of the operator A is closely related to the Witten index with the former defined as, Indeed the relation between the aforementioned two indices is, As we shall see shortly, the operators D and D † defined in relations (34) and (35) are Fredholm for the localized solutions (the localization entails specific boundary conditions for the operators which in the end render the operators to be Fredholm) around the vortex. The vector space ker D is given by the solutions of the equation D Φ = 0, with the solutions Φ being zero at spatial infinity. The last property is equivalent to searching for localized solutions around the vortex. As we have seen earlier, the solutions of the equation D Φ = 0, are given by the solutions of the equation (33), which are the two solutions we found earlier, namely, φ R and φ L and are explicitly given by equations (11) and (12). Hence the two localized solutions constitute the space ker D for the operator D . In the same line of reasoning, the localized solutions of the equation D † Φ = 0 are given by: in reference to the right-handed solution, while for the left handed one we have: In the same line of argument as in the D operator case, the operator D † is also Fredholm with the two solutions ϕ ' L and ϕ ' R constituting the space ker D † . To make contact with the N = 2 SUSY QM algebra, the supercharges are defined in terms of the operators D and D † and the corresponding zero modes are classified according to their P ± parity as follows: The parity odd zero modes are (that is the zero modes of the operator D ), while the parity even states are (the zero modes of D † ): Correspondingly, the zero modes of the Hamiltonian, H are | ψ + 0 〉 1 , | ψ + 0 〉 2 , | ψ -0 〉 1 , | ψ -0 〉 2 . Since the two operators D and D † are Fredholm owing to the finiteness of their corresponding kernels, the Fredholm index of the operator D is given by: Hence, the Witten index of the corresponding SUSY QM algebra is given by: /negationslash Based on the fact that ker D = ker D † as we found previously, the Witten index of the SUSY QM algebra is zero. Note however that n -= n + = 0 (using the previously deployed notation) a fact that implies unbroken supersymmetry (for physical systems exhibiting similar behavior, that is unbroken SUSY with zero Witten index and other interesting attributes, consult references [59-64]). Let us recapitulate what we found up to now. From the fermionic system around a vortex that is constructed by the zero modes solutions of the Bogoliubov-de Genne equation, we can form an N = 2 supersymmetric quantum mechanics algebra with no central charge. The supercharges are constructed by the operators D and D † which as we proved are Fredholm, in the case the zero mode solutions are localized around the vortex.", "pages": [ 4, 5, 6, 7, 8, 9, 10, 11 ] }, { "title": "1.1 A Brief Comment", "content": "Before closing this section, we will address the problem of finding the Witten index in the case we change the pairing gap ∆( x ). For example let the new pairing gap ∆( x ) ' be related to the old pairing gap by: with ∆ 1 ( x ) = e iθ | ∆ 1 ( r ) | and lim r →∞ | ∆ 1 ( r ) | > 0. At the beginning of this section we mentioned that the localized fermionic solutions around the vortex have some sort of universality, stemming from the fact that the pairing gap does not depend on z . This issue, has its impact on the Witten index, and in fact we shall prove that if we change the pairing gap according to relation (50), the Witten index remains invariant. Hence although the solutions might change, supersymmetry remains unbroken. To see this, we shall make use of a theorem which states that, the Fredholm index of a Fredholm operator D , namely ind D remains invariant if we add a symmetric odd operator C to this Fredholm operator, that is: In our case, since the new pairing gap obeys lim r →∞ | ∆ 1 ( r ) | > 0, the odd symmetric operator has the following representation: Hence, the Fredholm index of the operator D , defined in relation (34), is invariant with ind( D + C ) = ind D . Thereby, the Witten index ∆ = -ind D is also invariant, and hence the same results as in the case corresponding to ∆( x ) hold, that is, supersymmetry is unbroken.", "pages": [ 11 ] }, { "title": "2 N = 2 SUSY QM and Massless Dirac Fermion Quasinormal Modes in Reissner-Nordstrom-anti-de Sitter black hole spacetimes", "content": "In this section we shall present a system of Dirac fermions in a gravitational background, from which we can construct an N = 2 SUSY QM algebra. The gravitational background is that of an Reissner-Nordstrom-anti-de Sitter black hole spacetime. This background is a potential candidate spacetime that can describe color superconductivity. In view of the AdS/CFT correspondences between gauge theory and gravity, the fact that the aforementioned gravitational system and the color superconductor fermionic system have an underlying N = 2 SUSY QM is rather useful. Hence, although the two models are independent at first sight, they have a common underlying symmetry pattern which can be useful. To be more specific, the supersymmetry we shall present shortly, is very closely related to the quasinormal modes of the Dirac fermionic field in the Reissner-Nordstrom-anti-de Sitter black hole background. The perturbation of a black hole can be achieved either by directly perturbing the gravitational background or by simply adding matter or gauge fields in the black hole spacetime [22]. In the linear approximation, the fermionic field has no back-reaction on the metric. The metric in a d-dimensional Reissner-Nordstrom-anti-de Sitter spacetime is given by: where, f ( r ) is equal to: In the above equation, L is the AdS radius, Q is the black hole charge, and r 0 is related to the black hole mass M . The dΩ 2 d -2 ,k is the metric of constant curvature, with k characterizing the curvature. The value k > 0 characterizes the metric of an d -2 dimensional sphere, while the k = 0 describes R d -2 . Finally when k < 0 it describes H d -2 . We shall focus on the flat case in this paper, since we would like to make contact to a superconductor on a plane. In the 4-dimensional case, the zero curvature Reissner-Nordstrom-anti-de Sitter metric is, The corresponding spin connection ω ˆ a ˆ bc , is equal to: where, e ˆ ad denotes the tetrad field, while Γ d fc denotes the Christoffel connection. The Einstein-Maxwell action for the Dirac fermion field equals to [18]: In the above action (57), G 4 is the 4-dimensional gravitational constant, R is the corresponding Ricci scalar, N is a total coefficient characterizing matter fields, and q is the coupling constant between the fermion field and the abelian gauge field A a . Additionally, the operator D a is: with Σ ˆ c ˆ b = 1 4 [Γ ˆ c , Γ ˆ b ], and the Dirac gamma matrices are related to the vierbeins as, Γ b = e b ˆ a Γ ˆ a . A solution of the equations of motion corresponding to the action (57) is: In order to extract the quasinormal mode spectrum corresponding to the Reissner-Nordstromanti-de Sitter black hole spacetime, we consider the limit in which the fermionic field does not backreact on the metric and the abelian field, as we also mentioned at the beginning of this section. The wave function solution Ψ( r, x µ ) can be written in the following form [18]: with x µ = ( t, x, y ) and /vectorx = ( x, y ). Using the above form of the function (60), the Dirac equation can be cast into the following form [18]: where /vector k · Γ ˆ /vectorx = k x Γ ˆ x + k y Γ ˆ /vector y . The Dirac gamma matrices can be written in the following representation: with I the identity matrix and σ i the Pauli matrices, namely: For later convenience, we decompose the fermion field Hilbert space to the chirality operator subspaces, that is: and P ± Ψ = ± Ψ ± , with P ± = 1 ± Γ 5 and Γ 5 = i Γ t Γ x Γ y Γ r . Using the eigenstates Ψ ± , the Dirac equations of motion can be cast as [18]: and also with Ψ + = ψ + e -iωt + i /vector k/vectorx and Ψ -= ψ -e -iωt + i /vector k/vectorx . The set of the above equations (65) and (66) is invariant under the transformation: In the rest of this paper we shall be interested in the chiral limit m = 0. This will result to an unbroken chiral symmetry for the system, which proves to be very important and could be an underlying link between the fermionic gravitational system and the color superconductor around a vortex system. We focus on the quasinormal modes of ψ + . We set k y = 0. This is because the symmetry that the system possesses on the ( /vectorx, /vectory )-plane. Upon rewriting ψ + as ψ + = r -1 f -1 / 4 ˜ ψ , the equation (66) can be simplified to the following one: Decomposing the fermionic field ˜ ψ , as: the above equation (68), can be recast in the following form: The above equations are invariant under the symmetry: Using these equations, namely (70) we can construct an N = 2 supersymmetric algebra. This algebra is founded on the matrix: acting on the vector: It is obvious that the zero modes of the matrix (72) yield the solutions of equation (70) with respect to ω . But these solutions correspond to the zero modes of the Dirac fermionic system. Therefore the zero mode solutions of the matrix (72) and the quasinormal modes of the Dirac fermionic system are in bijective correspondence. Thereby, the existence of quasinormal modes guarantees the existence of zero modes for the aforementioned matrix. The adjoint of the matrix D RN is equal to: Correspondingly, the supercharges of the N = 2 algebra Q RN and Q † RN are equal to: Moreover, the quantum Hamiltonian is equal to, The supercharges (75) and the Hamiltonian and (76), satisfy the equations (17) and (19), namely Hence the algebraic structure of an N = 2 SUSY QM algebra, underlies this fermionic system that corresponds to the solution ψ + (recall that there is another identical system corresponding to ψ -, which we describe soon). Let's see if this underlying supersymmetry is broken or unbroken. The last strongly depends on the index of the operator D RN . But since the number of quasinormal modes is a discrete infinite set, and owing to the bijective correspondence between the zero modes of the operator D RN and the quasinormal modes, we conclude that the zero modes form a discrete infinite set. Therefore, the operator D RN is not Fredholm which means that the index of the operator and correspondingly the Witten index must be regularized. In order to do so, we shall make use of the heat-kernel regularized index [30-33], both for the operator D RN , denoted ind t D RN and for the Witten index, ∆ t , which are defined as: /negationslash /negationslash The parameter t , is positive number t > 0, and moreover the trace tr ± , stands for the trace in the subspaces H ± . The heat-kernel regularized index is defined for trace class operators [31]. In the regularized index case, the same hold in reference to supersymmetry breaking, that is if ∆ t = 0 supersymmetry is unbroken. When the Witten index is zero, if ker D RN = ker D † RN = 0, supersymmetry is broken, while when ker D RN = ker D † RN = 0 supersymmetry is unbroken. In the case at hand, supersymmetry is unbroken. We can see this without solving the zero mode equation of the D † RN operator. Indeed the existence of zero modes suffices to argue about supersymmetry. Since ker D RN = 0, the zero modes equation for the operator D † RN can yield two results. Either that ker D † RN = 0 or that ker D † RN = 0. If the second is true, then the Witten index is different than zero, ∆ t = 0, hence supersymmetry is unbroken. In the first case, ker D † RN = 0, it can either be that ker D † RN = ker D RN or that ker D † RN = ker D RN . In both cases supersymmetry is unbroken. Hence the system that is described by the ψ + function has an underlying unbroken N = 2 supersymmetry. /negationslash /negationslash Recall that there is another solution to the Dirac equation in this curved background, namely ψ -. The equations of motion corresponding to ψ -are equal to: /negationslash /negationslash /negationslash with, and ψ -being related to ψ -= r -1 f -1 / 4 ˜ ψ ' . By the same reasoning as in the ψ + case, the supersymmetric quantum algebra can be built on the matrix: acting on the vector: The supercharges of the new algebra are equal to and the Hamiltonian is Following the same line of argument as in the previous, we easily find an N = 2 underlying supersymmetry. Denoting the algebra corresponding to ψ -, N 2 and the one corresponding to ψ 1 , N 1 we have come to the result that the Dirac fermionic system in an ReissnerNordstrom-anti-de Sitter background, possesses an supersymmetry N , that is the direct sum of two N = 2 supersymmetries, namely: The total Hamiltonian of the system is H total = H RN ' + H RN . It is tempting to investigate if this supersymmetry N total results after the breaking of a larger supersymmetry, for example an N = 4 supersymmetry, or even the possibility that a central charge exists. In addition this N total supersymmetry could be the a sign of an underlying higher symmetry (for a quite similar situation but in a different context, consult reference [48] where two N = 2, d = 1 supersymmetries constitute a N = 4 supersymmetry). We defer this investigation to a future publication. Let us just mention that the N = 4 supersymmetric algebra is very important in string theory, since extended (with N = 4 , 6 ... ) supersymmetric quantum mechanics models are the resulting models from the dimensional reduction to one (temporal) dimension of N = 2 and N = 1 Super-Yang Mills theories [38-46]. In addition, extended supersymmetries serve as superextensions of integrable models like Calogero-Moser systems, Landau-type models [47] and also there exist interesting dualities between various supermultiplets with string theory origin (like T-duality) [66]. But the most salient feature of the extended supersymmetric quantum algebra is that it can be connected to a generalized harmonic superspace [38-46], with the last being a powerful tool for N ≥ 4 supersymmetric model building. These harmonic space structures are linked to supersymmetric linear models defined in the target space, for which the harmonic variables give rise to target space harmonics. Note that the Dirac solutions, actually the zero modes of the supercharges, are sections of the total spin bundle [67,68] over the Riemannian manifold M . Hence we can directly connect these sections to an extended supersymmetric sigma model in harmonic superspace. Although these issues are very interesting both in mathematical and physical aspects, we defer this work to a future article where we address formally all the aforementioned topics. The two N = 2 supersymmetries can be combined to a higher representation of a single N = 2, d = 1 supersymmetry. Indeed, the supercharges of this representation, which we denote Q T and Q † T is equal to: Accordingly, the Hamiltonian of the combined quantum system, which we denote H T reads, The operators (86) and (87), satisfy the N = 2, d = 1 supersymmetric quantum mechanics algebra, namely: In this case, the Witten parity operator is equal to: There exist equivalent higher dimensional representations for the combined N = 2, d = 1 algebra, which can be obtained from the above algebra, by making the following set of replacements:", "pages": [ 11, 12, 13, 14, 15, 16, 17 ] }, { "title": "2.1 A Global U (1) × U (1)", "content": "As we saw previously, the fermionic system in the Reissner-Nordstrom-anti-de Sitter black hole background can constitute a space with two N = 2 supersymmetric quantum mechanics algebras, described by the supercharges defined in relations (75) and (83). The two superalgebras are invariant under the transformations: Thus each quantum system is invariant under an R -symmetry of the form of an globalU (1). Correspondingly, the total system is invariant under an U (1) × U (1) symmetry. Each of the aforementioned U (1) symmetries is a symmetry of the Hilbert states corresponding to the spaces H RN + , H RN -and H RN ' + , H RN ' -. Let, ψ + RN and ψ -RN denote the Hilbert states corresponding to the spaces H + RN and H -RN . The U (1) transformation of the states is equal to, Obviously the parameters β + and β -are global parameters so that a = β + -β -. Accordingly, for the spaces H RN ' + , H RN ' -we have, with ψ + RN ' and ψ -RN ' the Hilbert states of the spaces H RN ' + , H RN ' -respectively. It worths mentioning that in some superconductors, such U (1) symmetries are realized. Particularly an initial U (1) × U (1) symmetry is broken to a single U (1) (see [65] for details).", "pages": [ 18 ] }, { "title": "Conclusions", "content": "In this paper we studied the supersymmetry structure underlying two physical fermionic systems, namely the color superconductor in the chiral limit, around the boundary vortex and a fermionic system in the Reissner-Nordstrom-anti-de Sitter black hole spacetime. For the first system we found by analyzing the Bogoliubov-de Genne equation that, in the chiral limit, the localized fermion zero modes of the color superconductor constitute an N = 2 supersymmetric quantum mechanics algebra with zero supercharge. Interestingly, by analyzing the quasinormal modes of the gravitational fermionic system in the ReissnerNordstrom-anti-de Sitter background, we found two unbroken N = 2 supersymmetries. We stressed the fact that this result is interesting from a mathematical point of view, owing to the fact that we can relate the fermionic gravitational system to a sigma model in harmonic superspace. Note that the unbroken supersymmetry of the system is guaranteed by the very own existence of fermionic quasinormal modes of the gravitational system. This, in turn has its own intrinsic appeal since quasinormal modes depend only on a few physical parameters of the black hole. Since these parameters enter the quantum algebra we have a supersymmetry depending on a few physical parameters and that depends on the existence of quasinormal modes. Moreover, the two N = 2, d = 1 algebras can be combined to form a higher dimensional reducible representation of an N = 2 supersymmetric quantum mechanics algebra. The two fermionic systems we presented in detail are believed to be interrelated, with the fermionic system in curved background being a promising candidate for describing the color superconductor. Thus, the supersymmetries we found, show us that under certain very general assumptions, these two systems have an underlying supersymmetric structure, of N = 2, d = 1 type. Hence, this common, in some way, underlying theme makes us believe that the two systems might be connected. But this is just an indication and not a direct correspondence between the two models. Additionally the supersymmetries of the two spaces have different Hilbert space structure, a fact that is seen easily from the operators being Fredholm in the color superconductor case, and non Fredholm in the other case. Moreover, the gravitational system has a much more rich structure, and both systems can be related to extended supersymmetric algebras. In addition, it would be of particular importance to investigate whether such a supersymmetric structure exists in the case when bosonic quasinormal modes [69-74] are studied. Supersymmetry for bosonic systems in curved background occurs in the context of local N = 2 supersymmetric backgrounds, where the fluctuations of the bosonic fields evolved in an Abelian Chern Simons vortex, have the same supersymmetric quantum mechanics algebra as the fermionic system [75].", "pages": [ 18, 19 ] } ]

[ { "title": "Generation of scale invariant density perturbations in a conformally invariant Inert Higgs doublet model", "content": "Moumita Das ∗ and Subhendra Mohanty † Physical Research Laboratory, Ahmedabad 380009, India", "pages": [ 1 ] }, { "title": "Abstract", "content": "If a Higgs field is conformally coupled to gravity, then it can give rise to the scale invariant density perturbations. We make use of this result in a realistic inert Higgs doublet model, where we have a pair of Higgs doublets conformally coupled to the gravity in the early universe. The perturbation of the inert Higgs is shown to be the scale invariant. This gives rise to the density perturbation observed through CMB by its couplings to the standard model Higgs and the subsequent decay. Loop corrections of this conformally coupled system gives rise to electroweak symmetry breaking. We constrain the couplings of the scalar potential by comparing with the amplitude and spectrum of CMB anisotropy measured by WMAP and this model leads to a prediction for the masses of the lightest Higgs and the other scalars.", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "It is well known that to generate the density perturbation of the CMB of the magnitude observed by COBE and WMAP, we need an inflationary period generated by the flat potential of a scalar field with coupling λ ∼ 10 -10 in a λφ 4 theory. For standard model Higgs, λ is approximately ∼ 1 and the Higgs can not be used as inflaton. A way out was proposed by Bezrukov and Shaposhnikov [1] who coupled the standard model Higgs with the Ricci scalar with a large coupling constant ξ ∼ 10 4 . This large coupling leads to problem with unitarity [2-4] of graviton-scalar scattering. Some attempts to solve the unitarity problem associated with the large Higgs curvature coupling are in [5-7] In this paper we follow a different approach for the generation of scale invariant density perturbations. It was shown by Rubakov and collaborators [8-10] that a conformally coupled field rolling down a quartic potential can generate scale invariant density perturbation. These perturbations can become superhorizon in an inflationary era or in a ekpyrotic scenario [11]. We work with the inert Higgs doublet (IDM) model [12, 13] with conformal couplings to the Ricci scalar. The mass terms which give rise to electroweak symmetry breaking are generated by the Coleman-Weinberg method [14]. The requirement of scale invariance at high energy scale and electro-weak symmetry breaking at low energies fixes the coupling constants of the theory. Specifically we find that the quartic coupling of the inert doublet, predicts a spectral index of the power spectrum of the perturbations to be consistent with observations. The amplitude of the power spectrum P ζ can be tuned to be consistent with the observations by choosing a suitable curvaton mechanism. This model specifies the mass of the Higgs boson to be m h = 291 GeV and the mass of the dark matter m A 0 = 550 GeV which can be tested respectively at the LHC and in cosmic ray observations. The main aim of this paper has been to show that a Higgs potential with not too small couplings can be a viable source of the observed scale invariant density perturbations. The scale invariant density perturbations become superhorizon during a phase of inflation at the electroweak scale. However other cosmological scenarios like a bounce models [11] of making the density perturbations superhorizon may be equally viable with our model. In Section-2 we describes the basics of the Inert Doublet Model (IDM). The one-loop correction to the potential and the calculation for running of coupling constant are briefly discussed in Section-3. We study the generation of the scale invariant density perturbation from the inert higgs doublet in Section-4. In Section-5 we list the scalar mass spectrum predicted by this model and identify the dark matter candidate.", "pages": [ 2, 3 ] }, { "title": "2. INERT DOUBLET MODEL", "content": "Inert Doublet Model (IDM) is a economical extension of Standard Model which solves the problem of naturalness [12] and it can also explain the electroweak symmetry breaking [13]. The lagrangian of this model respect the Z 2 symmetry, under which all Standard model particles including the SM Higgs H 1 are even and an extra scalar doublet H 2 is odd. Due to Z 2 symmetry, the cubic term and yukawa term for H 2 doublet are forbidden. This makes the inert doublet stable and its neutral component can be a candidate for dark matter. The two Higgs doublets H 1 and H 2 can be written in terms of their component fields as, The most general renormalisable potential will be, We consider the conformal case where µ 1 = µ 2 = 0. V c is the constant potential, which acts as cosmological constant and can be formed from the vev of different Higgs fields. We have chosen V c = 3 . 66 × 10 8 GeV 4 such that the minimum of the total potential becomes zero at present era. In the early universe the cosmological constant gives rise to an exponential expansion during which the scale invariant perturbations of the phase of the neutral component of H 2 become super-horizon. To achieve this we need that the potential is such that in the early universe, V ∼ -| λ 2 || H 2 | 4 and the neural component of H 2 rolls down this quartic potential while the minimum of H 1 is at 〈 H 1 〉 = 0. In the present era the potential should be such that the minima occurs at 〈 H 2 〉 = 0 and 〈 H 1 〉 = v = 246 GeV which gives rise to the electro-weak symmetry breaking. We show in the next section how this is achieved by radiative corrections starting from a scale invariant tree level potential.", "pages": [ 3 ] }, { "title": "3. COLEMAN-WEINBERG LOOP CORRECTION", "content": "We derive the one-loop correction to the potential (1) following Coleman-Weinberg formalism [14]. The generic one-loop correction to the potential can be written as [15], where J i is the spin of the fields and m i are the tree level masses, function of the Higgs field. The double derivative of the tree level potential (1) with respect to the fields give the the tree level masses, which are, where λ L,S ≡ λ 3 + λ 4 ± λ 5 . We regularize the divergent terms in Eq. (2) using the cut-off scale Λ and obtain The divergence in Eq. (4) can be removed by adding the counter terms in the potential of the form, where φ denote the scalar fields, considered in the model. We impose the regularization condition on the effective potential, such that at early era (with high µ value), the potential is scale invariant form (1) by choosing the counter terms as follows, where f ( m 2 i ) = log ( Λ 2 µ 2 + µ 2 m 2 i ) . With these counter terms the form of the effective potential turns out to be, where n i is degrees of freedom and m i are tree level masses, shown in Eq. (3). To get the correct electro-weak symmetry breaking in the present era and the scale invariant density perturbation in the early era, we have chosen a set of λ values in present epoch as shown in Table-(I). The values of λ i , where { i = 3to5 } are chosen such that we can get electro-weak symmetry breaking in the present era. Now we have study the running of couplings λ i , where { i = 1to5 } using the one-loop renormalization group equation for the inert doublet model [12]. From Fig (1), we can find the λ values in the early era µ glyph[similarequal] 10 4 are as follows, Only λ 2 at early universe is relevant for calculating the scale invariant density perturbation. The change in shape of the effective potential V eff ( H,h,µ ) in Eq. (8) from the early universe where we take µ = 2 . 2 × 10 4 to the present epoch where µ = 172 . 5 GeV is shown in Fig (2). We see that in the early universe for a given value of H the minima of V eff ( h ) (shown in Fig 3(a)) is at h = 0. We assume that in the early universe h = 0 and we see that V eff ( H ) is of the form as shown in Fig 3(b). The one loop correction of the potential has significance contribution in present era. When we take µ = 172 GeV then the potential (8) is of the form shown in Fig 4(a) and Fig 4(b). In this era, the V eff ( H ) has a minimum at H = 0 as shown in Fig 4(b). With H = 0 , the potential as a function of the field h has a minimum at h = v ∼ 246 GeV signifying the electroweak symmetry breaking. We note that the potential we calculate are at zero temperature which accurately describes the universe during inflation (when any prior temperature goes down exponentially in time) or in the present universe where the background temperature negligible compared to the electroweak scale. There is a radiation era after re-heating at the end of inflation. The effective potential at high temperature has been computed for the inert Higgs doublet model in [16], where the thermal evolution of the effective potential has been shown. In this paper we deal with the T = 0 case which is relevant during inflation and in the present universe.", "pages": [ 4, 5, 6, 7, 8 ] }, { "title": "4. GENERATION OF THE SCALE INVARIANT DENSITY PERTURBATION", "content": "We now turn to the question of the generation of density perturbations in the early era when V eff (8) simplifies to the form, where V c = 3 . 66 × 10 8 GeV 4 and λ 2 = -0 . 5. The Hubble parameter in this era can be calculated from Eq (9), We take the inert Higgs doublet to be conformally coupled to gravity and the action for this field can be written as, where H 2 contains the neutral part of the inert doublet i.e. H 2 = H 0 + i A 0 √ 2 and R is the scalar curvature, which conformally coupled with the field H 2 . The equation of the field H 2 will be, where a and H are the scale factor and Hubble constant respectively. Now defining H 2 = χ H 2 a and rewriting the Eq (13), we will get, where ' denotes the derivative with respect to conformal time η . We note that both a '' a and R 6 a 2 equal to 2 η 2 and the two terms in Eq. (14) cancel. So the equation for H 2 becomes, Expressing χ H 2 = ρ exp( i θ ), the conserved current will be of the form, Hence, the field rolls along the radial direction while the phase θ remains constant with the increase of ρ . Without loss of generality we can choose the fixed phase such that the field H 2 has only real component neutral component χ H 0 . The perturbations of H 2 will be along the imaginary axis and we can denote the full H 2 with the perturbations as χ H 2 = χ H 0 + i δχ A 0 , from Eq (15) the equation of motion of χ H 0 will be, Considering k glyph[lessmuch] 1 /η at late time, the solution will be, where √ -λ 2 is a real quantity as λ 2 is negative and η ∗ is a constant of integration. At the end of inflation when µ << 10 4 the shape of the potential changes, and H 0 starts rolling back to zero. Starting from (16) we see that the equation of motion of the perturbation, δχ A 0 is given by, Substituting χ H 0 from Eq (18), the equation becomes, This equation can be solved for early times and later times separately. At early time ( k ( η ∗ -η ) glyph[greatermuch] 1), third term can be neglected and the solution will be At later times, when ( k ( η ∗ -η ) glyph[lessmuch] 1), third term will dominate and in this case solution will look like, Hence, the super-horizon perturbations of the phase can be defined as, Therefore the perturbation of the phase δθ becomes, The power spectrum of δθ is scale invariant, If one considers the k dependence of the equation of motion of H 0 as discussed in [10] there will be a deviation from the scale free power spectrum (25) which will give rise to a non-zero spectral index, From Table-(II) we see that in the early universe λ 2 = -0 . 5 which gives the spectral index n s -1 = -0 . 04, which is consistent with the WMAP observation of n s = 0 . 967 ± 0 . 014 [17]. The perturbations of the phase δθ = δA 0 /H 0 can be converted to adiabatic perturbation by the decay of the A 0 field into standard model fields as in the curvaton mechanism [18]. The amplitude of adiabatic perturbation is related to the phase perturbation as where r is the ratio of the energy density in the A 0 field oscillations to the total energy density. Taking the unperturbed phase to be θ c ∼ π/ 2, and with λ 2 = -0 . 5 we see that r = 2 × 10 -4 to give the required P ζ = 10 -10 .", "pages": [ 9, 10, 11 ] }, { "title": "5. SCALAR MASS SPECTRUM", "content": "As the field H 2 has a zero vev in the present universe the lightest neutral components of H 2 will be stable and can be candidates for dark matter. We study the masses of the fields in present universe from the effective potential. Taking < H 1 > = 246GeV and < H 2 > = 0GeV and for λ i as in Table-(I) we find the mass spectrum of scalars in the present universe is as given in Table 3. We see that the field A 0 can be a candidate for heavy dark matter. We also see that the Higgs mass is predicted to be M h = 291 GeV which is not ruled out [19] and may be observed at the LHC.", "pages": [ 11 ] }, { "title": "6. CONCLUSIONS", "content": "The inert Higgs doublet model gives is a natural extension of the standard model and can be used for explaining the electroweak symmetry breaking by loop corrections [13] starting from a scale invariant tree level potential. We connect the scale invariance of the inert Higgs potential to the generation of scale invariant spectrum of a conformally coupled scalar as discussed by Rubakov and collaborators [8-10]. The requirement of scale invariance at high energy scale and electroweak symmetry breaking at low energies fixes the coupling constants of the theory. Specifically we find that the the quartic coupling of the inert doublet, λ 2 = -0 . 5 at µ = 10 4 GeV which predicts the spectral index of the power spectrum of the perturbations to be consistent with observations. The amplitude of the power spectrum P ζ can be tuned to be consistent with the observations by choosing a suitable curvaton mechanism. We make predictions for masses of the Higgs bosons and the dark matter (which is the lightest neutral component of the inert doublet) which can be tested in forthcoming experiments.", "pages": [ 12 ] } ]

[ { "title": "ABSTRACT", "content": "International Journal of Modern Physics A c © World Scientific Publishing Company", "pages": [ 1 ] }, { "title": "AN UPDATED ANALYSIS ON THE RISE OF THE HADRONIC TOTAL CROSS-SECTION AT THE LHC ENERGY REGION", "content": "M. J. MENON and P. V. R. G. SILVA Universidade Estadual de Campinas - UNICAMP Instituto de F'ısica Gleb Wataghin 13083-859 Campinas, SP, Brazil ([email protected], [email protected]) A forward amplitude analysis on pp and ¯ pp elastic scattering above 5 GeV is presented. The dataset includes the recent high-precision TOTEM measurements of the pp total and elastic (integrated) cross-sections at 7 TeV and 8 TeV. Following previous works, the leading high-energy contribution for the total cross-section ( σ tot ) is parametrized as ln γ ( s/s h ), where γ and s h are free real fit parameters. Singly-subtracted derivative dispersion relations are used to connect σ tot and the rho parameter ( ρ ) in an analytical way. Different fit procedures are considered, including individual fits to σ tot data, global fits to σ tot and ρ data, constrained and unconstrained data reductions. The results favor a rise of the σ tot faster than the log-squared bound by Froissart and Martin at the LHC energy region. The parametrization for σ tot is extended to fit the elastic cross-section ( σ el ) data with satisfactory results. The analysis indicates an asymptotic ratio σ el /σ tot consistent with 1/3 (as already obtained in a previous work). A critical discussion on the correlation, practical role and physical implications of the parameters γ and s h is presented. The discussion confronts the 2002 prediction of σ tot by the COMPETE Collaboration and the recent result by the Particle Data Group (2012 edition of the Review of Particle Physics). Some conjectures on possible implications of a fast rise of the proton-proton total cross-section at the highest energies are also presented. Keywords : Hadron-induced high- and super-high-energy interactions; Total crosssections; Asymptotic problems and properties PACS numbers: 13.85.-t, 13.85.Lg, 11.10.Jj", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The theoretical and phenomenological descriptions of the energy dependence of the hadronic total cross-section at high energies have been a fundamental and longstanding problem. Given that the optical theorem connects the total cross-section with the imaginary part of the forward elastic scattering amplitude, the main theoretical difficulty concerns the lack of a pure (model-independent) nonperturbative QCD description of the soft scattering states, in particular the elastic channel in the forward direction. In this section, we shortly recall some results on the rise of the total cross-section of interest in this paper, followed by the plan of the paper. Historical formal analyses by Froissart, Lukaszuk and Martin 1-4 have established", "pages": [ 1 ] }, { "title": "2 M. J. Menon & P. V. R. G. Silva", "content": "the famous asymptotic bound for the rise of the total cross-section, where s is the center-of-mass energy squared, c ≤ π/m 2 π ≈ 60 mb and s 0 is a constant . After the Martin derivation in the context of the axiomatic quantum field theory, 2 the log-squared bound has played a central role in model constructions, aimed to treat, interpret and describe strong interactions. Recently, Azimov has argued that it is not obvious that QCD can be considered an axiomatic theory and has also established new constraints for the rate of increase of σ tot ( s ). 5 The formalism allows to conclude that, depending on the behavior of the amplitude in the nonphysical region, the total hadronic cross-section may rise faster than the log-squared bound, without violation of unitarity 5 (see also Refs. 6 and 7). The dependence of the total cross-section on the energy has been usually investigated through forward amplitude analysis. This approach is characterized by different analytical parametrizations for σ tot ( s ) and global data reductions including the ρ parameter (ratio between the real and imaginary parts of the forward amplitude) by means of singly-subtracted dispersion relations. In 2002, the COMPETE Collaboration developed a detailed analysis on different functional forms, using a ranking procedure and including several reactions and different energy cutoffs. 8, 9 The analysis favored the asymptotic form ln 2 ( s/s 0 ), when compared with ln( s/s 0 ) or powers [ s/s 0 ] /epsilon1 with /epsilon1 > 0, a conclusion corroborated by subsequent works. 10-13 After that, data reductions with the COMPETE highest-rank parametrization, in agreement with the log-squared bound and all the experimental data available, became standard reference in the Review of Particle Physics by the Particle Data Group (PDG) up to the 2010 edition. 14 In the experimental context, the recent results from the LHC, reaching the highest energy region, opened new expectations in the phenomenological and theoretical contexts. However, the first results on pp elastic, diffractive and nondiffractive processes by the TOTEM Collaboration have indicated noticeable discrepancies with standard predictions from representative phenomenological models. 15-17 In general grounds, this novel experimental information seems yet unable to select or exclude classes of phenomenological approaches and/or theoretical pictures in a clear and conclusive way. In what concerns the pp total cross-section, the first 7 TeV TOTEM highprecision measurement has been obtained with small bunches and a luminositydependent method, indicating σ tot, 1 (7 TeV) = 98.3 ± 2.8 mb (small bunches / luminosity-dependent). 16 In particular and for our purposes, the COMPETE prediction (published 11 years ago) is in good agreement with this datum. 16 However, contrasting with this striking result, the 2012 edition of the Review of Particle Physics by the PDG has shown that with an updated dataset, including the above TOTEM datum, the fit with the highest-rank parametrization selected by the COMPETE Collaboration disagrees with the TOTEM result (Figure 46.10 in Ref. 18). Both results (COMPETE 2002 and PDG 2012) are displayed in Figure 1 and will be discussed along the paper. In 2011 and 2012, Fagundes, Menon and Silva 19,20 have revisited a parametrization for the total cross-section introduced by Amaldi et al. in the seventies, 21 characterized by using the exponent in the high-energy leading logarithm contribution as a free real fit parameter. Based on different fit procedures, variants and data ensembles from pp and ¯ pp forward scattering at √ s ≥ 5 GeV, the authors have shown that with the inclusion of the 7 TeV TOTEM result in the dataset, several statistically consistent solutions are obtained with the exponent greater than 2, suggesting, therefore, a faster-than-squared-logarithm rise for the total cross-section. 19, 20 More recently, the TOTEM Collaboration has obtained four new high-precision measurements for the total cross-section, with three data at 7 TeV and one datum at 8 TeV. 22-24 The results and experimental conditions can be summarized as follows: σ 2 (7 TeV) = 98.6 2.2 mb (large bunches / luminosity-dependent), 22 tot, ± σ tot, 3 (7 TeV) = 99.1 ± 4.3 mb ( ρ independent), 23 σ tot, 4 (7 TeV) = 98.0 ± 2.5 mb (luminosity-independent), 23 σ tot (8 TeV) = 101.7 ± 2.9 mb (luminosity-independent). 24 In this paper, we revisit, once more, the Amaldi et al. parametrization, now taking into account these recent measurements of the pp total cross-section at 7 TeV and 8 TeV. The general strategy follows the approach developed in our previous analysis. 19, 20 In addition to using the new data, we will also explore the practical role and the physical meaning of the free fit parameters associated with the highenergy leading contribution. In particular, we discuss the differences between the 2002 COMPETE prediction and the 2012 PDG result in the phenomenological context (Reggeon/Pomeron exchanges). As in Refs. 19 and 20, the analysis is also restricted to pp and ¯ pp data at √ s ≥ 5 GeV, the same energy cutoff used by the COMPETE Collaboration 8, 9 and in the last PDG versions. 14, 18 Our analysis and results favor, once more, a rise of the hadronic total cross-section faster than the logsquared bound at the LHC energy region. An extension of the parametrization to fit the elastic cross-section data allows us to infer the asymptotic ratio between the elastic and total cross-sections. The result is statistically consistent with a rational limit of 1/3 (as previously obtained in Ref. 20). Some comments and conjectures on the possible implications of a faster-than-squared-logarithm rise for the total cross-section are also presented. The paper is organized as follows. In section 2, we display the analytical parametrization for σ tot ( s ) and the formula connecting this quantity with ρ ( s ), using singly-subtracted derivative dispersion relations; 19 the fit procedures and strategies are also outlined. In section 3, we present the fit results with datasets up to √ s max = 7 TeV and √ s max = 8 TeV, in the cases of unconstrained and constrained data", "pages": [ 2, 3 ] }, { "title": "4 M. J. Menon & P. V. R. G. Silva", "content": "reductions, individual fits to σ tot data and global fits to σ tot and ρ data. A discussion on all the obtained results, with focus on the high-energy leading contribution parameters, is presented in section 4. The extension to the elastic cross-section data is treated in section 5 and discussions on the possibility of a remarkable fast rise of the total cross-section at the LHC energy region are presented in section 6. The conclusions and some final remarks are the contents of section 7.", "pages": [ 4 ] }, { "title": "2. Analytical Parametrization and Fit Procedures", "content": "In this section, after introducing the formulas to be employed in our data reductions, namely the parametrization for σ tot ( s ) and the analytical expression for ρ ( s ) from derivative dispersion relations, we outline some important points on the fit procedures and strategies.", "pages": [ 4 ] }, { "title": "2.1. Analytical parametrization and dispersion relation result", "content": "We consider the analytical parametrization for the pp and ¯ pp total cross-section introduced by Amaldi et al. in 1970s 21 and also employed by the UA4/2 Collaboration in 1990s. 25 It consists of two components, associated with low-energy ( LE ) and high-energy ( HE ) contributions: The first term accounts for the decrease of the total cross-section and the differences between pp and ¯ pp scattering at low energies. In this paper it is expressed by where τ = -1 (+1) for pp (¯ pp ) scattering, s l = 1 GeV 2 is fixed, and a 1 , b 1 , a 2 and b 2 are free fit parameters. The second term accounts for the rising of the cross-section at higher energies and is given by where α , β , γ and s h are free real parameters. For further reference, we briefly recall that, in the context of the Regge-Gribov theory, the decreasing σ LE ( s ) contribution is associated with Reggeon exchanges: b 1 and b 2 correspond to the intercept of the trajectories and a 1 , a 2 to the Reggeon strengths (residues). The σ HE ( s ) term simulates the rise of the total cross-section and is associated with the Pomeron exchange. For example, for γ = 1, the constant plus ln s terms correspond to a double pole at J = 1 and for γ = 2 a triple pole (expressed by ln 2 s , ln s and the constant terms). 8 For our purposes it is important to note that for γ = 2, parametrization (Eqs. (1)-(3)) has the same analytical structure of the highest-rank parametrization selected by the COMPETE Collaboration. 8, 9 An updated analysis on the rise of the hadronic total cross-section at the LHC energy region The analytical connection with the ρ parameter is obtained using singlysubtracted derivative dispersion relations in the operator expansion form introduced by Kang and Nicolescu 26 (also discussed in Ref. 27). In terms of the parametrization (1)-(3) for pp and ¯ pp scattering, the analytical results for ρ ( s ) read 19, 20 where K is the subtraction constant, and, as before, τ = -1 (+1) for pp (¯ pp ) scattering.", "pages": [ 4, 5 ] }, { "title": "2.2. Fit procedures and strategies", "content": "The fit procedures and methodology, as well as the role and applicability of the subtraction constant in the derivative dispersion relation approach, have been discussed in detail in Refs. 19 and 20. Here we recall and also outline some important points of interest in this work, especially the introduction of a representation for the parameter s h .", "pages": [ 5 ] }, { "title": "2.2.1. Dataset", "content": "Our goal is to investigate the rise of σ tot at the highest energy region and its asymptotic behavior. For that reason, we limit the analysis to particle-particle and antiparticle-particle collisions corresponding only to the largest energy interval with available data, namely pp and ¯ pp scattering. Although somewhat restrictive, the main point is that this choice allows the investigation of possible high-energy effects that may be unrelated to the trends of the lower energy data on other reactions (as the constraints dictated by a supposed universal behavior). The input dataset for fits concerns only accelerator data on σ tot and ρ from pp and ¯ pp scattering, covering the energy region from 5 GeV up to 8 TeV. The energy cutoff is the same employed in the COMPETE and PDG analyses. 8, 18 The data below 7 TeV have been collected from the PDG database, 18 without any kind of data selection or sieve procedure. Statistical and systematic errors have been added in quadrature. Estimations of the pp total cross-section from cosmic-ray experiments will be displayed in the figures as illustrative results. The TOTEM estimation for ρ at 7 TeV 23 is also displayed as illustration. All the references on these data and estimations can be found in Ref. 20.", "pages": [ 5 ] }, { "title": "2.2.2. Nonlinearity and feedback values", "content": "Because the nonlinearity of the fit demands a choice for the initial values (feedbacks) for all free parameters, 28 different choices have been tested and discussed in our previous analyses 19, 20 (see also Ref. 29). Here, to initialize our parametric set, we consider only the values of the fit results in the updated 2012 PDG version, obtained with the highest-rank COMPETE parametrization. 18 This choice is based on the arguments that follow. The PDG data reductions have been developed with γ fixed to 2. Due to the strong correlation among all the fit parameters (to be discussed in Subsec. 4.1), their final fit values are, therefore, consequences of this condition ( γ = 2). In this sense, initializing the parametrization with these values can be considered a 'conservative' choice. Moreover, since in the present analysis γ is treated as a free parameter in fits including the recent TOTEM measurements, this choice allows us to investigate possible departures from the standard/canonical assumption γ = 2. However, it should be noted that the PDG and COMPETE analyses include different collision processes (mesonp , baryonp , among others) and also tests on universality. Therefore our dataset, restricted to pp and ¯ pp scattering, corresponds to only a subset of the ensemble employed in the global analysis by the COMPETE and PDG. Despite this, we understand that with this choice for the feedbacks, we initialize our parametrization with a statistically and physically meaningful input, contributing to the search for a consistent fit solution (see also Subsec. 2.2.4 below). The values of the feedback parameters to be used in our fits are shown in the third column of Table 1. For further reference, we display in Table 1 the values of the parameters obtained with the highest-rank parametrization (Eqs. (1)-(3) with γ = 2) in both the 2002 COMPETE analysis (Table VIII in Ref. 8) and the 2012 PDG version (Table 46.2 in Ref. 18), which is based on an updated dataset including the first 7 TeV TOTEM datum. In the last case, the values of the parameters a 1 and a 2 correspond to our normalization of Eq. (2), namely s l = 1 GeV 2 fixed (which is different from the normalization adopted in Ref. 18). The corresponding results for σ tot ( s ) and uncertainty regions (evaluated through propagation from the errors in Table 1) are shown in Figure 1, together with the experimental information (in this figure only the first 7 TeV TOTEM measurement is displayed, as in Ref. 18). We shall return to these results along the paper.", "pages": [ 6 ] }, { "title": "2.2.3. Individual and global fits", "content": "Global fits to σ tot and ρ data demand the use of dispersion relations with one subtraction and therefore the introduction of one more free parameter, the subtraction constant. As already discussed in Refs. 19 and 20 this parameter does not have a physical interpretation as opposed to the parameters present in the total cross-section parametrization, which are associated with Reggeon and Pomeron exchanges. Moreover, several authors have criticized the usual methods to extract ρ (see Subsec. 2.2 in Ref. 19) and, in addition, due to the correlation among all the An updated analysis on the rise of the hadronic total cross-section at the LHC energy region 7 ± ± fit parameters, the ρ inclusion in global fit constraints the rise of the total crosssection, as demonstrated in Refs. 20, 30, 31 and references therein. Despite these disadvantages, we shall here consider both individual fits to σ tot data through Eqs. (1)-(3) and global fits to σ tot and ρ data using Eqs. (1)-(5).", "pages": [ 6, 7 ] }, { "title": "2.2.4. Minimization and statistics", "content": "The data reductions have been performed with the objects of the class TMinuit of the ROOT Framework. 32 We have employed the default MINUIT error analysis 33 with the selective criteria that follow. In the minimization program a Confidence Level of one standard deviation was adopted in all fits (UP = 1). In each test of fit, successive running of the MIGRAD have been considered (up to 5,000 calls), until full convergence has been reached, with the smallest FCN (chi-squared) and requiring Estimated Distance to Minimum (EDM) < 10 -4 , adequate for the one sigma CL. In some cases the MINOS algorithm and strategies 1 and 2 have also been employed to check the MIGRAD result. In addition, the error in the parameters should not exceed the central value. The error matrix provides the variances and covariances associated with each free parameter, which are used in the analytic evaluation of the uncertainty regions in the fitted and predicted quantities, by means of standard error propagation procedures. 28 To quantify goodness of fit we will resort to chi-square per degree of freedom ( χ 2 /DOF) and the corresponding integrated probability, P ( χ 2 ). 28 The goal is not to compare or select fit procedures or fit results but only to check the statistical consistence of the data reductions in a reasonable way.", "pages": [ 7 ] }, { "title": "2.2.5. Unconstrained and constrained fits", "content": "The consideration that the exponent γ in the leading logarithm component is a real (not integer) free fit parameter leads to some special consequences and particular conditions. These aspects, including the effects of the correlation between the free parameters γ and s h in data reductions, will be discussed in some detail in Sec. 4, after the presentation of our fit results. In order to implement and facilitate that discussion, we introduce here a useful representation for the high-energy scaling factor s h , which will lead us to distinguish between unconstrained and constrained fits, as explained in what follows. The representation is based on two arguments: (i) the reference to s h only as a (unknown) constant in the Froissart-Martin derivation of the bound; (ii) the reasonable physical conjecture that this factor might be proportional to the energy threshold for the scattering states (above the resonance region), namely s h ∝ ( m p + m p ) 2 , where m p is the proton mass. In this case, we can represent the scaling factor by where δ is a real dimensionless parameter and δ = 1 at the physical threshold. With this representation, we can distinguish two physical conditions in data reductions: either to consider δ indeed as a free fit parameter (equivalently, s h a free fit parameter) or to assume δ = 1 (equivalently, to fix s h = 4 m 2 p , the energy threshold). In what follows we shall denote these two variants by unconstrained fits ( δ free) and constrained fits ( δ = 1 fixed). Here, as in the COMPETE and PDG analyses, we shall treat s h as a constant (with the above representation). However, it should be noted that the possibility of a slow rise of s h with s (for large s ) has been discussed by some authors, as for example in Refs. 5 and 34.", "pages": [ 7, 8 ] }, { "title": "2.2.6. Ensembles and feedbacks", "content": "As mentioned earlier, our dataset consists of pp and ¯ pp accelerator data at √ s ≥ 5 GeV. In order to investigate the effect in the fits associated with the TOTEM measurement at 8 TeV, as compared with those at 7 TeV, we shall consider two data ensembles. The first one with data up to 7 TeV (including the four measurements) and a second one adding the 8 TeV datum. For reference, we will denote these two variants by √ s max = 7 TeV ensemble and √ s max = 8 TeV ensemble, respectively. For the √ s max = 7 TeV ensemble, we use as feedback the values of the parameters from the 2012 PDG version, displayed in the third column of Table 1. In the case of global fits to σ tot and ρ data, we consider K = 0 for the initial value of the subtraction constant. 18 The fit results with this ensemble are then used as feedback to initialize the parametrization with the √ s max = 8 TeV ensemble, in each one of the variants considered (individual, global, unconstrained and constrained cases).", "pages": [ 8 ] }, { "title": "3. Fit Results", "content": "Summarizing the variants discussed in the last section, we select two ensembles of accelerator data on σ tot and ρ at √ s ≥ 5 GeV, one up to √ s max = 7 TeV and another one up to √ s max = 8 TeV. In each case we consider both unconstrained fits ( s h or δ in Eq.(6) as free fit parameter) and constrained fits ( s h = 4 m 2 p or δ = 1 fixed), treating also both individual fits to σ tot data through Eqs. (1)-(3) and global fits to σ tot and ρ data, using Eqs. (1)-(5). In what follows we present the fit results and discuss only the accordance with the selective criteria outlined in Subsec. 2.2.4. The physical aspects and implications involved are addressed in Sec. 4.", "pages": [ 8, 9 ] }, { "title": "3.1. √ s max = 7 TeV ensemble", "content": "For this ensemble all data reductions presented satisfactory agreement with the selective criteria. The fit results and statistical information are displayed in Table 2. The curves, uncertainty regions (from error propagation) and experimental information in the case of global fits to σ tot and ρ data are shown in Fig. 2 for the unconstrained fit and in Fig. 3 for the constrained one.", "pages": [ 9 ] }, { "title": "3.2. √ s max = 8 TeV ensemble", "content": "For this ensemble, the unconstrained fits presented some disagreement with the selective criteria. In the case of global data reduction to σ tot and ρ the fit did not converge and therefore we have no solution for this case. In the individual fit to σ tot the data reduction converged but it should be noted that the corresponding error matrix is not positive definite. The fit results and statistical information are displayed in Table 3. The curves, uncertainty regions and experimental information in the case of the constrained global fit to σ tot and ρ data are shown in Fig. 4 (the unconstrained case did not converge).", "pages": [ 9 ] }, { "title": "4. Discussion", "content": "Using parametrizations (1)-(5), we are interested in a consistent quantitative description of the rise of the total cross-section at high energies. We have considered 8 variants of data reductions, obtaining full convergence in 7 cases. The fit results are displayed in Tables 2 and 3. Our main goal is to investigate if these results indicate a log-squared behavior or a rise faster than this bound. In what follows, for a given numerical result γ ± ∆ γ , we consider a result statistically consistent with a faster rise the cases in which γ -∆ γ > 2. In this case, for short, we will refer to a result statistically consistent with γ > 2. However, as remarked early in Subs. 2.2.5, to consider the exponent γ as a continuous real fit parameter has some special implications not present in the canonical assumption γ = 2 (fixed). These aspects are directly related to the high-energy scaling factor s h and have important consequences not only on the data reductions but also on the physical interpretation of the fit results. The goal of this section is to address these aspects. To this end, in Subsec. 4.1 we discuss the individual and global fit results, with focus on the value of the parameter γ and its relation with a faster-than-squaredlogarithm rise. That will lead us to our partial conclusions in favor of this faster rise. After that, in Subsec. 4.2 we address the practical role and physical implications associated with the correlation between γ and s h . This discussion will lead us in Subsec. 4.3 to a final conclusion in favor of the constrained fits in both physical and statistical contexts, indicating a rise of the total cross-section faster than the log-squared bound.", "pages": [ 9, 10 ] }, { "title": "4.1. Individual and global fits: partial conclusions", "content": "Based on the results displayed in Tables 2 and 3 we have the comments that follow. In the case of individual fits to σ tot data, all the results (constrained or unconstrained, √ s max = 7 or 8 TeV ensembles) are statistically consistent with γ > 2 (confirming, therefore, the conclusions first presented in Ref. 19). In all cases the integrated probability reads P ( χ 2 ) ≈ 0.8. The highest γ -values are associated with the √ s max = 8 TeV ensemble, indicating γ ≈ 2.5 (unconstrained fit) and γ ≈ 2.3 (constrained fit). The corresponding results for σ tot ( s ) and uncertainty regions are displayed in Fig. 5 for both the unconstrained and constrained fits, together with the experimental data. We notice that the agreement with the TOTEM measurements at 7 TeV is striking. In the case of the constrained fit the uncertainty region includes the central value at 8 TeV and also all the four 7 TeV central values. The numerical results and predictions for the total cross-section at some energies of interest are displayed in Table 4 (all the variants investigated in the individual fits). We also note that with the √ s max = 8 TeV ensemble, the predictions at 57 TeV, namely σ tot ∼ 142 - 143 mb are about 7 % larger than the central value of the result by the Pierre Auger Collaboration, namely 133 mb. 35 ± ± ± ± In the case of global fits to σ tot and ρ data, the results depend on the ensemble and on the constraint condition considered. In all cases of global convergent fits the integrated probability reads P ( χ 2 ) ≈ 0.2. For the constrained fits (both ensembles) the results are statistically consistent with γ > 2. In the unconstrained case and √ s max = 7 TeV ensemble the result may be considered barely consistent with γ > 2, since, up to 2 figures, γ lies in the interval 2.0 - 2.2. As stated before, for the √ s max = 8 TeV ensemble we did not obtain full convergence. The numerical results and predictions for the total cross-section at the energies of interest are displayed in Table 5. We note that at 57 TeV the results indicate σ tot ∼ 139 - 140 mb, which is about 5 % larger than the Auger central value. ± ± - ± It is important to note that, in going from the individual to global fits, the constraint imposed by the inclusion of the ρ information on the rise of σ tot is evident:", "pages": [ 10, 11 ] }, { "title": "12 M. J. Menon & P. V. R. G. Silva", "content": "in all cases the γ value decreases (Tables 2 and 3). In this respect, the subtraction constant plays a remarkable role due to its correlation with all the fit parameters in the nonlinear data reduction, specially those associated with the high-energy contribution, namely α , β , γ and s h . This effect can be illustrated by the correlation matrix in the MINUIT Code, which provides a measure of the correlation between each pair of free parameters through a coefficient with numerical limits ± 1 (full correlation) and 0 (no correlation). 28, 32 For example, the coefficients in the global fits with the √ s max = 7 TeV ensemble, in the cases of unconstrained fit (UF) and constrained fit (CF), are displayed in Table 6. In both cases the correlations between K and α , β or γ are around 0.8 - 0.9, affecting therefore the asymptotic behavior of σ tot . However, as already commented in Subsec. 2.2.3 (and in more detail in Ref. 20) this important parameter does not have a physical interpretation as is the case for those present in the parametrization of σ tot ( s ). We also stress that the integrated probabilities P ( χ 2 ) in the global fits are smaller than in the individual fits, ∼ 0 . 2 and ∼ 0 . 8, respectively. Despite the aforementioned constraint, based on the discussion concerning the γ values obtained in both individual and global fits, we conclude that our results favor a rise of the hadronic total cross-section faster than the log-squared behavior at the LHC energy region. By 'favor' we mean that, within the uncertainties, the fit results lead to γ values above 2 and not 2 or below 2. The results for the γ parameter from all the fully converged fits, within the uncertainties (Tables 2 and 3), are schematically displayed in Figure 6. Note that, in the case of constrained fits, the γ values lie in the interval 2.2 - 2.3, which indicates more stability than in the unconstrained cases and suggests, therefore, a support to the former variant. We shall return to this point in section 4.3.", "pages": [ 12 ] }, { "title": "4.2. The role and effects of the parameters γ and s h", "content": "From the example displayed in Table 6 we also notice a substantial negative correlation between the parameter γ and the energy scaling parameter s h . In fact, from Tables 2 and 3, in going from the unconstrained to the constrained case (and also from individual to global fits), a decrease in s h is associated with an increase in γ and vice versa. In this section, based on the previous results and discussion, we examine in some detail the important practical and physical role of the scale parameter s h , especially in what concerns data reductions with γ as a free real (not integer) parameter. In what follows, we first list five characteristics of interest involved, distinguishing the case of γ = 2, and then discuss the connections of these characteristics with our fit results, as well as with the COMPETE 2002 and PDG 2012 results (Table 1 and Figure 1). /negationslash where a = α + β ln 2 ( δ ) and b = -2 β ln( δ ), providing the explicit correlation among the high-energy parameters. The above expansion, however, is not possible in the case of γ real ( = 2) so that the correlations are somewhat hidden in the nonlinear data reductions. respect, as we will discuss, the energy cutoff for data reduction, √ s min , plays a central role, in connection with the corresponding energy scale, √ s h . Here, as in both COMPETE and PDG analyses, we have adopted √ s min = 5 GeV. Based on the above characteristics, it is expected that, depending on the values of s min and s h , different physical interpretations can emerge in the cases of γ = 2 (fixed) and γ real (not integer). Let us discuss these aspects and their connection with our fit results and those from the COMPETE 2002 and PDG 2012 analyses.", "pages": [ 13, 14 ] }, { "title": "· γ = 2 (fixed)", "content": "In this case, if s h < s min then in the region of experimental data ( √ s ≥ √ s min ), the high-energy component σ HE ( s ) increases with the energy, as expected in the standard concept of the soft Pomeron contribution. That is the case of the 2012 PDG version since, from Table 1: On the other hand, if s h > s min then in the interval √ s min ≤ √ s ≤ √ s h , the σ HE ( s ) component decreases as the energy increases, suggesting a physical disagreement with the standard soft Pomeron contribution. That, however, is the case of the 2002 COMPETE result, since, from Table 1: The dependence of σ HE ( s ) for the two cases above is shown in Fig. 7 in the energy interval 4 GeV ≤ √ s ≤ 7 GeV that includes the energy cutoff √ s min = 5 GeV. In this respect, according to the COMPETE Collaboration: 8 'One must note that in some processes, the falling ln 2 ( s/s 0 ) term from the triple pole at s < s 0 is important in restoring the degeneracy of the lower trajectories at low energy. Hence the squared logarithm manifests itself not only at very high energies, but also at energies below its zero.' However, even accepting this argument on a decreasing Pomeron contribution in the physical region considered, the fast rise of σ HE ( s ) as the energy decreases below √ s h (Figure 7) and above the physical threshold (2 m p ), remains, in our opinion, unexplained. Apractical or even pragmatic consequence of this COMPETE result is the agreement between their 2002 prediction (with γ = 2 in accordance with the FroissartMartin bound) and the 7 TeV TOTEM measurement. Or, in other words, this result is directly connected to the rather large value of s h . By contrast, the PDG result with γ = 2 and smaller s h , which is consistent with a rise of σ HE ( s ) in the whole interval of energy investigated, lies below the TOTEM datum (compare Figures 1 and 7). Summarizing, with γ = 2: (a) the COMPETE correctly describes the 7 TeV TOTEM datum, but with a decreasing σ HE ( s ) contribution above the cutoff s min up to s h and a increasing contribution as the energy decreases below s min (strictly divergent as s decreases); (b) in the PDG 2012 result, σ HE ( s ) increases in the whole energy-interval investigated (above the cutoff) but lies (within the uncertainty) below the TOTEM datum.", "pages": [ 14, 15 ] }, { "title": "· γ real (not integer)", "content": "For γ not integer, as commented before, the σ HE ( s ) component is not defined at √ s < √ s h . The contribution starts at √ s = √ s h with σ HE ( s h ) = α and from this point on it increases with the energy, as expected in the standard soft Pomeron concept. With respect to our fit results, the σ HE ( s ) component is well defined in the whole interval of energy investigated since √ s min = 5 GeV and in all data reductions s h < s min (Tables 2 and 3).", "pages": [ 15 ] }, { "title": "4.3. Conclusions on the best fit results", "content": "Based on the physical aspects related to the scale factor, we now examine our unconstrained and constrained fit results in connection with our representation (6). This discussion, together with that in Subsec. 4.1, will lead us to conclude that the best results are those obtained with the constrained variant. With the unconstrained fits ( δ or s h = δ 4 m 2 p free) we have obtained for the √ s max = 7 TeV ensemble s h ∼ 13 GeV 2 (individual fit), s h ∼ 16 GeV 2 (global fit) and for the √ s max = 8 TeV ensemble s h ∼ 0.6 GeV 2 (individual fit). In these cases it seems difficult to devise a physical meaning for the onset of the σ HE ( s ) component, because its value depends on the variant considered and data analyzed. Moreover, as mentioned in Subsec. 3.2, with the √ s max = 8 TeV ensemble, the error matrix is not positive definite in the individual fit and no convergence was obtained in the global fit. On the other hand, in the case of constrained fits ( s h = 4 m 2 p fixed) the σ HE ( s ) component starts at this threshold with σ HE ( s h ) = α and from this point on it increases with the energy, namely σ HE ( s ) ≥ α at √ s ≥ 2 m p . This threshold, √ s h ∼ 2 GeV, is also below the energy cutoff, √ s min = 5 GeV. This situation seems physically meaningful to us in both phenomenological and theoretical contexts. Furthermore, in all cases investigated with the constraint condition, especially with the √ s max = 8 TeV ensemble, we have obtained full convergence in the data reductions and consistent statistical results. As shown in Figure 6, the values of γ are also consistent (stable) and lie around 2.2 - 2.3, in all cases investigated with the constrained variant. This discussion and the points raised in the previous sections favor, therefore, the results obtained with the constrained fits. Among them, we select as our best results the individual and global fits with the √ s max = 8 TeV ensemble. Both indicate a rise of the total cross-section that is faster than the log-squared behavior.", "pages": [ 15 ] }, { "title": "5. Fits to Elastic Cross-Section Data", "content": "The total cross-section is related to the elastic cross-section in the forward direction via the optical theorem. That has led us 20 to explore the possibility of extending the same analytical parametrization of the total cross-section, Eqs. (1)-(3), to the elastic (integrated) cross-section data, σ el . Based on unitarity, the same value of the exponent γ obtained for σ tot ( s ) is assumed for σ el ( s ). For a detailed discussion on this assumption see section 3 in Ref. 20.", "pages": [ 16 ] }, { "title": "5.1. Fit and results", "content": "The experimental data on pp and ¯ pp scattering below 7 TeV have been extracted from the PDG database, 18 without any kind of data selection or sieve procedure. The dataset includes also the recent TOTEM results (four points at 7 TeV (Ref. 23) and one point at 8 TeV (Ref. 24)). Statistical and systematic errors have been added in quadrature. As feedback for initializing the parametrization of σ el data we consider here the values of the parameters from our selected fit results to σ tot with both γ and s h fixed (constrained fits in Table 3). We notice that the data reductions using as initial values the results from either the individual or global fits are similar. In what follows we focus mainly on the global case. The results with γ = 2.23 and s h = 3.521 GeV 2 (fixed) are displayed in Table 7 and in Figure 8 (up) together with the evaluated uncertainty region. From Table 7, the value of the a 2 parameter is statistically consistent with zero. This is a consequence of the equality of the pp and ¯ pp elastic cross-sections data at low energies. We have checked that letting a 2 = 0, the same fit result is obtained. However, we notice that the statistical quality of the fit is not so good: large reduced χ 2 and small integrated probability. Moreover, from Figure 8, the fit uncertainty region barely reaches the extremum of the lower error bar of the TOTEM result at 8 TeV. On statistical grounds, since this point constitutes a high-precision measurement (defining the experimental information at the highest energy), the somewhat low fit quality may be associated with the underestimation of this datum by the fit result. We shall return to this important point related to the 8 TeV TOTEM data in section 6.1. Despite the limitations on the statistical quality of the fit, from Figure 8 (up), the global description of the experimental data seems satisfactory, including, within the uncertainties, the lower error bars of the four TOTEM results at 7 TeV. If we accept this data reduction as a reasonable description of the experimental data, the results, together with that obtained for the total cross-section, allow us to predict the ratio between the elastic and total cross-section as function of the energy. The result, within the uncertainties, is shown in Figure 8 (down), together with the experimental data. Using the s -channel unitarity, we have also included in this figure the result from the estimations of the total cross-section and the inelastic cross-section ( σ inel ) at 57 TeV by the Auger Collaboration. 35", "pages": [ 16, 17 ] }, { "title": "5.2. Asymptotic ratios", "content": "The asymptotic ratio between the elastic and total cross-sections can be evaluated from parametrization (1)-(3). Denoting the parameters β associated with the σ tot and σ el fits by the corresponding indexes, for s →∞ , we have From Tables 3 and 7 and the s -channel unitarity, we obtain a result which is not in agreement with the naive black-disk model (limit 1/2), but statistically consistent, within the uncertainties, with rational limits as already obtained in our previous analysis, 20 where only the first TOTEM measurement at 7 TeV has been included. We note that from the individual fit to σ tot ( γ = 2.30 and s h = 3.521 GeV 2 fixed) we obtain σ el /σ tot → 0.301 ± 0.098, a result also in agreement with (8), within the uncertainties. These rational limits contrast with the prediction from the model-dependent amplitude analysis by Block and Halzen, which indicates the black-disk limit for both ratios. 40 However, the rational limits are not in disagreement with a saturation of the Pumplim bound, 41,42", "pages": [ 17 ] }, { "title": "18 M. J. Menon & P. V. R. G. Silva", "content": "where σ diff is the cross-section associated with the soft diffractive processes (single and double dissociation). This saturation and the rational limits corroborate the recent phenomenological arguments by Grau et al . who attribute the black-disk limit to the combination of the elastic and diffractive processes, calling also the attention to the possibility of the limiting value 1/3. 43 If that is the case, our results predict", "pages": [ 18 ] }, { "title": "6. On a Fast Rise of the Total Cross-Section", "content": "Our results with both ensembles ( √ s max =7TeV and 8 TeV) indicate the possibility of a rise of σ tot faster than the log-squared behavior. In addition, there seems to be some interesting aspects related to the data at the highest LHC energy that deserve further comments. In this section we first discuss some features of the 8 TeV TOTEM data ( σ tot and σ el ), as compared with those at 7 TeV and in the region below this energy. After that, we present a few conjectures related to the possibility of a fast increase of the total cross-section at the LHC energy region and beyond.", "pages": [ 18 ] }, { "title": "6.1. The 8 TeV TOTEM data", "content": "From Tables 2 and 3, in going from the √ s max = 7 TeV ensemble to the √ s max = 8 TeV ensemble, we can note a slight increase in the value of the γ parameter (although consistent within the uncertainties in the constrained case). That may suggest a rise of σ tot in the 7 - 8 TeV region faster than that observed up to 7 TeV. In this respect we draw the attention to the results that follows. Therefore, even with γ greater than 2 and the scaling parameter fixed or free, the fit results are not in statistical agreement with the 8 TeV TOTEM data on σ tot and σ el : the fits somewhat underestimate the high-precision experimental values, suggesting a rise faster than expected. In this respect, some quantitative inferences may be instructive, even if only in a limited context, as discussed in what follows. First, although associated with different variants, the results here obtained for the γ parameter ( γ i ± ∆ γ i ) can be used to provide a quantitative information on typical values associated, separately, with the ensembles √ s max = 7 TeV (four points) and √ s max = 8 TeV (three points). We have considered two evaluations, either a weighted mean (with weights 1 / [∆ γ i ] 2 ) or a fit by a constant function (MINUIT). The results are displayed in Table 8 showing that, from ensemble √ s max = 7 TeV to √ s max = 8 TeV, both evaluations indicate an increase in the value of γ around 16 %. ± ± Second, in order to get some quantitative information directly related to the highest energy region, we have developed fits to σ tot data with only the high-energy parametrization σ HE , Eq. (3), but applied to datasets with different energy cutoffs: √ s min = 62.5 GeV (CERN-ISR), 546 GeV (CERN-Collider) and 1.8 TeV (Fermilab). As already selected, we have considered the constrained variant ( s h = 4 m 2 p fixed) with √ s max = 8 TeV. In this case the parametrization has only three free fit parameters, namely α , β and γ . For the first cutoff ( √ s min = 62.5 GeV) we have used as feedback the values of the parameters obtained in the individual fit to σ tot data in the constrained case and ensemble √ s max = 8 TeV (fourth column in Table 3). Then, the fit result has been used as feedback for the second cutoff ( √ s min = 546 GeV) and the same process for the third one. The results with the first two cutoffs are displayed in Table 9 and Figure 9 (for √ s min = 1.8 TeV the results are similar to those obtained with √ s min = 546 GeV). We understand that all the aforementioned results and discussions seem to suggest a fast unexpected rise of the cross-sections from 7 to 8 TeV as compared with the region below 7 TeV. From Figure 9 we note that in the case of √ s min = 62.5 GeV the fit result with only three parameters is in plenty agreement with all the pp and ¯ pp experimental data above ∼ 30 GeV, describing also the pp data at lower energies. The TOTEM data at 7 and 8 TeV are also described within the uncertainties and in this case γ ≈ 2.5 (Table 9). With the cutoff √ s min = 546 GeV, the fit is in agreement with the high-energy data (above ∼ 100 GeV) and the TOTEM data is quite well described (especially at 8 TeV); however, in this case γ ≈ 3.3 (Table 9).", "pages": [ 18, 19 ] }, { "title": "6.2. Some conjectures", "content": "At this point, we could conjecture (if not speculate) on the implication of a possible increase of σ tot faster than ln 2 s . One possibility points to a power-like behavior s /epsilon1 , /epsilon1 > 0, which has always been an important and representative approach. 44-47 Predictions from unitarized models, developed nearly 10 years ago and consistent with the first 7 TeV TOTEM datum, are discussed, for example, in Refs. 48-52. A faster-than-squared-logarithm rise could also indicate the onset of some new physics effect at the LHC energy region. For example, one possible explanation for the short penetration depth, recently observed in ultra-high-energy cosmic rays (UHECRs) around 100 TeV, is just an increase of the proton cross-section faster than the extrapolations from models, which have been tested only at lower energies (see, for example, Refs. 53 and 54 and references therein). These conjectures are not in disagreement with the recent theoretical arguments by Azimov. 5-7 At last, in contrast to an effective violation of the Froissart-Martin bound, a fast rise of the total cross-section may also be associated with some local effect at the LHC energy region and/or beyond, so that, asymptotically, the bound might remain valid. In that case, γ could represent a kind of effective exponent, depending on the energy and, possibly, associated with sums of different high-energy contributions. However, if constituting only a local effect our asymptotic results for the ratios involving the cross-sections might not be valid.", "pages": [ 19, 20 ] }, { "title": "7. Conclusions and Final Remarks", "content": "In 2002, Barone and Predazzi stated (Ref. 36, page 140): 'The issue of the exact growth with energy of the total cross-sections is both delicate and unresolved; the mild growth of total cross-sections could be simulated by essentially any form and logarithmic physics is exceedingly difficult to resolve in a clear cut way.' The aim of this paper has been to take one more step in our investigation on the rise of the total hadronic cross-section at high energies. As in our previous analyses 19,20 we have employed the analytical parametrization introduced by Amaldi et al ., with the exponent γ in the high-energy leading logarithm contribution treated as a free real parameter in nonlinear data reductions. Here, the main points consisted in an updated analysis (including in the dataset the recent high-precision TOTEM measurements at 7 TeV and 8 TeV) and a discussion on the correlation, practical role and physical meaning associated to the exponent γ and the energy scale factor s h . As in our previous works, we have considered different variants, involving two ensembles, individual/global fits and unconstrained/constrained fits. As feedbacks for the nonlinear data reductions we have used the 'conservative' values obtained by the PDG in the recent 2012 Review of Particle Physics edition. In section 4 we have discussed all the fit results, indicating the advantages of the constrained fits ( s h = 4 m 2 p fixed) in both phenomenological and theoretical contexts and noticing also the statistical consistence of the fit results. In particular, we have concluded that the constrained fits with the √ s max = 8 TeV ensemble (individual and global cases) represent our best results (Table 3, fourth and fifth columns and Figures 4 and 5). They indicate a rise of the total hadronic cross-section faster than the log-squared bound at the LHC energy region. The results and predictions for the pp total cross-section at energies of interest are displayed in the last columns of Tables 4 and 5. A critical discussion on the COMPETE 2002 prediction for the total cross-section and the recent 2012 PDG result has been also presented. With the selected results mentioned above, extensions of the parametrization to fit the elastic cross-section data, with fixed γ , have led to almost satisfactory results. Asymptotic limits for the ratios between elastic/total and inelastic/total cross-sections indicate consistence with 1/3 and 2/3, respectively (in agreement with our previous result 20 ). We have called the attention to a possible fast rise of the cross-section between 7 TeV and 8 TeV, according to the TOTEM results. We have conjectured that a fast rise might be connected with the onset of some new phenomena and have also speculated on the possible connection with the short penetration depth recently observed in UHECRs. If these effects have a local character (finite energies), there might be no contradiction with the Froissart-Martin bound, since it has been derived for the asymptotic energy limit, s →∞ . At last, we mention our recently updated comparative analysis, 55 which includes also fits with either γ =2(fixed) or a simple pole parametrization for the high-energy contributions (namely s /epsilon1 , /epsilon1 > 0). Beyond further discussions on the effects associated with the parameters γ and s h (not present in the simple pole parametrization), the results complement and corroborate those presented here and in the previous works. 19, 20 Our final conclusion, as we have stressed, is that the rise of the total hadronic cross-section at the highest energies still constitutes an open problem, demanding, therefore, further and detailed investigation. Updated amplitude analyses by other authors including in the dataset all the high precision TOTEM measurements at 7 TeV and 8 TeV can provide further checks on the results we have obtained and the conclusions we have drawn.", "pages": [ 20, 21 ] }, { "title": "Acknowledgments", "content": "We are grateful to an anonymous referee for valuable comments and suggestions, especially in respect to section 6.1. We are thankfull to D.A. Fagundes and D.D. Chinellato for useful discussions. Research supported by FAPESP (Contracts Nos. 11/15016-4, 09/50180-0).", "pages": [ 22 ] }, { "title": "References", "content": "An updated analysis on the rise of the hadronic total cross-section at the LHC energy region 23 http://root.cern.ch/root/html/TMinuit.html. ρ", "pages": [ 23, 25 ] }, { "title": "26 M. J. Menon & P. V. R. G. Silva", "content": "ρ ρ 1.8 34.9 PDG (2012) (mb) tot σ 34.8 34.7 5 6 7 (GeV) s", "pages": [ 26, 27, 29, 30 ] } ]

[ { "title": "Lagrangian Formulation of a Magnetostatic Field in the Presence of a Minimal Length Scale Based on the Kempf Algebra", "content": "S. K. Moayedi a ∗ , M. R. Setare b † , B. Khosropour a ‡ Department of Physics, Faculty of Sciences, Arak University, Arak 38156-8-8349, Iran b Department of Science, Campus of Bijar, University of Kurdistan, Bijar, Iran", "pages": [ 1 ] }, { "title": "Abstract", "content": "In the 1990s, Kempf and his collaborators Mangano and Mann introduced a D -dimensional ( β, β ' )-two-parameter deformed Heisenberg algebra which leads to an isotropic minimal length ( /triangle X i ) min = ¯ h √ Dβ + β ' , ∀ i ∈ { 1 , 2 , · · · , D } . In this work, the Lagrangian formulation of a magnetostatic field in three spatial dimensions ( D = 3) described by Kempf algebra is presented in the special case of β ' = 2 β up to the first order over β . We show that at the classical level there is a similarity between magnetostatics in the presence of a minimal length scale (modified magnetostatics) and the magnetostatic sector of the Abelian Lee-Wick model in three spatial dimensions. The integral form of Ampere's law and the energy density of a magnetostatic field in the modified magnetostatics are obtained. Also, the Biot-Savart law in the modified magnetostatics is found. By studying the effect of minimal length corrections to the gyromagnetic moment of the muon, we conclude that the upper bound on the isotropic minimal length scale in three spatial dimensions is 4 . 42 × 10 -19 m . The relationship between magnetostatics with a minimal length and the Gaete-Spallucci non-local magnetostatics (J. Phys. A: Math. Theor. 45 , 065401 (2012)) is investigated. Keywords: Phenomenology of quantum gravity; Generalized uncertainty principle; Minimal length; Classical field theories; Classical electromagnetism; Quantum electrodynamics; Noncommutative field theory PACS: 04.60.Bc, 03.50.-z, 03.50.De, 12.20.-m, 11.10.Nx", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "One of the most important problems in theoretical physics is the unification between the Einstein's general theory of relativity and the Standard Model of particle physics [1]. According to Ref. [1], two important predictions of this unification are the following: ( i ) the existence of extra dimensions; and ( ii ) the existence of a minimal length scale on the order of the Planck length. Studies in string theory and loop quantum gravity emphasize that there is a minimal length scale in nature. Today's theoretical physicists know that the existence of a minimal length scale leads to a modification of Heisenberg uncertainty principle. This modified uncertainty principle can be written as where /lscript P is the Planck length and a i , ∀ i ∈ { 1 , 2 , · · · } , are positive numerical constants [2-4]. By keeping only the first two terms on the right-hand side of Eq. (1), we obtain the usual generalized uncertainty principle (GUP) as follows: It is clear that in Eq. (2), /triangle X is always larger than ( /triangle X ) min = √ a 1 /lscript P . At the present time, theoretical physicists believe that reformulation of quantum field theory in the presence of a minimal length scale leads to a divergenceless quantum field theory [5-7]. During recent years, reformulation of quantum mechanics, gravity, and quantum field theory in the presence of a minimal length scale have been studied extensively [5-21]. H. S. Snyder was the first who formulated the electromagnetic field in quantized spacetime [22]. There are many papers about electrodynamics in the presence of a minimal length scale. For a review, we refer the reader to Refs. [12,13,14,15,16,19,20]. In our previous work [15], we studied formulation of electrodynamics with an external source in the presence of a minimal measurable length. In this work, we study formulation of a magnetostatic field with an external current density in the presence of a minimal length scale based on the Kempf algebra. This paper is organized as follows. In Section 2, the D -dimensional ( β, β ' )-two-parameter deformed Heisenberg algebra introduced by Kempf and his co-workers is studied and it is shown that the Kempf algebra leads to a minimal length scale [2325]. In Section 3, the Lagrangian formulation of a magnetostatic field in three spatial dimensions described by Kempf algebra is introduced in the case of β ' = 2 β , whereas the position operators commute to the first order in β . It is shown that at the classical level there is a similarity between magnetostatics in the presence of a minimal length scale and the magnetostatic sector of the Abelian Lee-Wick model in three spatial dimensions. The Ampere's law and the energy density of a magnetostatic field in the presence of a minimal length scale are obtained. In Section 4, the Biot-Savart law in the presence of a minimal length scale is found. We show that at large spatial distances the modified Biot-Savart law becomes the Biot-Savart law in usual magnetostatics. In Section 5, we study the effect of minimal length corrections to the gyromagnetic moment of the muon. From this study we conclude that the upper bound on the isotropic minimal length scale in three spatial dimensions is 4 . 42 × 10 -19 m . This value for the isotropic minimal length scale is close to the electroweak length scale ( /lscript electroweak ∼ 10 -18 m ). In Section 6, the relationship between magnetostatics in the presence of a minimal length scale and a particular class of nonlocal magnetostatic field is investigated. Our conclusions are presented in Section 7. We use SI units throughout this paper.", "pages": [ 2, 3 ] }, { "title": "2 Modified Commutation Relations with a Minimal Length Scale", "content": "Kempf and co-workers have introduced a modified Heisenberg algebra which describes a D -dimensional quantized space [23-25]. The Kempf algebra in a D -dimensional space is characterized by the following modified commutation relations 0). In Eqs. Also, in the above where i, j = 1 , 2 , ..., D and β, β ' are two non-negative deformation parameters ( β, β ' ≥ (3) and (4), β and β ' are constant parameters with dimension ( momentum ) -2 . equations X i and P i are position and momentum operators in the deformed space. An immediate consequence of Eq. (3) is the appearance of an isotropic minimal length scale which is given by [26] In Ref. [27], Stetsko and Tkachuk introduced a representation which satisfies the modified Heisenberg algebra (3)-(5) up to the first order in deformation parameters β and β ' . The Stetsko-Tkachuk representations for the position and momentum operators in the deformed space can be written as follows: where x i and p i = i ¯ h∂ i = i ¯ h ∂ ∂x i are position and momentum operators in ordinary quantum mechanics, and p 2 = ∑ D i =1 p i p i . In this article, we study the special case of β ' = 2 β , in which the position operators commute to the first order in deformation parameter β , i.e., [ X i , X j ] = 0 and thus a diagonal representation for the position operator in the deformed space can be obtained. For this linear approximation, the modified Heisenberg algebra (3)-(5) becomes In 1999, Brau [28] showed that the following representations satisfy (9)-(11), in the first order in β : It is necessary to note that the Stetsko-Tkachuk representations (7),(8) and the Brau representations (12),(13) coincide when β ' = 2 β . Benczik has shown that the energy spectrum of some quantum systems in the deformed space with a minimal length are representation-independent [29]. It seems that the laws of physics in the presence of a minimal length must be representationindependent.", "pages": [ 3, 4 ] }, { "title": "3 Lagrangian Formulation of a Magnetostatic Field with an External Current Density in the Presence of a Minimal Length Scale Based on the Kempf Algebra", "content": "The Lagrangian density for a magnetostatic field with an external current density J ( x ) = ( J 1 ( x ) , J 2 ( x ) , J 3 ( x )) in three spatial dimensions ( D = 3) can be written as follows [30]: where i, j = 1 , 2 , 3 , F ij ( x ) = ∂ i A j ( x ) -∂ j A i ( x ) and A ( x ) = ( A 1 ( x ) , A 2 ( x ) , A 3 ( x )) are the electromagnetic field tensor and the vector potential respectively. The Euler-Lagrange equation for the components of the vector potential is If we substitute (14) into (15), we will obtain the following field equation for the magnetostatic field The electromagnetic field tensor F ij ( x ) satisfies the Bianchi identity The three-dimensional magnetic induction vector B ( x ) is defined as follows [31]: where Using Eqs. (18) and (19), Eqs. (16) and (17) can be written in the vector form as follows: ∇ B ( x ) = µ 0 J ( x ) , (20) × The above equations are the basic equations of magnetostatics [30]. An immediate consequence of Eq. (21) is that B ( x ) can be written as follows: Now, we want to obtain the Lagrangian density for a magnetostatic field in the peresence of a minimal length scale based on the Kempf algebra. For this purpose, we must replace the ordinary position and derivative operators with the deformed position and derivative operators according to Eqs. (12) and (13), i.e., where ∇ 2 := ∂ i ∂ i is the Laplace operator. Using Eqs. (23) and (24) the electromagnetic field tensor in the presence of a minimal length scale becomes It should be mentioned that the above modification of the electromagnetic field tensor has been introduced earlier by Hossenfelder and co-workers in order to study the minimal length effects in quantum electrodynamics in Ref. [16]. If we use Eqs. (23), (24), and (25), we obtain the or Lagrangian density for a magnetostatic field in the deformed space as follows 1 : The term 1 4 µ 0 (¯ h √ 2 β ) 2 F ij ( x ) ∇ 2 F ij ( x ) in Eq. (26) can be considered as a minimal length effect. After neglecting terms of order (¯ h √ 2 β ) 4 and higher in Eq. (26) we obtain The Lagrangian density (27) is similar to the magnetostatic sector of the Abelian Lee-Wick model which was introduced by Lee and Wick as a finite theory of quantum electrodynamics [32-36]. Eq. (27) can be written as where After dropping the total derivative term ∂ n Λ n ( x ), the Lagrangian density (28) will be equivalent to the following Lagrangian density: where a := ¯ h √ 2 β is a constant parameter which is called Podolsky's characteristic length [37-41]. The Euler-Lagrange equation for the Lagrangian density (30) is [42-44] If we substitute (30) into (31), we obtain the following field equation for the magnetostatic field in the deformed space 2 Using Eqs. (18) and (19), Eqs. (17) and (32) can be written in the vector form as follows: Equations (33) and (34) are fundamental equations of Podolsky's magnetostatics [45-48]. It should be noted that Eqs. (30), (33), and (34) can be obtained as the magnetostatic limit of Eqs. (20), (26), and (27) in our previous paper [15]. Using Stokes's theorem the integral form of Eq. (33) can be written in the form: where I is the total current passing though the closed curve C . Equation (35) is Ampere's law in the presence of a minimal length scale. It is clear that for ¯ h √ 2 β → 0, the modified Ampere's law in Eq. (35) becomes the usual Ampere's law. Now, let us obtain the energy density of a magnetostatic field in the presence of a minimal length scale. The energy density of a magnetostatic field in the usual magnetostatics is given by [30] Using Eqs. (23) and (24) the energy density of a magnetostatic field under the influence of a minimal length scale becomes or where φ i 1 ··· i k := ∂ i 1 · · · ∂ i k φ . This definition has been used by Moeller and Zwiebach in Ref. [44]. where we use the abbreviation ML for the minimal length. If we use the vector identities together with Eq. (34), the modified energy density u ML B can be written in the form where After dropping the total divergence term ∇ · Ω ( x ), the modified energy density (40) will be equivalent to the following modified energy density: The term 1 2 µ 0 (¯ h √ 2 β ) 2 ( ∇ × B ( x )) · ( ∇ × B ( x )) in Eq. (42) shows the effect of minimal length corrections.", "pages": [ 4, 5, 6, 7, 8 ] }, { "title": "4 Green's Function for a Magnetostatic Field in the Presence of a Minimal Length Scale", "content": "Substituting Eq. (22) into Eq. (33) and using the vector identity (38) we obtain In the Coulomb gauge ( ∇ · A ( x ) = 0), Eq. (43) can be written as The solution of Eq. (44) in terms of the Green's function, G ( x , x ' ), is given by where A 0 ( x ) and G ( x , x ' ) satisfy the equations and Now, let us solve Eq. (47) by writting G ( x , x ' ) and δ ( x -x ' ) in terms of Fourier integrals as follows: If we substitute Eqs. (48) and (49) into Eq. (47), we obtain the functional form of ˜ G ( k ) as follows: If Eq. (50) is inserted into Eq. (48), the Green's function G ( x , x ' ) becomes This type of Green's function has been considered in electrodynamics to avoid divergences associated with point charges [38,45,49,50]. Using Eqs. (45) and (51) the particular solution of Eq. (44), which vanishes at infinity is The vector potential (52) satisfies the Coulomb gauge condition ∇ · A ( x ) = 0. The expression (52) can be applied to current circuits by making the substitution: J ( x ' ) d 3 x ' → Id l ' . Thus where C is the contour defined by the wire. If we use Eqs. (22) and (52), we obtain the magnetic induction vector B ( x ) as follows: or Equation (54) is the Biot-Savart law in the presence of a minimal length scale. In the limit a = ¯ h √ 2 β → 0, the modified Biot-Savart law in (54) smoothly becomes the usual Biot-Savart law, i.e.,", "pages": [ 8, 9, 10 ] }, { "title": "5 Upper Bound Estimation of the Minimal Length Scale in Modified Magnetostatics", "content": "Now, let us estimate the upper bounds on the isotropic minimal length scale in modified magnetostatics. By putting β ' = 2 β into (6) the isotropic minimal length scale becomes The isotropic minimal length scale (56) in three spatial dimensions is given by where a = ¯ h √ 2 β . In a series of papers, Sprenger and co-workers [51,52] have concluded that the minimal length scale ( /triangle X i ) min in Eq. (57) might lie anywhere between the Planck length scale ( /lscript P ∼ 10 -35 m ) and the electroweak length scale ( /lscript electroweak ∼ 10 -18 m ), i.e., According to above statements, the upper bound on the isotropic minimal length scale in three spatial dimensions becomes Inserting (59) into (57), we find In a series of papers, Accioly et al. [34, 36, 37] have estimated an upper bound on Podolsky's characteristic length a by computing the anomalous magnetic moment of the electron in the framework of Podolsky's electrodynamics. This upper bound on a is Note that the upper bound on the Podolsky's characteristic length a in Eq. (60) is near to the upper bound on the Podolsky's characteristic length in Eq. (61). Another upper bound on the minimal length scale has been obtained in Ref. [53] by considering minimal length corrections to the gyromagnetic moment of electrons and muons. If we compare Eq. (13) in this work with Eq. (40) in Ref. [16], we obtain where L f is the minimal length scale in Refs. [16,53]. If we substitute (62) into (56), we will obtain the isotropic minimal length in three spatial dimensions as follows: The minimal length scale L f in Eqs. (62) and (63) can be written as where M f is a new fundamental mass scale [16,53]. Inserting Eq. (64) into Eq. (63), we find In Ref. [53] it was shown that the effect of minimal length corrections to the gyromagnetic moment of the muon leads to the following lower bound on the fundamental mass scale of the theory: Substituting Eq. (66) into Eq. (65), the isotropic minimal length scale in three spatial dimensions becomes If we insert Eq. (67) into Eq. (57), we will find It is interesting to note that the numerical value of the upper bound on a in Eq. (68) and the numerical value of the upper bound on a in Eq. (60) are close to each other.", "pages": [ 10, 11, 12 ] }, { "title": "6 The Equivalence between the Gaete-Spallucci Non-Local Magnetostatics and Magnetostatics in the Presence of a Minimal Length Scale", "content": "Smailagic and Spallucci have proposed an approach to formulate quantum field theory in the presence of a minimal length scale [54-56]. Using the Smailagic-Spallucci approach, Gaete and Spallucci have introduced a U (1) gauge field with a non-local kinetic term whose magnetostatic sector is where θ is a constant parameter with dimension of ( length ) 2 [57]. The function exp ( -θ ∇ 2 ) in Eq. (69) can be expanded in a formal power series as follows: where ( ∇ 2 ) l denotes the ∇ 2 operator applied l times [58]. After inserting Eq. (70) into Eq. (69), we obtain the following Lagrangian density: If we neglect terms of order θ 2 and higher in Eq. (71), we find A comparison between Eqs. (27) and (72) clearly shows that there is an equivalence between the Gaete-Spallucci non-local magnetostatics to the first order in θ and the magnetostatic sector of the Abelian Lee-Wick model (or magnetostatics in the presence of a minimal length scale). The relationship between the non-commutative constant parameter θ in Eq. (72) and a = ¯ h √ 2 β in Eq. (27) is According to Eq. (73), a = √ θ plays the role of the minimal length in the Gaete-Spallucci nonlocal magnetostatics [57,59]. If we insert Eq. (73) into Eq. (57), we find Using Eq. (68) in Eq. (73), we obtain the following upper bound for the non-commutative parameter θ : where we use the abbreviation MLCGMM for the minimal length corrections to the gyromagnetic moment of the muon. Chaichian and his collaborators have investigated the Lamb shift in noncommutative quantum electrodynamics ( NCQED ) [60,61]. They found the following upper bound for the non-commutative parameter θ : θ NCQED ≤ (10 GeV ) , or For a review of the phenomenology of non-commutative geometry see Ref. [62]. The upper bound (75) is about two orders of magnitude larger than the upper bound (76), i.e., If we insert (61) into (73), we obtain the following upper bound for θ : where we use the abbreviation MLCGME for the minimal length corrections to the gyromagnetic moment of the electron. The upper bound (78) is about four orders of magnitude larger than the upper bound (76), i.e., 4 - 2 A comparison between Eq. (77) and Eq. (79) shows that θ MLCGMM is nearer to θ NCQED . It should be emphasized that the magnetostatics in the presence of a minimal length scale is only correct to the first order in the deformation parameter β , while the Gaete-Spallucci non-local magnetostatics is valid to all orders in the non-commutative parameter θ .", "pages": [ 12, 13, 14 ] }, { "title": "7 Conclusions", "content": "After the appearance of quantum field theory many theoretical physicists have attempted to reformulate quantum field theory in the presence of a minimal length scale [63,64]. The hope was that the introduction of such a minimal length scale leads to a divergenceless quantum field theory [65]. Recent studies in perturbative string theory and quantum gravity suggest that there is a minimal length scale in nature [1]. Today's we know that the existence of a minimal length scale leads to a generalization of Heisenberg uncertainty principle. An immediate consequence of the GUP is that the usual position and derivative operators must be replaced by the modified position and derivative operators according to Eqs. (23) and (24) for β ' = 2 β . We have formulated magnetostatics in the presence of a minimal length scale based on the Kempf algebra. It was shown that there is a similarity between magnetostatics in the presence of a minimal length scale and the magnetostatic sector of the Abelian Lee-Wick model. The integral form of Ampere's law and the energy density of a magnetostatic field in the presence of a minimal length scale have been obtained. Also, the Biot-Savart law in the presence of a minimal length scale has been found. We have shown that in the limit ¯ h √ 2 β → 0, the modified Ampere and Biot-Savart laws become the usual Ampere and Biot-Savart laws. It is necessary to note that the upper bounds on the isotropic minimal length scale in Eqs. (59) and (67) are close to the electroweak length scale ( /lscript electroweak ∼ 10 -18 m ). We have demonstrated the equivalence between the Gaete-Spallucci non-local magnetostatics up to the first order over θ and magnetostatics with a minimal length up to the first order over the deformation parameter β . Recently, Romero and collaborators have formulated a higher-derivative electrodynamics [66]. In this work we have formulated a higher-derivative magnetostatics in the framework of Kempf algebra whereas the authors of [66] have studied an electrodynamics consistent with anisotropic transformations of spacetime with an arbitrary dynamic exponent z .", "pages": [ 14 ] }, { "title": "Acknowledgments", "content": "We are grateful to S. Meljanac and J. M. Romero for their interest in this work and for drawing our attention to the references [8,19,66]. Also, we would like to thank the referee for useful comments and suggestions.", "pages": [ 15 ] } ]

[ { "title": "ABSTRACT", "content": "International Journal of Modern Physics D c © World Scientific Publishing Company", "pages": [ 1 ] }, { "title": "BACK-REACTION IN RELATIVISTIC COSMOLOGY", "content": "Timothy Clifton School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, UK. [email protected] We introduce the concept of back-reaction in relativistic cosmological modeling. Roughly speaking, this can be thought of as the difference between the large-scale behaviour of an inhomogeneous cosmological solution of Einstein's equations, and a homogeneous and isotropic solution that is a best-fit to either the average of observables or dynamics in the inhomogeneous solution. This is sometimes paraphrased as 'the effect that structure has of the large-scale evolution of the universe'. Various different approaches have been taken in the literature in order to try and understand back-reaction in cosmology. We provide a brief and critical summary of some of them, highlighting recent progress that has been made in each case. Keywords : Inhomogeneous cosmology; Large-scale structure PACS numbers: 98.80.Jk", "pages": [ 1 ] }, { "title": "1. What is Back-Reaction?", "content": "The term 'back-reaction' is often used in cosmology to mean 'the effect that structure has on the large-scale evolution of the universe, and observations made within it'. Implicit within this statement are a number of fundamental problems that have yet to be fully understood. These include: Let us now briefly consider each of these points, before moving on to discuss recent attempts to understand them. Point 1 above alludes to the fact that in relativistic theories what we mean by the spatial separation of any two astrophysical objects depends on how we choose to foliate the universe with hyper-surfaces of constant time. In a spatially homogeneous universe, or a universe with an irrotational matter content, natural-looking choices might present themselves. In general, however, we should be free to make any number of choices. This then presents a problem: If the distance between any two astrophysical objects is in general foliation dependent, and we have no preferred foliation, then how should we go about defining the rate of change of distance between objects, and hence the expansion of the universe? In the end, the answer to this question will depend on exactly what one is trying to achieve, and is complicated considerably by the fact that in cosmology one is often interested in non-local averages (a notoriously difficult concept to define in general relativity). Below we will consider several different cases of interest. Point 2 is a subsequent problem that needs to be addressed, once a concept of 'large-scale expansion' exists that one is prepared to consider. It is not in general the case that observations made in an inhomogeneous geometry will have a straightforward correspondence with the observables that one would measure in a spatially homogeneous and isotropic universe with the same rate of expansion on large scales. That is, even if one succeeds in finding a good description for the large-scale expansion of the universe, then one still needs to do further work in order to relate this to observations made in the underlying inhomogeneous space-time. Once again, this is complicated considerably by the fact that we are often interested in the average of observables. This is in general a highly non-trivial problem, and below we will review some recent progress towards understanding it. Finally, point 3 is related to the fact that in order to test proposed solutions to the problems posed in points 1 and 2 it is of considerable interest to have cosmological models that are sophisticated enough to allow at least some of the interesting behavior that we expect in general. This is an extremely difficult problem. Although many inhomogeneous cosmological solutions to Einstein's equations are known, 1 most of these solutions are restricted either because they are required to exhibit a high degree of symmetry, or because they are algebraically special. Constructions such as the 'Swiss cheese' models allow some potential progress to be made, but are themselves severely restricted by the boundary conditions at the edge of each 'hole'. New approaches are required to make further progress in this area, and, once again, we will discuss some recent progress below. In Section 2 we consider approaches based on averaging over a set of prescribed spatial hyper-surfaces. In Section 3 we consider approaches based on averaging in four dimensions. Section 4 contains a discussion of some models that may be of use for studying averaging, and in Section 5 we provide a few closing comments.", "pages": [ 1, 2 ] }, { "title": "2. Spatial Averaging Approach", "content": "One way to proceed with the study of back-reaction is to consider the expansion of regions of space in a given foliation. The equations that govern this expansion can then be found, and compared to the Friedmann equations. This often leads one to consider the volume-weighted average of quantities such as energy density and pressure. The equations that result are therefore often referred to as the 'averaged field equations'. While simple, this approach has a number of obvious drawbacks. Firstly, it is manifestly not foliation invariant. Secondly, there is a freedom in how one chooses to specify that two spatial volumes at different times are the same region. And thirdly, the averaging of quantities over the spatial volume being considered is often only well defined for scalars. One can specify choices for the first and second of these that may initially appear natural, but that could in the end lead one to consider hyper-surfaces in the inhomogeneous space-time that become arbitrarily, and increasingly, distorted. The third of these problems is of more fundamental difficulty, as tensors cannot in general be compared at different points. Nevertheless, this approach provides a useful framework to investigate, and can be shown to give a straightforward correspondance to the average of observables in some situations.", "pages": [ 2, 3 ] }, { "title": "2.1. Buchert's Equations", "content": "The most well studied set of averaged equations that result from this approach are those found by Buchert after averaging the Hamiltonian, Raychaudhuri and conservation equations: 2 where a D and 〈 (3) R 〉 are the 'scale factor' and average Ricci curvature of the region of space D being considered, angular brackets denote a volume average throughout that region, and Q D is the back-reaction term that quantifies differences from the Friedmann equations that one might otherwise construct from these quantities. These are defined as where Θ and σ are the expansion and volume-preserving shear of the set of curves orthogonal to the hyper-surfaces containing D , and t and X i are the proper time measured along this set of curves and the spatial coordinates in the hyper-surfaces of constant t , respectively. The quantity t 0 is the value of t on some reference hypersurface (usually taken to be the one that contains us at present). One may note that equations (1)-(3) do not form a closed set. Extra information is therefore required, which can be given by specifying Q D = Q D ( t ). Presumably this requires either extra equations, or some knowledge of the inhomogeneous spacetime being averaged. As previously stated, one may also note that the averaging procedure given here by the angular brackets is foliation dependent and only applicable to scalars (this is particularly problematic for the term 〈 σ 2 〉 D in equation (6), as the evolution equation for σ 2 will contain tensors). Finally, while the expansion of the spatial domain D may not itself be directly observable, we will explain below that in some cases it can be linked to observables.", "pages": [ 3, 4 ] }, { "title": "2.2. Links to Observables", "content": "The term 'observables' can cover a wide array of different possibilities in cosmology. Here we will mainly be concerned with the luminosity distance-redshift relation. This is itself a direct observable of considerable interest for the interpretation of, for example, supernova observations. Beyond this, it is also often required in the interpretation of other observables as it is very often the case that one needs to transform from 'redshift space' to some concept of position space (i.e. the position of astrophysical objects on some spatial hyper-surface). The usual method for calculating luminosity distances in an inhomogeneous space-time is to first find the angular diameter distance to the emitting object as a function of some affine parameter, measuring distance along past-directed null geodesics. This can be achieved by integrating the Sachs optical equations. 3 In these equations the Ricci curvature of the space-time sources the evolution of the expansion of the past-directed null geodesics, and the Weyl curvature sources the evolution of their volume-preserving shear (which itself acts as a source for their expansion). The angular-diameter distance can then be straightforwardly related to the luminosity distance, 4 and the redshift can be calculated as a function of the affine distance (once the world-lines of the objects emitting the radiation have been specified). This then provides the luminosity distance as a function of redshift at all points on an observer's past-light cone, provided that geometric optics remains a good approximation, and that the light emitted from the distant object is not obscured by some intermediate matter before it reaches the observer. Although the method outlined above is, in general, a complicated problem involving a number of subtleties, it was recently shown by Rasanen 5 that progress can be made in space-times that display statistical homogeneity and isotropy on large scales. In this case one can estimate the average luminosity distance as a function of the average redshift that an observer in such a space-time may expect to reconstruct from observations made over cosmologically interesting distances. Assuming that the matter content is irrotational, that the shear in the null trajectories can safely be assumed to be small, that structures evolve slowly, and that hyper-surfaces of constant proper time can also be taken to be the same hyper-surfaces that display statistical homogeneity and isotropy, Rasanen made a convincing case that the av-erag minosity distance-redshift relation in the inhomogeneous space-time should be well approximated by observables calculated in a homogeneous and isotropic model with a scale factor that evolves according to equations (1)-(3). An alternative approach to this problem was taken by Clarkson and Umeh. 6 These authors considered expressions for measures of distance expanded as a power series in redshift, as derived for general space-times by Kristian and Sachs. 7 They then performed a decomposition into spherical harmonics, and constructed the following deceleration parameter, based on an analogy between the monopole of this expansion and the corresponding relations in a Friedmann universe: where H = Θ / 3 is the isotropic part of the Hubble rate, and subscript '0' denotes a quantity evaluated at z = 0. Using this expression they could consider the average deceleration within either a region of space, or a region of space-time. However, for matter obeying the strong-energy condition it can be seen from equation (7) that the average of q 0 will always be non-negative, and so the space-time (according to this measure) will always be inferred to be decelerating (in the absence of Λ). This is in contrast to the averaged evolutions possible from equations (1)-(3), and at first glance would appear to contradict the results of Rasanen described above. In fact, there is no contradiction between these two sets of results. 8 That is, the observable calculated by Clarkson and Umeh should be expected to be a good approximation to the deceleration that one would infer from observations made within a small region around an observer. This measure is closely related to the acceleration of space within that region, as specified by Einstein's equations (as long as shear is small), and not by equations (1)-(3). The observational measures considered by Rasanen, however, are only expected to approach the evolution described by equations (1)-(3) when the distances over which observations are made are much larger than the homogeneity scale of the space-time under consideration. This is, of course, the regime in which cosmological observations are usually made. Using example space-times it has been explicitly demonstrated that it is entirely possible for a set of observers in a given region of the universe to infer deceleration from Clarkson and Umeh's measure, while inferring acceleration from Buchert's measure. 8 This clearly demonstrates that the acceleration inferred from cosmological observations does not have to be closely related to the local acceleration of space itself. It also demonstrates that quantities that are uniquely defined in an exactly homogeneous and isotropic universe (such as q 0 ) can bifurcate into multiple different quantities in space-times that are only statistically homogeneous and isotropic, and that in general these new quantities can take very different values from each other. One must therefore proceed with care.", "pages": [ 4, 5 ] }, { "title": "3. Space-Time Averaging Approach", "content": "An alternative approach to considering the volume weighted average of quantities within 3-dimensional spatial regions is to instead consider averaging geometric quantities within 4-dimensional regions of space-time. Such a process is in general difficult to define in a covariant way, and so far has required the application of bi-local operators. These allow tensors to be compared at different points by transporting them along prescribed sets of curves. This then leads to the problems of how the curves in question should be prescribed, and exactly which transport method should be used. Various proposals exist as to the best way to address these issues. 9 While complicated, the idea of averaging quantities in 4-dimensional regions of space-time inherently avoids any foliation dependence. These approaches are also often aimed at averaging tensors directly, rather than just scalars. This has obvious advantages for gravitational theories constructed from tensors, such as general relativity.", "pages": [ 6 ] }, { "title": "3.1. Zalaletdinov's Equations", "content": "Probably the most well known attempt at averaging in space-time, and constructing a set of effective field equations that the averages should obey, is that of Zalaletdinov. 10 The first step in this approach is to construct the following average for a tensor p α... β... : where primed coordinates are those used in the 4-dimensional region Σ, which is the averaging domain associated with the point x . The quantities A α µ ' are the bi-local operators, which are functions of both x and x ' , and the quantity V Σ = ∫ Σ √ -g ' d 4 x ' is the volume of Σ. Each point, x , is expected to have associated with it its own averaging domain, Σ, which is related to other averaging domains by being transported around the manifold. By applying this averaging technique to the connection, and by using some 'splitting rules', Zalaletdinov is able to use Einstein's equations to derive a set of field equations that the averaged connection must obey. The are called the Macroscopic Field Equations, and are written 10 where bars denote averaged quantities, and M γβ = M α γαβ and Q µν = Z α µνα and Z α µνβ = 2 Z α glyph[epsilon1] µ [ glyph[epsilon1] νβ ] , where and where underlined indices are not included in symmetrization operations. The tensor Z α µ βγ νσ is known as the 2-point correlation tensor, and obeys its own algebraic and differential constraints. 10 The Macroscopic Field Equations (9) can be used to describe the behavior of a particular inhomogeneous space-time after averaging has been performed, but they can also be used as a set of field equations to which one can look for solutions directly. This latter approach has so far been taken in the cases of macroscopic geometries, ¯ g µν , that are spatially homogeneous and isotropic, 11,12 and geometries that are spherically symmetric and static. 13 This work has allowed some possible behaviors of averaged space-times to be found without specifying the underlying microscopic geometry. However, it has also so far required a number of assumptions to be made about the correlations that are present. These include the vanishing of the three-point and four-point correlation tensors, and the vanishing of the 'electric' part of the 2-point correlation tensor. 12 The particular situations in which these assumptions are valid remains to be determined, as is also the case for the assumptions that go into the derivation of the Macroscopic Field Equations (9). Nevertheless, this is an interesting approach that deserves further study.", "pages": [ 6, 7 ] }, { "title": "3.2. Links to Observables", "content": "Under the assumption that the macroscopic geometry is spatially homogeneous and isotropic (that is, after the averaging procedure has been applied, and the 'averaged' geometry displays these symmetries), then Coley, Pelavas and Zalaletdinov find the following to be a solution of the Macroscopic Field Equations (9): 11 where k g is a constant, and where a ( t ) and ρ obeys the Friedmann-like equations where k d is a constant (not necessarily equal to k g ), and where ρ and p are the macroscopic energy density and pressure (obtained after averaging the right-hand side of Einstein's equations). Superficially, the geometry given in equations (12)-(14) looks a lot like the spatially homogeneous and isotropic solutions of Einstein's equations in the presence of a perfect fluid. There is, however, a very significant difference: The spatial curvature constant that appears in the macroscopic geometry, k g , is not in general the same as the term that looks like spatial curvature in the Friedmann-like equation (13) (i.e. the one that contains k d ). That is, spatially curvature can take different values depending on the situation being considered. If one measured the angles at the corners of triangle, and determined the curvature of space in this way, then this would give a different result to that which would be obtained by measuring the recessional velocities of astrophysical objects and inferring the spatial curvature through the dynamical (Friedmann-like) equation (13). This behavior is impossible within the spatially homogeneous and isotropic solutions of Einstein's equations, and so could provide some potentially observable phenomena that could be used to test this approach. The difference between k g and k d is determined by terms that appear in the correlation tensor, Z α µ βγ νσ , and so by attempting to determine the difference between k g and k d observationally we could attempt to constrain Z α µ βγ νσ , and hence some of the possible effects of averaging. A first step towards investigating this possibility has recently been taken. 14 The authors of this work assume that average observables are determined by null trajectories in the average geometry, as specified in equation (12), and that redshifts are represented by the average scale factor, a ( t ). They then find that luminosity distances are given by the following equation: where the matter content of the macroscopic space-time has been assumed to be well approximated by non-interacting dust and Λ, where H 0 = ˙ a/a | z =0 , and where the Ω i are defined as where ρ m, 0 is the present energy density in dust, and 8 πGρ Λ = Λ is the effective energy density in Λ. The expression for luminosity distance given in equation (15) can now be used to interpret cosmological observations, and to obtain constraints on the Ω i . glyph[negationslash] Using data from the Hubble Space Telescope (HST), 15 the Wilkinson Microwave Anisotropy Probe (WMAP), 16 observations of the Baryon Acoustic Oscillations (BAOs), 17 and the Union2 18 and SDSS 19 supernova data sets, the parameters Ω k d , Ω k g and Ω Λ were constrained to take the values given in Table 1 below. 14 The additional freedom of allowing Ω k d = Ω k g in this analysis means that the CMB+ H 0 is now no longer sufficient to constrain the spatial curvature of the universe significantly. Observations of the CMB+ H 0 alone are also no longer sufficient to require Λ = 0. This simple extra degree of freedom therefore undermines two of the most important results of modern observational cosmology. By adding further data sets the constraints on Ω k d and Ω k g are improved, but still remain much weaker than in the standard Friedmann models that satisfy Einstein's equations. Even so, however, it was still found that the results of using all available observables were sufficient to require Ω Λ = 0 to high confidence, and that a spatially flat universe was consistent with observations. Finally, although the combination of some data sets excluded the possibility Ω k d = Ω k g at the 95% confidence level, it was found that the special case Ω k d = Ω k g was compatible with most combinations of these data sets. glyph[negationslash] glyph[negationslash] In general one might also consider the possibility of not just allowing Ω k d and Ω k g to be different, but also allowing them to functions of scale. Such a result might arise, for example, from performing averaging over domains of different sizes, a process which is implicitly carried out when consider different cosmological observables. Such a possibility allows for considerable extra freedom. 14", "pages": [ 7, 8, 9 ] }, { "title": "4. Constructing Inhomogeneous Models", "content": "We have so far considered attempts to describe the large-scale behavior of the universe by averaging over regions of space or space-time. In the end, the particular approach that one should use when performing this type of operation should probably be guided by the phenomena that one is trying to create a model to interpret. Different observable phenomena may require different approaches, and so one needs to know the limits of any particular approach, as well as the situations in which it reliably reproduces the required results. For this it is useful to have inhomogeneous cosmological models that are of sufficient generality to allow some of the interesting behavior that is expected in general. Such models can then be used to test ideas about averaging, back-reaction, and the large-scale evolution of space. Unfortunately it is extremely difficult to construct such models. This does not mean that there is an absence of any interesting behavior to study, only that we need to become more sophisticated in our model building to quantify and constrain the different possibilities in a reliable way. Some of the principal difficulties involved with this are how to model over-dense regions of the universe without having to deal with the rapid formation of singularities, how to introduce structure into the universe without assuming a Friedmann background or matching onto a Friedmann model at a boundary, and how to allow structure to form on different scales without assuming linearity in the field equations. For further discussion of inhomogeneous cosmological solutions the reader is referred to the contribution to these proceedings by Krasi'nski, 20 and to the comprehensive texts. 1, 21 It is currently almost beyond hope to construct a model that allows for all of the possibilities discussed above, while simultaneously maintaining sufficiently generality to model realistic distributions of matter. We are therefore forced to investigate toy models that we hope may reflect some of the features of the real universe, even if they are not realistic in every way. Once toy models have been constructed we can then consider the averaging problem by applying some of the methods discussed above to them, or by fitting or comparing them to Friedmann models directly. Their existence also makes more advanced models a more realistic proposition. It is for these reasons that it is of interest to consider simple n -body solutions of Einstein's equations. Such solutions, if they can be found, will allow over-dense regions to be studied without rapid collapse occurring, and without recourse to the assumption of a Friedmann background or linearity in the gravitational field equations. This will be the subject of Section 4.1.", "pages": [ 9, 10 ] }, { "title": "4.1. A Lattice of Black Holes", "content": "The simplest configuration of n bodies that one can imagine is a regularly arranged set of points. Although such a configuration limits the behaviors that are possible, it does allow for the most straightforward possible comparison to smoothed-out Friedmann-like universes. That is, by 'zooming out' in order to consider large numbers of points, and by performing some kind of coarse graining or smoothing, one could easily imagine such a situation looking more and more like a spatially homogeneous and isotropic universe, which could then be compared to the Friedmann solutions of Einstein's equations. Regularity of the distribution also provides a limited number of preferred spatial planes and curves that can have their area and length compared to those of the Friedmann solutions. Here we will consider spatially closed universes. These are known to admit hypersurfaces of time symmetry at the maximum of expansion of the space-time that allow the constraint equations to be solved in a particularly simple way. 22 The method that we will deploy to ensure that our massive bodies are regularly arranged is to tile the hyper-surface of maximum expansion with a number of regular polyhedra. A mass is then placed at the center of each polyhedron, which by symmetry will be an equal distance from each of its nearest neighbors. These polyhedra will be referred to in what follows as 'cells'. There are seven such tilings that are possible in three spatial dimensions, as listed in Table 2. Also displayed in this table are the Schlafli symbols of the polychora that these tilings constitute. 23 Once the arrangement of masses has been chosen, the geometry of the hypersurface of maximum-of-expansion can be found. With the exception of the 2-cell, this has been done for each of the structures described above. 24 The 2-cell is special in that the time-symmetric geometry at the maximum of expansion of this structure is simply a slice through the global Schwarzschild solution. The geometry of the full space-time is therefore already known exactly in this case, and is not of cosmological interest here (by including a non-zero Λ, however, other structures are also possible 25 ). An illustration of the geometry at the maximum of expansion in the case of the 8-cell and the 120-cell is given in Figure 1, below. Each of the illustrations here corresponds to a single 2-dimensional slice through the 3-dimensional geometry. In the case of the 8-cell this slice contains 6 masses, while in the 120-cell it contains many more (although not all 120). The geometry of the 3-space of maximum of expansion in each case is conformally related to the geometry of a 3-sphere, with a scale factor that is a function of position. The distance from the origin in the illustrations in Figure 1 is proportional to this scale factor, and it can be seen that as the number of the masses in the lattice is increased, the bulk of the space approaches homogeneity. It is only in the vicinity of the masses themselves that inhomogeneities exist (as depicted by the tube-like structures). (a) A slice through the 8-cell. Once we have the geometry of the hyper-surface of maximum of expansion, we can take a measure of the scale of the solution, and compare this to the scale of a spatially closed Friedmann universe that contains the same amount of 'proper mass 24 '. The Friedmann solutions will, of course, have this mass evenly distributed throughout space, and so by comparing to the scale of the inhomogeneous geometry we can obtain a measure of back-reaction. For the choice of scale in the inhomoge- neous space on could choose a number of different measures. Here we consider the proper length of the edge of a cell. This corresponds to the scale of curvature for the sphere that appears to emerge when the number of masses becomes large (as can be seen from the illustration in Figure 1b). The difference in scale in each case is given in the last column of Table 2. It can be seen that the broad trend is for the scale of the homogeneous and inhomogeneous space-times to approach each other as the number of cells becomes large. However, for only a small number of cells ( ∼ 5 to 24) the difference in scale can be of order 10%. In any case, this method provides an exact quantification of back-reaction, and provides an arena for testing formalisms designed for more general configurations of energy and momentum. Anumerical evolution of the 8-cell has now been performed, 26 and other methods have also been used to address problem of understand the evolution of this type of structure. 27-29", "pages": [ 10, 11, 12 ] }, { "title": "5. Discussion and Outlook", "content": "Various approaches to back-reaction and averaging already exist in the literature, but much work remains to be done if we are to fully understand their observational consequences in the real universe. Motivation for taking these problems seriously comes from the apparent necessity of including dark energy when we interpret observations within a linearly perturbed Friedmann model, as well as the requirement to understand all possible sources of error and uncertainty in precision cosmology. To fully address this problem it is likely that we will need to develop more sophisticated models of inhomogeneous space-times, as well as developing a more sophisticated understanding of averaging in general relativity. Research in this area should be considered exceptionally timely, with large amounts of resources currently being invested into observational probes designed to improve our understanding of dark energy, and the universe around us.", "pages": [ 12 ] }, { "title": "Acknowledgments", "content": "I acknowledge the support of the STFC.", "pages": [ 12 ] } ]

[ { "title": "ABSTRACT", "content": "What happens at the horizon? 1", "pages": [ 1 ] }, { "title": "Samir D. Mathur", "content": "Department of Physics, The Ohio State University, Columbus, OH 43210, USA [email protected] March 31, 2013", "pages": [ 1 ] }, { "title": "Abstract", "content": "The Schwarzschild metric has an apparent singularity at the horizon r = 2 M . What really happens there? If physics at the horizon is 'normal' laboratory physics, then we run into Hawking's information paradox. If we want nontrivial structure at the horizon, then we need a mechanism to generate this structure that evades the 'no hair' conjectures of the past. Further, if we have such structure, then what would the role of the traditional black hole metric which continues smoothly past the horizon? Recent work has provided an answer to these questions, and in the process revealed a beautiful tie-up between gravity, string theory and thermodynamics. One of the most basic solutions to Einstein's equations is the Schwarzschild metric corresponding to a point source Near infinity this metric reproduces the expected weak field effects of a mass M placed at r = 0. But moving inwards, we encounter a singularity at r = 2 M . What happens there? This question has led physicists through several twists and turns, and at the end, has led to a deep insight into the nature of quantum gravity. In this essay we recount this fascinating story, which has only recently reached its conclusion. First iteration: Particles falling in from infinity appear to slow down and freeze as they approach the horizon r = 2 M . Thus they never cross into the region r < 2 M even if we wait till t →∞ . This suggests the possibility that we may never need to talk about the region inside the horizon; physics should somehow be complete in the region r > 2 M . In the quantized theory of gravity, 't Hooft argued that there would be a 'brick wall' at the horizon that scatters infalling quanta back to infinity [1]. Susskind and his collaborators argued that quantum gravity effects would create an effective membrane at a 'stretched horizon' just outside r = 2 M , where infalling quanta will be absorbed and reemitted [2]. If these views were correct, there would indeed be a complete description of black hole physics with no interior region r < 2 M . Second iteration: But the above picture soon runs into trouble. The singularity at r = 2 M is just a coordinate singularity, and the metric can be continued smoothly across the horizon using Kruskal coordinates. Infalling particles appear to freeze at the horizon only because the time coordinate t does not cover their full trajectory; when we switch to Kruskal coordinates then the particle trajectories continue through the horizon and reach r = 0. The arguments for nontrivial effects at r = 2 M in the quantum theory also turned out to be flawed. In the Schwarzschild coordinate system we have g tt → 0 as we approach the horizon, and the corresponding time dilation leads to large quantum fluctuations for all fields. In particular the gravitational field has large fluctuations, and a naive analysis can suggest that something nontrivial is happening at the horizon. But since the horizon is a normal place in Kruskal coordinates, the effects of these violent quantum fluctuations should really all cancel out, leaving no 'reflecting barrier' at the horizon. A closer analysis of the above mentioned quantum gravitational computations indeed indicates that they are coordinate artifacts; see [3] for an example where the cancellations were demonstrated for a particular example. Should we therefore accept that the horizon is a 'normal place'? The problem with this conclusion is of course the information paradox pointed out by Hawking [4]. If the vicinity of the horizon is a normal place where the local fields are in the vacuum, then we have a progressive creation of entangled Hawking pairs. As the black hole evaporates, the radiation at infinity gets progressively more entangled with the hole left behind. We then encounter a sharp problem near the endpoint of evaporation where a tiny planck sized remnant must somehow be able to carry an arbitrarily large entanglement with the radiation at infinity. Most string theorists had not worried too much about the information paradox, for a reason which turned out to be incorrect. Since the number N of emitted Hawking quanta is large, they assumed that small quantum gravity corrections of order /epsilon1 /lessmuch 1 to the state of each created pair would be enough to make the overall state of the radiation unentangled from the remnant. But in [5] it was shown (using strong subaddditivity) that this belief was wrong; the reduction in entanglement entropy due to small corrections is bounded by Thus we can evade the information problem only by finding order unity corrections to evolution at the horizon. We get a 'theorem': if the horizon is a 'normal place' where lab physics holds to leading order, then we cannot evade the information paradox. Third iteration: The solution to the information problem is found by actually constructing the states in string theory with mass M . It turns out that the geometry obtained is not the Schwarzschild one, but has the following structure (fig.1). For a hole in 3+1 dimensions, we have 6 compact directions, which we take to be small circles. One of these circles 'pinches off' before reaching the horizon, so that spacetime ends there. There is an additional twist that makes the metric near this pinch-off the metric of a KK monopole. The microstates of the hole are given by different configurations of KK monopoles and antimonopoles distributed around the rough location r = 2 M where the horizon would have been, but in each case there is no horizon or 'interior' of the horizon. These configurations are termed fuzzballs, and for simple holes it has been shown that all states of the hole are of this form [6]. Thus we have finally found 'real' structure at the horizon, not a coordinate artifact. There are ergoregions between the KK monopoles, which radiate at exactly the rate expected from Hawking radiation [7]. But there is no information paradox, since particle creation is not happening by the Hawking process where one member of the created pair fall through a horizon. This discussion suggests that the interior region r < 2 M of the traditional black hole has no role at all. Susskind had postulated that there might be a 'complementary' description of the dynamics of the stretched horizon; in this description an infalling observer would see the traditional interior of the black hole [2]. In [8] the authors used the inequality (2) to argue that such a complementarity would not be possible; the order unity corrections required at the horizon would create a 'firewall' that cannot be consistent with a smooth continuation of the metric to r < 2 M . But as we will see now, there is a further twist to the story, so that the interior of the hole has a role even though no microstate actually has such an interior. Fourth iteration: Consider probing the complicated surface of a generic fuzzball as shown in fig.2(a); this corresponds a 2-point function measured in a highly excited quantum gravity state. We will argue that, for suitable operators, we can get a good approximation to such a correlator by using instead the traditional metric (1) of the hole, as shown in fig.2(b). The latter geometry has none of the KK monopole excitations of the fuzzball surface but it does have the interior region r < 2 M . This is the sense in which we will find the role of the black hole interior. To motivate this proposal, recall the a scalar field φ on Minkowski space can be decomposed into fields in the right and left Rindler wedges. Following arguments of Israel [9], Maldacena [10] and van Raamsdonk [11], we expect a similar decomposition where the state of an eternal black hole can be written as an entangled sum of gravitational states in the right and left quadrants: The states | g k 〉 R live in the right wedge, and go to the vacuum at infinity. This is just the nature we observed for the fuzzball states, which asymptote to flat infinity and end just before reaching the horizon where a compact circle pinches off. Thus we conjecture that the states | g k 〉 are in fact the fuzzball states which describe microstates of the black hole [12]. We now make two observations (fig.3): (i) The expectation value in the eternal black hole state of an operator in the right wedge is given by a thermal average over fuzzball states (ii) A given black hole is in one fuzzball state. But for a generic fuzzball state, and for suitable operators ˆ O R , we can approximate the expectation value by the ensemble average over all fuzzballs where in the second step we have used (4). But this is just the statement in fig.2, where the expectation value in one fuzzball is approximated by a black hole geometry which does have a region past the horizon. To summarize, we have arrived at the following picture. The microstates of the hole are the fuzzballs depicted in fig.1(c); these states have no horizon and no 'interior' region analogous to the region r < 2 M . The fuzzball radiates at the Hawking temperature T from its surface just like any normal body, so there is no information problem. Probing the fuzzball at energies E /greatermuch kT excites collective modes of the fuzzball which can be well approximated by an ensemble average over fuzzballs, and this average is reproduced by the traditional black hole geometry.", "pages": [ 1, 2, 3, 4, 5, 6 ] } ]

[ { "title": "CosMIn: The Solution to the Cosmological Constant Problem", "content": "Hamsa Padmanabhan, T. Padmanabhan IUCAA, Post Bag 4, Ganeshkhind, Pune - 411 007, India. email: [email protected], [email protected]", "pages": [ 1 ] }, { "title": "Abstract", "content": "The current acceleration of the universe can be modelled in terms of a cosmological constant Λ. We show that the extremely small value of Λ L 2 P ≈ 3 . 4 × 10 -122 , the holy grail of theoretical physics, can be understood in terms of a new, dimensionless, conserved number CosMIn, which counts the number of modes crossing the Hubble radius during the three phases of evolution of the universe. Theoretical considerations suggest that N ≈ 4 π . This single postulate leads us to the correct, observed numerical value of the cosmological constant! This approach also provides a unified picture of cosmic evolution relating the early inflationary phase to the late accelerating phase. 1 Keywords: cosmological constant, inflation, holographic equipartition PACs: 95.36.+x 04.60.-m 98.80.Es Introduction: Our description of the cosmos is very tantalizing! It has three distinct phases of evolution, bearing no apparent relation to each other: An early inflationary phase, driven possibly by a scalar field, a late-time accelerated phase, dominated by dark energy, and a transient phase in between, dominated by radiation and matter. The first and the last phases are approximately de Sitter, with Hubble radii H -1 inf and H -1 Λ , characterized by two dimensionless ratios β -1 ≡ H inf L P and Λ L 2 P , where L P ≡ ( G /planckover2pi1 /c 3 ) 1 / 2 is the Planck length. If inflation took place at GUTs scale ( ∼ 10 15 GeV), then β ≈ 3 . 8 × 10 7 , while observations [1, 2] suggest that Λ L 2 P ≈ 3 . 4 × 10 -122 ≈ 3 × e -281 . It is expected that physics at, say, the GUTs scale will (eventually) determine β . But no fundamental principle has been suggested to explain the extremely small value of Λ L 2 P , which is related (directly or indirectly) to the cosmological constant problem. Understanding this issue [3, 4] from first principles is considered very important in theoretical physics today. In this essay, we will describe an approach which tackles this problem (for more details, see [5]) and also provides a unified picture of cosmic evolution. We show that ln(Λ L 2 P ) is related to a dimensionless number ('Cosmic Mode Index', dominated or CosMIn, N c ) that counts the number of modes within the Hubble volume that cross the Hubble radius between the end of inflation and the beginning of late-time acceleration. CosMIn is a characteristic number for our universe and it is possible to argue [9] that the natural value for N c is about 4 π ; i.e., N c = 4 πµ with µ ∼ 1. This single postulate allows us to determine the numerical value of Λ L 2 P ! We obtain Λ L 2 P = Cβ -2 exp( -24 π 2 µ ), where C depends on n γ /n m , the ratio between the number densities of photons and matter. This leads to the correct observed value of the cosmological constant for a GUTs scale inflation and the range of C permitted by cosmological observations. CosMIn and the cosmological constant: Aproper length scale λ prop ( a ) ≡ a/k (labelled by a co-moving wave number, k ) crosses the Hubble radius whenever the equation λ prop ( a ) = H -1 ( a ), i.e., k = aH ( a ) is satisfied. For a generic mode (see Fig.1; line marked ABC ), this equation has three solutions: a = a A (during the inflationary phase; at A ), a = a B (during the radiation/matter dominated phase; at B ), a = a C (during the late-time accelerating phase; at C ). But modes with k < k -exit during the inflationary phase and never re-enter. Similarly, modes with k > k + remain inside the Hubble radius and only exit during the late-time acceleration phase. The modes with comoving wavenumbers in the range ( k, k + dk ) where k = aH ( a ) and dk = [ d ( aH ) /da ] da cross the Hubble radius during the interval ( a, a + da ). The number of modes in a comoving Hubble volume V com = (4 πH -3 / 3 a 3 ) with wave numbers in the interval ( k, k + dk ) is dN = V com d 3 k/ (2 π ) 3 . Hence, the number of modes that cross the Hubble radius in the interval ( a 1 < a < a 2 ) is given by where we have used V com = 4 π/ 3 H 3 a 3 and k = Ha . All the modes which exit the Hubble radius during a A < a < a X enter the Hubble radius during a X < a < a B (and again exit during a Y < a < a Q .) So the number of modes which do this during PX , XY or Y Q is a characteristic, 'conserved' number ('CosMIn') for our universe, say N c . Its importance is related to the the cosmic parallelogram PXQY (Fig.1) which arises only in a universe having three distinct phases. The epochs P and Q , limiting the otherwise semi-eternal de Sitter phases, now have a special significance [6, 7, 8]. Modes which exit the Hubble radius before a = a P never re-enter. On the other hand, the epoch a = a Q denotes (approximately) the time when the CMBR temperature falls below the de Sitter temperature [6, 7, 8]. The special role of PXQY makes the value of CosMIn significant. As shown in Fig.1, these modes in PXQY (with k -< k < k + ) cross the Planck length during a -< a < a + . Based on holographic considerations, it is possible to argue [9] that Planck scale physics imposes the condition N c = N ( a -, a + ) ≈ 4 π at this stage. So, by computing CosMIn for the universe, and equating to 4 π , we can determine Λ L 2 P . As a quick check on the paradigm N c ≈ 4 π , let us approximate the intermediate phase of the universe as purely radiation dominated ( H ( a ) ∝ a -2 ) and assume Planck scale inflation ( β = 1), thereby eliminating all free parameters. The above procedure now [9] gives: Thus, Λ L 2 P is directly related to CosMIn and, in this simple model, there are no other adjustable parameters . Eq. (2) leads to the observed value Λ L 2 P = 3 . 4 × 10 -122 when µ = 1 . 18, showing we are clearly on the right track! /negationslash The presence of matter and the fact that the inflationary scale may not be the Planck scale in our universe ( β = 1), surprisingly, make the postulate of N c = 4 π work better in the real universe and reproduce the observed value of the cosmological constant. In this case, it is simpler to express Λ in terms of N c , β and a variable σ defined through σ 4 ≡ (Ω 3 R / Ω 4 m )[1 -Ω m -Ω R ] . Even though the values for Ω R and Ω m depend on the epoch t = t ∗ at which they are measured, the value of σ is the same at all epochs. (It is an example of an epoch-invariant parameter and, of course, the value of Λ L 2 P can only depend on such epoch-invariant parameters [5]). Determining N c in terms of Λ L 2 P , β and σ , and expressing Λ L 2 P in terms of the other parameters, we get: where C ( σ ) = 12( σr ) 4 (3 r + 4) -2 and r satisfies the quartic equation σ 4 r 4 = (1 / 2) r +1. Given the numerical value of σ , the inflation scale determined by β , and our postulate µ = 1, we can calculate the value of Λ L 2 P from Eq. (3). The result in Eq. (3) is summarized in Fig.2. The thick black curve is obtained from Eq. (3) if we take µ = 1 and β = 3 . 83 × 10 7 (corresponding to the inflationary energy scale of V 1 / 4 inf = 1 . 16 × 10 15 GeV) and leads to the observed (mean) value of Λ L 2 P = 3 . 39 × 10 -122 (horizontal unbroken, blue line). Observational constraints [1, 2] lead to σ = 0 . 003 +0 . 004 -0 . 001 (three vertical, red lines) and Λ L 2 P = (3 . 03 -3 . 77) × 10 -122 (horizontal, broken blue lines). This cosmologically allowed range in σ and Λ L 2 P is bracketed by the two broken black curves obtained by varying β in the range (2 . 64 -7 . 29) × 10 7 (i.e, V 1 / 4 inf = (0 . 84 -1 . 4) × 10 15 GeV). So, for an acceptable range of energy scales of inflation, and for the range of σ allowed by cosmological observations, our postulate N c = 4 π gives the correct value for the cosmological constant. Since our results only depend on the combination β -2 exp( -24 π 2 µ ), the same set of curves arise in a Planck scale inflationary model ( β = 1) with µ in the range (1 . 144 -1 . 153). There are three conceptually attractive features about Planck scale inflation with β = 1. First, it eliminates one free parameter, β , and gives a direct relation between the two scales Λ and L 2 P which occur in the Einstein-Hilbert action. (The dependence of the result on σ is weak and can be thought of as a matter of detail, like, for example, the fine structure correction to spectral lines beyond Bohr's model). Second, we can think of the intermediate phase as a mere transient connecting two de Sitter phases (the chicken is just the egg's way of making another egg!), both of which are semi-eternal. Since the de Sitter universe is time-translationally invariant, it is a natural candidate to describe the geometry of the universe dominated by a single length scale L P in the initial phase and Λ -1 / 2 in the final phase. The quantum instability of the de Sitter phase at the Planck scale can lead to cosmogenesis and the transient radiation/matter dominated phase, which gives way, eventually, to the late-time acceleration phase. Finally, the argument for N c ≈ 4 π is quite natural with Planck scale inflation. The transition at X , entrenched in Planck scale physics in such a model, can easily account for deviations of µ from unity. An integrated view of cosmology: The standard approach to cosmology treats the evolution of the universe in a fragmentary manner, with Planck scale physics, the inflationary era, the matter sector properties and the late-time acceleration each introducing their own parameters - like L 2 P , E inf , ( n m /n γ ) , Λ - all independently specified, bearing no relation with each other. Even if GUTs scale physics (eventually) determines E inf and ( n m /n γ ), there is still no link between these parameters, L P and Λ. In striking contrast, our paradigm, the postulate N c = 4 π acts as the connecting thread leading to a unified, holistic approach to cosmic evolution. In fact, when σ /lessmuch 1, one can write Eq. (3) (with µ = 1) as: where K ≡ ( π 11 / 2 /ζ (3) 2 )(1 / 6 3 10 3 / 2 ) ≈ 0 . 055 , ρ m ≡ mn m with m being the (mean) mass of the particle contributing to matter density, and M P being the Planck mass. In a consistent quantum theory of gravity, we expect inflation (which determines β ) and genesis of matter (which determines m and n m /n γ ) to be related to Planck scale physics such that our fundamental relation in Eq. (4) holds. Solving the cosmological constant problem by actually determining its numerical value has not been attempted before. This approach is similar in spirit to the Bohr model of the hydrogen atom, which used the postulate J = n /planckover2pi1 to explain the hydrogen spectrum. Here, our postulate N c = 4 π , captures the essence and explains the value of Λ L 2 P . This is simpler and more elegant than many other ad-hoc assumptions made in the literature [3, 4] to solve the cosmological constant problem. More importantly, we do know that this postulate is correct! The value of CosMIn can be determined directly from the observed value of Λ as well as other cosmological parameters. We would then find that it is indeed very close to 4 π . Why does it work? Recent work [9] has shown that cosmic evolution can be thought of as a quest for holographic equilibrium . One can associate with the proper Hubble volume V prop ≡ 4 π/ 3 H 3 , the numbers, which count the surface and bulk degrees of freedom, where E = ( ρ +3 p ) V prop is the Komar energy, T = H/ 2 π is the analogue of the horizon temperature and /epsilon1 = ± 1 is chosen to keep N bulk positive. Clearly, | E | = (1 / 2) N bulk k B T denotes equipartition of energy. Holographic equipartition is the demand that N sur = N bulk , which holds in a de Sitter universe with p = -ρ, /epsilon1 = 1 , H 2 = (8 π/ 3) ρ . When the universe is not pure de Sitter, we expect the holographic discrepancy between N sur and N bulk to drive the expansion of the universe, which suggests [8] the law: Incredibly, this leads to the standard Friedmann equation for cosmic expansion, but now obtained without using the field equations of general relativity! The right hand side is (nearly) zero in the initial and final phases (with V prop ≈ constant) and cosmic expansion in the transient phase can be interpreted as a quest towards holographic equipartition. It is then natural to associate a number N sur = 4 πL 2 P /L 2 P = 4 π with the modes, when they cross the Planck scale. Clearly, such a quantum gravitational imprint has far reaching consequences, culminating in the solution to the cosmological constant problem itself. Acknowledgements: T.P's research is partially supported by the J. C. Bose research grant of DST, India. H.P's research is supported by the SPM research grant of CSIR, India. We thank Sunu Engineer for useful comments.", "pages": [ 1, 2, 3, 4, 5, 6 ] } ]

[ { "title": "Entanglement entropy from surface terms in general relativity", "content": "Arpan Bhattacharyya and Aninda Sinha email: [email protected], [email protected] 1 Centre for High Energy Physics, Indian Institute of Science, Bangalore 560 012, India. September 25, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "Entanglement entropy in local quantum field theories is typically ultraviolet divergent due to short distance effects in the neighbourhood of the entangling region. In the context of gauge/gravity duality, we show that surface terms in general relativity are able to capture this entanglement entropy. In particular, we demonstrate that for 1+1 dimensional CFTs at finite temperature whose gravity dual is the BTZ black hole, the Gibbons-Hawking-York term precisely reproduces the entanglement entropy which can be computed independently in the field theory. Essay awarded honourable mention in the Gravity Research Foundation 2013 Awards for Essays on Gravitation. The Einstein-Hilbert action for gravity needs to be supplemented by the Gibbons-Hawking-York surface term [1, 2] to make the variational (Dirichlet boundary value) problem well defined. Namely, the total action (in Euclidean signature) is given by where g is the determinant of the bulk metric, R is the scalar curvature for the bulk space time Λ is the cosmological constant, I GHY is the Gibbons-Hawking-York surface term and K is the extrinsic curvature defined on the boundary surface with the determinant of the induced metric given by h . Quite remarkably, even before Gibbons and Hawking or York, Einstein had written down a first order lagrangian for gravity given by g ab (Γ n ma Γ m nb -Γ m mn Γ n ab ) which differs from I tot by a surface term [3]. The surface terms in general relativity are crucial not only to produce a well defined variational principle but also to produce correct black hole thermodynamics. These terms are also important to compute the correct Noether charges arising from diffeomorphism invariance [4]. Furthermore, evaluating these terms on a black hole horizon, one reproduces black hole entropy [5]. In this essay, we will argue that the Gibbons-Hawking-York surface term gives the entanglement entropy in gauge/gravity duality. Entanglement entropy is a useful non-local probe of how much the degrees of freedom in a region of spacetime are entangled with the rest. The original motivation for considering entanglement entropy was the hope that such considerations would shed light on the microscopic origin of black hole entropy [6, 7]. However, entanglement entropy is a useful tool in other areas of physics also, such as condensed matter systems. Recently, it has been used to shed light on time dependent physics as well where direct computational techniques are not available. It is known that entanglement entropy in conformal field theories in even d dimensions take the form Here l is a length scale parametrizing the size of the entangling region and glyph[epsilon1] is a short-distance cutoff. The leading l d -2 term gives the famous area law with a non-universal proportionality constant-when d = 2 the leading term is the log term. In even dimensions, the coefficient of the log term is a universal quantity typically related to a function of the conformal anomalies in the theory while in odd dimensions the log term is replaced by a constant which is considered to be a measure of the degrees of freedom [8]. In the context of quantum field theories, a direct computation of entanglement entropy is hard and has been possible only in very specific examples [9, 10]. The leading term in S EE is proportional to the area and gives the famous area-law. This was the main motivation for trying to relate black hole entropy with entanglement entropy of quantum fields. The gauge/gravity correspondence or the AdS/CFT correspondence gives a way to relate a quantum (typically conformal) field theory in d dimensions to a theory of gravity in anti de Sitter backgrounds in d +1 dimensions [11]. The prescription in gauge/gravity duality to compute entanglement entropy in the CFT is the following. The conformal field theory is supposed to live on the boundary of the AdS space. One considers a t = 0 slice of this boundary. Then a spatial region ∂ N of this boundary is considered and a minimal surface M extending into the bulk is found which ends on ∂ N . Ryu and Takayanagi (RT) proposed [12] that the area of this minimal surface is the entanglement entropy. To determine this minimal surface one considers an 'area functional' which is then minimized. This functional when evaluated on the black hole horizon would lead to the black hole entropy. At the onset we should emphasise that this is a prescription, which passes several checks, with no general proof. Unlike the Wald entropy formula which is valid for a general theory of gravity, no such analogue exists for the entanglement entropy. This is an unfortunate state of affairs which needs remedy in the near future since holographic methods are becoming a popular tool to gain intuition about physics at strong coupling [13]. For computational purposes one typically uses Einstein gravity in a weakly curved anti-de Sitter (AdS) background which corresponds to a strongly coupled conformal field theory (CFT). According to the RT prescription, in order to derive the holographic entanglement entropy for a d dimensional quantum field theory, one has to minimize the following entropy functional on a d -1 dimensional hypersurface (a bulk co-dimension two surface), where glyph[lscript] P is the Planck length and h is the induced metric on the hypersurface. The resulting surface is a minimal surface with vanishing extrinsice curvature. The gravity dual theory is simply Einstein gravity with a negative cosmological constant Λ = -d ( d -1) / (2 L 2 ) where L is the AdS radius. This procedure can be also used to compute entanglement entropy for the finite temperature field theory. It corresponds to the presence of a black hole in the bulk space time. We will consider the BTZ black hole as the result for the corresponding 1 + 1 dimensional CFT is well known. Let us consider the following metric for the (non-rotating) BTZ black hole, where, r = r H is the horizon. Next we put φ = f ( r ) into the metric and evaluate S . After finding the Euler-Lagrange equation for f ( r ) from S we can determine f ( r ) i.e., how the entangling surface extends into the bulk space time (see fig. 1): where r 0 = r H coth r H l/L 2 , l being the length of the entangling surface in the field theory. Then using this solution, assuming r H l/L 2 glyph[greatermuch] 1 which corresponds to the high temperature phase of the field theory, and evaluating S one gets the following well known result for the log part of the entanglement entropy: where c = 12 πL/glyph[lscript] P is the central charge of the two dimensional CFT, glyph[epsilon1] is the UV cut-off, l is the length of the entangling surface and β = 1 / ( LT ) = 2 πL/r H is identified as the periodicity in the time coordinate which is related to the inverse temperature T of the field theory. An independent calculation in 1+1d produces exactly this result [9] which is taken to be strong evidence for the validity of the minimal area prescription. Any proposal for the entanglement entropy should agree with this. Although this prescription passes certain non-trivial checks such as the strong sub-additivity condition, how does one reproduce the entangling surface in field theory? According to the AdS/CFT dictionary the AdS radius is to be identified with an RG scale. So naively one would expect that the radius of the entangling region should become a function of the RG scale. But what are the rules? Furthermore, what observable do we use in field theory to probe entanglement entropy? The RT prescription does not appear to provide direct answers to these questions. We can partially remedy this with the following observation. Consider field theory on the bulk co-dimension one slice φ = f ( r ). We will not set t = 0. Since the time direction is a direct product with the rest, the trace of the extrinsic curvature satisfies ( d ) K a a = ( d ) K t t + ( d -1) K i i . Thus ( d ) K t t -h t t ( d ) K a a = 0 leads to ( d -1) K i i = 0 which is the same as the minimal surface condition for the d -1 slice used in the RT calculation. But the combination ( d ) K t t -h t t ( d ) K a a is nothing but the tt component of the usual Brown-York (holographic [14]) stress tensor evaluated on the co-dimension one slice in d -dimensions! Here h tt is simply the tt component of the pullback metric on the co-dimension one slice-in the Brown-York tensor the indices are raised and lowered using this metric. Thus an alternate way to compute entanglement entropy in gauge/gravity duality presents itself [15]. We first compute the time-time component of the Brown-York stress tensor on the co-dimension one entangling surface given by φ = f ( r ). Set this to zero and determine f ( r ). The above argument guarantees that f ( r ) will work out to be the same as what follows from the RT prescription. Now it is intuitive, that since entanglement entropy is related to the common boundary between the degrees of freedom living in the two regions, one of which is traced over, it must be related to the surface terms in general relativity. Now recall that the RT area functional was such that evaluated on the black hole horizon, we got the black hole entropy. The Gibbons-Hawking-York term evaluated on the horizon of a black hole is also known to yield black hole entropy [5]. Let us work out the GibbonsHawking-York surface term in our case explicitly. Unlike the RT area functional, there is a time integral in this case. But we know that time has to be periodic, with period β = 1 /T , T being the temperature of the BTZ black hole. After some straightforward algebra, we get where in going to the second line we have assumed that the field theory is in the high temperature phase which makes r 0 ≈ r H . Thus the RT result is identical to what comes from the GibbonsHawking-York surface term. This agreement can be shown to hold even at zero temperatures if one makes time periodic with the periodicity related to the inverse Unruh temperature. Also this connection holds for any dimensions not just 1+1d. For the computations in higher dimensions readers are referred to [15, 16] . This method can also be applied for the stationary cases such as rotating BTZ. The explicitly time-dependent situations are left for future work.", "pages": [ 1, 2, 3, 4, 5 ] }, { "title": "Conclusions", "content": "We have shown that entanglement entropy for field theories having holographic duals are related to the surface terms arising in general relativity. This may point at a more systematic way of computing entanglement entropy by relating it to Noether charges in general relativity. Since the procedure for computing entanglement entropy was given in terms of the Brown-York stress tensor, this naturally suggests a possible way to find out about the entangling surface using field theory methods. Some evidence for this has been presented in [15].", "pages": [ 5 ] }, { "title": "Acknowledgments", "content": "We thank Gautam Mandal, Rob Myers and Tadashi Takayanagi for useful discussions and correspondence.", "pages": [ 5 ] } ]

[ { "title": "SPACETIMES WITH ALL SCALAR CURVATURE INVARIANTS IN TERMS OF A COSMOLOGICAL CONSTANT", "content": "Abstract. In this letter we provide an invariant characterization for all spacetimes with all polynomial scalar invariants constructed from the Riemann tensor and its covariant derivatives vanishing except those zeroth order curvature invariants expressed as polynomials in Λ, the cosmological constant. Using this invariant description we provide explicit forms for the metric.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Given a metric for a spacetime, one may construct scalar curvature invariants by contracting the curvature tensor with various copies of itself. Curvature invariants of order n > 1 are produced by contracting polynomials of the curvature tensor with its covariant derivatives up to order n . Extending the arguments made in [1] the entire collection of spacetimes with vanishing scalar invariants were identified [2] by using the GHP formalism and the boost-weight decomposition to define balanced scalars and balanced spinors which in turn produce tensors which vanish upon contraction. Spacetimes with this property are said to be V SI , these spacetimes are a subclass of the CSI spacetimes in which all polynomial scalar curvature invariants are constant [3] /negationslash It was noted that this approach could be extended to the case with non-zero cosmological constant, Λ = 0 [2]. This produces a subclass of the CSI spacetimes for which all scalar curvature invariants either vanish or are polynomials in terms of Λ, denoted as the CSI Λ spacetimes. These are of interest as they are a natural and simple step from the V SI and CSI spacetimes revealing the interconnection between the two. /negationslash As an example consider the plane-fronted gravitational waves, which constitute the entirety of the Petrov type N CSI Λ spacetimes. These were originally derived by Kundt [5] in 1961 with vanishing cosmological constant. At the time, this was a reasonable constraint as it produced the simplest pure radiation solutions admitting a twist-free and non-expanding null congruence. Although the plausibility of a nonvanishing cosmological constant had been considered in [6], it was not until the 1981 that the Petrov type N solutions with cosmological constant were determined [7, 8] and the plane-fronted gravitational waves in spacetimes with Λ = 0 were identified [9]. The resulting class of KN (Λ)[ α, β ] metrics were classified by the sign of the cosmological constant Λ = 0 and another invariant κ ' = 1 3 Λ α 2 +2 β ¯ β arising from /negationslash the metric, Excluding, the Λ = 0 cases, this produces four canonical classes, which were shown to have a canonical form by setting α and β to specific values using the appropriate coordinate transformations [10]: The physical interpretation of each of these subclasses is examined in [11] using the equations of geodesic deviation relative to an arbitrary timelike geodesic; these may be interpreted as exact transverse gravitational waves with two polarization modes propagating on either Minkowski, de Sitter or anti-de Sitter space. /negationslash The plane-fronted gravitational waves, with Λ = 0, belong to the V SI class of spacetimes [2], by adding a non-vanishing cosmological constant we have produced four distinct classes of CSI spacetimes. It is reasonable to ask how many new distinct CSI spacetimes are produced by adding Λ = 0 to each of the V SI spacetimes. In light of the results of [3] we may classify the above solutions by examining the Segre type and comparing to the metric forms in [3]. These spacetimes are of interest in quantum gravity. In the case of the vacuum plane wave spacetimes, it was shown that the vanishing of all scalar curvature invariants lead to all quantum corrections vanishing [13, 14]. Spacetimes for which all quantum corrections are a multiple of the metric are called universal [15]; such spacetimes are important as they are solutions to the quantum theory, despite our lack of knowledge of the particular theory. Recently it was proven that any universal spacetime in four dimensions must have constant scalar curvature invariants [16]. The CSI Λ spacetimes are a special subcase of the CSI universal spacetimes. /negationslash The goal of this letter will be to derive necessary and sufficient conditions on the Newman-Penrose scalar [4] for the class of CSI spacetimes with all non-zero scalar curvature invariants expressed in terms of the cosmological constant Λ = 0, as a parallel to the result in [2]. In the following section we state the theorem and break up the proof of the necessity and sufficiency of the conditions into two subsections. The third section employs the invariant characterization of the CSI Λ spacetimes along with the exhaustive list of CSI spacetimes to identify all of the metrics for the CSI Λ spacetimes. Finally in the last section we discuss the relevance of the CSI Λ spacetimes to the equivalence problem for Lorentzian manifolds.", "pages": [ 1, 2 ] }, { "title": "2. The CSI Λ Theorem", "content": "We wish to provide a simple set of conditions for spacetimes in which the Ricci Scalar is constant, and the only curvature invariants which are non-zero are the zeroth order invariants expressed as various polynomials of the cosmological constant Λ. Theorem 2.1. Given a spacetime, all invariants constructed from the traceless Ricci tensor, Weyl tensor and their covariant derivatives vanish, if and only if the following conditions are satisfied: These spacetimes belong to the CSI class of spacetimes and we will say they are CSI Λ spacetimes. We choose the tangent vector to the null congruence to be /lscript a and a spin basis so that o A ¯ o ˙ A ↔ /lscript a . The analytic conditions of theorem 2.1 (1) for this spin basis is expressed in terms of the vanishing of the spin coefficients [4]: and the second condition of theorem 2.1 may be expressed as Following the work done for V SI spacetimes, the definitions and results given in the Necessity and Sufficiency proof of [2] may be used to generalize the case where Λ = 0 is constant. /negationslash with the usual non-zero scalar products -/lscript a n a = m a ¯ m a = 1. The spinorial form of the Riemann tensor R abcd is where and Λ = R/ 24 with R the Ricci scalar. The Weyl spinor Ψ ABCD = Ψ ( ABCD ) is related to the Weyl tensor by Taking projections of this tensor onto the basis spinors o A , ι A give five complex scalar quantities Ψ i , i ∈ [0 , 4]. Similarly the Ricci Spinor Φ AB ˙ C ˙ D = Φ ( AB )( ˙ C ˙ D ) = ¯ Φ ˙ ˙ is connected to the traceless Ricci tensor S ab = R ab 1 Rg ab A BCD -4 We denote the projections of Φ AB ˙ C ˙ D onto o A , ι A by Φ 00 = ¯ Φ 00 , Φ 01 = ¯ Φ 10 , Φ 02 = ¯ Φ 20 , Φ 11 = ¯ Φ 11 , Φ 12 = ¯ Φ 21 and Φ 22 = ¯ Φ 22 The analytic expressions of theorem 2.1 (1) , (2) imply Following the convention used in [2] we will say a scalar η is a weighted quantity of type { p, q } if for every non-vanishing scalar field λ , a transformation of the form ↦ ↦ representing a boost in the /lscript a -n a plane and a spatial rotation in the m a -¯ m a plane transformations η in the following manner The boost weight, b, of a weighted quantity is defined by b = 1 2 ( p + q ). The frame derivatives are defined as and so the covariant derivative may be expressed in terms of the frame, The GHP formalism introduces new derivative operators ð , þ , ð ' and þ ' which are additive and obey the Leibniz rule. By including the spin-coefficient β in the expression for these operators, they act on scalars, spinors and tensors η of type { p, q } as follows: To show the sufficiency conditions we assume the analytic conditions of theorem 2.1 hold along with the requirement that o A , ι A are parallely propogated along /lscript a as well. Due to (2) we have the following relations on the spin coefficients The spin-coefficient equations, the Bianchi identities and commutator relations [4] are greatly simplified by imposing (4), (3), (5). The non-trivial relations that apply to proving the theorem are: To proceed we analyze the boost weights of the quantities involved in these relations. In particular we will use the idea of a balanced scalar. Definition 2.2. Given a weighted scalar η with boost-weight b , we shall say it is balanced if þ -b η = 0 for b < 0 and η = 0 for b ≥ 0. Many of the lemmas as given in [2] follow without change despite Λ's non-vanishing. The proof of lemma 4 requires some modification due to (17). For that reason, we will state each lemma leading to the main result without proof, unless there is some required change due to Λ = 0: /negationslash Lemma 2.3. If η is a balanced scalar then ¯ η is also balanced. Lemma 2.4. If η is a balanced scalar then, are all balanced as well. Proof. Let b be the boost-weight of a balanced scalar η . Using table 1 it is clear that the scalars listed in the first row have boost-weights b, b -1 , b -1 , b -2, respectively.To show these are balanced we must prove that the following must vanish: while for the second row we require that four more quantities vanish to match their boost-weight: As the equations (17) and (23) are the only that differ from the V SI case, we must only check to see if two conditions still hold and the remaining six conditions hold automatically. The first condition follows using the Leibniz rule and equations (17) and the fact that þ 2 ρ ' = 0. Since we may expand this as As η is a balanced scalar for which b < 0, these last two terms vanish. To prove the second condition, we use the commutator relation (23) and the constancy of Λ to get Using induction one may show that þ 1 -b þ ' η = þ ' þ 1 -b η = 0. /square Lemma 2.5. If η 1 , η 2 are balanced scalars both of type { p, q } then η 1 + η 2 is a balanced scalar of type { p, q } as well. Lemma 2.6. If η 1 and η 2 are balanced scalars then η 1 η 2 is also balanced. Definition 2.7. A balanced spinor is a weighted spinor of type { 0 , 0 } whose components are all balanced scalars. Lemma 2.8. If S 1 and S 2 are balanced spinors then S 1 S 2 is also a balanced spinor Lemma 2.9. A covariant derivative of an arbitrary order of a balanced sinpor S is again a balanced spinor Proof. Applying the covariant derivative to a balanced spinor S , From table 1 in [2] it follows that ∇ A ˙ A S is a weighted spinor of type { 0 , 0 } . By virtue of how þ , ð , þ ' and ð ' act on the basis vectors, the components may be shown to be balanced scalars using lemmas 2.3, 2.4 and 2.5. /square Lemma 2.10. A scalar constructed as a contraction of a balanced spinor is equal to zero. From table 1 in [2], and equations (19)- (22) it follows that the Weyl spinor and Ricci spinor and their complex conjugates are balanced spinors (lemma 2.3). Their product and covariant derivatives of arbitrary orders are balanced spinors as well (lemmas 2.8 and 2.9). At this point to prove the sufficiency of the conditions of theorem 2.1 we must state two more results: Lemma 2.11. The product of a balanced spinor and a weighted constant of type { 0 , 0 } is a balanced spinor. Lemma 2.12. A scalar constructed as a contraction from the product of a balanced spinor, /epsilon1 AB , /epsilon1 AB and their conjugates is equal to zero. With these observations, and equations (7), (8), (9) and (10) imply that any contraction of the product of N copies of the Riemann tensor with itself must vanish except for the contraction of the term built exclusively out of the product of N copies of Λ( /epsilon1 AC /epsilon1 BD + /epsilon1 AD /epsilon1 BD ¯ /epsilon1 ˙ A ˙ B ¯ /epsilon1 ˙ C ˙ D ) . Lemma 2.11 ensures all other terms are the products of balanced spinors, /epsilon1 's and ¯ /epsilon1 's; these terms must vanish when contracted by lemmas 2.10 and 2.12. To show that all non-zero curvature invariants appear at zeroth order, we note that the n th covariant derivative of the Riemann tensor is a balanced spinor for n > 0, as ∇ /epsilon1 AB = 0 and Λ is a constant. Thus any product of the Riemann tensor with its n th covariant derivative must vanish upon contraction by lemma 2.12, while any contraction of the product of the n th and m th covariant derivative of the Riemann tensor must vanish necessarily by lemma 2.10.", "pages": [ 3, 4, 5, 6, 7 ] }, { "title": "3. Local description of all CSI Λ spacetimes", "content": "To the author's knowledge, an explicit list of metrics has yet to be given for the CSI Λ spacetimes However, portions of the CSI Λ spacetimes have been studied under other pretenses. For example, the whole of the CSI Λ spacetimes with Λ = 0 are known; these are the V SI spacetimes [2]. For non-zero Λ we list known examples by Petrov Type. The plane-fronted gravitational waves constitute all of the Petrov type N CSI Λ spacetimes. Metrics for these spacetimes were found in [9] and these were further classified into canonical forms in [10]. As a generalization of the Kundt waves, all Petrov Type III CSI Λ spacetimes admitting pure radiation (Φ 12 = 0) were listed in [17]. Lastly in the case of Petrov Type O, all spacetimes with Segre type { (2 , 11) } [18] have been invariantly classified, implying that the CSI Λ spacetimes in this subcase are all known. We concentrate on the complete list of all CSI spacetimes given in [3], by imposing the necessary conditions on the curvature scalars [4] we can identify those CSI metrics which belong to the CSI Λ case. We may interpret the vanishing or non-vanishing of Φ 22 , Φ 02 and Φ 20 in terms of the Segre type, [12], to determine the metric forms permitted for the CSI Λ spacetimes in [3]: { (1 , 111) } , { (2 , 11) } and { (3 , 1) } . Employing Kundt coordinates, any CSI spacetime may be expressed as ds 2 = 2 du [ dv + H ( v, u, x k ) du + W i ( v, u, x k ) dx i ] + g ij ( x k ) dx i dx j where dS 2 H = g ij ( x k ) dx i dx j is the locally homogeneous metric of the transverse space and the metric functions H and W i are functions of the form where σ is a constant. As the transverse space must be a locally homogeneous two-dimensional space, up to scaling, there are (locally) only the sphere S 2 , flat space and the Hyperbolic plane H 2 . Exploiting this fact we list all of the CSI Λ spacetimes by the transverse metric and the one-form W (1) = W (1) i dx i . The constant σ will be specified in each case. 3.1. The Sphere S 2 . For those metrics with Segre type { (1 , 111) } , { (2 , 11) } and { (3 , 1) } , one must have σ > 0 and the transverse metric expressed as where Depending on the form of H (1) , H (0) and W (0) these are of Petrov type III,N or O. 3.2. The Euclidean plane E 2 . For those metrics with Segre type { (1 , 111) } , { (2 , 11) } and { (3 , 1) } , the transverse metric will be where the σ and the one-form are now This is the VSI case and so these are of Petrov type III, N or O. 3.3. The Hyperbolic plane H 2 . Segre type for these metrics are { (1 , 111) } , { (2 , 11) } and { (3 , 1) } . We require that σ < 0 and set σ = -q 2 , depending on the case we will use different coordinates for the Hyperbolic plane. For all of these cases the Petrov type is III, N or O.", "pages": [ 7, 8 ] }, { "title": "4. Discussion", "content": "In this paper the results of [2] were extended to produce an invariant characterization of the class of spacetimes whose non-zero scalar curvature invariants are expressed as polynomials of Λ ( CSI Λ spacetimes) by determining necessary and sufficient conditions on the Newman-Penrose curvature scalars [4]. Then by employing the exhaustive list of metrics for the CSI spacetimes [3] all of the CSI Λ spacetimes are found by comparing Segre Type, and the sign of the cosmological constant Λ. These invariants determine the metrics whose range of Petrov types: III, N, and O which complete the necessary and sufficient conditions for these spacetimes to be CSI Λ . In this sense all of the CSI Λ spacetimes have been given a local description in terms of a metric, however the conditions on the metrics for the subcases arising from the Petrov classification are not known. Equivalently, the interconnection between the Cartan invariants and the specialization in Petrov type III → N → O is unknown. In order to answer such a question one must apply the Karlhede algorithm to the entirety of the CSI Λ metrics. Such a task which would be an effort to implement but entirely plausible; recently all vacuum Petrov type N V SI spacetimes have been invariantly classified using the Cartan invariants arising from the application of the Karlhede algorithm to the vacuum PP-waves [22], and the vacuum Kundt waves [23]. Such a classification would give insight into the application of the Karlhede algorithm to CSI spacetimes, which is an important question related to the equivalence problem for these spacetimes. Furthermore the collection of CSI metrics have higher-dimensional analogues [19, 20, 21]. By comparing the four dimensional subcases with their higher dimensional counterparts it is hoped an analogue of the Karlhede algorithm could be implemented for all CSI Λ spacetimes.", "pages": [ 9 ] } ]

[ { "title": "ABSTRACT", "content": "1", "pages": [ 1 ] }, { "title": "Signature change by GUP", "content": "T. Ghaneh 1 ∗ , F. Darabi 2 † , and H. Motavalli 1 ‡ Department of Theoretical Physics and Astrophysics, University of Tabriz, 51666-16471, Tabriz, Iran. 2 Department of Physics, Azarbaijan Shahid Madani University, 53714-161, Tabriz, Iran. September 5, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "We revisit the issue of continuous signature transition from Euclidean to Lorentzian metrics in a cosmological model described by FRW metric minimally coupled with a self interacting massive scalar field. Then, using a noncommutative phase space of dynamical variables deformed by Generalized Uncertainty Principle (GUP) we show that the signature transition occurs even for a model described by FRW metric minimally coupled with a free massless scalar field accompanied by a cosmological constant. This indicates that the continuous signature transition might have been easily occurred at early universe just by a free massless scalar field, a cosmological constant and a noncommutative phase space deformed by GUP, without resorting to a massive scalar field having an ad hoc complicate potential. We also study the quantum cosmology of the model and obtain a solution of Wheeler-DeWitt equation which shows a good correspondence with the classical path. PACS Nos: 98.80.Qc; 03.65.Fd; 03.65.-w; 03.65.Ge; 11.30.Pb Keywords: GUP, Noncommutative, Signature Change", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The idea of noncommuting coordinates firstly was proposed by Wigner [1] and separately by Snyder [2]. This idea has been followed by Connes[3] and Woronowicz [4] as noncommutative (NC) geometry, leading to a new formulation of quantum gravity through NC differential calculus [5]. The link between NC geometry and string theory has also become evident by Seiberg and Witten [6], which resulted in NC field theories via the NC algebra based on the Moyal product [7]. Riemannian geometry of noncommutative surfaces has extensively been studied by Chaichian et al where they have developed a Riemannian geometry of noncommutative surfaces as a first step towards the construction of a consistent noncommutative gravitational theory [39], which is relevant to the present paper. Possible effect of spacetime noncommutativity on primordial gravitational waves in inflationary cosmology has also been studied [40]. Moreover, the fact that spacetime noncommutativity could suppress quantum fluctuations of matter fields, and dramatically constrain the random walking regime of the inflaton field at high energy scale is shown in [41]. In recent years, the existence of a minimal observable length has been predicted by different aspects in merging gravity with quantum theory of fields [8, 9, 10, 11, 12, 13, 14, 15, 16]. First, it was derived from string theory [10, 15, 17]. In the spirit of perturbative string theory, this comes from the fact that strings can not probe distances smaller than the string size. This is the natural cut-off length at which the quantum effects of gravitation become considerable in comparison with the electroweak and strong interactions and the transparent smooth view of the very notion of the space-time becomes opaque. When the energy of a string reaches the Planck mass, the excitations of string may cause a nonzero extension [15]. But creative calculations [18] show that this prediction is more reliable in quantum gravity and is not necessarily related to high energy or short distance behavior of the strings [12, 19] (examples of some other techniques can be found in [20, 21, 22, 23, 24]). There are other approaches to quantum gravity like the recently proposed Doubly Special Relativity (DSR) theories which suggest the presence of maximum observable momenta [25, 26, 27, 28], connecting to minimum positions. Other branches of high energy physics such as the very early universe, or strong gravitational fields in black hole physics are also concerned about the minimal length [18]. In fact, the usual Heisenberg Uncertainty Principle (HUP) fails for energies near the Planck scale, when the Schwarzschild radius is comparable to the Compton wavelength and both are close to the Planck length. This problem is resolved by revising the characteristic scale through the modification of HUP to what is known as the Generalized Uncertainty Principle (GUP) [29, 30]. Among all complicated footprints of GUP, the most elegant description follows from the simple deductions of Newtonian and quantum gravity [31], by considering a quantum particle such as electron, to be observed by photon in a thought instrument like the Heisenberg microscope. This elegancy explains why all of the arguments such as gedanken string collisions [10, 19], the thought experiment of black holes [18, 32], de Sitter space [2], the symmetry of massless particle [33] and wave packets [34], agree that GUP holds at all scales as [10, 13, 18] Motivated by the above arguments, in this paper we try to study the influences of GUP on a Friedmann-Robertson-Walker (FRW) model of Hartle-Hawking universe. The application of Einstein's field equations to the system of universe always faces with the problem of initial conditions. The Big Bang singularity is such a well-known problem in the standard model of cosmology. However, one can remove this problem by presenting a physical realization for the philosophical concept of a universe with no beginning. This presentation was firstly made by Hartle and Hawking [35], where they showed that in the quantum interpretation of the very early universe, it is not possible to express quantum amplitudes by 4-manifolds with globally Lorentzian geometries, instead they should be Euclidean compact manifolds with where p 2 = ∑ D j =1 p 2 j , D is dimension of space, β ∼ l 2 pl / 2 /planckover2pi1 2 , l pl is Planck Length and β ' is a constant. boundaries just located at a signature-changing hypersurface understood as the beginning of our Lorentzian universe. This is well known as the no boundary proposal . In this direction of thinking about quantum interpretation of the early universe, many works have also been accomplished on different cosmological models to study whether it is possible to realize a classical signature change [36, 44, 45, 46, 47] or not. Some of them have also considered the quantization of their models [44, 45, 46, 48, 49, 50]. In a recent work [51], the special attention has been paid for the case where the phase space coordinates are noncommutaive via the Moyal product approach. In the present work, we aim to study the effects of noncommutativity through the GUP approach in the phase space of a cosmological model which exhibits the signature change at the classical and quantum levels in the commutative case. We start with a FRW type metric and use a scalar field as the matter source of Einstein's field equations. Then, we apply the noncommutativity to the minisuperspace of corresponding effective action by the use of GUP approach in deforming the Poisson bracket. The conditions for which the classical signature change is possible are then investigated. Also, we study the quantum cosmology of this noncommutative signature changing model and find the perturbative solutions of the corresponding Wheeler-DeWitt equation. Finally, we investigate the interesting issue of classical-quantum correspondence in this model.", "pages": [ 1, 2, 3 ] }, { "title": "2 Classical Signature Dynamics", "content": "We consider a model of universe with the metric [36] where R ( /pi1 ) is the scale factor, k = -1 , 0 , 1 determines the spatial curvature. The sign of /pi1 is responsible for the geometry to be Lorentzian or Euclidian and the hypersurface of signature change is identified by /pi1 = 0. The cosmic time t is related to /pi1 via t = 2 3 /pi1 3 / 2 when /pi1 is definitely positive. One common way to treat the signature change problem is to obtain the exact solutions in Lorentzian region ( /pi1 > 0) and extrapolate them in Euclidian region continuously. In Lorentzian region, the line element (2) takes the form where k = 0 is set in agreement with the current observations. We also assume an scalar field with interacting potential U ( φ ) as the matter source. The corresponding action leads to the following point like Lagrangian where the units are adopted so that κ ≡ 1 and the York-Gibbons-Hawking boundary term S Y GH is canceled by the surface terms 1 . A change of dynamical variables defined by (0 ≤ R < ∞ , -∞ < φ < + ∞ ) casts the Lagrangian into a more convenient form where α 2 = 3 8 , and a coefficient ' -2 α 2 ' is ignored by using the zero energy condition 2 . Now, we choose the potential U ( φ ) [36] in which a 1 , a 2 and b are constant parameters. Using (6) and (7), the potential is expressed in terms of φ where the physical parameters are defined as the cosmological constant and the mass of scalar field, respectively. The Hamiltonian of system becomes where p 1 , p 2 are the momenta conjugate to x 1 , x 2 , respectively. The dynamical equations ˙ x i = { x i , H} , ( i = 1 , 2) are then written as [36] where In the normal mode basis V = S -1 ξ = ( q 1 q 2 ) for diagonalization of M as S -1 MS = D = diag ( m + , m -) we find and the solutions under initial conditions ˙ V (0) = 0 are found as where A 1 , A 2 ∈ R . These solutions remain real when the phase of ( √ m + t ) changes by π/ 2, so they are good candidates for real signature changing geometries. Note that the constants A 1 and A 2 are correlated by the zero energy condition [36] where I = S T JMS and J = ( 1 0 0 - 1 ) . The equation (18) is quadratic for the ratio χ = A 1 / A 2 and its roots χ ± are determined by the parameters of λ, m 2 , b . By choosing A 2 = 1, the solutions fall into two following classes where and At last, the original variables R and φ are recovered from x 1 and x 2 via (6) and (7) as We conclude that: i) for both eigenvalues of M being positive, no signature transition occurs, ii) for the product of the eigenvalues less than zero, the constraint (18) is not satisfied with a real solution for the amplitude χ , and iii) for both eigenvalues being negative, x 1 ( β ) , x 2 ( β ) exhibit bounded oscillations in the region β > 0 and are unbounded for β < 0 (see Fig.1 [36]). Such behaviour is translated into the solutions for R and φ (see Fig.2 [36]). Therefore, it is possible to choose parameters so that the manifold becomes Euclidean for a finite range of β < 0 and undergoes a transition at β = 0 to become Lorentzian for a further finite range of β > 0 [36].", "pages": [ 3, 4, 5 ] }, { "title": "3 Noncommutativity via deformation", "content": "The study of noncommutativity between phase space variables is based on the replacing of usual product between the variables with the star-product; and in flat Euclidian spaces all the star-products are c-equivalent to the so called Moyal product [37]. Let us assume f ( x 1 , .., x n ; p 1 , .., p n ) , g ( x 1 , .., x n ; p 1 , .., p n ) to be two arbitrary functions. Then, the Moyal product is defined as such that and θ ij , ¯ θ ij are antisymmetric N × N matrices. Then, the deformed Poisson brackets read as Therefore, the coordinates of a phase space equipped with Moyal product satisfy Considering the following transformations [38] one finds that ( x ' i , p ' j ) fulfill the same commutation relations as (27) with respect to the usual Poisson brackets provided that ( x i , p j ) follows the usual commutation relations This approach is so called noncommutativity via deformation .", "pages": [ 5, 6 ] }, { "title": "4 Phase Space Deformation via GUP", "content": "In this section, we aim to study the effects of noncommutativity in the phase space via deformation by GUP approach. The equation (1) represents a modification of Heisenberg algebra as [ x ' i , p ' j ] = i /planckover2pi1 ( δ ij (1 + βp ' 2 ) + β ' p ' i p ' j ) , (31) where β , β ' are taken to be small up to the first order. Then the ansatz of classical-quantum correspondence, [ , ] → i /planckover2pi1 { , } , introduces the deformed poisson bracket of position coordinates and momenta [52] where primes on x, p denotes the modified coordinates. Assuming { p ' i , p ' j } = 0, the Jacobi identity almost uniquely specifies that [29, 53] Remembering the usual (non-modified) algebra { x i , p j } = δ ij , the relations (32)-(33) can be realized by considering the following transformations γ being an arbitrary constant given by γ = β + β ' ( D +1 2 ) [54] .", "pages": [ 6, 7 ] }, { "title": "5 Signature Change in Deformed Phase Space", "content": "Let us follow the 2-dimensional model explained initially in section 2. The Hamiltonian of the deformed system is It can be described in terms of commutative coordinates by the use the transformations (34) as where x i , p j reads the common Poisson algebra, and It is usual to set β ' = 2 β [55, 56, 57, 58] to make the shape of Z ( p ) more refined as Z ( P ), P := p 1 + p 2 . As is shown for a non-deformed system [36] or the system deformed by moyal product approach [51], the existence of a non-zero cross-term parameter b in U ( φ ) is the only way to break the symmetry of the system under φ →-φ and make the change of signature happen. However, we show that in contrary to the moyal product approach, in GUP approach b is not the only parameter responsible for signature change. To this end, we explicitly set b = 0. On the other hand, to show that for a continuous signature transition we need not choose a massive scalar field we take a massless scalar field (i.e a 2 = -a 1 ). By this set up we are going to assert that a very specific scalar field potential of the form (10) is not needed for a continuous signature transition. This makes continuous signature transition much easier than the model introduced in [36] because the justification of the complicate potential (10) at early universe is not a simple task. In the present model, however, we just need the elements i) a free massless scalar field, ii) a cosmological constant, and iii) GUP which are supposed to be trivial in the conditions at early universe. The classical equations of motion ˙ x i = { x i , H ' } , i = 1 , 2, are then obtained as Also, the dynamical equations of momenta, ˙ p i = { p i , H ' } , yield where a dot denotes differentiation with respect to t . To decouple these equations, we merge (38) with (39) first, and then compute the summation and subtraction of the results. This procedure leads to the following equations Eq.(40) is a differential equation with linear symmetry and it can be solved by order reduction via it's symmetry generators. Then the particular solution is obtained as or equivalently where Π( ν ; ϑ, κ ) is the incomplete elliptic integral of the third kind, C ± = C 1 C 2 ± a 1 , and C 1 , C 2 , C 3 are constants to be detected by initial conditions. One can check that any such particular solution still remains a solution of (40) if it is multiplied by a minus sign, or (and) if any of the transformations t →-t or (and) t → it is applied . A simplified result is obtained at the special case where C 1 C 2 = a 1 Physical values of λ and β ought to satisfy ¯ R (0) = 0 and must also yield a positive ¯ R ( /pi1 ) at the right neighborhood of /pi1 = 0, the area which can be called as Lorentzian region . The least requirement we expect is that the imaginary part of the physical functions ¯ R , ¯ φ and ¯ R vanish at that area. Fig.1 and Fig.2 show the signature transition by real solutions from Euclidean to Lorentzian regions for a possible set of values 3 .", "pages": [ 7, 8, 9 ] }, { "title": "6 Quantum Cosmology", "content": "The high energy and small scale of very early universe provides the possibility of having noncommutativity and GUP in the minisuperspace configurations of the model discussed here. But, in such small scale the quantum behavior is inevitable. Thus, it is necessary to study the quantized model and check if the quantization results are consistent with the classical solutions of dynamical equations. Introducing the momentum quantum operators ˆ p 1 = -i∂/∂x 1 , ˆ p 2 = -i∂/∂x 2 and applying the Weyl symmetrization rule to (36) to construct the Hamiltonian operator, leads to the Wheeler-DeWitt(WD) equation of the form ˆ H ' Ψ( x 1 , x 2 ) = 0. Defining the real and imaginary parts of the wave function as Ψ = ψ r + iψ i splits WD equation in to two parts where, In order to obtain a quantum criterion to test the classical results of previous section, we consider the special case ψ r = A ψ i ≡ F ( x 1 , x 2 ), A being a constant. This converts (46) into H 1 F = 0 and H 2 F = 0, the second of which is automatically satisfied if F = F ( x 1 x 2 ), and the first one becomes where X := x 1 x 2 and x 1 is regarded as a parameter. The solution of (48) is an expression of Generalized Hypergeometric Functions as where A 1 , A 2 are two constants and As figure 3 shows, the density plot of the quantum solution (50) is in good agreement with the classical solution obtained in the previous section.", "pages": [ 9, 10, 11 ] }, { "title": "7 Conclusions", "content": "Using a noncommutative phase space of dynamical variables deformed by Generalized Uncertainty Principle we have shown that continuous signature transition from Euclidean to Lorentzian may occurs for a model described by FRW metric minimally coupled with a free massless scalar field φ accompanied by a cosmological constant. The transformations of GUP in deforming the phase space breaks the symmetry of Hamiltonian under φ → -φ causing a possible continuous change of signature. This indicates that for a signature transition to happen, instead of a massive scalar field having an ad hoc and complicate potential, we just need a free massless scalar field, a cosmological constant and a noncommutative phase space deformed by GUP. These elements are supposed to be trivial in the extreme conditions at early universe. In commutative [36] as well as moyal transformed noncommutative Hamiltonian [51], we need a coupling b in the scalar field potential to trigger the signature transition. However, using GUP in the absence of such potential and coupling, we have the expression β ' p 1 p 2 in Hamiltonian (36) coming directly from the special structure of GUP deformations (34) which means that the GUP noncommutativity can cause a change of signature by itself. In other words, GUP accompanied by noncommutativity may establish a general framework for a continuous change of signature. Moreover, in principle, the signature transition is possible for both negative and positive cosmological constants. This significantly differs from the moyal approach [51] in which only the negative values of cosmological constant are acceptable. We have also studied the quantum cosmology of this model and obtained a solution of Wheeler-DeWitt equation showing a good correspondence with the classical path.", "pages": [ 11, 12 ] } ]

[ { "title": "Dynamics of tachyon field with an inverse square potential in loop quantum cosmology", "content": "Fei Huang and Jian-Yang Zhu ∗ Department of Physics, Beijing Normal University, Beijing 100875, China", "pages": [ 1 ] }, { "title": "Kui Xiao †", "content": "Department of Basic Teaching, Hunan Institute of Technology, Hengyang 421002, China (Dated: October 29, 2019) The dynamical behavior of tachyon field with an inverse potential is investigated in loop quantum cosmology. It reveals that the late time behavior of tachyon field with this potential leads to a power-law expansion. In addition, an additional barotropic perfect fluid with the adiabatic index 0 < γ < 2 is added, and the dynamical system is shown to be an autonomous one. The stability of this autonomous system is discussed using phase plane analysis. There exist up to five fixed points with only two of them possibly stable. The two stable node (attractor) solutions are specified, and their cosmological indications are discussed. For the tachyon dominated solution, the further discussion is stretched to the possibility of considering tachyon field as a combination of two parts which respectively behave like dark matter and dark energy. PACS numbers: 98.80.Cq", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "The tachyon field with various potential has been studied a lot in cosmology. Many models have been built by treating tachyon field as inflaton field [1], candidate of dark energy [2, 3], or a dual role of the two [4]. But Ref.[4] also argued that, for the class of potentials which V ( φ ) → 0 as φ → ∞ , radiation domination will never commence since the tachyon field energy density ρ φ can at best scale as a -3 . The radiation energy density would always redshift faster than the tachyon field. The tachyon field with an inverse square potential is shown to be able to produce a power-law expansion [5]. Coupled with a barotropic perfect fluid, the dynamical behavior of this potential has been studied in classical cosmology[6]. However, the tracking solution, in which the energy density of the field and the barotropic fluid scales as a same power of a , may not be viable because its constraints on the adiabatic index γ . Our work of tachyon field cosmology is performed under the framework of loop quantum cosmology. LQC is a canonical quantization of homogeneous spacetime using the techniques developed in loop quantum gravity (LQG). The loop quantum effects can be very well described by the effective theory of LQC. A modified Friedmann equation is proposed and two corrections are often considered: the inverse volume correction and the holonomy correction. However, the holonomy correction dominates over the inverse volume correction for a universe with a large scale factor, and thus the latter can be neglected without harm. Therefore, we only consider the holonomy correction in this paper. Currently, tachyon matter has not been thoroughly investigated in loop quantum cosmology(LQC). A. A. Sen [7] generalized the description of tachyon matter in standard cosmology to LQC under inverse volume correction. Xiong and Zhu [1] investigated the inflation scenario of a pure tachyon field with an exponential potential under the holonomy correction. Xiao and Zhu [8] performed a phenomenological analysis of tachyon warm inflation, in which the tachyon field with an exponential potential is coupled with radiation and interaction between the two matters were considered. However, the dynamics of tachyon field in LQC has not been investigated, yet. In this paper, we discuss the dynamics of tachyon field coupled with a barotropic perfect fluid in LQC. We focus on the inverse square potential, and try to explore the possibility of a viable dark energy model. The organization of this paper is as follows. In Sec.II, we try to derive the expansion law for a pure tachyonic matter in LQC. In Sec.III, we couple the field with a barotropic perfect fluid and analyze the dynamics of the autonomous system. The cosmological implications of the phase plane analysis are presented in Secs.IV A and IVB. The conclusions are made in Sec. V.", "pages": [ 1 ] }, { "title": "II. TACHYON MATTER IN LOOP QUANTUM COSMOLOGY", "content": "Based on the holonomy correction in loop quantum cosmology, the modified Friedmann equation of a flat( k = 0) Friedmann-Robertson-Walker (FRW) cosmological model is where H is the Hubble parameter, ρ and ρ c denote the matter density and critical density, respectively. We also set 8 πG = 1 for convenience. The energy conservation equation is the same as the classical one, where p is the pressure. Differentiate Friedmann eqaution with respect to time, we have where 'dot' denotes the derivative with respect to the cosmological time t . Therefore the conditions for superinflation ( ˙ H > 0) are where ω = p/ρ is the equation of state. It's easy to see the existence of superinflation in LQC is purely an effect of quantum geometry, because it originates from the time derivative of the modification term (1 -ρ/ρ c ). The Raychaudhuri equation then becomes which indicates the conditions for a > 0:  The regions for ˙ H > 0 and a > 0 are portrayed explicitly in Fig.1. Obviously, as the matter density decreases, the correction term becomes less and less important, and the Friedmann equation as well as the conditions for a > 0 and ˙ H > 0 are consistent with classical cosmology in an asymptotical way. Now we consider the model with only tachyon field. Note that the tachyon field referred in this paper is just a scalar field with an nonquadratic kinetic term. We don't claim any identification with the tachyon in string theory. According to Sen [9, 10], the energy density and pressure of the tachyon field in a flat FRW cosmology can be expressed as where φ is the tachyon field, V ( φ ) denotes its potential, and the equation of state is We see ω φ ranges smoothly from -1 to 0. Using the energy conservation Eq. (2), the evolution equation of tachyon field can be written explicitly as where V ' = dV/dφ . Here, we are interested in the inverse square potential with β > 0. According to [5] this potential is able to produce a power law expansion in classical cosmology. In classical cosmology, one can use the Hubble parameter instead of the field φ as a fundamental quantity by employing the Hamilton-Jacobi formulation. By using Eqs. (1) and (3) and dropping the ρ/ρ c terms, one can obtain which indicates ˙ H ≤ 0, and therefore superinflation will not occur in classical cosmology. Divide both sides by ˙ φ , we have and thus For the inverse square potential, an exact solution H ∼ φ -1 is found, and after some algebra one can arrive at where The solution is inflationary when n > 1, or β > √ 4 / 3. Although one can have arbitrarily fast expansion with an arbitrarily big n or β , the condition for an accelerated expansion already calls for an energy scale close to a Plank mass [2, 3]. Refs.[4, 11] also discussed the problem of tachyon model in reheating. Therefore this tachyon model is more suitable as a dark energy model rather than an inflaton model. However, it is difficult to find an exact solution like this in LQC due to the existence of the quantum modification term. Combining equation (1) and (3), we have Obviously, ˙ H > 0 when (1 -2 ρ φ /ρ c ) < 0, which means superinflation will naturally happen and it's purely an effect of quantum geometry as we said before. Divide both sides by ˙ φ , we obtain and therefore Fortunately, we are still able to analyze its asymptotical behavior. When ρ φ /lessmuch ρ c , we can neglect the derivatives of the correction terms since these terms, since (1 -ρ φ /ρ c ) and (1 -2 ρ φ /ρ c ) will not change significantly. Then, an approximated power-law expansion can be obtained, with So we have Obviously, m → n as ρ φ /ρ c → 0. The quantum geometry results in a different evolution of tachyon field and the scale factor,but the evolution will converge to the classical one at late time when the quantum effects vanishes.", "pages": [ 1, 2, 3 ] }, { "title": "III. WITH BAROTROPIC FLUID", "content": "In order to obtain a viable dark energy model, we add a barotropic perfect fluid in our model, for which the equation of state is p γ = ( γ -1) ρ γ , where γ is a constant.Therefore, the equation of state for the whole is where and Ω φ and Ω γ are the fractional densities defined as If we simply assume there is no interaction between the tachyon field and barotropic fluid, then their evolution equations are and Eq.(3) now becomes Combining Eqs. (22)-(24) and the Friedmann equation, we can construct a 4-dementional autonomous system. To see this, we usually introduce 4 convenient variables [6, 12]: Moreover, we use N = ln a 3 instead of the cosmological time t as an independent variable, therefore for any timedependent function f , we have Here, one should be careful that this treatment may not be practicable when ˙ a = 0, however, at that point, one can always switch back to t without causing any problem. With the help of new variables, the Eqs. (1) and (22)(24) can now be expressed respectively as follows: Using the above equations, differentiate x , y , z , and λ with respect to N , the nonlinear dynamical system is shown in an obvious autonomous form: where can be fixed if the potential is specified. Note that, the autonomous system in LQC has one more dimension than the one in classical cosmology, because of the modification term ρ/ρ c [12]. For the inverse square potential V ( φ ) = βφ -2 investigated in this paper, Γ = 3 / 2. Moreover, we find λ = -αy for this potential, where which means the system can be reduced into a 3dimensional one: The equilibrium points or fixed points ( x e , y e , z e , λ e ) are solutions acquired by setting Before we analyze the autonomous system described by Eq.(31), we should state several physical restrictions. First, the density should be limited and real-valued, therefore -1 < x < 1. The potential is required to be positive, so we have y > 0. In the analysis, we add the three end points of x and y for convenience, i.e. we set x ∈ [ -1 , 1], y ∈ [0 , + ∞ ]. Moreover, the total density should be nonnegative and lower than the critical density, i.e. 0 ≤ z ≤ 1. Moreover, we apply a normal restriction for γ , that is 0 < γ < 2. Therefore, ω will range from -1 to 1 due to the choice of γ . Since neither of the two parts should have a negative density, we have Ω φ , Ω γ ∈ [0 , 1]. This restriction leads to (1 -z ) 2 y 4 + x 2 ≤ 1, and the equal sign is for Ω φ = 1. This indicates the physically possible phase plane projection on the x -y plane shrinks as z decreases from 1 to 0. In fact, as z → 0 the phase plane projection becomes y 4 + x 2 ≤ 1 asymptotically. According to Lyapunov's theory of stability, the stability of a fixed point can be determined by the property of the linearized system about it. The linearization is done by expanding Eq.(31) about the fixed points and keeping only the linear parts. After that, one can obtain a matrix , at each fixed point. If the all the eigenvalues of (33) for a fixed point have negative real parts, the fixed point is (locally) exponentially stable. However, if there exist one or more eigenvalues which have positive real parts, then the fixed point is unstable. In both stable and unstable cases, the fixed point is a node if the eigenvalues are all real, otherwise it will be a spiral point. Furthermore, the fixed point is called saddle point if the eigenvalues have both positive and negative real parts. We found up to 5 fixed points for Eq.(31) , their properties are listed in Table I, where The physical requirements mentioned before also set restrictions on the existence and stability of these fixed points.By considering γ/α 2 / √ 1 -γ ≤ 1, we found where f ( γ ) = 4 α 2 -20 γα 2 +17 γ 2 α 2 +16 γ 2 √ 1 -γ , which indicates the two eigenvalues of P 3 with seemingly complicated square root part are definitely real and nonpositive. Note that, when γ > α 2 y 2 1 , P 3 doesn't exist. There are only four fixed points in the system, and P 4 , as a stable node, is the only attractor in the system. When γ = α 2 y 2 1 , a bifurcation occurs as the fifth fixed point, P 3 , emerges and coincides with P 4 . As γ decreases from α 2 y 2 1 to 0, P 4 turns into a unstable saddle and P 3 becomes a new attractor. The position of P 3 is located on a straight line on the x-y plane described by ( x, y ) = ( √ γ, √ γ/α ) and in the limit case γ = 0, it will coincide with P 1 .", "pages": [ 3, 4 ] }, { "title": "IV. LATE TIME EVOLUTION", "content": "From TABLE I we can see that all the fixed points locate at z = 0, where the energy density vanishes and the LQC modification term ρ/ρ c becomes unimportant. We have proved in last section that the the physically possible phase plane projection on x -y plane becomes y 4 + x 2 ≤ 1 asymptotically. Therefore for the late time evolution, it is sufficient for us to investigate the phase projection on the x -y plane at small z by noting that z monotonously decreases as a or N = ln a 3 increases, which we can easily read from Eq. (29). In the previous section, we have proved that there is at most one stable point in our model, then nearly all solutions will end there (see Figs.2 and 3). This attracting behavior allows us to investigate the properties of late time evolution near the attractor, regardless of the history before it. The finetuning problem can thus be waived since the same ending occurs for a wide range of initial conditions. Here we analyze two different solutions, the tachyon dominated solutions and tracker solutions.", "pages": [ 4, 5 ] }, { "title": "A. Tracker solutions", "content": "When γ < α 2 y 2 1 , P 4 becomes an unstable saddle while P 3 ( √ γ, √ γ/α, 0) comes out as the only attractor in our system. The fractional density is a constant at P 3 , which means both matter scale as the same power of a . Therefore, we have near the attractor, where φ 0 is an integration constant. For a same β or α , this solution produces a faster expansion than the one with tachyon as the only matter. It will lead to an eternal acceleration if γ < min { 2 / 3 , α 2 y 2 1 } , while if 2 / 3 < γ < α 2 y 2 1 , a deceleration phase will definitely occur at late time. Due to the choice of γ and α , Ω φ can range from 0 to 1. However, Refs. [2, 3] argues that this solution cannot be a viable one, since its existence requires γ < α 2 y 2 1 < 1.", "pages": [ 5 ] }, { "title": "B. Tachyon dominated solutions", "content": "In the case γ > α 2 y 2 1 , P 4 does not exist. As the solutions converge to the only attractor P 3 ( αy 1 , y 1 , 0) in late time evolution, tachyon will become dominant while both matters are decreasing. In fact, near the attractor we will have This result is consistent with the result we derived in the second section. It is easy to see that if α 2 y 2 1 < 2 / 3, the solutions depict an eternal acceleration. On the other hand, the universe will end up in deceleration if α 2 y 2 1 > 2 / 3. An interesting possibility is to see tachyon field as a combination of two parts [13, 14], which behave like a pressureless dust (dark matter, denoted by lower case DM ) and a cosmological constant(dark energy, denoted by lower case Λ), respectively: where and In this way, dark matter and dark energy originate from a same scalar field, and the dynamics of tachyon field becomes a description of their dynamics and interaction. The ratio between the two parts is The proportion of dark matter rises as ˙ φ 2 increases. If the barotropic fluid is radiation( γ = 4 / 3), then, by the virtue of the attractor solution, it is possible to have a trajectory that goes from the radiation dominated era to the matter dominated era, and then to the dark energy dominated regime. Radiation should dominate first, that is Ω γ ≈ 1. Then, to have the universe dominated by matters described by tachyon field after the era dominated by radiation, that is Ω φ ≈ 1, we need the trajectory to stay close to the boundary of the phase plane, i.e. (1 -z ) 2 y 4 + x 2 ≈ 1, or y 4 + x 2 ≈ 1 because z is very small. To have a sufficient long matter dominated era before dark energy take over, we just need the trajectory to stay close to the saddle point P 2 ± after radiation's domination. Therefore we will rule out the trajectories which are far away from the boundary of the phase plane in following discussion. The negative/positive branch, which starts near P 2 -/ P 2+ , will show a quite different evolution, because the attractor only lies in the right half of the phase plane and therefore the negative branch will have to get across to arrive at the attractor. The cosmological consequence will also be quite different. For the positive branch, the ratio of dark energy will increase as ˙ φ decreases. If the attractor is inflationary ( α 2 y 2 1 < 2 / 3), the allowed trajectories can go naturally into the dark energy dominated regime that leads to acceleration. While if the attractor is not inflationary( α 2 y 2 1 > 2 / 3), acceleration will not take place because ˙ φ decreases almost monotonously for the feasible trajectories which are close to the boundary. On the other hand, for the negative branch, the universe will definitely enter a dark energy dominated regime that leads to acceleration when ˙ φ 2 < 2 / 3, or -2 / 3 < ˙ φ < 2 / 3, since the trajectories will get across from the left half to the right half. Note that, for the negative branch, the universe is not expanding in the power-law way we described before when it first gets into the acceleration regime -2 / 3 < ˙ φ < 0. When ˙ φ = 0, dark energy dominates completely. After that, the ratio of matter increases again, and the final state of universe will(or will not) be inflationary if α 2 y 2 1 < 2 / 3(or α 2 y 2 1 > 2 / 3). So there is possibility that we are just currently living in a transitory accelerating period and the acceleration rate can change according to the dynamics of the tachyon field.", "pages": [ 5, 6 ] }, { "title": "V. CONCLUSION", "content": "Previous works on tachyon cosmology in spatially flat FRW universe showed a purely tachyonic matter with an inverse square potential V = βφ -2 leads to a powerlaw expansion [5]. In LQC scenario, although it is hard to find a exact solution, the expansion of universe is nearly a power-law one when ρ/ρ c is small. When the tachyon field is coupled with a barotropic perfect fluid with 0 < γ < 2, we are able to find two kinds of stable nodes which exist exclusively to each other and represent different cosmological situations. The tracker solution exists for γ < α 2 y 2 1 , while the tachyon dominated solution exists for γ > α 2 y 2 1 . Refs.[2, 3] argued that the tracker solution cannot be a viable one, since its existence requires γ < α 2 y 2 1 < 1. Therefore we focused on the tachyon dominated solution. We considered the tachyon field as a combination of two parts which respectively behave like dark matter and dark energy. Because of the existence of stable node(attractor), we found when γ = 4 / 3, it's possible to have a trajectory in which the universe evolves naturally from radiation dominated regime to matter dominated regime, and then into the current dark energy dominated regime through qualitative discussion. The negative and positive branches, which respectively start from close to ˙ φ < 0 and ˙ φ > 0, can be interpreted into different cosmological evolutions and it is possible that we are just living in a transitory accelerating period, while the final stage of universe will be identical for the same α or β . However, this perspective to see tachyon field as a combination of two parts revives the need of fine tuning. We also avoided the discussion of the period before radiation dominated era. Further work can be done to bridge the gap.", "pages": [ 6, 7 ] }, { "title": "Acknowledgments", "content": "This work was supported by the National Natural Science Foundation of China (Grant Nos. 11175019 and 11235003) and the Fundamental Research Funds for the Central Universities.", "pages": [ 7 ] } ]

[ { "title": "F. Nemes", "content": "ELTE Department of Atomic Physics 1117 Budapest P'azm'any P'eter street 1/A and", "pages": [ 1 ] }, { "title": "B. Mik'oczi", "content": "MTA Wigner FK, Research Institute for Particle and Nuclear Physics Budapest 114, P.O. Box 49, H-1525, Hungary November 2, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "The motion of binary star systems is re-examined in the presence of perturbations from the theory of general relativity. To handle the singularity of the Kepler problem, the equation of motion is regularized and linearized with quaternions. In this way first order perturbation results are derived using the quaternion based approach.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "In this paper gravitational effects as perturbations of the Kepler problem are examined with post-Newtonian approximation. Gravitational effects become strong when the components of the binary are close to each other, and the orbital separation is small. 1 The Kepler problem is singular when the separation is zero, therefore to study gravitational effects the desingularization - or regularization - of the equation of motion would be a substantial step. It is well known that Kustaanheimo (1964) solved the regularization of the three-dimensional Kepler problem with spinors[1], which was reformulated by Stiefel[2]. In their method - the KS method for short - the regularization was carried out in four dimensions, and it was proved that the three-dimensional Kepler problem can only be regularized using four-dimensional linear spaces. We follow another approach developed by J. Vrbik. In his work the mentioned four-dimensional space is the linear space of quaternions and the regularization is calculated with quaternion algebra. He applied his method with success to the Lunar problem[3], and several perturbative forces were studied in details[4, 5, 6]. In the present work we use his method to examine gravitational effects analytically with quaternions. The leading order correction of general relativity to classical mechanics is calculated first. The formula for the precession of the pericentre is derived based on the Vrbik's quaternion formulae. Then the gravitational radiation reaction is analyzed, where the famous Peters-Mathews formula is proved[7]. In this calculation we manage to remove the residual coordinate gauge freedom of the gravitational reaction from the quaternionic equation of motion. In addition using a one-dimensional model we demonstrate that the regularization can lead to different results depending on that the Sundman transformation is employed with the perturbed or unperturbed orbital separation. The regularization is defined with four-dimensional spaces, thus an additional geometrical constraint - a gauge - have to be applied to describe the three-dimensional spatial Kepler problem. In the KS method the so-called bilinear relation is defined, which is an excellent gauge for numerical calculations. Vrbik proposes another constraint to provide an analytic perturbative method, since - according to Vrbik - the bilinear relation is too restrictive to build an analytic perturbative method. This constraint is the major difference between the KS method and Vrbik's work. The Laplace vector is a constant of motion of the Kepler problem, which is a consequence of the hidden symmetry of the problem[8]. This symmetry becomes manifest in four dimensions, which shows that the Kepler problem has another interesting connection with the four-dimensional space. This connection has far reaching consequences[9, 10]. Quaternions were first applied to regularize the Kepler problem by Chelnokov who successfully regularized the Kepler problem to describe rotating coordinate systems[11]. Moreover he was able to apply his results to describe the optimal control problem of a spacecraft[12]. Later it was shown by Vivarelli (1983) in a general mathematical sense that the KS method can be transformed identically into quaternion algebra[13]. Quaternion algebra proved to be very useful to derive the central ideas of the KS method. Remarkably the bilinear relation is described as a fibration of the quaternion space. More recently Waldvogel showed that the spatial Kepler motion can be elegantly formulated with quaternions using a novel star conjugation operator[14]. The star conjugation is especially useful to handle the bilinear relation. The interesting connection with the Birkhoff transformation is also shown [15]. Quaternions turned to be useful in case of three and N-body applications[16]. It has to be emphasized that the mentioned quaternion approaches exclusively apply the 'bilinear relation' as a gauge to reduce the dimensions from four to three, while Vrbik apply his special gauge. This paper is organized as follows: a short outline of Vrbik's approach is provided in Section 2 and 3, where we describe the transformation of the Kepler problem into quaternion differential equation. Then the solution is given in terms of ordinary differential equations of orbital elements. The advantages of Vrbik's calculus compared with the KS method are highlighted. In Section 2.5 a one-dimensional example is given where we demonstrate that the result of the regularization depends on whether the Sundman transformation is applied with perturbed or unperturbed orbital separation. In Section 4 Vrbik's method is applied to two perturbations. First of all, the leading order correction of general relativity to classical mechanics is examined. The formula for the precession of the pericentre is derived. Then the gravitational radiation reaction is analyzed, where the famous Peters-Mathews formula is proved using the quaternion approach [7]. In this calculation we solved to cancel the residual coordinate gauge freedom of the gravitational radiation reaction in the quaternionic equation of motion. The conclusion and the outlook is given in Section 5 followed by the Appendix.", "pages": [ 1, 2, 3 ] }, { "title": "2.1 Quaternion algebra basics", "content": "The quaternion algebra has three imaginary units, generally called i , j and k , where i 2 = j 2 = k 2 = -1. Any of them anticommute while the real unit 1 commutes with each of them. The four units together form the algebra's generators. Thus any element of the algebra can be written as 2 Quaternion conjugation reverses the sign of the imaginary units The magnitude of a quaternion is defined as Any quaternion can be written in the form A + a ˆ a , where a is the magnitude and ˆ a is the unit direction of a . Since ˆ a 2 = -1 the exponential on any quaternion can be expressed with Euler's formula", "pages": [ 3 ] }, { "title": "2.1.1 Representation of spatial vectors and rotations", "content": "Spatial vectors are represented with pure quaternions, which has no real part where the z -axis is associated with the i unit. With this interpretation quaternion multiplication can be expressed as By substituting A = B = 0 into this expression (7), the anticommutative cross product can be expressed as Let us introduce a vector w . It is straightforward to show that a rotation around the vector w with magnitude | w | can be written as[4] where R = e w 2 . Note that R ¯ R = 1, hence x = R ˜ x ¯ R is the inverse rotation. Let us suppose that the rotation is parameter dependent R ( s ), and differentiate it with respect to s where definition (9) of ˜ x was applied. With the help of the identity ( ¯ RR ) ' = ¯ R ' R + ¯ RR ' = 0 and the cross product (8) this can be further written where Z = 2 ¯ RR ' . Consequently Z is the instantaneous angular velocity of ˜ x with respect to s . With an inverse rotation the angular velocity Z can be expressed in a special coordinate system - in the Kepler frame - where ˜ x is instantaneously at rest where the subscript indicates the Kepler frame.", "pages": [ 4 ] }, { "title": "2.2 The Kepler problem with quaternions", "content": "Equipped with the quaternion formulae we turn to regularize the Kepler problem with quaternions. The perturbed Kepler problem in the G = c = 1 system is given by the equation where r is the orbital separation vector of the orbiting bodies ε is the small parameter of the perturbation, and m = m 1 + m 2 is the sum of the two masses. To regularize the Kepler problem the separation is defined by the following quaternionic equation where U = U + U 3 i + U 2 j + U 1 k is a general quaternion. The conjugate of r is where the AB = ¯ B ¯ A , ¯ ¯ A = A and ¯ k = -k properties were used. Therefore r has no real part and can be written as r = z i + y j + x k . A direct calculation from (15) tells us that and the real part UU 1 -U 3 U 2 + U 2 U 3 -U 1 U = 0 indeed vanishes. The obtained transformation (17) is just the KS transformation with the U →-U convention. [2] Transformation (15) maps the four-dimensional quaternion space into the three-dimensional space of spatial vectors. Therefore the solution to a given three-dimensional r in terms of four-dimensional U is not unique. From (15) it is clear that the transformation where α is an arbitrary real number, is a continuous symmetry of (15). It follows that we have a one-dimensional compact manifold - a fibre - of U s for a given r , and (18) defines a fibration of the space of U s. The geometrical background of this transformation is elegantly described in Waldvogel (2005) [14]. This additional degree of freedom will be constrained in a careful manner. To complete the regularization the time has to be also transformed. The Sundman transformation is given by where s is the modified time and a is a real and at this point arbitrary function of s (it will be chosen such that it simplifies the solution). From now the operator ' indicates differentiation with respect to the modified time s . Inserting the definition of the orbital separation (15) into the equation of motion (13) while transforming the original time into the modified one using (19) lead us to the following quaternionic differential equation [4] where which is a scalar in the sense that it is invariant under conjugation. This quantity is the four-dimensional scalar product of the tangent vector of the U ( s ) curve and the tangent of the fibre at that point multiplied by two 3 . Let us define a condition which means that the trajectory U ( s ) intersects the fibres under right angles. It is indeed a condition as transforming U with symmetry transformation (18) Γ transforms as hence condition (22) can be satisfied by solving a first order differential equation for α ( s ). The geometrical constraint (22) is equivalent with the so-called ' bilinear relation ', which plays an essential role in the KS method[2]. It can be proved that (18) is a dynamical symmetry, since the transformed U solves the equation of motion. Furthermore if condition (22) is satisfied then Γ ' also vanishes, which means that one can maintain this condition by finding the proper initial conditions U (0) and U ' (0) which satisfy the 'bilinear relation' (22)[4].", "pages": [ 4, 5, 6 ] }, { "title": "2.3 Solving the unperturbed case: Kepler orbits", "content": "The unperturbed situation with condition (22) reduces the perturbed equation of motion (20) to the following equation The coefficient of U is where h is a constant of motion. Let us fix the parameter a which means that a is the semimajor axis of the elliptical motion. With this choice the equation of motion is the harmonic oscillator with constant frequency where the subscript 0 indicates the unperturbed case. The general solution of this second order differential quaternion equation has six free parameters as it is constrained with (22) and has a redundant phase (18). The trial solution of the unperturbed case can be parametrized as follows where β ± = 1 ± β 2 and q = e i ω 2 with ω = 2( s -s p ). In the next paragraph it is shown that the trial solution describes an elliptical Kepler orbit. Let us set R = 1 for the moment and substitute U 0 into the definition of the separation (15) r 0 = aβ -1 + k ( z + β 2 z -1 +2 β ) , (29) where z = q 2 and the identity q k = k q -1 was applied. This formula can be further expanded using the z = cos ω + i sin ω identity which means that according to (6) r 0 describes the following parametric curve with e = 2 ββ -1 + . These equations describe an ellipse in the ( x, y ) coordinateplane, with semimajor axis a , and eccentricity e . From equations (31) it follows that ω = 0 parametrizes the apocenter, thus ω is equivalent with the eccentric anomaly, except that the latter is zero at the pericenter. This tells us that s p is the time advance of apocenter passage measured in modified time. In the general case R is obviously the rotation between the orbital and reference frames, where the rotation according to (9) can be given with Euler angles The formulae which provide the connection between the a , β , s p and angular parameters - the orbital elements - and the quaternion components are collected in the Appendix. Despite of the great advantages of the gauge condition (22) - which is especially fine for numerical calculations - for perturbative calculations another geometrical condition is proposed.", "pages": [ 6, 7 ] }, { "title": "2.4 The perturbed case", "content": "The trial solution for the perturbed equation of motion (20), is just the unperturbed solution form (28) completed with general ε proportional terms [4] where both the D and S quaternions are O ( ε ) quantities, and complex in the sense that they are in the subspace spanned by the units 1 and i , while the b quantity is real. Using the definition of the separation (15) the perturbed separation vector in the Kepler frame is where the ε 2 terms were neglected. The result tells us that parameter b describes a translation along the i unit which is a translation along the z -axis of the Kepler frame according to (6). By considering (30) it parametrizes a translation perpendicular to the orbital plane. The same is true for the imaginary part of S , while the real part of S has no physical effect. Parameter D is complex and it is multiplied with k , thus the result is in the subspace spanned by the units j and k . These units are associated (6) with the ( x, y ) coordinate plane of the Kepler frame, which is the orbital plane according to (30). After introducing the trial solution in the perturbed case we fix the gauge. Vrbik's condition is that the real part of S must vanish[4] where the operator ∗ conjugates its complex quaternion argument. In this case the trial solution (33) has no k proportional part. Transformation (18) has a simple geometrical interpretation. It describes a double rotation, one in the (1, k ) and another one in the ( i , j ) subspace. Hence a transformation (18) on U (33) whose tangent is the coefficient of the k part of the trial solution (33) divided by its real part cancels the coefficient of k . In the leading ε order this rotation is which can be fulfilled without solving any differential equation for α in contrary to (23).", "pages": [ 7, 8 ] }, { "title": "2.5 Example for the regularization", "content": "The one-dimensional two-body problem is considered with the following special force f = ε ˙ x 2 , therefore We have chosen this kind of special force since the equation of motion has a constant of motion 4 . Eq. (37) can be regularized with the following transformations Then Eq. (37) is In case of unperturbed motion ( ε = 0) the energy is the constant of motion ( E 0 = 2( y ' ) 2 /y 2 -m/y 2 ) and Eq. (40) is the equation of the harmonic oscillator where clearly for bounded motion E 0 < 0. The general solution for Eq. (41) is y 0 = C 1 e i Ω s + C 2 e -i Ω s , where Ω = | E 0 | / 2 is the orbital frequency. √ We assume that for perturbed motion ( ε = 0) the form of the solution is y ε = y 0 + εδ . Then Eq. (40) in the leading order of δ is /negationslash and using the y 0 = A cos(Ω s ) solution of the unperturbed motion in (42) we get where the sign of Ω 2 δ is positive, since E 0 < 0. Numerical solutions for Eq. (43) can be seen on Fig. 1. It can be seen that the numerical solution δ ( s ) of these two examples are well-behaving, bounded functions for various initial values. So far we have represented the regularization of the one-dimensional perturbed two-body problem with a heuristic special force. Let us consider the one-dimensional model using the generalized Sundman transformation where ˜ x is not fixed yet. Eq. (37) can be regularized using transformations (38) and (44) If ˜ x = x ( desingularized in perturbed orbit) we obtain Eq. (42). If ˜ x = x 0 ( desingularized in unperturbed orbit) the result is Substituting the y ε = y 0 + εδ 0 perturbed solution, one obtains where E ( y 0 ) = [2 ( y ' 0 ) 2 + 5 m ] /y 2 0 is not the constant of motion (note that the coefficient of the linear term is the constant of motion in case ˜ x = x (Eq. (42)). In total two types of desingularization (˜ x = x, x 0 ) using the y 0 = A cos(Ω s ) (we have fixed the frequency Ω = 1) unperturbed solution can be given The numerical analyzis of these two equations with different initial values can be seen on Fig. 2.5. It can be seen that in this one-dimensional perturbed two-body problem the two types of desingularization methods lead to quite different solutions. Therefore the Sundman transformation is generally nontrivial in perturbed equations.", "pages": [ 8, 9, 10 ] }, { "title": "3 Orbital elements with quaternions", "content": "Before explaining the quaternion approach the classical equations are described in order to explain the relationship between the two different methods. The equation system of the classical two-body problem is of total order six, hence it can be described with six first integrals, which are also called orbital element s. These elements are the semi-major axis a , the eccentricity e , the inclination θ , longitude of the ascending node φ , the argument of the pericenter ψ 5 and the mean anomaly at the epoch l 0 (or time of pericenter passage t 0 ) related to the dynamics. The Lagrange planetary equation s in the standard perturbed two-body problem are[17] where χ is the true anomaly , ξ is the eccentric anomaly and r is the parametrization of the osculating orbit and l is the mean anomaly , which can be defined by the Kepler equation and n = m 1 / 2 a -3 / 2 is the mean motion . The S , T quantities are the projections of the perturbing force to the orbital plane, while W is the projection to the normal vector of the orbital plane ˆ k To derive quaternion differential equations for the orbital elements the trial solution (33) has to be substituted into the equation of motion (20). To simplify the calculation the quaternion equation of motion (20) is decoupled into two complex equations. The derivation of the complex equations are given in details[4, 18] and the most important steps are briefly outlined in our Appendix. Here only the solution and the necessary definitions are presented. The following auxiliary quaternion quantities have to be defined where the operator Cx is a projector, which projects its quaternion argument to the complex subspace spanned by the units 1 and i . The additional subscript 0 indicates the unperturbed value of the symbol. The subscript K is omitted in r and r 0 as they are scalars and have the same value in every frame. To point out the relationship between the quaternion formulae and the classical equations (50) note that the real and imaginary part of the complex Q quantity is proportional to the previously introduced S and T (53) respectively, while W is proportional to W . The quaternion coefficients (54) can be expanded into Laurent series[4] together with D and S from (33) /negationslash The Laurent series are given in powers of z . From the definition of the orbital separation r (15) follows that this is enough as the expansion of the separation contains every power of q . The coefficients D -1 , D 0 and S -1 , S 1 were left out from the expansion of D and S since they would only duplicate the q and q -1 terms of the solution (33), while S 0 was explicitly separated as b . Substituting expansions (55) and (56) into the complex equations (92) and (93) the differential equations for the orbital elements can be extracted by matching the coefficients of z with the same power on both side of the equation. The obtained differential equations are the following[4, 18] and the two additional formulae for D and S is given in the Appendix. The Z i quantities are the components of Z K , where the Kepler frame subscript was dropped to simplify the notation. Note that equation (61) is singular in β , which shows that in the circular orbit limit the ordinary sense of the rotation no longer valid. The coefficients Q n or W n of the Laurent series can be obtained with a contour integral, where C 0 is the unit circle Note that the Laurent expansion (56) of D and S has simplified the form of the differential equations (57)-(63) with respect to the Lagrange's planetary equations (50). The Laurent series of D and S absorbed the 'short' term oscillatory part of the equation. The remaining differential equations contain only the adiabatic, 'long' term part, which might be easier to solve. We have to amend the equations above with the transformation of the angular velocity from the comoving Kepler frame to the inertial system, which are the following This transformation is familiar from classical mechanics, in deriving the Euler equations of the the rigid body.", "pages": [ 10, 11, 12, 13 ] }, { "title": "4 GR perturbations", "content": "In this section perturbations calculated from the general relativity are examined using the described quaternion approach. The perturbations are examined with post-Newtonian approximation. The post-Newtonian approximation applies an expansion of corrections to the Newtonian gravitational theory with an expansion parameter ε ≈ v 2 ≈ m/r , which is supposed to be small, where v is the velocity. We use equations up to ε 5 / 2 , (post) 5 / 2 -Newtonian order, which is the order where the dominant gravitational radiation damping forces occur. First of all, the (post) 1 -Newtonian correction to the classical mechanics will be examined in the first section. This is followed by the (post) 5 / 2 -Newtonian analysis of gravitational radiation where we rederive the classical Peters-Mathews formula.", "pages": [ 14 ] }, { "title": "4.1 Planar assumption", "content": "The mentioned perturbations are planar perturbations, in the sense that the force lies within the orbital plane. In this case obviously S = b = 0 and the trial solution (33) contains only perturbations within the orbital plane It follows that in case of planar forces the S ∗ = -S condition (35) is true. Remarkably the Γ = 0 condition is also satisfied[4]. To show this we need the derivative of U expressed with Kepler frame quantities Therefore and since the rotation can be dropped under the real part operator In the planar case U K is a complex number, therefore both ¯ U K k U ' K and r K are in the orbital plane spanned by the j and k units. In the planar case the orbital plane is preserved, therefore Z K must be perpendicular to this j , k subspace. It means that Z K has only i part. It is easy to see from equation (69) that the argument of the operator Re has no real part. Therefore in case of planar forces Γ vanishes. Consequently the equation of motion (20) simplifies Let us introduce a τ parameter by rescaling the modified time dτ = 2 a 1 / 2 m -1 / 2 ds . With the help of the τ parameter the equation of motion is just the perturbed harmonic oscillator In the planar case the equation of motion substantially simplified and identical with the equation of Waldvogel[14]. 6 In the planar case the perturbations D can be expressed in a more conventional way. Let us introduce a = a 0 + δa and β = β 0 + δβ in the trial solution (28) where δa and δβ are first order quantities. In this case by matching the first order part of Eqs. (28) and (66) one obtaines the following important relations where A = 1 + β 0 z and B = z -β 0 (1 -β 0 ) 2 A .", "pages": [ 14, 15 ] }, { "title": "4.2 The classical post-Newtonian effect", "content": "In this section the leading contribution of general relativity to classical Newtonian mechanics is examined in details. The force is given by numerous authors [19] where the subscript PN denotes the post-Newtonian term, ˆ n = r /r , η = ( m 1 m 2 ) /m 2 and v = | v | is the absolute value of the orbital velocity v = d r /dt . After transforming it to quaternion expression This result (74) have to be substituted into the definition of Q (54), where the separation r K , its magnitude r and their derivatives are treated as functions of z according to (29). Therefore the result is a function of z Applying the contour integral (64) the coefficients Q n can be computed as follows. β < 1 therefore the only singularity of Q inside C 0 is at -β . The other pole at -1 /β lies outside the unit circle. Q can be expanded around its pole at -β and keeping the coefficient of the ( z + β ) -1 part, the result is In the same way with Q /z one finds Q 0 = -2 ma -1 β -4 -β + ( -3 -8 β 2 -β 4 + η +17 β 2 η +3 β 4 η ) . (77) Both of the coefficients Q -1 and Q 0 are real. The differential equation for the semimajor axis is proportional to their imaginary part (57), therefore a ' = 0. The remaining differential equations can be calculated in the same way. The resulting nontrivial differential equations in modified time for the orbital parameters are as follows[18]. The equation for the argument of the pericenter and for the modified time at apocenter Using the transformation to the modified time (19) in the leading order from (78) it follows that which is the known expression for the precession of the pericenter[20]. The remaining differential equations have zero on the right hand side of the equation and the corresponding orbital element is constant.", "pages": [ 15, 16 ] }, { "title": "4.3 Gravitational radiation reaction", "content": "Gravitational radiation damping has been recognized as a process with very important observable consequences: the PSR 1913+16 system has given the first evidence that gravitational waves exist[21], and other systems are of high importance as well [22, 23]. The equation of motion is given by [24] where the subscript RR indicates the radiation reaction term and 7 The γ and ρ parameters in (82) represent the residue of gauge freedom that has not been fixed by the energy balance method and that has no physical meaning. It is known that these arbitrariness is equivalent with a coordinate transformation whose effect on the two-body separation vector is We use transformation (83) to remove the gauge dependency from the quaternion equation (20), after substituting (81) as the perturbing force. In order to apply transformation (83) on the quaternion equation of motion (20) we have to rewrite it in quaternion form using modified time (19). The definition of the modified time (19) contains the separation r therefore any gauge dependent transformation of the separation, like (83), leads to gauge dependent modified time s ( γ, ρ ). Consequently the transformation of any real time derivative involves a new gauge dependent contribution It follows that transformation (83) in its original form does not cancel these new gauge dependent contributions and needs to be reparametrized. The reparametrized transformation in quaternion form is the following where K , L and N are unknown coefficients. To obtain them consider that Q (54) is Laurent series in z and gauge independence requires that every ρ and γ proportional term in the coefficients must vanish. E.g. the coefficients of z 2 γ , z 2 ρ and z 3 β 2 ρ of Q (54) after simplification lead to a linear equation system which determines that N = 3 and K = L = 1 [18]. In the leading order according to (15) this is equivalent with the following real time vectorial transformation which is slightly different from (83). The result from formula (57)-(63) for the semimajor axis is now gauge independent[18] and also for the modified eccentricity The remaining differential equations are trivial, with a zero on the right hand side, and the corresponding orbital elements remain constant. The gauge independent value of parameter D is given in [18], while S is zero. Now we are in the position that the latter result for the semi major axis (87) and also the expression for the eccentricity (88) can be easily verified. They must be equal with the two corresponding classical formula derived from the well known Peters-Mathews formula [7], which describes the effect of gravitational radiation. After substituting the expression e = 2 ββ -1 + into (89) and (90) together with the transformation rule between the real and modified time (19) one can derive the two equation below which are indeed identical with the two formula derived from the Peters-Mathews equation [25].", "pages": [ 16, 17, 18 ] }, { "title": "5 Conclusion and outlook", "content": "In this paper general relativity perturbations were examined using a new approach where the regularization of the Kepler problem is given with quaternions. This approach is based on the usual Kustaanheimo-Stiefel method which is defined with matrices. With the new calculus the differential equations of the orbital parameters were derived in case when the perturbation is the leading (post) 1 -Newtonian order correction of general relativity. To test the new method the precession of the pericentre is rederived. Then the gravitational radiation reaction was analyzed, where the famous Peters-Mathews formula was reproved using the quaternion approach [7]. We have studied the gauge dependence of the equations of motion and we managed to remove the residual gauge freedom from the quaternionic equation of motion. The new quaternionic approach is easy to implement with program code. Quaternions can be represented with pairs of complex numbers, then the equations can be calculated and solved with the help of complex analysis. This feature makes this method to a very efficient calculus for symbolic computations. With the quaternion based regularization the spin-orbit and spin-spin interactions can be examined as well[26]. It is foreseen that these spin interaction related calculations provide the next step of our studies.", "pages": [ 18 ] }, { "title": "Acknowledgment", "content": "We would like to thank Prof. J. Vrbik to explain and interpret some points of his method. F. N. especially would like to thank Prof. P. Forg'acs for the initiation of this study and for his help during the whole work. The authors would like to thank M. Vas'uth for giving valuable information and advices.", "pages": [ 19 ] }, { "title": "A The components of U expressed with orbital elements", "content": "These formulae can be straightforwardly derived from the unperturbed solution (28) using the expression of the rotation (32) with the rotation angles where ω ± = ( φ ± ψ ) / 2.", "pages": [ 19 ] }, { "title": "B Decoupling the equation of motion", "content": "For convenience the quaternion equation of motion (20) can be decoupled into two complex equations. Premultiplying (20) with ( -1 -β 2 ) ¯ U K / (2 a ) and also postmultiplying it by ¯ R while keeping only the 1, i part in O ( ε ) one obtains[4] where z ± = z ± z -1 and Z n are the components of the angular velocity vector (12) Similarly premultiplying equation (20) with ( 1 + β 2 ) ¯ U K k /a and then keeping only the complex part in O ( ε ) the second complex equation is the following", "pages": [ 19 ] }, { "title": "C The solution for D and S", "content": "Similarly by pairing the powers of z in the complex equations (92) and (93) two additional equation for D and S can be gained /negationslash", "pages": [ 20 ] }, { "title": "D The D and S quantity in case of the (post) 1 -newtonian effect", "content": "The complicated quantity D is given only up to second β order while S is zero.", "pages": [ 20 ] }, { "title": "E Gravitational radiation: D and S", "content": "The fairly complicated D quantity is given in second β order while S is zero.", "pages": [ 20 ] } ]

[ { "title": "Quintessential and phantom power-law solutions in scalar tensor model of dark energy", "content": "L. N. Granda ∗ D.F. Jimenez † , and C. Sanchez ‡ Departamento de Fisica, Universidad del Valle A.A. 25360, Cali, Colombia", "pages": [ 1 ] }, { "title": "Abstract", "content": "We consider a scalar-tensor model of dark energy with kinetic and Gauss Bonnet couplings. We study the conditions for the existence of quintessential and phantom power-law expansion, and also analyze these conditions in absence of potential (closely related to string theory). A mechanism to avoid the Big Rip singularity in various asymptotic limits of the model has been studied. It was found that the kinetic and Gauss-Bonnet couplings might prevent the Big Rip singularity in a phantom scenario. The autonomous system for the model has been used to study the stability properties of the power-law solution, and the centre manifold analysis was used to treat zero eigenvalues. PACS 98.80.-k, 95.36+x, 04.50.kd", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "According to recent observations the current universe undergoes a phase of accelerated expansion, due to domination of dark energy (DE) over the matter content of the universe(usual barionic and dark matter) [1, 2, 3, 4, 5, 6]. The astrophysical observations indicate that the equation of state parameter for dark energy lies in a narrow region around w = -1, which might include values smaller than -1. The current observational data are in agreement with the simplest possibility of the cosmological constant as the source of DE, but there is no mechanism to explain its smallness (expressed in Planck units) in contradiction with the expected value as the vacuum energy in particle physics [7], [8]; and observational data also show a better fit for a redshift dependent equation of state. Despite the variety of DE models, it is however difficult to fulfill all observational requirements like the observed value of the equation of state parameter (EoS) of DE, w ≈ -1, the current content of DE relative to that of dark matter (known as coincidence problem), and the estimated redshift transition between decelerated and accelerated phases, among others. Among the models used to explain the DE (for review see [7, 8, 9, 10, 11, 12]), the scalar-tensor theories are some of the most studied, not only because they contain direct couplings of the scalar field to the curvature, many of them predicted by fundamental theories like Kaluza-Klein and string theories [13], [14], but also because the scalar-tensor models fulfill in principle many of the above requirements. In the present work we consider a string and higher-dimensional gravity inspired scalar-tensor model, with non minimal kinetic and Gauss Bonnet (GB) couplings, to study late time cosmological dynamics and board some issues of dark energy. These terms are present in the next to leading α ' corrections in the string effective action (where the coupling coefficients are functions of the scalar field) [15], [16] and have the notorious advantage that lead to second order differential equations, preserving the theory ghost free. Some late time cosmological aspects of scalar field model with derivative couplings to curvature have been considered in [17], [18, 19, 20], [21], [22]. On the other hand, the GB invariant coupled to scalar field has been extensively studied. In [23] the GB correction was proposed to study the dynamics of dark energy, where it was found that quintessence or phantom phase may occur in the late time universe. Different aspects of accelerating cosmologies with GB correction have been also discussed in [24], [25], [26], [27], [28], [29]. The modified GB theory applied to dark energy has been suggested in [30], and further studies of the modified GB model applied to late time acceleration, have been considered among others, in [31], [32], [33], [34], [35]. The combined effect of GB and kinetic coupling to curvature in the context of dark energy, has been considered in [36, 37, 38, 39]. In [36] solutions with Big Rip and Little Rip singularities have been considered, in [37, 38] the reconstruction of different cosmological scenarios, including known phenomenological models has been studied, and in [39] some exact solutions have been found. In the present paper we focus on the power-law solutions of the quintessence and phantom types, in the case of late-time cosmology with scalar field dominance. We analyze the conditions to avoid the Big Rip singularity presented in phantom powerlaw solutions. The autonomous system for the model has been considered to find the restrictions on the parameter space of the model satisfying the conditions of stability. In section II we introduce the model and give the general equations, which are then expanded on the FRW metric to study the different power-law solutions. In section III we analyze the conditions to evade the Big Rip singularity in different scenarios. In section IV we introduce the dynamical variables and analyze the stability of the power-law solutions. Concluding remarks are given in section V.", "pages": [ 1, 2, 3 ] }, { "title": "2 The model and power-law solutions", "content": "We consider the following action that contains the Gauss Bonnet coupling to the scalar field and kinetic couplings to curvature (such terms are present in the leading α ' correction to the string effective action [16]). where γ = ± 1 (+1 for the standard scalar field and -1 for the phantom scalar), G µν = R µν -1 2 g µν R , G is the 4-dimensional GB invariant G = R 2 -4 R µν R µν + R µνρσ R µνρσ . The coupling F 1 ( φ ) has dimension of ( length ) 2 , and the coupling F 2 ( φ ) is dimensionless. Note that compared with the more general action that leads to the second-order equations of motion (in metric and scalar field) [40], we are neglecting derivative terms that are not directly coupled to curvature, of the form glyph[square] φ∂ µ φ∂ µ φ and ( ∂ µ φ∂ µ φ ) 2 , which is is acceptable in a cosmological scenario with accelerated expansion. The properties of the GB invariant guarantee the absence of ghost terms in the theory. Hence, the equations derived from this action contain only second derivatives of the metric and the scalar field. By varying Eq. (2.1) with respect to metric we derive the gravitational field equations given by the expressions where κ 2 = 8 πG , T m µν is the usual energy-momentum tensor for the matter component, the tensor T µν represents the variation of the terms which depend on the scalar field φ and can be written as where T φ µν corresponds to the variations of the standard minimally coupled terms, T K µν comes from the kinetic coupling, and T GB µν comes from the variation of the coupling with GB. Due to the kinetic coupling with curvature and the GB coupling, the quantities derived from this energy-momentum tensors will be considered as effective ones. The respective components of the energy-momentum tensor (2.3) are given by and In this last expression the properties of the 4-dimensional GB invariant have been used (see [41], [42]). By varying with respect to the scalar field gives us the equation of motion Let us consider the spatially-flat Friedmann-Robertson-Walker (FRW) metric, where a ( t ) is the scale factor. Replacing this metric in Eqs. (2.2)-(2.8) we obtain the set of equations describing the dynamical evolution of the FRW background and the scalar field in the present model: In general the couplings F 1 ( φ ) and F 2 ( φ ) could be arbitrary functions of the scalar field, which gives more general character to the model (2.1) where the couplings should be constrained by known observational limits. General couplings also allow to increase the number of phenomenologically viable solutions to the DE problem. Based on effective limits of fundamental theories like super gravity or string theory, the kinetic and GB couplings appear as exponentials of the scalar field (in leading α ' correction in the case of string theory), but if take into consideration higher order corrections in α ' expansion in the effective string theory, the couplings should change. Here we will consider the string inspired model with exponential couplings of the form where we also consider an exponential potential that allows to study power-law solutions. In this section we will consider only the standard scalar field corresponding to γ = 1. Lets propose the solution for quintessential power-law (QPL) expansion, and for phantom power-law expansion (PPL). By replacing (2.13) and (2.14) in the Friedmann equation (2.9), one obtains the restrictions and from the equation of motion (2.11) we find where the lower minus sign follows for the phantom solution. Let's consider the two cases separately.", "pages": [ 3, 4, 5, 6 ] }, { "title": "Quintessential power-law expansion", "content": "From Eq. (2.16) and (2.17) for the quintessence expansion one finds V 0 as For γ = 1, the conditions ξ > 0 and p > 1 are enough to give positive potential, i.e. V 0 > 0. Thus in the regime of accelerated expansion, V 0 > 0. One might consider negative kinetic coupling ξ < 0, which leads to the following condition by demanding positivity of the potential Assuming ξ = -bt 2 1 ( b > 0), this condition reduces to Taking for instance b = 1, it is clear that in the regime of accelerated expansion ( p > 1) this condition can be satisfied. On the other hand, considering the phantom model ( γ = -1), then in the regime of accelerated expansion ( p > 1) the condition allows quintessence PL. Let's consider only the effect of the kinetic coupling (the effect of the GB coupling has been considered in [42]). Setting η = 0 in Eqs. (2.16) and (2.17) and solving with respect to V 0 and ξ one finds A reasonable assumption for φ 0 could be κ 2 φ 2 0 = 1 which leads to In the region of interest that is p > 1 (for both QPL and PPL), we can see that independently of γ the potential is positive. Therefore we have QPL solution for the case of pure kinetic coupling ( η = 0). A closely related to string theory is the case where V ( φ ) = 0. To cancel the potential, it follows from (2.18) that Considering γ = 1, we note that if ξ > 0 then the sufficient condition to have φ 2 0 > 0 is that 1 / 3 < p < 1. In this case in absence of potential we can not have accelerated expansion. More interesting is the case when ξ < 0. Let's represent ξ = -bt 2 1 , where b > 0. Then the condition to cancel the potential becomes If we take for instance b = 1, then there are two regions of the p -parameter line that satisfy the requirement φ 2 0 > 0: 0 . 182 < p < 0 . 633 which is applicable to early-time cosmology, and p > 1 which gives accelerated expansion. Then in absence of potential it is possible to describe late time QPL expansion in the frame of the present model. Considering γ = -1, if ξ < 0, then from (2.24) follows that for p > 1, φ 2 0 > 0, and if ξ > 0, then the consistency of (2.24) demands that p > 1 and ξ < (5 p -1) t 2 1 6 p 2 (3 p -1) . In the particular case of the model (2.1) without GB coupling, the condition to cancel the potential leads to in the case γ = 1, the first equality has sense only for 1 / 6 < p < 2 / 3 (see eq. (3.8) in [22]), which is appropriate for early-time cosmology. In the case γ = -1, the first equality is consistent for p > 2 / 3, which allows accelerated expansion for p > 1.", "pages": [ 6, 7, 8 ] }, { "title": "Phantom power-law expansion", "content": "For the case of PPL expansion one finds from (2.16) and (2.17) (lower sign) for V 0 For γ = 1, the potential is always positive for ξ > 0 and p > 1. For negative kinetic coupling, assuming for instance ξ = -bt 2 1 ( b > 0) one finds the condition for positive potential taking b = 1 for instance, this restriction is consistent for any p > 1. Assuming γ = -1, then for ξ > 0 the potential is positive provided that p > 1 and (setting ξ = bt 2 1 , b > 0) For ξ < 0 and p > 1 (setting ξ = -bt 2 1 ) one finds Considering only the effect of the kinetic coupling (i.e. η = 0) one finds For γ = 1, this potential is positive for p > 1 / 3 and in this case the kinetic coupling becomes negative ( ξ < 0). Hence, in the case η = 0 the PPL expansion takes place for negative kinetic coupling. For γ = -1, the potential is positive for p > 1 / 3 and κ 2 φ 2 0 > 6 p 2 (3 p +2) 6 p +1 . It is interesting also to study the conditions to cancel the potential in the case of PPL. From (2.27) follows For γ = 1, it follows that for ξ > 0 there is not way to cancel the potential for p > 0. But if we consider the negative coupling ξ < 0, then (setting ξ = -bt 2 1 , b > 0) the condition to cancel the potential leads to taking for instance b = 1, it follows that (2.33) is consistent for p > 0 . 483. Therefore it is possible to have PPL in absence of potential. Taking γ = -1 in (2.32), then for ξ < 0 there is always possible to have PPL for any p , and for ξ > 0, the restriction (2.32) is consistent provided that ξ < (5 p +1) t 2 1 6 p 2 (3 p +1) . If we limit the model and consider only the effect of the kinetic coupling ( η = 0), then from (2.31) for γ = 1 follows that there is not way to have V 0 = 0. Hence, in order to have PPL expansion in this case, it is necessary to have a potential. In the case of γ = -1, the condition to cancel the potential leads to which is consistent for any p > 0. Another important solution of the model (2.1) with the potential and couplings as given by (2.12) is the de Sitter solution. In fact, if we consider the scalar field and the Hubble parameter as constants, i.e. φ = const. = c and H = const. = H 1 , then by replacing in (2.9) and (2.11) we find Note that in this solution the kinetic coupling is irrelevant since ˙ φ = 0 (the solution is possible for negative η ). In the important the case of V ( φ ) = 0, as follows from (2.25) and (2.33) there is an asymptotic de Sitter solution at p →∞ where", "pages": [ 8, 9, 10 ] }, { "title": "3 Possible mechanisms to avoid the BR singularity", "content": "The PPL solution suffers the well known problem of the future BR singularity at t = t s [43, 44, 45, 46]. We may use the fact that in the frame of the present model, the PPL can be obtained in the absence of potential, and that in the case of dominance of potential over the other interaction terms the model presents asymptotic quintom behavior. This fact could provide a mechanism to evade the future BR singularity as follows. Let's focus on the PPL in the case when we can neglect the effect of the potential. To this end we propose the following model It is clear that the power-law is not a solution to this model, unless δ = 0. Nevertheless, we can make some qualitative analysis based on asymptotic behavior of the model. In this case, when the curvature is small we assume that the solution behaves as (2.14), which in absence of potential leads to the condition (2.33), that follows for PPL expansion. Small curvature means that ( t s -t ) is large, and the potential that behaves like V ∝ 1 / ( t s -t ) 2(1+ δ ) , can be neglected compared to the kinetic and GB couplings that behave as 1 / ( t s -t ) 2 . But as the universe evolves, the difference ( t s -t ) becomes smaller and the curvature increases as t → t s . At this stage the potential becomes dominant over the GB and kinetic couplings and the model becomes dominated by pure quintessential scalar field. In this case the only possible power-law solution for the potential (3.1) is of the form which leads to the known conditions giving an EoS parameter w > -1, avoiding in this way the future BR singularity. As the universe continue evolving after the dominance of the potential, the curvature turns again to small values and the interacting terms start dominating again. Nevertheless, when the potential and the couplings are present there is an important solution, namely the de Sitter solution. In fact, if we assume for the model (3.1) then, by replacing in (2.9) and (2.11) this solution gives This is an alternative to the approach presented in [42] where the GB coupling and phantom scalar field were considered. If we consider the phantom version of the model (i.e. with γ = -1), then there are more alternatives to avoid the BR singularity.", "pages": [ 10, 11 ] }, { "title": "Phantom scalar field γ = -1", "content": "Bellow we consider the phantom kinetic term in the model (2.1) (i.e. assuming γ = -1), which gives more possible ways of evading the BR singularity, namely", "pages": [ 11 ] }, { "title": "1. In absence of kinetic coupling ( ξ = 0 ).", "content": "In this case, the model reproduces the same results presented in [42].", "pages": [ 12 ] }, { "title": "2. In absence of GB coupling ( η = 0 ).", "content": "One may consider the situation when initially at low curvature (large time) the potential term dominates during PPL expansion, and then when the solution is closer to the BR singularity the kinetic coupling becomes dominant, allowing the possibility of QPL expansion with EoS parameter w > -1. To this end we propose the model Assuming that the solution behaves as (2.14), then neglecting the kinetic coupling, from (2.9) and (2.11) (for γ = -1) follow the known conditions for existence of PPL solution (2.14) as the curvature increases when t → t s , the kinetic coupling becomes dominating and the potential could be neglected. In this case we may assume the same solution (3.2) for the quintessential expansion, that being replaced in (2.9) and (2.11) (setting V = 0 and F 2 = 0) leads to the conditions which is consistent for p > 2 / 3. So there is possible to change the effective EoS from w < -1 to w > -1 avoiding the BR singularity. Hence, when the term with kinetic coupling becomes dominant, the BR singularity might be prevented. So the kinetic coupling may play a role similar to the GB coupling [42] in working against the singularity. For the power-law solution of the form (3.2), after the dominance of the kinetic coupling the curvature becomes small again, and the potential term recovers his dominance, but there exists a solution when both, the potential and the kinetic coupling are present which corresponds to a de Sitter phase. If we assume the solution (3.4), then replacing in (2.9) and (2.11) with (3.6), one finds which is valid for arbitrary c , since ˙ φ = 0, making this solution independent of the kinetic coupling. Therefore, there is a possibility that the universe enters in a de Sitter phase after domination of the kinetic coupling.", "pages": [ 12, 13 ] }, { "title": "3. In absence of potential ( V = 0 ).", "content": "Here we consider the following model Assuming the phantom solution (2.14), we see that at low curvature when t s -t is large, the GB coupling behaves as 1 / ( t s -t ) 2+2 δ and can be neglected with respect to the term with kinetic coupling. In this case, by solving Eqs. (2.9) and (2.11) one finds the conditions which corresponds to phantom phase with effective EoS w < -1. When the curvature turns to large values at t → t s , the term with kinetic coupling could be neglected giving rise to the dominance of the GB term. Neglecting the kinetic coupling in (3.10) and assuming a solution of the form given by eq. (3.2), one finds from (2.9) and (2.11) which is consistent for p > 1, leading to quintessential expansion with w > -1. Note that the role of the kinetic and GB couplings could be changed, i.e. one starts with domination of the GB term and ends with domination of the kinetic coupling, by proposing In this case, at low curvature the kinetic coupling can be neglected, and replacing the PPL solution (2.14) in (2.9) and (2.11) one finds the conditions which is consistent for any p > 0 and negative η . At large curvature when t → t s , the kinetic term becomes dominant and we can propose the solution (3.2), which being replaced in (2.9) and (2.11) gives the restrictions which is consistent for accelerated expansion with p > 1 and effective EoS w > -1. However in the model without potential, there is not de Sitter solution corresponding to constant scalar field.", "pages": [ 13, 14 ] }, { "title": "4. With all the terms.", "content": "We may consider the phantom scalar model with the potential and couplings given by If we neglect the couplings F 1 and F 2 at low curvature, then the only possible powerlaw solution is the phantom one, given by (2.14). When the curvature becomes large at t → t s , the interacting terms become relevant and (neglecting the potential) there is a power-law solution of the form (3.2) which leads to which is positive whenever t 2 1 > 6 ξp 2 (3 p -1) / (5 p -1). For negative ξ this condition is satisfied for any p > 1, leading to w > -1 and avoiding the BR singularity. Note that the solution (3.17) exists in the asymptotic de Sitter limit at p →∞ ( w →-1), given by valid for ξ < 0 (in this limit η → 0). After the dominance of the coupling terms the curvature begins to decrease again, but there exists a de Sitter solution (3.4) when the three terms in (3.16) are present So it is possible to implement the asymptotic mechanism to avoid the BR singularity as proposed in [42], in different variants of the model depending on the correlation between the kinetic coupling, the GB coupling and the potential. As has been shown, this mechanism can be implemented in the standard and phantom version of the scalar field. It is worth mentioning that the account of quantum effects near the singularity were also considered to moderate the BR singularity [47]. Another interesting alternative to avoid the future BR singularity is provided by the solutions known as 'Little Rip' (LR) [48, 49, 50, 36], which are free of future singularity. The LR solutions produce late-time cosmological effects similar to that of the BR solutions, as the rapid expansion in the near future with an EoS w < -1, but the scale factor and density remain finite in finite time. As in the case of BR, the LR solutions also lead to the dissolution of all bound structures in the universe in the future.", "pages": [ 14, 15 ] }, { "title": "4 Stability of the power-law solution for the string motivated model", "content": "We use the dynamical system approach in order to analyze the stability of the above power-law solutions, in the specific case of V = 0, that apart from simplifying the dynamical system is also closely related to string theory. Let's introduce the following dimensionless variables: In fact, these variables are related with the density parameters for the different sectors of the model where The Eq. (2.9) imply the following restriction on the density parameters and the effective equation of state (EoS) can be written as Introducing the e-folding variable N = log a , in terms of the variables (3.1), the Eqs. (2.9)-(2.12) can be transformed into the following first-order autonomous system where ' ' ' denotes derivative with respect to N and γ = ± 1 is the sign of the free kinetic term. Note that the last three Eqs. come from the explicit form of the potential and the couplings given in (2.12). From Eq. (4.8) follows the expression for the slow-roll parameter glyph[epsilon1] It is easy to check that the power-law solutions (2.13) and (2.14) are critical points of the system, i.e., if we write the dynamical variables (4.1) for H and φ given by (2.13) or (2.14) as where the '-' sign is for the PPL, then these variables satisfy the equations: x ' 0 = k ' 0 = g ' 0 = 0. So we will consider small perturbations and check the stability around the critical point ( x 0 , k 0 , g 0 ). Solving the system (4.6)(4.11) with respect to x ' , k ' , g ' one can write for small perturbations, x ' , k ' , g ' suffer the change where the matrix is valuated at the fixed point ( x 0 , k 0 , g 0 ) given by (4.12). The stability under small perturbations demand that the eigenvalues of the above matrix be negative or complex with negative real component. We will analyze the stability for two different cases. In the first case we consider the model with only kinetic coupling ( g = 0), and in the second case we consider both couplings (the case with only GB coupling was considered in [42]).", "pages": [ 15, 16, 17 ] }, { "title": "The model with non-minimally coupled kinetic term.", "content": "By setting g = 0 in (4.6)-(4.11), it reduces to a two dimensional system for x and k , and for small perturbations we may write where the two dimensional matrix M is evaluated at the critical point ( x 0 , k 0 ), which gives the components where the lower sign is assigned to the phantom model ( γ = -1, see Eqs. (2.9) and (2.11)). Replacing x 0 and k 0 from (4.12) and using the restrictions (2.26) for QPL, we find the following eigenvalues for the matrix M where p should obey the above restriction for consistency, according to (2.26). In the interval 1 / 6 < p < 2 / 3, λ 2 > 0 and therefore the power-law solution of the form H = p/t is unstable for the model in absence of potential and GB term. Note that the PPL solution with only kinetic coupling is not possible as was demonstrated above (see eq. (2.31) for V 0 = 0). On the other hand, if we consider the phantom model that obeys the Eqs. (2.9) and (2.11) ( γ = -1), then taking into account the lower sign in the components (4.17) one finds the following eigenvalues for the QPL where the last inequality follows from the consistency of the solution of Eqs. (2.9) and (2.11) with V = 0 and η = 0, for QPL (see eq. (3.8)). Therefore, for the phantom model with kinetic coupling, the power-law solution (2.13) is a stable fixed point provided p > 2 / 3. If we consider the phantom model with PPL (2.14), then using the lower sign in (4.17) and in (4.12) one finds the following eigenvalues which is valid for any p > 0, as follows from Eqs. (2.9) and (2.11) with V = 0 and η = 0, for PPL. Then the power-law solution (2.14) is a stable fixed point for the phantom model with V = 0 and η = 0.", "pages": [ 17, 18 ] }, { "title": "The model with kinetic and GB couplings", "content": "Here we consider the stability of power-law solution for the three dimensional autonomous system (4.6)-(4.11) in the following cases: Quintessence and phantom power-law for γ = 1 . Evaluating the matrix elements of eq. (4.15) for the fixed point (4.12) (upper sign) and for the QPL (upper sign), we find the following eigenvalues note that λ 2 and λ 3 are negative for p > 2 / 3. This restriction is compatible with the condition of consistency for the QPL solution (2.13), as can be seen in (2.25). For the PPL we have found the eigenvalues where λ 2 and λ 3 are negative for any p > 0, which is compatible with the restrictions Quintessence and phantom power-law for γ = -1 . Evaluating the matrix elements in (4.15) for the fixed point (4.12) (lower sign) and for the QPL (upper sign), we find the following eigenvalues which is compatible with the solutions in absence of potential for p > For PPL we have the eigenvalues 1 (see 3.17). valid for ξ < 0 and p > 0. Note that the eigenvalues are independent of γ . So in the presence of both couplings the stability of the power-law solution does not depend on the standard or phantom character of the model. But a problem appears in both cases due to the presence of zero eigenvalues. In this case the linear expansion fails to provide information on the stability of the fixed point. We need to consider higher order corrections to study the stability of perturbations along the zero eigenvalue direction.", "pages": [ 18, 19 ] }, { "title": "The centre manifold analysis.", "content": "To analyze the stability in the presence of zero eigenvalues we use the approach of the central manifold [51], [52], [53], which reduces the dimensionality of the system near the critical point, and limits the stability analysis to the reduced system. Thus the stability properties of the system become determined by the (in)stability of the reduced system. To this end we need to translate the fixed point (4.12) to the origin, by introducing the variables (we keep the same symbols) The simplest case takes place for one zero eigenvalue, which leads to one-dimensional reduced system. Composing the matrix M 0 with the eigenvectors of the Hessian in the new defined fixed point (0 , 0 , 0), we introduce a new set of coordinates In these coordinates the dynamical equations can be written in the form where the last column represents the non-linear terms. Note that in our case the variable u is actually the same variable x . Then the system can be written as where We now turn to the definition of centre manifold: The space where Dh is the matrix of first derivatives of the vector valued function y = h ( x ), is called the centre manifold for the system (4.28). Since y = h ( x ), the dynamics of the system becomes reduced to the centre manifold in the neighborhood of x , and the stability properties of the full dynamical system depend on the analysis in the centre manifold. Using y = h ( x ), the system (4.28) leads to This differential equation can be used to find h ( u ), and then by replacing h ( u ) into the first equation (4.28) (i.e. u ' = ˜ f 1 ( u, h ( u ))) we can analyze the stability of the reduced system. Near the critical point we can Taylor expand h ( u ) in powers of u and calculate the coefficients of the first non-trivial terms from (4.31). In our case we assume h of the form Using the restrictions on ξ and η for the QPL solution in absence of potential, the critical point (4.12) takes the form where α is given by (2.15). To avoid large analytical expressions we will limit the analysis to specific values of p . The first interesting value of p corresponds to the de Sitter limit, which gives the critical point ( x 0 = 0 , k 0 = -1 , g 0 = 2) and the corresponding eigenvalues from (4.21) are ( λ 1 = 0 , λ 2 = -3 , λ 3 = -3). Applying the above central manifold analysis, and after large but straightforward calculations we find the following equation for the reduced system by replacing h ( α ) = 9 16 α 2 -3 64 and integrating this equation one obtains where u 0 is the initial perturbation along the zero eigenvalue direction. In order for this initial perturbation to decay it follows from (4.35) that independently of the sign of u 0 , the coefficient h ( α ) must be negative. Therefore if h ( α ) < 0 the critical point will be stable. Turning to eq. (4.34) we see that if α 2 > 12 then h < 0 and the critical point is a stable attractor. Another reasonable value is p = 200 / 9, which gives the EoS parameter w = -0 . 97. In this case the critical point is ( x 0 = 9 100 α , k 0 = -191 197 -26757 1970000 α 2 , g 0 = 388 197 + 10719 985000 α 2 ) and the eigenvalues are ( λ 1 = 0 , λ 2 = -591 / 200 , λ 3 = -291 / 100). The centre manifold analysis gives but the analytical expression for µ ( α ) is too large to be displayed here. We limit ourselves to numerical evaluation for some values of α : ( α = 2, µ = 0 . 809), ( α = 3, µ = 0 . 03526), ( α = 12, µ = -0 . 03592) and ( α = 14, µ = -0 . 03549). Note that for the last two values of α the critical point is stable. One can also perform the same analysis for the PPL (lower sing in (4.12) with eigenvalues given by (4.22). For the specific value of p = 80 / 3 giving the EoS parameter w = -1 . 025, the centre manifold analysis gives evaluating ν ( α ) for some values of α , gives:( α = 1, ν = 0 . 055), ( α = 3, ν = -0 . 008), ( α = 12, ν = -0 . 049), ( α = 14, ν = -0 . 051). The last three values of α lead to stability.", "pages": [ 19, 20, 21, 22 ] }, { "title": "5 Discussion", "content": "We studied late time power-law cosmological solutions based on string spired scalartensor model including a coupling to the Gauss-Bonnet invariant and kinetic couplings to curvature. The model allows quintessential and phantom power-law expansion in a variety of scenarios that involve different asymptotic limits. In the case with potential the conditions (2.18) (2.19) and (2.21) allow QPL expansion, and the conditions (2.27), (2.28), (2.29) and (2.30) allow PPL expansion. In absence of the GB coupling, the restrictions (2.22) allow QPL and the restrictions (2.31) allow PPL. When we neglect the potential (which is closely related to string theory), the condition (2.24) allows QPL expansion and (2.32) allows PPL. If in addition to the potential we neglect the GB term, then the model continues to have power-law solutions, where in this case for γ = 1, the condition (2.26) allows power-law but in the range of decelerated expansion. And for the case of γ = -1, it follows from (2.26) that the model has QPL solution. Concerning PPL solutions in this limit (i.e. V = 0, η = 0), then from (2.31) for γ = 1 it follows that there is not PPL without potential, and in the case of γ = -1, the condition (2.34) allows PPL for any p > 0. The model also exhibit de Sitter solution in various scenarios: considering φ = const. and H = const. , then from (2.9) and (2.11) follows the solution (2.35) and in absence of potential the asymptotic de Sitter solution (2.36) takes place. We also investigated the possible mechanism to avoid the Big Rip singularity in the case of PPL expansion. To this end we made a qualitative analysis, by proposing different asymptotic scenarios where one or two interacting terms (including the potential) in (2.1) are relevant at low curvature, while the remaining terms become relevant at large curvature (when t → t s ), providing a quintessential solution and evading in this way the singularity (see [42]). After that, the universe might evolve asymptotically towards a de Sitter solution as was shown in different scenarios. The mechanism was implemented in the standard and phantom version of the scalar field (i.e. γ = ± 1). Of special interest is the first case (3.1) ( γ = 1) where at t → t s the dynamics becomes dominated by pure quintessential scalar field, which is the only possibility. We have performed the stability analysis for the string inspired model (i.e. in absence of potential) and have found that the power-law solution (2.13) (or (2.14)) is a critical point of the model and is stable fixed point in different scenarios: we first considered the case where the GB coupling is neglected (i.e. V = 0, η = 0). In this case, for γ = 1, the critical point is unstable in the allowable range of p (1 / 6 < p < 2 / 3, see (2.26)). For γ = -1, the critical point (2.13) is stable attractor, provided that p > 2 / 3. The PPL solution (2.14 is a stable attractor for the phantom model ( γ = -1) for any p > 0. The scalar field with GB correction (i.e. with V = 0, ξ = 0) was considered in [42]. Next we analyzed the stability of the model with GB and kinetic coupling terms. In the model with γ = 1, for QPL we have found the eigenvalues (0 , (1 -3 p ) /p, (2 -3 p ) /p ) and for PPL (0 , -(2 + 3 p ) /p, -(1 + 3 p ) /p ). For γ = -1, we have found the same eigenvalues. Due to the presence of zero eigenvalues, the linear expansion fails to provide information on the stability of the fixed point. We used the centre manifold analysis to determine the stability properties of the critical point, and found that in the important limit of de Sitter solution the critical point is a stable attractor, under certain restriction coming from the expression (4.34). Numerical evaluation was also done for concrete values of p and it was found stability for some cases. Resuming, the presence of the kinetic coupling besides the GB coupling, extends the number of possible scenarios to realize cosmological solutions with BR singularity, compared to the model with only GB correction. Additionally, the different asymptotic cases of the present model not only extend such possibilities, but also provide a number of alternatives to avoid the BR singularity leading to an universe that might evolve towards a de Sitter phase. Of special interest is the first case given by the model (3.1), in which the BR solution is obtained without appealing to phantom field (i.e. γ = 1), and near the singularity the dynamics becomes dominated by purely quintessential scalar field with EoS w > -1. Note that in this scenario the terms with couplings produce the PPL and the scalar potential acts against the BR singularity, while in the other cases ( γ = -1, see Eqs. (3.6), (3.10), (3.16)) the GB and kinetic couplings might prevent the BR singularity. The above results show that the string effects could play significant role in late time cosmology.", "pages": [ 22, 23, 24 ] }, { "title": "Acknowledgments", "content": "This work was supported by Universidad del Valle under project CI 7883.", "pages": [ 24 ] } ]

[ { "title": "Modified holographic Ricci dark energy model and statefinder diagnosis in flat universe.", "content": "Titus K Mathew 1 , Jishnu Suresh 2 and Divya Divakaran 3 Department of Physics, Cochin University of Science and Technology, Kochi-22, India . E-mail: 1 [email protected], [email protected], 2 [email protected], 3 [email protected].", "pages": [ 1 ] }, { "title": "Abstract", "content": "Evolution of the universe with modified holographic Ricci dark energy model is considered. Dependency of the equation of state parameter and deceleration parameter on the redshift and model parameters are obtained. It is shown that the density evolution of both the non-relativistic matter and dark energy are same until recent times. The evolutionary trajectories of the model for different model parameters are obtained in the statefinder planes, r -s and r -q planes. The present statefinder parameters are obtained for different model parameter values, using that the model is differentiated from other standard models like ΛCDM model etc. We have also shown that the evolutionary trajectories are depending on the model parameters, and at past times the dark energy is behaving like cold dark matter, with equation of state equal to zero. Keywords : Dark energy, Holographic model, Statefinder diagnostic, Cosmological evolution. PACS numbers : 98.80.Cq, 98.65.Dx", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Observations of distant type Ia supernovae (SNIa) and cosmic microwave background anisotropy have shown that the present universe is accelerating [1]. This expansion may be driven by a component with negative pressure, called dark energy. The simplest model of dark energy is the cosmological constant Λ which can fit the observations in a fair way [2, 3], whose equation of state is ω Λ = -1 . during the evolution of the universe. However there are two serious problems with cosmological constant model, namely the fine tuning and the cosmic coincidence [4]. To solve these problems different dynamic dark energy models have been proposed, with varying equation of state during the expansion of the universe. Holographic dark energy (HDE) is one among them [5, 6, 7]. HDE is constructed based on the holographic principle, that in quantum gravity, the entropy of a system scales not with its volume but with its surface area L 2 , analogically the cosmological constant in Einstein's theory also is inverse of some length squared. It was shown that [5] in effective quantum field theory, the zero point energy of the system with size L should not exceed the mass of a black hole with the same size, thus L 3 ρ Λ ≤ LM 2 P , where ρ Λ is the quantum zero-point energy and M P = 1 / √ 8 πG , is the reduced Plank mass. This inequality relation implies a link between the ultraviolet (UV) cut-off, defined through ρ Λ and the infrared (IR) cut-off encoded in the scale L. In the context of cosmology one can take the dark energy density of the universe ρ X as the same as the vacuum energy, i.e. ρ x = ρ Λ . The largest IR cut-off L is chosen by saturating the inequality, so that the holographic energy density can be written as where c is numerical constant. In the current literature, the IR cut-off has been taken as the Hubble horizon [6, 7], particle horizon and event horizon [7] or some generalized IR cut off [8, 9, 10]. The HDE models with Hubble horizon or particle horizon as the IR cut-off, cannot lead to the current accelerated expansion [6] of the universe. When the event horizon is taken as the length scale, the model is suffered from the following disadvantage. Future event horizon is a global concept of space-time. On the other hand density of dark energy is a local quantity. So the relation between them will pose challenges to the concept of causality. These leads to the introduction new HDE, where the length scale is given by the average radius of the Ricci scalar curvature, R -1 / 2 . The holographic Ricci dark energy model introduced by Granda and Oliveros [11] based on the space-time scalar curvature, is fairly good in fitting with the observational data. This model have the following advantages. First, the fine tuning problem can be avoided in this model. Moreover, the presence of event horizon is not presumed in this model, so that the causality problem can be avoided. The coincidence problem can also be solved effectively in this model. Recently a modified form of Ricci dark energy was studied [12] in connection with the dark matter interaction, and analyses the model using Om diagnostic. In this paper we have considered the evolution of the universe in Modified Holographic Ricci Dark Energy (MHRDE) model and obtain the statefinder parameters to discriminate this model with other standard dark energy models. Statefinder parameters is a sensitive and diagnostic tool used to discriminate various dark energy models. The Hubble parameter H and deceleration parameter q alone cannot discriminate various dark energy models because of the degeneracy on these parameters. Hence Sahni et al. [13] introduces a set of parameters { r, s } called statefinder parameters, defined as, where a is the scale factor of the expanding universe and Ω total is the total energy density containing dark energy, energy corresponds to curvature and also matter (we are neglecting the radiation part in our analysis). In general statefinder parameter is a geometrical diagnostic such that it depends upon the expansion factor and hence on the metric describing space-time. The r -s plot of dark energy models can help to differentiate and discriminate various models. For the well known ΛCDM model, the r -s trajectory is corresponds to fixed point, with r = 1 and s = 0 [13]. The cosmological behavior of various dark models including holographic dark energy model, were studied and differentiated in the recent literature using statefinder parameters [14, 15, 16, 17]. The paper is organized as follows. In section 2, we have studied the cosmological behavior of the MHRDE model and in section 3 we have considered the statefinder diagnostic analysis followed by the conclusions in section 4.", "pages": [ 2, 3 ] }, { "title": "2 The MHRDE model", "content": "The universe is described by the Friedmann-Robertson-Walker metric given by where ( r, θ, φ ) are the co-moving coordinates, k is the curvature parameter with values, k = 1 , 0 , -1 for closed, flat and open universes respectively and a ( t ) is the scale factor, with a 0 = 1 , is taken as its present value. The Friedmann equation describing the evolution of the universe is where we have taken 8 πG = 1, the summation includes the energy densities of non-relativistic matter and dark energy, i.e. ∑ i ρ i = ρ m + ρ x . The modified holographic Ricci dark energy can be expressed by taking the IR cutoff with the modified Ricci radius in terms of ˙ H and H 2 as [11, 12] where ˙ H is the time derivative of the Hubble parameter, α and β are free constants, the model parameters. Chimento et. al. studied this type of dark energy in interaction with the dark matter with Chaplygin gas [12], but our analysis is mainly concentrated on the cosmological evolution of MHRDE and analyses with statefinder diagnostic. Substituting the dark energy density as the MHRDE in the Friedmann equation, and changing the variable form cosmic time t to x = ln a we get Introducing the normalized Hubble parameter as h = H/H 0 and Ω k = -k/H 2 0 , where H 0 is the Hubble parameter for x = 0, the above equation become, where Ω mo = ρ m 0 / 3 H 2 0 is the current density parameter of non-relativistic matter ( we will take 0.27 as its values for our analysis throughout.) with current density ρ m 0 and Ω ko is the present relative density parameter of the curvature. We will consider only flat universe, where Ω k 0 = 0 in our further analysis. Solving the first order differential equation (7) we obtain the dimensionless Hubble parameter h as, Comparing this with the standard Friedman equation, the dark energy density can be identified as This shows that similar to the result obtained in references [8, 15] for Ricci dark energy, the MHRDE density has one part which evolves like non-relativistic matter ( ∼ e -3 x ) and the other part is slowly increasing with the decrease in redshift. The pressure corresponding the dark energy can be calculated as, Form the conservation equation, we can obtain the corresponding equation of state parameter for the flat universe, using equation (9) as, This equation of state implies the possibility of transit form ω x > -1 to ω x < -1 , corresponds to the phantom model [18, 19] for suitable model parameter values. Recent observational evidences shows that the dark energy equation of state parameter can crosses the value -1 [20]. In a universe dominated with MHRDE, where the contribution from the non-relativistic matter behavior term is negligible in the dark energy density, the equation state parameter become, So if β is less than zero, the equation of state can crosses the phantom divide. In the far future of the universe, when redshift z →-1 also, the equation of state parameter reduces to the form given in equation (12). So the behavior of the dark energy is depending strongly on the model parameter β. We have plotted the evolution of the equation of state parameter of MHRDE with redshift in figure 1, using the best fit values of the model parameters α and β as [12] ( α, β ) = (1 . 01 , 0 . 15) and Ω m 0 =0.27. The evolution of ω x of dark energy shows that in the remote past of the universe, that is at large redshift, the equation of state parameter is near zero, implies that the dark energy behaves like the cold dark matter in the remote past. The plot also shows that at far future of the universe as z → -1 , the equation of state parameter approaches a saturation value. The L H L H H L H L present value of the equation of state parameter according to this plot is negative, and is around ω x = -0 . 7 . For other values of the model parameters ( α, β ) [12] as ( α, β ) = (1.01, 0.05), (1.01,0.1), (1.2,-0.05), (1.2,-0.1), (4/3,0.1) and (4/3, -0.1) the behavior of the equation of state parameter is given figure 2. For a given value of α the saturation value of ω x in the future universe decreases as | β | increases. In figure 3, we have shown a comparison of the evolution of non-relativistic matter density and MHRDE density in logarithmic scale. Here we have neglected phase transitions, transitions from non-relativistic to relativistic particles at high temperatures and new degrees of freedom etc. It is expected that these would not make much qualitative difference in the result. The plot shows that in the present model the densities of non-relativistic matter and dark energy were comparable with each other in the past universe that is at high redshift. The acceleration began at low redshifts, which solves the coincidence problem. The deceleration parameter q for the MHRDE model can obtained using the relation This equation can be expressed in terms of the dimensionless Hubble parameter h as Using equation (8 ) the above equation can be written as This equation shows the dependence of the deceleration parameter on the model parameters α and β. As an approximation, if we neglect the contribution form the first terms in both numerator and denominator (since they are negligibly small) the deceleration parameter will become q = (3 β -2) / 2 . Which shows, as β increases form from zero, the parameter q increases form -1, that is the universe enter the acceleration phase at successively later times. In figure 4 and figure 5 we have plotted the evolution of the deceleration parameter with redshift. Figure 4 is for the best fitting model parameters α =1.01, β =0.15 and figure 5 is for the remaining model parameter values. The plots shows that at large redshift, the deceleration parameter approaches 0.5. The universe is entering the acceleration in the recent past at z < 1 . The plot also shows that as α increase, the entry to the accelerating phase is occurring at relatively lower values of redshift, that is the universe entering the accelerating phase at relatively later times as the parameter α L H L H H L redshift z H L increases. The transition of the universe from deceleration to the accelerating phase is occurred at the the redshift Z T = 0 . 76, for the best fit model parameters . For comparison the combined analysis of SNe+CMB data with ΛCDM model gives the range Z T (Λ CDM ) = 0.50 - 0.73 [20, 24]. For taking consideration of the entire model parametric range, the transition to the accelerating phase can be obtained, as in figure 5 as Z T ( MHRDE )=0.50 - 0.76. The comparison of the two ranges shows that in the MHRDE model the universe entering the accelerating expansion phase earlier than in the ΛCDM model. The present value of the deceleration parameter for the best fit model parameters α =1.01, β =0.15 is q 0 = -0 . 45 as from figure 4.", "pages": [ 3, 4, 5, 6, 7 ] }, { "title": "3 Statefinder diagnostic", "content": "We have calculated the statefinder parameters r and s , as defined earlier in equation (2). Statefinder parameters can provide us with a diagnosis which should unambiguously probe the properties of various classes of dark energy models. Equation (2) for r and s can be rewrite in terms of h 2 as, and On substituting the relation for h 2 from equation (8), the above equations for a flat universe (in which Ω k = 0) become and s is become, From equations (18) and (19), it is evident that r = 1 , s = 0 if β = 0 and no matter what value α is, and this point in the r -s plane is corresponds to the ΛCDM model. This point is a very fixed point, thus statefinder diagnostic fails to discriminate between ΛCDM model and MHRDE model for the model parameter value β = 0 . Since s is a constant for flat universe in this model, the trajectory in the r -s plane is a vertical segment, with constant s during the evolution of the universe, while r is monotonically decreasing form 1, if β is positive and monotonically increasing if β is assuming negative values. For a simple understanding, let us assume that Ω m contribution is negligible small, when dark energy is dominating, then the equation (18) reduces to In this case for β =0.05, 0.1 and 0.15 the corresponding values of r are 0.79, 0.60, 0.43 respectively. Bur when β assumes the negative values -0.05 and -0.10, the corresponding values of r become 1.24 and 1.5 respectively. So at the outset the MHRDE model gives a r -s trajectory, as r starting form 1 and due to evolution of the universe the r will decreases to 1 -9 β (1 -β ) 2 if β is positive and increases to 1 + 9 β ( β -1) 2 , if β is negative. The r -s evolutionary trajectory in the MHRDE model in flat universe for the best fit model parameters α =1.01 and β =0.15, is given in figure 6. In this plot as the universe expands, the trajectory in the r -s plane starts form left to right. The standard ΛCDM model is corresponds to r = 1 , s = 0 is denoted. In this model the parameter r first decrease very slowly with s , then after around s =0 r decreases steeply. The today's value of the statefinder parameter ( r 0 =0.59, s 0 =0.15) is denoted in the plot. For other model parameters, the r -s plots are given in figure 7. These plots also shows the same behavior of figure 6, but the separation between ΛCDM model and MHRDE model in the r -s plane is increasing as β increases. The respective todays universe corresponds r 0 , s 0 = 0 . 71 , 0 . 1 and r 0 , s 0 = 0 . 85 , 0 . 05 . For negative values of β , the evolutionary characteristics is plotted in figure 8 for model parameters α, β = 1 , 2 , -0 . 10; 4 / 3 , -0 . 10 . Here also the evolution in the r -s plane is from left to right. In this case the behavior is different form that for the positive β value, in the sense that as s increases The r is increasing to vales greater than one. The increase is very slowly at first then increases steeply as the universe evolves. The today's value in these cases are r 0 =1.325, s 0 =-0.10 when ( α, β )= (1.2,-0.10) and r 0 =1.321, s 0 =-0.10 for ( α, β )=(4/3, -0.10) respectively. The difference between MHRDE model for these model parameters and ΛCDM can be noted. The statefinder diagnostic can discriminate this model with other models. As example, for the quintessence model the r -s trajectory is lying in the region s > 0 , r < 1 and for Chaplygin gas the trajectory is in the region s < 0 , r > 1 . Holographic dark energy with the future event horizon as IR cutoff, starts its evolution form s = 2 / 3 , r = 1 and ends on at ΛCDM model fixed point in the future [16, 27]. In order to confirm the r -s behavior of MHRDE model, we have plotted the behavior in r -z plane, in figure 9. For MHRDE model, the r value is commencing from 1 irrespective of the values of α and β at remote past and as the universe evolves, r is decreasing, if β is positive and increasing, if β is negative. We have studied the evolutionary behavior in the r -q plane also. For positive value of β the plot is shown in figure 10 for the standard values α = 1 . 01 , β = 0 . 15 . The figure shows that both ΛCDM model and MHRDE model commence evolving from the same point in the s s H L r past corresponds to r = 1 , q = 0 . 5 , which corresponds to a matter dominated SCDM universe. In ΛCDM model the trajectory will end their evolution at q = -1 , r = 1 which corresponds to de Sitter model, while in MHRDE model the behavior is different from this. The statefinder trajectory in holographic dark energy model with future event horizon has the same starting point and the same end point as ΛCDM model [25, 26]. Thus MHRDE model is also different form holographic dark energy with event horizon form the statefinder viewpoint. For negative values β the plot is as given in figure 11. The evolution of the trajectory is starting from left to right. Note that the r value is at the increase from one as the universe evolves. It is evident from the plot that the present position of the model corresponds to r 0 =1.325 and q 0 = -0.63.", "pages": [ 7, 8, 9, 10, 11 ] }, { "title": "4 Conclusions", "content": "We have studied the modified holographic Ricci dark energy (MHRDE) in flat universe, where the IR cutoff is given by the modified Ricci scalar, and the dark energy become ρ x = 2( ˙ H + 3 αH 2 / 2) / ( α -β ) where α and β are model parameters. We have calculated the relevant cosmological parameters and their evolution and also analyzed the model form the statefinder view point for discriminating it from other models. The importance of the model is that it depends on the local quantities and thus avoids the causality problem. The density of MHRDE is comparable with the non-relativistic matter at high redshift as shown in figure 3 and began to dominate at low redshifts, thus the model is free from the coincidence problem. The evolution of equation of state parameter is studied. The equation of state parameter is nearly zero at high redshift, implies that in the past universe MHRDE behaves like cold dark matter. Further evolution of equation of state is strongly depending on the model parameter β. If the β parameter is positive the equation of state is greater than -1. For negative values of β , the equation of state cross the phantom divide ω x < -1 . In this model the deceleration parameter starts form around 0.5 at the early times and and starts to become negative when the redshift z < 1 . . In general we have found that in MHRDE model the universe entering the accelerating phase at times earlier (for allowed range of parameters α and β ), than in the ΛCDM model. But in particular as the model parameter α increases, the universe enter the accelerating phase at relatively later times. We have applied the statefinder diagnostic to the MHRDE and plot the trajectories in the r -s and r -q planes. The statefinder diagnostic is a crucial tool for discriminating different dark energy models. The statefinder trajectories are depending on the model parameters α and β. For positive values of β the r values will decreases from one and for negative β the r will increases form one as the universe evolves. The values of α and β are constrained using observational data in reference [12], the best fit value is α =1.01, β =0.15. The present value of ( r, s ) can be viewed as a discriminator for testing different dark energy models. For the ΛCDM model statefinder is a fixed point r =1, s =0. For positive values β parameter the r -s and r -q plots of MHRDE shows that, the evolutionary trajectories starts form r = 1 and q = 0 . 5 , in the past universe (for the best fit model parameters), which reveals that the MHRDE is behaving like cold dark matter in the past. The further evolution of MHRDE in the r -s plane shows that the present position of MHRDE model in the r -s plane for the best fit parameter is r 0 =0.59, s 0 =0.15 and in the r -q plane is r 0 =0.59, q 0 =-0.45. The difference between the MHRDE and ΛCDM models is in the evolution of the equation of state parameter, which is -1 in the ΛCDM model and a time-dependent variable in MHRDE model. A further comparison can be made with the new HDE model [14], which gives the present values r 0 ( HDE ) = 1 . 357 , s 0 ( HDE ) = -0 . 102 and r 0 ( HDE ) = 1 . 357 , q 0 ( HDE ) = -0 . 590 . So in the r -s plane the distance of the MHRDE model form the ΛCDM fixed point is slightly larger compared to the new HDE model for positive values of β parameter. However in the case of MHRDE model the starting point in r -s plane and r -q plane is ( r = 1 , s = 0 and r = 1 , q = 0 . 5) is same as that in the ΛCDM model. For negative values of the β the r -s trajectory we have plotted is different compared to that of positive β values. For negative β values the r value can attains values greater than one as s increases. The present status of the evolution in the r -s plane is r 0 =1.325, s 0 =-0.10 for model parameters α =1.2, β =-0.10 and r 0 =1.321, s 0 =-0.10. The r -q for α =1.2 and β =-0.10 shows that the present state of the MHRDE model is corresponds to r 0 = 1 . 325 and q 0 = - 0.63. These values shows that the MHRDE model is different form ΛCDM model for the present time when β parameter is negative also. But compared the new HDE model, the present MHRDE model doesn't show much deviation, shows that for negative values the behavior of MHRDE model is almost similar to new HDE model. Irrespective of whether β is positive or negative the MHRDE model is commence to evolve from SCDM model. When β is positive 1 is the maximum value of r , on the other hand when β is negative 1 is the minimum value of r. However the exact discrimination of the dark energy models is possible only if we can obtain the present r -s values in a model independent way form the observational data. It is expected that the future high-precision SNAP-type observations can lead to the present statefinder parameters, which could help us to find the right dark energy models.", "pages": [ 11, 12 ] } ]